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

Optimal. Leaf size=104 \[ -\frac{4 (e x)^{3/2} (4 A b-a B)}{9 a^3 e^4 \sqrt{a+b x^3}}-\frac{2 (e x)^{3/2} (4 A b-a B)}{9 a^2 e^4 \left (a+b x^3\right )^{3/2}}-\frac{2 A}{3 a e (e x)^{3/2} \left (a+b x^3\right )^{3/2}} \]

[Out]

(-2*A)/(3*a*e*(e*x)^(3/2)*(a + b*x^3)^(3/2)) - (2*(4*A*b - a*B)*(e*x)^(3/2))/(9*
a^2*e^4*(a + b*x^3)^(3/2)) - (4*(4*A*b - a*B)*(e*x)^(3/2))/(9*a^3*e^4*Sqrt[a + b
*x^3])

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Rubi [A]  time = 0.15937, antiderivative size = 104, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.115 \[ -\frac{4 (e x)^{3/2} (4 A b-a B)}{9 a^3 e^4 \sqrt{a+b x^3}}-\frac{2 (e x)^{3/2} (4 A b-a B)}{9 a^2 e^4 \left (a+b x^3\right )^{3/2}}-\frac{2 A}{3 a e (e x)^{3/2} \left (a+b x^3\right )^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

(-2*A)/(3*a*e*(e*x)^(3/2)*(a + b*x^3)^(3/2)) - (2*(4*A*b - a*B)*(e*x)^(3/2))/(9*
a^2*e^4*(a + b*x^3)^(3/2)) - (4*(4*A*b - a*B)*(e*x)^(3/2))/(9*a^3*e^4*Sqrt[a + b
*x^3])

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Rubi in Sympy [A]  time = 14.19, size = 97, normalized size = 0.93 \[ - \frac{2 A}{3 a e \left (e x\right )^{\frac{3}{2}} \left (a + b x^{3}\right )^{\frac{3}{2}}} - \frac{2 \left (e x\right )^{\frac{3}{2}} \left (4 A b - B a\right )}{9 a^{2} e^{4} \left (a + b x^{3}\right )^{\frac{3}{2}}} - \frac{4 \left (e x\right )^{\frac{3}{2}} \left (4 A b - B a\right )}{9 a^{3} e^{4} \sqrt{a + b x^{3}}} \]

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)**(5/2),x)

[Out]

-2*A/(3*a*e*(e*x)**(3/2)*(a + b*x**3)**(3/2)) - 2*(e*x)**(3/2)*(4*A*b - B*a)/(9*
a**2*e**4*(a + b*x**3)**(3/2)) - 4*(e*x)**(3/2)*(4*A*b - B*a)/(9*a**3*e**4*sqrt(
a + b*x**3))

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Mathematica [A]  time = 0.0997221, size = 65, normalized size = 0.62 \[ \frac{x \left (-6 a^2 \left (A-B x^3\right )+4 a b x^3 \left (B x^3-6 A\right )-16 A b^2 x^6\right )}{9 a^3 (e x)^{5/2} \left (a+b x^3\right )^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

(x*(-16*A*b^2*x^6 - 6*a^2*(A - B*x^3) + 4*a*b*x^3*(-6*A + B*x^3)))/(9*a^3*(e*x)^
(5/2)*(a + b*x^3)^(3/2))

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Maple [A]  time = 0.009, size = 62, normalized size = 0.6 \[ -{\frac{2\,x \left ( 8\,A{b}^{2}{x}^{6}-2\,Bab{x}^{6}+12\,aAb{x}^{3}-3\,B{a}^{2}{x}^{3}+3\,A{a}^{2} \right ) }{9\,{a}^{3}} \left ( b{x}^{3}+a \right ) ^{-{\frac{3}{2}}} \left ( ex \right ) ^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/9*x*(8*A*b^2*x^6-2*B*a*b*x^6+12*A*a*b*x^3-3*B*a^2*x^3+3*A*a^2)/(b*x^3+a)^(3/2
)/a^3/(e*x)^(5/2)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.214399, size = 126, normalized size = 1.21 \[ \frac{2 \,{\left (2 \,{\left (B a b - 4 \, A b^{2}\right )} x^{6} + 3 \,{\left (B a^{2} - 4 \, A a b\right )} x^{3} - 3 \, A a^{2}\right )} \sqrt{b x^{3} + a} \sqrt{e x}}{9 \,{\left (a^{3} b^{2} e^{3} x^{8} + 2 \, a^{4} b e^{3} x^{5} + a^{5} e^{3} x^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/9*(2*(B*a*b - 4*A*b^2)*x^6 + 3*(B*a^2 - 4*A*a*b)*x^3 - 3*A*a^2)*sqrt(b*x^3 + a
)*sqrt(e*x)/(a^3*b^2*e^3*x^8 + 2*a^4*b*e^3*x^5 + a^5*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)**(5/2),x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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