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

Optimal. Leaf size=650 \[ -\frac{27\ 3^{3/4} \left (1-\sqrt{3}\right ) a^{7/3} \sqrt{e x} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\sqrt [3]{a}+\left (1+\sqrt{3}\right ) \sqrt [3]{b} x\right )^2}} (a B+20 A b) \text{EllipticF}\left (\cos ^{-1}\left (\frac{\sqrt [3]{a}+\left (1-\sqrt{3}\right ) \sqrt [3]{b} x}{\sqrt [3]{a}+\left (1+\sqrt{3}\right ) \sqrt [3]{b} x}\right ),\frac{1}{4} \left (2+\sqrt{3}\right )\right )}{896 b^{2/3} e^2 \sqrt{\frac{\sqrt [3]{b} x \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\sqrt [3]{a}+\left (1+\sqrt{3}\right ) \sqrt [3]{b} x\right )^2}} \sqrt{a+b x^3}}+\frac{81 \left (1+\sqrt{3}\right ) a^2 \sqrt{e x} \sqrt{a+b x^3} (a B+20 A b)}{448 b^{2/3} e^2 \left (\sqrt [3]{a}+\left (1+\sqrt{3}\right ) \sqrt [3]{b} x\right )}-\frac{81 \sqrt [4]{3} a^{7/3} \sqrt{e x} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\sqrt [3]{a}+\left (1+\sqrt{3}\right ) \sqrt [3]{b} x\right )^2}} (a B+20 A b) E\left (\cos ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) \sqrt [3]{b} x+\sqrt [3]{a}}{\left (1+\sqrt{3}\right ) \sqrt [3]{b} x+\sqrt [3]{a}}\right )|\frac{1}{4} \left (2+\sqrt{3}\right )\right )}{448 b^{2/3} e^2 \sqrt{\frac{\sqrt [3]{b} x \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\sqrt [3]{a}+\left (1+\sqrt{3}\right ) \sqrt [3]{b} x\right )^2}} \sqrt{a+b x^3}}+\frac{(e x)^{5/2} \left (a+b x^3\right )^{5/2} (a B+20 A b)}{10 a e^4}+\frac{3 (e x)^{5/2} \left (a+b x^3\right )^{3/2} (a B+20 A b)}{28 e^4}+\frac{27 a (e x)^{5/2} \sqrt{a+b x^3} (a B+20 A b)}{224 e^4}-\frac{2 A \left (a+b x^3\right )^{7/2}}{a e \sqrt{e x}} \]

[Out]

(27*a*(20*A*b + a*B)*(e*x)^(5/2)*Sqrt[a + b*x^3])/(224*e^4) + (81*(1 + Sqrt[3])*a^2*(20*A*b + a*B)*Sqrt[e*x]*S
qrt[a + b*x^3])/(448*b^(2/3)*e^2*(a^(1/3) + (1 + Sqrt[3])*b^(1/3)*x)) + (3*(20*A*b + a*B)*(e*x)^(5/2)*(a + b*x
^3)^(3/2))/(28*e^4) + ((20*A*b + a*B)*(e*x)^(5/2)*(a + b*x^3)^(5/2))/(10*a*e^4) - (2*A*(a + b*x^3)^(7/2))/(a*e
*Sqrt[e*x]) - (81*3^(1/4)*a^(7/3)*(20*A*b + a*B)*Sqrt[e*x]*(a^(1/3) + b^(1/3)*x)*Sqrt[(a^(2/3) - a^(1/3)*b^(1/
3)*x + b^(2/3)*x^2)/(a^(1/3) + (1 + Sqrt[3])*b^(1/3)*x)^2]*EllipticE[ArcCos[(a^(1/3) + (1 - Sqrt[3])*b^(1/3)*x
)/(a^(1/3) + (1 + Sqrt[3])*b^(1/3)*x)], (2 + Sqrt[3])/4])/(448*b^(2/3)*e^2*Sqrt[(b^(1/3)*x*(a^(1/3) + b^(1/3)*
x))/(a^(1/3) + (1 + Sqrt[3])*b^(1/3)*x)^2]*Sqrt[a + b*x^3]) - (27*3^(3/4)*(1 - Sqrt[3])*a^(7/3)*(20*A*b + a*B)
*Sqrt[e*x]*(a^(1/3) + b^(1/3)*x)*Sqrt[(a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2)/(a^(1/3) + (1 + Sqrt[3])*b^(
1/3)*x)^2]*EllipticF[ArcCos[(a^(1/3) + (1 - Sqrt[3])*b^(1/3)*x)/(a^(1/3) + (1 + Sqrt[3])*b^(1/3)*x)], (2 + Sqr
t[3])/4])/(896*b^(2/3)*e^2*Sqrt[(b^(1/3)*x*(a^(1/3) + b^(1/3)*x))/(a^(1/3) + (1 + Sqrt[3])*b^(1/3)*x)^2]*Sqrt[
a + b*x^3])

