\(\int \frac {x^{3/2} (c+d x^2)^3}{a+b x^2} \, dx\) [442]

Optimal result

Integrand size = 24, antiderivative size = 326 \[ \int \frac {x^{3/2} \left (c+d x^2\right )^3}{a+b x^2} \, dx=\frac {2 (b c-a d)^3 \sqrt {x}}{b^4}+\frac {2 d \left (3 b^2 c^2-3 a b c d+a^2 d^2\right ) x^{5/2}}{5 b^3}+\frac {2 d^2 (3 b c-a d) x^{9/2}}{9 b^2}+\frac {2 d^3 x^{13/2}}{13 b}+\frac {\sqrt [4]{a} (b c-a d)^3 \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} b^{17/4}}-\frac {\sqrt [4]{a} (b c-a d)^3 \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} b^{17/4}}+\frac {\sqrt [4]{a} (b c-a d)^3 \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{2 \sqrt {2} b^{17/4}}-\frac {\sqrt [4]{a} (b c-a d)^3 \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{2 \sqrt {2} b^{17/4}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.21 (sec) , antiderivative size = 326, 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, 217, 1179, 642, 1176, 631, 210} \[ \int \frac {x^{3/2} \left (c+d x^2\right )^3}{a+b x^2} \, dx=\frac {2 d x^{5/2} \left (a^2 d^2-3 a b c d+3 b^2 c^2\right )}{5 b^3}+\frac {\sqrt [4]{a} \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right ) (b c-a d)^3}{\sqrt {2} b^{17/4}}-\frac {\sqrt [4]{a} \arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right ) (b c-a d)^3}{\sqrt {2} b^{17/4}}+\frac {\sqrt [4]{a} (b c-a d)^3 \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{2 \sqrt {2} b^{17/4}}-\frac {\sqrt [4]{a} (b c-a d)^3 \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{2 \sqrt {2} b^{17/4}}+\frac {2 \sqrt {x} (b c-a d)^3}{b^4}+\frac {2 d^2 x^{9/2} (3 b c-a d)}{9 b^2}+\frac {2 d^3 x^{13/2}}{13 b} \]

[In]

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

[Out]

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

Mathematica [A] (verified)

Time = 0.24 (sec) , antiderivative size = 231, normalized size of antiderivative = 0.71 \[ \int \frac {x^{3/2} \left (c+d x^2\right )^3}{a+b x^2} \, dx=\frac {2 \sqrt {x} \left (-585 a^3 d^3+117 a^2 b d^2 \left (15 c+d x^2\right )-13 a b^2 d \left (135 c^2+27 c d x^2+5 d^2 x^4\right )+3 b^3 \left (195 c^3+117 c^2 d x^2+65 c d^2 x^4+15 d^3 x^6\right )\right )}{585 b^4}-\frac {\sqrt [4]{a} (-b c+a d)^3 \arctan \left (\frac {\sqrt {a}-\sqrt {b} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}\right )}{\sqrt {2} b^{17/4}}+\frac {\sqrt [4]{a} (-b c+a d)^3 \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a}+\sqrt {b} x}\right )}{\sqrt {2} b^{17/4}} \]

[In]

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

[Out]

(2*Sqrt[x]*(-585*a^3*d^3 + 117*a^2*b*d^2*(15*c + d*x^2) - 13*a*b^2*d*(135*c^2 + 27*c*d*x^2 + 5*d^2*x^4) + 3*b^
3*(195*c^3 + 117*c^2*d*x^2 + 65*c*d^2*x^4 + 15*d^3*x^6)))/(585*b^4) - (a^(1/4)*(-(b*c) + a*d)^3*ArcTan[(Sqrt[a
] - Sqrt[b]*x)/(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])])/(Sqrt[2]*b^(17/4)) + (a^(1/4)*(-(b*c) + a*d)^3*ArcTanh[(Sqr
t[2]*a^(1/4)*b^(1/4)*Sqrt[x])/(Sqrt[a] + Sqrt[b]*x)])/(Sqrt[2]*b^(17/4))

Maple [A] (verified)

