3.559 \(\int \frac{(e x)^{7/2} \left (A+B x^3\right )}{\left (a+b x^3\right )^{5/2}} \, dx\)

Optimal. Leaf size=114 \[ \frac{2 (e x)^{9/2} (A b-a B)}{9 a b e \left (a+b x^3\right )^{3/2}}+\frac{2 B e^{7/2} \tanh ^{-1}\left (\frac{\sqrt{b} (e x)^{3/2}}{e^{3/2} \sqrt{a+b x^3}}\right )}{3 b^{5/2}}-\frac{2 B e^2 (e x)^{3/2}}{3 b^2 \sqrt{a+b x^3}} \]

[Out]

(2*(A*b - a*B)*(e*x)^(9/2))/(9*a*b*e*(a + b*x^3)^(3/2)) - (2*B*e^2*(e*x)^(3/2))/
(3*b^2*Sqrt[a + b*x^3]) + (2*B*e^(7/2)*ArcTanh[(Sqrt[b]*(e*x)^(3/2))/(e^(3/2)*Sq
rt[a + b*x^3])])/(3*b^(5/2))

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Rubi [A]  time = 0.232017, antiderivative size = 114, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ \frac{2 (e x)^{9/2} (A b-a B)}{9 a b e \left (a+b x^3\right )^{3/2}}+\frac{2 B e^{7/2} \tanh ^{-1}\left (\frac{\sqrt{b} (e x)^{3/2}}{e^{3/2} \sqrt{a+b x^3}}\right )}{3 b^{5/2}}-\frac{2 B e^2 (e x)^{3/2}}{3 b^2 \sqrt{a+b x^3}} \]

Antiderivative was successfully verified.

[In]  Int[((e*x)^(7/2)*(A + B*x^3))/(a + b*x^3)^(5/2),x]

[Out]

(2*(A*b - a*B)*(e*x)^(9/2))/(9*a*b*e*(a + b*x^3)^(3/2)) - (2*B*e^2*(e*x)^(3/2))/
(3*b^2*Sqrt[a + b*x^3]) + (2*B*e^(7/2)*ArcTanh[(Sqrt[b]*(e*x)^(3/2))/(e^(3/2)*Sq
rt[a + b*x^3])])/(3*b^(5/2))

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Rubi in Sympy [A]  time = 24.381, size = 104, normalized size = 0.91 \[ - \frac{2 B e^{2} \left (e x\right )^{\frac{3}{2}}}{3 b^{2} \sqrt{a + b x^{3}}} + \frac{2 B e^{\frac{7}{2}} \operatorname{atanh}{\left (\frac{\sqrt{b} \left (e x\right )^{\frac{3}{2}}}{e^{\frac{3}{2}} \sqrt{a + b x^{3}}} \right )}}{3 b^{\frac{5}{2}}} + \frac{2 \left (e x\right )^{\frac{9}{2}} \left (A b - B a\right )}{9 a b e \left (a + b x^{3}\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x)**(7/2)*(B*x**3+A)/(b*x**3+a)**(5/2),x)

[Out]

-2*B*e**2*(e*x)**(3/2)/(3*b**2*sqrt(a + b*x**3)) + 2*B*e**(7/2)*atanh(sqrt(b)*(e
*x)**(3/2)/(e**(3/2)*sqrt(a + b*x**3)))/(3*b**(5/2)) + 2*(e*x)**(9/2)*(A*b - B*a
)/(9*a*b*e*(a + b*x**3)**(3/2))

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Mathematica [A]  time = 0.321678, size = 99, normalized size = 0.87 \[ \frac{2 e^3 \sqrt{e x} \left (\frac{\sqrt{b} x^{3/2} \left (-3 a^2 B-4 a b B x^3+A b^2 x^3\right )}{a \left (a+b x^3\right )^{3/2}}+3 B \tanh ^{-1}\left (\frac{\sqrt{b} x^{3/2}}{\sqrt{a+b x^3}}\right )\right )}{9 b^{5/2} \sqrt{x}} \]

Antiderivative was successfully verified.