________________________________________________________________________________________

Rubi [A]  time = 0.698379, antiderivative size = 650, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {453, 279, 329, 308, 225, 1881} \[ \frac{81 \left (1+\sqrt{3}\right ) a^2 \sqrt{e x} \sqrt{a+b x^3} (a B+20 A b)}{448 b^{2/3} e^2 \left (\sqrt [3]{a}+\left (1+\sqrt{3}\right ) \sqrt [3]{b} x\right )}-\frac{27\ 3^{3/4} \left (1-\sqrt{3}\right ) a^{7/3} \sqrt{e x} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\sqrt [3]{a}+\left (1+\sqrt{3}\right ) \sqrt [3]{b} x\right )^2}} (a B+20 A b) F\left (\cos ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) \sqrt [3]{b} x+\sqrt [3]{a}}{\left (1+\sqrt{3}\right ) \sqrt [3]{b} x+\sqrt [3]{a}}\right )|\frac{1}{4} \left (2+\sqrt{3}\right )\right )}{896 b^{2/3} e^2 \sqrt{\frac{\sqrt [3]{b} x \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\sqrt [3]{a}+\left (1+\sqrt{3}\right ) \sqrt [3]{b} x\right )^2}} \sqrt{a+b x^3}}-\frac{81 \sqrt [4]{3} a^{7/3} \sqrt{e x} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\sqrt [3]{a}+\left (1+\sqrt{3}\right ) \sqrt [3]{b} x\right )^2}} (a B+20 A b) E\left (\cos ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) \sqrt [3]{b} x+\sqrt [3]{a}}{\left (1+\sqrt{3}\right ) \sqrt [3]{b} x+\sqrt [3]{a}}\right )|\frac{1}{4} \left (2+\sqrt{3}\right )\right )}{448 b^{2/3} e^2 \sqrt{\frac{\sqrt [3]{b} x \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\sqrt [3]{a}+\left (1+\sqrt{3}\right ) \sqrt [3]{b} x\right )^2}} \sqrt{a+b x^3}}+\frac{(e x)^{5/2} \left (a+b x^3\right )^{5/2} (a B+20 A b)}{10 a e^4}+\frac{3 (e x)^{5/2} \left (a+b x^3\right )^{3/2} (a B+20 A b)}{28 e^4}+\frac{27 a (e x)^{5/2} \sqrt{a+b x^3} (a B+20 A b)}{224 e^4}-\frac{2 A \left (a+b x^3\right )^{7/2}}{a e \sqrt{e x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(27*a*(20*A*b + a*B)*(e*x)^(5/2)*Sqrt[a + b*x^3])/(224*e^4) + (81*(1 + Sqrt[3])*a^2*(20*A*b + a*B)*Sqrt[e*x]*S
qrt[a + b*x^3])/(448*b^(2/3)*e^2*(a^(1/3) + (1 + Sqrt[3])*b^(1/3)*x)) + (3*(20*A*b + a*B)*(e*x)^(5/2)*(a + b*x
^3)^(3/2))/(28*e^4) + ((20*A*b + a*B)*(e*x)^(5/2)*(a + b*x^3)^(5/2))/(10*a*e^4) - (2*A*(a + b*x^3)^(7/2))/(a*e
*Sqrt[e*x]) - (81*3^(1/4)*a^(7/3)*(20*A*b + a*B)*Sqrt[e*x]*(a^(1/3) + b^(1/3)*x)*Sqrt[(a^(2/3) - a^(1/3)*b^(1/
3)*x + b^(2/3)*x^2)/(a^(1/3) + (1 + Sqrt[3])*b^(1/3)*x)^2]*EllipticE[ArcCos[(a^(1/3) + (1 - Sqrt[3])*b^(1/3)*x
)/(a^(1/3) + (1 + Sqrt[3])*b^(1/3)*x)], (2 + Sqrt[3])/4])/(448*b^(2/3)*e^2*Sqrt[(b^(1/3)*x*(a^(1/3) + b^(1/3)*
x))/(a^(1/3) + (1 + Sqrt[3])*b^(1/3)*x)^2]*Sqrt[a + b*x^3]) - (27*3^(3/4)*(1 - Sqrt[3])*a^(7/3)*(20*A*b + a*B)
*Sqrt[e*x]*(a^(1/3) + b^(1/3)*x)*Sqrt[(a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2)/(a^(1/3) + (1 + Sqrt[3])*b^(
1/3)*x)^2]*EllipticF[ArcCos[(a^(1/3) + (1 - Sqrt[3])*b^(1/3)*x)/(a^(1/3) + (1 + Sqrt[3])*b^(1/3)*x)], (2 + Sqr
t[3])/4])/(896*b^(2/3)*e^2*Sqrt[(b^(1/3)*x*(a^(1/3) + b^(1/3)*x))/(a^(1/3) + (1 + Sqrt[3])*b^(1/3)*x)^2]*Sqrt[
a + b*x^3])