Time = 2.70 (sec) , antiderivative size = 260, normalized size of antiderivative = 0.80

method	result	size
risch	\(-\frac {2 \left (-45 b^{3} d^{3} x^{6}+65 a \,b^{2} d^{3} x^{4}-195 b^{3} c \,d^{2} x^{4}-117 x^{2} a^{2} b \,d^{3}+351 x^{2} a \,b^{2} c \,d^{2}-351 x^{2} b^{3} c^{2} d +585 a^{3} d^{3}-1755 a^{2} b c \,d^{2}+1755 a \,b^{2} c^{2} d -585 b^{3} c^{3}\right ) \sqrt {x}}{585 b^{4}}+\frac {\left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{4 b^{4}}\)	\(260\)
derivativedivides	\(-\frac {2 \left (-\frac {d^{3} x^{\frac {13}{2}} b^{3}}{13}+\frac {a \,b^{2} d^{3} x^{\frac {9}{2}}}{9}-\frac {b^{3} c \,d^{2} x^{\frac {9}{2}}}{3}-\frac {a^{2} b \,d^{3} x^{\frac {5}{2}}}{5}+\frac {3 a \,b^{2} c \,d^{2} x^{\frac {5}{2}}}{5}-\frac {3 b^{3} c^{2} d \,x^{\frac {5}{2}}}{5}+a^{3} d^{3} \sqrt {x}-3 a^{2} b c \,d^{2} \sqrt {x}+3 a \,b^{2} c^{2} d \sqrt {x}-b^{3} c^{3} \sqrt {x}\right )}{b^{4}}+\frac {\left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{4 b^{4}}\)	\(268\)
default	\(-\frac {2 \left (-\frac {d^{3} x^{\frac {13}{2}} b^{3}}{13}+\frac {a \,b^{2} d^{3} x^{\frac {9}{2}}}{9}-\frac {b^{3} c \,d^{2} x^{\frac {9}{2}}}{3}-\frac {a^{2} b \,d^{3} x^{\frac {5}{2}}}{5}+\frac {3 a \,b^{2} c \,d^{2} x^{\frac {5}{2}}}{5}-\frac {3 b^{3} c^{2} d \,x^{\frac {5}{2}}}{5}+a^{3} d^{3} \sqrt {x}-3 a^{2} b c \,d^{2} \sqrt {x}+3 a \,b^{2} c^{2} d \sqrt {x}-b^{3} c^{3} \sqrt {x}\right )}{b^{4}}+\frac {\left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{4 b^{4}}\)	\(268\)

[In]

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

[Out]

-2/585*(-45*b^3*d^3*x^6+65*a*b^2*d^3*x^4-195*b^3*c*d^2*x^4-117*a^2*b*d^3*x^2+351*a*b^2*c*d^2*x^2-351*b^3*c^2*d
*x^2+585*a^3*d^3-1755*a^2*b*c*d^2+1755*a*b^2*c^2*d-585*b^3*c^3)*x^(1/2)/b^4+1/4*(a^3*d^3-3*a^2*b*c*d^2+3*a*b^2
*c^2*d-b^3*c^3)/b^4*(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 = 0.28 (sec) , antiderivative size = 1695, normalized size of antiderivative = 5.20 \[ \int \frac {x^{3/2} \left (c+d x^2\right )^3}{a+b x^2} \, dx=\text {Too large to display} \]

[In]

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

[Out]

