3.173 \(\int \frac{x^{3/2} (A+B x^3)}{(a+b x^3)^3} \, dx\)

Optimal. Leaf size=327 \[ \frac{(5 a B+7 A b) \log \left (-\sqrt{3} \sqrt [6]{a} \sqrt [6]{b} \sqrt{x}+\sqrt [3]{a}+\sqrt [3]{b} x\right )}{144 \sqrt{3} a^{13/6} b^{11/6}}-\frac{(5 a B+7 A b) \log \left (\sqrt{3} \sqrt [6]{a} \sqrt [6]{b} \sqrt{x}+\sqrt [3]{a}+\sqrt [3]{b} x\right )}{144 \sqrt{3} a^{13/6} b^{11/6}}-\frac{(5 a B+7 A b) \tan ^{-1}\left (\sqrt{3}-\frac{2 \sqrt [6]{b} \sqrt{x}}{\sqrt [6]{a}}\right )}{216 a^{13/6} b^{11/6}}+\frac{(5 a B+7 A b) \tan ^{-1}\left (\frac{2 \sqrt [6]{b} \sqrt{x}}{\sqrt [6]{a}}+\sqrt{3}\right )}{216 a^{13/6} b^{11/6}}+\frac{(5 a B+7 A b) \tan ^{-1}\left (\frac{\sqrt [6]{b} \sqrt{x}}{\sqrt [6]{a}}\right )}{108 a^{13/6} b^{11/6}}+\frac{x^{5/2} (5 a B+7 A b)}{36 a^2 b \left (a+b x^3\right )}+\frac{x^{5/2} (A b-a B)}{6 a b \left (a+b x^3\right )^2} \]

[Out]

((A*b - a*B)*x^(5/2))/(6*a*b*(a + b*x^3)^2) + ((7*A*b + 5*a*B)*x^(5/2))/(36*a^2*b*(a + b*x^3)) - ((7*A*b + 5*a
*B)*ArcTan[Sqrt[3] - (2*b^(1/6)*Sqrt[x])/a^(1/6)])/(216*a^(13/6)*b^(11/6)) + ((7*A*b + 5*a*B)*ArcTan[Sqrt[3] +
(2*b^(1/6)*Sqrt[x])/a^(1/6)])/(216*a^(13/6)*b^(11/6)) + ((7*A*b + 5*a*B)*ArcTan[(b^(1/6)*Sqrt[x])/a^(1/6)])/(
108*a^(13/6)*b^(11/6)) + ((7*A*b + 5*a*B)*Log[a^(1/3) - Sqrt[3]*a^(1/6)*b^(1/6)*Sqrt[x] + b^(1/3)*x])/(144*Sqr
t[3]*a^(13/6)*b^(11/6)) - ((7*A*b + 5*a*B)*Log[a^(1/3) + Sqrt[3]*a^(1/6)*b^(1/6)*Sqrt[x] + b^(1/3)*x])/(144*Sq
rt[3]*a^(13/6)*b^(11/6))

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Rubi [A]  time = 0.597, antiderivative size = 327, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 9, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.409, Rules used = {457, 290, 329, 295, 634, 618, 204, 628, 205} \[ \frac{(5 a B+7 A b) \log \left (-\sqrt{3} \sqrt [6]{a} \sqrt [6]{b} \sqrt{x}+\sqrt [3]{a}+\sqrt [3]{b} x\right )}{144 \sqrt{3} a^{13/6} b^{11/6}}-\frac{(5 a B+7 A b) \log \left (\sqrt{3} \sqrt [6]{a} \sqrt [6]{b} \sqrt{x}+\sqrt [3]{a}+\sqrt [3]{b} x\right )}{144 \sqrt{3} a^{13/6} b^{11/6}}-\frac{(5 a B+7 A b) \tan ^{-1}\left (\sqrt{3}-\frac{2 \sqrt [6]{b} \sqrt{x}}{\sqrt [6]{a}}\right )}{216 a^{13/6} b^{11/6}}+\frac{(5 a B+7 A b) \tan ^{-1}\left (\frac{2 \sqrt [6]{b} \sqrt{x}}{\sqrt [6]{a}}+\sqrt{3}\right )}{216 a^{13/6} b^{11/6}}+\frac{(5 a B+7 A b) \tan ^{-1}\left (\frac{\sqrt [6]{b} \sqrt{x}}{\sqrt [6]{a}}\right )}{108 a^{13/6} b^{11/6}}+\frac{x^{5/2} (5 a B+7 A b)}{36 a^2 b \left (a+b x^3\right )}+\frac{x^{5/2} (A b-a B)}{6 a b \left (a+b x^3\right )^2} \]

Antiderivative was successfully verified.

