\(\int \frac {x^{5/2} (c+d x^2)^3}{a+b x^2} \, dx\) [441]

Optimal result

Integrand size = 24, antiderivative size = 328 \[ \int \frac {x^{5/2} \left (c+d x^2\right )^3}{a+b x^2} \, dx=\frac {2 (b c-a d)^3 x^{3/2}}{3 b^4}+\frac {2 d \left (3 b^2 c^2-3 a b c d+a^2 d^2\right ) x^{7/2}}{7 b^3}+\frac {2 d^2 (3 b c-a d) x^{11/2}}{11 b^2}+\frac {2 d^3 x^{15/2}}{15 b}+\frac {a^{3/4} (b c-a d)^3 \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} b^{19/4}}-\frac {a^{3/4} (b c-a d)^3 \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} b^{19/4}}-\frac {a^{3/4} (b c-a d)^3 \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{2 \sqrt {2} b^{19/4}}+\frac {a^{3/4} (b c-a d)^3 \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{2 \sqrt {2} b^{19/4}} \]

[Out]

2/3*(-a*d+b*c)^3*x^(3/2)/b^4+2/7*d*(a^2*d^2-3*a*b*c*d+3*b^2*c^2)*x^(7/2)/b^3+2/11*d^2*(-a*d+3*b*c)*x^(11/2)/b^
2+2/15*d^3*x^(15/2)/b+1/2*a^(3/4)*(-a*d+b*c)^3*arctan(1-b^(1/4)*2^(1/2)*x^(1/2)/a^(1/4))/b^(19/4)*2^(1/2)-1/2*
a^(3/4)*(-a*d+b*c)^3*arctan(1+b^(1/4)*2^(1/2)*x^(1/2)/a^(1/4))/b^(19/4)*2^(1/2)-1/4*a^(3/4)*(-a*d+b*c)^3*ln(a^
(1/2)+x*b^(1/2)-a^(1/4)*b^(1/4)*2^(1/2)*x^(1/2))/b^(19/4)*2^(1/2)+1/4*a^(3/4)*(-a*d+b*c)^3*ln(a^(1/2)+x*b^(1/2
)+a^(1/4)*b^(1/4)*2^(1/2)*x^(1/2))/b^(19/4)*2^(1/2)

Rubi [A] (verified)

Time = 0.22 (sec) , antiderivative size = 328, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {472, 327, 335, 303, 1176, 631, 210, 1179, 642} \[ \int \frac {x^{5/2} \left (c+d x^2\right )^3}{a+b x^2} \, dx=\frac {a^{3/4} \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right ) (b c-a d)^3}{\sqrt {2} b^{19/4}}-\frac {a^{3/4} \arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right ) (b c-a d)^3}{\sqrt {2} b^{19/4}}-\frac {a^{3/4} (b c-a d)^3 \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{2 \sqrt {2} b^{19/4}}+\frac {a^{3/4} (b c-a d)^3 \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{2 \sqrt {2} b^{19/4}}+\frac {2 d x^{7/2} \left (a^2 d^2-3 a b c d+3 b^2 c^2\right )}{7 b^3}+\frac {2 x^{3/2} (b c-a d)^3}{3 b^4}+\frac {2 d^2 x^{11/2} (3 b c-a d)}{11 b^2}+\frac {2 d^3 x^{15/2}}{15 b} \]

[In]

Int[(x^(5/2)*(c + d*x^2)^3)/(a + b*x^2),x]

[Out]

(2*(b*c - a*d)^3*x^(3/2))/(3*b^4) + (2*d*(3*b^2*c^2 - 3*a*b*c*d + a^2*d^2)*x^(7/2))/(7*b^3) + (2*d^2*(3*b*c -
a*d)*x^(11/2))/(11*b^2) + (2*d^3*x^(15/2))/(15*b) + (a^(3/4)*(b*c - a*d)^3*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x]
)/a^(1/4)])/(Sqrt[2]*b^(19/4)) - (a^(3/4)*(b*c - a*d)^3*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(Sqrt[2
]*b^(19/4)) - (a^(3/4)*(b*c - a*d)^3*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(2*Sqrt[2]*b^
(19/4)) + (a^(3/4)*(b*c - a*d)^3*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(2*Sqrt[2]*b^(19/
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 303

