3.1310 \(\int \frac{1}{x^{17/2} \sqrt{a+b x^5}} \, dx\)

Optimal. Leaf size=48 \[ \frac{4 b \sqrt{a+b x^5}}{15 a^2 x^{5/2}}-\frac{2 \sqrt{a+b x^5}}{15 a x^{15/2}} \]

[Out]

(-2*Sqrt[a + b*x^5])/(15*a*x^(15/2)) + (4*b*Sqrt[a + b*x^5])/(15*a^2*x^(5/2))

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Rubi [A]  time = 0.0427456, antiderivative size = 48, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118 \[ \frac{4 b \sqrt{a+b x^5}}{15 a^2 x^{5/2}}-\frac{2 \sqrt{a+b x^5}}{15 a x^{15/2}} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^(17/2)*Sqrt[a + b*x^5]),x]

[Out]

(-2*Sqrt[a + b*x^5])/(15*a*x^(15/2)) + (4*b*Sqrt[a + b*x^5])/(15*a^2*x^(5/2))

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Rubi in Sympy [A]  time = 4.53086, size = 42, normalized size = 0.88 \[ - \frac{2 \sqrt{a + b x^{5}}}{15 a x^{\frac{15}{2}}} + \frac{4 b \sqrt{a + b x^{5}}}{15 a^{2} x^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**(17/2)/(b*x**5+a)**(1/2),x)

[Out]

-2*sqrt(a + b*x**5)/(15*a*x**(15/2)) + 4*b*sqrt(a + b*x**5)/(15*a**2*x**(5/2))

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Mathematica [A]  time = 0.0351076, size = 31, normalized size = 0.65 \[ -\frac{2 \left (a-2 b x^5\right ) \sqrt{a+b x^5}}{15 a^2 x^{15/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^(17/2)*Sqrt[a + b*x^5]),x]

[Out]

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

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Maple [A]  time = 0.007, size = 26, normalized size = 0.5 \[ -{\frac{-4\,b{x}^{5}+2\,a}{15\,{a}^{2}}\sqrt{b{x}^{5}+a}{x}^{-{\frac{15}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^(17/2)/(b*x^5+a)^(1/2),x)

[Out]

-2/15*(b*x^5+a)^(1/2)*(-2*b*x^5+a)/x^(15/2)/a^2

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Maxima [A]  time = 1.43893, size = 47, normalized size = 0.98 \[ \frac{2 \,{\left (\frac{3 \, \sqrt{b x^{5} + a} b}{x^{\frac{5}{2}}} - \frac{{\left (b x^{5} + a\right )}^{\frac{3}{2}}}{x^{\frac{15}{2}}}\right )}}{15 \, a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(b*x^5 + a)*x^(17/2)),x, algorithm="maxima")

[Out]

2/15*(3*sqrt(b*x^5 + a)*b/x^(5/2) - (b*x^5 + a)^(3/2)/x^(15/2))/a^2

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Fricas [A]  time = 0.228314, size = 36, normalized size = 0.75 \[ \frac{2 \,{\left (2 \, b x^{5} - a\right )} \sqrt{b x^{5} + a}}{15 \, a^{2} x^{\frac{15}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(b*x^5 + a)*x^(17/2)),x, algorithm="fricas")

[Out]

2/15*(2*b*x^5 - a)*sqrt(b*x^5 + a)/(a^2*x^(15/2))

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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(1/x**(17/2)/(b*x**5+a)**(1/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.232102, size = 49, normalized size = 1.02 \[ -\frac{4 \, b^{\frac{3}{2}}}{15 \, a^{2}} - \frac{2 \,{\left ({\left (b + \frac{a}{x^{5}}\right )}^{\frac{3}{2}} - 3 \, \sqrt{b + \frac{a}{x^{5}}} b\right )}}{15 \, a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(b*x^5 + a)*x^(17/2)),x, algorithm="giac")

[Out]

-4/15*b^(3/2)/a^2 - 2/15*((b + a/x^5)^(3/2) - 3*sqrt(b + a/x^5)*b)/a^2