\(\int \frac {x^{3/2} (a+b x^2)^2}{(c+d x^2)^3} \, dx\) [435]

Optimal result

Integrand size = 24, antiderivative size = 402 \[ \int \frac {x^{3/2} \left (a+b x^2\right )^2}{\left (c+d x^2\right )^3} \, dx=-\frac {\left (10 a b-\frac {45 b^2 c}{d}+\frac {3 a^2 d}{c}\right ) \sqrt {x}}{16 c d^2}+\frac {(b c-a d)^2 x^{5/2}}{4 c d^2 \left (c+d x^2\right )^2}-\frac {(b c-a d) (13 b c+3 a d) x^{5/2}}{16 c^2 d^2 \left (c+d x^2\right )}+\frac {\left (45 b^2 c^2-10 a b c d-3 a^2 d^2\right ) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{32 \sqrt {2} c^{7/4} d^{13/4}}-\frac {\left (45 b^2 c^2-10 a b c d-3 a^2 d^2\right ) \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{32 \sqrt {2} c^{7/4} d^{13/4}}+\frac {\left (45 b^2 c^2-10 a b c d-3 a^2 d^2\right ) \log \left (\sqrt {c}-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{64 \sqrt {2} c^{7/4} d^{13/4}}-\frac {\left (45 b^2 c^2-10 a b c d-3 a^2 d^2\right ) \log \left (\sqrt {c}+\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{64 \sqrt {2} c^{7/4} d^{13/4}} \]

[Out]

1/4*(-a*d+b*c)^2*x^(5/2)/c/d^2/(d*x^2+c)^2-1/16*(-a*d+b*c)*(3*a*d+13*b*c)*x^(5/2)/c^2/d^2/(d*x^2+c)+1/64*(-3*a
^2*d^2-10*a*b*c*d+45*b^2*c^2)*arctan(1-d^(1/4)*2^(1/2)*x^(1/2)/c^(1/4))/c^(7/4)/d^(13/4)*2^(1/2)-1/64*(-3*a^2*
d^2-10*a*b*c*d+45*b^2*c^2)*arctan(1+d^(1/4)*2^(1/2)*x^(1/2)/c^(1/4))/c^(7/4)/d^(13/4)*2^(1/2)+1/128*(-3*a^2*d^
2-10*a*b*c*d+45*b^2*c^2)*ln(c^(1/2)+x*d^(1/2)-c^(1/4)*d^(1/4)*2^(1/2)*x^(1/2))/c^(7/4)/d^(13/4)*2^(1/2)-1/128*
(-3*a^2*d^2-10*a*b*c*d+45*b^2*c^2)*ln(c^(1/2)+x*d^(1/2)+c^(1/4)*d^(1/4)*2^(1/2)*x^(1/2))/c^(7/4)/d^(13/4)*2^(1
/2)-1/16*(10*a*b-45*b^2*c/d+3*a^2*d/c)*x^(1/2)/c/d^2

Rubi [A] (verified)

Time = 0.23 (sec) , antiderivative size = 402, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {474, 468, 327, 335, 217, 1179, 642, 1176, 631, 210} \[ \int \frac {x^{3/2} \left (a+b x^2\right )^2}{\left (c+d x^2\right )^3} \, dx=\frac {\left (-3 a^2 d^2-10 a b c d+45 b^2 c^2\right ) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{32 \sqrt {2} c^{7/4} d^{13/4}}-\frac {\left (-3 a^2 d^2-10 a b c d+45 b^2 c^2\right ) \arctan \left (\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}+1\right )}{32 \sqrt {2} c^{7/4} d^{13/4}}+\frac {\left (-3 a^2 d^2-10 a b c d+45 b^2 c^2\right ) \log \left (-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{64 \sqrt {2} c^{7/4} d^{13/4}}-\frac {\left (-3 a^2 d^2-10 a b c d+45 b^2 c^2\right ) \log \left (\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{64 \sqrt {2} c^{7/4} d^{13/4}}-\frac {\sqrt {x} \left (\frac {3 a^2 d}{c}+10 a b-\frac {45 b^2 c}{d}\right )}{16 c d^2}-\frac {x^{5/2} (b c-a d) (3 a d+13 b c)}{16 c^2 d^2 \left (c+d x^2\right )}+\frac {x^{5/2} (b c-a d)^2}{4 c d^2 \left (c+d x^2\right )^2} \]

[In]

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

[Out]

-1/16*((10*a*b - (45*b^2*c)/d + (3*a^2*d)/c)*Sqrt[x])/(c*d^2) + ((b*c - a*d)^2*x^(5/2))/(4*c*d^2*(c + d*x^2)^2
) - ((b*c - a*d)*(13*b*c + 3*a*d)*x^(5/2))/(16*c^2*d^2*(c + d*x^2)) + ((45*b^2*c^2 - 10*a*b*c*d - 3*a^2*d^2)*A
rcTan[1 - (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/(32*Sqrt[2]*c^(7/4)*d^(13/4)) - ((45*b^2*c^2 - 10*a*b*c*d - 3*a^
2*d^2)*ArcTan[1 + (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/(32*Sqrt[2]*c^(7/4)*d^(13/4)) + ((45*b^2*c^2 - 10*a*b*c*
d - 3*a^2*d^2)*Log[Sqrt[c] - Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/(64*Sqrt[2]*c^(7/4)*d^(13/4)) - ((4
5*b^2*c^2 - 10*a*b*c*d - 3*a^2*d^2)*Log[Sqrt[c] + Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/(64*Sqrt[2]*c^
(7/4)*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 468

