3.5.15 \(\int \frac {x^{7/2} (a+b x^2)^2}{c+d x^2} \, dx\) [415]

Optimal. Leaf size=311 \[ -\frac {2 c (b c-a d)^2 \sqrt {x}}{d^4}+\frac {2 (b c-a d)^2 x^{5/2}}{5 d^3}-\frac {2 b (b c-2 a d) x^{9/2}}{9 d^2}+\frac {2 b^2 x^{13/2}}{13 d}-\frac {c^{5/4} (b c-a d)^2 \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{\sqrt {2} d^{17/4}}+\frac {c^{5/4} (b c-a d)^2 \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{\sqrt {2} d^{17/4}}-\frac {c^{5/4} (b c-a d)^2 \log \left (\sqrt {c}-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{2 \sqrt {2} d^{17/4}}+\frac {c^{5/4} (b c-a d)^2 \log \left (\sqrt {c}+\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{2 \sqrt {2} d^{17/4}} \]

[Out]

2/5*(-a*d+b*c)^2*x^(5/2)/d^3-2/9*b*(-2*a*d+b*c)*x^(9/2)/d^2+2/13*b^2*x^(13/2)/d-1/2*c^(5/4)*(-a*d+b*c)^2*arcta
n(1-d^(1/4)*2^(1/2)*x^(1/2)/c^(1/4))/d^(17/4)*2^(1/2)+1/2*c^(5/4)*(-a*d+b*c)^2*arctan(1+d^(1/4)*2^(1/2)*x^(1/2
)/c^(1/4))/d^(17/4)*2^(1/2)-1/4*c^(5/4)*(-a*d+b*c)^2*ln(c^(1/2)+x*d^(1/2)-c^(1/4)*d^(1/4)*2^(1/2)*x^(1/2))/d^(
17/4)*2^(1/2)+1/4*c^(5/4)*(-a*d+b*c)^2*ln(c^(1/2)+x*d^(1/2)+c^(1/4)*d^(1/4)*2^(1/2)*x^(1/2))/d^(17/4)*2^(1/2)-
2*c*(-a*d+b*c)^2*x^(1/2)/d^4

________________________________________________________________________________________

Rubi [A]
time = 0.22, antiderivative size = 311, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 9, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {472, 327, 335, 217, 1179, 642, 1176, 631, 210} \begin {gather*} -\frac {c^{5/4} (b c-a d)^2 \text {ArcTan}\left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{\sqrt {2} d^{17/4}}+\frac {c^{5/4} (b c-a d)^2 \text {ArcTan}\left (\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}+1\right )}{\sqrt {2} d^{17/4}}-\frac {c^{5/4} (b c-a d)^2 \log \left (-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{2 \sqrt {2} d^{17/4}}+\frac {c^{5/4} (b c-a d)^2 \log \left (\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{2 \sqrt {2} d^{17/4}}-\frac {2 c \sqrt {x} (b c-a d)^2}{d^4}+\frac {2 x^{5/2} (b c-a d)^2}{5 d^3}-\frac {2 b x^{9/2} (b c-2 a d)}{9 d^2}+\frac {2 b^2 x^{13/2}}{13 d} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(x^(7/2)*(a + b*x^2)^2)/(c + d*x^2),x]

[Out]

(-2*c*(b*c - a*d)^2*Sqrt[x])/d^4 + (2*(b*c - a*d)^2*x^(5/2))/(5*d^3) - (2*b*(b*c - 2*a*d)*x^(9/2))/(9*d^2) + (
2*b^2*x^(13/2))/(13*d) - (c^(5/4)*(b*c - a*d)^2*ArcTan[1 - (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/(Sqrt[2]*d^(17/
4)) + (c^(5/4)*(b*c - a*d)^2*ArcTan[1 + (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/(Sqrt[2]*d^(17/4)) - (c^(5/4)*(b*c
- a*d)^2*Log[Sqrt[c] - Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/(2*Sqrt[2]*d^(17/4)) + (c^(5/4)*(b*c - a
*d)^2*Log[Sqrt[c] + Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/(2*Sqrt[2]*d^(17/4))

