3.5.27 \(\int \frac {x^{3/2} (a+b x^2)^2}{(c+d x^2)^2} \, dx\) [427]

Optimal. Leaf size=346 \[ -\frac {(b c-a d) (9 b c-a d) \sqrt {x}}{2 c d^3}+\frac {2 b^2 x^{5/2}}{5 d^2}+\frac {(b c-a d)^2 x^{5/2}}{2 c d^2 \left (c+d x^2\right )}-\frac {(b c-a d) (9 b c-a d) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{4 \sqrt {2} c^{3/4} d^{13/4}}+\frac {(b c-a d) (9 b c-a d) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{4 \sqrt {2} c^{3/4} d^{13/4}}-\frac {(b c-a d) (9 b c-a d) \log \left (\sqrt {c}-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{8 \sqrt {2} c^{3/4} d^{13/4}}+\frac {(b c-a d) (9 b c-a d) \log \left (\sqrt {c}+\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{8 \sqrt {2} c^{3/4} d^{13/4}} \]

[Out]

2/5*b^2*x^(5/2)/d^2+1/2*(-a*d+b*c)^2*x^(5/2)/c/d^2/(d*x^2+c)-1/8*(-a*d+b*c)*(-a*d+9*b*c)*arctan(1-d^(1/4)*2^(1
/2)*x^(1/2)/c^(1/4))/c^(3/4)/d^(13/4)*2^(1/2)+1/8*(-a*d+b*c)*(-a*d+9*b*c)*arctan(1+d^(1/4)*2^(1/2)*x^(1/2)/c^(
1/4))/c^(3/4)/d^(13/4)*2^(1/2)-1/16*(-a*d+b*c)*(-a*d+9*b*c)*ln(c^(1/2)+x*d^(1/2)-c^(1/4)*d^(1/4)*2^(1/2)*x^(1/
2))/c^(3/4)/d^(13/4)*2^(1/2)+1/16*(-a*d+b*c)*(-a*d+9*b*c)*ln(c^(1/2)+x*d^(1/2)+c^(1/4)*d^(1/4)*2^(1/2)*x^(1/2)
)/c^(3/4)/d^(13/4)*2^(1/2)-1/2*(-a*d+b*c)*(-a*d+9*b*c)*x^(1/2)/c/d^3

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Rubi [A]
time = 0.21, antiderivative size = 346, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 10, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {474, 470, 327, 335, 217, 1179, 642, 1176, 631, 210} \begin {gather*} -\frac {(b c-a d) (9 b c-a d) \text {ArcTan}\left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{4 \sqrt {2} c^{3/4} d^{13/4}}+\frac {(b c-a d) (9 b c-a d) \text {ArcTan}\left (\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}+1\right )}{4 \sqrt {2} c^{3/4} d^{13/4}}-\frac {(b c-a d) (9 b c-a d) \log \left (-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{8 \sqrt {2} c^{3/4} d^{13/4}}+\frac {(b c-a d) (9 b c-a d) \log \left (\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{8 \sqrt {2} c^{3/4} d^{13/4}}-\frac {\sqrt {x} (b c-a d) (9 b c-a d)}{2 c d^3}+\frac {x^{5/2} (b c-a d)^2}{2 c d^2 \left (c+d x^2\right )}+\frac {2 b^2 x^{5/2}}{5 d^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(x^(3/2)*(a + b*x^2)^2)/(c + d*x^2)^2,x]

[Out]

-1/2*((b*c - a*d)*(9*b*c - a*d)*Sqrt[x])/(c*d^3) + (2*b^2*x^(5/2))/(5*d^2) + ((b*c - a*d)^2*x^(5/2))/(2*c*d^2*
(c + d*x^2)) - ((b*c - a*d)*(9*b*c - a*d)*ArcTan[1 - (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/(4*Sqrt[2]*c^(3/4)*d^
(13/4)) + ((b*c - a*d)*(9*b*c - a*d)*ArcTan[1 + (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/(4*Sqrt[2]*c^(3/4)*d^(13/4
)) - ((b*c - a*d)*(9*b*c - a*d)*Log[Sqrt[c] - Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/(8*Sqrt[2]*c^(3/4)
*d^(13/4)) + ((b*c - a*d)*(9*b*c - a*d)*Log[Sqrt[c] + Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/(8*Sqrt[2]
*c^(3/4)*d^(13/4))

Rule 210

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^(-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])
], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 217

Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[a/b, 2]], s = Denominator[Rt[a/b, 2]]}, Di
st[1/(2*r), Int[(r - s*x^2)/(a + b*x^4), x], x] + Dist[1/(2*r), Int[(r + s*x^2)/(a + b*x^4), x], x]] /; FreeQ[
{a, b}, x] && (GtQ[a/b, 0] || (PosQ[a/b] && AtomQ[SplitProduct[SumBaseQ, a]] && AtomQ[SplitProduct[SumBaseQ, b
]]))

Rule 327

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[c^(n - 1)*(c*x)^(m - n + 1)*((a + b*x^n
)^(p + 1)/(b*(m + n*p + 1))), x] - Dist[a*c^n*((m - n + 1)/(b*(m + n*p + 1))), Int[(c*x)^(m - n)*(a + b*x^n)^p
, x], x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b,
c, n, m, p, x]

Rule 335

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = Denominator[m]}, Dist[k/c, Subst[I
nt[x^(k*(m + 1) - 1)*(a + b*(x^(k*n)/c^n))^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0]
&& FractionQ[m] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 470

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Simp[d*(e*x)^(m +
1)*((a + b*x^n)^(p + 1)/(b*e*(m + n*(p + 1) + 1))), x] - Dist[(a*d*(m + 1) - b*c*(m + n*(p + 1) + 1))/(b*(m +
n*(p + 1) + 1)), Int[(e*x)^m*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, c, d, e, m, n, p}, x] && NeQ[b*c - a*d, 0]
&& NeQ[m + n*(p + 1) + 1, 0]

Rule 474

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^2, x_Symbol] :> Simp[(-(b*c - a*
d)^2)*(e*x)^(m + 1)*((a + b*x^n)^(p + 1)/(a*b^2*e*n*(p + 1))), x] + Dist[1/(a*b^2*n*(p + 1)), Int[(e*x)^m*(a +
b*x^n)^(p + 1)*Simp[(b*c - a*d)^2*(m + 1) + b^2*c^2*n*(p + 1) + a*b*d^2*n*(p + 1)*x^n, x], x], x] /; FreeQ[{a
, b, c, d, e, m, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && LtQ[p, -1]

Rule 631

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[a*(c/b^2)]}, Dist[-2/b, Sub
st[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b)], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 642

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[d*(Log[RemoveContent[a + b*x +
c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rule 1176

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[2*(d/e), 2]}, Dist[e/(2*c), Int[1/S
imp[d/e + q*x + x^2, x], x], x] + Dist[e/(2*c), Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; FreeQ[{a, c, d, e},
x] && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]

Rule 1179

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[-2*(d/e), 2]}, Dist[e/(2*c*q), Int[
(q - 2*x)/Simp[d/e + q*x - x^2, x], x], x] + Dist[e/(2*c*q), Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /
; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]