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Simp[(-(b*c - a*d
))*(e*x)^(m + 1)*((a + b*x^n)^(p + 1)/(a*b*e*n*(p + 1))), x] - Dist[(a*d*(m + 1) - b*c*(m + n*(p + 1) + 1))/(a
*b*n*(p + 1)), Int[(e*x)^m*(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && NeQ[b*c - a*d, 0]
&& LtQ[p, -1] && (( !IntegerQ[p + 1/2] && NeQ[p, -5/4]) ||  !RationalQ[m] || (IGtQ[n, 0] && ILtQ[p + 1/2, 0]
&& LeQ[-1, m, (-n)*(p + 1)]))

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*} \text {integral}& = \frac {(b c-a d)^2 x^{5/2}}{4 c d^2 \left (c+d x^2\right )^2}-\frac {\int \frac {x^{3/2} \left (\frac {1}{2} \left (-8 a^2 d^2+5 (b c-a d)^2\right )-4 b^2 c d x^2\right )}{\left (c+d x^2\right )^2} \, dx}{4 c d^2} \\ & = \frac {(b c-a d)^2 x^{5/2}}{4 c d^2 \left (c+d x^2\right )^2}-\frac {(b c-a d) (13 b c+3 a d) x^{5/2}}{16 c^2 d^2 \left (c+d x^2\right )}+\frac {\left (45 b^2 c^2-10 a b c d-3 a^2 d^2\right ) \int \frac {x^{3/2}}{c+d x^2} \, dx}{32 c^2 d^2} \\ & = \frac {\left (45 b^2 c^2-10 a b c d-3 a^2 d^2\right ) \sqrt {x}}{16 c^2 d^3}+\frac {(b c-a d)^2 x^{5/2}}{4 c d^2 \left (c+d x^2\right )^2}-\frac {(b c-a d) (13 b c+3 a d) x^{5/2}}{16 c^2 d^2 \left (c+d x^2\right )}-\frac {\left (45 b^2 c^2-10 a b c d-3 a^2 d^2\right ) \int \frac {1}{\sqrt {x} \left (c+d x^2\right )} \, dx}{32 c d^3} \\ & = \frac {\left (45 b^2 c^2-10 a b c d-3 a^2 d^2\right ) \sqrt {x}}{16 c^2 d^3}+\frac {(b c-a d)^2 x^{5/2}}{4 c d^2 \left (c+d x^2\right )^2}-\frac {(b c-a d) (13 b c+3 a d) x^{5/2}}{16 c^2 d^2 \left (c+d x^2\right )}-\frac {\left (45 b^2 c^2-10 a b c d-3 a^2 d^2\right ) \text {Subst}\left (\int \frac {1}{c+d x^4} \, dx,x,\sqrt {x}\right )}{16 c d^3} \\ & = \frac {\left (45 b^2 c^2-10 a b c d-3 a^2 d^2\right ) \sqrt {x}}{16 c^2 d^3}+\frac {(b c-a d)^2 x^{5/2}}{4 c d^2 \left (c+d x^2\right )^2}-\frac {(b c-a d) (13 b c+3 a d) x^{5/2}}{16 c^2 d^2 \left (c+d x^2\right )}-\frac {\left (45 b^2 c^2-10 a b c d-3 a^2 d^2\right ) \text {Subst}\left (\int \frac {\sqrt {c}-\sqrt {d} x^2}{c+d x^4} \, dx,x,\sqrt {x}\right )}{32 c^{3/2} d^3}-\frac {\left (45 b^2 c^2-10 a b c d-3 a^2 d^2\right ) \text {Subst}\left (\int \frac {\sqrt {c}+\sqrt {d} x^2}{c+d x^4} \, dx,x,\sqrt {x}\right )}{32 c^{3/2} d^3} \\ & = \frac {\left (45 b^2 c^2-10 a b c d-3 a^2 d^2\right ) \sqrt {x}}{16 c^2 d^3}+\frac {(b c-a d)^2 x^{5/2}}{4 c d^2 \left (c+d x^2\right )^2}-\frac {(b c-a d) (13 b c+3 a d) x^{5/2}}{16 c^2 d^2 \left (c+d x^2\right )}-\frac {\left (45 b^2 c^2-10 a b c d-3 a^2 d^2\right ) \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 )}{64 c^{3/2} d^{7/2}}-\frac {\left (45 b^2 c^2-10 a b c d-3 a^2 d^2\right ) \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 )}{64 c^{3/2} d^{7/2}}+\frac {\left (45 b^2 c^2-10 a b c d-3 a^2 d^2\right ) \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 )}{64 \sqrt {2} c^{7/4} d^{13/4}}+\frac {\left (45 b^2 c^2-10 a b c d-3 a^2 d^2\right ) \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 )}{64 \sqrt {2} c^{7/4} d^{13/4}} \\ & = \frac {\left (45 b^2 c^2-10 a b c d-3 a^2 d^2\right ) \sqrt {x}}{16 c^2 d^3}+\frac {(b c-a d)^2 x^{5/2}}{4 c d^2 \left (c+d x^2\right )^2}-\frac {(b c-a d) (13 b c+3 a d) x^{5/2}}{16 c^2 d^2 \left (c+d x^2\right )}+\frac {\left (45 b^2 c^2-10 a b c d-3 a^2 d^2\right ) \log \left (\sqrt {c}-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{64 \sqrt {2} c^{7/4} d^{13/4}}-\frac {\left (45 b^2 c^2-10 a b c d-3 a^2 d^2\right ) \log \left (\sqrt {c}+\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{64 \sqrt {2} c^{7/4} d^{13/4}}-\frac {\left (45 b^2 c^2-10 a b c d-3 a^2 d^2\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{32 \sqrt {2} c^{7/4} d^{13/4}}+\frac {\left (45 b^2 c^2-10 a b c d-3 a^2 d^2\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{32 \sqrt {2} c^{7/4} d^{13/4}} \\ & = \frac {\left (45 b^2 c^2-10 a b c d-3 a^2 d^2\right ) \sqrt {x}}{16 c^2 d^3}+\frac {(b c-a d)^2 x^{5/2}}{4 c d^2 \left (c+d x^2\right )^2}-\frac {(b c-a d) (13 b c+3 a d) x^{5/2}}{16 c^2 d^2 \left (c+d x^2\right )}+\frac {\left (45 b^2 c^2-10 a b c d-3 a^2 d^2\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{32 \sqrt {2} c^{7/4} d^{13/4}}-\frac {\left (45 b^2 c^2-10 a b c d-3 a^2 d^2\right ) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{32 \sqrt {2} c^{7/4} d^{13/4}}+\frac {\left (45 b^2 c^2-10 a b c d-3 a^2 d^2\right ) \log \left (\sqrt {c}-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{64 \sqrt {2} c^{7/4} d^{13/4}}-\frac {\left (45 b^2 c^2-10 a b c d-3 a^2 d^2\right ) \log \left (\sqrt {c}+\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{64 \sqrt {2} c^{7/4} d^{13/4}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.84 (sec) , antiderivative size = 232, normalized size of antiderivative = 0.58 \[ \int \frac {x^{3/2} \left (a+b x^2\right )^2}{\left (c+d x^2\right )^3} \, dx=\frac {\frac {4 c^{3/4} \sqrt [4]{d} \sqrt {x} \left (a^2 d^2 \left (-3 c+d x^2\right )-2 a b c d \left (5 c+9 d x^2\right )+b^2 c \left (45 c^2+81 c d x^2+32 d^2 x^4\right )\right )}{\left (c+d x^2\right )^2}+\sqrt {2} \left (45 b^2 c^2-10 a b c d-3 a^2 d^2\right ) \arctan \left (\frac {\sqrt {c}-\sqrt {d} x}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}}\right )-\sqrt {2} \left (45 b^2 c^2-10 a b c d-3 a^2 d^2\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}}{\sqrt {c}+\sqrt {d} x}\right )}{64 c^{7/4} d^{13/4}} \]

