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

Optimal. Leaf size=661 \[ -\frac{27\ 3^{3/4} \left (1-\sqrt{3}\right ) a^{10/3} e \sqrt{e x} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\sqrt [3]{a}+\left (1+\sqrt{3}\right ) \sqrt [3]{b} x\right )^2}} (26 A b-5 a B) F\left (\cos ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) \sqrt [3]{b} x+\sqrt [3]{a}}{\left (1+\sqrt{3}\right ) \sqrt [3]{b} x+\sqrt [3]{a}}\right )|\frac{1}{4} \left (2+\sqrt{3}\right )\right )}{23296 b^{5/3} \sqrt{\frac{\sqrt [3]{b} x \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\sqrt [3]{a}+\left (1+\sqrt{3}\right ) \sqrt [3]{b} x\right )^2}} \sqrt{a+b x^3}}-\frac{81 \sqrt [4]{3} a^{10/3} e \sqrt{e x} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\sqrt [3]{a}+\left (1+\sqrt{3}\right ) \sqrt [3]{b} x\right )^2}} (26 A b-5 a B) E\left (\cos ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) \sqrt [3]{b} x+\sqrt [3]{a}}{\left (1+\sqrt{3}\right ) \sqrt [3]{b} x+\sqrt [3]{a}}\right )|\frac{1}{4} \left (2+\sqrt{3}\right )\right )}{11648 b^{5/3} \sqrt{\frac{\sqrt [3]{b} x \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\sqrt [3]{a}+\left (1+\sqrt{3}\right ) \sqrt [3]{b} x\right )^2}} \sqrt{a+b x^3}}+\frac{81 \left (1+\sqrt{3}\right ) a^3 e \sqrt{e x} \sqrt{a+b x^3} (26 A b-5 a B)}{11648 b^{5/3} \left (\sqrt [3]{a}+\left (1+\sqrt{3}\right ) \sqrt [3]{b} x\right )}+\frac{27 a^2 (e x)^{5/2} \sqrt{a+b x^3} (26 A b-5 a B)}{5824 b e}+\frac{(e x)^{5/2} \left (a+b x^3\right )^{5/2} (26 A b-5 a B)}{260 b e}+\frac{3 a (e x)^{5/2} \left (a+b x^3\right )^{3/2} (26 A b-5 a B)}{728 b e}+\frac{B (e x)^{5/2} \left (a+b x^3\right )^{7/2}}{13 b e} \]

[Out]

(27*a^2*(26*A*b - 5*a*B)*(e*x)^(5/2)*Sqrt[a + b*x^3])/(5824*b*e) + (81*(1 + Sqrt
[3])*a^3*(26*A*b - 5*a*B)*e*Sqrt[e*x]*Sqrt[a + b*x^3])/(11648*b^(5/3)*(a^(1/3) +
(1 + Sqrt[3])*b^(1/3)*x)) + (3*a*(26*A*b - 5*a*B)*(e*x)^(5/2)*(a + b*x^3)^(3/2)
)/(728*b*e) + ((26*A*b - 5*a*B)*(e*x)^(5/2)*(a + b*x^3)^(5/2))/(260*b*e) + (B*(e
*x)^(5/2)*(a + b*x^3)^(7/2))/(13*b*e) - (81*3^(1/4)*a^(10/3)*(26*A*b - 5*a*B)*e*
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])/(11648*b^(5
/3)*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^(10/3)*(26*A*b - 5*a*B)*e*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 + Sqrt[3])/4])/(23296*b^(5/3)*S
qrt[(b^(1/3)*x*(a^(1/3) + b^(1/3)*x))/(a^(1/3) + (1 + Sqrt[3])*b^(1/3)*x)^2]*Sqr
t[a + b*x^3])