1/1170*(585*b^4*(-(a*b^12*c^12 - 12*a^2*b^11*c^11*d + 66*a^3*b^10*c^10*d^2 - 220*a^4*b^9*c^9*d^3 + 495*a^5*b^8
*c^8*d^4 - 792*a^6*b^7*c^7*d^5 + 924*a^7*b^6*c^6*d^6 - 792*a^8*b^5*c^5*d^7 + 495*a^9*b^4*c^4*d^8 - 220*a^10*b^
3*c^3*d^9 + 66*a^11*b^2*c^2*d^10 - 12*a^12*b*c*d^11 + a^13*d^12)/b^17)^(1/4)*log(b^4*(-(a*b^12*c^12 - 12*a^2*b
^11*c^11*d + 66*a^3*b^10*c^10*d^2 - 220*a^4*b^9*c^9*d^3 + 495*a^5*b^8*c^8*d^4 - 792*a^6*b^7*c^7*d^5 + 924*a^7*
b^6*c^6*d^6 - 792*a^8*b^5*c^5*d^7 + 495*a^9*b^4*c^4*d^8 - 220*a^10*b^3*c^3*d^9 + 66*a^11*b^2*c^2*d^10 - 12*a^1
2*b*c*d^11 + a^13*d^12)/b^17)^(1/4) - (b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*sqrt(x)) + 585*I*b^4
*(-(a*b^12*c^12 - 12*a^2*b^11*c^11*d + 66*a^3*b^10*c^10*d^2 - 220*a^4*b^9*c^9*d^3 + 495*a^5*b^8*c^8*d^4 - 792*
a^6*b^7*c^7*d^5 + 924*a^7*b^6*c^6*d^6 - 792*a^8*b^5*c^5*d^7 + 495*a^9*b^4*c^4*d^8 - 220*a^10*b^3*c^3*d^9 + 66*
a^11*b^2*c^2*d^10 - 12*a^12*b*c*d^11 + a^13*d^12)/b^17)^(1/4)*log(I*b^4*(-(a*b^12*c^12 - 12*a^2*b^11*c^11*d +
66*a^3*b^10*c^10*d^2 - 220*a^4*b^9*c^9*d^3 + 495*a^5*b^8*c^8*d^4 - 792*a^6*b^7*c^7*d^5 + 924*a^7*b^6*c^6*d^6 -
792*a^8*b^5*c^5*d^7 + 495*a^9*b^4*c^4*d^8 - 220*a^10*b^3*c^3*d^9 + 66*a^11*b^2*c^2*d^10 - 12*a^12*b*c*d^11 +
a^13*d^12)/b^17)^(1/4) - (b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*sqrt(x)) - 585*I*b^4*(-(a*b^12*c^
12 - 12*a^2*b^11*c^11*d + 66*a^3*b^10*c^10*d^2 - 220*a^4*b^9*c^9*d^3 + 495*a^5*b^8*c^8*d^4 - 792*a^6*b^7*c^7*d
^5 + 924*a^7*b^6*c^6*d^6 - 792*a^8*b^5*c^5*d^7 + 495*a^9*b^4*c^4*d^8 - 220*a^10*b^3*c^3*d^9 + 66*a^11*b^2*c^2*
d^10 - 12*a^12*b*c*d^11 + a^13*d^12)/b^17)^(1/4)*log(-I*b^4*(-(a*b^12*c^12 - 12*a^2*b^11*c^11*d + 66*a^3*b^10*
c^10*d^2 - 220*a^4*b^9*c^9*d^3 + 495*a^5*b^8*c^8*d^4 - 792*a^6*b^7*c^7*d^5 + 924*a^7*b^6*c^6*d^6 - 792*a^8*b^5
*c^5*d^7 + 495*a^9*b^4*c^4*d^8 - 220*a^10*b^3*c^3*d^9 + 66*a^11*b^2*c^2*d^10 - 12*a^12*b*c*d^11 + a^13*d^12)/b
^17)^(1/4) - (b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*sqrt(x)) - 585*b^4*(-(a*b^12*c^12 - 12*a^2*b^
11*c^11*d + 66*a^3*b^10*c^10*d^2 - 220*a^4*b^9*c^9*d^3 + 495*a^5*b^8*c^8*d^4 - 792*a^6*b^7*c^7*d^5 + 924*a^7*b
^6*c^6*d^6 - 792*a^8*b^5*c^5*d^7 + 495*a^9*b^4*c^4*d^8 - 220*a^10*b^3*c^3*d^9 + 66*a^11*b^2*c^2*d^10 - 12*a^12
*b*c*d^11 + a^13*d^12)/b^17)^(1/4)*log(-b^4*(-(a*b^12*c^12 - 12*a^2*b^11*c^11*d + 66*a^3*b^10*c^10*d^2 - 220*a
^4*b^9*c^9*d^3 + 495*a^5*b^8*c^8*d^4 - 792*a^6*b^7*c^7*d^5 + 924*a^7*b^6*c^6*d^6 - 792*a^8*b^5*c^5*d^7 + 495*a
^9*b^4*c^4*d^8 - 220*a^10*b^3*c^3*d^9 + 66*a^11*b^2*c^2*d^10 - 12*a^12*b*c*d^11 + a^13*d^12)/b^17)^(1/4) - (b^
3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*sqrt(x)) + 4*(45*b^3*d^3*x^6 + 585*b^3*c^3 - 1755*a*b^2*c^2*d
+ 1755*a^2*b*c*d^2 - 585*a^3*d^3 + 65*(3*b^3*c*d^2 - a*b^2*d^3)*x^4 + 117*(3*b^3*c^2*d - 3*a*b^2*c*d^2 + a^2*
b*d^3)*x^2)*sqrt(x))/b^4

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 736 vs. \(2 (309) = 618\).

