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

Optimal. Leaf size=161 \[ -\frac{a^2 e^{7/2} (2 A b-a B) \tanh ^{-1}\left (\frac{\sqrt{b} (e x)^{3/2}}{e^{3/2} \sqrt{a+b x^3}}\right )}{24 b^{5/2}}+\frac{a e^2 (e x)^{3/2} \sqrt{a+b x^3} (2 A b-a B)}{24 b^2}+\frac{(e x)^{9/2} \sqrt{a+b x^3} (2 A b-a B)}{12 b e}+\frac{B (e x)^{9/2} \left (a+b x^3\right )^{3/2}}{9 b e} \]

[Out]

(a*(2*A*b - a*B)*e^2*(e*x)^(3/2)*Sqrt[a + b*x^3])/(24*b^2) + ((2*A*b - a*B)*(e*x
)^(9/2)*Sqrt[a + b*x^3])/(12*b*e) + (B*(e*x)^(9/2)*(a + b*x^3)^(3/2))/(9*b*e) -
(a^2*(2*A*b - a*B)*e^(7/2)*ArcTanh[(Sqrt[b]*(e*x)^(3/2))/(e^(3/2)*Sqrt[a + b*x^3
])])/(24*b^(5/2))

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Rubi [A]  time = 0.341719, antiderivative size = 161, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.269 \[ -\frac{a^2 e^{7/2} (2 A b-a B) \tanh ^{-1}\left (\frac{\sqrt{b} (e x)^{3/2}}{e^{3/2} \sqrt{a+b x^3}}\right )}{24 b^{5/2}}+\frac{a e^2 (e x)^{3/2} \sqrt{a+b x^3} (2 A b-a B)}{24 b^2}+\frac{(e x)^{9/2} \sqrt{a+b x^3} (2 A b-a B)}{12 b e}+\frac{B (e x)^{9/2} \left (a+b x^3\right )^{3/2}}{9 b e} \]

Antiderivative was successfully verified.

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

[Out]

(a*(2*A*b - a*B)*e^2*(e*x)^(3/2)*Sqrt[a + b*x^3])/(24*b^2) + ((2*A*b - a*B)*(e*x
)^(9/2)*Sqrt[a + b*x^3])/(12*b*e) + (B*(e*x)^(9/2)*(a + b*x^3)^(3/2))/(9*b*e) -
(a^2*(2*A*b - a*B)*e^(7/2)*ArcTanh[(Sqrt[b]*(e*x)^(3/2))/(e^(3/2)*Sqrt[a + b*x^3
])])/(24*b^(5/2))

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Rubi in Sympy [A]  time = 28.9976, size = 141, normalized size = 0.88 \[ \frac{B \left (e x\right )^{\frac{9}{2}} \left (a + b x^{3}\right )^{\frac{3}{2}}}{9 b e} - \frac{a^{2} e^{\frac{7}{2}} \left (A b - \frac{B a}{2}\right ) \operatorname{atanh}{\left (\frac{\sqrt{b} \left (e x\right )^{\frac{3}{2}}}{e^{\frac{3}{2}} \sqrt{a + b x^{3}}} \right )}}{12 b^{\frac{5}{2}}} + \frac{a e^{2} \left (e x\right )^{\frac{3}{2}} \sqrt{a + b x^{3}} \left (2 A b - B a\right )}{24 b^{2}} + \frac{\left (e x\right )^{\frac{9}{2}} \sqrt{a + b x^{3}} \left (A b - \frac{B a}{2}\right )}{6 b e} \]

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)*(a + b*x**3)**(3/2)/(9*b*e) - a**2*e**(7/2)*(A*b - B*a/2)*atanh(s
qrt(b)*(e*x)**(3/2)/(e**(3/2)*sqrt(a + b*x**3)))/(12*b**(5/2)) + a*e**2*(e*x)**(
3/2)*sqrt(a + b*x**3)*(2*A*b - B*a)/(24*b**2) + (e*x)**(9/2)*sqrt(a + b*x**3)*(A
*b - B*a/2)/(6*b*e)

