3.164 \(\int \frac{x^{5/2} (A+B x^3)}{(a+b x^3)^2} \, dx\)

Optimal. Leaf size=312 \[ -\frac{(A b-7 a B) \log \left (-\sqrt{3} \sqrt [6]{a} \sqrt [6]{b} \sqrt{x}+\sqrt [3]{a}+\sqrt [3]{b} x\right )}{12 \sqrt{3} a^{5/6} b^{13/6}}+\frac{(A b-7 a B) \log \left (\sqrt{3} \sqrt [6]{a} \sqrt [6]{b} \sqrt{x}+\sqrt [3]{a}+\sqrt [3]{b} x\right )}{12 \sqrt{3} a^{5/6} b^{13/6}}-\frac{(A b-7 a B) \tan ^{-1}\left (\sqrt{3}-\frac{2 \sqrt [6]{b} \sqrt{x}}{\sqrt [6]{a}}\right )}{18 a^{5/6} b^{13/6}}+\frac{(A b-7 a B) \tan ^{-1}\left (\frac{2 \sqrt [6]{b} \sqrt{x}}{\sqrt [6]{a}}+\sqrt{3}\right )}{18 a^{5/6} b^{13/6}}+\frac{(A b-7 a B) \tan ^{-1}\left (\frac{\sqrt [6]{b} \sqrt{x}}{\sqrt [6]{a}}\right )}{9 a^{5/6} b^{13/6}}-\frac{\sqrt{x} (A b-7 a B)}{3 a b^2}+\frac{x^{7/2} (A b-a B)}{3 a b \left (a+b x^3\right )} \]

[Out]

-((A*b - 7*a*B)*Sqrt[x])/(3*a*b^2) + ((A*b - a*B)*x^(7/2))/(3*a*b*(a + b*x^3)) - ((A*b - 7*a*B)*ArcTan[Sqrt[3]
- (2*b^(1/6)*Sqrt[x])/a^(1/6)])/(18*a^(5/6)*b^(13/6)) + ((A*b - 7*a*B)*ArcTan[Sqrt[3] + (2*b^(1/6)*Sqrt[x])/a
^(1/6)])/(18*a^(5/6)*b^(13/6)) + ((A*b - 7*a*B)*ArcTan[(b^(1/6)*Sqrt[x])/a^(1/6)])/(9*a^(5/6)*b^(13/6)) - ((A*
b - 7*a*B)*Log[a^(1/3) - Sqrt[3]*a^(1/6)*b^(1/6)*Sqrt[x] + b^(1/3)*x])/(12*Sqrt[3]*a^(5/6)*b^(13/6)) + ((A*b -
7*a*B)*Log[a^(1/3) + Sqrt[3]*a^(1/6)*b^(1/6)*Sqrt[x] + b^(1/3)*x])/(12*Sqrt[3]*a^(5/6)*b^(13/6))

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Rubi [A]  time = 0.501587, antiderivative size = 312, 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, 321, 329, 209, 634, 618, 204, 628, 205} \[ -\frac{(A b-7 a B) \log \left (-\sqrt{3} \sqrt [6]{a} \sqrt [6]{b} \sqrt{x}+\sqrt [3]{a}+\sqrt [3]{b} x\right )}{12 \sqrt{3} a^{5/6} b^{13/6}}+\frac{(A b-7 a B) \log \left (\sqrt{3} \sqrt [6]{a} \sqrt [6]{b} \sqrt{x}+\sqrt [3]{a}+\sqrt [3]{b} x\right )}{12 \sqrt{3} a^{5/6} b^{13/6}}-\frac{(A b-7 a B) \tan ^{-1}\left (\sqrt{3}-\frac{2 \sqrt [6]{b} \sqrt{x}}{\sqrt [6]{a}}\right )}{18 a^{5/6} b^{13/6}}+\frac{(A b-7 a B) \tan ^{-1}\left (\frac{2 \sqrt [6]{b} \sqrt{x}}{\sqrt [6]{a}}+\sqrt{3}\right )}{18 a^{5/6} b^{13/6}}+\frac{(A b-7 a B) \tan ^{-1}\left (\frac{\sqrt [6]{b} \sqrt{x}}{\sqrt [6]{a}}\right )}{9 a^{5/6} b^{13/6}}-\frac{\sqrt{x} (A b-7 a B)}{3 a b^2}+\frac{x^{7/2} (A b-a B)}{3 a b \left (a+b x^3\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((A*b - 7*a*B)*Sqrt[x])/(3*a*b^2) + ((A*b - a*B)*x^(7/2))/(3*a*b*(a + b*x^3)) - ((A*b - 7*a*B)*ArcTan[Sqrt[3]
- (2*b^(1/6)*Sqrt[x])/a^(1/6)])/(18*a^(5/6)*b^(13/6)) + ((A*b - 7*a*B)*ArcTan[Sqrt[3] + (2*b^(1/6)*Sqrt[x])/a
^(1/6)])/(18*a^(5/6)*b^(13/6)) + ((A*b - 7*a*B)*ArcTan[(b^(1/6)*Sqrt[x])/a^(1/6)])/(9*a^(5/6)*b^(13/6)) - ((A*
b - 7*a*B)*Log[a^(1/3) - Sqrt[3]*a^(1/6)*b^(1/6)*Sqrt[x] + b^(1/3)*x])/(12*Sqrt[3]*a^(5/6)*b^(13/6)) + ((A*b -
7*a*B)*Log[a^(1/3) + Sqrt[3]*a^(1/6)*b^(1/6)*Sqrt[x] + b^(1/3)*x])/(12*Sqrt[3]*a^(5/6)*b^(13/6))