[In]  Integrate[((e*x)^(7/2)*(A + B*x^3))/(a + b*x^3)^(5/2),x]

[Out]

(2*e^3*Sqrt[e*x]*((Sqrt[b]*x^(3/2)*(-3*a^2*B + A*b^2*x^3 - 4*a*b*B*x^3))/(a*(a +
b*x^3)^(3/2)) + 3*B*ArcTanh[(Sqrt[b]*x^(3/2))/Sqrt[a + b*x^3]]))/(9*b^(5/2)*Sqr
t[x])

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Maple [C]  time = 0.086, size = 7081, normalized size = 62.1 \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x)^(7/2)*(B*x^3+A)/(b*x^3+a)^(5/2),x)

[Out]

result too large to display

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (B x^{3} + A\right )} \left (e x\right )^{\frac{7}{2}}}{{\left (b x^{3} + a\right )}^{\frac{5}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^3 + A)*(e*x)^(7/2)/(b*x^3 + a)^(5/2),x, algorithm="maxima")

[Out]

integrate((B*x^3 + A)*(e*x)^(7/2)/(b*x^3 + a)^(5/2), x)

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Fricas [A]  time = 0.401526, size = 1, normalized size = 0.01 \[ \left [\frac{3 \,{\left (B a b^{2} e^{3} x^{6} + 2 \, B a^{2} b e^{3} x^{3} + B a^{3} e^{3}\right )} \sqrt{\frac{e}{b}} \log \left (-8 \, b^{2} e x^{6} - 8 \, a b e x^{3} - a^{2} e - 4 \,{\left (2 \, b^{2} x^{4} + a b x\right )} \sqrt{b x^{3} + a} \sqrt{e x} \sqrt{\frac{e}{b}}\right ) - 4 \,{\left ({\left (4 \, B a b - A b^{2}\right )} e^{3} x^{4} + 3 \, B a^{2} e^{3} x\right )} \sqrt{b x^{3} + a} \sqrt{e x}}{18 \,{\left (a b^{4} x^{6} + 2 \, a^{2} b^{3} x^{3} + a^{3} b^{2}\right )}}, \frac{3 \,{\left (B a b^{2} e^{3} x^{6} + 2 \, B a^{2} b e^{3} x^{3} + B a^{3} e^{3}\right )} \sqrt{-\frac{e}{b}} \arctan \left (\frac{2 \, \sqrt{b x^{3} + a} \sqrt{e x} x}{{\left (2 \, b x^{3} + a\right )} \sqrt{-\frac{e}{b}}}\right ) - 2 \,{\left ({\left (4 \, B a b - A b^{2}\right )} e^{3} x^{4} + 3 \, B a^{2} e^{3} x\right )} \sqrt{b x^{3} + a} \sqrt{e x}}{9 \,{\left (a b^{4} x^{6} + 2 \, a^{2} b^{3} x^{3} + a^{3} b^{2}\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^3 + A)*(e*x)^(7/2)/(b*x^3 + a)^(5/2),x, algorithm="fricas")

[Out]

[1/18*(3*(B*a*b^2*e^3*x^6 + 2*B*a^2*b*e^3*x^3 + B*a^3*e^3)*sqrt(e/b)*log(-8*b^2*
e*x^6 - 8*a*b*e*x^3 - a^2*e - 4*(2*b^2*x^4 + a*b*x)*sqrt(b*x^3 + a)*sqrt(e*x)*sq
rt(e/b)) - 4*((4*B*a*b - A*b^2)*e^3*x^4 + 3*B*a^2*e^3*x)*sqrt(b*x^3 + a)*sqrt(e*
x))/(a*b^4*x^6 + 2*a^2*b^3*x^3 + a^3*b^2), 1/9*(3*(B*a*b^2*e^3*x^6 + 2*B*a^2*b*e
^3*x^3 + B*a^3*e^3)*sqrt(-e/b)*arctan(2*sqrt(b*x^3 + a)*sqrt(e*x)*x/((2*b*x^3 +
a)*sqrt(-e/b))) - 2*((4*B*a*b - A*b^2)*e^3*x^4 + 3*B*a^2*e^3*x)*sqrt(b*x^3 + a)*
sqrt(e*x))/(a*b^4*x^6 + 2*a^2*b^3*x^3 + a^3*b^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((e*x)**(7/2)*(B*x**3+A)/(b*x**3+a)**(5/2),x)

[Out]

Timed out

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GIAC/XCAS [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)^(7/2)/(b*x^3 + a)^(5/2),x, algorithm="giac")

[Out]

Timed out