Rule 453

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

Rule 279

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[((c*x)^(m + 1)*(a + b*x^n)^p)/(c*(m +
n*p + 1)), x] + Dist[(a*n*p)/(m + n*p + 1), Int[(c*x)^m*(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b, c, m}, x]
&& IGtQ[n, 0] && GtQ[p, 0] && 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 308

Int[(x_)^4/Sqrt[(a_) + (b_.)*(x_)^6], x_Symbol] :> With[{r = Numer[Rt[b/a, 3]], s = Denom[Rt[b/a, 3]]}, Dist[(
(Sqrt[3] - 1)*s^2)/(2*r^2), Int[1/Sqrt[a + b*x^6], x], x] - Dist[1/(2*r^2), Int[((Sqrt[3] - 1)*s^2 - 2*r^2*x^4
)/Sqrt[a + b*x^6], x], x]] /; FreeQ[{a, b}, x]

Rule 225

Int[1/Sqrt[(a_) + (b_.)*(x_)^6], x_Symbol] :> With[{r = Numer[Rt[b/a, 3]], s = Denom[Rt[b/a, 3]]}, Simp[(x*(s
+ r*x^2)*Sqrt[(s^2 - r*s*x^2 + r^2*x^4)/(s + (1 + Sqrt[3])*r*x^2)^2]*EllipticF[ArcCos[(s + (1 - Sqrt[3])*r*x^2
)/(s + (1 + Sqrt[3])*r*x^2)], (2 + Sqrt[3])/4])/(2*3^(1/4)*s*Sqrt[a + b*x^6]*Sqrt[(r*x^2*(s + r*x^2))/(s + (1
+ Sqrt[3])*r*x^2)^2]), x]] /; FreeQ[{a, b}, x]

Rule 1881

Int[((c_) + (d_.)*(x_)^4)/Sqrt[(a_) + (b_.)*(x_)^6], x_Symbol] :> With[{r = Numer[Rt[b/a, 3]], s = Denom[Rt[b/
a, 3]]}, Simp[((1 + Sqrt[3])*d*s^3*x*Sqrt[a + b*x^6])/(2*a*r^2*(s + (1 + Sqrt[3])*r*x^2)), x] - Simp[(3^(1/4)*
d*s*x*(s + r*x^2)*Sqrt[(s^2 - r*s*x^2 + r^2*x^4)/(s + (1 + Sqrt[3])*r*x^2)^2]*EllipticE[ArcCos[(s + (1 - Sqrt[
3])*r*x^2)/(s + (1 + Sqrt[3])*r*x^2)], (2 + Sqrt[3])/4])/(2*r^2*Sqrt[(r*x^2*(s + r*x^2))/(s + (1 + Sqrt[3])*r*
x^2)^2]*Sqrt[a + b*x^6]), x]] /; FreeQ[{a, b, c, d}, x] && EqQ[2*Rt[b/a, 3]^2*c - (1 - Sqrt[3])*d, 0]