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 321

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 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 209

Int[((a_) + (b_.)*(x_)^(n_))^(-1), x_Symbol] :> Module[{r = Numerator[Rt[a/b, n]], s = Denominator[Rt[a/b, n]]
, k, u, v}, Simp[u = Int[(r - s*Cos[((2*k - 1)*Pi)/n]*x)/(r^2 - 2*r*s*Cos[((2*k - 1)*Pi)/n]*x + s^2*x^2), x] +
Int[(r + s*Cos[((2*k - 1)*Pi)/n]*x)/(r^2 + 2*r*s*Cos[((2*k - 1)*Pi)/n]*x + s^2*x^2), x]; (2*r^2*Int[1/(r^2 +
s^2*x^2), x])/(a*n) + Dist[(2*r)/(a*n), Sum[u, {k, 1, (n - 2)/4}], x], x]] /; FreeQ[{a, b}, x] && IGtQ[(n - 2)
/4, 0] && 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^{5/2} \left (A+B x^3\right )}{\left (a+b x^3\right )^2} \, dx &=\frac{(A b-a B) x^{7/2}}{3 a b \left (a+b x^3\right )}+\frac{\left (-\frac{A b}{2}+\frac{7 a B}{2}\right ) \int \frac{x^{5/2}}{a+b x^3} \, dx}{3 a b}\\ &=-\frac{(A b-7 a B) \sqrt{x}}{3 a b^2}+\frac{(A b-a B) x^{7/2}}{3 a b \left (a+b x^3\right )}+\frac{(A b-7 a B) \int \frac{1}{\sqrt{x} \left (a+b x^3\right )} \, dx}{6 b^2}\\ &=-\frac{(A b-7 a B) \sqrt{x}}{3 a b^2}+\frac{(A b-a B) x^{7/2}}{3 a b \left (a+b x^3\right )}+\frac{(A b-7 a B) \operatorname{Subst}\left (\int \frac{1}{a+b x^6} \, dx,x,\sqrt{x}\right )}{3 b^2}\\ &=-\frac{(A b-7 a B) \sqrt{x}}{3 a b^2}+\frac{(A b-a B) x^{7/2}}{3 a b \left (a+b x^3\right )}+\frac{(A b-7 a B) \operatorname{Subst}\left (\int \frac{\sqrt [6]{a}-\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 )}{9 a^{5/6} b^2}+\frac{(A b-7 a B) \operatorname{Subst}\left (\int \frac{\sqrt [6]{a}+\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 )}{9 a^{5/6} b^2}+\frac{(A b-7 a B) \operatorname{Subst}\left (\int \frac{1}{\sqrt [3]{a}+\sqrt [3]{b} x^2} \, dx,x,\sqrt{x}\right )}{9 a^{2/3} b^2}\\ &=-\frac{(A b-7 a B) \sqrt{x}}{3 a b^2}+\frac{(A b-a B) x^{7/2}}{3 a b \left (a+b x^3\right )}+\frac{(A b-7 a B) \tan ^{-1}\left (\frac{\sqrt [6]{b} \sqrt{x}}{\sqrt [6]{a}}\right )}{9 a^{5/6} b^{13/6}}-\frac{(A b-7 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 )}{12 \sqrt{3} a^{5/6} b^{13/6}}+\frac{(A b-7 