Int[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[a/b, 2]], s = Denominator[Rt[a/b, 2]]},
Dist[1/(2*s), Int[(r + s*x^2)/(a + b*x^4), x], x] - Dist[1/(2*s), Int[(r - s*x^2)/(a + b*x^4), x], x]] /; Free
Q[{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*} \text {integral}& = \int \left (\frac {d \left (3 b^2 c^2-3 a b c d+a^2 d^2\right ) x^{5/2}}{b^3}+\frac {d^2 (3 b c-a d) x^{9/2}}{b^2}+\frac {d^3 x^{13/2}}{b}+\frac {\left (b^3 c^3-3 a b^2 c^2 d+3 a^2 b c d^2-a^3 d^3\right ) x^{5/2}}{b^3 \left (a+b x^2\right )}\right ) \, dx \\ & = \frac {2 d \left (3 b^2 c^2-3 a b c d+a^2 d^2\right ) x^{7/2}}{7 b^3}+\frac {2 d^2 (3 b c-a d) x^{11/2}}{11 b^2}+\frac {2 d^3 x^{15/2}}{15 b}+\frac {(b c-a d)^3 \int \frac {x^{5/2}}{a+b x^2} \, dx}{b^3} \\ & = \frac {2 (b c-a d)^3 x^{3/2}}{3 b^4}+\frac {2 d \left (3 b^2 c^2-3 a b c d+a^2 d^2\right ) x^{7/2}}{7 b^3}+\frac {2 d^2 (3 b c-a d) x^{11/2}}{11 b^2}+\frac {2 d^3 x^{15/2}}{15 b}-\frac {\left (a (b c-a d)^3\right ) \int \frac {\sqrt {x}}{a+b x^2} \, dx}{b^4} \\ & = \frac {2 (b c-a d)^3 x^{3/2}}{3 b^4}+\frac {2 d \left (3 b^2 c^2-3 a b c d+a^2 d^2\right ) x^{7/2}}{7 b^3}+\frac {2 d^2 (3 b c-a d) x^{11/2}}{11 b^2}+\frac {2 d^3 x^{15/2}}{15 b}-\frac {\left (2 a (b c-a d)^3\right ) \text {Subst}\left (\int \frac {x^2}{a+b x^4} \, dx,x,\sqrt {x}\right )}{b^4} \\ & = \frac {2 (b c-a d)^3 x^{3/2}}{3 b^4}+\frac {2 d \left (3 b^2 c^2-3 a b c d+a^2 d^2\right ) x^{7/2}}{7 b^3}+\frac {2 d^2 (3 b c-a d) x^{11/2}}{11 b^2}+\frac {2 d^3 x^{15/2}}{15 b}+\frac {\left (a (b c-a d)^3\right ) \text {Subst}\left (\int \frac {\sqrt {a}-\sqrt {b} x^2}{a+b x^4} \, dx,x,\sqrt {x}\right )}{b^{9/2}}-\frac {\left (a (b c-a d)^3\right ) \text {Subst}\left (\int \frac {\sqrt {a}+\sqrt {b} x^2}{a+b x^4} \, dx,x,\sqrt {x}\right )}{b^{9/2}} \\ & = \frac {2 (b c-a d)^3 x^{3/2}}{3 b^4}+\frac {2 d \left (3 b^2 c^2-3 a b c d+a^2 d^2\right ) x^{7/2}}{7 b^3}+\frac {2 d^2 (3 b c-a d) x^{11/2}}{11 b^2}+\frac {2 d^3 x^{15/2}}{15 b}-\frac {\left (a (b c-a d)^3\right ) \text {Subst}\left (\int \frac {1}{\frac {\sqrt {a}}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}+x^2} \, dx,x,\sqrt {x}\right )}{2 b^5}-\frac {\left (a (b c-a d)^3\right ) \text {Subst}\left (\int \frac {1}{\frac {\sqrt {a}}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}+x^2} \, dx,x,\sqrt {x}\right )}{2 b^5}-\frac {\left (a^{3/4} (b c-a d)^3\right ) \text {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt [4]{a}}{\sqrt [4]{b}}+2 x}{-\frac {\sqrt {a}}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}-x^2} \, dx,x,\sqrt {x}\right )}{2 \sqrt {2} b^{19/4}}-\frac {\left (a^{3/4} (b c-a d)^3\right ) \text {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt [4]{a}}{\sqrt [4]{b}}-2 x}{-\frac {\sqrt {a}}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}-x^2} \, dx,x,\sqrt {x}\right )}{2 \sqrt {2} b^{19/4}} \\ & = \frac {2 (b c-a d)^3 x^{3/2}}{3 b^4}+\frac {2 d \left (3 b^2 c^2-3 a b c d+a^2 d^2\right ) x^{7/2}}{7 b^3}+\frac {2 d^2 (3 b c-a d) x^{11/2}}{11 b^2}+\frac {2 d^3 x^{15/2}}{15 b}-\frac {a^{3/4} (b c-a d)^3 \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{2 \sqrt {2} b^{19/4}}+\frac {a^{3/4} (b c-a d)^3 \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{2 \sqrt {2} b^{19/4}}-\frac {\left (a^{3/4} (b c-a d)^3\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} b^{19/4}}+\frac {\left (a^{3/4} (b c-a d)^3\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} b^{19/4}} \\ & = \frac {2 (b c-a d)^3 x^{3/2}}{3 b^4}+\frac {2 d \left (3 b^2 c^2-3 a b c d+a^2 d^2\right ) x^{7/2}}{7 b^3}+\frac {2 d^2 (3 b c-a d) x^{11/2}}{11 b^2}+\frac {2 d^3 x^{15/2}}{15 b}+\frac {a^{3/4} (b c-a d)^3 \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} b^{19/4}}-\frac {a^{3/4} (b c-a d)^3 \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} b^{19/4}}-\frac {a^{3/4} (b c-a d)^3 \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{2 \sqrt {2} b^{19/4}}+\frac {a^{3/4} (b c-a d)^3 \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{2 \sqrt {2} b^{19/4}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.26 (sec) , antiderivative size = 230, normalized size of antiderivative = 0.70 \[ \int \frac {x^{5/2} \left (c+d x^2\right )^3}{a+b x^2} \, dx=\frac {2 x^{3/2} \left (-385 a^3 d^3+165 a^2 b d^2 \left (7 c+d x^2\right )-15 a b^2 d \left (77 c^2+33 c d x^2+7 d^2 x^4\right )+b^3 \left (385 c^3+495 c^2 d x^2+315 c d^2 x^4+77 d^3 x^6\right )\right )}{1155 b^4}-\frac {a^{3/4} (-b c+a d)^3 \arctan \left (\frac {\sqrt {a}-\sqrt {b} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}\right )}{\sqrt {2} b^{19/4}}+\frac {a^{3/4} (b c-a d)^3 \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a}+\sqrt {b} x}\right )}{\sqrt {2} b^{19/4}} \]

[In]

Integrate[(x^(5/2)*(c + d*x^2)^3)/(a + b*x^2),x]

[Out]