Time = 36.67 (sec) , antiderivative size = 736, normalized size of antiderivative = 2.26 \[ \int \frac {x^{3/2} \left (c+d x^2\right )^3}{a+b x^2} \, dx=\begin {cases} \tilde {\infty } \left (2 c^{3} \sqrt {x} + \frac {6 c^{2} d x^{\frac {5}{2}}}{5} + \frac {2 c d^{2} x^{\frac {9}{2}}}{3} + \frac {2 d^{3} x^{\frac {13}{2}}}{13}\right ) & \text {for}\: a = 0 \wedge b = 0 \\\frac {\frac {2 c^{3} x^{\frac {5}{2}}}{5} + \frac {2 c^{2} d x^{\frac {9}{2}}}{3} + \frac {6 c d^{2} x^{\frac {13}{2}}}{13} + \frac {2 d^{3} x^{\frac {17}{2}}}{17}}{a} & \text {for}\: b = 0 \\\frac {2 c^{3} \sqrt {x} + \frac {6 c^{2} d x^{\frac {5}{2}}}{5} + \frac {2 c d^{2} x^{\frac {9}{2}}}{3} + \frac {2 d^{3} x^{\frac {13}{2}}}{13}}{b} & \text {for}\: a = 0 \\- \frac {2 a^{3} d^{3} \sqrt {x}}{b^{4}} - \frac {a^{3} d^{3} \sqrt [4]{- \frac {a}{b}} \log {\left (\sqrt {x} - \sqrt [4]{- \frac {a}{b}} \right )}}{2 b^{4}} + \frac {a^{3} d^{3} \sqrt [4]{- \frac {a}{b}} \log {\left (\sqrt {x} + \sqrt [4]{- \frac {a}{b}} \right )}}{2 b^{4}} + \frac {a^{3} d^{3} \sqrt [4]{- \frac {a}{b}} \operatorname {atan}{\left (\frac {\sqrt {x}}{\sqrt [4]{- \frac {a}{b}}} \right )}}{b^{4}} + \frac {6 a^{2} c d^{2} \sqrt {x}}{b^{3}} + \frac {3 a^{2} c d^{2} \sqrt [4]{- \frac {a}{b}} \log {\left (\sqrt {x} - \sqrt [4]{- \frac {a}{b}} \right )}}{2 b^{3}} - \frac {3 a^{2} c d^{2} \sqrt [4]{- \frac {a}{b}} \log {\left (\sqrt {x} + \sqrt [4]{- \frac {a}{b}} \right )}}{2 b^{3}} - \frac {3 a^{2} c d^{2} \sqrt [4]{- \frac {a}{b}} \operatorname {atan}{\left (\frac {\sqrt {x}}{\sqrt [4]{- \frac {a}{b}}} \right )}}{b^{3}} + \frac {2 a^{2} d^{3} x^{\frac {5}{2}}}{5 b^{3}} - \frac {6 a c^{2} d \sqrt {x}}{b^{2}} - \frac {3 a c^{2} d \sqrt [4]{- \frac {a}{b}} \log {\left (\sqrt {x} - \sqrt [4]{- \frac {a}{b}} \right )}}{2 b^{2}} + \frac {3 a c^{2} d \sqrt [4]{- \frac {a}{b}} \log {\left (\sqrt {x} + \sqrt [4]{- \frac {a}{b}} \right )}}{2 b^{2}} + \frac {3 a c^{2} d \sqrt [4]{- \frac {a}{b}} \operatorname {atan}{\left (\frac {\sqrt {x}}{\sqrt [4]{- \frac {a}{b}}} \right )}}{b^{2}} - \frac {6 a c d^{2} x^{\frac {5}{2}}}{5 b^{2}} - \frac {2 a d^{3} x^{\frac {9}{2}}}{9 b^{2}} + \frac {2 c^{3} \sqrt {x}}{b} + \frac {c^{3} \sqrt [4]{- \frac {a}{b}} \log {\left (\sqrt {x} - \sqrt [4]{- \frac {a}{b}} \right )}}{2 b} - \frac {c^{3} \sqrt [4]{- \frac {a}{b}} \log {\left (\sqrt {x} + \sqrt [4]{- \frac {a}{b}} \right )}}{2 b} - \frac {c^{3} \sqrt [4]{- \frac {a}{b}} \operatorname {atan}{\left (\frac {\sqrt {x}}{\sqrt [4]{- \frac {a}{b}}} \right )}}{b} + \frac {6 c^{2} d x^{\frac {5}{2}}}{5 b} + \frac {2 c d^{2} x^{\frac {9}{2}}}{3 b} + \frac {2 d^{3} x^{\frac {13}{2}}}{13 b} & \text {otherwise} \end {cases} \]