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 )}{12 \sqrt{3} a^{5/6} b^{13/6}}+\frac{(A b-7 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 )}{36 a^{2/3} b^2}+\frac{(A b-7 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 )}{36 a^{2/3} b^2}\\ &=-\frac{(A b-7 a B) \sqrt{x}}{3 a b^2}+\frac{(A b-a B) x^{7/2}}{3 a b \left (a+b x^3\right )}+\frac{(A b-7 a B) \tan ^{-1}\left (\frac{\sqrt [6]{b} \sqrt{x}}{\sqrt [6]{a}}\right )}{9 a^{5/6} b^{13/6}}-\frac{(A b-7 a B) \log \left (\sqrt [3]{a}-\sqrt{3} \sqrt [6]{a} \sqrt [6]{b} \sqrt{x}+\sqrt [3]{b} x\right )}{12 \sqrt{3} a^{5/6} b^{13/6}}+\frac{(A b-7 a B) \log \left (\sqrt [3]{a}+\sqrt{3} \sqrt [6]{a} \sqrt [6]{b} \sqrt{x}+\sqrt [3]{b} x\right )}{12 \sqrt{3} a^{5/6} b^{13/6}}+\frac{(A b-7 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 )}{18 \sqrt{3} a^{5/6} b^{13/6}}-\frac{(A b-7 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 )}{18 \sqrt{3} a^{5/6} b^{13/6}}\\ &=-\frac{(A b-7 a B) \sqrt{x}}{3 a b^2}+\frac{(A b-a B) x^{7/2}}{3 a b \left (a+b x^3\right )}-\frac{(A b-7 a B) \tan ^{-1}\left (\sqrt{3}-\frac{2 \sqrt [6]{b} \sqrt{x}}{\sqrt [6]{a}}\right )}{18 a^{5/6} b^{13/6}}+\frac{(A b-7 a B) \tan ^{-1}\left (\sqrt{3}+\frac{2 \sqrt [6]{b} \sqrt{x}}{\sqrt [6]{a}}\right )}{18 a^{5/6} b^{13/6}}+\frac{(A b-7 a B) \tan ^{-1}\left (\frac{\sqrt [6]{b} \sqrt{x}}{\sqrt [6]{a}}\right )}{9 a^{5/6} b^{13/6}}-\frac{(A b-7 a B) \log \left (\sqrt [3]{a}-\sqrt{3} \sqrt [6]{a} \sqrt [6]{b} \sqrt{x}+\sqrt [3]{b} x\right )}{12 \sqrt{3} a^{5/6} b^{13/6}}+\frac{(A b-7 a B) \log \left (\sqrt [3]{a}+\sqrt{3} \sqrt [6]{a} \sqrt [6]{b} \sqrt{x}+\sqrt [3]{b} x\right )}{12 \sqrt{3} a^{5/6} b^{13/6}}\\ \end{align*}

Mathematica [C]  time = 0.0761386, size = 76, normalized size = 0.24 \[ \frac{\sqrt{x} \left (\left (a+b x^3\right ) (A b-7 a B) \, _2F_1\left (\frac{1}{6},1;\frac{7}{6};-\frac{b x^3}{a}\right )+a \left (7 a B-A b+6 b B x^3\right )\right )}{3 a b^2 \left (a+b x^3\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[x]*(a*(-(A*b) + 7*a*B + 6*b*B*x^3) + (A*b - 7*a*B)*(a + b*x^3)*Hypergeometric2F1[1/6, 1, 7/6, -((b*x^3)/
a)]))/(3*a*b^2*(a + b*x^3))