(2*x^(3/2)*(-385*a^3*d^3 + 165*a^2*b*d^2*(7*c + d*x^2) - 15*a*b^2*d*(77*c^2 + 33*c*d*x^2 + 7*d^2*x^4) + b^3*(3
85*c^3 + 495*c^2*d*x^2 + 315*c*d^2*x^4 + 77*d^3*x^6)))/(1155*b^4) - (a^(3/4)*(-(b*c) + a*d)^3*ArcTan[(Sqrt[a]
- Sqrt[b]*x)/(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])])/(Sqrt[2]*b^(19/4)) + (a^(3/4)*(b*c - a*d)^3*ArcTanh[(Sqrt[2]*
a^(1/4)*b^(1/4)*Sqrt[x])/(Sqrt[a] + Sqrt[b]*x)])/(Sqrt[2]*b^(19/4))

Maple [A] (verified)

Time = 2.96 (sec) , antiderivative size = 259, normalized size of antiderivative = 0.79

method	result	size
derivativedivides	\(-\frac {2 \left (-\frac {d^{3} x^{\frac {15}{2}} b^{3}}{15}+\frac {\left (a \,b^{2} d^{3}-3 b^{3} c \,d^{2}\right ) x^{\frac {11}{2}}}{11}+\frac {\left (-a^{2} b \,d^{3}+3 a \,b^{2} c \,d^{2}-3 b^{3} c^{2} d \right ) x^{\frac {7}{2}}}{7}+\frac {\left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) x^{\frac {3}{2}}}{3}\right )}{b^{4}}+\frac {a \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) \sqrt {2}\, \left (\ln \left (\frac {x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{4 b^{5} \left (\frac {a}{b}\right )^{\frac {1}{4}}}\)	\(259\)
default	\(-\frac {2 \left (-\frac {d^{3} x^{\frac {15}{2}} b^{3}}{15}+\frac {\left (a \,b^{2} d^{3}-3 b^{3} c \,d^{2}\right ) x^{\frac {11}{2}}}{11}+\frac {\left (-a^{2} b \,d^{3}+3 a \,b^{2} c \,d^{2}-3 b^{3} c^{2} d \right ) x^{\frac {7}{2}}}{7}+\frac {\left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) x^{\frac {3}{2}}}{3}\right )}{b^{4}}+\frac {a \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) \sqrt {2}\, \left (\ln \left (\frac {x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{4 b^{5} \left (\frac {a}{b}\right )^{\frac {1}{4}}}\)	\(259\)
risch	\(-\frac {2 x^{\frac {3}{2}} \left (-77 b^{3} d^{3} x^{6}+105 a \,b^{2} d^{3} x^{4}-315 b^{3} c \,d^{2} x^{4}-165 x^{2} a^{2} b \,d^{3}+495 x^{2} a \,b^{2} c \,d^{2}-495 x^{2} b^{3} c^{2} d +385 a^{3} d^{3}-1155 a^{2} b c \,d^{2}+1155 a \,b^{2} c^{2} d -385 b^{3} c^{3}\right )}{1155 b^{4}}+\frac {a \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) \sqrt {2}\, \left (\ln \left (\frac {x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{4 b^{5} \left (\frac {a}{b}\right )^{\frac {1}{4}}}\)	\(261\)

[In]

int(x^(5/2)*(d*x^2+c)^3/(b*x^2+a),x,method=_RETURNVERBOSE)

[Out]

-2/b^4*(-1/15*d^3*x^(15/2)*b^3+1/11*(a*b^2*d^3-3*b^3*c*d^2)*x^(11/2)+1/7*(-a^2*b*d^3+3*a*b^2*c*d^2-3*b^3*c^2*d
)*x^(7/2)+1/3*(a^3*d^3-3*a^2*b*c*d^2+3*a*b^2*c^2*d-b^3*c^3)*x^(3/2))+1/4*a*(a^3*d^3-3*a^2*b*c*d^2+3*a*b^2*c^2*
d-b^3*c^3)/b^5/(a/b)^(1/4)*2^(1/2)*(ln((x-(a/b)^(1/4)*x^(1/2)*2^(1/2)+(a/b)^(1/2))/(x+(a/b)^(1/4)*x^(1/2)*2^(1
/2)+(a/b)^(1/2)))+2*arctan(2^(1/2)/(a/b)^(1/4)*x^(1/2)+1)+2*arctan(2^(1/2)/(a/b)^(1/4)*x^(1/2)-1))

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.30 (sec) , antiderivative size = 2070, normalized size of antiderivative = 6.31 \[ \int \frac {x^{5/2} \left (c+d x^2\right )^3}{a+b x^2} \, dx=\text {Too large to display} \]

[In]

integrate(x^(5/2)*(d*x^2+c)^3/(b*x^2+a),x, algorithm="fricas")

[Out]