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 472

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

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^{7/2} \left (a+b x^2\right )^2}{c+d x^2} \, dx &=\int \left (-\frac {b (b c-2 a d) x^{7/2}}{d^2}+\frac {b^2 x^{11/2}}{d}+\frac {\left (b^2 c^2-2 a b c d+a^2 d^2\right ) x^{7/2}}{d^2 \left (c+d x^2\right )}\right ) \, dx\\ &=-\frac {2 b (b c-2 a d) x^{9/2}}{9 d^2}+\frac {2 b^2 x^{13/2}}{13 d}+\frac {(b c-a d)^2 \int \frac {x^{7/2}}{c+d x^2} \, dx}{d^2}\\ &=\frac {2 (b c-a d)^2 x^{5/2}}{5 d^3}-\frac {2 b (b c-2 a d) x^{9/2}}{9 d^2}+\frac {2 b^2 x^{13/2}}{13 d}-\frac {\left (c (b c-a d)^2\right ) \int \frac {x^{3/2}}{c+d x^2} \, dx}{d^3}\\ &=-\frac {2 c (b c-a d)^2 \sqrt {x}}{d^4}+\frac {2 (b c-a d)^2 x^{5/2}}{5 d^3}-\frac {2 b (b c-2 a d) x^{9/2}}{9 d^2}+\frac {2 b^2 x^{13/2}}{13 d}+\frac {\left (c^2 (b c-a d)^2\right ) \int \frac {1}{\sqrt {x} \left (c+d x^2\right )} \, dx}{d^4}\\ &=-\frac {2 c (b c-a d)^2 \sqrt {x}}{d^4}+\frac {2 (b c-a d)^2 x^{5/2}}{5 d^3}-\frac {2 b (b c-2 a d) x^{9/2}}{9 d^2}+\frac {2 b^2 x^{13/2}}{13 d}+\frac {\left (2 c^2 (b c-a d)^2\right ) \text {Subst}\left (\int \frac {1}{c+d x^4} \, dx,x,\sqrt {x}\right )}{d^4}\\ &=-\frac {2 c (b c-a d)^2 \sqrt {x}}{d^4}+\frac {2 (b c-a d)^2 x^{5/2}}{5 d^3}-\frac {2 b (b c-2 a d) x^{9/2}}{9 d^2}+\frac {2 b^2 x^{13/2}}{13 d}+\frac {\left (c^{3/2} (b c-a d)^2\right ) \text {Subst}\left (\int \frac {\sqrt {c}-\sqrt {d} x^2}{c+d x^4} \, dx,x,\sqrt {x}\right )}{d^4}+\frac {\left (c^{3/2} (b c-a d)^2\right ) \text {Subst}\left (\int \frac {\sqrt {c}+\sqrt {d} x^2}{c+d x^4} \, dx,x,\sqrt {x}\right )}{d^4}\\ &=-\frac {2 c (b c-a d)^2 \sqrt {x}}{d^4}+\frac {2 (b c-a d)^2 x^{5/2}}{5 d^3}-\frac {2 b (b c-2 a d) x^{9/2}}{9 d^2}+\frac {2 b^2 x^{13/2}}{13 d}+\frac {\left (c^{3/2} (b c-a 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 )}{2 d^{9/2}}+\frac {\left (c^{3/2} (b c-a 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 )}{2 d^{9/2}}-\frac {\left (c^{5/4} (b c-a 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 )}{2 \sqrt {2} d^{17/4}}-\frac {\left (c^{5/4} (b c-a 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 )}{2 \sqrt {2} d^{17/4}}\\ &=-\frac {2 c (b c-a d)^2 \sqrt {x}}{d^4}+\frac {2 (b c-a d)^2 x^{5/2}}{5 d^3}-\frac {2 b (b c-2 a d) x^{9/2}}{9 d^2}+\frac {2 b^2 x^{13/2}}{13 d}-\frac {c^{5/4} (b c-a d)^2 \log \left (\sqrt {c}-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{2 \sqrt {2} d^{17/4}}+\frac {c^{5/4} (b c-a d)^2 \log \left (\sqrt {c}+\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{2 \sqrt {2} d^{17/4}}+\frac {\left (c^{5/4} (b c-a 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 )}{\sqrt {2} d^{17/4}}-\frac {\left (c^{5/4} (b c-a 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 )}{\sqrt {2} d^{17/4}}\\ &=-\frac {2 c (b c-a d)^2 \sqrt {x}}{d^4}+\frac {2 (b c-a d)^2 x^{5/2}}{5 d^3}-\frac {2 b (b c-2 a d) x^{9/2}}{9 d^2}+\frac {2 b^2 x^{13/2}}{13 d}-\frac {c^{5/4} (b c-a d)^2 \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{\sqrt {2} d^{17/4}}+\frac {c^{5/4} (b c-a d)^2 \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{\sqrt {2} d^{17/4}}-\frac {c^{5/4} (b c-a d)^2 \log \left (\sqrt {c}-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{2 \sqrt {2} d^{17/4}}+\frac {c^{5/4} (b c-a d)^2 \log \left (\sqrt {c}+\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{2 \sqrt {2} d^{17/4}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 0.24, size = 219, normalized size = 0.70 \begin {gather*} \frac {2 \sqrt {x} \left (117 a^2 d^2 \left (-5 c+d x^2\right )+26 a b d \left (45 c^2-9 c d x^2+5 d^2 x^4\right )+b^2 \left (-585 c^3+117 c^2 d x^2-65 c d^2 x^4+45 d^3 x^6\right )\right )}{585 d^4}-\frac {c^{5/4} (b c-a d)^2 \tan ^{-1}\left (\frac {\sqrt {c}-\sqrt {d} x}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}}\right )}{\sqrt {2} d^{17/4}}+\frac {c^{5/4} (b c-a d)^2 \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}}{\sqrt {c}+\sqrt {d} x}\right )}{\sqrt {2} d^{17/4}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(x^(7/2)*(a + b*x^2)^2)/(c + d*x^2),x]

[Out]

(2*Sqrt[x]*(117*a^2*d^2*(-5*c + d*x^2) + 26*a*b*d*(45*c^2 - 9*c*d*x^2 + 5*d^2*x^4) + b^2*(-585*c^3 + 117*c^2*d
*x^2 - 65*c*d^2*x^4 + 45*d^3*x^6)))/(585*d^4) - (c^(5/4)*(b*c - a*d)^2*ArcTan[(Sqrt[c] - Sqrt[d]*x)/(Sqrt[2]*c
^(1/4)*d^(1/4)*Sqrt[x])])/(Sqrt[2]*d^(17/4)) + (c^(5/4)*(b*c - a*d)^2*ArcTanh[(Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x]
)/(Sqrt[c] + Sqrt[d]*x)])/(Sqrt[2]*d^(17/4))