Rubi steps

\begin {align*} \int \frac {x^{3/2} \left (a+b x^2\right )^2}{\left (c+d x^2\right )^2} \, dx &=\frac {(b c-a d)^2 x^{5/2}}{2 c d^2 \left (c+d x^2\right )}-\frac {\int \frac {x^{3/2} \left (\frac {1}{2} \left (-4 a^2 d^2+5 (b c-a d)^2\right )-2 b^2 c d x^2\right )}{c+d x^2} \, dx}{2 c d^2}\\ &=\frac {2 b^2 x^{5/2}}{5 d^2}+\frac {(b c-a d)^2 x^{5/2}}{2 c d^2 \left (c+d x^2\right )}-\frac {((b c-a d) (9 b c-a d)) \int \frac {x^{3/2}}{c+d x^2} \, dx}{4 c d^2}\\ &=-\frac {(b c-a d) (9 b c-a d) \sqrt {x}}{2 c d^3}+\frac {2 b^2 x^{5/2}}{5 d^2}+\frac {(b c-a d)^2 x^{5/2}}{2 c d^2 \left (c+d x^2\right )}+\frac {((b c-a d) (9 b c-a d)) \int \frac {1}{\sqrt {x} \left (c+d x^2\right )} \, dx}{4 d^3}\\ &=-\frac {(b c-a d) (9 b c-a d) \sqrt {x}}{2 c d^3}+\frac {2 b^2 x^{5/2}}{5 d^2}+\frac {(b c-a d)^2 x^{5/2}}{2 c d^2 \left (c+d x^2\right )}+\frac {((b c-a d) (9 b c-a d)) \text {Subst}\left (\int \frac {1}{c+d x^4} \, dx,x,\sqrt {x}\right )}{2 d^3}\\ &=-\frac {(b c-a d) (9 b c-a d) \sqrt {x}}{2 c d^3}+\frac {2 b^2 x^{5/2}}{5 d^2}+\frac {(b c-a d)^2 x^{5/2}}{2 c d^2 \left (c+d x^2\right )}+\frac {((b c-a d) (9 b c-a d)) \text {Subst}\left (\int \frac {\sqrt {c}-\sqrt {d} x^2}{c+d x^4} \, dx,x,\sqrt {x}\right )}{4 \sqrt {c} d^3}+\frac {((b c-a d) (9 b c-a d)) \text {Subst}\left (\int \frac {\sqrt {c}+\sqrt {d} x^2}{c+d x^4} \, dx,x,\sqrt {x}\right )}{4 \sqrt {c} d^3}\\ &=-\frac {(b c-a d) (9 b c-a d) \sqrt {x}}{2 c d^3}+\frac {2 b^2 x^{5/2}}{5 d^2}+\frac {(b c-a d)^2 x^{5/2}}{2 c d^2 \left (c+d x^2\right )}+\frac {((b c-a d) (9 b c-a d)) \text {Subst}\left (\int \frac {1}{\frac {\sqrt {c}}{\sqrt {d}}-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{d}}+x^2} \, dx,x,\sqrt {x}\right )}{8 \sqrt {c} d^{7/2}}+\frac {((b c-a d) (9 b c-a d)) \text {Subst}\left (\int \frac {1}{\frac {\sqrt {c}}{\sqrt {d}}+\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{d}}+x^2} \, dx,x,\sqrt {x}\right )}{8 \sqrt {c} d^{7/2}}-\frac {((b c-a d) (9 b c-a d)) \text {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt [4]{c}}{\sqrt [4]{d}}+2 x}{-\frac {\sqrt {c}}{\sqrt {d}}-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{d}}-x^2} \, dx,x,\sqrt {x}\right )}{8 \sqrt {2} c^{3/4} d^{13/4}}-\frac {((b c-a d) (9 b c-a d)) \text {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt [4]{c}}{\sqrt [4]{d}}-2 x}{-\frac {\sqrt {c}}{\sqrt {d}}+\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{d}}-x^2} \, dx,x,\sqrt {x}\right )}{8 \sqrt {2} c^{3/4} d^{13/4}}\\ &=-\frac {(b c-a d) (9 b c-a d) \sqrt {x}}{2 c d^3}+\frac {2 b^2 x^{5/2}}{5 d^2}+\frac {(b c-a d)^2 x^{5/2}}{2 c d^2 \left (c+d x^2\right )}-\frac {(b c-a d) (9 b c-a d) \log \left (\sqrt {c}-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{8 \sqrt {2} c^{3/4} d^{13/4}}+\frac {(b c-a d) (9 b c-a d) \log \left (\sqrt {c}+\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{8 \sqrt {2} c^{3/4} d^{13/4}}+\frac {((b c-a d) (9 b c-a d)) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{4 \sqrt {2} c^{3/4} d^{13/4}}-\frac {((b c-a d) (9 b c-a d)) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{4 \sqrt {2} c^{3/4} d^{13/4}}\\ &=-\frac {(b c-a d) (9 b c-a d) \sqrt {x}}{2 c d^3}+\frac {2 b^2 x^{5/2}}{5 d^2}+\frac {(b c-a d)^2 x^{5/2}}{2 c d^2 \left (c+d x^2\right )}-\frac {(b c-a d) (9 b c-a d) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{4 \sqrt {2} c^{3/4} d^{13/4}}+\frac {(b c-a d) (9 b c-a d) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{4 \sqrt {2} c^{3/4} d^{13/4}}-\frac {(b c-a d) (9 b c-a d) \log \left (\sqrt {c}-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{8 \sqrt {2} c^{3/4} d^{13/4}}+\frac {(b c-a d) (9 b c-a d) \log \left (\sqrt {c}+\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{8 \sqrt {2} c^{3/4} d^{13/4}}\\ \end {align*}

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Mathematica [A]
time = 0.58, size = 221, normalized size = 0.64 \begin {gather*} \frac {\frac {4 \sqrt [4]{d} \sqrt {x} \left (-5 a^2 d^2+10 a b d \left (5 c+4 d x^2\right )+b^2 \left (-45 c^2-36 c d x^2+4 d^2 x^4\right )\right )}{c+d x^2}-\frac {5 \sqrt {2} \left (9 b^2 c^2-10 a b c d+a^2 d^2\right ) \tan ^{-1}\left (\frac {\sqrt {c}-\sqrt {d} x}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}}\right )}{c^{3/4}}+\frac {5 \sqrt {2} \left (9 b^2 c^2-10 a b c d+a^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}}{\sqrt {c}+\sqrt {d} x}\right )}{c^{3/4}}}{40 d^{13/4}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(x^(3/2)*(a + b*x^2)^2)/(c + d*x^2)^2,x]

[Out]

((4*d^(1/4)*Sqrt[x]*(-5*a^2*d^2 + 10*a*b*d*(5*c + 4*d*x^2) + b^2*(-45*c^2 - 36*c*d*x^2 + 4*d^2*x^4)))/(c + d*x
^2) - (5*Sqrt[2]*(9*b^2*c^2 - 10*a*b*c*d + a^2*d^2)*ArcTan[(Sqrt[c] - Sqrt[d]*x)/(Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt
[x])])/c^(3/4) + (5*Sqrt[2]*(9*b^2*c^2 - 10*a*b*c*d + a^2*d^2)*ArcTanh[(Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x])/(Sqrt
[c] + Sqrt[d]*x)])/c^(3/4))/(40*d^(13/4))