[In]

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

[Out]

((4*c^(3/4)*d^(1/4)*Sqrt[x]*(a^2*d^2*(-3*c + d*x^2) - 2*a*b*c*d*(5*c + 9*d*x^2) + b^2*c*(45*c^2 + 81*c*d*x^2 +
32*d^2*x^4)))/(c + d*x^2)^2 + Sqrt[2]*(45*b^2*c^2 - 10*a*b*c*d - 3*a^2*d^2)*ArcTan[(Sqrt[c] - Sqrt[d]*x)/(Sqr
t[2]*c^(1/4)*d^(1/4)*Sqrt[x])] - Sqrt[2]*(45*b^2*c^2 - 10*a*b*c*d - 3*a^2*d^2)*ArcTanh[(Sqrt[2]*c^(1/4)*d^(1/4
)*Sqrt[x])/(Sqrt[c] + Sqrt[d]*x)])/(64*c^(7/4)*d^(13/4))

Maple [A] (verified)

Time = 2.77 (sec) , antiderivative size = 216, normalized size of antiderivative = 0.54

method	result	size
derivativedivides	\(\frac {2 b^{2} \sqrt {x}}{d^{3}}+\frac {\frac {2 \left (\frac {d \left (a^{2} d^{2}-18 a b c d +17 b^{2} c^{2}\right ) x^{\frac {5}{2}}}{32 c}+\left (-\frac {3}{32} a^{2} d^{2}-\frac {5}{16} a b c d +\frac {13}{32} b^{2} c^{2}\right ) \sqrt {x}\right )}{\left (d \,x^{2}+c \right )^{2}}+\frac {\left (3 a^{2} d^{2}+10 a b c d -45 b^{2} c^{2}\right ) \left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}{x -\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}-1\right )\right )}{128 c^{2}}}{d^{3}}\)	\(216\)
default	\(\frac {2 b^{2} \sqrt {x}}{d^{3}}+\frac {\frac {2 \left (\frac {d \left (a^{2} d^{2}-18 a b c d +17 b^{2} c^{2}\right ) x^{\frac {5}{2}}}{32 c}+\left (-\frac {3}{32} a^{2} d^{2}-\frac {5}{16} a b c d +\frac {13}{32} b^{2} c^{2}\right ) \sqrt {x}\right )}{\left (d \,x^{2}+c \right )^{2}}+\frac {\left (3 a^{2} d^{2}+10 a b c d -45 b^{2} c^{2}\right ) \left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}{x -\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}-1\right )\right )}{128 c^{2}}}{d^{3}}\)	\(216\)
risch	\(\frac {2 b^{2} \sqrt {x}}{d^{3}}+\frac {\frac {\frac {d \left (a^{2} d^{2}-18 a b c d +17 b^{2} c^{2}\right ) x^{\frac {5}{2}}}{16 c}+2 \left (-\frac {3}{32} a^{2} d^{2}-\frac {5}{16} a b c d +\frac {13}{32} b^{2} c^{2}\right ) \sqrt {x}}{\left (d \,x^{2}+c \right )^{2}}+\frac {\left (3 a^{2} d^{2}+10 a b c d -45 b^{2} c^{2}\right ) \left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}{x -\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}-1\right )\right )}{128 c^{2}}}{d^{3}}\)	\(216\)