[In]

Int[(x^(3/2)*(A + B*x^3))/(a + b*x^3)^3,x]

[Out]

((A*b - a*B)*x^(5/2))/(6*a*b*(a + b*x^3)^2) + ((7*A*b + 5*a*B)*x^(5/2))/(36*a^2*b*(a + b*x^3)) - ((7*A*b + 5*a
*B)*ArcTan[Sqrt[3] - (2*b^(1/6)*Sqrt[x])/a^(1/6)])/(216*a^(13/6)*b^(11/6)) + ((7*A*b + 5*a*B)*ArcTan[Sqrt[3] +
(2*b^(1/6)*Sqrt[x])/a^(1/6)])/(216*a^(13/6)*b^(11/6)) + ((7*A*b + 5*a*B)*ArcTan[(b^(1/6)*Sqrt[x])/a^(1/6)])/(
108*a^(13/6)*b^(11/6)) + ((7*A*b + 5*a*B)*Log[a^(1/3) - Sqrt[3]*a^(1/6)*b^(1/6)*Sqrt[x] + b^(1/3)*x])/(144*Sqr
t[3]*a^(13/6)*b^(11/6)) - ((7*A*b + 5*a*B)*Log[a^(1/3) + Sqrt[3]*a^(1/6)*b^(1/6)*Sqrt[x] + b^(1/3)*x])/(144*Sq
rt[3]*a^(13/6)*b^(11/6))

Rule 457

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

Rule 290

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

Rule 329

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 295

Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Module[{r = Numerator[Rt[a/b, n]], s = Denominator[Rt[a/
b, n]], k, u}, Simp[u = Int[(r*Cos[((2*k - 1)*m*Pi)/n] - s*Cos[((2*k - 1)*(m + 1)*Pi)/n]*x)/(r^2 - 2*r*s*Cos[(
(2*k - 1)*Pi)/n]*x + s^2*x^2), x] + Int[(r*Cos[((2*k - 1)*m*Pi)/n] + s*Cos[((2*k - 1)*(m + 1)*Pi)/n]*x)/(r^2 +
2*r*s*Cos[((2*k - 1)*Pi)/n]*x + s^2*x^2), x]; (2*(-1)^(m/2)*r^(m + 2)*Int[1/(r^2 + s^2*x^2), x])/(a*n*s^m) +
Dist[(2*r^(m + 1))/(a*n*s^m), Sum[u, {k, 1, (n - 2)/4}], x], x]] /; FreeQ[{a, b}, x] && IGtQ[(n - 2)/4, 0] &&
IGtQ[m, 0] && LtQ[m, n - 1] && PosQ[a/b]

Rule 634

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[(2*c*d - b*e)/(2*c), Int[1/(a +
b*x + c*x^2), x], x] + Dist[e/(2*c), Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] &
& NeQ[2*c*d - b*e, 0] && NeQ[b^2 - 4*a*c, 0] &&  !NiceSqrtQ[b^2 - 4*a*c]