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Maple [A]
time = 0.12, size = 199, normalized size = 0.58

method	result	size
derivativedivides	\(\frac {2 b \left (\frac {b d \,x^{\frac {5}{2}}}{5}+2 a d \sqrt {x}-2 b c \sqrt {x}\right )}{d^{3}}+\frac {\frac {2 \left (-\frac {1}{4} a^{2} d^{2}+\frac {1}{2} a b c d -\frac {1}{4} b^{2} c^{2}\right ) \sqrt {x}}{d \,x^{2}+c}+\frac {\left (a^{2} d^{2}-10 a b c d +9 b^{2} c^{2}\right ) \left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}{x -\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}-1\right )\right )}{16 c}}{d^{3}}\)	\(199\)
default	\(\frac {2 b \left (\frac {b d \,x^{\frac {5}{2}}}{5}+2 a d \sqrt {x}-2 b c \sqrt {x}\right )}{d^{3}}+\frac {\frac {2 \left (-\frac {1}{4} a^{2} d^{2}+\frac {1}{2} a b c d -\frac {1}{4} b^{2} c^{2}\right ) \sqrt {x}}{d \,x^{2}+c}+\frac {\left (a^{2} d^{2}-10 a b c d +9 b^{2} c^{2}\right ) \left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}{x -\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}-1\right )\right )}{16 c}}{d^{3}}\)	\(199\)
risch	\(\frac {2 b \left (b d \,x^{2}+10 a d -10 b c \right ) \sqrt {x}}{5 d^{3}}-\frac {\sqrt {x}\, a^{2}}{2 d \left (d \,x^{2}+c \right )}+\frac {\sqrt {x}\, a b c}{d^{2} \left (d \,x^{2}+c \right )}-\frac {\sqrt {x}\, b^{2} c^{2}}{2 d^{3} \left (d \,x^{2}+c \right )}+\frac {\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}-1\right ) a^{2}}{8 d c}-\frac {5 \left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}-1\right ) a b}{4 d^{2}}+\frac {9 \left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}-1\right ) b^{2} c}{8 d^{3}}+\frac {\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, \ln \left (\frac {x +\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}{x -\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}\right ) a^{2}}{16 d c}-\frac {5 \left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, \ln \left (\frac {x +\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}{x -\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}\right ) a b}{8 d^{2}}+\frac {9 \left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, \ln \left (\frac {x +\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}{x -\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}\right ) b^{2} c}{16 d^{3}}+\frac {\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}+1\right ) a^{2}}{8 d c}-\frac {5 \left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}+1\right ) a b}{4 d^{2}}+\frac {9 \left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}+1\right ) b^{2} c}{8 d^{3}}\)	\(514\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(3/2)*(b*x^2+a)^2/(d*x^2+c)^2,x,method=_RETURNVERBOSE)

[Out]

2*b/d^3*(1/5*b*d*x^(5/2)+2*a*d*x^(1/2)-2*b*c*x^(1/2))+2/d^3*((-1/4*a^2*d^2+1/2*a*b*c*d-1/4*b^2*c^2)*x^(1/2)/(d
*x^2+c)+1/32*(a^2*d^2-10*a*b*c*d+9*b^2*c^2)*(c/d)^(1/4)/c*2^(1/2)*(ln((x+(c/d)^(1/4)*x^(1/2)*2^(1/2)+(c/d)^(1/
2))/(x-(c/d)^(1/4)*x^(1/2)*2^(1/2)+(c/d)^(1/2)))+2*arctan(2^(1/2)/(c/d)^(1/4)*x^(1/2)+1)+2*arctan(2^(1/2)/(c/d
)^(1/4)*x^(1/2)-1)))

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Maxima [A]
time = 0.54, size = 336, normalized size = 0.97 \begin {gather*} -\frac {{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \sqrt {x}}{2 \, {\left (d^{4} x^{2} + c d^{3}\right )}} + \frac {2 \, {\left (b^{2} d x^{\frac {5}{2}} - 10 \, {\left (b^{2} c - a b d\right )} \sqrt {x}\right )}}{5 \, d^{3}} + \frac {\frac {2 \, \sqrt {2} {\left (9 \, b^{2} c^{2} - 10 \, a b c d + a^{2} d^{2}\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} + 2 \, \sqrt {d} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {c} \sqrt {d}}}\right )}{\sqrt {c} \sqrt {\sqrt {c} \sqrt {d}}} + \frac {2 \, \sqrt {2} {\left (9 \, b^{2} c^{2} - 10 \, a b c d + a^{2} d^{2}\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} - 2 \, \sqrt {d} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {c} \sqrt {d}}}\right )}{\sqrt {c} \sqrt {\sqrt {c} \sqrt {d}}} + \frac {\sqrt {2} {\left (9 \, b^{2} c^{2} - 10 \, a b c d + a^{2} d^{2}\right )} \log \left (\sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} \sqrt {x} + \sqrt {d} x + \sqrt {c}\right )}{c^{\frac {3}{4}} d^{\frac {1}{4}}} - \frac {\sqrt {2} {\left (9 \, b^{2} c^{2} - 10 \, a b c d + a^{2} d^{2}\right )} \log \left (-\sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} \sqrt {x} + \sqrt {d} x + \sqrt {c}\right )}{c^{\frac {3}{4}} d^{\frac {1}{4}}}}{16 \, d^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(3/2)*(b*x^2+a)^2/(d*x^2+c)^2,x, algorithm="maxima")

[Out]

-1/2*(b^2*c^2 - 2*a*b*c*d + a^2*d^2)*sqrt(x)/(d^4*x^2 + c*d^3) + 2/5*(b^2*d*x^(5/2) - 10*(b^2*c - a*b*d)*sqrt(
x))/d^3 + 1/16*(2*sqrt(2)*(9*b^2*c^2 - 10*a*b*c*d + a^2*d^2)*arctan(1/2*sqrt(2)*(sqrt(2)*c^(1/4)*d^(1/4) + 2*s
qrt(d)*sqrt(x))/sqrt(sqrt(c)*sqrt(d)))/(sqrt(c)*sqrt(sqrt(c)*sqrt(d))) + 2*sqrt(2)*(9*b^2*c^2 - 10*a*b*c*d + a
^2*d^2)*arctan(-1/2*sqrt(2)*(sqrt(2)*c^(1/4)*d^(1/4) - 2*sqrt(d)*sqrt(x))/sqrt(sqrt(c)*sqrt(d)))/(sqrt(c)*sqrt
(sqrt(c)*sqrt(d))) + sqrt(2)*(9*b^2*c^2 - 10*a*b*c*d + a^2*d^2)*log(sqrt(2)*c^(1/4)*d^(1/4)*sqrt(x) + sqrt(d)*
x + sqrt(c))/(c^(3/4)*d^(1/4)) - sqrt(2)*(9*b^2*c^2 - 10*a*b*c*d + a^2*d^2)*log(-sqrt(2)*c^(1/4)*d^(1/4)*sqrt(
x) + sqrt(d)*x + sqrt(c))/(c^(3/4)*d^(1/4)))/d^3