[In]

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

[Out]

2*b^2/d^3*x^(1/2)+2/d^3*((1/32*d*(a^2*d^2-18*a*b*c*d+17*b^2*c^2)/c*x^(5/2)+(-3/32*a^2*d^2-5/16*a*b*c*d+13/32*b
^2*c^2)*x^(1/2))/(d*x^2+c)^2+1/256*(3*a^2*d^2+10*a*b*c*d-45*b^2*c^2)/c^2*(c/d)^(1/4)*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)))

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.28 (sec) , antiderivative size = 1310, normalized size of antiderivative = 3.26 \[ \int \frac {x^{3/2} \left (a+b x^2\right )^2}{\left (c+d x^2\right )^3} \, dx=\text {Too large to display} \]

[In]

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

[Out]

1/64*((c*d^5*x^4 + 2*c^2*d^4*x^2 + c^3*d^3)*(-(4100625*b^8*c^8 - 3645000*a*b^7*c^7*d + 121500*a^2*b^6*c^6*d^2
+ 549000*a^3*b^5*c^5*d^3 - 42650*a^4*b^4*c^4*d^4 - 36600*a^5*b^3*c^3*d^5 + 540*a^6*b^2*c^2*d^6 + 1080*a^7*b*c*
d^7 + 81*a^8*d^8)/(c^7*d^13))^(1/4)*log(c^2*d^3*(-(4100625*b^8*c^8 - 3645000*a*b^7*c^7*d + 121500*a^2*b^6*c^6*
d^2 + 549000*a^3*b^5*c^5*d^3 - 42650*a^4*b^4*c^4*d^4 - 36600*a^5*b^3*c^3*d^5 + 540*a^6*b^2*c^2*d^6 + 1080*a^7*
b*c*d^7 + 81*a^8*d^8)/(c^7*d^13))^(1/4) - (45*b^2*c^2 - 10*a*b*c*d - 3*a^2*d^2)*sqrt(x)) - (-I*c*d^5*x^4 - 2*I
*c^2*d^4*x^2 - I*c^3*d^3)*(-(4100625*b^8*c^8 - 3645000*a*b^7*c^7*d + 121500*a^2*b^6*c^6*d^2 + 549000*a^3*b^5*c
^5*d^3 - 42650*a^4*b^4*c^4*d^4 - 36600*a^5*b^3*c^3*d^5 + 540*a^6*b^2*c^2*d^6 + 1080*a^7*b*c*d^7 + 81*a^8*d^8)/
(c^7*d^13))^(1/4)*log(I*c^2*d^3*(-(4100625*b^8*c^8 - 3645000*a*b^7*c^7*d + 121500*a^2*b^6*c^6*d^2 + 549000*a^3
*b^5*c^5*d^3 - 42650*a^4*b^4*c^4*d^4 - 36600*a^5*b^3*c^3*d^5 + 540*a^6*b^2*c^2*d^6 + 1080*a^7*b*c*d^7 + 81*a^8
*d^8)/(c^7*d^13))^(1/4) - (45*b^2*c^2 - 10*a*b*c*d - 3*a^2*d^2)*sqrt(x)) - (I*c*d^5*x^4 + 2*I*c^2*d^4*x^2 + I*
c^3*d^3)*(-(4100625*b^8*c^8 - 3645000*a*b^7*c^7*d + 121500*a^2*b^6*c^6*d^2 + 549000*a^3*b^5*c^5*d^3 - 42650*a^
4*b^4*c^4*d^4 - 36600*a^5*b^3*c^3*d^5 + 540*a^6*b^2*c^2*d^6 + 1080*a^7*b*c*d^7 + 81*a^8*d^8)/(c^7*d^13))^(1/4)
*log(-I*c^2*d^3*(-(4100625*b^8*c^8 - 3645000*a*b^7*c^7*d + 121500*a^2*b^6*c^6*d^2 + 549000*a^3*b^5*c^5*d^3 - 4
2650*a^4*b^4*c^4*d^4 - 36600*a^5*b^3*c^3*d^5 + 540*a^6*b^2*c^2*d^6 + 1080*a^7*b*c*d^7 + 81*a^8*d^8)/(c^7*d^13)
)^(1/4) - (45*b^2*c^2 - 10*a*b*c*d - 3*a^2*d^2)*sqrt(x)) - (c*d^5*x^4 + 2*c^2*d^4*x^2 + c^3*d^3)*(-(4100625*b^
8*c^8 - 3645000*a*b^7*c^7*d + 121500*a^2*b^6*c^6*d^2 + 549000*a^3*b^5*c^5*d^3 - 42650*a^4*b^4*c^4*d^4 - 36600*
a^5*b^3*c^3*d^5 + 540*a^6*b^2*c^2*d^6 + 1080*a^7*b*c*d^7 + 81*a^8*d^8)/(c^7*d^13))^(1/4)*log(-c^2*d^3*(-(41006
25*b^8*c^8 - 3645000*a*b^7*c^7*d + 121500*a^2*b^6*c^6*d^2 + 549000*a^3*b^5*c^5*d^3 - 42650*a^4*b^4*c^4*d^4 - 3
6600*a^5*b^3*c^3*d^5 + 540*a^6*b^2*c^2*d^6 + 1080*a^7*b*c*d^7 + 81*a^8*d^8)/(c^7*d^13))^(1/4) - (45*b^2*c^2 -
10*a*b*c*d - 3*a^2*d^2)*sqrt(x)) + 4*(32*b^2*c*d^2*x^4 + 45*b^2*c^3 - 10*a*b*c^2*d - 3*a^2*c*d^2 + (81*b^2*c^2
*d - 18*a*b*c*d^2 + a^2*d^3)*x^2)*sqrt(x))/(c*d^5*x^4 + 2*c^2*d^4*x^2 + c^3*d^3)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2302 vs. \(2 (403) = 806\).