1/2310*(1155*b^4*(-(a^3*b^12*c^12 - 12*a^4*b^11*c^11*d + 66*a^5*b^10*c^10*d^2 - 220*a^6*b^9*c^9*d^3 + 495*a^7*
b^8*c^8*d^4 - 792*a^8*b^7*c^7*d^5 + 924*a^9*b^6*c^6*d^6 - 792*a^10*b^5*c^5*d^7 + 495*a^11*b^4*c^4*d^8 - 220*a^
12*b^3*c^3*d^9 + 66*a^13*b^2*c^2*d^10 - 12*a^14*b*c*d^11 + a^15*d^12)/b^19)^(1/4)*log(b^14*(-(a^3*b^12*c^12 -
12*a^4*b^11*c^11*d + 66*a^5*b^10*c^10*d^2 - 220*a^6*b^9*c^9*d^3 + 495*a^7*b^8*c^8*d^4 - 792*a^8*b^7*c^7*d^5 +
924*a^9*b^6*c^6*d^6 - 792*a^10*b^5*c^5*d^7 + 495*a^11*b^4*c^4*d^8 - 220*a^12*b^3*c^3*d^9 + 66*a^13*b^2*c^2*d^1
0 - 12*a^14*b*c*d^11 + a^15*d^12)/b^19)^(3/4) - (a^2*b^9*c^9 - 9*a^3*b^8*c^8*d + 36*a^4*b^7*c^7*d^2 - 84*a^5*b
^6*c^6*d^3 + 126*a^6*b^5*c^5*d^4 - 126*a^7*b^4*c^4*d^5 + 84*a^8*b^3*c^3*d^6 - 36*a^9*b^2*c^2*d^7 + 9*a^10*b*c*
d^8 - a^11*d^9)*sqrt(x)) - 1155*I*b^4*(-(a^3*b^12*c^12 - 12*a^4*b^11*c^11*d + 66*a^5*b^10*c^10*d^2 - 220*a^6*b
^9*c^9*d^3 + 495*a^7*b^8*c^8*d^4 - 792*a^8*b^7*c^7*d^5 + 924*a^9*b^6*c^6*d^6 - 792*a^10*b^5*c^5*d^7 + 495*a^11
*b^4*c^4*d^8 - 220*a^12*b^3*c^3*d^9 + 66*a^13*b^2*c^2*d^10 - 12*a^14*b*c*d^11 + a^15*d^12)/b^19)^(1/4)*log(I*b
^14*(-(a^3*b^12*c^12 - 12*a^4*b^11*c^11*d + 66*a^5*b^10*c^10*d^2 - 220*a^6*b^9*c^9*d^3 + 495*a^7*b^8*c^8*d^4 -
792*a^8*b^7*c^7*d^5 + 924*a^9*b^6*c^6*d^6 - 792*a^10*b^5*c^5*d^7 + 495*a^11*b^4*c^4*d^8 - 220*a^12*b^3*c^3*d^
9 + 66*a^13*b^2*c^2*d^10 - 12*a^14*b*c*d^11 + a^15*d^12)/b^19)^(3/4) - (a^2*b^9*c^9 - 9*a^3*b^8*c^8*d + 36*a^4
*b^7*c^7*d^2 - 84*a^5*b^6*c^6*d^3 + 126*a^6*b^5*c^5*d^4 - 126*a^7*b^4*c^4*d^5 + 84*a^8*b^3*c^3*d^6 - 36*a^9*b^
2*c^2*d^7 + 9*a^10*b*c*d^8 - a^11*d^9)*sqrt(x)) + 1155*I*b^4*(-(a^3*b^12*c^12 - 12*a^4*b^11*c^11*d + 66*a^5*b^
10*c^10*d^2 - 220*a^6*b^9*c^9*d^3 + 495*a^7*b^8*c^8*d^4 - 792*a^8*b^7*c^7*d^5 + 924*a^9*b^6*c^6*d^6 - 792*a^10
*b^5*c^5*d^7 + 495*a^11*b^4*c^4*d^8 - 220*a^12*b^3*c^3*d^9 + 66*a^13*b^2*c^2*d^10 - 12*a^14*b*c*d^11 + a^15*d^
12)/b^19)^(1/4)*log(-I*b^14*(-(a^3*b^12*c^12 - 12*a^4*b^11*c^11*d + 66*a^5*b^10*c^10*d^2 - 220*a^6*b^9*c^9*d^3
+ 495*a^7*b^8*c^8*d^4 - 792*a^8*b^7*c^7*d^5 + 924*a^9*b^6*c^6*d^6 - 792*a^10*b^5*c^5*d^7 + 495*a^11*b^4*c^4*d
^8 - 220*a^12*b^3*c^3*d^9 + 66*a^13*b^2*c^2*d^10 - 12*a^14*b*c*d^11 + a^15*d^12)/b^19)^(3/4) - (a^2*b^9*c^9 -
9*a^3*b^8*c^8*d + 36*a^4*b^7*c^7*d^2 - 84*a^5*b^6*c^6*d^3 + 126*a^6*b^5*c^5*d^4 - 126*a^7*b^4*c^4*d^5 + 84*a^8
*b^3*c^3*d^6 - 36*a^9*b^2*c^2*d^7 + 9*a^10*b*c*d^8 - a^11*d^9)*sqrt(x)) - 1155*b^4*(-(a^3*b^12*c^12 - 12*a^4*b
^11*c^11*d + 66*a^5*b^10*c^10*d^2 - 220*a^6*b^9*c^9*d^3 + 495*a^7*b^8*c^8*d^4 - 792*a^8*b^7*c^7*d^5 + 924*a^9*
b^6*c^6*d^6 - 792*a^10*b^5*c^5*d^7 + 495*a^11*b^4*c^4*d^8 - 220*a^12*b^3*c^3*d^9 + 66*a^13*b^2*c^2*d^10 - 12*a
^14*b*c*d^11 + a^15*d^12)/b^19)^(1/4)*log(-b^14*(-(a^3*b^12*c^12 - 12*a^4*b^11*c^11*d + 66*a^5*b^10*c^10*d^2 -
220*a^6*b^9*c^9*d^3 + 495*a^7*b^8*c^8*d^4 - 792*a^8*b^7*c^7*d^5 + 924*a^9*b^6*c^6*d^6 - 792*a^10*b^5*c^5*d^7
+ 495*a^11*b^4*c^4*d^8 - 220*a^12*b^3*c^3*d^9 + 66*a^13*b^2*c^2*d^10 - 12*a^14*b*c*d^11 + a^15*d^12)/b^19)^(3/
4) - (a^2*b^9*c^9 - 9*a^3*b^8*c^8*d + 36*a^4*b^7*c^7*d^2 - 84*a^5*b^6*c^6*d^3 + 126*a^6*b^5*c^5*d^4 - 126*a^7*
b^4*c^4*d^5 + 84*a^8*b^3*c^3*d^6 - 36*a^9*b^2*c^2*d^7 + 9*a^10*b*c*d^8 - a^11*d^9)*sqrt(x)) + 4*(77*b^3*d^3*x^
7 + 105*(3*b^3*c*d^2 - a*b^2*d^3)*x^5 + 165*(3*b^3*c^2*d - 3*a*b^2*c*d^2 + a^2*b*d^3)*x^3 + 385*(b^3*c^3 - 3*a
*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*x)*sqrt(x))/b^4

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 743 vs. \(2 (311) = 622\).