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1334 vs. \(2 (262) = 524\).
time = 1.53, size = 1334, normalized size = 3.86 \begin {gather*} \frac {20 \, {\left (d^{4} x^{2} + c d^{3}\right )} \left (-\frac {6561 \, b^{8} c^{8} - 29160 \, a b^{7} c^{7} d + 51516 \, a^{2} b^{6} c^{6} d^{2} - 45720 \, a^{3} b^{5} c^{5} d^{3} + 21286 \, a^{4} b^{4} c^{4} d^{4} - 5080 \, a^{5} b^{3} c^{3} d^{5} + 636 \, a^{6} b^{2} c^{2} d^{6} - 40 \, a^{7} b c d^{7} + a^{8} d^{8}}{c^{3} d^{13}}\right )^{\frac {1}{4}} \arctan \left (\frac {\sqrt {c^{2} d^{6} \sqrt {-\frac {6561 \, b^{8} c^{8} - 29160 \, a b^{7} c^{7} d + 51516 \, a^{2} b^{6} c^{6} d^{2} - 45720 \, a^{3} b^{5} c^{5} d^{3} + 21286 \, a^{4} b^{4} c^{4} d^{4} - 5080 \, a^{5} b^{3} c^{3} d^{5} + 636 \, a^{6} b^{2} c^{2} d^{6} - 40 \, a^{7} b c d^{7} + a^{8} d^{8}}{c^{3} d^{13}}} + {\left (81 \, b^{4} c^{4} - 180 \, a b^{3} c^{3} d + 118 \, a^{2} b^{2} c^{2} d^{2} - 20 \, a^{3} b c d^{3} + a^{4} d^{4}\right )} x} c^{2} d^{10} \left (-\frac {6561 \, b^{8} c^{8} - 29160 \, a b^{7} c^{7} d + 51516 \, a^{2} b^{6} c^{6} d^{2} - 45720 \, a^{3} b^{5} c^{5} d^{3} + 21286 \, a^{4} b^{4} c^{4} d^{4} - 5080 \, a^{5} b^{3} c^{3} d^{5} + 636 \, a^{6} b^{2} c^{2} d^{6} - 40 \, a^{7} b c d^{7} + a^{8} d^{8}}{c^{3} d^{13}}\right )^{\frac {3}{4}} - {\left (9 \, b^{2} c^{4} d^{10} - 10 \, a b c^{3} d^{11} + a^{2} c^{2} d^{12}\right )} \sqrt {x} \left (-\frac {6561 \, b^{8} c^{8} - 29160 \, a b^{7} c^{7} d + 51516 \, a^{2} b^{6} c^{6} d^{2} - 45720 \, a^{3} b^{5} c^{5} d^{3} + 21286 \, a^{4} b^{4} c^{4} d^{4} - 5080 \, a^{5} b^{3} c^{3} d^{5} + 636 \, a^{6} b^{2} c^{2} d^{6} - 40 \, a^{7} b c d^{7} + a^{8} d^{8}}{c^{3} d^{13}}\right )^{\frac {3}{4}}}{6561 \, b^{8} c^{8} - 29160 \, a b^{7} c^{7} d + 51516 \, a^{2} b^{6} c^{6} d^{2} - 45720 \, a^{3} b^{5} c^{5} d^{3} + 21286 \, a^{4} b^{4} c^{4} d^{4} - 5080 \, a^{5} b^{3} c^{3} d^{5} + 636 \, a^{6} b^{2} c^{2} d^{6} - 40 \, a^{7} b c d^{7} + a^{8} d^{8}}\right ) + 5 \, {\left (d^{4} x^{2} + c d^{3}\right )} \left (-\frac {6561 \, b^{8} c^{8} - 29160 \, a b^{7} c^{7} d + 51516 \, a^{2} b^{6} c^{6} d^{2} - 45720 \, a^{3} b^{5} c^{5} d^{3} + 21286 \, a^{4} b^{4} c^{4} d^{4} - 5080 \, a^{5} b^{3} c^{3} d^{5} + 636 \, a^{6} b^{2} c^{2} d^{6} - 40 \, a^{7} b c d^{7} + a^{8} d^{8}}{c^{3} d^{13}}\right )^{\frac {1}{4}} \log \left (c d^{3} \left (-\frac {6561 \, b^{8} c^{8} - 29160 \, a b^{7} c^{7} d + 51516 \, a^{2} b^{6} c^{6} d^{2} - 45720 \, a^{3} b^{5} c^{5} d^{3} + 21286 \, a^{4} b^{4} c^{4} d^{4} - 5080 \, a^{5} b^{3} c^{3} d^{5} + 636 \, a^{6} b^{2} c^{2} d^{6} - 40 \, a^{7} b c d^{7} + a^{8} d^{8}}{c^{3} d^{13}}\right )^{\frac {1}{4}} + {\left (9 \, b^{2} c^{2} - 10 \, a b c d + a^{2} d^{2}\right )} \sqrt {x}\right ) - 5 \, {\left (d^{4} x^{2} + c d^{3}\right )} \left (-\frac {6561 \, b^{8} c^{8} - 29160 \, a b^{7} c^{7} d + 51516 \, a^{2} b^{6} c^{6} d^{2} - 45720 \, a^{3} b^{5} c^{5} d^{3} + 21286 \, a^{4} b^{4} c^{4} d^{4} - 5080 \, a^{5} b^{3} c^{3} d^{5} + 636 \, a^{6} b^{2} c^{2} d^{6} - 40 \, a^{7} b c d^{7} + a^{8} d^{8}}{c^{3} d^{13}}\right )^{\frac {1}{4}} \log \left (-c d^{3} \left (-\frac {6561 \, b^{8} c^{8} - 29160 \, a b^{7} c^{7} d + 51516 \, a^{2} b^{6} c^{6} d^{2} - 45720 \, a^{3} b^{5} c^{5} d^{3} + 21286 \, a^{4} b^{4} c^{4} d^{4} - 5080 \, a^{5} b^{3} c^{3} d^{5} + 636 \, a^{6} b^{2} c^{2} d^{6} - 40 \, a^{7} b c d^{7} + a^{8} d^{8}}{c^{3} d^{13}}\right )^{\frac {1}{4}} + {\left (9 \, b^{2} c^{2} - 10 \, a b c d + a^{2} d^{2}\right )} \sqrt {x}\right ) + 4 \, {\left (4 \, b^{2} d^{2} x^{4} - 45 \, b^{2} c^{2} + 50 \, a b c d - 5 \, a^{2} d^{2} - 4 \, {\left (9 \, b^{2} c d - 10 \, a b d^{2}\right )} x^{2}\right )} \sqrt {x}}{40 \, {\left (d^{4} x^{2} + c d^{3}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(3/2)*(b*x^2+a)^2/(d*x^2+c)^2,x, algorithm="fricas")

[Out]

1/40*(20*(d^4*x^2 + c*d^3)*(-(6561*b^8*c^8 - 29160*a*b^7*c^7*d + 51516*a^2*b^6*c^6*d^2 - 45720*a^3*b^5*c^5*d^3
+ 21286*a^4*b^4*c^4*d^4 - 5080*a^5*b^3*c^3*d^5 + 636*a^6*b^2*c^2*d^6 - 40*a^7*b*c*d^7 + a^8*d^8)/(c^3*d^13))^
(1/4)*arctan((sqrt(c^2*d^6*sqrt(-(6561*b^8*c^8 - 29160*a*b^7*c^7*d + 51516*a^2*b^6*c^6*d^2 - 45720*a^3*b^5*c^5
*d^3 + 21286*a^4*b^4*c^4*d^4 - 5080*a^5*b^3*c^3*d^5 + 636*a^6*b^2*c^2*d^6 - 40*a^7*b*c*d^7 + a^8*d^8)/(c^3*d^1
3)) + (81*b^4*c^4 - 180*a*b^3*c^3*d + 118*a^2*b^2*c^2*d^2 - 20*a^3*b*c*d^3 + a^4*d^4)*x)*c^2*d^10*(-(6561*b^8*
c^8 - 29160*a*b^7*c^7*d + 51516*a^2*b^6*c^6*d^2 - 45720*a^3*b^5*c^5*d^3 + 21286*a^4*b^4*c^4*d^4 - 5080*a^5*b^3
*c^3*d^5 + 636*a^6*b^2*c^2*d^6 - 40*a^7*b*c*d^7 + a^8*d^8)/(c^3*d^13))^(3/4) - (9*b^2*c^4*d^10 - 10*a*b*c^3*d^
11 + a^2*c^2*d^12)*sqrt(x)*(-(6561*b^8*c^8 - 29160*a*b^7*c^7*d + 51516*a^2*b^6*c^6*d^2 - 45720*a^3*b^5*c^5*d^3
+ 21286*a^4*b^4*c^4*d^4 - 5080*a^5*b^3*c^3*d^5 + 636*a^6*b^2*c^2*d^6 - 40*a^7*b*c*d^7 + a^8*d^8)/(c^3*d^13))^
(3/4))/(6561*b^8*c^8 - 29160*a*b^7*c^7*d + 51516*a^2*b^6*c^6*d^2 - 45720*a^3*b^5*c^5*d^3 + 21286*a^4*b^4*c^4*d
^4 - 5080*a^5*b^3*c^3*d^5 + 636*a^6*b^2*c^2*d^6 - 40*a^7*b*c*d^7 + a^8*d^8)) + 5*(d^4*x^2 + c*d^3)*(-(6561*b^8
*c^8 - 29160*a*b^7*c^7*d + 51516*a^2*b^6*c^6*d^2 - 45720*a^3*b^5*c^5*d^3 + 21286*a^4*b^4*c^4*d^4 - 5080*a^5*b^
3*c^3*d^5 + 636*a^6*b^2*c^2*d^6 - 40*a^7*b*c*d^7 + a^8*d^8)/(c^3*d^13))^(1/4)*log(c*d^3*(-(6561*b^8*c^8 - 2916
0*a*b^7*c^7*d + 51516*a^2*b^6*c^6*d^2 - 45720*a^3*b^5*c^5*d^3 + 21286*a^4*b^4*c^4*d^4 - 5080*a^5*b^3*c^3*d^5 +
636*a^6*b^2*c^2*d^6 - 40*a^7*b*c*d^7 + a^8*d^8)/(c^3*d^13))^(1/4) + (9*b^2*c^2 - 10*a*b*c*d + a^2*d^2)*sqrt(x
)) - 5*(d^4*x^2 + c*d^3)*(-(6561*b^8*c^8 - 29160*a*b^7*c^7*d + 51516*a^2*b^6*c^6*d^2 - 45720*a^3*b^5*c^5*d^3 +
21286*a^4*b^4*c^4*d^4 - 5080*a^5*b^3*c^3*d^5 + 636*a^6*b^2*c^2*d^6 - 40*a^7*b*c*d^7 + a^8*d^8)/(c^3*d^13))^(1
/4)*log(-c*d^3*(-(6561*b^8*c^8 - 29160*a*b^7*c^7*d + 51516*a^2*b^6*c^6*d^2 - 45720*a^3*b^5*c^5*d^3 + 21286*a^4
*b^4*c^4*d^4 - 5080*a^5*b^3*c^3*d^5 + 636*a^6*b^2*c^2*d^6 - 40*a^7*b*c*d^7 + a^8*d^8)/(c^3*d^13))^(1/4) + (9*b
^2*c^2 - 10*a*b*c*d + a^2*d^2)*sqrt(x)) + 4*(4*b^2*d^2*x^4 - 45*b^2*c^2 + 50*a*b*c*d - 5*a^2*d^2 - 4*(9*b^2*c*
d - 10*a*b*d^2)*x^2)*sqrt(x))/(d^4*x^2 + c*d^3)

