3.1306 \(\int \frac{x^{23/2}}{\sqrt{a+b x^5}} \, dx\)

Optimal. Leaf size=83 \[ \frac{3 a^2 \tanh ^{-1}\left (\frac{\sqrt{b} x^{5/2}}{\sqrt{a+b x^5}}\right )}{20 b^{5/2}}-\frac{3 a x^{5/2} \sqrt{a+b x^5}}{20 b^2}+\frac{x^{15/2} \sqrt{a+b x^5}}{10 b} \]

[Out]

(-3*a*x^(5/2)*Sqrt[a + b*x^5])/(20*b^2) + (x^(15/2)*Sqrt[a + b*x^5])/(10*b) + (3
*a^2*ArcTanh[(Sqrt[b]*x^(5/2))/Sqrt[a + b*x^5]])/(20*b^(5/2))

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Rubi [A]  time = 0.117107, antiderivative size = 83, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.294 \[ \frac{3 a^2 \tanh ^{-1}\left (\frac{\sqrt{b} x^{5/2}}{\sqrt{a+b x^5}}\right )}{20 b^{5/2}}-\frac{3 a x^{5/2} \sqrt{a+b x^5}}{20 b^2}+\frac{x^{15/2} \sqrt{a+b x^5}}{10 b} \]

Antiderivative was successfully verified.

[In]  Int[x^(23/2)/Sqrt[a + b*x^5],x]

[Out]

(-3*a*x^(5/2)*Sqrt[a + b*x^5])/(20*b^2) + (x^(15/2)*Sqrt[a + b*x^5])/(10*b) + (3
*a^2*ArcTanh[(Sqrt[b]*x^(5/2))/Sqrt[a + b*x^5]])/(20*b^(5/2))

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Rubi in Sympy [A]  time = 12.0729, size = 75, normalized size = 0.9 \[ \frac{3 a^{2} \operatorname{atanh}{\left (\frac{\sqrt{b} x^{\frac{5}{2}}}{\sqrt{a + b x^{5}}} \right )}}{20 b^{\frac{5}{2}}} - \frac{3 a x^{\frac{5}{2}} \sqrt{a + b x^{5}}}{20 b^{2}} + \frac{x^{\frac{15}{2}} \sqrt{a + b x^{5}}}{10 b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**(23/2)/(b*x**5+a)**(1/2),x)

[Out]

3*a**2*atanh(sqrt(b)*x**(5/2)/sqrt(a + b*x**5))/(20*b**(5/2)) - 3*a*x**(5/2)*sqr
t(a + b*x**5)/(20*b**2) + x**(15/2)*sqrt(a + b*x**5)/(10*b)

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Mathematica [A]  time = 0.118315, size = 70, normalized size = 0.84 \[ \frac{3 a^2 \tanh ^{-1}\left (\frac{\sqrt{b} x^{5/2}}{\sqrt{a+b x^5}}\right )+\sqrt{b} x^{5/2} \sqrt{a+b x^5} \left (2 b x^5-3 a\right )}{20 b^{5/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^(23/2)/Sqrt[a + b*x^5],x]

[Out]

(Sqrt[b]*x^(5/2)*Sqrt[a + b*x^5]*(-3*a + 2*b*x^5) + 3*a^2*ArcTanh[(Sqrt[b]*x^(5/
2))/Sqrt[a + b*x^5]])/(20*b^(5/2))

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Maple [F]  time = 0.076, size = 0, normalized size = 0. \[ \int{1{x}^{{\frac{23}{2}}}{\frac{1}{\sqrt{b{x}^{5}+a}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(23/2)/(b*x^5+a)^(1/2),x)

[Out]

int(x^(23/2)/(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^(23/2)/sqrt(b*x^5 + a),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.733749, size = 1, normalized size = 0.01 \[ \left [\frac{3 \, a^{2} \log \left (-4 \,{\left (2 \, b^{2} x^{7} + a b x^{2}\right )} \sqrt{b x^{5} + a} \sqrt{x} -{\left (8 \, b^{2} x^{10} + 8 \, a b x^{5} + a^{2}\right )} \sqrt{b}\right ) + 4 \,{\left (2 \, b x^{7} - 3 \, a x^{2}\right )} \sqrt{b x^{5} + a} \sqrt{b} \sqrt{x}}{80 \, b^{\frac{5}{2}}}, \frac{3 \, a^{2} \arctan \left (\frac{2 \, \sqrt{b x^{5} + a} \sqrt{-b} x^{\frac{5}{2}}}{2 \, b x^{5} + a}\right ) + 2 \,{\left (2 \, b x^{7} - 3 \, a x^{2}\right )} \sqrt{b x^{5} + a} \sqrt{-b} \sqrt{x}}{40 \, \sqrt{-b} b^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^(23/2)/sqrt(b*x^5 + a),x, algorithm="fricas")

[Out]

[1/80*(3*a^2*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)) + 4*(2*b*x^7 - 3*a*x^2)*sqrt(b*x^5 + a)*sqrt(b)*sqrt
(x))/b^(5/2), 1/40*(3*a^2*arctan(2*sqrt(b*x^5 + a)*sqrt(-b)*x^(5/2)/(2*b*x^5 + a
)) + 2*(2*b*x^7 - 3*a*x^2)*sqrt(b*x^5 + a)*sqrt(-b)*sqrt(x))/(sqrt(-b)*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(x**(23/2)/(b*x**5+a)**(1/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.258996, size = 78, normalized size = 0.94 \[ \frac{1}{20} \, \sqrt{b x^{5} + a}{\left (\frac{2 \, x^{5}}{b} - \frac{3 \, a}{b^{2}}\right )} x^{\frac{5}{2}} - \frac{3 \, a^{2}{\rm ln}\left ({\left | -\sqrt{b} x^{\frac{5}{2}} + \sqrt{b x^{5} + a} \right |}\right )}{20 \, b^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^(23/2)/sqrt(b*x^5 + a),x, algorithm="giac")

[Out]

1/20*sqrt(b*x^5 + a)*(2*x^5/b - 3*a/b^2)*x^(5/2) - 3/20*a^2*ln(abs(-sqrt(b)*x^(5
/2) + sqrt(b*x^5 + a)))/b^(5/2)