Rule 618

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 204

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

Rule 628

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 205

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

Rubi steps

\begin{align*} \int \frac{x^{3/2} \left (A+B x^3\right )}{\left (a+b x^3\right )^3} \, dx &=\frac{(A b-a B) x^{5/2}}{6 a b \left (a+b x^3\right )^2}+\frac{\left (\frac{7 A b}{2}+\frac{5 a B}{2}\right ) \int \frac{x^{3/2}}{\left (a+b x^3\right )^2} \, dx}{6 a b}\\ &=\frac{(A b-a B) x^{5/2}}{6 a b \left (a+b x^3\right )^2}+\frac{(7 A b+5 a B) x^{5/2}}{36 a^2 b \left (a+b x^3\right )}+\frac{(7 A b+5 a B) \int \frac{x^{3/2}}{a+b x^3} \, dx}{72 a^2 b}\\ &=\frac{(A b-a B) x^{5/2}}{6 a b \left (a+b x^3\right )^2}+\frac{(7 A b+5 a B) x^{5/2}}{36 a^2 b \left (a+b x^3\right )}+\frac{(7 A b+5 a B) \operatorname{Subst}\left (\int \frac{x^4}{a+b x^6} \, dx,x,\sqrt{x}\right )}{36 a^2 b}\\ &=\frac{(A b-a B) x^{5/2}}{6 a b \left (a+b x^3\right )^2}+\frac{(7 A b+5 a B) x^{5/2}}{36 a^2 b \left (a+b x^3\right )}+\frac{(7 A b+5 a B) \operatorname{Subst}\left (\int \frac{-\frac{\sqrt [6]{a}}{2}+\frac{1}{2} \sqrt{3} \sqrt [6]{b} x}{\sqrt [3]{a}-\sqrt{3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{b} x^2} \, dx,x,\sqrt{x}\right )}{108 a^{13/6} b^{5/3}}+\frac{(7 A b+5 a B) \operatorname{Subst}\left (\int \frac{-\frac{\sqrt [6]{a}}{2}-\frac{1}{2} \sqrt{3} \sqrt [6]{b} x}{\sqrt [3]{a}+\sqrt{3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{b} x^2} \, dx,x,\sqrt{x}\right )}{108 a^{13/6} b^{5/3}}+\frac{(7 A b+5 a B) \operatorname{Subst}\left (\int \frac{1}{\sqrt [3]{a}+\sqrt [3]{b} x^2} \, dx,x,\sqrt{x}\right )}{108 a^2 b^{5/3}}\\ &=\frac{(A b-a B) x^{5/2}}{6 a b \left (a+b x^3\right )^2}+\frac{(7 A b+5 a B) x^{5/2}}{36 a^2 b \left (a+b x^3\right )}+\frac{(7 A b+5 a B) \tan ^{-1}\left (\frac{\sqrt [6]{b} \sqrt{x}}{\sqrt [6]{a}}\right )}{108 a^{13/6} b^{11/6}}+\frac{(7 A b+5 a B) \operatorname{Subst}\left (\int \frac{-\sqrt{3} \sqrt [6]{a} \sqrt [6]{b}+2 \sqrt [3]{b} x}{\sqrt [3]{a}-\sqrt{3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{b} x^2} \, dx,x,\sqrt{x}\right )}{144 \sqrt{3} a^{13/6} b^{11/6}}-\frac{(7 A b+5 a B) \operatorname{Subst}\left (\int \frac{\sqrt{3} \sqrt [6]{a} \sqrt [6]{b}+2 \sqrt [3]{b} x}{\sqrt [3]{a}+\sqrt{3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{b} x^2} \, dx,x,\sqrt{x}\right )}{144 \sqrt{3} a^{13/6} b^{11/6}}+\frac{(7 A b+5 a B) \operatorname{Subst}\left (\int \frac{1}{\sqrt [3]{a}-\sqrt{3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{b} x^2} \, dx,x,\sqrt{x}\right )}{432 a^2 b^{5/3}}+\frac{(7 A b+5 a B) \operatorname{Subst}\left (\int \frac{1}{\sqrt [3]{a}+\sqrt{3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{b} x^2} \, dx,x,\sqrt{x}\right )}{432 a^2 b^{5/3}}\\ &=\frac{(A b-a B) x^{5/2}}{6 a b \left (a+b x^3\right )^2}+\frac{(7 A b+5 a B) x^{5/2}}{36 a^2 b \left (a+b x^3\right )}+\frac{(7 A b+5 a B) \tan ^{-1}\left (\frac{\sqrt [6]{b} \sqrt{x}}{\sqrt [6]{a}}\right )}{108 a^{13/6} b^{11/6}}+\frac{(7 A b+5 a B) \log \left (\sqrt [3]{a}-\sqrt{3} \sqrt [6]{a} \sqrt [6]{b} \sqrt{x}+\sqrt [3]{b} x\right )}{144 \sqrt{3} a^{13/6} b^{11/6}}-\frac{(7 A b+5 a B) \log \left (\sqrt [3]{a}+\sqrt{3} \sqrt [6]{a} \sqrt [6]{b} \sqrt{x}+\sqrt [3]{b} x\right )}{144 \sqrt{3} a^{13/6} b^{11/6}}+\frac{(7 A b+5 a B) \operatorname{Subst}\left (\int \frac{1}{-\frac{1}{3}-x^2} \, dx,x,1-\frac{2 \sqrt [6]{b} \sqrt{x}}{\sqrt{3} \sqrt [6]{a}}\right )}{216 \sqrt{3} a^{13/6} b^{11/6}}-\frac{(7 A b+5 a B) \operatorname{Subst}\left (\int \frac{1}{-\frac{1}{3}-x^2} \, dx,x,1+\frac{2 \sqrt [6]{b} \sqrt{x}}{\sqrt{3} \sqrt [6]{a}}\right )}{216 \sqrt{3} a^{13/6} b^{11/6}}\\ &=\frac{(A b-a B) x^{5/2}}{6 a b \left (a+b x^3\right )^2}+\frac{(7 A b+5 a B) x^{5/2}}{36 a^2 b \left (a+b x^3\right )}-\frac{(7 A b+5 a B) \tan ^{-1}\left (\sqrt{3}-\frac{2 \sqrt [6]{b} \sqrt{x}}{\sqrt [6]{a}}\right )}{216 a^{13/6} b^{11/6}}+\frac{(7 A b+5 a B) \tan ^{-1}\left (\sqrt{3}+\frac{2 \sqrt [6]{b} \sqrt{x}}{\sqrt [6]{a}}\right )}{216 a^{13/6} b^{11/6}}+\frac{(7 A b+5 a B) \tan ^{-1}\left (\frac{\sqrt [6]{b} \sqrt{x}}{\sqrt [6]{a}}\right )}{108 a^{13/6} b^{11/6}}+\frac{(7 A b+5 a B) \log \left (\sqrt [3]{a}-\sqrt{3} \sqrt [6]{a} \sqrt [6]{b} \sqrt{x}+\sqrt [3]{b} x\right )}{144 \sqrt{3} a^{13/6} b^{11/6}}-\frac{(7 A b+5 a B) \log \left (\sqrt [3]{a}+\sqrt{3} \sqrt [6]{a} \sqrt [6]{b} \sqrt{x}+\sqrt [3]{b} x\right )}{144 \sqrt{3} a^{13/6} b^{11/6}}\\ \end{align*}

Mathematica [C]  time = 0.0538124, size = 62, normalized size = 0.19 \[ \frac{2 x^{5/2} \left ((A b-a B) \, _2F_1\left (\frac{5}{6},3;\frac{11}{6};-\frac{b x^3}{a}\right )+a B \, _2F_1\left (\frac{5}{6},2;\frac{11}{6};-\frac{b x^3}{a}\right )\right )}{5 a^3 b} \]

Antiderivative was successfully verified.