Rubi steps

\begin{align*} \int \frac{\left (a+b x^3\right )^{5/2} \left (A+B x^3\right )}{(e x)^{3/2}} \, dx &=-\frac{2 A \left (a+b x^3\right )^{7/2}}{a e \sqrt{e x}}+\frac{(20 A b+a B) \int (e x)^{3/2} \left (a+b x^3\right )^{5/2} \, dx}{a e^3}\\ &=\frac{(20 A b+a B) (e x)^{5/2} \left (a+b x^3\right )^{5/2}}{10 a e^4}-\frac{2 A \left (a+b x^3\right )^{7/2}}{a e \sqrt{e x}}+\frac{(3 (20 A b+a B)) \int (e x)^{3/2} \left (a+b x^3\right )^{3/2} \, dx}{4 e^3}\\ &=\frac{3 (20 A b+a B) (e x)^{5/2} \left (a+b x^3\right )^{3/2}}{28 e^4}+\frac{(20 A b+a B) (e x)^{5/2} \left (a+b x^3\right )^{5/2}}{10 a e^4}-\frac{2 A \left (a+b x^3\right )^{7/2}}{a e \sqrt{e x}}+\frac{(27 a (20 A b+a B)) \int (e x)^{3/2} \sqrt{a+b x^3} \, dx}{56 e^3}\\ &=\frac{27 a (20 A b+a B) (e x)^{5/2} \sqrt{a+b x^3}}{224 e^4}+\frac{3 (20 A b+a B) (e x)^{5/2} \left (a+b x^3\right )^{3/2}}{28 e^4}+\frac{(20 A b+a B) (e x)^{5/2} \left (a+b x^3\right )^{5/2}}{10 a e^4}-\frac{2 A \left (a+b x^3\right )^{7/2}}{a e \sqrt{e x}}+\frac{\left (81 a^2 (20 A b+a B)\right ) \int \frac{(e x)^{3/2}}{\sqrt{a+b x^3}} \, dx}{448 e^3}\\ &=\frac{27 a (20 A b+a B) (e x)^{5/2} \sqrt{a+b x^3}}{224 e^4}+\frac{3 (20 A b+a B) (e x)^{5/2} \left (a+b x^3\right )^{3/2}}{28 e^4}+\frac{(20 A b+a B) (e x)^{5/2} \left (a+b x^3\right )^{5/2}}{10 a e^4}-\frac{2 A \left (a+b x^3\right )^{7/2}}{a e \sqrt{e x}}+\frac{\left (81 a^2 (20 A b+a B)\right ) \operatorname{Subst}\left (\int \frac{x^4}{\sqrt{a+\frac{b x^6}{e^3}}} \, dx,x,\sqrt{e x}\right )}{224 e^4}\\ &=\frac{27 a (20 A b+a B) (e x)^{5/2} \sqrt{a+b x^3}}{224 e^4}+\frac{3 (20 A b+a B) (e x)^{5/2} \left (a+b x^3\right )^{3/2}}{28 e^4}+\frac{(20 A b+a B) (e x)^{5/2} \left (a+b x^3\right )^{5/2}}{10 a e^4}-\frac{2 A \left (a+b x^3\right )^{7/2}}{a e \sqrt{e x}}-\frac{\left (81 a^2 (20 A b+a B)\right ) \operatorname{Subst}\left (\int \frac{\left (-1+\sqrt{3}\right ) a^{2/3} e^2-2 b^{2/3} x^4}{\sqrt{a+\frac{b x^6}{e^3}}} \, dx,x,\sqrt{e x}\right )}{448 b^{2/3} e^4}-\frac{\left (81 \left (1-\sqrt{3}\right ) a^{8/3} (20 A b+a B)\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+\frac{b x^6}{e^3}}} \, dx,x,\sqrt{e x}\right )}{448 b^{2/3} e^2}\\ &=\frac{27 a (20 A b+a B) (e x)^{5/2} \sqrt{a+b x^3}}{224 e^4}+\frac{81 \left (1+\sqrt{3}\right ) a^2 (20 A b+a B) \sqrt{e x} \sqrt{a+b x^3}}{448 b^{2/3} e^2 \left (\sqrt [3]{a}+\left (1+\sqrt{3}\right ) \sqrt [3]{b} x\right )}+\frac{3 (20 A b+a B) (e x)^{5/2} \left (a+b x^3\right )^{3/2}}{28 e^4}+\frac{(20 A b+a B) (e x)^{5/2} \left (a+b x^3\right )^{5/2}}{10 a e^4}-\frac{2 A \left (a+b x^3\right )^{7/2}}{a e \sqrt{e x}}-\frac{81 \sqrt [4]{3} a^{7/3} (20 A b+a B) \sqrt{e x} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\sqrt [3]{a}+\left (1+\sqrt{3}\right ) \sqrt [3]{b} x\right )^2}} E\left (\cos ^{-1}\left (\frac{\sqrt [3]{a}+\left (1-\sqrt{3}\right ) \sqrt [3]{b} x}{\sqrt [3]{a}+\left (1+\sqrt{3}\right ) \sqrt [3]{b} x}\right )|\frac{1}{4} \left (2+\sqrt{3}\right )\right )}{448 b^{2/3} e^2 \sqrt{\frac{\sqrt [3]{b} x \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\sqrt [3]{a}+\left (1+\sqrt{3}\right ) \sqrt [3]{b} x\right )^2}} \sqrt{a+b x^3}}-\frac{27\ 3^{3/4} \left (1-\sqrt{3}\right ) a^{7/3} (20 A b+a B) \sqrt{e x} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\sqrt [3]{a}+\left (1+\sqrt{3}\right ) \sqrt [3]{b} x\right )^2}} F\left (\cos ^{-1}\left (\frac{\sqrt [3]{a}+\left (1-\sqrt{3}\right ) \sqrt [3]{b} x}{\sqrt [3]{a}+\left (1+\sqrt{3}\right ) \sqrt [3]{b} x}\right )|\frac{1}{4} \left (2+\sqrt{3}\right )\right )}{896 b^{2/3} e^2 \sqrt{\frac{\sqrt [3]{b} x \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\sqrt [3]{a}+\left (1+\sqrt{3}\right ) \sqrt [3]{b} x\right )^2}} \sqrt{a+b x^3}}\\ \end{align*}