Time = 155.05 (sec) , antiderivative size = 2302, normalized size of antiderivative = 5.73 \[ \int \frac {x^{3/2} \left (a+b x^2\right )^2}{\left (c+d x^2\right )^3} \, dx=\text {Too large to display} \]

[In]

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

[Out]

Piecewise((zoo*(-2*a**2/(7*x**(7/2)) - 4*a*b/(3*x**(3/2)) + 2*b**2*sqrt(x)), Eq(c, 0) & Eq(d, 0)), ((2*a**2*x*
*(5/2)/5 + 4*a*b*x**(9/2)/9 + 2*b**2*x**(13/2)/13)/c**3, Eq(d, 0)), ((-2*a**2/(7*x**(7/2)) - 4*a*b/(3*x**(3/2)
) + 2*b**2*sqrt(x))/d**3, Eq(c, 0)), (-12*a**2*c**2*d**2*sqrt(x)/(64*c**4*d**3 + 128*c**3*d**4*x**2 + 64*c**2*
d**5*x**4) - 3*a**2*c**2*d**2*(-c/d)**(1/4)*log(sqrt(x) - (-c/d)**(1/4))/(64*c**4*d**3 + 128*c**3*d**4*x**2 +
64*c**2*d**5*x**4) + 3*a**2*c**2*d**2*(-c/d)**(1/4)*log(sqrt(x) + (-c/d)**(1/4))/(64*c**4*d**3 + 128*c**3*d**4
*x**2 + 64*c**2*d**5*x**4) + 6*a**2*c**2*d**2*(-c/d)**(1/4)*atan(sqrt(x)/(-c/d)**(1/4))/(64*c**4*d**3 + 128*c*
*3*d**4*x**2 + 64*c**2*d**5*x**4) + 4*a**2*c*d**3*x**(5/2)/(64*c**4*d**3 + 128*c**3*d**4*x**2 + 64*c**2*d**5*x
**4) - 6*a**2*c*d**3*x**2*(-c/d)**(1/4)*log(sqrt(x) - (-c/d)**(1/4))/(64*c**4*d**3 + 128*c**3*d**4*x**2 + 64*c
**2*d**5*x**4) + 6*a**2*c*d**3*x**2*(-c/d)**(1/4)*log(sqrt(x) + (-c/d)**(1/4))/(64*c**4*d**3 + 128*c**3*d**4*x
**2 + 64*c**2*d**5*x**4) + 12*a**2*c*d**3*x**2*(-c/d)**(1/4)*atan(sqrt(x)/(-c/d)**(1/4))/(64*c**4*d**3 + 128*c
**3*d**4*x**2 + 64*c**2*d**5*x**4) - 3*a**2*d**4*x**4*(-c/d)**(1/4)*log(sqrt(x) - (-c/d)**(1/4))/(64*c**4*d**3
+ 128*c**3*d**4*x**2 + 64*c**2*d**5*x**4) + 3*a**2*d**4*x**4*(-c/d)**(1/4)*log(sqrt(x) + (-c/d)**(1/4))/(64*c
**4*d**3 + 128*c**3*d**4*x**2 + 64*c**2*d**5*x**4) + 6*a**2*d**4*x**4*(-c/d)**(1/4)*atan(sqrt(x)/(-c/d)**(1/4)
)/(64*c**4*d**3 + 128*c**3*d**4*x**2 + 64*c**2*d**5*x**4) - 40*a*b*c**3*d*sqrt(x)/(64*c**4*d**3 + 128*c**3*d**
4*x**2 + 64*c**2*d**5*x**4) - 10*a*b*c**3*d*(-c/d)**(1/4)*log(sqrt(x) - (-c/d)**(1/4))/(64*c**4*d**3 + 128*c**
3*d**4*x**2 + 64*c**2*d**5*x**4) + 10*a*b*c**3*d*(-c/d)**(1/4)*log(sqrt(x) + (-c/d)**(1/4))/(64*c**4*d**3 + 12
8*c**3*d**4*x**2 + 64*c**2*d**5*x**4) + 20*a*b*c**3*d*(-c/d)**(1/4)*atan(sqrt(x)/(-c/d)**(1/4))/(64*c**4*d**3
+ 128*c**3*d**4*x**2 + 64*c**2*d**5*x**4) - 72*a*b*c**2*d**2*x**(5/2)/(64*c**4*d**3 + 128*c**3*d**4*x**2 + 64*
c**2*d**5*x**4) - 20*a*b*c**2*d**2*x**2*(-c/d)**(1/4)*log(sqrt(x) - (-c/d)**(1/4))/(64*c**4*d**3 + 128*c**3*d*
*4*x**2 + 64*c**2*d**5*x**4) + 20*a*b*c**2*d**2*x**2*(-c/d)**(1/4)*log(sqrt(x) + (-c/d)**(1/4))/(64*c**4*d**3
+ 128*c**3*d**4*x**2 + 64*c**2*d**5*x**4) + 40*a*b*c**2*d**2*x**2*(-c/d)**(1/4)*atan(sqrt(x)/(-c/d)**(1/4))/(6
4*c**4*d**3 + 128*c**3*d**4*x**2 + 64*c**2*d**5*x**4) - 10*a*b*c*d**3*x**4*(-c/d)**(1/4)*log(sqrt(x) - (-c/d)*
*(1/4))/(64*c**4*d**3 + 128*c**3*d**4*x**2 + 64*c**2*d**5*x**4) + 10*a*b*c*d**3*x**4*(-c/d)**(1/4)*log(sqrt(x)
+ (-c/d)**(1/4))/(64*c**4*d**3 + 128*c**3*d**4*x**2 + 64*c**2*d**5*x**4) + 20*a*b*c*d**3*x**4*(-c/d)**(1/4)*a
tan(sqrt(x)/(-c/d)**(1/4))/(64*c**4*d**3 + 128*c**3*d**4*x**2 + 64*c**2*d**5*x**4) + 180*b**2*c**4*sqrt(x)/(64
*c**4*d**3 + 128*c**3*d**4*x**2 + 64*c**2*d**5*x**4) + 45*b**2*c**4*(-c/d)**(1/4)*log(sqrt(x) - (-c/d)**(1/4))
/(64*c**4*d**3 + 128*c**3*d**4*x**2 + 64*c**2*d**5*x**4) - 45*b**2*c**4*(-c/d)**(1/4)*log(sqrt(x) + (-c/d)**(1
/4))/(64*c**4*d**3 + 128*c**3*d**4*x**2 + 64*c**2*d**5*x**4) - 90*b**2*c**4*(-c/d)**(1/4)*atan(sqrt(x)/(-c/d)*
*(1/4))/(64*c**4*d**3 + 128*c**3*d**4*x**2 + 64*c**2*d**5*x**4) + 324*b**2*c**3*d*x**(5/2)/(64*c**4*d**3 + 128
*c**3*d**4*x**2 + 64*c**2*d**5*x**4) + 90*b**2*c**3*d*x**2*(-c/d)**(1/4)*log(sqrt(x) - (-c/d)**(1/4))/(64*c**4
*d**3 + 128*c**3*d**4*x**2 + 64*c**2*d**5*x**4) - 90*b**2*c**3*d*x**2*(-c/d)**(1/4)*log(sqrt(x) + (-c/d)**(1/4
))/(64*c**4*d**3 + 128*c**3*d**4*x**2 + 64*c**2*d**5*x**4) - 180*b**2*c**3*d*x**2*(-c/d)**(1/4)*atan(sqrt(x)/(
-c/d)**(1/4))/(64*c**4*d**3 + 128*c**3*d**4*x**2 + 64*c**2*d**5*x**4) + 128*b**2*c**2*d**2*x**(9/2)/(64*c**4*d
**3 + 128*c**3*d**4*x**2 + 64*c**2*d**5*x**4) + 45*b**2*c**2*d**2*x**4*(-c/d)**(1/4)*log(sqrt(x) - (-c/d)**(1/
4))/(64*c**4*d**3 + 128*c**3*d**4*x**2 + 64*c**2*d**5*x**4) - 45*b**2*c**2*d**2*x**4*(-c/d)**(1/4)*log(sqrt(x)
+ (-c/d)**(1/4))/(64*c**4*d**3 + 128*c**3*d**4*x**2 + 64*c**2*d**5*x**4) - 90*b**2*c**2*d**2*x**4*(-c/d)**(1/
4)*atan(sqrt(x)/(-c/d)**(1/4))/(64*c**4*d**3 + 128*c**3*d**4*x**2 + 64*c**2*d**5*x**4), True))