[In]

Integrate[(x^(3/2)*(A + B*x^3))/(a + b*x^3)^3,x]

[Out]

(2*x^(5/2)*(a*B*Hypergeometric2F1[5/6, 2, 11/6, -((b*x^3)/a)] + (A*b - a*B)*Hypergeometric2F1[5/6, 3, 11/6, -(
(b*x^3)/a)]))/(5*a^3*b)

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Maple [A]  time = 0.042, size = 411, normalized size = 1.3 \begin{align*} 2\,{\frac{1}{ \left ( b{x}^{3}+a \right ) ^{2}} \left ({\frac{ \left ( 7\,Ab+5\,Ba \right ){x}^{11/2}}{72\,{a}^{2}}}+{\frac{ \left ( 13\,Ab-Ba \right ){x}^{5/2}}{72\,ab}} \right ) }+{\frac{7\,A}{108\,{a}^{2}b}\arctan \left ({\sqrt{x}{\frac{1}{\sqrt [6]{{\frac{a}{b}}}}}} \right ){\frac{1}{\sqrt [6]{{\frac{a}{b}}}}}}+{\frac{5\,B}{108\,a{b}^{2}}\arctan \left ({\sqrt{x}{\frac{1}{\sqrt [6]{{\frac{a}{b}}}}}} \right ){\frac{1}{\sqrt [6]{{\frac{a}{b}}}}}}+{\frac{7\,\sqrt{3}A}{432\,{a}^{3}} \left ({\frac{a}{b}} \right ) ^{{\frac{5}{6}}}\ln \left ( x-\sqrt{3}\sqrt [6]{{\frac{a}{b}}}\sqrt{x}+\sqrt [3]{{\frac{a}{b}}} \right ) }+{\frac{5\,\sqrt{3}B}{432\,{a}^{2}b} \left ({\frac{a}{b}} \right ) ^{{\frac{5}{6}}}\ln \left ( x-\sqrt{3}\sqrt [6]{{\frac{a}{b}}}\sqrt{x}+\sqrt [3]{{\frac{a}{b}}} \right ) }+{\frac{7\,A}{216\,{a}^{2}b}\arctan \left ( 2\,{\sqrt{x}{\frac{1}{\sqrt [6]{{\frac{a}{b}}}}}}-\sqrt{3} \right ){\frac{1}{\sqrt [6]{{\frac{a}{b}}}}}}+{\frac{5\,B}{216\,a{b}^{2}}\arctan \left ( 2\,{\sqrt{x}{\frac{1}{\sqrt [6]{{\frac{a}{b}}}}}}-\sqrt{3} \right ){\frac{1}{\sqrt [6]{{\frac{a}{b}}}}}}-{\frac{7\,\sqrt{3}A}{432\,{a}^{3}} \left ({\frac{a}{b}} \right ) ^{{\frac{5}{6}}}\ln \left ( x+\sqrt{3}\sqrt [6]{{\frac{a}{b}}}\sqrt{x}+\sqrt [3]{{\frac{a}{b}}} \right ) }-{\frac{5\,\sqrt{3}B}{432\,{a}^{2}b} \left ({\frac{a}{b}} \right ) ^{{\frac{5}{6}}}\ln \left ( x+\sqrt{3}\sqrt [6]{{\frac{a}{b}}}\sqrt{x}+\sqrt [3]{{\frac{a}{b}}} \right ) }+{\frac{7\,A}{216\,{a}^{2}b}\arctan \left ( 2\,{\sqrt{x}{\frac{1}{\sqrt [6]{{\frac{a}{b}}}}}}+\sqrt{3} \right ){\frac{1}{\sqrt [6]{{\frac{a}{b}}}}}}+{\frac{5\,B}{216\,a{b}^{2}}\arctan \left ( 2\,{\sqrt{x}{\frac{1}{\sqrt [6]{{\frac{a}{b}}}}}}+\sqrt{3} \right ){\frac{1}{\sqrt [6]{{\frac{a}{b}}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(3/2)*(B*x^3+A)/(b*x^3+a)^3,x)

[Out]

2*(1/72*(7*A*b+5*B*a)/a^2*x^(11/2)+1/72*(13*A*b-B*a)/a/b*x^(5/2))/(b*x^3+a)^2+7/108/a^2/b/(a/b)^(1/6)*arctan(x
^(1/2)/(a/b)^(1/6))*A+5/108/a/b^2/(a/b)^(1/6)*arctan(x^(1/2)/(a/b)^(1/6))*B+7/432/a^3*3^(1/2)*(a/b)^(5/6)*ln(x
-3^(1/2)*(a/b)^(1/6)*x^(1/2)+(a/b)^(1/3))*A+5/432/a^2/b*3^(1/2)*(a/b)^(5/6)*ln(x-3^(1/2)*(a/b)^(1/6)*x^(1/2)+(
a/b)^(1/3))*B+7/216/a^2/b/(a/b)^(1/6)*arctan(2*x^(1/2)/(a/b)^(1/6)-3^(1/2))*A+5/216/a/b^2/(a/b)^(1/6)*arctan(2
*x^(1/2)/(a/b)^(1/6)-3^(1/2))*B-7/432/a^3*3^(1/2)*(a/b)^(5/6)*ln(x+3^(1/2)*(a/b)^(1/6)*x^(1/2)+(a/b)^(1/3))*A-
5/432/a^2/b*3^(1/2)*(a/b)^(5/6)*ln(x+3^(1/2)*(a/b)^(1/6)*x^(1/2)+(a/b)^(1/3))*B+7/216/a^2/b/(a/b)^(1/6)*arctan
(2*x^(1/2)/(a/b)^(1/6)+3^(1/2))*A+5/216/a/b^2/(a/b)^(1/6)*arctan(2*x^(1/2)/(a/b)^(1/6)+3^(1/2))*B