________________________________________________________________________________________

Maple [A]
time = 0.13, size = 230, normalized size = 0.74

method	result	size
derivativedivides	\(-\frac {2 \left (-\frac {b^{2} x^{\frac {13}{2}} d^{3}}{13}+\frac {\left (-\left (a d -b c \right ) b \,d^{2}-a b \,d^{3}\right ) x^{\frac {9}{2}}}{9}+\frac {\left (-\left (a d -b c \right ) a \,d^{2}+b d \left (a c d -b \,c^{2}\right )\right ) x^{\frac {5}{2}}}{5}+\left (a d -b c \right ) \left (a c d -b \,c^{2}\right ) \sqrt {x}\right )}{d^{4}}+\frac {c \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}{x -\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}-1\right )\right )}{4 d^{4}}\)	\(230\)
default	\(-\frac {2 \left (-\frac {b^{2} x^{\frac {13}{2}} d^{3}}{13}+\frac {\left (-\left (a d -b c \right ) b \,d^{2}-a b \,d^{3}\right ) x^{\frac {9}{2}}}{9}+\frac {\left (-\left (a d -b c \right ) a \,d^{2}+b d \left (a c d -b \,c^{2}\right )\right ) x^{\frac {5}{2}}}{5}+\left (a d -b c \right ) \left (a c d -b \,c^{2}\right ) \sqrt {x}\right )}{d^{4}}+\frac {c \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}{x -\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}-1\right )\right )}{4 d^{4}}\)	\(230\)
risch	\(-\frac {2 \left (-45 b^{2} d^{3} x^{6}-130 a b \,d^{3} x^{4}+65 b^{2} c \,d^{2} x^{4}-117 a^{2} d^{3} x^{2}+234 a b c \,d^{2} x^{2}-117 b^{2} c^{2} d \,x^{2}+585 a^{2} c \,d^{2}-1170 a b \,c^{2} d +585 b^{2} c^{3}\right ) \sqrt {x}}{585 d^{4}}+\frac {c \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}}{2 d^{2}}-\frac {c^{2} \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}{d^{3}}+\frac {c^{3} \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}}{2 d^{4}}+\frac {c \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}}{2 d^{2}}-\frac {c^{2} \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}{d^{3}}+\frac {c^{3} \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}}{2 d^{4}}+\frac {c \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}}{4 d^{2}}-\frac {c^{2} \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}{2 d^{3}}+\frac {c^{3} \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}}{4 d^{4}}\)	\(538\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(7/2)*(b*x^2+a)^2/(d*x^2+c),x,method=_RETURNVERBOSE)

[Out]

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

________________________________________________________________________________________

Maxima [A]
time = 0.52, size = 360, normalized size = 1.16 \begin {gather*} \frac {{\left (\frac {2 \, \sqrt {2} {\left (b^{2} c^{2} - 2 \, 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 (b^{2} c^{2} - 2 \, 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 (b^{2} c^{2} - 2 \, 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 (b^{2} c^{2} - 2 \, 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}}}\right )} c^{2}}{4 \, d^{4}} + \frac {2 \, {\left (45 \, b^{2} d^{3} x^{\frac {13}{2}} - 65 \, {\left (b^{2} c d^{2} - 2 \, a b d^{3}\right )} x^{\frac {9}{2}} + 117 \, {\left (b^{2} c^{2} d - 2 \, a b c d^{2} + a^{2} d^{3}\right )} x^{\frac {5}{2}} - 585 \, {\left (b^{2} c^{3} - 2 \, a b c^{2} d + a^{2} c d^{2}\right )} \sqrt {x}\right )}}{585 \, d^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(7/2)*(b*x^2+a)^2/(d*x^2+c),x, algorithm="maxima")

[Out]

1/4*(2*sqrt(2)*(b^2*c^2 - 2*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))) + 2*sqrt(2)*(b^2*c^2 - 2*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)*(b^2*c^2 - 2*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)*(b^2*c^2 - 2*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)))*c^2/d^4 + 2/585*(45*b^2*d^3*x^(13/2) - 65*(b^2*c*d^2 - 2*a*b*d^3)*x^(9/2) + 117*(b^2*c^2
*d - 2*a*b*c*d^2 + a^2*d^3)*x^(5/2) - 585*(b^2*c^3 - 2*a*b*c^2*d + a^2*c*d^2)*sqrt(x))/d^4

