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

Optimal. Leaf size=67 \[ -\frac{2 (e x)^{3/2} (2 A b-a B)}{3 a^2 e^4 \sqrt{a+b x^3}}-\frac{2 A}{3 a e (e x)^{3/2} \sqrt{a+b x^3}} \]

[Out]

(-2*A)/(3*a*e*(e*x)^(3/2)*Sqrt[a + b*x^3]) - (2*(2*A*b - a*B)*(e*x)^(3/2))/(3*a^
2*e^4*Sqrt[a + b*x^3])

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Rubi [A]  time = 0.110773, antiderivative size = 67, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077 \[ -\frac{2 (e x)^{3/2} (2 A b-a B)}{3 a^2 e^4 \sqrt{a+b x^3}}-\frac{2 A}{3 a e (e x)^{3/2} \sqrt{a+b x^3}} \]

Antiderivative was successfully verified.

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

[Out]

(-2*A)/(3*a*e*(e*x)^(3/2)*Sqrt[a + b*x^3]) - (2*(2*A*b - a*B)*(e*x)^(3/2))/(3*a^
2*e^4*Sqrt[a + b*x^3])

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Rubi in Sympy [A]  time = 9.81044, size = 61, normalized size = 0.91 \[ - \frac{2 A}{3 a e \left (e x\right )^{\frac{3}{2}} \sqrt{a + b x^{3}}} - \frac{4 \left (e x\right )^{\frac{3}{2}} \left (A b - \frac{B a}{2}\right )}{3 a^{2} e^{4} \sqrt{a + b x^{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*A/(3*a*e*(e*x)**(3/2)*sqrt(a + b*x**3)) - 4*(e*x)**(3/2)*(A*b - B*a/2)/(3*a**
2*e**4*sqrt(a + b*x**3))

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Mathematica [A]  time = 0.0706414, size = 45, normalized size = 0.67 \[ \frac{x \left (-2 a A+2 a B x^3-4 A b x^3\right )}{3 a^2 (e x)^{5/2} \sqrt{a+b x^3}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.009, size = 39, normalized size = 0.6 \[ -{\frac{2\,x \left ( 2\,A{x}^{3}b-Ba{x}^{3}+Aa \right ) }{3\,{a}^{2}}{\frac{1}{\sqrt{b{x}^{3}+a}}} \left ( ex \right ) ^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/3*x*(2*A*b*x^3-B*a*x^3+A*a)/(b*x^3+a)^(1/2)/a^2/(e*x)^(5/2)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.214847, size = 77, normalized size = 1.15 \[ \frac{2 \,{\left ({\left (B a - 2 \, A b\right )} x^{3} - A a\right )} \sqrt{b x^{3} + a} \sqrt{e x}}{3 \,{\left (a^{2} b e^{3} x^{5} + a^{3} e^{3} x^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/3*((B*a - 2*A*b)*x^3 - A*a)*sqrt(b*x^3 + a)*sqrt(e*x)/(a^2*b*e^3*x^5 + a^3*e^3
*x^2)

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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