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [B]  time = 3.24809, size = 10242, normalized size = 31.32 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/432*(4*sqrt(3)*(a^2*b^3*x^6 + 2*a^3*b^2*x^3 + a^4*b)*(-(15625*B^6*a^6 + 131250*A*B^5*a^5*b + 459375*A^2*B^4
*a^4*b^2 + 857500*A^3*B^3*a^3*b^3 + 900375*A^4*B^2*a^2*b^4 + 504210*A^5*B*a*b^5 + 117649*A^6*b^6)/(a^13*b^11))
^(1/6)*arctan(1/3*(2*sqrt(3)*sqrt((3125*B^5*a^16*b^9 + 21875*A*B^4*a^15*b^10 + 61250*A^2*B^3*a^14*b^11 + 85750
*A^3*B^2*a^13*b^12 + 60025*A^4*B*a^12*b^13 + 16807*A^5*a^11*b^14)*sqrt(x)*(-(15625*B^6*a^6 + 131250*A*B^5*a^5*
b + 459375*A^2*B^4*a^4*b^2 + 857500*A^3*B^3*a^3*b^3 + 900375*A^4*B^2*a^2*b^4 + 504210*A^5*B*a*b^5 + 117649*A^6
*b^6)/(a^13*b^11))^(5/6) + (9765625*B^10*a^10 + 136718750*A*B^9*a^9*b + 861328125*A^2*B^8*a^8*b^2 + 3215625000
*A^3*B^7*a^7*b^3 + 7878281250*A^4*B^6*a^6*b^4 + 13235512500*A^5*B^5*a^5*b^5 + 15441431250*A^6*B^4*a^4*b^6 + 12
353145000*A^7*B^3*a^3*b^7 + 6485401125*A^8*B^2*a^2*b^8 + 2017680350*A^9*B*a*b^9 + 282475249*A^10*b^10)*x - (15
625*B^6*a^15*b^7 + 131250*A*B^5*a^14*b^8 + 459375*A^2*B^4*a^13*b^9 + 857500*A^3*B^3*a^12*b^10 + 900375*A^4*B^2
*a^11*b^11 + 504210*A^5*B*a^10*b^12 + 117649*A^6*a^9*b^13)*(-(15625*B^6*a^6 + 131250*A*B^5*a^5*b + 459375*A^2*
B^4*a^4*b^2 + 857500*A^3*B^3*a^3*b^3 + 900375*A^4*B^2*a^2*b^4 + 504210*A^5*B*a*b^5 + 117649*A^6*b^6)/(a^13*b^1
1))^(2/3))*a^2*b^2*(-(15625*B^6*a^6 + 131250*A*B^5*a^5*b + 459375*A^2*B^4*a^4*b^2 + 857500*A^3*B^3*a^3*b^3 + 9
00375*A^4*B^2*a^2*b^4 + 504210*A^5*B*a*b^5 + 117649*A^6*b^6)/(a^13*b^11))^(1/6) - 2*sqrt(3)*(3125*B^5*a^7*b^2
+ 21875*A*B^4*a^6*b^3 + 61250*A^2*B^3*a^5*b^4 + 85750*A^3*B^2*a^4*b^5 + 60025*A^4*B*a^3*b^6 + 16807*A^5*a^2*b^
7)*sqrt(x)*(-(15625*B^6*a^6 + 131250*A*B^5*a^5*b + 459375*A^2*B^4*a^4*b^2 + 857500*A^3*B^3*a^3*b^3 + 900375*A^
4*B^2*a^2*b^4 + 504210*A^5*B*a*b^5 + 117649*A^6*b^6)/(a^13*b^11))^(1/6) + sqrt(3)*(15625*B^6*a^6 + 131250*A*B^
5*a^5*b + 459375*A^2*B^4*a^4*b^2 + 857500*A^3*B^3*a^3*b^3 + 900375*A^4*B^2*a^2*b^4 + 504210*A^5*B*a*b^5 + 1176
49*A^6*b^6))/(15625*B^6*a^6 + 131250*A*B^5*a^5*b + 459375*A^2*B^4*a^4*b^2 + 857500*A^3*B^3*a^3*b^3 + 900375*A^
4*B^2*a^2*b^4 + 504210*A^5*B*a*b^5 + 117649*A^6*b^6)) + 4*sqrt(3)*(a^2*b^3*x^6 + 2*a^3*b^2*x^3 + a^4*b)*(-(156
25*B^6*a^6 + 131250*A*B^5*a^5*b + 459375*A^2*B^4*a^4*b^2 + 857500*A^3*B^3*a^3*b^3 + 900375*A^4*B^2*a^2*b^4 + 5
04210*A^5*B*a*b^5 + 117649*A^6*b^6)/(a^13*b^11))^(1/6)*arctan(1/3*(2*sqrt(3)*sqrt(-(3125*B^5*a^16*b^9 + 21875*
A*B^4*a^15*b^10 + 61250*A^2*B^3*a^14*b^11 + 85750*A^3*B^2*a^13*b^12 + 60025*A^4*B*a^12*b^13 + 16807*A^5*a^11*b