________________________________________________________________________________________

Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1334 vs. \(2 (230) = 460\).
time = 0.63, size = 1334, normalized size = 4.29 \begin {gather*} \frac {2340 \, d^{4} \left (-\frac {b^{8} c^{13} - 8 \, a b^{7} c^{12} d + 28 \, a^{2} b^{6} c^{11} d^{2} - 56 \, a^{3} b^{5} c^{10} d^{3} + 70 \, a^{4} b^{4} c^{9} d^{4} - 56 \, a^{5} b^{3} c^{8} d^{5} + 28 \, a^{6} b^{2} c^{7} d^{6} - 8 \, a^{7} b c^{6} d^{7} + a^{8} c^{5} d^{8}}{d^{17}}\right )^{\frac {1}{4}} \arctan \left (\frac {\sqrt {d^{8} \sqrt {-\frac {b^{8} c^{13} - 8 \, a b^{7} c^{12} d + 28 \, a^{2} b^{6} c^{11} d^{2} - 56 \, a^{3} b^{5} c^{10} d^{3} + 70 \, a^{4} b^{4} c^{9} d^{4} - 56 \, a^{5} b^{3} c^{8} d^{5} + 28 \, a^{6} b^{2} c^{7} d^{6} - 8 \, a^{7} b c^{6} d^{7} + a^{8} c^{5} d^{8}}{d^{17}}} + {\left (b^{4} c^{6} - 4 \, a b^{3} c^{5} d + 6 \, a^{2} b^{2} c^{4} d^{2} - 4 \, a^{3} b c^{3} d^{3} + a^{4} c^{2} d^{4}\right )} x} d^{13} \left (-\frac {b^{8} c^{13} - 8 \, a b^{7} c^{12} d + 28 \, a^{2} b^{6} c^{11} d^{2} - 56 \, a^{3} b^{5} c^{10} d^{3} + 70 \, a^{4} b^{4} c^{9} d^{4} - 56 \, a^{5} b^{3} c^{8} d^{5} + 28 \, a^{6} b^{2} c^{7} d^{6} - 8 \, a^{7} b c^{6} d^{7} + a^{8} c^{5} d^{8}}{d^{17}}\right )^{\frac {3}{4}} - {\left (b^{2} c^{3} d^{13} - 2 \, a b c^{2} d^{14} + a^{2} c d^{15}\right )} \sqrt {x} \left (-\frac {b^{8} c^{13} - 8 \, a b^{7} c^{12} d + 28 \, a^{2} b^{6} c^{11} d^{2} - 56 \, a^{3} b^{5} c^{10} d^{3} + 70 \, a^{4} b^{4} c^{9} d^{4} - 56 \, a^{5} b^{3} c^{8} d^{5} + 28 \, a^{6} b^{2} c^{7} d^{6} - 8 \, a^{7} b c^{6} d^{7} + a^{8} c^{5} d^{8}}{d^{17}}\right )^{\frac {3}{4}}}{b^{8} c^{13} - 8 \, a b^{7} c^{12} d + 28 \, a^{2} b^{6} c^{11} d^{2} - 56 \, a^{3} b^{5} c^{10} d^{3} + 70 \, a^{4} b^{4} c^{9} d^{4} - 56 \, a^{5} b^{3} c^{8} d^{5} + 28 \, a^{6} b^{2} c^{7} d^{6} - 8 \, a^{7} b c^{6} d^{7} + a^{8} c^{5} d^{8}}\right ) + 585 \, d^{4} \left (-\frac {b^{8} c^{13} - 8 \, a b^{7} c^{12} d + 28 \, a^{2} b^{6} c^{11} d^{2} - 56 \, a^{3} b^{5} c^{10} d^{3} + 70 \, a^{4} b^{4} c^{9} d^{4} - 56 \, a^{5} b^{3} c^{8} d^{5} + 28 \, a^{6} b^{2} c^{7} d^{6} - 8 \, a^{7} b c^{6} d^{7} + a^{8} c^{5} d^{8}}{d^{17}}\right )^{\frac {1}{4}} \log \left (d^{4} \left (-\frac {b^{8} c^{13} - 8 \, a b^{7} c^{12} d + 28 \, a^{2} b^{6} c^{11} d^{2} - 56 \, a^{3} b^{5} c^{10} d^{3} + 70 \, a^{4} b^{4} c^{9} d^{4} - 56 \, a^{5} b^{3} c^{8} d^{5} + 28 \, a^{6} b^{2} c^{7} d^{6} - 8 \, a^{7} b c^{6} d^{7} + a^{8} c^{5} d^{8}}{d^{17}}\right )^{\frac {1}{4}} + {\left (b^{2} c^{3} - 2 \, a b c^{2} d + a^{2} c d^{2}\right )} \sqrt {x}\right ) - 585 \, d^{4} \left (-\frac {b^{8} c^{13} - 8 \, a b^{7} c^{12} d + 28 \, a^{2} b^{6} c^{11} d^{2} - 56 \, a^{3} b^{5} c^{10} d^{3} + 70 \, a^{4} b^{4} c^{9} d^{4} - 56 \, a^{5} b^{3} c^{8} d^{5} + 28 \, a^{6} b^{2} c^{7} d^{6} - 8 \, a^{7} b c^{6} d^{7} + a^{8} c^{5} d^{8}}{d^{17}}\right )^{\frac {1}{4}} \log \left (-d^{4} \left (-\frac {b^{8} c^{13} - 8 \, a b^{7} c^{12} d + 28 \, a^{2} b^{6} c^{11} d^{2} - 56 \, a^{3} b^{5} c^{10} d^{3} + 70 \, a^{4} b^{4} c^{9} d^{4} - 56 \, a^{5} b^{3} c^{8} d^{5} + 28 \, a^{6} b^{2} c^{7} d^{6} - 8 \, a^{7} b c^{6} d^{7} + a^{8} c^{5} d^{8}}{d^{17}}\right )^{\frac {1}{4}} + {\left (b^{2} c^{3} - 2 \, a b c^{2} d + a^{2} c d^{2}\right )} \sqrt {x}\right ) + 4 \, {\left (45 \, b^{2} d^{3} x^{6} - 585 \, b^{2} c^{3} + 1170 \, a b c^{2} d - 585 \, a^{2} c d^{2} - 65 \, {\left (b^{2} c d^{2} - 2 \, a b d^{3}\right )} x^{4} + 117 \, {\left (b^{2} c^{2} d - 2 \, a b c d^{2} + a^{2} d^{3}\right )} x^{2}\right )} \sqrt {x}}{1170 \, d^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(7/2)*(b*x^2+a)^2/(d*x^2+c),x, algorithm="fricas")

[Out]