Maxima [A] (verification not implemented)

none

Time = 0.32 (sec) , antiderivative size = 374, normalized size of antiderivative = 0.93 \[ \int \frac {x^{3/2} \left (a+b x^2\right )^2}{\left (c+d x^2\right )^3} \, dx=\frac {{\left (17 \, b^{2} c^{2} d - 18 \, a b c d^{2} + a^{2} d^{3}\right )} x^{\frac {5}{2}} + {\left (13 \, b^{2} c^{3} - 10 \, a b c^{2} d - 3 \, a^{2} c d^{2}\right )} \sqrt {x}}{16 \, {\left (c d^{5} x^{4} + 2 \, c^{2} d^{4} x^{2} + c^{3} d^{3}\right )}} + \frac {2 \, b^{2} \sqrt {x}}{d^{3}} - \frac {\frac {2 \, \sqrt {2} {\left (45 \, b^{2} c^{2} - 10 \, a b c d - 3 \, a^{2} d^{2}\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} + 2 \, \sqrt {d} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {c} \sqrt {d}}}\right )}{\sqrt {c} \sqrt {\sqrt {c} \sqrt {d}}} + \frac {2 \, \sqrt {2} {\left (45 \, b^{2} c^{2} - 10 \, a b c d - 3 \, a^{2} d^{2}\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} - 2 \, \sqrt {d} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {c} \sqrt {d}}}\right )}{\sqrt {c} \sqrt {\sqrt {c} \sqrt {d}}} + \frac {\sqrt {2} {\left (45 \, b^{2} c^{2} - 10 \, a b c d - 3 \, a^{2} d^{2}\right )} \log \left (\sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} \sqrt {x} + \sqrt {d} x + \sqrt {c}\right )}{c^{\frac {3}{4}} d^{\frac {1}{4}}} - \frac {\sqrt {2} {\left (45 \, b^{2} c^{2} - 10 \, a b c d - 3 \, a^{2} d^{2}\right )} \log \left (-\sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} \sqrt {x} + \sqrt {d} x + \sqrt {c}\right )}{c^{\frac {3}{4}} d^{\frac {1}{4}}}}{128 \, c d^{3}} \]