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Maple [A]  time = 0.039, size = 399, normalized size = 1.3 \begin{align*} 2\,{\frac{B\sqrt{x}}{{b}^{2}}}-{\frac{A}{3\,b \left ( b{x}^{3}+a \right ) }\sqrt{x}}+{\frac{Ba}{3\,{b}^{2} \left ( b{x}^{3}+a \right ) }\sqrt{x}}-{\frac{7\,B}{9\,{b}^{2}}\sqrt [6]{{\frac{a}{b}}}\arctan \left ({\sqrt{x}{\frac{1}{\sqrt [6]{{\frac{a}{b}}}}}} \right ) }+{\frac{7\,B\sqrt{3}}{36\,{b}^{2}}\sqrt [6]{{\frac{a}{b}}}\ln \left ( x-\sqrt{3}\sqrt [6]{{\frac{a}{b}}}\sqrt{x}+\sqrt [3]{{\frac{a}{b}}} \right ) }-{\frac{7\,B}{18\,{b}^{2}}\sqrt [6]{{\frac{a}{b}}}\arctan \left ( 2\,{\sqrt{x}{\frac{1}{\sqrt [6]{{\frac{a}{b}}}}}}-\sqrt{3} \right ) }-{\frac{7\,B\sqrt{3}}{36\,{b}^{2}}\sqrt [6]{{\frac{a}{b}}}\ln \left ( x+\sqrt{3}\sqrt [6]{{\frac{a}{b}}}\sqrt{x}+\sqrt [3]{{\frac{a}{b}}} \right ) }-{\frac{7\,B}{18\,{b}^{2}}\sqrt [6]{{\frac{a}{b}}}\arctan \left ( 2\,{\sqrt{x}{\frac{1}{\sqrt [6]{{\frac{a}{b}}}}}}+\sqrt{3} \right ) }+{\frac{A}{9\,ab}\sqrt [6]{{\frac{a}{b}}}\arctan \left ({\sqrt{x}{\frac{1}{\sqrt [6]{{\frac{a}{b}}}}}} \right ) }-{\frac{A\sqrt{3}}{36\,ab}\sqrt [6]{{\frac{a}{b}}}\ln \left ( x-\sqrt{3}\sqrt [6]{{\frac{a}{b}}}\sqrt{x}+\sqrt [3]{{\frac{a}{b}}} \right ) }+{\frac{A}{18\,ab}\sqrt [6]{{\frac{a}{b}}}\arctan \left ( 2\,{\sqrt{x}{\frac{1}{\sqrt [6]{{\frac{a}{b}}}}}}-\sqrt{3} \right ) }+{\frac{A\sqrt{3}}{36\,ab}\sqrt [6]{{\frac{a}{b}}}\ln \left ( x+\sqrt{3}\sqrt [6]{{\frac{a}{b}}}\sqrt{x}+\sqrt [3]{{\frac{a}{b}}} \right ) }+{\frac{A}{18\,ab}\sqrt [6]{{\frac{a}{b}}}\arctan \left ( 2\,{\sqrt{x}{\frac{1}{\sqrt [6]{{\frac{a}{b}}}}}}+\sqrt{3} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(5/2)*(B*x^3+A)/(b*x^3+a)^2,x)

[Out]

2*B/b^2*x^(1/2)-1/3/b*x^(1/2)/(b*x^3+a)*A+1/3/b^2*x^(1/2)/(b*x^3+a)*B*a-7/9/b^2*B*(a/b)^(1/6)*arctan(x^(1/2)/(
a/b)^(1/6))+7/36/b^2*B*3^(1/2)*(a/b)^(1/6)*ln(x-3^(1/2)*(a/b)^(1/6)*x^(1/2)+(a/b)^(1/3))-7/18/b^2*B*(a/b)^(1/6
)*arctan(2*x^(1/2)/(a/b)^(1/6)-3^(1/2))-7/36/b^2*B*3^(1/2)*(a/b)^(1/6)*ln(x+3^(1/2)*(a/b)^(1/6)*x^(1/2)+(a/b)^
(1/3))-7/18/b^2*B*(a/b)^(1/6)*arctan(2*x^(1/2)/(a/b)^(1/6)+3^(1/2))+1/9/b*A/a*(a/b)^(1/6)*arctan(x^(1/2)/(a/b)
^(1/6))-1/36/b*A/a*3^(1/2)*(a/b)^(1/6)*ln(x-3^(1/2)*(a/b)^(1/6)*x^(1/2)+(a/b)^(1/3))+1/18/b*A/a*(a/b)^(1/6)*ar
ctan(2*x^(1/2)/(a/b)^(1/6)-3^(1/2))+1/36/b*A/a*3^(1/2)*(a/b)^(1/6)*ln(x+3^(1/2)*(a/b)^(1/6)*x^(1/2)+(a/b)^(1/3
))+1/18/b*A/a*(a/b)^(1/6)*arctan(2*x^(1/2)/(a/b)^(1/6)+3^(1/2))