1/1170*(2340*d^4*(-(b^8*c^13 - 8*a*b^7*c^12*d + 28*a^2*b^6*c^11*d^2 - 56*a^3*b^5*c^10*d^3 + 70*a^4*b^4*c^9*d^4
- 56*a^5*b^3*c^8*d^5 + 28*a^6*b^2*c^7*d^6 - 8*a^7*b*c^6*d^7 + a^8*c^5*d^8)/d^17)^(1/4)*arctan((sqrt(d^8*sqrt(
-(b^8*c^13 - 8*a*b^7*c^12*d + 28*a^2*b^6*c^11*d^2 - 56*a^3*b^5*c^10*d^3 + 70*a^4*b^4*c^9*d^4 - 56*a^5*b^3*c^8*
d^5 + 28*a^6*b^2*c^7*d^6 - 8*a^7*b*c^6*d^7 + a^8*c^5*d^8)/d^17) + (b^4*c^6 - 4*a*b^3*c^5*d + 6*a^2*b^2*c^4*d^2
- 4*a^3*b*c^3*d^3 + a^4*c^2*d^4)*x)*d^13*(-(b^8*c^13 - 8*a*b^7*c^12*d + 28*a^2*b^6*c^11*d^2 - 56*a^3*b^5*c^10
*d^3 + 70*a^4*b^4*c^9*d^4 - 56*a^5*b^3*c^8*d^5 + 28*a^6*b^2*c^7*d^6 - 8*a^7*b*c^6*d^7 + a^8*c^5*d^8)/d^17)^(3/
4) - (b^2*c^3*d^13 - 2*a*b*c^2*d^14 + a^2*c*d^15)*sqrt(x)*(-(b^8*c^13 - 8*a*b^7*c^12*d + 28*a^2*b^6*c^11*d^2 -
56*a^3*b^5*c^10*d^3 + 70*a^4*b^4*c^9*d^4 - 56*a^5*b^3*c^8*d^5 + 28*a^6*b^2*c^7*d^6 - 8*a^7*b*c^6*d^7 + a^8*c^
5*d^8)/d^17)^(3/4))/(b^8*c^13 - 8*a*b^7*c^12*d + 28*a^2*b^6*c^11*d^2 - 56*a^3*b^5*c^10*d^3 + 70*a^4*b^4*c^9*d^
4 - 56*a^5*b^3*c^8*d^5 + 28*a^6*b^2*c^7*d^6 - 8*a^7*b*c^6*d^7 + a^8*c^5*d^8)) + 585*d^4*(-(b^8*c^13 - 8*a*b^7*
c^12*d + 28*a^2*b^6*c^11*d^2 - 56*a^3*b^5*c^10*d^3 + 70*a^4*b^4*c^9*d^4 - 56*a^5*b^3*c^8*d^5 + 28*a^6*b^2*c^7*
d^6 - 8*a^7*b*c^6*d^7 + a^8*c^5*d^8)/d^17)^(1/4)*log(d^4*(-(b^8*c^13 - 8*a*b^7*c^12*d + 28*a^2*b^6*c^11*d^2 -
56*a^3*b^5*c^10*d^3 + 70*a^4*b^4*c^9*d^4 - 56*a^5*b^3*c^8*d^5 + 28*a^6*b^2*c^7*d^6 - 8*a^7*b*c^6*d^7 + a^8*c^5
*d^8)/d^17)^(1/4) + (b^2*c^3 - 2*a*b*c^2*d + a^2*c*d^2)*sqrt(x)) - 585*d^4*(-(b^8*c^13 - 8*a*b^7*c^12*d + 28*a
^2*b^6*c^11*d^2 - 56*a^3*b^5*c^10*d^3 + 70*a^4*b^4*c^9*d^4 - 56*a^5*b^3*c^8*d^5 + 28*a^6*b^2*c^7*d^6 - 8*a^7*b
*c^6*d^7 + a^8*c^5*d^8)/d^17)^(1/4)*log(-d^4*(-(b^8*c^13 - 8*a*b^7*c^12*d + 28*a^2*b^6*c^11*d^2 - 56*a^3*b^5*c
^10*d^3 + 70*a^4*b^4*c^9*d^4 - 56*a^5*b^3*c^8*d^5 + 28*a^6*b^2*c^7*d^6 - 8*a^7*b*c^6*d^7 + a^8*c^5*d^8)/d^17)^
(1/4) + (b^2*c^3 - 2*a*b*c^2*d + a^2*c*d^2)*sqrt(x)) + 4*(45*b^2*d^3*x^6 - 585*b^2*c^3 + 1170*a*b*c^2*d - 585*
a^2*c*d^2 - 65*(b^2*c*d^2 - 2*a*b*d^3)*x^4 + 117*(b^2*c^2*d - 2*a*b*c*d^2 + a^2*d^3)*x^2)*sqrt(x))/d^4

________________________________________________________________________________________

Sympy [A]
time = 93.27, size = 561, normalized size = 1.80 \begin {gather*} \begin {cases} \tilde {\infty } \left (\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}\right ) & \text {for}\: c = 0 \wedge d = 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}}{d} & \text {for}\: c = 0 \\\frac {\frac {2 a^{2} x^{\frac {9}{2}}}{9} + \frac {4 a b x^{\frac {13}{2}}}{13} + \frac {2 b^{2} x^{\frac {17}{2}}}{17}}{c} & \text {for}\: d = 0 \\- \frac {2 a^{2} c \sqrt {x}}{d^{2}} - \frac {a^{2} c \sqrt [4]{- \frac {c}{d}} \log {\left (\sqrt {x} - \sqrt [4]{- \frac {c}{d}} \right )}}{2 d^{2}} + \frac {a^{2} c \sqrt [4]{- \frac {c}{d}} \log {\left (\sqrt {x} + \sqrt [4]{- \frac {c}{d}} \right )}}{2 d^{2}} + \frac {a^{2} c \sqrt [4]{- \frac {c}{d}} \operatorname {atan}{\left (\frac {\sqrt {x}}{\sqrt [4]{- \frac {c}{d}}} \right )}}{d^{2}} + \frac {2 a^{2} x^{\frac {5}{2}}}{5 d} + \frac {4 a b c^{2} \sqrt {x}}{d^{3}} + \frac {a b c^{2} \sqrt [4]{- \frac {c}{d}} \log {\left (\sqrt {x} - \sqrt [4]{- \frac {c}{d}} \right )}}{d^{3}} - \frac {a b c^{2} \sqrt [4]{- \frac {c}{d}} \log {\left (\sqrt {x} + \sqrt [4]{- \frac {c}{d}} \right )}}{d^{3}} - \frac {2 a b c^{2} \sqrt [4]{- \frac {c}{d}} \operatorname {atan}{\left (\frac {\sqrt {x}}{\sqrt [4]{- \frac {c}{d}}} \right )}}{d^{3}} - \frac {4 a b c x^{\frac {5}{2}}}{5 d^{2}} + \frac {4 a b x^{\frac {9}{2}}}{9 d} - \frac {2 b^{2} c^{3} \sqrt {x}}{d^{4}} - \frac {b^{2} c^{3} \sqrt [4]{- \frac {c}{d}} \log {\left (\sqrt {x} - \sqrt [4]{- \frac {c}{d}} \right )}}{2 d^{4}} + \frac {b^{2} c^{3} \sqrt [4]{- \frac {c}{d}} \log {\left (\sqrt {x} + \sqrt [4]{- \frac {c}{d}} \right )}}{2 d^{4}} + \frac {b^{2} c^{3} \sqrt [4]{- \frac {c}{d}} \operatorname {atan}{\left (\frac {\sqrt {x}}{\sqrt [4]{- \frac {c}{d}}} \right )}}{d^{4}} + \frac {2 b^{2} c^{2} x^{\frac {5}{2}}}{5 d^{3}} - \frac {2 b^{2} c x^{\frac {9}{2}}}{9 d^{2}} + \frac {2 b^{2} x^{\frac {13}{2}}}{13 d} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**(7/2)*(b*x**2+a)**2/(d*x**2+c),x)

[Out]