[In]

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

[Out]

Piecewise((zoo*(2*c**3*sqrt(x) + 6*c**2*d*x**(5/2)/5 + 2*c*d**2*x**(9/2)/3 + 2*d**3*x**(13/2)/13), Eq(a, 0) &
Eq(b, 0)), ((2*c**3*x**(5/2)/5 + 2*c**2*d*x**(9/2)/3 + 6*c*d**2*x**(13/2)/13 + 2*d**3*x**(17/2)/17)/a, Eq(b, 0
)), ((2*c**3*sqrt(x) + 6*c**2*d*x**(5/2)/5 + 2*c*d**2*x**(9/2)/3 + 2*d**3*x**(13/2)/13)/b, Eq(a, 0)), (-2*a**3
*d**3*sqrt(x)/b**4 - a**3*d**3*(-a/b)**(1/4)*log(sqrt(x) - (-a/b)**(1/4))/(2*b**4) + a**3*d**3*(-a/b)**(1/4)*l
og(sqrt(x) + (-a/b)**(1/4))/(2*b**4) + a**3*d**3*(-a/b)**(1/4)*atan(sqrt(x)/(-a/b)**(1/4))/b**4 + 6*a**2*c*d**
2*sqrt(x)/b**3 + 3*a**2*c*d**2*(-a/b)**(1/4)*log(sqrt(x) - (-a/b)**(1/4))/(2*b**3) - 3*a**2*c*d**2*(-a/b)**(1/
4)*log(sqrt(x) + (-a/b)**(1/4))/(2*b**3) - 3*a**2*c*d**2*(-a/b)**(1/4)*atan(sqrt(x)/(-a/b)**(1/4))/b**3 + 2*a*
*2*d**3*x**(5/2)/(5*b**3) - 6*a*c**2*d*sqrt(x)/b**2 - 3*a*c**2*d*(-a/b)**(1/4)*log(sqrt(x) - (-a/b)**(1/4))/(2
*b**2) + 3*a*c**2*d*(-a/b)**(1/4)*log(sqrt(x) + (-a/b)**(1/4))/(2*b**2) + 3*a*c**2*d*(-a/b)**(1/4)*atan(sqrt(x
)/(-a/b)**(1/4))/b**2 - 6*a*c*d**2*x**(5/2)/(5*b**2) - 2*a*d**3*x**(9/2)/(9*b**2) + 2*c**3*sqrt(x)/b + c**3*(-
a/b)**(1/4)*log(sqrt(x) - (-a/b)**(1/4))/(2*b) - c**3*(-a/b)**(1/4)*log(sqrt(x) + (-a/b)**(1/4))/(2*b) - c**3*
(-a/b)**(1/4)*atan(sqrt(x)/(-a/b)**(1/4))/b + 6*c**2*d*x**(5/2)/(5*b) + 2*c*d**2*x**(9/2)/(3*b) + 2*d**3*x**(1
3/2)/(13*b), True))

Maxima [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 437, normalized size of antiderivative = 1.34 \[ \int \frac {x^{3/2} \left (c+d x^2\right )^3}{a+b x^2} \, dx=-\frac {{\left (\frac {2 \, \sqrt {2} {\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} + 2 \, \sqrt {b} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {b}}} + \frac {2 \, \sqrt {2} {\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} - 2 \, \sqrt {b} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {b}}} + \frac {\sqrt {2} {\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \log \left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {x} + \sqrt {b} x + \sqrt {a}\right )}{a^{\frac {3}{4}} b^{\frac {1}{4}}} - \frac {\sqrt {2} {\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \log \left (-\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {x} + \sqrt {b} x + \sqrt {a}\right )}{a^{\frac {3}{4}} b^{\frac {1}{4}}}\right )} a}{4 \, b^{4}} + \frac {2 \, {\left (45 \, b^{3} d^{3} x^{\frac {13}{2}} + 65 \, {\left (3 \, b^{3} c d^{2} - a b^{2} d^{3}\right )} x^{\frac {9}{2}} + 117 \, {\left (3 \, b^{3} c^{2} d - 3 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} x^{\frac {5}{2}} + 585 \, {\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \sqrt {x}\right )}}{585 \, b^{4}} \]

