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

Optimal. Leaf size=553 \[ \frac{\left (1-\sqrt{3}\right ) e \sqrt{e x} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\sqrt [3]{a}+\left (1+\sqrt{3}\right ) \sqrt [3]{b} x\right )^2}} (2 A b-5 a B) F\left (\cos ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) \sqrt [3]{b} x+\sqrt [3]{a}}{\left (1+\sqrt{3}\right ) \sqrt [3]{b} x+\sqrt [3]{a}}\right )|\frac{1}{4} \left (2+\sqrt{3}\right )\right )}{6 \sqrt [4]{3} a^{2/3} b^{5/3} \sqrt{\frac{\sqrt [3]{b} x \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\sqrt [3]{a}+\left (1+\sqrt{3}\right ) \sqrt [3]{b} x\right )^2}} \sqrt{a+b x^3}}+\frac{e \sqrt{e x} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\sqrt [3]{a}+\left (1+\sqrt{3}\right ) \sqrt [3]{b} x\right )^2}} (2 A b-5 a B) E\left (\cos ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) \sqrt [3]{b} x+\sqrt [3]{a}}{\left (1+\sqrt{3}\right ) \sqrt [3]{b} x+\sqrt [3]{a}}\right )|\frac{1}{4} \left (2+\sqrt{3}\right )\right )}{3^{3/4} a^{2/3} b^{5/3} \sqrt{\frac{\sqrt [3]{b} x \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\sqrt [3]{a}+\left (1+\sqrt{3}\right ) \sqrt [3]{b} x\right )^2}} \sqrt{a+b x^3}}-\frac{\left (1+\sqrt{3}\right ) e \sqrt{e x} \sqrt{a+b x^3} (2 A b-5 a B)}{3 a b^{5/3} \left (\sqrt [3]{a}+\left (1+\sqrt{3}\right ) \sqrt [3]{b} x\right )}+\frac{2 (e x)^{5/2} (A b-a B)}{3 a b e \sqrt{a+b x^3}} \]

[Out]

(2*(A*b - a*B)*(e*x)^(5/2))/(3*a*b*e*Sqrt[a + b*x^3]) - ((1 + Sqrt[3])*(2*A*b -
5*a*B)*e*Sqrt[e*x]*Sqrt[a + b*x^3])/(3*a*b^(5/3)*(a^(1/3) + (1 + Sqrt[3])*b^(1/3
)*x)) + ((2*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[Arc
Cos[(a^(1/3) + (1 - Sqrt[3])*b^(1/3)*x)/(a^(1/3) + (1 + Sqrt[3])*b^(1/3)*x)], (2
+ Sqrt[3])/4])/(3^(3/4)*a^(2/3)*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]) + ((1 - Sqrt[3])*(2*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])/
(6*3^(1/4)*a^(2/3)*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])

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Rubi [A]  time = 1.20035, antiderivative size = 553, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.192 \[ \frac{\left (1-\sqrt{3}\right ) e \sqrt{e x} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\sqrt [3]{a}+\left (1+\sqrt{3}\right ) \sqrt [3]{b} x\right )^2}} (2 A b-5 a B) F\left (\cos ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) \sqrt [3]{b} x+\sqrt [3]{a}}{\left (1+\sqrt{3}\right ) \sqrt [3]{b} x+\sqrt [3]{a}}\right )|\frac{1}{4} \left (2+\sqrt{3}\right )\right )}{6 \sqrt [4]{3} a^{2/3} b^{5/3} \sqrt{\frac{\sqrt [3]{b} x \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\sqrt [3]{a}+\left (1+\sqrt{3}\right ) \sqrt [3]{b} x\right )^2}} \sqrt{a+b x^3}}+\frac{e \sqrt{e x} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\sqrt [3]{a}+\left (1+\sqrt{3}\right ) \sqrt [3]{b} x\right )^2}} (2 A b-5 a B) E\left (\cos ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) \sqrt [3]{b} x+\sqrt [3]{a}}{\left (1+\sqrt{3}\right ) \sqrt [3]{b} x+\sqrt [3]{a}}\right )|\frac{1}{4} \left (2+\sqrt{3}\right )\right )}{3^{3/4} a^{2/3} b^{5/3} \sqrt{\frac{\sqrt [3]{b} x \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\sqrt [3]{a}+\left (1+\sqrt{3}\right ) \sqrt [3]{b} x\right )^2}} \sqrt{a+b x^3}}-\frac{\left (1+\sqrt{3}\right ) e \sqrt{e x} \sqrt{a+b x^3} (2 A b-5 a B)}{3 a b^{5/3} \left (\sqrt [3]{a}+\left (1+\sqrt{3}\right ) \sqrt [3]{b} x\right )}+\frac{2 (e x)^{5/2} (A b-a B)}{3 a b e \sqrt{a+b x^3}} \]