Piecewise((zoo*(2*a**2*x**(5/2)/5 + 4*a*b*x**(9/2)/9 + 2*b**2*x**(13/2)/13), 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)/d, Eq(c, 0)), ((2*a**2*x**(9/2)/9 + 4*a*b*x**(13/2)/13 + 2*
b**2*x**(17/2)/17)/c, Eq(d, 0)), (-2*a**2*c*sqrt(x)/d**2 - a**2*c*(-c/d)**(1/4)*log(sqrt(x) - (-c/d)**(1/4))/(
2*d**2) + a**2*c*(-c/d)**(1/4)*log(sqrt(x) + (-c/d)**(1/4))/(2*d**2) + a**2*c*(-c/d)**(1/4)*atan(sqrt(x)/(-c/d
)**(1/4))/d**2 + 2*a**2*x**(5/2)/(5*d) + 4*a*b*c**2*sqrt(x)/d**3 + a*b*c**2*(-c/d)**(1/4)*log(sqrt(x) - (-c/d)
**(1/4))/d**3 - a*b*c**2*(-c/d)**(1/4)*log(sqrt(x) + (-c/d)**(1/4))/d**3 - 2*a*b*c**2*(-c/d)**(1/4)*atan(sqrt(
x)/(-c/d)**(1/4))/d**3 - 4*a*b*c*x**(5/2)/(5*d**2) + 4*a*b*x**(9/2)/(9*d) - 2*b**2*c**3*sqrt(x)/d**4 - b**2*c*
*3*(-c/d)**(1/4)*log(sqrt(x) - (-c/d)**(1/4))/(2*d**4) + b**2*c**3*(-c/d)**(1/4)*log(sqrt(x) + (-c/d)**(1/4))/
(2*d**4) + b**2*c**3*(-c/d)**(1/4)*atan(sqrt(x)/(-c/d)**(1/4))/d**4 + 2*b**2*c**2*x**(5/2)/(5*d**3) - 2*b**2*c
*x**(9/2)/(9*d**2) + 2*b**2*x**(13/2)/(13*d), True))

________________________________________________________________________________________

Giac [A]
time = 1.07, size = 436, normalized size = 1.40 \begin {gather*} \frac {\sqrt {2} {\left (\left (c d^{3}\right )^{\frac {1}{4}} b^{2} c^{3} - 2 \, \left (c d^{3}\right )^{\frac {1}{4}} a b c^{2} d + \left (c d^{3}\right )^{\frac {1}{4}} a^{2} c 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 )}{2 \, d^{5}} + \frac {\sqrt {2} {\left (\left (c d^{3}\right )^{\frac {1}{4}} b^{2} c^{3} - 2 \, \left (c d^{3}\right )^{\frac {1}{4}} a b c^{2} d + \left (c d^{3}\right )^{\frac {1}{4}} a^{2} c 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 )}{2 \, d^{5}} + \frac {\sqrt {2} {\left (\left (c d^{3}\right )^{\frac {1}{4}} b^{2} c^{3} - 2 \, \left (c d^{3}\right )^{\frac {1}{4}} a b c^{2} d + \left (c d^{3}\right )^{\frac {1}{4}} a^{2} c d^{2}\right )} \log \left (\sqrt {2} \sqrt {x} \left (\frac {c}{d}\right )^{\frac {1}{4}} + x + \sqrt {\frac {c}{d}}\right )}{4 \, d^{5}} - \frac {\sqrt {2} {\left (\left (c d^{3}\right )^{\frac {1}{4}} b^{2} c^{3} - 2 \, \left (c d^{3}\right )^{\frac {1}{4}} a b c^{2} d + \left (c d^{3}\right )^{\frac {1}{4}} a^{2} c d^{2}\right )} \log \left (-\sqrt {2} \sqrt {x} \left (\frac {c}{d}\right )^{\frac {1}{4}} + x + \sqrt {\frac {c}{d}}\right )}{4 \, d^{5}} + \frac {2 \, {\left (45 \, b^{2} d^{12} x^{\frac {13}{2}} - 65 \, b^{2} c d^{11} x^{\frac {9}{2}} + 130 \, a b d^{12} x^{\frac {9}{2}} + 117 \, b^{2} c^{2} d^{10} x^{\frac {5}{2}} - 234 \, a b c d^{11} x^{\frac {5}{2}} + 117 \, a^{2} d^{12} x^{\frac {5}{2}} - 585 \, b^{2} c^{3} d^{9} \sqrt {x} + 1170 \, a b c^{2} d^{10} \sqrt {x} - 585 \, a^{2} c d^{11} \sqrt {x}\right )}}{585 \, d^{13}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(7/2)*(b*x^2+a)^2/(d*x^2+c),x, algorithm="giac")

[Out]

1/2*sqrt(2)*((c*d^3)^(1/4)*b^2*c^3 - 2*(c*d^3)^(1/4)*a*b*c^2*d + (c*d^3)^(1/4)*a^2*c*d^2)*arctan(1/2*sqrt(2)*(
sqrt(2)*(c/d)^(1/4) + 2*sqrt(x))/(c/d)^(1/4))/d^5 + 1/2*sqrt(2)*((c*d^3)^(1/4)*b^2*c^3 - 2*(c*d^3)^(1/4)*a*b*c
^2*d + (c*d^3)^(1/4)*a^2*c*d^2)*arctan(-1/2*sqrt(2)*(sqrt(2)*(c/d)^(1/4) - 2*sqrt(x))/(c/d)^(1/4))/d^5 + 1/4*s
qrt(2)*((c*d^3)^(1/4)*b^2*c^3 - 2*(c*d^3)^(1/4)*a*b*c^2*d + (c*d^3)^(1/4)*a^2*c*d^2)*log(sqrt(2)*sqrt(x)*(c/d)
^(1/4) + x + sqrt(c/d))/d^5 - 1/4*sqrt(2)*((c*d^3)^(1/4)*b^2*c^3 - 2*(c*d^3)^(1/4)*a*b*c^2*d + (c*d^3)^(1/4)*a
^2*c*d^2)*log(-sqrt(2)*sqrt(x)*(c/d)^(1/4) + x + sqrt(c/d))/d^5 + 2/585*(45*b^2*d^12*x^(13/2) - 65*b^2*c*d^11*
x^(9/2) + 130*a*b*d^12*x^(9/2) + 117*b^2*c^2*d^10*x^(5/2) - 234*a*b*c*d^11*x^(5/2) + 117*a^2*d^12*x^(5/2) - 58
5*b^2*c^3*d^9*sqrt(x) + 1170*a*b*c^2*d^10*sqrt(x) - 585*a^2*c*d^11*sqrt(x))/d^13