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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^(5/2)*(B*x^3+A)/(b*x^3+a)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 2.52257, size = 5975, normalized size = 19.15 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/36*(4*sqrt(3)*(b^3*x^3 + a*b^2)*(-(117649*B^6*a^6 - 100842*A*B^5*a^5*b + 36015*A^2*B^4*a^4*b^2 - 6860*A^3*B^
3*a^3*b^3 + 735*A^4*B^2*a^2*b^4 - 42*A^5*B*a*b^5 + A^6*b^6)/(a^5*b^13))^(1/6)*arctan(1/3*(2*sqrt(3)*sqrt(a^2*b
^4*(-(117649*B^6*a^6 - 100842*A*B^5*a^5*b + 36015*A^2*B^4*a^4*b^2 - 6860*A^3*B^3*a^3*b^3 + 735*A^4*B^2*a^2*b^4
- 42*A^5*B*a*b^5 + A^6*b^6)/(a^5*b^13))^(1/3) + (49*B^2*a^2 - 14*A*B*a*b + A^2*b^2)*x + (7*B*a^2*b^2 - A*a*b^
3)*sqrt(x)*(-(117649*B^6*a^6 - 100842*A*B^5*a^5*b + 36015*A^2*B^4*a^4*b^2 - 6860*A^3*B^3*a^3*b^3 + 735*A^4*B^2
*a^2*b^4 - 42*A^5*B*a*b^5 + A^6*b^6)/(a^5*b^13))^(1/6))*a^4*b^11*(-(117649*B^6*a^6 - 100842*A*B^5*a^5*b + 3601
5*A^2*B^4*a^4*b^2 - 6860*A^3*B^3*a^3*b^3 + 735*A^4*B^2*a^2*b^4 - 42*A^5*B*a*b^5 + A^6*b^6)/(a^5*b^13))^(5/6) +
2*sqrt(3)*(7*B*a^5*b^11 - A*a^4*b^12)*sqrt(x)*(-(117649*B^6*a^6 - 100842*A*B^5*a^5*b + 36015*A^2*B^4*a^4*b^2
- 6860*A^3*B^3*a^3*b^3 + 735*A^4*B^2*a^2*b^4 - 42*A^5*B*a*b^5 + A^6*b^6)/(a^5*b^13))^(5/6) - sqrt(3)*(117649*B
^6*a^6 - 100842*A*B^5*a^5*b + 36015*A^2*B^4*a^4*b^2 - 6860*A^3*B^3*a^3*b^3 + 735*A^4*B^2*a^2*b^4 - 42*A^5*B*a*
b^5 + A^6*b^6))/(117649*B^6*a^6 - 100842*A*B^5*a^5*b + 36015*A^2*B^4*a^4*b^2 - 6860*A^3*B^3*a^3*b^3 + 735*A^4*
B^2*a^2*b^4 - 42*A^5*B*a*b^5 + A^6*b^6)) + 4*sqrt(3)*(b^3*x^3 + a*b^2)*(-(117649*B^6*a^6 - 100842*A*B^5*a^5*b
+ 36015*A^2*B^4*a^4*b^2 - 6860*A^3*B^3*a^3*b^3 + 735*A^4*B^2*a^2*b^4 - 42*A^5*B*a*b^5 + A^6*b^6)/(a^5*b^13))^(
1/6)*arctan(1/3*(2*sqrt(3)*sqrt(a^2*b^4*(-(117649*B^6*a^6 - 100842*A*B^5*a^5*b + 36015*A^2*B^4*a^4*b^2 - 6860*
A^3*B^3*a^3*b^3 + 735*A^4*B^2*a^2*b^4 - 42*A^5*B*a*b^5 + A^6*b^6)/(a^5*b^13))^(1/3) + (49*B^2*a^2 - 14*A*B*a*b
+ A^2*b^2)*x - (7*B*a^2*b^2 - A*a*b^3)*sqrt(x)*(-(117649*B^6*a^6 - 100842*A*B^5*a^5*b + 36015*A^2*B^4*a^4*b^2
- 6860*A^3*B^3*a^3*b^3 + 735*A^4*B^2*a^2*b^4 - 42*A^5*B*a*b^5 + A^6*b^6)/(a^5*b^13))^(1/6))*a^4*b^11*(-(11764
9*B^6*a^6 - 100842*A*B^5*a^5*b + 36015*A^2*B^4*a^4*b^2 - 6860*A^3*B^3*a^3*b^3 + 735*A^4*B^2*a^2*b^4 - 42*A^5*B
*a*b^5 + A^6*b^6)/(a^5*b^13))^(5/6) + 2*sqrt(3)*(7*B*a^5*b^11 - A*a^4*b^12)*sqrt(x)*(-(117649*B^6*a^6 - 100842