[In]

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

[Out]

-1/4*(2*sqrt(2)*(b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*arctan(1/2*sqrt(2)*(sqrt(2)*a^(1/4)*b^(1/4
) + 2*sqrt(b)*sqrt(x))/sqrt(sqrt(a)*sqrt(b)))/(sqrt(a)*sqrt(sqrt(a)*sqrt(b))) + 2*sqrt(2)*(b^3*c^3 - 3*a*b^2*c
^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*arctan(-1/2*sqrt(2)*(sqrt(2)*a^(1/4)*b^(1/4) - 2*sqrt(b)*sqrt(x))/sqrt(sqrt(a)
*sqrt(b)))/(sqrt(a)*sqrt(sqrt(a)*sqrt(b))) + sqrt(2)*(b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*log(s
qrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x + sqrt(a))/(a^(3/4)*b^(1/4)) - sqrt(2)*(b^3*c^3 - 3*a*b^2*c^2*d + 3
*a^2*b*c*d^2 - a^3*d^3)*log(-sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x + sqrt(a))/(a^(3/4)*b^(1/4)))*a/b^4 +
2/585*(45*b^3*d^3*x^(13/2) + 65*(3*b^3*c*d^2 - a*b^2*d^3)*x^(9/2) + 117*(3*b^3*c^2*d - 3*a*b^2*c*d^2 + a^2*b*
d^3)*x^(5/2) + 585*(b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*sqrt(x))/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 = 0.29 (sec) , antiderivative size = 531, normalized size of antiderivative = 1.63 \[ \int \frac {x^{3/2} \left (c+d x^2\right )^3}{a+b x^2} \, dx=-\frac {\sqrt {2} {\left (\left (a b^{3}\right )^{\frac {1}{4}} b^{3} c^{3} - 3 \, \left (a b^{3}\right )^{\frac {1}{4}} a b^{2} c^{2} d + 3 \, \left (a b^{3}\right )^{\frac {1}{4}} a^{2} b c d^{2} - \left (a b^{3}\right )^{\frac {1}{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^{5}} - \frac {\sqrt {2} {\left (\left (a b^{3}\right )^{\frac {1}{4}} b^{3} c^{3} - 3 \, \left (a b^{3}\right )^{\frac {1}{4}} a b^{2} c^{2} d + 3 \, \left (a b^{3}\right )^{\frac {1}{4}} a^{2} b c d^{2} - \left (a b^{3}\right )^{\frac {1}{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^{5}} - \frac {\sqrt {2} {\left (\left (a b^{3}\right )^{\frac {1}{4}} b^{3} c^{3} - 3 \, \left (a b^{3}\right )^{\frac {1}{4}} a b^{2} c^{2} d + 3 \, \left (a b^{3}\right )^{\frac {1}{4}} a^{2} b c d^{2} - \left (a b^{3}\right )^{\frac {1}{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^{5}} + \frac {\sqrt {2} {\left (\left (a b^{3}\right )^{\frac {1}{4}} b^{3} c^{3} - 3 \, \left (a b^{3}\right )^{\frac {1}{4}} a b^{2} c^{2} d + 3 \, \left (a b^{3}\right )^{\frac {1}{4}} a^{2} b c d^{2} - \left (a b^{3}\right )^{\frac {1}{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^{5}} + \frac {2 \, {\left (45 \, b^{12} d^{3} x^{\frac {13}{2}} + 195 \, b^{12} c d^{2} x^{\frac {9}{2}} - 65 \, a b^{11} d^{3} x^{\frac {9}{2}} + 351 \, b^{12} c^{2} d x^{\frac {5}{2}} - 351 \, a b^{11} c d^{2} x^{\frac {5}{2}} + 117 \, a^{2} b^{10} d^{3} x^{\frac {5}{2}} + 585 \, b^{12} c^{3} \sqrt {x} - 1755 \, a b^{11} c^{2} d \sqrt {x} + 1755 \, a^{2} b^{10} c d^{2} \sqrt {x} - 585 \, a^{3} b^{9} d^{3} \sqrt {x}\right )}}{585 \, b^{13}} \]