________________________________________________________________________________________

Mupad [B]
time = 0.23, size = 1202, normalized size = 3.86 \begin {gather*} x^{5/2}\,\left (\frac {2\,a^2}{5\,d}+\frac {c\,\left (\frac {2\,b^2\,c}{d^2}-\frac {4\,a\,b}{d}\right )}{5\,d}\right )-x^{9/2}\,\left (\frac {2\,b^2\,c}{9\,d^2}-\frac {4\,a\,b}{9\,d}\right )+\frac {2\,b^2\,x^{13/2}}{13\,d}-\frac {c\,\sqrt {x}\,\left (\frac {2\,a^2}{d}+\frac {c\,\left (\frac {2\,b^2\,c}{d^2}-\frac {4\,a\,b}{d}\right )}{d}\right )}{d}+\frac {{\left (-c\right )}^{5/4}\,\mathrm {atan}\left (\frac {\frac {{\left (-c\right )}^{5/4}\,{\left (a\,d-b\,c\right )}^2\,\left (\frac {16\,\sqrt {x}\,\left (a^4\,c^4\,d^4-4\,a^3\,b\,c^5\,d^3+6\,a^2\,b^2\,c^6\,d^2-4\,a\,b^3\,c^7\,d+b^4\,c^8\right )}{d^5}-\frac {16\,{\left (-c\right )}^{5/4}\,{\left (a\,d-b\,c\right )}^2\,\left (a^2\,c^3\,d^2-2\,a\,b\,c^4\,d+b^2\,c^5\right )}{d^{21/4}}\right )\,1{}\mathrm {i}}{2\,d^{17/4}}+\frac {{\left (-c\right )}^{5/4}\,{\left (a\,d-b\,c\right )}^2\,\left (\frac {16\,\sqrt {x}\,\left (a^4\,c^4\,d^4-4\,a^3\,b\,c^5\,d^3+6\,a^2\,b^2\,c^6\,d^2-4\,a\,b^3\,c^7\,d+b^4\,c^8\right )}{d^5}+\frac {16\,{\left (-c\right )}^{5/4}\,{\left (a\,d-b\,c\right )}^2\,\left (a^2\,c^3\,d^2-2\,a\,b\,c^4\,d+b^2\,c^5\right )}{d^{21/4}}\right )\,1{}\mathrm {i}}{2\,d^{17/4}}}{\frac {{\left (-c\right )}^{5/4}\,{\left (a\,d-b\,c\right )}^2\,\left (\frac {16\,\sqrt {x}\,\left (a^4\,c^4\,d^4-4\,a^3\,b\,c^5\,d^3+6\,a^2\,b^2\,c^6\,d^2-4\,a\,b^3\,c^7\,d+b^4\,c^8\right )}{d^5}-\frac {16\,{\left (-c\right )}^{5/4}\,{\left (a\,d-b\,c\right )}^2\,\left (a^2\,c^3\,d^2-2\,a\,b\,c^4\,d+b^2\,c^5\right )}{d^{21/4}}\right )}{2\,d^{17/4}}-\frac {{\left (-c\right )}^{5/4}\,{\left (a\,d-b\,c\right )}^2\,\left (\frac {16\,\sqrt {x}\,\left (a^4\,c^4\,d^4-4\,a^3\,b\,c^5\,d^3+6\,a^2\,b^2\,c^6\,d^2-4\,a\,b^3\,c^7\,d+b^4\,c^8\right )}{d^5}+\frac {16\,{\left (-c\right )}^{5/4}\,{\left (a\,d-b\,c\right )}^2\,\left (a^2\,c^3\,d^2-2\,a\,b\,c^4\,d+b^2\,c^5\right )}{d^{21/4}}\right )}{2\,d^{17/4}}}\right )\,{\left (a\,d-b\,c\right )}^2\,1{}\mathrm {i}}{d^{17/4}}+\frac {{\left (-c\right )}^{5/4}\,\mathrm {atan}\left (\frac {\frac {{\left (-c\right )}^{5/4}\,{\left (a\,d-b\,c\right )}^2\,\left (\frac {16\,\sqrt {x}\,\left (a^4\,c^4\,d^4-4\,a^3\,b\,c^5\,d^3+6\,a^2\,b^2\,c^6\,d^2-4\,a\,b^3\,c^7\,d+b^4\,c^8\right )}{d^5}-\frac {{\left (-c\right )}^{5/4}\,{\left (a\,d-b\,c\right )}^2\,\left (a^2\,c^3\,d^2-2\,a\,b\,c^4\,d+b^2\,c^5\right )\,16{}\mathrm {i}}{d^{21/4}}\right )}{2\,d^{17/4}}+\frac {{\left (-c\right )}^{5/4}\,{\left (a\,d-b\,c\right )}^2\,\left (\frac {16\,\sqrt {x}\,\left (a^4\,c^4\,d^4-4\,a^3\,b\,c^5\,d^3+6\,a^2\,b^2\,c^6\,d^2-4\,a\,b^3\,c^7\,d+b^4\,c^8\right )}{d^5}+\frac {{\left (-c\right )}^{5/4}\,{\left (a\,d-b\,c\right )}^2\,\left (a^2\,c^3\,d^2-2\,a\,b\,c^4\,d+b^2\,c^5\right )\,16{}\mathrm {i}}{d^{21/4}}\right )}{2\,d^{17/4}}}{\frac {{\left (-c\right )}^{5/4}\,{\left (a\,d-b\,c\right )}^2\,\left (\frac {16\,\sqrt {x}\,\left (a^4\,c^4\,d^4-4\,a^3\,b\,c^5\,d^3+6\,a^2\,b^2\,c^6\,d^2-4\,a\,b^3\,c^7\,d+b^4\,c^8\right )}{d^5}-\frac {{\left (-c\right )}^{5/4}\,{\left (a\,d-b\,c\right )}^2\,\left (a^2\,c^3\,d^2-2\,a\,b\,c^4\,d+b^2\,c^5\right )\,16{}\mathrm {i}}{d^{21/4}}\right )\,1{}\mathrm {i}}{2\,d^{17/4}}-\frac {{\left (-c\right )}^{5/4}\,{\left (a\,d-b\,c\right )}^2\,\left (\frac {16\,\sqrt {x}\,\left (a^4\,c^4\,d^4-4\,a^3\,b\,c^5\,d^3+6\,a^2\,b^2\,c^6\,d^2-4\,a\,b^3\,c^7\,d+b^4\,c^8\right )}{d^5}+\frac {{\left (-c\right )}^{5/4}\,{\left (a\,d-b\,c\right )}^2\,\left (a^2\,c^3\,d^2-2\,a\,b\,c^4\,d+b^2\,c^5\right )\,16{}\mathrm {i}}{d^{21/4}}\right )\,1{}\mathrm {i}}{2\,d^{17/4}}}\right )\,{\left (a\,d-b\,c\right )}^2}{d^{17/4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^(7/2)*(a + b*x^2)^2)/(c + d*x^2),x)