*A*B^5*a^5*b + 36015*A^2*B^4*a^4*b^2 - 6860*A^3*B^3*a^3*b^3 + 735*A^4*B^2*a^2*b^4 - 42*A^5*B*a*b^5 + A^6*b^6)/
(a^5*b^13))^(5/6) + sqrt(3)*(117649*B^6*a^6 - 100842*A*B^5*a^5*b + 36015*A^2*B^4*a^4*b^2 - 6860*A^3*B^3*a^3*b^
3 + 735*A^4*B^2*a^2*b^4 - 42*A^5*B*a*b^5 + A^6*b^6))/(117649*B^6*a^6 - 100842*A*B^5*a^5*b + 36015*A^2*B^4*a^4*
b^2 - 6860*A^3*B^3*a^3*b^3 + 735*A^4*B^2*a^2*b^4 - 42*A^5*B*a*b^5 + A^6*b^6)) - (b^3*x^3 + a*b^2)*(-(117649*B^
6*a^6 - 100842*A*B^5*a^5*b + 36015*A^2*B^4*a^4*b^2 - 6860*A^3*B^3*a^3*b^3 + 735*A^4*B^2*a^2*b^4 - 42*A^5*B*a*b
^5 + A^6*b^6)/(a^5*b^13))^(1/6)*log(a^2*b^4*(-(117649*B^6*a^6 - 100842*A*B^5*a^5*b + 36015*A^2*B^4*a^4*b^2 - 6
860*A^3*B^3*a^3*b^3 + 735*A^4*B^2*a^2*b^4 - 42*A^5*B*a*b^5 + A^6*b^6)/(a^5*b^13))^(1/3) + (49*B^2*a^2 - 14*A*B
*a*b + A^2*b^2)*x + (7*B*a^2*b^2 - A*a*b^3)*sqrt(x)*(-(117649*B^6*a^6 - 100842*A*B^5*a^5*b + 36015*A^2*B^4*a^4
*b^2 - 6860*A^3*B^3*a^3*b^3 + 735*A^4*B^2*a^2*b^4 - 42*A^5*B*a*b^5 + A^6*b^6)/(a^5*b^13))^(1/6)) + (b^3*x^3 +
a*b^2)*(-(117649*B^6*a^6 - 100842*A*B^5*a^5*b + 36015*A^2*B^4*a^4*b^2 - 6860*A^3*B^3*a^3*b^3 + 735*A^4*B^2*a^2
*b^4 - 42*A^5*B*a*b^5 + A^6*b^6)/(a^5*b^13))^(1/6)*log(a^2*b^4*(-(117649*B^6*a^6 - 100842*A*B^5*a^5*b + 36015*
A^2*B^4*a^4*b^2 - 6860*A^3*B^3*a^3*b^3 + 735*A^4*B^2*a^2*b^4 - 42*A^5*B*a*b^5 + A^6*b^6)/(a^5*b^13))^(1/3) + (
49*B^2*a^2 - 14*A*B*a*b + A^2*b^2)*x - (7*B*a^2*b^2 - A*a*b^3)*sqrt(x)*(-(117649*B^6*a^6 - 100842*A*B^5*a^5*b
+ 36015*A^2*B^4*a^4*b^2 - 6860*A^3*B^3*a^3*b^3 + 735*A^4*B^2*a^2*b^4 - 42*A^5*B*a*b^5 + A^6*b^6)/(a^5*b^13))^(
1/6)) + 2*(b^3*x^3 + a*b^2)*(-(117649*B^6*a^6 - 100842*A*B^5*a^5*b + 36015*A^2*B^4*a^4*b^2 - 6860*A^3*B^3*a^3*
b^3 + 735*A^4*B^2*a^2*b^4 - 42*A^5*B*a*b^5 + A^6*b^6)/(a^5*b^13))^(1/6)*log(a*b^2*(-(117649*B^6*a^6 - 100842*A
*B^5*a^5*b + 36015*A^2*B^4*a^4*b^2 - 6860*A^3*B^3*a^3*b^3 + 735*A^4*B^2*a^2*b^4 - 42*A^5*B*a*b^5 + A^6*b^6)/(a
^5*b^13))^(1/6) - (7*B*a - A*b)*sqrt(x)) - 2*(b^3*x^3 + a*b^2)*(-(117649*B^6*a^6 - 100842*A*B^5*a^5*b + 36015*
A^2*B^4*a^4*b^2 - 6860*A^3*B^3*a^3*b^3 + 735*A^4*B^2*a^2*b^4 - 42*A^5*B*a*b^5 + A^6*b^6)/(a^5*b^13))^(1/6)*log
(-a*b^2*(-(117649*B^6*a^6 - 100842*A*B^5*a^5*b + 36015*A^2*B^4*a^4*b^2 - 6860*A^3*B^3*a^3*b^3 + 735*A^4*B^2*a^
2*b^4 - 42*A^5*B*a*b^5 + A^6*b^6)/(a^5*b^13))^(1/6) - (7*B*a - A*b)*sqrt(x)) + 12*(6*B*b*x^3 + 7*B*a - A*b)*sq
rt(x))/(b^3*x^3 + a*b^2)