Time = 83.48 (sec) , antiderivative size = 743, normalized size of antiderivative = 2.27 \[ \int \frac {x^{5/2} \left (c+d x^2\right )^3}{a+b x^2} \, dx=\begin {cases} \tilde {\infty } \left (\frac {2 c^{3} x^{\frac {3}{2}}}{3} + \frac {6 c^{2} d x^{\frac {7}{2}}}{7} + \frac {6 c d^{2} x^{\frac {11}{2}}}{11} + \frac {2 d^{3} x^{\frac {15}{2}}}{15}\right ) & \text {for}\: a = 0 \wedge b = 0 \\\frac {\frac {2 c^{3} x^{\frac {7}{2}}}{7} + \frac {6 c^{2} d x^{\frac {11}{2}}}{11} + \frac {2 c d^{2} x^{\frac {15}{2}}}{5} + \frac {2 d^{3} x^{\frac {19}{2}}}{19}}{a} & \text {for}\: b = 0 \\\frac {\frac {2 c^{3} x^{\frac {3}{2}}}{3} + \frac {6 c^{2} d x^{\frac {7}{2}}}{7} + \frac {6 c d^{2} x^{\frac {11}{2}}}{11} + \frac {2 d^{3} x^{\frac {15}{2}}}{15}}{b} & \text {for}\: a = 0 \\- \frac {2 a^{3} d^{3} x^{\frac {3}{2}}}{3 b^{4}} - \frac {a^{3} d^{3} \left (- \frac {a}{b}\right )^{\frac {3}{4}} \log {\left (\sqrt {x} - \sqrt [4]{- \frac {a}{b}} \right )}}{2 b^{4}} + \frac {a^{3} d^{3} \left (- \frac {a}{b}\right )^{\frac {3}{4}} \log {\left (\sqrt {x} + \sqrt [4]{- \frac {a}{b}} \right )}}{2 b^{4}} - \frac {a^{3} d^{3} \left (- \frac {a}{b}\right )^{\frac {3}{4}} \operatorname {atan}{\left (\frac {\sqrt {x}}{\sqrt [4]{- \frac {a}{b}}} \right )}}{b^{4}} + \frac {2 a^{2} c d^{2} x^{\frac {3}{2}}}{b^{3}} + \frac {3 a^{2} c d^{2} \left (- \frac {a}{b}\right )^{\frac {3}{4}} \log {\left (\sqrt {x} - \sqrt [4]{- \frac {a}{b}} \right )}}{2 b^{3}} - \frac {3 a^{2} c d^{2} \left (- \frac {a}{b}\right )^{\frac {3}{4}} \log {\left (\sqrt {x} + \sqrt [4]{- \frac {a}{b}} \right )}}{2 b^{3}} + \frac {3 a^{2} c d^{2} \left (- \frac {a}{b}\right )^{\frac {3}{4}} \operatorname {atan}{\left (\frac {\sqrt {x}}{\sqrt [4]{- \frac {a}{b}}} \right )}}{b^{3}} + \frac {2 a^{2} d^{3} x^{\frac {7}{2}}}{7 b^{3}} - \frac {2 a c^{2} d x^{\frac {3}{2}}}{b^{2}} - \frac {3 a c^{2} d \left (- \frac {a}{b}\right )^{\frac {3}{4}} \log {\left (\sqrt {x} - \sqrt [4]{- \frac {a}{b}} \right )}}{2 b^{2}} + \frac {3 a c^{2} d \left (- \frac {a}{b}\right )^{\frac {3}{4}} \log {\left (\sqrt {x} + \sqrt [4]{- \frac {a}{b}} \right )}}{2 b^{2}} - \frac {3 a c^{2} d \left (- \frac {a}{b}\right )^{\frac {3}{4}} \operatorname {atan}{\left (\frac {\sqrt {x}}{\sqrt [4]{- \frac {a}{b}}} \right )}}{b^{2}} - \frac {6 a c d^{2} x^{\frac {7}{2}}}{7 b^{2}} - \frac {2 a d^{3} x^{\frac {11}{2}}}{11 b^{2}} + \frac {2 c^{3} x^{\frac {3}{2}}}{3 b} + \frac {c^{3} \left (- \frac {a}{b}\right )^{\frac {3}{4}} \log {\left (\sqrt {x} - \sqrt [4]{- \frac {a}{b}} \right )}}{2 b} - \frac {c^{3} \left (- \frac {a}{b}\right )^{\frac {3}{4}} \log {\left (\sqrt {x} + \sqrt [4]{- \frac {a}{b}} \right )}}{2 b} + \frac {c^{3} \left (- \frac {a}{b}\right )^{\frac {3}{4}} \operatorname {atan}{\left (\frac {\sqrt {x}}{\sqrt [4]{- \frac {a}{b}}} \right )}}{b} + \frac {6 c^{2} d x^{\frac {7}{2}}}{7 b} + \frac {6 c d^{2} x^{\frac {11}{2}}}{11 b} + \frac {2 d^{3} x^{\frac {15}{2}}}{15 b} & \text {otherwise} \end {cases} \]

[In]

integrate(x**(5/2)*(d*x**2+c)**3/(b*x**2+a),x)

[Out]