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 1280 vs. \(2 (321) = 642\).
time = 70.04, size = 1280, normalized size = 3.70 \begin {gather*} \begin {cases} \tilde {\infty } \left (- \frac {2 a^{2}}{3 x^{\frac {3}{2}}} + 4 a b \sqrt {x} + \frac {2 b^{2} x^{\frac {5}{2}}}{5}\right ) & \text {for}\: c = 0 \wedge d = 0 \\\frac {- \frac {2 a^{2}}{3 x^{\frac {3}{2}}} + 4 a b \sqrt {x} + \frac {2 b^{2} x^{\frac {5}{2}}}{5}}{d^{2}} & \text {for}\: c = 0 \\\frac {\frac {2 a^{2} x^{\frac {5}{2}}}{5} + \frac {4 a b x^{\frac {9}{2}}}{9} + \frac {2 b^{2} x^{\frac {13}{2}}}{13}}{c^{2}} & \text {for}\: d = 0 \\- \frac {20 a^{2} c d^{2} \sqrt {x}}{40 c^{2} d^{3} + 40 c d^{4} x^{2}} - \frac {5 a^{2} c d^{2} \sqrt [4]{- \frac {c}{d}} \log {\left (\sqrt {x} - \sqrt [4]{- \frac {c}{d}} \right )}}{40 c^{2} d^{3} + 40 c d^{4} x^{2}} + \frac {5 a^{2} c d^{2} \sqrt [4]{- \frac {c}{d}} \log {\left (\sqrt {x} + \sqrt [4]{- \frac {c}{d}} \right )}}{40 c^{2} d^{3} + 40 c d^{4} x^{2}} + \frac {10 a^{2} c d^{2} \sqrt [4]{- \frac {c}{d}} \operatorname {atan}{\left (\frac {\sqrt {x}}{\sqrt [4]{- \frac {c}{d}}} \right )}}{40 c^{2} d^{3} + 40 c d^{4} x^{2}} - \frac {5 a^{2} d^{3} x^{2} \sqrt [4]{- \frac {c}{d}} \log {\left (\sqrt {x} - \sqrt [4]{- \frac {c}{d}} \right )}}{40 c^{2} d^{3} + 40 c d^{4} x^{2}} + \frac {5 a^{2} d^{3} x^{2} \sqrt [4]{- \frac {c}{d}} \log {\left (\sqrt {x} + \sqrt [4]{- \frac {c}{d}} \right )}}{40 c^{2} d^{3} + 40 c d^{4} x^{2}} + \frac {10 a^{2} d^{3} x^{2} \sqrt [4]{- \frac {c}{d}} \operatorname {atan}{\left (\frac {\sqrt {x}}{\sqrt [4]{- \frac {c}{d}}} \right )}}{40 c^{2} d^{3} + 40 c d^{4} x^{2}} + \frac {200 a b c^{2} d \sqrt {x}}{40 c^{2} d^{3} + 40 c d^{4} x^{2}} + \frac {50 a b c^{2} d \sqrt [4]{- \frac {c}{d}} \log {\left (\sqrt {x} - \sqrt [4]{- \frac {c}{d}} \right )}}{40 c^{2} d^{3} + 40 c d^{4} x^{2}} - \frac {50 a b c^{2} d \sqrt [4]{- \frac {c}{d}} \log {\left (\sqrt {x} + \sqrt [4]{- \frac {c}{d}} \right )}}{40 c^{2} d^{3} + 40 c d^{4} x^{2}} - \frac {100 a b c^{2} d \sqrt [4]{- \frac {c}{d}} \operatorname {atan}{\left (\frac {\sqrt {x}}{\sqrt [4]{- \frac {c}{d}}} \right )}}{40 c^{2} d^{3} + 40 c d^{4} x^{2}} + \frac {160 a b c d^{2} x^{\frac {5}{2}}}{40 c^{2} d^{3} + 40 c d^{4} x^{2}} + \frac {50 a b c d^{2} x^{2} \sqrt [4]{- \frac {c}{d}} \log {\left (\sqrt {x} - \sqrt [4]{- \frac {c}{d}} \right )}}{40 c^{2} d^{3} + 40 c d^{4} x^{2}} - \frac {50 a b c d^{2} x^{2} \sqrt [4]{- \frac {c}{d}} \log {\left (\sqrt {x} + \sqrt [4]{- \frac {c}{d}} \right )}}{40 c^{2} d^{3} + 40 c d^{4} x^{2}} - \frac {100 a b c d^{2} x^{2} \sqrt [4]{- \frac {c}{d}} \operatorname {atan}{\left (\frac {\sqrt {x}}{\sqrt [4]{- \frac {c}{d}}} \right )}}{40 c^{2} d^{3} + 40 c d^{4} x^{2}} - \frac {180 b^{2} c^{3} \sqrt {x}}{40 c^{2} d^{3} + 40 c d^{4} x^{2}} - \frac {45 b^{2} c^{3} \sqrt [4]{- \frac {c}{d}} \log {\left (\sqrt {x} - \sqrt [4]{- \frac {c}{d}} \right )}}{40 c^{2} d^{3} + 40 c d^{4} x^{2}} + \frac {45 b^{2} c^{3} \sqrt [4]{- \frac {c}{d}} \log {\left (\sqrt {x} + \sqrt [4]{- \frac {c}{d}} \right )}}{40 c^{2} d^{3} + 40 c d^{4} x^{2}} + \frac {90 b^{2} c^{3} \sqrt [4]{- \frac {c}{d}} \operatorname {atan}{\left (\frac {\sqrt {x}}{\sqrt [4]{- \frac {c}{d}}} \right )}}{40 c^{2} d^{3} + 40 c d^{4} x^{2}} - \frac {144 b^{2} c^{2} d x^{\frac {5}{2}}}{40 c^{2} d^{3} + 40 c d^{4} x^{2}} - \frac {45 b^{2} c^{2} d x^{2} \sqrt [4]{- \frac {c}{d}} \log {\left (\sqrt {x} - \sqrt [4]{- \frac {c}{d}} \right )}}{40 c^{2} d^{3} + 40 c d^{4} x^{2}} + \frac {45 b^{2} c^{2} d x^{2} \sqrt [4]{- \frac {c}{d}} \log {\left (\sqrt {x} + \sqrt [4]{- \frac {c}{d}} \right )}}{40 c^{2} d^{3} + 40 c d^{4} x^{2}} + \frac {90 b^{2} c^{2} d x^{2} \sqrt [4]{- \frac {c}{d}} \operatorname {atan}{\left (\frac {\sqrt {x}}{\sqrt [4]{- \frac {c}{d}}} \right )}}{40 c^{2} d^{3} + 40 c d^{4} x^{2}} + \frac {16 b^{2} c d^{2} x^{\frac {9}{2}}}{40 c^{2} d^{3} + 40 c d^{4} x^{2}} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**(3/2)*(b*x**2+a)**2/(d*x**2+c)**2,x)