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Mathematica [A]  time = 0.264325, size = 124, normalized size = 0.77 \[ \frac{e^2 (e x)^{3/2} \left (\sqrt{b} \left (a+b x^3\right ) \left (-3 a^2 B+2 a b \left (3 A+B x^3\right )+4 b^2 x^3 \left (3 A+2 B x^3\right )\right )+3 a^2 \sqrt{\frac{a}{x^3}+b} (a B-2 A b) \tanh ^{-1}\left (\frac{\sqrt{\frac{a}{x^3}+b}}{\sqrt{b}}\right )\right )}{72 b^{5/2} \sqrt{a+b x^3}} \]

Antiderivative was successfully verified.

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

[Out]

(e^2*(e*x)^(3/2)*(Sqrt[b]*(a + b*x^3)*(-3*a^2*B + 2*a*b*(3*A + B*x^3) + 4*b^2*x^
3*(3*A + 2*B*x^3)) + 3*a^2*(-2*A*b + a*B)*Sqrt[b + a/x^3]*ArcTanh[Sqrt[b + a/x^3
]/Sqrt[b]]))/(72*b^(5/2)*Sqrt[a + b*x^3])

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Maple [C]  time = 0.312, size = 7293, normalized size = 45.3 \[ \text{output too large to display} \]

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)*sqrt(b*x^3 + a)*(e*x)^(7/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.692477, size = 1, normalized size = 0.01 \[ \left [-\frac{3 \,{\left (B a^{3} - 2 \, A a^{2} b\right )} e^{3} \sqrt{\frac{e}{b}} \log \left (-8 \, b^{2} e x^{6} - 8 \, a b e x^{3} - a^{2} e + 4 \,{\left (2 \, b^{2} x^{4} + a b x\right )} \sqrt{b x^{3} + a} \sqrt{e x} \sqrt{\frac{e}{b}}\right ) - 4 \,{\left (8 \, B b^{2} e^{3} x^{7} + 2 \,{\left (B a b + 6 \, A b^{2}\right )} e^{3} x^{4} - 3 \,{\left (B a^{2} - 2 \, A a b\right )} e^{3} x\right )} \sqrt{b x^{3} + a} \sqrt{e x}}{288 \, b^{2}}, \frac{3 \,{\left (B a^{3} - 2 \, A a^{2} b\right )} e^{3} \sqrt{-\frac{e}{b}} \arctan \left (\frac{2 \, \sqrt{b x^{3} + a} \sqrt{e x} x}{{\left (2 \, b x^{3} + a\right )} \sqrt{-\frac{e}{b}}}\right ) + 2 \,{\left (8 \, B b^{2} e^{3} x^{7} + 2 \,{\left (B a b + 6 \, A b^{2}\right )} e^{3} x^{4} - 3 \,{\left (B a^{2} - 2 \, A a b\right )} e^{3} x\right )} \sqrt{b x^{3} + a} \sqrt{e x}}{144 \, b^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[-1/288*(3*(B*a^3 - 2*A*a^2*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*(8*B*b^2*e^
3*x^7 + 2*(B*a*b + 6*A*b^2)*e^3*x^4 - 3*(B*a^2 - 2*A*a*b)*e^3*x)*sqrt(b*x^3 + a)
*sqrt(e*x))/b^2, 1/144*(3*(B*a^3 - 2*A*a^2*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*(8*B*b^2*e^3*x^7 + 2*(B*a*b +
6*A*b^2)*e^3*x^4 - 3*(B*a^2 - 2*A*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]  time = 0., size = 0, normalized size = 0. \[ \int{\left (B x^{3} + A\right )} \sqrt{b x^{3} + a} \left (e x\right )^{\frac{7}{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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