[In]

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

[Out]

-1/2*sqrt(2)*((a*b^3)^(1/4)*b^3*c^3 - 3*(a*b^3)^(1/4)*a*b^2*c^2*d + 3*(a*b^3)^(1/4)*a^2*b*c*d^2 - (a*b^3)^(1/4
)*a^3*d^3)*arctan(1/2*sqrt(2)*(sqrt(2)*(a/b)^(1/4) + 2*sqrt(x))/(a/b)^(1/4))/b^5 - 1/2*sqrt(2)*((a*b^3)^(1/4)*
b^3*c^3 - 3*(a*b^3)^(1/4)*a*b^2*c^2*d + 3*(a*b^3)^(1/4)*a^2*b*c*d^2 - (a*b^3)^(1/4)*a^3*d^3)*arctan(-1/2*sqrt(
2)*(sqrt(2)*(a/b)^(1/4) - 2*sqrt(x))/(a/b)^(1/4))/b^5 - 1/4*sqrt(2)*((a*b^3)^(1/4)*b^3*c^3 - 3*(a*b^3)^(1/4)*a
*b^2*c^2*d + 3*(a*b^3)^(1/4)*a^2*b*c*d^2 - (a*b^3)^(1/4)*a^3*d^3)*log(sqrt(2)*sqrt(x)*(a/b)^(1/4) + x + sqrt(a
/b))/b^5 + 1/4*sqrt(2)*((a*b^3)^(1/4)*b^3*c^3 - 3*(a*b^3)^(1/4)*a*b^2*c^2*d + 3*(a*b^3)^(1/4)*a^2*b*c*d^2 - (a
*b^3)^(1/4)*a^3*d^3)*log(-sqrt(2)*sqrt(x)*(a/b)^(1/4) + x + sqrt(a/b))/b^5 + 2/585*(45*b^12*d^3*x^(13/2) + 195
*b^12*c*d^2*x^(9/2) - 65*a*b^11*d^3*x^(9/2) + 351*b^12*c^2*d*x^(5/2) - 351*a*b^11*c*d^2*x^(5/2) + 117*a^2*b^10
*d^3*x^(5/2) + 585*b^12*c^3*sqrt(x) - 1755*a*b^11*c^2*d*sqrt(x) + 1755*a^2*b^10*c*d^2*sqrt(x) - 585*a^3*b^9*d^
3*sqrt(x))/b^13

Mupad [B] (verification not implemented)

Time = 0.23 (sec) , antiderivative size = 1564, normalized size of antiderivative = 4.80 \[ \int \frac {x^{3/2} \left (c+d x^2\right )^3}{a+b x^2} \, dx=\text {Too large to display} \]

[In]

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

[Out]