[Out]

Piecewise((zoo*(-2*a**2/(3*x**(3/2)) + 4*a*b*sqrt(x) + 2*b**2*x**(5/2)/5), Eq(c, 0) & Eq(d, 0)), ((-2*a**2/(3*
x**(3/2)) + 4*a*b*sqrt(x) + 2*b**2*x**(5/2)/5)/d**2, Eq(c, 0)), ((2*a**2*x**(5/2)/5 + 4*a*b*x**(9/2)/9 + 2*b**
2*x**(13/2)/13)/c**2, Eq(d, 0)), (-20*a**2*c*d**2*sqrt(x)/(40*c**2*d**3 + 40*c*d**4*x**2) - 5*a**2*c*d**2*(-c/
d)**(1/4)*log(sqrt(x) - (-c/d)**(1/4))/(40*c**2*d**3 + 40*c*d**4*x**2) + 5*a**2*c*d**2*(-c/d)**(1/4)*log(sqrt(
x) + (-c/d)**(1/4))/(40*c**2*d**3 + 40*c*d**4*x**2) + 10*a**2*c*d**2*(-c/d)**(1/4)*atan(sqrt(x)/(-c/d)**(1/4))
/(40*c**2*d**3 + 40*c*d**4*x**2) - 5*a**2*d**3*x**2*(-c/d)**(1/4)*log(sqrt(x) - (-c/d)**(1/4))/(40*c**2*d**3 +
40*c*d**4*x**2) + 5*a**2*d**3*x**2*(-c/d)**(1/4)*log(sqrt(x) + (-c/d)**(1/4))/(40*c**2*d**3 + 40*c*d**4*x**2)
+ 10*a**2*d**3*x**2*(-c/d)**(1/4)*atan(sqrt(x)/(-c/d)**(1/4))/(40*c**2*d**3 + 40*c*d**4*x**2) + 200*a*b*c**2*
d*sqrt(x)/(40*c**2*d**3 + 40*c*d**4*x**2) + 50*a*b*c**2*d*(-c/d)**(1/4)*log(sqrt(x) - (-c/d)**(1/4))/(40*c**2*
d**3 + 40*c*d**4*x**2) - 50*a*b*c**2*d*(-c/d)**(1/4)*log(sqrt(x) + (-c/d)**(1/4))/(40*c**2*d**3 + 40*c*d**4*x*
*2) - 100*a*b*c**2*d*(-c/d)**(1/4)*atan(sqrt(x)/(-c/d)**(1/4))/(40*c**2*d**3 + 40*c*d**4*x**2) + 160*a*b*c*d**
2*x**(5/2)/(40*c**2*d**3 + 40*c*d**4*x**2) + 50*a*b*c*d**2*x**2*(-c/d)**(1/4)*log(sqrt(x) - (-c/d)**(1/4))/(40
*c**2*d**3 + 40*c*d**4*x**2) - 50*a*b*c*d**2*x**2*(-c/d)**(1/4)*log(sqrt(x) + (-c/d)**(1/4))/(40*c**2*d**3 + 4
0*c*d**4*x**2) - 100*a*b*c*d**2*x**2*(-c/d)**(1/4)*atan(sqrt(x)/(-c/d)**(1/4))/(40*c**2*d**3 + 40*c*d**4*x**2)
- 180*b**2*c**3*sqrt(x)/(40*c**2*d**3 + 40*c*d**4*x**2) - 45*b**2*c**3*(-c/d)**(1/4)*log(sqrt(x) - (-c/d)**(1
/4))/(40*c**2*d**3 + 40*c*d**4*x**2) + 45*b**2*c**3*(-c/d)**(1/4)*log(sqrt(x) + (-c/d)**(1/4))/(40*c**2*d**3 +
40*c*d**4*x**2) + 90*b**2*c**3*(-c/d)**(1/4)*atan(sqrt(x)/(-c/d)**(1/4))/(40*c**2*d**3 + 40*c*d**4*x**2) - 14
4*b**2*c**2*d*x**(5/2)/(40*c**2*d**3 + 40*c*d**4*x**2) - 45*b**2*c**2*d*x**2*(-c/d)**(1/4)*log(sqrt(x) - (-c/d
)**(1/4))/(40*c**2*d**3 + 40*c*d**4*x**2) + 45*b**2*c**2*d*x**2*(-c/d)**(1/4)*log(sqrt(x) + (-c/d)**(1/4))/(40
*c**2*d**3 + 40*c*d**4*x**2) + 90*b**2*c**2*d*x**2*(-c/d)**(1/4)*atan(sqrt(x)/(-c/d)**(1/4))/(40*c**2*d**3 + 4
0*c*d**4*x**2) + 16*b**2*c*d**2*x**(9/2)/(40*c**2*d**3 + 40*c*d**4*x**2), True))

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Giac [A]
time = 0.62, size = 408, normalized size = 1.18 \begin {gather*} \frac {\sqrt {2} {\left (9 \, \left (c d^{3}\right )^{\frac {1}{4}} b^{2} c^{2} - 10 \, \left (c d^{3}\right )^{\frac {1}{4}} a b c d + \left (c d^{3}\right )^{\frac {1}{4}} a^{2} d^{2}\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {c}{d}\right )^{\frac {1}{4}} + 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {c}{d}\right )^{\frac {1}{4}}}\right )}{8 \, c d^{4}} + \frac {\sqrt {2} {\left (9 \, \left (c d^{3}\right )^{\frac {1}{4}} b^{2} c^{2} - 10 \, \left (c d^{3}\right )^{\frac {1}{4}} a b c d + \left (c d^{3}\right )^{\frac {1}{4}} a^{2} d^{2}\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {c}{d}\right )^{\frac {1}{4}} - 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {c}{d}\right )^{\frac {1}{4}}}\right )}{8 \, c d^{4}} + \frac {\sqrt {2} {\left (9 \, \left (c d^{3}\right )^{\frac {1}{4}} b^{2} c^{2} - 10 \, \left (c d^{3}\right )^{\frac {1}{4}} a b c d + \left (c d^{3}\right )^{\frac {1}{4}} a^{2} d^{2}\right )} \log \left (\sqrt {2} \sqrt {x} \left (\frac {c}{d}\right )^{\frac {1}{4}} + x + \sqrt {\frac {c}{d}}\right )}{16 \, c d^{4}} - \frac {\sqrt {2} {\left (9 \, \left (c d^{3}\right )^{\frac {1}{4}} b^{2} c^{2} - 10 \, \left (c d^{3}\right )^{\frac {1}{4}} a b c d + \left (c d^{3}\right )^{\frac {1}{4}} a^{2} d^{2}\right )} \log \left (-\sqrt {2} \sqrt {x} \left (\frac {c}{d}\right )^{\frac {1}{4}} + x + \sqrt {\frac {c}{d}}\right )}{16 \, c d^{4}} - \frac {b^{2} c^{2} \sqrt {x} - 2 \, a b c d \sqrt {x} + a^{2} d^{2} \sqrt {x}}{2 \, {\left (d x^{2} + c\right )} d^{3}} + \frac {2 \, {\left (b^{2} d^{8} x^{\frac {5}{2}} - 10 \, b^{2} c d^{7} \sqrt {x} + 10 \, a b d^{8} \sqrt {x}\right )}}{5 \, d^{10}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(3/2)*(b*x^2+a)^2/(d*x^2+c)^2,x, algorithm="giac")