[Out]

x^(5/2)*((2*a^2)/(5*d) + (c*((2*b^2*c)/d^2 - (4*a*b)/d))/(5*d)) - x^(9/2)*((2*b^2*c)/(9*d^2) - (4*a*b)/(9*d))
+ (2*b^2*x^(13/2))/(13*d) - (c*x^(1/2)*((2*a^2)/d + (c*((2*b^2*c)/d^2 - (4*a*b)/d))/d))/d + ((-c)^(5/4)*atan((
((-c)^(5/4)*(a*d - b*c)^2*((16*x^(1/2)*(b^4*c^8 + a^4*c^4*d^4 - 4*a^3*b*c^5*d^3 + 6*a^2*b^2*c^6*d^2 - 4*a*b^3*
c^7*d))/d^5 - (16*(-c)^(5/4)*(a*d - b*c)^2*(b^2*c^5 + a^2*c^3*d^2 - 2*a*b*c^4*d))/d^(21/4))*1i)/(2*d^(17/4)) +
((-c)^(5/4)*(a*d - b*c)^2*((16*x^(1/2)*(b^4*c^8 + a^4*c^4*d^4 - 4*a^3*b*c^5*d^3 + 6*a^2*b^2*c^6*d^2 - 4*a*b^3
*c^7*d))/d^5 + (16*(-c)^(5/4)*(a*d - b*c)^2*(b^2*c^5 + a^2*c^3*d^2 - 2*a*b*c^4*d))/d^(21/4))*1i)/(2*d^(17/4)))
/(((-c)^(5/4)*(a*d - b*c)^2*((16*x^(1/2)*(b^4*c^8 + a^4*c^4*d^4 - 4*a^3*b*c^5*d^3 + 6*a^2*b^2*c^6*d^2 - 4*a*b^
3*c^7*d))/d^5 - (16*(-c)^(5/4)*(a*d - b*c)^2*(b^2*c^5 + a^2*c^3*d^2 - 2*a*b*c^4*d))/d^(21/4)))/(2*d^(17/4)) -
((-c)^(5/4)*(a*d - b*c)^2*((16*x^(1/2)*(b^4*c^8 + a^4*c^4*d^4 - 4*a^3*b*c^5*d^3 + 6*a^2*b^2*c^6*d^2 - 4*a*b^3*
c^7*d))/d^5 + (16*(-c)^(5/4)*(a*d - b*c)^2*(b^2*c^5 + a^2*c^3*d^2 - 2*a*b*c^4*d))/d^(21/4)))/(2*d^(17/4))))*(a
*d - b*c)^2*1i)/d^(17/4) + ((-c)^(5/4)*atan((((-c)^(5/4)*(a*d - b*c)^2*((16*x^(1/2)*(b^4*c^8 + a^4*c^4*d^4 - 4
*a^3*b*c^5*d^3 + 6*a^2*b^2*c^6*d^2 - 4*a*b^3*c^7*d))/d^5 - ((-c)^(5/4)*(a*d - b*c)^2*(b^2*c^5 + a^2*c^3*d^2 -
2*a*b*c^4*d)*16i)/d^(21/4)))/(2*d^(17/4)) + ((-c)^(5/4)*(a*d - b*c)^2*((16*x^(1/2)*(b^4*c^8 + a^4*c^4*d^4 - 4*
a^3*b*c^5*d^3 + 6*a^2*b^2*c^6*d^2 - 4*a*b^3*c^7*d))/d^5 + ((-c)^(5/4)*(a*d - b*c)^2*(b^2*c^5 + a^2*c^3*d^2 - 2
*a*b*c^4*d)*16i)/d^(21/4)))/(2*d^(17/4)))/(((-c)^(5/4)*(a*d - b*c)^2*((16*x^(1/2)*(b^4*c^8 + a^4*c^4*d^4 - 4*a
^3*b*c^5*d^3 + 6*a^2*b^2*c^6*d^2 - 4*a*b^3*c^7*d))/d^5 - ((-c)^(5/4)*(a*d - b*c)^2*(b^2*c^5 + a^2*c^3*d^2 - 2*
a*b*c^4*d)*16i)/d^(21/4))*1i)/(2*d^(17/4)) - ((-c)^(5/4)*(a*d - b*c)^2*((16*x^(1/2)*(b^4*c^8 + a^4*c^4*d^4 - 4
*a^3*b*c^5*d^3 + 6*a^2*b^2*c^6*d^2 - 4*a*b^3*c^7*d))/d^5 + ((-c)^(5/4)*(a*d - b*c)^2*(b^2*c^5 + a^2*c^3*d^2 -
2*a*b*c^4*d)*16i)/d^(21/4))*1i)/(2*d^(17/4))))*(a*d - b*c)^2)/d^(17/4)

________________________________________________________________________________________