^14)*sqrt(x)*(-(15625*B^6*a^6 + 131250*A*B^5*a^5*b + 459375*A^2*B^4*a^4*b^2 + 857500*A^3*B^3*a^3*b^3 + 900375*
A^4*B^2*a^2*b^4 + 504210*A^5*B*a*b^5 + 117649*A^6*b^6)/(a^13*b^11))^(5/6) + (9765625*B^10*a^10 + 136718750*A*B
^9*a^9*b + 861328125*A^2*B^8*a^8*b^2 + 3215625000*A^3*B^7*a^7*b^3 + 7878281250*A^4*B^6*a^6*b^4 + 13235512500*A
^5*B^5*a^5*b^5 + 15441431250*A^6*B^4*a^4*b^6 + 12353145000*A^7*B^3*a^3*b^7 + 6485401125*A^8*B^2*a^2*b^8 + 2017
680350*A^9*B*a*b^9 + 282475249*A^10*b^10)*x - (15625*B^6*a^15*b^7 + 131250*A*B^5*a^14*b^8 + 459375*A^2*B^4*a^1
3*b^9 + 857500*A^3*B^3*a^12*b^10 + 900375*A^4*B^2*a^11*b^11 + 504210*A^5*B*a^10*b^12 + 117649*A^6*a^9*b^13)*(-
(15625*B^6*a^6 + 131250*A*B^5*a^5*b + 459375*A^2*B^4*a^4*b^2 + 857500*A^3*B^3*a^3*b^3 + 900375*A^4*B^2*a^2*b^4
+ 504210*A^5*B*a*b^5 + 117649*A^6*b^6)/(a^13*b^11))^(2/3))*a^2*b^2*(-(15625*B^6*a^6 + 131250*A*B^5*a^5*b + 45
9375*A^2*B^4*a^4*b^2 + 857500*A^3*B^3*a^3*b^3 + 900375*A^4*B^2*a^2*b^4 + 504210*A^5*B*a*b^5 + 117649*A^6*b^6)/
(a^13*b^11))^(1/6) - 2*sqrt(3)*(3125*B^5*a^7*b^2 + 21875*A*B^4*a^6*b^3 + 61250*A^2*B^3*a^5*b^4 + 85750*A^3*B^2
*a^4*b^5 + 60025*A^4*B*a^3*b^6 + 16807*A^5*a^2*b^7)*sqrt(x)*(-(15625*B^6*a^6 + 131250*A*B^5*a^5*b + 459375*A^2
*B^4*a^4*b^2 + 857500*A^3*B^3*a^3*b^3 + 900375*A^4*B^2*a^2*b^4 + 504210*A^5*B*a*b^5 + 117649*A^6*b^6)/(a^13*b^
11))^(1/6) - sqrt(3)*(15625*B^6*a^6 + 131250*A*B^5*a^5*b + 459375*A^2*B^4*a^4*b^2 + 857500*A^3*B^3*a^3*b^3 + 9
00375*A^4*B^2*a^2*b^4 + 504210*A^5*B*a*b^5 + 117649*A^6*b^6))/(15625*B^6*a^6 + 131250*A*B^5*a^5*b + 459375*A^2
*B^4*a^4*b^2 + 857500*A^3*B^3*a^3*b^3 + 900375*A^4*B^2*a^2*b^4 + 504210*A^5*B*a*b^5 + 117649*A^6*b^6)) - 2*(a^
2*b^3*x^6 + 2*a^3*b^2*x^3 + a^4*b)*(-(15625*B^6*a^6 + 131250*A*B^5*a^5*b + 459375*A^2*B^4*a^4*b^2 + 857500*A^3
*B^3*a^3*b^3 + 900375*A^4*B^2*a^2*b^4 + 504210*A^5*B*a*b^5 + 117649*A^6*b^6)/(a^13*b^11))^(1/6)*log(a^11*b^9*(
-(15625*B^6*a^6 + 131250*A*B^5*a^5*b + 459375*A^2*B^4*a^4*b^2 + 857500*A^3*B^3*a^3*b^3 + 900375*A^4*B^2*a^2*b^
4 + 504210*A^5*B*a*b^5 + 117649*A^6*b^6)/(a^13*b^11))^(5/6) + (3125*B^5*a^5 + 21875*A*B^4*a^4*b + 61250*A^2*B^
3*a^3*b^2 + 85750*A^3*B^2*a^2*b^3 + 60025*A^4*B*a*b^4 + 16807*A^5*b^5)*sqrt(x)) + 2*(a^2*b^3*x^6 + 2*a^3*b^2*x
^3 + a^4*b)*(-(15625*B^6*a^6 + 131250*A*B^5*a^5*b + 459375*A^2*B^4*a^4*b^2 + 857500*A^3*B^3*a^3*b^3 + 900375*A
^4*B^2*a^2*b^4 + 504210*A^5*B*a*b^5 + 117649*A^6*b^6)/(a^13*b^11))^(1/6)*log(-a^11*b^9*(-(15625*B^6*a^6 + 1312
50*A*B^5*a^5*b + 459375*A^2*B^4*a^4*b^2 + 857500*A^3*B^3*a^3*b^3 + 900375*A^4*B^2*a^2*b^4 + 504210*A^5*B*a*b^5
+ 117649*A^6*b^6)/(a^13*b^11))^(5/6) + (3125*B^5*a^5 + 21875*A*B^4*a^4*b + 61250*A^2*B^3*a^3*b^2 + 85750*A^3*