x^(1/2)*((2*c^3)/b - (a*((6*c^2*d)/b + (a*((2*a*d^3)/b^2 - (6*c*d^2)/b))/b))/b) - x^(9/2)*((2*a*d^3)/(9*b^2) -
(2*c*d^2)/(3*b)) + x^(5/2)*((6*c^2*d)/(5*b) + (a*((2*a*d^3)/b^2 - (6*c*d^2)/b))/(5*b)) + (2*d^3*x^(13/2))/(13
*b) + ((-a)^(1/4)*atan((((-a)^(1/4)*((16*x^(1/2)*(a^8*d^6 + a^2*b^6*c^6 - 6*a^3*b^5*c^5*d + 15*a^4*b^4*c^4*d^2
- 20*a^5*b^3*c^3*d^3 + 15*a^6*b^2*c^2*d^4 - 6*a^7*b*c*d^5))/b^5 - (16*(-a)^(1/4)*(a*d - b*c)^3*(a^5*d^3 - a^2
*b^3*c^3 + 3*a^3*b^2*c^2*d - 3*a^4*b*c*d^2))/b^(21/4))*(a*d - b*c)^3*1i)/(2*b^(17/4)) + ((-a)^(1/4)*((16*x^(1/
2)*(a^8*d^6 + a^2*b^6*c^6 - 6*a^3*b^5*c^5*d + 15*a^4*b^4*c^4*d^2 - 20*a^5*b^3*c^3*d^3 + 15*a^6*b^2*c^2*d^4 - 6
*a^7*b*c*d^5))/b^5 + (16*(-a)^(1/4)*(a*d - b*c)^3*(a^5*d^3 - a^2*b^3*c^3 + 3*a^3*b^2*c^2*d - 3*a^4*b*c*d^2))/b
^(21/4))*(a*d - b*c)^3*1i)/(2*b^(17/4)))/(((-a)^(1/4)*((16*x^(1/2)*(a^8*d^6 + a^2*b^6*c^6 - 6*a^3*b^5*c^5*d +
15*a^4*b^4*c^4*d^2 - 20*a^5*b^3*c^3*d^3 + 15*a^6*b^2*c^2*d^4 - 6*a^7*b*c*d^5))/b^5 - (16*(-a)^(1/4)*(a*d - b*c
)^3*(a^5*d^3 - a^2*b^3*c^3 + 3*a^3*b^2*c^2*d - 3*a^4*b*c*d^2))/b^(21/4))*(a*d - b*c)^3)/(2*b^(17/4)) - ((-a)^(
1/4)*((16*x^(1/2)*(a^8*d^6 + a^2*b^6*c^6 - 6*a^3*b^5*c^5*d + 15*a^4*b^4*c^4*d^2 - 20*a^5*b^3*c^3*d^3 + 15*a^6*
b^2*c^2*d^4 - 6*a^7*b*c*d^5))/b^5 + (16*(-a)^(1/4)*(a*d - b*c)^3*(a^5*d^3 - a^2*b^3*c^3 + 3*a^3*b^2*c^2*d - 3*
a^4*b*c*d^2))/b^(21/4))*(a*d - b*c)^3)/(2*b^(17/4))))*(a*d - b*c)^3*1i)/b^(17/4) + ((-a)^(1/4)*atan((((-a)^(1/
4)*((16*x^(1/2)*(a^8*d^6 + a^2*b^6*c^6 - 6*a^3*b^5*c^5*d + 15*a^4*b^4*c^4*d^2 - 20*a^5*b^3*c^3*d^3 + 15*a^6*b^
2*c^2*d^4 - 6*a^7*b*c*d^5))/b^5 - ((-a)^(1/4)*(a*d - b*c)^3*(a^5*d^3 - a^2*b^3*c^3 + 3*a^3*b^2*c^2*d - 3*a^4*b
*c*d^2)*16i)/b^(21/4))*(a*d - b*c)^3)/(2*b^(17/4)) + ((-a)^(1/4)*((16*x^(1/2)*(a^8*d^6 + a^2*b^6*c^6 - 6*a^3*b
^5*c^5*d + 15*a^4*b^4*c^4*d^2 - 20*a^5*b^3*c^3*d^3 + 15*a^6*b^2*c^2*d^4 - 6*a^7*b*c*d^5))/b^5 + ((-a)^(1/4)*(a
*d - b*c)^3*(a^5*d^3 - a^2*b^3*c^3 + 3*a^3*b^2*c^2*d - 3*a^4*b*c*d^2)*16i)/b^(21/4))*(a*d - b*c)^3)/(2*b^(17/4
)))/(((-a)^(1/4)*((16*x^(1/2)*(a^8*d^6 + a^2*b^6*c^6 - 6*a^3*b^5*c^5*d + 15*a^4*b^4*c^4*d^2 - 20*a^5*b^3*c^3*d
^3 + 15*a^6*b^2*c^2*d^4 - 6*a^7*b*c*d^5))/b^5 - ((-a)^(1/4)*(a*d - b*c)^3*(a^5*d^3 - a^2*b^3*c^3 + 3*a^3*b^2*c
^2*d - 3*a^4*b*c*d^2)*16i)/b^(21/4))*(a*d - b*c)^3*1i)/(2*b^(17/4)) - ((-a)^(1/4)*((16*x^(1/2)*(a^8*d^6 + a^2*
b^6*c^6 - 6*a^3*b^5*c^5*d + 15*a^4*b^4*c^4*d^2 - 20*a^5*b^3*c^3*d^3 + 15*a^6*b^2*c^2*d^4 - 6*a^7*b*c*d^5))/b^5
+ ((-a)^(1/4)*(a*d - b*c)^3*(a^5*d^3 - a^2*b^3*c^3 + 3*a^3*b^2*c^2*d - 3*a^4*b*c*d^2)*16i)/b^(21/4))*(a*d - b
*c)^3*1i)/(2*b^(17/4))))*(a*d - b*c)^3)/b^(17/4)