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

Optimal. Leaf size=121 \[ -\frac{a e^{7/2} (4 A b-3 a B) \tanh ^{-1}\left (\frac{\sqrt{b} (e x)^{3/2}}{e^{3/2} \sqrt{a+b x^3}}\right )}{12 b^{5/2}}+\frac{e^2 (e x)^{3/2} \sqrt{a+b x^3} (4 A b-3 a B)}{12 b^2}+\frac{B (e x)^{9/2} \sqrt{a+b x^3}}{6 b e} \]

[Out]

((4*A*b - 3*a*B)*e^2*(e*x)^(3/2)*Sqrt[a + b*x^3])/(12*b^2) + (B*(e*x)^(9/2)*Sqrt
[a + b*x^3])/(6*b*e) - (a*(4*A*b - 3*a*B)*e^(7/2)*ArcTanh[(Sqrt[b]*(e*x)^(3/2))/
(e^(3/2)*Sqrt[a + b*x^3])])/(12*b^(5/2))

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Rubi [A]  time = 0.271242, antiderivative size = 121, 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 \[ -\frac{a e^{7/2} (4 A b-3 a B) \tanh ^{-1}\left (\frac{\sqrt{b} (e x)^{3/2}}{e^{3/2} \sqrt{a+b x^3}}\right )}{12 b^{5/2}}+\frac{e^2 (e x)^{3/2} \sqrt{a+b x^3} (4 A b-3 a B)}{12 b^2}+\frac{B (e x)^{9/2} \sqrt{a+b x^3}}{6 b e} \]

Antiderivative was successfully verified.

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

[Out]

((4*A*b - 3*a*B)*e^2*(e*x)^(3/2)*Sqrt[a + b*x^3])/(12*b^2) + (B*(e*x)^(9/2)*Sqrt
[a + b*x^3])/(6*b*e) - (a*(4*A*b - 3*a*B)*e^(7/2)*ArcTanh[(Sqrt[b]*(e*x)^(3/2))/
(e^(3/2)*Sqrt[a + b*x^3])])/(12*b^(5/2))

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Rubi in Sympy [A]  time = 23.5521, size = 110, normalized size = 0.91 \[ \frac{B \left (e x\right )^{\frac{9}{2}} \sqrt{a + b x^{3}}}{6 b e} - \frac{a e^{\frac{7}{2}} \left (4 A b - 3 B a\right ) \operatorname{atanh}{\left (\frac{\sqrt{b} \left (e x\right )^{\frac{3}{2}}}{e^{\frac{3}{2}} \sqrt{a + b x^{3}}} \right )}}{12 b^{\frac{5}{2}}} + \frac{e^{2} \left (e x\right )^{\frac{3}{2}} \sqrt{a + b x^{3}} \left (4 A b - 3 B a\right )}{12 b^{2}} \]

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)**(1/2),x)

[Out]

B*(e*x)**(9/2)*sqrt(a + b*x**3)/(6*b*e) - a*e**(7/2)*(4*A*b - 3*B*a)*atanh(sqrt(
b)*(e*x)**(3/2)/(e**(3/2)*sqrt(a + b*x**3)))/(12*b**(5/2)) + e**2*(e*x)**(3/2)*s
qrt(a + b*x**3)*(4*A*b - 3*B*a)/(12*b**2)

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Mathematica [A]  time = 0.219016, size = 100, normalized size = 0.83 \[ \frac{e^2 (e x)^{3/2} \left (\sqrt{b} \left (a+b x^3\right ) \left (-3 a B+4 A b+2 b B x^3\right )+a \sqrt{\frac{a}{x^3}+b} (3 a B-4 A b) \tanh ^{-1}\left (\frac{\sqrt{\frac{a}{x^3}+b}}{\sqrt{b}}\right )\right )}{12 b^{5/2} \sqrt{a+b x^3}} \]

Antiderivative was successfully verified.

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

[Out]

(e^2*(e*x)^(3/2)*(Sqrt[b]*(a + b*x^3)*(4*A*b - 3*a*B + 2*b*B*x^3) + a*(-4*A*b +
3*a*B)*Sqrt[b + a/x^3]*ArcTanh[Sqrt[b + a/x^3]/Sqrt[b]]))/(12*b^(5/2)*Sqrt[a + b
*x^3])

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Maple [C]  time = 0.067, size = 6861, normalized size = 56.7 \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x)^(7/2)*(B*x^3+A)/(b*x^3+a)^(1/2),x)

[Out]

result too large to display

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.64966, size = 1, normalized size = 0.01 \[ \left [-\frac{{\left (3 \, B a^{2} - 4 \, A a b\right )} e^{3} \sqrt{\frac{e}{b}} \log \left (-8 \, b^{2} e x^{6} - 8 \, a b e x^{3} - a^{2} e + 4 \,{\left (2 \, b^{2} x^{4} + a b x\right )} \sqrt{b x^{3} + a} \sqrt{e x} \sqrt{\frac{e}{b}}\right ) - 4 \,{\left (2 \, B b e^{3} x^{4} -{\left (3 \, B a - 4 \, A b\right )} e^{3} x\right )} \sqrt{b x^{3} + a} \sqrt{e x}}{48 \, b^{2}}, \frac{{\left (3 \, B a^{2} - 4 \, A a b\right )} e^{3} \sqrt{-\frac{e}{b}} \arctan \left (\frac{2 \, \sqrt{b x^{3} + a} \sqrt{e x} x}{{\left (2 \, b x^{3} + a\right )} \sqrt{-\frac{e}{b}}}\right ) + 2 \,{\left (2 \, B b e^{3} x^{4} -{\left (3 \, B a - 4 \, A b\right )} e^{3} x\right )} \sqrt{b x^{3} + a} \sqrt{e x}}{24 \, b^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[-1/48*((3*B*a^2 - 4*A*a*b)*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)*sqrt(e/b)) - 4*(2*B*b*e^3*x^4
- (3*B*a - 4*A*b)*e^3*x)*sqrt(b*x^3 + a)*sqrt(e*x))/b^2, 1/24*((3*B*a^2 - 4*A*a
*b)*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*(2*B*b*e^3*x^4 - (3*B*a - 4*A*b)*e^3*x)*sqrt(b*x^3 + a)*sqrt(e*x))/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)**(1/2),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)/sqrt(b*x^3 + a),x, algorithm="giac")

[Out]

Timed out