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Rubi [A]  time = 1.67917, antiderivative size = 661, 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 \[ -\frac{27\ 3^{3/4} \left (1-\sqrt{3}\right ) a^{10/3} e \sqrt{e x} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\sqrt [3]{a}+\left (1+\sqrt{3}\right ) \sqrt [3]{b} x\right )^2}} (26 A b-5 a B) F\left (\cos ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) \sqrt [3]{b} x+\sqrt [3]{a}}{\left (1+\sqrt{3}\right ) \sqrt [3]{b} x+\sqrt [3]{a}}\right )|\frac{1}{4} \left (2+\sqrt{3}\right )\right )}{23296 b^{5/3} \sqrt{\frac{\sqrt [3]{b} x \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\sqrt [3]{a}+\left (1+\sqrt{3}\right ) \sqrt [3]{b} x\right )^2}} \sqrt{a+b x^3}}-\frac{81 \sqrt [4]{3} a^{10/3} e \sqrt{e x} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\sqrt [3]{a}+\left (1+\sqrt{3}\right ) \sqrt [3]{b} x\right )^2}} (26 A b-5 a B) E\left (\cos ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) \sqrt [3]{b} x+\sqrt [3]{a}}{\left (1+\sqrt{3}\right ) \sqrt [3]{b} x+\sqrt [3]{a}}\right )|\frac{1}{4} \left (2+\sqrt{3}\right )\right )}{11648 b^{5/3} \sqrt{\frac{\sqrt [3]{b} x \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\sqrt [3]{a}+\left (1+\sqrt{3}\right ) \sqrt [3]{b} x\right )^2}} \sqrt{a+b x^3}}+\frac{81 \left (1+\sqrt{3}\right ) a^3 e \sqrt{e x} \sqrt{a+b x^3} (26 A b-5 a B)}{11648 b^{5/3} \left (\sqrt [3]{a}+\left (1+\sqrt{3}\right ) \sqrt [3]{b} x\right )}+\frac{27 a^2 (e x)^{5/2} \sqrt{a+b x^3} (26 A b-5 a B)}{5824 b e}+\frac{(e x)^{5/2} \left (a+b x^3\right )^{5/2} (26 A b-5 a B)}{260 b e}+\frac{3 a (e x)^{5/2} \left (a+b x^3\right )^{3/2} (26 A b-5 a B)}{728 b e}+\frac{B (e x)^{5/2} \left (a+b x^3\right )^{7/2}}{13 b e} \]

Antiderivative was successfully verified.

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

[Out]

(27*a^2*(26*A*b - 5*a*B)*(e*x)^(5/2)*Sqrt[a + b*x^3])/(5824*b*e) + (81*(1 + Sqrt
[3])*a^3*(26*A*b - 5*a*B)*e*Sqrt[e*x]*Sqrt[a + b*x^3])/(11648*b^(5/3)*(a^(1/3) +
(1 + Sqrt[3])*b^(1/3)*x)) + (3*a*(26*A*b - 5*a*B)*(e*x)^(5/2)*(a + b*x^3)^(3/2)
)/(728*b*e) + ((26*A*b - 5*a*B)*(e*x)^(5/2)*(a + b*x^3)^(5/2))/(260*b*e) + (B*(e
*x)^(5/2)*(a + b*x^3)^(7/2))/(13*b*e) - (81*3^(1/4)*a^(10/3)*(26*A*b - 5*a*B)*e*
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])/(11648*b^(5
/3)*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^(10/3)*(26*A*b - 5*a*B)*e*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 + Sqrt[3])/4])/(23296*b^(5/3)*S
qrt[(b^(1/3)*x*(a^(1/3) + b^(1/3)*x))/(a^(1/3) + (1 + Sqrt[3])*b^(1/3)*x)^2]*Sqr
t[a + b*x^3])

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Rubi in Sympy [A]  time = 91.7133, size = 605, normalized size = 0.92 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

B*(e*x)**(5/2)*(a + b*x**3)**(7/2)/(13*b*e) - 81*3**(1/4)*a**(10/3)*e*sqrt(e*x)*
sqrt((a**(2/3) - a**(1/3)*b**(1/3)*x + b**(2/3)*x**2)/(a**(1/3) + b**(1/3)*x*(1
+ sqrt(3)))**2)*(a**(1/3) + b**(1/3)*x)*(26*A*b - 5*B*a)*elliptic_e(acos((a**(1/
3) + b**(1/3)*x*(-sqrt(3) + 1))/(a**(1/3) + b**(1/3)*x*(1 + sqrt(3)))), sqrt(3)/
4 + 1/2)/(11648*b**(5/3)*sqrt(b**(1/3)*x*(a**(1/3) + b**(1/3)*x)/(a**(1/3) + b**
(1/3)*x*(1 + sqrt(3)))**2)*sqrt(a + b*x**3)) - 27*3**(3/4)*a**(10/3)*e*sqrt(e*x)
*sqrt((a**(2/3) - a**(1/3)*b**(1/3)*x + b**(2/3)*x**2)/(a**(1/3) + b**(1/3)*x*(1
+ sqrt(3)))**2)*(-sqrt(3) + 1)*(a**(1/3) + b**(1/3)*x)*(26*A*b - 5*B*a)*ellipti
c_f(acos((a**(1/3) + b**(1/3)*x*(-sqrt(3) + 1))/(a**(1/3) + b**(1/3)*x*(1 + sqrt
(3)))), sqrt(3)/4 + 1/2)/(23296*b**(5/3)*sqrt(b**(1/3)*x*(a**(1/3) + b**(1/3)*x)
/(a**(1/3) + b**(1/3)*x*(1 + sqrt(3)))**2)*sqrt(a + b*x**3)) + a**3*e*sqrt(e*x)*
(81/5824 + 81*sqrt(3)/5824)*sqrt(a + b*x**3)*(26*A*b - 5*B*a)/(2*b**(5/3)*(a**(1
/3) + b**(1/3)*x*(1 + sqrt(3)))) + 27*a**2*(e*x)**(5/2)*sqrt(a + b*x**3)*(26*A*b
- 5*B*a)/(5824*b*e) + 3*a*(e*x)**(5/2)*(a + b*x**3)**(3/2)*(26*A*b - 5*B*a)/(72
8*b*e) + (e*x)**(5/2)*(a + b*x**3)**(5/2)*(26*A*b - 5*B*a)/(260*b*e)

