3.1307 \(\int \frac{x^{13/2}}{\sqrt{a+b x^5}} \, dx\)

Optimal. Leaf size=57 \[ \frac{x^{5/2} \sqrt{a+b x^5}}{5 b}-\frac{a \tanh ^{-1}\left (\frac{\sqrt{b} x^{5/2}}{\sqrt{a+b x^5}}\right )}{5 b^{3/2}} \]

[Out]

(x^(5/2)*Sqrt[a + b*x^5])/(5*b) - (a*ArcTanh[(Sqrt[b]*x^(5/2))/Sqrt[a + b*x^5]])
/(5*b^(3/2))

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Rubi [A]  time = 0.0885204, antiderivative size = 57, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.294 \[ \frac{x^{5/2} \sqrt{a+b x^5}}{5 b}-\frac{a \tanh ^{-1}\left (\frac{\sqrt{b} x^{5/2}}{\sqrt{a+b x^5}}\right )}{5 b^{3/2}} \]

Antiderivative was successfully verified.

[In]  Int[x^(13/2)/Sqrt[a + b*x^5],x]

[Out]

(x^(5/2)*Sqrt[a + b*x^5])/(5*b) - (a*ArcTanh[(Sqrt[b]*x^(5/2))/Sqrt[a + b*x^5]])
/(5*b^(3/2))

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Rubi in Sympy [A]  time = 8.75827, size = 48, normalized size = 0.84 \[ - \frac{a \operatorname{atanh}{\left (\frac{\sqrt{b} x^{\frac{5}{2}}}{\sqrt{a + b x^{5}}} \right )}}{5 b^{\frac{3}{2}}} + \frac{x^{\frac{5}{2}} \sqrt{a + b x^{5}}}{5 b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**(13/2)/(b*x**5+a)**(1/2),x)

[Out]

-a*atanh(sqrt(b)*x**(5/2)/sqrt(a + b*x**5))/(5*b**(3/2)) + x**(5/2)*sqrt(a + b*x
**5)/(5*b)

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Mathematica [A]  time = 0.10126, size = 57, normalized size = 1. \[ \frac{x^{5/2} \sqrt{a+b x^5}}{5 b}-\frac{a \tanh ^{-1}\left (\frac{\sqrt{b} x^{5/2}}{\sqrt{a+b x^5}}\right )}{5 b^{3/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^(13/2)/Sqrt[a + b*x^5],x]

[Out]

(x^(5/2)*Sqrt[a + b*x^5])/(5*b) - (a*ArcTanh[(Sqrt[b]*x^(5/2))/Sqrt[a + b*x^5]])
/(5*b^(3/2))

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Maple [F]  time = 0.053, size = 0, normalized size = 0. \[ \int{1{x}^{{\frac{13}{2}}}{\frac{1}{\sqrt{b{x}^{5}+a}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(13/2)/(b*x^5+a)^(1/2),x)

[Out]

int(x^(13/2)/(b*x^5+a)^(1/2),x)

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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(x^(13/2)/sqrt(b*x^5 + a),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.720871, size = 1, normalized size = 0.02 \[ \left [\frac{4 \, \sqrt{b x^{5} + a} \sqrt{b} x^{\frac{5}{2}} + a \log \left (4 \,{\left (2 \, b^{2} x^{7} + a b x^{2}\right )} \sqrt{b x^{5} + a} \sqrt{x} -{\left (8 \, b^{2} x^{10} + 8 \, a b x^{5} + a^{2}\right )} \sqrt{b}\right )}{20 \, b^{\frac{3}{2}}}, \frac{2 \, \sqrt{b x^{5} + a} \sqrt{-b} x^{\frac{5}{2}} - a \arctan \left (\frac{2 \, \sqrt{b x^{5} + a} \sqrt{-b} x^{\frac{5}{2}}}{2 \, b x^{5} + a}\right )}{10 \, \sqrt{-b} b}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^(13/2)/sqrt(b*x^5 + a),x, algorithm="fricas")

[Out]

[1/20*(4*sqrt(b*x^5 + a)*sqrt(b)*x^(5/2) + a*log(4*(2*b^2*x^7 + a*b*x^2)*sqrt(b*
x^5 + a)*sqrt(x) - (8*b^2*x^10 + 8*a*b*x^5 + a^2)*sqrt(b)))/b^(3/2), 1/10*(2*sqr
t(b*x^5 + a)*sqrt(-b)*x^(5/2) - a*arctan(2*sqrt(b*x^5 + a)*sqrt(-b)*x^(5/2)/(2*b
*x^5 + a)))/(sqrt(-b)*b)]

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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(x**(13/2)/(b*x**5+a)**(1/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.250401, size = 59, normalized size = 1.04 \[ \frac{\sqrt{b x^{5} + a} x^{\frac{5}{2}}}{5 \, b} + \frac{a{\rm ln}\left ({\left | -\sqrt{b} x^{\frac{5}{2}} + \sqrt{b x^{5} + a} \right |}\right )}{5 \, b^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^(13/2)/sqrt(b*x^5 + a),x, algorithm="giac")

[Out]

1/5*sqrt(b*x^5 + a)*x^(5/2)/b + 1/5*a*ln(abs(-sqrt(b)*x^(5/2) + sqrt(b*x^5 + a))
)/b^(3/2)