Mathematica [C]  time = 0.0439877, size = 87, normalized size = 0.13 \[ \frac{2 x \sqrt{a+b x^3} \left (\frac{a^2 x^3 (a B+20 A b) \, _2F_1\left (-\frac{5}{2},\frac{5}{6};\frac{11}{6};-\frac{b x^3}{a}\right )}{\sqrt{\frac{b x^3}{a}+1}}-5 A \left (a+b x^3\right )^3\right )}{5 a (e x)^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*x*Sqrt[a + b*x^3]*(-5*A*(a + b*x^3)^3 + (a^2*(20*A*b + a*B)*x^3*Hypergeometric2F1[-5/2, 5/6, 11/6, -((b*x^3
)/a)])/Sqrt[1 + (b*x^3)/a]))/(5*a*(e*x)^(3/2))

________________________________________________________________________________________

Maple [C]  time = 0.056, size = 6530, normalized size = 10.1 \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

result too large to display

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (B x^{3} + A\right )}{\left (b x^{3} + a\right )}^{\frac{5}{2}}}{\left (e x\right )^{\frac{3}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (B b^{2} x^{9} +{\left (2 \, B a b + A b^{2}\right )} x^{6} +{\left (B a^{2} + 2 \, A a b\right )} x^{3} + A a^{2}\right )} \sqrt{b x^{3} + a} \sqrt{e x}}{e^{2} x^{2}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((B*b^2*x^9 + (2*B*a*b + A*b^2)*x^6 + (B*a^2 + 2*A*a*b)*x^3 + A*a^2)*sqrt(b*x^3 + a)*sqrt(e*x)/(e^2*x^
2), x)