[Out]

1/8*sqrt(2)*(9*(c*d^3)^(1/4)*b^2*c^2 - 10*(c*d^3)^(1/4)*a*b*c*d + (c*d^3)^(1/4)*a^2*d^2)*arctan(1/2*sqrt(2)*(s
qrt(2)*(c/d)^(1/4) + 2*sqrt(x))/(c/d)^(1/4))/(c*d^4) + 1/8*sqrt(2)*(9*(c*d^3)^(1/4)*b^2*c^2 - 10*(c*d^3)^(1/4)
*a*b*c*d + (c*d^3)^(1/4)*a^2*d^2)*arctan(-1/2*sqrt(2)*(sqrt(2)*(c/d)^(1/4) - 2*sqrt(x))/(c/d)^(1/4))/(c*d^4) +
1/16*sqrt(2)*(9*(c*d^3)^(1/4)*b^2*c^2 - 10*(c*d^3)^(1/4)*a*b*c*d + (c*d^3)^(1/4)*a^2*d^2)*log(sqrt(2)*sqrt(x)
*(c/d)^(1/4) + x + sqrt(c/d))/(c*d^4) - 1/16*sqrt(2)*(9*(c*d^3)^(1/4)*b^2*c^2 - 10*(c*d^3)^(1/4)*a*b*c*d + (c*
d^3)^(1/4)*a^2*d^2)*log(-sqrt(2)*sqrt(x)*(c/d)^(1/4) + x + sqrt(c/d))/(c*d^4) - 1/2*(b^2*c^2*sqrt(x) - 2*a*b*c
*d*sqrt(x) + a^2*d^2*sqrt(x))/((d*x^2 + c)*d^3) + 2/5*(b^2*d^8*x^(5/2) - 10*b^2*c*d^7*sqrt(x) + 10*a*b*d^8*sqr
t(x))/d^10

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Mupad [B]
time = 0.14, size = 1238, normalized size = 3.58 \begin {gather*} \frac {2\,b^2\,x^{5/2}}{5\,d^2}-\frac {\sqrt {x}\,\left (\frac {a^2\,d^2}{2}-a\,b\,c\,d+\frac {b^2\,c^2}{2}\right )}{d^4\,x^2+c\,d^3}-\sqrt {x}\,\left (\frac {4\,b^2\,c}{d^3}-\frac {4\,a\,b}{d^2}\right )+\frac {\mathrm {atan}\left (\frac {\frac {\left (\frac {\sqrt {x}\,\left (a^4\,d^4-20\,a^3\,b\,c\,d^3+118\,a^2\,b^2\,c^2\,d^2-180\,a\,b^3\,c^3\,d+81\,b^4\,c^4\right )}{d^3}-\frac {\left (a\,d-b\,c\right )\,\left (a\,d-9\,b\,c\right )\,\left (8\,a^2\,c\,d^2-80\,a\,b\,c^2\,d+72\,b^2\,c^3\right )}{8\,{\left (-c\right )}^{3/4}\,d^{13/4}}\right )\,\left (a\,d-b\,c\right )\,\left (a\,d-9\,b\,c\right )\,1{}\mathrm {i}}{8\,{\left (-c\right )}^{3/4}\,d^{13/4}}+\frac {\left (\frac {\sqrt {x}\,\left (a^4\,d^4-20\,a^3\,b\,c\,d^3+118\,a^2\,b^2\,c^2\,d^2-180\,a\,b^3\,c^3\,d+81\,b^4\,c^4\right )}{d^3}+\frac {\left (a\,d-b\,c\right )\,\left (a\,d-9\,b\,c\right )\,\left (8\,a^2\,c\,d^2-80\,a\,b\,c^2\,d+72\,b^2\,c^3\right )}{8\,{\left (-c\right )}^{3/4}\,d^{13/4}}\right )\,\left (a\,d-b\,c\right )\,\left (a\,d-9\,b\,c\right )\,1{}\mathrm {i}}{8\,{\left (-c\right )}^{3/4}\,d^{13/4}}}{\frac {\left (\frac {\sqrt {x}\,\left (a^4\,d^4-20\,a^3\,b\,c\,d^3+118\,a^2\,b^2\,c^2\,d^2-180\,a\,b^3\,c^3\,d+81\,b^4\,c^4\right )}{d^3}-\frac {\left (a\,d-b\,c\right )\,\left (a\,d-9\,b\,c\right )\,\left (8\,a^2\,c\,d^2-80\,a\,b\,c^2\,d+72\,b^2\,c^3\right )}{8\,{\left (-c\right )}^{3/4}\,d^{13/4}}\right )\,\left (a\,d-b\,c\right )\,\left (a\,d-9\,b\,c\right )}{8\,{\left (-c\right )}^{3/4}\,d^{13/4}}-\frac {\left (\frac {\sqrt {x}\,\left (a^4\,d^4-20\,a^3\,b\,c\,d^3+118\,a^2\,b^2\,c^2\,d^2-180\,a\,b^3\,c^3\,d+81\,b^4\,c^4\right )}{d^3}+\frac {\left (a\,d-b\,c\right )\,\left (a\,d-9\,b\,c\right )\,\left (8\,a^2\,c\,d^2-80\,a\,b\,c^2\,d+72\,b^2\,c^3\right )}{8\,{\left (-c\right )}^{3/4}\,d^{13/4}}\right )\,\left (a\,d-b\,c\right )\,\left (a\,d-9\,b\,c\right )}{8\,{\left (-c\right )}^{3/4}\,d^{13/4}}}\right )\,\left (a\,d-b\,c\right )\,\left (a\,d-9\,b\,c\right )\,1{}\mathrm {i}}{4\,{\left (-c\right )}^{3/4}\,d^{13/4}}+\frac {\mathrm {atan}\left (\frac {\frac {\left (\frac {\sqrt {x}\,\left (a^4\,d^4-20\,a^3\,b\,c\,d^3+118\,a^2\,b^2\,c^2\,d^2-180\,a\,b^3\,c^3\,d+81\,b^4\,c^4\right )}{d^3}-\frac {\left (a\,d-b\,c\right )\,\left (a\,d-9\,b\,c\right )\,\left (8\,a^2\,c\,d^2-80\,a\,b\,c^2\,d+72\,b^2\,c^3\right )\,1{}\mathrm {i}}{8\,{\left (-c\right )}^{3/4}\,d^{13/4}}\right )\,\left (a\,d-b\,c\right )\,\left (a\,d-9\,b\,c\right )}{8\,{\left (-c\right )}^{3/4}\,d^{13/4}}+\frac {\left (\frac {\sqrt {x}\,\left (a^4\,d^4-20\,a^3\,b\,c\,d^3+118\,a^2\,b^2\,c^2\,d^2-180\,a\,b^3\,c^3\,d+81\,b^4\,c^4\right )}{d^3}+\frac {\left (a\,d-b\,c\right )\,\left (a\,d-9\,b\,c\right )\,\left (8\,a^2\,c\,d^2-80\,a\,b\,c^2\,d+72\,b^2\,c^3\right )\,1{}\mathrm {i}}{8\,{\left (-c\right )}^{3/4}\,d^{13/4}}\right )\,\left (a\,d-b\,c\right )\,\left (a\,d-9\,b\,c\right )}{8\,{\left (-c\right )}^{3/4}\,d^{13/4}}}{\frac {\left (\frac {\sqrt {x}\,\left (a^4\,d^4-20\,a^3\,b\,c\,d^3+118\,a^2\,b^2\,c^2\,d^2-180\,a\,b^3\,c^3\,d+81\,b^4\,c^4\right )}{d^3}-\frac {\left (a\,d-b\,c\right )\,\left (a\,d-9\,b\,c\right )\,\left (8\,a^2\,c\,d^2-80\,a\,b\,c^2\,d+72\,b^2\,c^3\right )\,1{}\mathrm {i}}{8\,{\left (-c\right )}^{3/4}\,d^{13/4}}\right )\,\left (a\,d-b\,c\right )\,\left (a\,d-9\,b\,c\right )\,1{}\mathrm {i}}{8\,{\left (-c\right )}^{3/4}\,d^{13/4}}-\frac {\left (\frac {\sqrt {x}\,\left (a^4\,d^4-20\,a^3\,b\,c\,d^3+118\,a^2\,b^2\,c^2\,d^2-180\,a\,b^3\,c^3\,d+81\,b^4\,c^4\right )}{d^3}+\frac {\left (a\,d-b\,c\right )\,\left (a\,d-9\,b\,c\right )\,\left (8\,a^2\,c\,d^2-80\,a\,b\,c^2\,d+72\,b^2\,c^3\right )\,1{}\mathrm {i}}{8\,{\left (-c\right )}^{3/4}\,d^{13/4}}\right )\,\left (a\,d-b\,c\right )\,\left (a\,d-9\,b\,c\right )\,1{}\mathrm {i}}{8\,{\left (-c\right )}^{3/4}\,d^{13/4}}}\right )\,\left (a\,d-b\,c\right )\,\left (a\,d-9\,b\,c\right )}{4\,{\left (-c\right )}^{3/4}\,d^{13/4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^(3/2)*(a + b*x^2)^2)/(c + d*x^2)^2,x)