Piecewise((zoo*(2*c**3*x**(3/2)/3 + 6*c**2*d*x**(7/2)/7 + 6*c*d**2*x**(11/2)/11 + 2*d**3*x**(15/2)/15), Eq(a,
0) & Eq(b, 0)), ((2*c**3*x**(7/2)/7 + 6*c**2*d*x**(11/2)/11 + 2*c*d**2*x**(15/2)/5 + 2*d**3*x**(19/2)/19)/a, E
q(b, 0)), ((2*c**3*x**(3/2)/3 + 6*c**2*d*x**(7/2)/7 + 6*c*d**2*x**(11/2)/11 + 2*d**3*x**(15/2)/15)/b, Eq(a, 0)
), (-2*a**3*d**3*x**(3/2)/(3*b**4) - a**3*d**3*(-a/b)**(3/4)*log(sqrt(x) - (-a/b)**(1/4))/(2*b**4) + a**3*d**3
*(-a/b)**(3/4)*log(sqrt(x) + (-a/b)**(1/4))/(2*b**4) - a**3*d**3*(-a/b)**(3/4)*atan(sqrt(x)/(-a/b)**(1/4))/b**
4 + 2*a**2*c*d**2*x**(3/2)/b**3 + 3*a**2*c*d**2*(-a/b)**(3/4)*log(sqrt(x) - (-a/b)**(1/4))/(2*b**3) - 3*a**2*c
*d**2*(-a/b)**(3/4)*log(sqrt(x) + (-a/b)**(1/4))/(2*b**3) + 3*a**2*c*d**2*(-a/b)**(3/4)*atan(sqrt(x)/(-a/b)**(
1/4))/b**3 + 2*a**2*d**3*x**(7/2)/(7*b**3) - 2*a*c**2*d*x**(3/2)/b**2 - 3*a*c**2*d*(-a/b)**(3/4)*log(sqrt(x) -
(-a/b)**(1/4))/(2*b**2) + 3*a*c**2*d*(-a/b)**(3/4)*log(sqrt(x) + (-a/b)**(1/4))/(2*b**2) - 3*a*c**2*d*(-a/b)*
*(3/4)*atan(sqrt(x)/(-a/b)**(1/4))/b**2 - 6*a*c*d**2*x**(7/2)/(7*b**2) - 2*a*d**3*x**(11/2)/(11*b**2) + 2*c**3
*x**(3/2)/(3*b) + c**3*(-a/b)**(3/4)*log(sqrt(x) - (-a/b)**(1/4))/(2*b) - c**3*(-a/b)**(3/4)*log(sqrt(x) + (-a
/b)**(1/4))/(2*b) + c**3*(-a/b)**(3/4)*atan(sqrt(x)/(-a/b)**(1/4))/b + 6*c**2*d*x**(7/2)/(7*b) + 6*c*d**2*x**(
11/2)/(11*b) + 2*d**3*x**(15/2)/(15*b), True))

Maxima [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 331, normalized size of antiderivative = 1.01 \[ \int \frac {x^{5/2} \left (c+d x^2\right )^3}{a+b x^2} \, dx=-\frac {{\left (a b^{3} c^{3} - 3 \, a^{2} b^{2} c^{2} d + 3 \, a^{3} b c d^{2} - a^{4} d^{3}\right )} {\left (\frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} + 2 \, \sqrt {b} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {\sqrt {a} \sqrt {b}} \sqrt {b}} + \frac {2 \, \sqrt {2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} - 2 \, \sqrt {b} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {\sqrt {a} \sqrt {b}} \sqrt {b}} - \frac {\sqrt {2} \log \left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {x} + \sqrt {b} x + \sqrt {a}\right )}{a^{\frac {1}{4}} b^{\frac {3}{4}}} + \frac {\sqrt {2} \log \left (-\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {x} + \sqrt {b} x + \sqrt {a}\right )}{a^{\frac {1}{4}} b^{\frac {3}{4}}}\right )}}{4 \, b^{4}} + \frac {2 \, {\left (77 \, b^{3} d^{3} x^{\frac {15}{2}} + 105 \, {\left (3 \, b^{3} c d^{2} - a b^{2} d^{3}\right )} x^{\frac {11}{2}} + 165 \, {\left (3 \, b^{3} c^{2} d - 3 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} x^{\frac {7}{2}} + 385 \, {\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} x^{\frac {3}{2}}\right )}}{1155 \, b^{4}} \]

[In]

integrate(x^(5/2)*(d*x^2+c)^3/(b*x^2+a),x, algorithm="maxima")

[Out]

-1/4*(a*b^3*c^3 - 3*a^2*b^2*c^2*d + 3*a^3*b*c*d^2 - a^4*d^3)*(2*sqrt(2)*arctan(1/2*sqrt(2)*(sqrt(2)*a^(1/4)*b^
(1/4) + 2*sqrt(b)*sqrt(x))/sqrt(sqrt(a)*sqrt(b)))/(sqrt(sqrt(a)*sqrt(b))*sqrt(b)) + 2*sqrt(2)*arctan(-1/2*sqrt
(2)*(sqrt(2)*a^(1/4)*b^(1/4) - 2*sqrt(b)*sqrt(x))/sqrt(sqrt(a)*sqrt(b)))/(sqrt(sqrt(a)*sqrt(b))*sqrt(b)) - sqr
t(2)*log(sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x + sqrt(a))/(a^(1/4)*b^(3/4)) + sqrt(2)*log(-sqrt(2)*a^(1/
4)*b^(1/4)*sqrt(x) + sqrt(b)*x + sqrt(a))/(a^(1/4)*b^(3/4)))/b^4 + 2/1155*(77*b^3*d^3*x^(15/2) + 105*(3*b^3*c*
d^2 - a*b^2*d^3)*x^(11/2) + 165*(3*b^3*c^2*d - 3*a*b^2*c*d^2 + a^2*b*d^3)*x^(7/2) + 385*(b^3*c^3 - 3*a*b^2*c^2
*d + 3*a^2*b*c*d^2 - a^3*d^3)*x^(3/2))/b^4

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 531 vs. \(2 (245) = 490\).