________________________________________________________________________________________

Sympy [C]  time = 159.126, size = 311, normalized size = 0.48 \begin{align*} \frac{A a^{\frac{5}{2}} \Gamma \left (- \frac{1}{6}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, - \frac{1}{6} \\ \frac{5}{6} \end{matrix}\middle |{\frac{b x^{3} e^{i \pi }}{a}} \right )}}{3 e^{\frac{3}{2}} \sqrt{x} \Gamma \left (\frac{5}{6}\right )} + \frac{2 A a^{\frac{3}{2}} b x^{\frac{5}{2}} \Gamma \left (\frac{5}{6}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, \frac{5}{6} \\ \frac{11}{6} \end{matrix}\middle |{\frac{b x^{3} e^{i \pi }}{a}} \right )}}{3 e^{\frac{3}{2}} \Gamma \left (\frac{11}{6}\right )} + \frac{A \sqrt{a} b^{2} x^{\frac{11}{2}} \Gamma \left (\frac{11}{6}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, \frac{11}{6} \\ \frac{17}{6} \end{matrix}\middle |{\frac{b x^{3} e^{i \pi }}{a}} \right )}}{3 e^{\frac{3}{2}} \Gamma \left (\frac{17}{6}\right )} + \frac{B a^{\frac{5}{2}} x^{\frac{5}{2}} \Gamma \left (\frac{5}{6}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, \frac{5}{6} \\ \frac{11}{6} \end{matrix}\middle |{\frac{b x^{3} e^{i \pi }}{a}} \right )}}{3 e^{\frac{3}{2}} \Gamma \left (\frac{11}{6}\right )} + \frac{2 B a^{\frac{3}{2}} b x^{\frac{11}{2}} \Gamma \left (\frac{11}{6}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, \frac{11}{6} \\ \frac{17}{6} \end{matrix}\middle |{\frac{b x^{3} e^{i \pi }}{a}} \right )}}{3 e^{\frac{3}{2}} \Gamma \left (\frac{17}{6}\right )} + \frac{B \sqrt{a} b^{2} x^{\frac{17}{2}} \Gamma \left (\frac{17}{6}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, \frac{17}{6} \\ \frac{23}{6} \end{matrix}\middle |{\frac{b x^{3} e^{i \pi }}{a}} \right )}}{3 e^{\frac{3}{2}} \Gamma \left (\frac{23}{6}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

A*a**(5/2)*gamma(-1/6)*hyper((-1/2, -1/6), (5/6,), b*x**3*exp_polar(I*pi)/a)/(3*e**(3/2)*sqrt(x)*gamma(5/6)) +
2*A*a**(3/2)*b*x**(5/2)*gamma(5/6)*hyper((-1/2, 5/6), (11/6,), b*x**3*exp_polar(I*pi)/a)/(3*e**(3/2)*gamma(11
/6)) + A*sqrt(a)*b**2*x**(11/2)*gamma(11/6)*hyper((-1/2, 11/6), (17/6,), b*x**3*exp_polar(I*pi)/a)/(3*e**(3/2)
*gamma(17/6)) + B*a**(5/2)*x**(5/2)*gamma(5/6)*hyper((-1/2, 5/6), (11/6,), b*x**3*exp_polar(I*pi)/a)/(3*e**(3/
2)*gamma(11/6)) + 2*B*a**(3/2)*b*x**(11/2)*gamma(11/6)*hyper((-1/2, 11/6), (17/6,), b*x**3*exp_polar(I*pi)/a)/
(3*e**(3/2)*gamma(17/6)) + B*sqrt(a)*b**2*x**(17/2)*gamma(17/6)*hyper((-1/2, 17/6), (23/6,), b*x**3*exp_polar(
I*pi)/a)/(3*e**(3/2)*gamma(23/6))

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (B x^{3} + A\right )}{\left (b x^{3} + a\right )}^{\frac{5}{2}}}{\left (e x\right )^{\frac{3}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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