[Out]

(2*b^2*x^(5/2))/(5*d^2) - (x^(1/2)*((a^2*d^2)/2 + (b^2*c^2)/2 - a*b*c*d))/(c*d^3 + d^4*x^2) - x^(1/2)*((4*b^2*
c)/d^3 - (4*a*b)/d^2) + (atan(((((x^(1/2)*(a^4*d^4 + 81*b^4*c^4 + 118*a^2*b^2*c^2*d^2 - 180*a*b^3*c^3*d - 20*a
^3*b*c*d^3))/d^3 - ((a*d - b*c)*(a*d - 9*b*c)*(72*b^2*c^3 + 8*a^2*c*d^2 - 80*a*b*c^2*d))/(8*(-c)^(3/4)*d^(13/4
)))*(a*d - b*c)*(a*d - 9*b*c)*1i)/(8*(-c)^(3/4)*d^(13/4)) + (((x^(1/2)*(a^4*d^4 + 81*b^4*c^4 + 118*a^2*b^2*c^2
*d^2 - 180*a*b^3*c^3*d - 20*a^3*b*c*d^3))/d^3 + ((a*d - b*c)*(a*d - 9*b*c)*(72*b^2*c^3 + 8*a^2*c*d^2 - 80*a*b*
c^2*d))/(8*(-c)^(3/4)*d^(13/4)))*(a*d - b*c)*(a*d - 9*b*c)*1i)/(8*(-c)^(3/4)*d^(13/4)))/((((x^(1/2)*(a^4*d^4 +
81*b^4*c^4 + 118*a^2*b^2*c^2*d^2 - 180*a*b^3*c^3*d - 20*a^3*b*c*d^3))/d^3 - ((a*d - b*c)*(a*d - 9*b*c)*(72*b^
2*c^3 + 8*a^2*c*d^2 - 80*a*b*c^2*d))/(8*(-c)^(3/4)*d^(13/4)))*(a*d - b*c)*(a*d - 9*b*c))/(8*(-c)^(3/4)*d^(13/4
)) - (((x^(1/2)*(a^4*d^4 + 81*b^4*c^4 + 118*a^2*b^2*c^2*d^2 - 180*a*b^3*c^3*d - 20*a^3*b*c*d^3))/d^3 + ((a*d -
b*c)*(a*d - 9*b*c)*(72*b^2*c^3 + 8*a^2*c*d^2 - 80*a*b*c^2*d))/(8*(-c)^(3/4)*d^(13/4)))*(a*d - b*c)*(a*d - 9*b
*c))/(8*(-c)^(3/4)*d^(13/4))))*(a*d - b*c)*(a*d - 9*b*c)*1i)/(4*(-c)^(3/4)*d^(13/4)) + (atan(((((x^(1/2)*(a^4*
d^4 + 81*b^4*c^4 + 118*a^2*b^2*c^2*d^2 - 180*a*b^3*c^3*d - 20*a^3*b*c*d^3))/d^3 - ((a*d - b*c)*(a*d - 9*b*c)*(
72*b^2*c^3 + 8*a^2*c*d^2 - 80*a*b*c^2*d)*1i)/(8*(-c)^(3/4)*d^(13/4)))*(a*d - b*c)*(a*d - 9*b*c))/(8*(-c)^(3/4)
*d^(13/4)) + (((x^(1/2)*(a^4*d^4 + 81*b^4*c^4 + 118*a^2*b^2*c^2*d^2 - 180*a*b^3*c^3*d - 20*a^3*b*c*d^3))/d^3 +
((a*d - b*c)*(a*d - 9*b*c)*(72*b^2*c^3 + 8*a^2*c*d^2 - 80*a*b*c^2*d)*1i)/(8*(-c)^(3/4)*d^(13/4)))*(a*d - b*c)
*(a*d - 9*b*c))/(8*(-c)^(3/4)*d^(13/4)))/((((x^(1/2)*(a^4*d^4 + 81*b^4*c^4 + 118*a^2*b^2*c^2*d^2 - 180*a*b^3*c
^3*d - 20*a^3*b*c*d^3))/d^3 - ((a*d - b*c)*(a*d - 9*b*c)*(72*b^2*c^3 + 8*a^2*c*d^2 - 80*a*b*c^2*d)*1i)/(8*(-c)
^(3/4)*d^(13/4)))*(a*d - b*c)*(a*d - 9*b*c)*1i)/(8*(-c)^(3/4)*d^(13/4)) - (((x^(1/2)*(a^4*d^4 + 81*b^4*c^4 + 1
18*a^2*b^2*c^2*d^2 - 180*a*b^3*c^3*d - 20*a^3*b*c*d^3))/d^3 + ((a*d - b*c)*(a*d - 9*b*c)*(72*b^2*c^3 + 8*a^2*c
*d^2 - 80*a*b*c^2*d)*1i)/(8*(-c)^(3/4)*d^(13/4)))*(a*d - b*c)*(a*d - 9*b*c)*1i)/(8*(-c)^(3/4)*d^(13/4))))*(a*d
- b*c)*(a*d - 9*b*c))/(4*(-c)^(3/4)*d^(13/4))

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