3.216 \(\int \frac{A+B x^3}{x \sqrt{a+b x^3}} \, dx\)

Optimal. Leaf size=48 \[ \frac{2 B \sqrt{a+b x^3}}{3 b}-\frac{2 A \tanh ^{-1}\left (\frac{\sqrt{a+b x^3}}{\sqrt{a}}\right )}{3 \sqrt{a}} \]

[Out]

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

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Rubi [A]  time = 0.122553, antiderivative size = 48, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ \frac{2 B \sqrt{a+b x^3}}{3 b}-\frac{2 A \tanh ^{-1}\left (\frac{\sqrt{a+b x^3}}{\sqrt{a}}\right )}{3 \sqrt{a}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 10.174, size = 42, normalized size = 0.88 \[ - \frac{2 A \operatorname{atanh}{\left (\frac{\sqrt{a + b x^{3}}}{\sqrt{a}} \right )}}{3 \sqrt{a}} + \frac{2 B \sqrt{a + b x^{3}}}{3 b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.12216, size = 61, normalized size = 1.27 \[ \frac{2 \left (B \left (a+b x^3\right )-A b \sqrt{\frac{b x^3}{a}+1} \tanh ^{-1}\left (\sqrt{\frac{b x^3}{a}+1}\right )\right )}{3 b \sqrt{a+b x^3}} \]

Antiderivative was successfully verified.

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

[Out]

(2*(B*(a + b*x^3) - A*b*Sqrt[1 + (b*x^3)/a]*ArcTanh[Sqrt[1 + (b*x^3)/a]]))/(3*b*
Sqrt[a + b*x^3])

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Maple [A]  time = 0.011, size = 37, normalized size = 0.8 \[ -{\frac{2\,A}{3}{\it Artanh} \left ({1\sqrt{b{x}^{3}+a}{\frac{1}{\sqrt{a}}}} \right ){\frac{1}{\sqrt{a}}}}+{\frac{2\,B}{3\,b}\sqrt{b{x}^{3}+a}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/3*A*arctanh((b*x^3+a)^(1/2)/a^(1/2))/a^(1/2)+2/3*B*(b*x^3+a)^(1/2)/b

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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)*x),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.265625, size = 1, normalized size = 0.02 \[ \left [\frac{A b \log \left (\frac{{\left (b x^{3} + 2 \, a\right )} \sqrt{a} - 2 \, \sqrt{b x^{3} + a} a}{x^{3}}\right ) + 2 \, \sqrt{b x^{3} + a} B \sqrt{a}}{3 \, \sqrt{a} b}, \frac{2 \,{\left (A b \arctan \left (\frac{a}{\sqrt{b x^{3} + a} \sqrt{-a}}\right ) + \sqrt{b x^{3} + a} B \sqrt{-a}\right )}}{3 \, \sqrt{-a} b}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/3*(A*b*log(((b*x^3 + 2*a)*sqrt(a) - 2*sqrt(b*x^3 + a)*a)/x^3) + 2*sqrt(b*x^3
+ a)*B*sqrt(a))/(sqrt(a)*b), 2/3*(A*b*arctan(a/(sqrt(b*x^3 + a)*sqrt(-a))) + sqr
t(b*x^3 + a)*B*sqrt(-a))/(sqrt(-a)*b)]

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Sympy [A]  time = 7.97656, size = 143, normalized size = 2.98 \[ \frac{2 A \left (\begin{cases} \frac{\operatorname{atan}{\left (\frac{1}{\sqrt{- \frac{1}{a}} \sqrt{a + b x^{3}}} \right )}}{a \sqrt{- \frac{1}{a}}} & \text{for}\: - \frac{1}{a} > 0 \\- \frac{\operatorname{acoth}{\left (\frac{1}{\sqrt{a + b x^{3}} \sqrt{\frac{1}{a}}} \right )}}{a \sqrt{\frac{1}{a}}} & \text{for}\: - \frac{1}{a} < 0 \wedge \frac{1}{a} < \frac{1}{a + b x^{3}} \\- \frac{\operatorname{atanh}{\left (\frac{1}{\sqrt{a + b x^{3}} \sqrt{\frac{1}{a}}} \right )}}{a \sqrt{\frac{1}{a}}} & \text{for}\: \frac{1}{a} > \frac{1}{a + b x^{3}} \wedge - \frac{1}{a} < 0 \end{cases}\right )}{3} + \frac{2 B \sqrt{a + b x^{3}}}{3 b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*A*Piecewise((atan(1/(sqrt(-1/a)*sqrt(a + b*x**3)))/(a*sqrt(-1/a)), -1/a > 0),
(-acoth(1/(sqrt(a + b*x**3)*sqrt(1/a)))/(a*sqrt(1/a)), (-1/a < 0) & (1/a < 1/(a
+ b*x**3))), (-atanh(1/(sqrt(a + b*x**3)*sqrt(1/a)))/(a*sqrt(1/a)), (-1/a < 0) &
(1/a > 1/(a + b*x**3))))/3 + 2*B*sqrt(a + b*x**3)/(3*b)

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GIAC/XCAS [A]  time = 0.217321, size = 54, normalized size = 1.12 \[ \frac{2 \, A \arctan \left (\frac{\sqrt{b x^{3} + a}}{\sqrt{-a}}\right )}{3 \, \sqrt{-a}} + \frac{2 \, \sqrt{b x^{3} + a} B}{3 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/3*A*arctan(sqrt(b*x^3 + a)/sqrt(-a))/sqrt(-a) + 2/3*sqrt(b*x^3 + a)*B/b