B^2*a^2*b^3 + 60025*A^4*B*a*b^4 + 16807*A^5*b^5)*sqrt(x)) - (a^2*b^3*x^6 + 2*a^3*b^2*x^3 + a^4*b)*(-(15625*B^6
*a^6 + 131250*A*B^5*a^5*b + 459375*A^2*B^4*a^4*b^2 + 857500*A^3*B^3*a^3*b^3 + 900375*A^4*B^2*a^2*b^4 + 504210*
A^5*B*a*b^5 + 117649*A^6*b^6)/(a^13*b^11))^(1/6)*log((3125*B^5*a^16*b^9 + 21875*A*B^4*a^15*b^10 + 61250*A^2*B^
3*a^14*b^11 + 85750*A^3*B^2*a^13*b^12 + 60025*A^4*B*a^12*b^13 + 16807*A^5*a^11*b^14)*sqrt(x)*(-(15625*B^6*a^6
+ 131250*A*B^5*a^5*b + 459375*A^2*B^4*a^4*b^2 + 857500*A^3*B^3*a^3*b^3 + 900375*A^4*B^2*a^2*b^4 + 504210*A^5*B
*a*b^5 + 117649*A^6*b^6)/(a^13*b^11))^(5/6) + (9765625*B^10*a^10 + 136718750*A*B^9*a^9*b + 861328125*A^2*B^8*a
^8*b^2 + 3215625000*A^3*B^7*a^7*b^3 + 7878281250*A^4*B^6*a^6*b^4 + 13235512500*A^5*B^5*a^5*b^5 + 15441431250*A
^6*B^4*a^4*b^6 + 12353145000*A^7*B^3*a^3*b^7 + 6485401125*A^8*B^2*a^2*b^8 + 2017680350*A^9*B*a*b^9 + 282475249
*A^10*b^10)*x - (15625*B^6*a^15*b^7 + 131250*A*B^5*a^14*b^8 + 459375*A^2*B^4*a^13*b^9 + 857500*A^3*B^3*a^12*b^
10 + 900375*A^4*B^2*a^11*b^11 + 504210*A^5*B*a^10*b^12 + 117649*A^6*a^9*b^13)*(-(15625*B^6*a^6 + 131250*A*B^5*
a^5*b + 459375*A^2*B^4*a^4*b^2 + 857500*A^3*B^3*a^3*b^3 + 900375*A^4*B^2*a^2*b^4 + 504210*A^5*B*a*b^5 + 117649
*A^6*b^6)/(a^13*b^11))^(2/3)) + (a^2*b^3*x^6 + 2*a^3*b^2*x^3 + a^4*b)*(-(15625*B^6*a^6 + 131250*A*B^5*a^5*b +
459375*A^2*B^4*a^4*b^2 + 857500*A^3*B^3*a^3*b^3 + 900375*A^4*B^2*a^2*b^4 + 504210*A^5*B*a*b^5 + 117649*A^6*b^6
)/(a^13*b^11))^(1/6)*log(-(3125*B^5*a^16*b^9 + 21875*A*B^4*a^15*b^10 + 61250*A^2*B^3*a^14*b^11 + 85750*A^3*B^2
*a^13*b^12 + 60025*A^4*B*a^12*b^13 + 16807*A^5*a^11*b^14)*sqrt(x)*(-(15625*B^6*a^6 + 131250*A*B^5*a^5*b + 4593
75*A^2*B^4*a^4*b^2 + 857500*A^3*B^3*a^3*b^3 + 900375*A^4*B^2*a^2*b^4 + 504210*A^5*B*a*b^5 + 117649*A^6*b^6)/(a
^13*b^11))^(5/6) + (9765625*B^10*a^10 + 136718750*A*B^9*a^9*b + 861328125*A^2*B^8*a^8*b^2 + 3215625000*A^3*B^7
*a^7*b^3 + 7878281250*A^4*B^6*a^6*b^4 + 13235512500*A^5*B^5*a^5*b^5 + 15441431250*A^6*B^4*a^4*b^6 + 1235314500
0*A^7*B^3*a^3*b^7 + 6485401125*A^8*B^2*a^2*b^8 + 2017680350*A^9*B*a*b^9 + 282475249*A^10*b^10)*x - (15625*B^6*
a^15*b^7 + 131250*A*B^5*a^14*b^8 + 459375*A^2*B^4*a^13*b^9 + 857500*A^3*B^3*a^12*b^10 + 900375*A^4*B^2*a^11*b^
11 + 504210*A^5*B*a^10*b^12 + 117649*A^6*a^9*b^13)*(-(15625*B^6*a^6 + 131250*A*B^5*a^5*b + 459375*A^2*B^4*a^4*
b^2 + 857500*A^3*B^3*a^3*b^3 + 900375*A^4*B^2*a^2*b^4 + 504210*A^5*B*a*b^5 + 117649*A^6*b^6)/(a^13*b^11))^(2/3
)) - 12*((5*B*a*b + 7*A*b^2)*x^5 - (B*a^2 - 13*A*a*b)*x^2)*sqrt(x))/(a^2*b^3*x^6 + 2*a^3*b^2*x^3 + a^4*b)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**(3/2)*(B*x**3+A)/(b*x**3+a)**3,x)