Antiderivative was successfully verified.

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

[Out]

(2*(A*b - a*B)*(e*x)^(5/2))/(3*a*b*e*Sqrt[a + b*x^3]) - ((1 + Sqrt[3])*(2*A*b -
5*a*B)*e*Sqrt[e*x]*Sqrt[a + b*x^3])/(3*a*b^(5/3)*(a^(1/3) + (1 + Sqrt[3])*b^(1/3
)*x)) + ((2*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[Arc
Cos[(a^(1/3) + (1 - Sqrt[3])*b^(1/3)*x)/(a^(1/3) + (1 + Sqrt[3])*b^(1/3)*x)], (2
+ Sqrt[3])/4])/(3^(3/4)*a^(2/3)*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]) + ((1 - Sqrt[3])*(2*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])/
(6*3^(1/4)*a^(2/3)*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])

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Rubi in Sympy [A]  time = 63.85, size = 503, normalized size = 0.91 \[ \frac{2 \left (e x\right )^{\frac{5}{2}} \left (A b - B a\right )}{3 a b e \sqrt{a + b x^{3}}} - \frac{e \sqrt{e x} \left (\frac{2}{3} + \frac{2 \sqrt{3}}{3}\right ) \sqrt{a + b x^{3}} \left (2 A b - 5 B a\right )}{2 a b^{\frac{5}{3}} \left (\sqrt [3]{a} + \sqrt [3]{b} x \left (1 + \sqrt{3}\right )\right )} + \frac{\sqrt [4]{3} e \sqrt{e x} \sqrt{\frac{a^{\frac{2}{3}} - \sqrt [3]{a} \sqrt [3]{b} x + b^{\frac{2}{3}} x^{2}}{\left (\sqrt [3]{a} + \sqrt [3]{b} x \left (1 + \sqrt{3}\right )\right )^{2}}} \left (\sqrt [3]{a} + \sqrt [3]{b} x\right ) \left (2 A b - 5 B a\right ) E\left (\operatorname{acos}{\left (\frac{\sqrt [3]{a} + \sqrt [3]{b} x \left (- \sqrt{3} + 1\right )}{\sqrt [3]{a} + \sqrt [3]{b} x \left (1 + \sqrt{3}\right )} \right )}\middle | \frac{\sqrt{3}}{4} + \frac{1}{2}\right )}{3 a^{\frac{2}{3}} b^{\frac{5}{3}} \sqrt{\frac{\sqrt [3]{b} x \left (\sqrt [3]{a} + \sqrt [3]{b} x\right )}{\left (\sqrt [3]{a} + \sqrt [3]{b} x \left (1 + \sqrt{3}\right )\right )^{2}}} \sqrt{a + b x^{3}}} + \frac{3^{\frac{3}{4}} e \sqrt{e x} \sqrt{\frac{a^{\frac{2}{3}} - \sqrt [3]{a} \sqrt [3]{b} x + b^{\frac{2}{3}} x^{2}}{\left (\sqrt [3]{a} + \sqrt [3]{b} x \left (1 + \sqrt{3}\right )\right )^{2}}} \left (- \sqrt{3} + 1\right ) \left (\sqrt [3]{a} + \sqrt [3]{b} x\right ) \left (2 A b - 5 B a\right ) F\left (\operatorname{acos}{\left (\frac{\sqrt [3]{a} + \sqrt [3]{b} x \left (- \sqrt{3} + 1\right )}{\sqrt [3]{a} + \sqrt [3]{b} x \left (1 + \sqrt{3}\right )} \right )}\middle | \frac{\sqrt{3}}{4} + \frac{1}{2}\right )}{18 a^{\frac{2}{3}} b^{\frac{5}{3}} \sqrt{\frac{\sqrt [3]{b} x \left (\sqrt [3]{a} + \sqrt [3]{b} x\right )}{\left (\sqrt [3]{a} + \sqrt [3]{b} x \left (1 + \sqrt{3}\right )\right )^{2}}} \sqrt{a + b x^{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*(e*x)**(5/2)*(A*b - B*a)/(3*a*b*e*sqrt(a + b*x**3)) - e*sqrt(e*x)*(2/3 + 2*sqr
t(3)/3)*sqrt(a + b*x**3)*(2*A*b - 5*B*a)/(2*a*b**(5/3)*(a**(1/3) + b**(1/3)*x*(1
+ sqrt(3)))) + 3**(1/4)*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)*(2*
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)/(3*a**(2/3)*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)) + 3**(3/4)*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)*(2*A*b - 5*B*a)*elliptic_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)/(18*a**(2/3)*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))