[In]

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

[Out]

1/16*((17*b^2*c^2*d - 18*a*b*c*d^2 + a^2*d^3)*x^(5/2) + (13*b^2*c^3 - 10*a*b*c^2*d - 3*a^2*c*d^2)*sqrt(x))/(c*
d^5*x^4 + 2*c^2*d^4*x^2 + c^3*d^3) + 2*b^2*sqrt(x)/d^3 - 1/128*(2*sqrt(2)*(45*b^2*c^2 - 10*a*b*c*d - 3*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))) + 2*sqrt(2)*(45*b^2*c^2 - 10*a*b*c*d - 3*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)*(45*b^2*c^2 - 10*a*b*c*d -
3*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)*(45*b^2*c^2
- 10*a*b*c*d - 3*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)))/(c*d^
3)

Giac [A] (verification not implemented)

none

Time = 0.32 (sec) , antiderivative size = 426, normalized size of antiderivative = 1.06 \[ \int \frac {x^{3/2} \left (a+b x^2\right )^2}{\left (c+d x^2\right )^3} \, dx=\frac {2 \, b^{2} \sqrt {x}}{d^{3}} - \frac {\sqrt {2} {\left (45 \, \left (c d^{3}\right )^{\frac {1}{4}} b^{2} c^{2} - 10 \, \left (c d^{3}\right )^{\frac {1}{4}} a b c d - 3 \, \left (c d^{3}\right )^{\frac {1}{4}} a^{2} d^{2}\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {c}{d}\right )^{\frac {1}{4}} + 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {c}{d}\right )^{\frac {1}{4}}}\right )}{64 \, c^{2} d^{4}} - \frac {\sqrt {2} {\left (45 \, \left (c d^{3}\right )^{\frac {1}{4}} b^{2} c^{2} - 10 \, \left (c d^{3}\right )^{\frac {1}{4}} a b c d - 3 \, \left (c d^{3}\right )^{\frac {1}{4}} a^{2} d^{2}\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {c}{d}\right )^{\frac {1}{4}} - 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {c}{d}\right )^{\frac {1}{4}}}\right )}{64 \, c^{2} d^{4}} - \frac {\sqrt {2} {\left (45 \, \left (c d^{3}\right )^{\frac {1}{4}} b^{2} c^{2} - 10 \, \left (c d^{3}\right )^{\frac {1}{4}} a b c d - 3 \, \left (c d^{3}\right )^{\frac {1}{4}} a^{2} d^{2}\right )} \log \left (\sqrt {2} \sqrt {x} \left (\frac {c}{d}\right )^{\frac {1}{4}} + x + \sqrt {\frac {c}{d}}\right )}{128 \, c^{2} d^{4}} + \frac {\sqrt {2} {\left (45 \, \left (c d^{3}\right )^{\frac {1}{4}} b^{2} c^{2} - 10 \, \left (c d^{3}\right )^{\frac {1}{4}} a b c d - 3 \, \left (c d^{3}\right )^{\frac {1}{4}} a^{2} d^{2}\right )} \log \left (-\sqrt {2} \sqrt {x} \left (\frac {c}{d}\right )^{\frac {1}{4}} + x + \sqrt {\frac {c}{d}}\right )}{128 \, c^{2} d^{4}} + \frac {17 \, b^{2} c^{2} d x^{\frac {5}{2}} - 18 \, a b c d^{2} x^{\frac {5}{2}} + a^{2} d^{3} x^{\frac {5}{2}} + 13 \, b^{2} c^{3} \sqrt {x} - 10 \, a b c^{2} d \sqrt {x} - 3 \, a^{2} c d^{2} \sqrt {x}}{16 \, {\left (d x^{2} + c\right )}^{2} c d^{3}} \]

[In]

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

[Out]

2*b^2*sqrt(x)/d^3 - 1/64*sqrt(2)*(45*(c*d^3)^(1/4)*b^2*c^2 - 10*(c*d^3)^(1/4)*a*b*c*d - 3*(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^2*d^4) - 1/64*sqrt(2)*(45*(c*d^3)^(1/4
)*b^2*c^2 - 10*(c*d^3)^(1/4)*a*b*c*d - 3*(c*d^3)^(1/4)*a^2*d^2)*arctan(-1/2*sqrt(2)*(sqrt(2)*(c/d)^(1/4) - 2*s
qrt(x))/(c/d)^(1/4))/(c^2*d^4) - 1/128*sqrt(2)*(45*(c*d^3)^(1/4)*b^2*c^2 - 10*(c*d^3)^(1/4)*a*b*c*d - 3*(c*d^3
)^(1/4)*a^2*d^2)*log(sqrt(2)*sqrt(x)*(c/d)^(1/4) + x + sqrt(c/d))/(c^2*d^4) + 1/128*sqrt(2)*(45*(c*d^3)^(1/4)*
b^2*c^2 - 10*(c*d^3)^(1/4)*a*b*c*d - 3*(c*d^3)^(1/4)*a^2*d^2)*log(-sqrt(2)*sqrt(x)*(c/d)^(1/4) + x + sqrt(c/d)
)/(c^2*d^4) + 1/16*(17*b^2*c^2*d*x^(5/2) - 18*a*b*c*d^2*x^(5/2) + a^2*d^3*x^(5/2) + 13*b^2*c^3*sqrt(x) - 10*a*
b*c^2*d*sqrt(x) - 3*a^2*c*d^2*sqrt(x))/((d*x^2 + c)^2*c*d^3)