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Mathematica [C]  time = 2.33076, size = 337, normalized size = 0.51 \[ \frac{e^2 \left (2 (-a)^{2/3} b x^3 \left (a+b x^3\right ) \left (a^2 (405 a B+9542 A b)+112 b^2 x^6 (55 a B+26 A b)+8 a b x^3 (625 a B+1118 A b)+2240 b^3 B x^9\right )+135 a^3 (26 A b-5 a B) \left (3 (-a)^{2/3} \left (a+b x^3\right )+(-1)^{2/3} 3^{3/4} a b^{2/3} x^2 \sqrt{\frac{(-1)^{5/6} \left (\sqrt [3]{-a}-\sqrt [3]{b} x\right )}{\sqrt [3]{b} x}} \sqrt{\frac{\frac{(-a)^{2/3}}{b^{2/3}}+\frac{\sqrt [3]{-a} x}{\sqrt [3]{b}}+x^2}{x^2}} \left ((-1)^{5/6} F\left (\sin ^{-1}\left (\frac{\sqrt{-\frac{i \sqrt [3]{-a}}{\sqrt [3]{b} x}-(-1)^{5/6}}}{\sqrt [4]{3}}\right )|\sqrt [3]{-1}\right )+\sqrt{3} E\left (\sin ^{-1}\left (\frac{\sqrt{-\frac{i \sqrt [3]{-a}}{\sqrt [3]{b} x}-(-1)^{5/6}}}{\sqrt [4]{3}}\right )|\sqrt [3]{-1}\right )\right )\right )\right )}{58240 (-a)^{2/3} b^2 \sqrt{e x} \sqrt{a+b x^3}} \]

Warning: Unable to verify antiderivative.

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

[Out]

(e^2*(2*(-a)^(2/3)*b*x^3*(a + b*x^3)*(a^2*(9542*A*b + 405*a*B) + 8*a*b*(1118*A*b
+ 625*a*B)*x^3 + 112*b^2*(26*A*b + 55*a*B)*x^6 + 2240*b^3*B*x^9) + 135*a^3*(26*
A*b - 5*a*B)*(3*(-a)^(2/3)*(a + b*x^3) + (-1)^(2/3)*3^(3/4)*a*b^(2/3)*x^2*Sqrt[(
(-1)^(5/6)*((-a)^(1/3) - b^(1/3)*x))/(b^(1/3)*x)]*Sqrt[((-a)^(2/3)/b^(2/3) + ((-
a)^(1/3)*x)/b^(1/3) + x^2)/x^2]*(Sqrt[3]*EllipticE[ArcSin[Sqrt[-(-1)^(5/6) - (I*
(-a)^(1/3))/(b^(1/3)*x)]/3^(1/4)], (-1)^(1/3)] + (-1)^(5/6)*EllipticF[ArcSin[Sqr
t[-(-1)^(5/6) - (I*(-a)^(1/3))/(b^(1/3)*x)]/3^(1/4)], (-1)^(1/3)]))))/(58240*(-a
)^(2/3)*b^2*Sqrt[e*x]*Sqrt[a + b*x^3])

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Maple [C]  time = 0.076, size = 6202, normalized size = 9.4 \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

result too large to display

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left ({\left (B b^{2} e x^{10} +{\left (2 \, B a b + A b^{2}\right )} e x^{7} +{\left (B a^{2} + 2 \, A a b\right )} e x^{4} + A a^{2} e x\right )} \sqrt{b x^{3} + a} \sqrt{e x}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((B*b^2*e*x^10 + (2*B*a*b + A*b^2)*e*x^7 + (B*a^2 + 2*A*a*b)*e*x^4 + A*a
^2*e*x)*sqrt(b*x^3 + a)*sqrt(e*x), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (B x^{3} + A\right )}{\left (b x^{3} + a\right )}^{\frac{5}{2}} \left (e x\right )^{\frac{3}{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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