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Mathematica [C]  time = 5.23835, size = 266, normalized size = 0.48 \[ \frac{x (e x)^{3/2} \left (6 b (A b-a B)-(5 a B-2 A b) \left (-3 \left (\frac{a}{x^3}+b\right )+\frac{\sqrt [6]{-1} 3^{3/4} a b^{2/3} \sqrt{\frac{(-1)^{5/6} \left (\sqrt [3]{-a}-\sqrt [3]{b} x\right )}{\sqrt [3]{b} x}} \sqrt{\frac{\frac{(-a)^{2/3}}{b^{2/3}}+\frac{\sqrt [3]{-a} x}{\sqrt [3]{b}}+x^2}{x^2}} \left (\sqrt [3]{-1} F\left (\sin ^{-1}\left (\frac{\sqrt{-\frac{i \sqrt [3]{-a}}{\sqrt [3]{b} x}-(-1)^{5/6}}}{\sqrt [4]{3}}\right )|\sqrt [3]{-1}\right )-i \sqrt{3} E\left (\sin ^{-1}\left (\frac{\sqrt{-\frac{i \sqrt [3]{-a}}{\sqrt [3]{b} x}-(-1)^{5/6}}}{\sqrt [4]{3}}\right )|\sqrt [3]{-1}\right )\right )}{(-a)^{2/3} x}\right )\right )}{9 a b^2 \sqrt{a+b x^3}} \]

Warning: Unable to verify antiderivative.

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

[Out]

(x*(e*x)^(3/2)*(6*b*(A*b - a*B) - (-2*A*b + 5*a*B)*(-3*(b + a/x^3) + ((-1)^(1/6)
*3^(3/4)*a*b^(2/3)*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]*((-I)*Sqrt[3]*EllipticE
[ArcSin[Sqrt[-(-1)^(5/6) - (I*(-a)^(1/3))/(b^(1/3)*x)]/3^(1/4)], (-1)^(1/3)] + (
-1)^(1/3)*EllipticF[ArcSin[Sqrt[-(-1)^(5/6) - (I*(-a)^(1/3))/(b^(1/3)*x)]/3^(1/4
)], (-1)^(1/3)]))/((-a)^(2/3)*x))))/(9*a*b^2*Sqrt[a + b*x^3])

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Maple [C]  time = 0.076, size = 5392, normalized size = 9.8 \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

result too large to display

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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