Mupad [B] (verification not implemented)

Time = 0.29 (sec) , antiderivative size = 1236, normalized size of antiderivative = 3.07 \[ \int \frac {x^{3/2} \left (a+b x^2\right )^2}{\left (c+d x^2\right )^3} \, dx=\text {Too large to display} \]

[In]

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

[Out]

(2*b^2*x^(1/2))/d^3 - (x^(1/2)*((3*a^2*d^2)/16 - (13*b^2*c^2)/16 + (5*a*b*c*d)/8) - (x^(5/2)*(a^2*d^3 + 17*b^2
*c^2*d - 18*a*b*c*d^2))/(16*c))/(c^2*d^3 + d^5*x^4 + 2*c*d^4*x^2) + (atan(((((3*a^2*d^2 - 45*b^2*c^2 + 10*a*b*
c*d)^2/(64*(-c)^(7/4)*d^(13/4)) - (x^(1/2)*(9*a^4*d^4 + 2025*b^4*c^4 - 170*a^2*b^2*c^2*d^2 - 900*a*b^3*c^3*d +
60*a^3*b*c*d^3))/(64*c^2*d^3))*(3*a^2*d^2 - 45*b^2*c^2 + 10*a*b*c*d)*1i)/(64*(-c)^(7/4)*d^(13/4)) - (((3*a^2*
d^2 - 45*b^2*c^2 + 10*a*b*c*d)^2/(64*(-c)^(7/4)*d^(13/4)) + (x^(1/2)*(9*a^4*d^4 + 2025*b^4*c^4 - 170*a^2*b^2*c
^2*d^2 - 900*a*b^3*c^3*d + 60*a^3*b*c*d^3))/(64*c^2*d^3))*(3*a^2*d^2 - 45*b^2*c^2 + 10*a*b*c*d)*1i)/(64*(-c)^(
7/4)*d^(13/4)))/((((3*a^2*d^2 - 45*b^2*c^2 + 10*a*b*c*d)^2/(64*(-c)^(7/4)*d^(13/4)) - (x^(1/2)*(9*a^4*d^4 + 20
25*b^4*c^4 - 170*a^2*b^2*c^2*d^2 - 900*a*b^3*c^3*d + 60*a^3*b*c*d^3))/(64*c^2*d^3))*(3*a^2*d^2 - 45*b^2*c^2 +
10*a*b*c*d))/(64*(-c)^(7/4)*d^(13/4)) + (((3*a^2*d^2 - 45*b^2*c^2 + 10*a*b*c*d)^2/(64*(-c)^(7/4)*d^(13/4)) + (
x^(1/2)*(9*a^4*d^4 + 2025*b^4*c^4 - 170*a^2*b^2*c^2*d^2 - 900*a*b^3*c^3*d + 60*a^3*b*c*d^3))/(64*c^2*d^3))*(3*
a^2*d^2 - 45*b^2*c^2 + 10*a*b*c*d))/(64*(-c)^(7/4)*d^(13/4))))*(3*a^2*d^2 - 45*b^2*c^2 + 10*a*b*c*d)*1i)/(32*(
-c)^(7/4)*d^(13/4)) + (atan((((((3*a^2*d^2 - 45*b^2*c^2 + 10*a*b*c*d)^2*1i)/(64*(-c)^(7/4)*d^(13/4)) - (x^(1/2
)*(9*a^4*d^4 + 2025*b^4*c^4 - 170*a^2*b^2*c^2*d^2 - 900*a*b^3*c^3*d + 60*a^3*b*c*d^3))/(64*c^2*d^3))*(3*a^2*d^
2 - 45*b^2*c^2 + 10*a*b*c*d))/(64*(-c)^(7/4)*d^(13/4)) - ((((3*a^2*d^2 - 45*b^2*c^2 + 10*a*b*c*d)^2*1i)/(64*(-
c)^(7/4)*d^(13/4)) + (x^(1/2)*(9*a^4*d^4 + 2025*b^4*c^4 - 170*a^2*b^2*c^2*d^2 - 900*a*b^3*c^3*d + 60*a^3*b*c*d
^3))/(64*c^2*d^3))*(3*a^2*d^2 - 45*b^2*c^2 + 10*a*b*c*d))/(64*(-c)^(7/4)*d^(13/4)))/(((((3*a^2*d^2 - 45*b^2*c^
2 + 10*a*b*c*d)^2*1i)/(64*(-c)^(7/4)*d^(13/4)) - (x^(1/2)*(9*a^4*d^4 + 2025*b^4*c^4 - 170*a^2*b^2*c^2*d^2 - 90
0*a*b^3*c^3*d + 60*a^3*b*c*d^3))/(64*c^2*d^3))*(3*a^2*d^2 - 45*b^2*c^2 + 10*a*b*c*d)*1i)/(64*(-c)^(7/4)*d^(13/
4)) + ((((3*a^2*d^2 - 45*b^2*c^2 + 10*a*b*c*d)^2*1i)/(64*(-c)^(7/4)*d^(13/4)) + (x^(1/2)*(9*a^4*d^4 + 2025*b^4
*c^4 - 170*a^2*b^2*c^2*d^2 - 900*a*b^3*c^3*d + 60*a^3*b*c*d^3))/(64*c^2*d^3))*(3*a^2*d^2 - 45*b^2*c^2 + 10*a*b
*c*d)*1i)/(64*(-c)^(7/4)*d^(13/4))))*(3*a^2*d^2 - 45*b^2*c^2 + 10*a*b*c*d))/(32*(-c)^(7/4)*d^(13/4))