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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**(5/2)*(B*x**3+A)/(b*x**3+a)**2,x)

[Out]

Timed out

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Giac [A]  time = 1.20197, size = 423, normalized size = 1.36 \begin{align*} \frac{2 \, B \sqrt{x}}{b^{2}} - \frac{\sqrt{3}{\left (7 \, \left (a b^{5}\right )^{\frac{1}{6}} B a - \left (a b^{5}\right )^{\frac{1}{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 )}{36 \, a b^{3}} + \frac{\sqrt{3}{\left (7 \, \left (a b^{5}\right )^{\frac{1}{6}} B a - \left (a b^{5}\right )^{\frac{1}{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 )}{36 \, a b^{3}} + \frac{B a \sqrt{x} - A b \sqrt{x}}{3 \,{\left (b x^{3} + a\right )} b^{2}} - \frac{{\left (7 \, \left (a b^{5}\right )^{\frac{1}{6}} B a - \left (a b^{5}\right )^{\frac{1}{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 )}{18 \, a b^{3}} - \frac{{\left (7 \, \left (a b^{5}\right )^{\frac{1}{6}} B a - \left (a b^{5}\right )^{\frac{1}{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 )}{18 \, a b^{3}} - \frac{{\left (7 \, \left (a b^{5}\right )^{\frac{1}{6}} B a - \left (a b^{5}\right )^{\frac{1}{6}} A b\right )} \arctan \left (\frac{\sqrt{x}}{\left (\frac{a}{b}\right )^{\frac{1}{6}}}\right )}{9 \, a b^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*B*sqrt(x)/b^2 - 1/36*sqrt(3)*(7*(a*b^5)^(1/6)*B*a - (a*b^5)^(1/6)*A*b)*log(sqrt(3)*sqrt(x)*(a/b)^(1/6) + x +
(a/b)^(1/3))/(a*b^3) + 1/36*sqrt(3)*(7*(a*b^5)^(1/6)*B*a - (a*b^5)^(1/6)*A*b)*log(-sqrt(3)*sqrt(x)*(a/b)^(1/6
) + x + (a/b)^(1/3))/(a*b^3) + 1/3*(B*a*sqrt(x) - A*b*sqrt(x))/((b*x^3 + a)*b^2) - 1/18*(7*(a*b^5)^(1/6)*B*a -
(a*b^5)^(1/6)*A*b)*arctan((sqrt(3)*(a/b)^(1/6) + 2*sqrt(x))/(a/b)^(1/6))/(a*b^3) - 1/18*(7*(a*b^5)^(1/6)*B*a
- (a*b^5)^(1/6)*A*b)*arctan(-(sqrt(3)*(a/b)^(1/6) - 2*sqrt(x))/(a/b)^(1/6))/(a*b^3) - 1/9*(7*(a*b^5)^(1/6)*B*a
- (a*b^5)^(1/6)*A*b)*arctan(sqrt(x)/(a/b)^(1/6))/(a*b^3)