Time = 0.31 (sec) , antiderivative size = 531, normalized size of antiderivative = 1.62 \[ \int \frac {x^{5/2} \left (c+d x^2\right )^3}{a+b x^2} \, dx=-\frac {\sqrt {2} {\left (\left (a b^{3}\right )^{\frac {3}{4}} b^{3} c^{3} - 3 \, \left (a b^{3}\right )^{\frac {3}{4}} a b^{2} c^{2} d + 3 \, \left (a b^{3}\right )^{\frac {3}{4}} a^{2} b c d^{2} - \left (a b^{3}\right )^{\frac {3}{4}} a^{3} d^{3}\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a}{b}\right )^{\frac {1}{4}} + 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{2 \, b^{7}} - \frac {\sqrt {2} {\left (\left (a b^{3}\right )^{\frac {3}{4}} b^{3} c^{3} - 3 \, \left (a b^{3}\right )^{\frac {3}{4}} a b^{2} c^{2} d + 3 \, \left (a b^{3}\right )^{\frac {3}{4}} a^{2} b c d^{2} - \left (a b^{3}\right )^{\frac {3}{4}} a^{3} d^{3}\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a}{b}\right )^{\frac {1}{4}} - 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{2 \, b^{7}} + \frac {\sqrt {2} {\left (\left (a b^{3}\right )^{\frac {3}{4}} b^{3} c^{3} - 3 \, \left (a b^{3}\right )^{\frac {3}{4}} a b^{2} c^{2} d + 3 \, \left (a b^{3}\right )^{\frac {3}{4}} a^{2} b c d^{2} - \left (a b^{3}\right )^{\frac {3}{4}} a^{3} d^{3}\right )} \log \left (\sqrt {2} \sqrt {x} \left (\frac {a}{b}\right )^{\frac {1}{4}} + x + \sqrt {\frac {a}{b}}\right )}{4 \, b^{7}} - \frac {\sqrt {2} {\left (\left (a b^{3}\right )^{\frac {3}{4}} b^{3} c^{3} - 3 \, \left (a b^{3}\right )^{\frac {3}{4}} a b^{2} c^{2} d + 3 \, \left (a b^{3}\right )^{\frac {3}{4}} a^{2} b c d^{2} - \left (a b^{3}\right )^{\frac {3}{4}} a^{3} d^{3}\right )} \log \left (-\sqrt {2} \sqrt {x} \left (\frac {a}{b}\right )^{\frac {1}{4}} + x + \sqrt {\frac {a}{b}}\right )}{4 \, b^{7}} + \frac {2 \, {\left (77 \, b^{14} d^{3} x^{\frac {15}{2}} + 315 \, b^{14} c d^{2} x^{\frac {11}{2}} - 105 \, a b^{13} d^{3} x^{\frac {11}{2}} + 495 \, b^{14} c^{2} d x^{\frac {7}{2}} - 495 \, a b^{13} c d^{2} x^{\frac {7}{2}} + 165 \, a^{2} b^{12} d^{3} x^{\frac {7}{2}} + 385 \, b^{14} c^{3} x^{\frac {3}{2}} - 1155 \, a b^{13} c^{2} d x^{\frac {3}{2}} + 1155 \, a^{2} b^{12} c d^{2} x^{\frac {3}{2}} - 385 \, a^{3} b^{11} d^{3} x^{\frac {3}{2}}\right )}}{1155 \, b^{15}} \]

[In]

integrate(x^(5/2)*(d*x^2+c)^3/(b*x^2+a),x, algorithm="giac")

[Out]

-1/2*sqrt(2)*((a*b^3)^(3/4)*b^3*c^3 - 3*(a*b^3)^(3/4)*a*b^2*c^2*d + 3*(a*b^3)^(3/4)*a^2*b*c*d^2 - (a*b^3)^(3/4
)*a^3*d^3)*arctan(1/2*sqrt(2)*(sqrt(2)*(a/b)^(1/4) + 2*sqrt(x))/(a/b)^(1/4))/b^7 - 1/2*sqrt(2)*((a*b^3)^(3/4)*
b^3*c^3 - 3*(a*b^3)^(3/4)*a*b^2*c^2*d + 3*(a*b^3)^(3/4)*a^2*b*c*d^2 - (a*b^3)^(3/4)*a^3*d^3)*arctan(-1/2*sqrt(
2)*(sqrt(2)*(a/b)^(1/4) - 2*sqrt(x))/(a/b)^(1/4))/b^7 + 1/4*sqrt(2)*((a*b^3)^(3/4)*b^3*c^3 - 3*(a*b^3)^(3/4)*a
*b^2*c^2*d + 3*(a*b^3)^(3/4)*a^2*b*c*d^2 - (a*b^3)^(3/4)*a^3*d^3)*log(sqrt(2)*sqrt(x)*(a/b)^(1/4) + x + sqrt(a
/b))/b^7 - 1/4*sqrt(2)*((a*b^3)^(3/4)*b^3*c^3 - 3*(a*b^3)^(3/4)*a*b^2*c^2*d + 3*(a*b^3)^(3/4)*a^2*b*c*d^2 - (a
*b^3)^(3/4)*a^3*d^3)*log(-sqrt(2)*sqrt(x)*(a/b)^(1/4) + x + sqrt(a/b))/b^7 + 2/1155*(77*b^14*d^3*x^(15/2) + 31
5*b^14*c*d^2*x^(11/2) - 105*a*b^13*d^3*x^(11/2) + 495*b^14*c^2*d*x^(7/2) - 495*a*b^13*c*d^2*x^(7/2) + 165*a^2*
b^12*d^3*x^(7/2) + 385*b^14*c^3*x^(3/2) - 1155*a*b^13*c^2*d*x^(3/2) + 1155*a^2*b^12*c*d^2*x^(3/2) - 385*a^3*b^
11*d^3*x^(3/2))/b^15

Mupad [B] (verification not implemented)