[Out]

Timed out

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Giac [A]  time = 1.295, size = 443, normalized size = 1.35 \begin{align*} \frac{5 \, B a b x^{\frac{11}{2}} + 7 \, A b^{2} x^{\frac{11}{2}} - B a^{2} x^{\frac{5}{2}} + 13 \, A a b x^{\frac{5}{2}}}{36 \,{\left (b x^{3} + a\right )}^{2} a^{2} b} - \frac{\sqrt{3}{\left (5 \, \left (a b^{5}\right )^{\frac{5}{6}} B a + 7 \, \left (a b^{5}\right )^{\frac{5}{6}} A b\right )} \log \left (\sqrt{3} \sqrt{x} \left (\frac{a}{b}\right )^{\frac{1}{6}} + x + \left (\frac{a}{b}\right )^{\frac{1}{3}}\right )}{432 \, a^{3} b^{6}} + \frac{\sqrt{3}{\left (5 \, \left (a b^{5}\right )^{\frac{5}{6}} B a + 7 \, \left (a b^{5}\right )^{\frac{5}{6}} A b\right )} \log \left (-\sqrt{3} \sqrt{x} \left (\frac{a}{b}\right )^{\frac{1}{6}} + x + \left (\frac{a}{b}\right )^{\frac{1}{3}}\right )}{432 \, a^{3} b^{6}} + \frac{{\left (5 \, \left (a b^{5}\right )^{\frac{5}{6}} B a + 7 \, \left (a b^{5}\right )^{\frac{5}{6}} A b\right )} \arctan \left (\frac{\sqrt{3} \left (\frac{a}{b}\right )^{\frac{1}{6}} + 2 \, \sqrt{x}}{\left (\frac{a}{b}\right )^{\frac{1}{6}}}\right )}{216 \, a^{3} b^{6}} + \frac{{\left (5 \, \left (a b^{5}\right )^{\frac{5}{6}} B a + 7 \, \left (a b^{5}\right )^{\frac{5}{6}} A b\right )} \arctan \left (-\frac{\sqrt{3} \left (\frac{a}{b}\right )^{\frac{1}{6}} - 2 \, \sqrt{x}}{\left (\frac{a}{b}\right )^{\frac{1}{6}}}\right )}{216 \, a^{3} b^{6}} + \frac{{\left (5 \, \left (a b^{5}\right )^{\frac{5}{6}} B a + 7 \, \left (a b^{5}\right )^{\frac{5}{6}} A b\right )} \arctan \left (\frac{\sqrt{x}}{\left (\frac{a}{b}\right )^{\frac{1}{6}}}\right )}{108 \, a^{3} b^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/36*(5*B*a*b*x^(11/2) + 7*A*b^2*x^(11/2) - B*a^2*x^(5/2) + 13*A*a*b*x^(5/2))/((b*x^3 + a)^2*a^2*b) - 1/432*sq
rt(3)*(5*(a*b^5)^(5/6)*B*a + 7*(a*b^5)^(5/6)*A*b)*log(sqrt(3)*sqrt(x)*(a/b)^(1/6) + x + (a/b)^(1/3))/(a^3*b^6)
+ 1/432*sqrt(3)*(5*(a*b^5)^(5/6)*B*a + 7*(a*b^5)^(5/6)*A*b)*log(-sqrt(3)*sqrt(x)*(a/b)^(1/6) + x + (a/b)^(1/3
))/(a^3*b^6) + 1/216*(5*(a*b^5)^(5/6)*B*a + 7*(a*b^5)^(5/6)*A*b)*arctan((sqrt(3)*(a/b)^(1/6) + 2*sqrt(x))/(a/b
)^(1/6))/(a^3*b^6) + 1/216*(5*(a*b^5)^(5/6)*B*a + 7*(a*b^5)^(5/6)*A*b)*arctan(-(sqrt(3)*(a/b)^(1/6) - 2*sqrt(x
))/(a/b)^(1/6))/(a^3*b^6) + 1/108*(5*(a*b^5)^(5/6)*B*a + 7*(a*b^5)^(5/6)*A*b)*arctan(sqrt(x)/(a/b)^(1/6))/(a^3
*b^6)