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

Optimal. Leaf size=83 \[ \frac{\sqrt{e} (2 A b-a B) \tanh ^{-1}\left (\frac{\sqrt{b} (e x)^{3/2}}{e^{3/2} \sqrt{a+b x^3}}\right )}{3 b^{3/2}}+\frac{B (e x)^{3/2} \sqrt{a+b x^3}}{3 b e} \]

[Out]

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

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Rubi [A]  time = 0.202006, antiderivative size = 83, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.192 \[ \frac{\sqrt{e} (2 A b-a B) \tanh ^{-1}\left (\frac{\sqrt{b} (e x)^{3/2}}{e^{3/2} \sqrt{a+b x^3}}\right )}{3 b^{3/2}}+\frac{B (e x)^{3/2} \sqrt{a+b x^3}}{3 b e} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 19.0747, size = 73, normalized size = 0.88 \[ \frac{B \left (e x\right )^{\frac{3}{2}} \sqrt{a + b x^{3}}}{3 b e} + \frac{2 \sqrt{e} \left (A b - \frac{B a}{2}\right ) \operatorname{atanh}{\left (\frac{\sqrt{b} \left (e x\right )^{\frac{3}{2}}}{e^{\frac{3}{2}} \sqrt{a + b x^{3}}} \right )}}{3 b^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.170303, size = 82, normalized size = 0.99 \[ \frac{x \sqrt{e x} \left (\sqrt{\frac{a}{x^3}+b} (2 A b-a B) \tanh ^{-1}\left (\frac{\sqrt{\frac{a}{x^3}+b}}{\sqrt{b}}\right )+\sqrt{b} B \left (a+b x^3\right )\right )}{3 b^{3/2} \sqrt{a+b x^3}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [C]  time = 0.04, size = 6424, normalized size = 77.4 \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x^3+A)*(e*x)^(1/2)/(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(e*x)/sqrt(b*x^3 + a),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.634202, size = 1, normalized size = 0.01 \[ \left [\frac{4 \, \sqrt{b x^{3} + a} \sqrt{e x} B x -{\left (B a - 2 \, A b\right )} \sqrt{\frac{e}{b}} \log \left (-8 \, b^{2} e x^{6} - 8 \, a b e x^{3} - a^{2} e - 4 \,{\left (2 \, b^{2} x^{4} + a b x\right )} \sqrt{b x^{3} + a} \sqrt{e x} \sqrt{\frac{e}{b}}\right )}{12 \, b}, \frac{2 \, \sqrt{b x^{3} + a} \sqrt{e x} B x -{\left (B a - 2 \, A b\right )} \sqrt{-\frac{e}{b}} \arctan \left (\frac{2 \, \sqrt{b x^{3} + a} \sqrt{e x} x}{{\left (2 \, b x^{3} + a\right )} \sqrt{-\frac{e}{b}}}\right )}{6 \, b}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/12*(4*sqrt(b*x^3 + a)*sqrt(e*x)*B*x - (B*a - 2*A*b)*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)))/b, 1/6*(2*sqrt(b*x^3 + a)*sqrt(e*x)*B*x - (B*a - 2*A*b)*sqrt(-e/b)*arctan(
2*sqrt(b*x^3 + a)*sqrt(e*x)*x/((2*b*x^3 + a)*sqrt(-e/b))))/b]

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Sympy [A]  time = 8.51888, size = 107, normalized size = 1.29 \[ \frac{2 A \sqrt{e} \operatorname{asinh}{\left (\frac{\sqrt{b} \left (e x\right )^{\frac{3}{2}}}{\sqrt{a} e^{\frac{3}{2}}} \right )}}{3 \sqrt{b}} + \frac{B \sqrt{a} \left (e x\right )^{\frac{3}{2}} \sqrt{1 + \frac{b x^{3}}{a}}}{3 b e} - \frac{B a \sqrt{e} \operatorname{asinh}{\left (\frac{\sqrt{b} \left (e x\right )^{\frac{3}{2}}}{\sqrt{a} e^{\frac{3}{2}}} \right )}}{3 b^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*A*sqrt(e)*asinh(sqrt(b)*(e*x)**(3/2)/(sqrt(a)*e**(3/2)))/(3*sqrt(b)) + B*sqrt(
a)*(e*x)**(3/2)*sqrt(1 + b*x**3/a)/(3*b*e) - B*a*sqrt(e)*asinh(sqrt(b)*(e*x)**(3
/2)/(sqrt(a)*e**(3/2)))/(3*b**(3/2))

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

[Out]

Timed out