Time = 5.32 (sec) , antiderivative size = 634, normalized size of antiderivative = 1.93 \[ \int \frac {x^{5/2} \left (c+d x^2\right )^3}{a+b x^2} \, dx=x^{3/2}\,\left (\frac {2\,c^3}{3\,b}-\frac {a\,\left (\frac {6\,c^2\,d}{b}+\frac {a\,\left (\frac {2\,a\,d^3}{b^2}-\frac {6\,c\,d^2}{b}\right )}{b}\right )}{3\,b}\right )-x^{11/2}\,\left (\frac {2\,a\,d^3}{11\,b^2}-\frac {6\,c\,d^2}{11\,b}\right )+x^{7/2}\,\left (\frac {6\,c^2\,d}{7\,b}+\frac {a\,\left (\frac {2\,a\,d^3}{b^2}-\frac {6\,c\,d^2}{b}\right )}{7\,b}\right )+\frac {2\,d^3\,x^{15/2}}{15\,b}+\frac {{\left (-a\right )}^{3/4}\,\mathrm {atan}\left (\frac {{\left (-a\right )}^{3/4}\,b^{1/4}\,\sqrt {x}\,{\left (a\,d-b\,c\right )}^3\,\left (a^9\,d^6-6\,a^8\,b\,c\,d^5+15\,a^7\,b^2\,c^2\,d^4-20\,a^6\,b^3\,c^3\,d^3+15\,a^5\,b^4\,c^4\,d^2-6\,a^4\,b^5\,c^5\,d+a^3\,b^6\,c^6\right )}{a^{13}\,d^9-9\,a^{12}\,b\,c\,d^8+36\,a^{11}\,b^2\,c^2\,d^7-84\,a^{10}\,b^3\,c^3\,d^6+126\,a^9\,b^4\,c^4\,d^5-126\,a^8\,b^5\,c^5\,d^4+84\,a^7\,b^6\,c^6\,d^3-36\,a^6\,b^7\,c^7\,d^2+9\,a^5\,b^8\,c^8\,d-a^4\,b^9\,c^9}\right )\,{\left (a\,d-b\,c\right )}^3}{b^{19/4}}+\frac {{\left (-a\right )}^{3/4}\,\mathrm {atan}\left (\frac {{\left (-a\right )}^{3/4}\,b^{1/4}\,\sqrt {x}\,{\left (a\,d-b\,c\right )}^3\,\left (a^9\,d^6-6\,a^8\,b\,c\,d^5+15\,a^7\,b^2\,c^2\,d^4-20\,a^6\,b^3\,c^3\,d^3+15\,a^5\,b^4\,c^4\,d^2-6\,a^4\,b^5\,c^5\,d+a^3\,b^6\,c^6\right )\,1{}\mathrm {i}}{a^{13}\,d^9-9\,a^{12}\,b\,c\,d^8+36\,a^{11}\,b^2\,c^2\,d^7-84\,a^{10}\,b^3\,c^3\,d^6+126\,a^9\,b^4\,c^4\,d^5-126\,a^8\,b^5\,c^5\,d^4+84\,a^7\,b^6\,c^6\,d^3-36\,a^6\,b^7\,c^7\,d^2+9\,a^5\,b^8\,c^8\,d-a^4\,b^9\,c^9}\right )\,{\left (a\,d-b\,c\right )}^3\,1{}\mathrm {i}}{b^{19/4}} \]

[In]

int((x^(5/2)*(c + d*x^2)^3)/(a + b*x^2),x)

[Out]

x^(3/2)*((2*c^3)/(3*b) - (a*((6*c^2*d)/b + (a*((2*a*d^3)/b^2 - (6*c*d^2)/b))/b))/(3*b)) - x^(11/2)*((2*a*d^3)/
(11*b^2) - (6*c*d^2)/(11*b)) + x^(7/2)*((6*c^2*d)/(7*b) + (a*((2*a*d^3)/b^2 - (6*c*d^2)/b))/(7*b)) + (2*d^3*x^
(15/2))/(15*b) + ((-a)^(3/4)*atan(((-a)^(3/4)*b^(1/4)*x^(1/2)*(a*d - b*c)^3*(a^9*d^6 + a^3*b^6*c^6 - 6*a^4*b^5
*c^5*d + 15*a^5*b^4*c^4*d^2 - 20*a^6*b^3*c^3*d^3 + 15*a^7*b^2*c^2*d^4 - 6*a^8*b*c*d^5))/(a^13*d^9 - a^4*b^9*c^
9 + 9*a^5*b^8*c^8*d - 36*a^6*b^7*c^7*d^2 + 84*a^7*b^6*c^6*d^3 - 126*a^8*b^5*c^5*d^4 + 126*a^9*b^4*c^4*d^5 - 84
*a^10*b^3*c^3*d^6 + 36*a^11*b^2*c^2*d^7 - 9*a^12*b*c*d^8))*(a*d - b*c)^3)/b^(19/4) + ((-a)^(3/4)*atan(((-a)^(3
/4)*b^(1/4)*x^(1/2)*(a*d - b*c)^3*(a^9*d^6 + a^3*b^6*c^6 - 6*a^4*b^5*c^5*d + 15*a^5*b^4*c^4*d^2 - 20*a^6*b^3*c
^3*d^3 + 15*a^7*b^2*c^2*d^4 - 6*a^8*b*c*d^5)*1i)/(a^13*d^9 - a^4*b^9*c^9 + 9*a^5*b^8*c^8*d - 36*a^6*b^7*c^7*d^
2 + 84*a^7*b^6*c^6*d^3 - 126*a^8*b^5*c^5*d^4 + 126*a^9*b^4*c^4*d^5 - 84*a^10*b^3*c^3*d^6 + 36*a^11*b^2*c^2*d^7
- 9*a^12*b*c*d^8))*(a*d - b*c)^3*1i)/b^(19/4)