3.2780 \(\int \frac{(c x)^{-1-\frac{5 n}{2}}}{\sqrt{a+b x^n}} \, dx\)

Optimal. Leaf size=98 \[ -\frac{16 (c x)^{-5 n/2} \left (a+b x^n\right )^{5/2}}{15 a^3 c n}+\frac{8 (c x)^{-5 n/2} \left (a+b x^n\right )^{3/2}}{3 a^2 c n}-\frac{2 (c x)^{-5 n/2} \sqrt{a+b x^n}}{a c n} \]

[Out]

(-2*Sqrt[a + b*x^n])/(a*c*n*(c*x)^((5*n)/2)) + (8*(a + b*x^n)^(3/2))/(3*a^2*c*n*
(c*x)^((5*n)/2)) - (16*(a + b*x^n)^(5/2))/(15*a^3*c*n*(c*x)^((5*n)/2))

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Rubi [A]  time = 0.103744, antiderivative size = 98, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.087 \[ -\frac{16 (c x)^{-5 n/2} \left (a+b x^n\right )^{5/2}}{15 a^3 c n}+\frac{8 (c x)^{-5 n/2} \left (a+b x^n\right )^{3/2}}{3 a^2 c n}-\frac{2 (c x)^{-5 n/2} \sqrt{a+b x^n}}{a c n} \]

Antiderivative was successfully verified.

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

[Out]

(-2*Sqrt[a + b*x^n])/(a*c*n*(c*x)^((5*n)/2)) + (8*(a + b*x^n)^(3/2))/(3*a^2*c*n*
(c*x)^((5*n)/2)) - (16*(a + b*x^n)^(5/2))/(15*a^3*c*n*(c*x)^((5*n)/2))

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Rubi in Sympy [A]  time = 12.5213, size = 82, normalized size = 0.84 \[ - \frac{2 \left (c x\right )^{- \frac{5 n}{2}} \sqrt{a + b x^{n}}}{a c n} + \frac{8 \left (c x\right )^{- \frac{5 n}{2}} \left (a + b x^{n}\right )^{\frac{3}{2}}}{3 a^{2} c n} - \frac{16 \left (c x\right )^{- \frac{5 n}{2}} \left (a + b x^{n}\right )^{\frac{5}{2}}}{15 a^{3} c n} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*(c*x)**(-5*n/2)*sqrt(a + b*x**n)/(a*c*n) + 8*(c*x)**(-5*n/2)*(a + b*x**n)**(3
/2)/(3*a**2*c*n) - 16*(c*x)**(-5*n/2)*(a + b*x**n)**(5/2)/(15*a**3*c*n)

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Mathematica [A]  time = 0.0581889, size = 56, normalized size = 0.57 \[ -\frac{2 (c x)^{-5 n/2} \sqrt{a+b x^n} \left (3 a^2-4 a b x^n+8 b^2 x^{2 n}\right )}{15 a^3 c n} \]

Antiderivative was successfully verified.

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

[Out]

(-2*Sqrt[a + b*x^n]*(3*a^2 - 4*a*b*x^n + 8*b^2*x^(2*n)))/(15*a^3*c*n*(c*x)^((5*n
)/2))

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Maple [F]  time = 0.083, size = 0, normalized size = 0. \[ \int{1 \left ( cx \right ) ^{-1-{\frac{5\,n}{2}}}{\frac{1}{\sqrt{a+b{x}^{n}}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

int((c*x)^(-1-5/2*n)/(a+b*x^n)^(1/2),x)

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Maxima [A]  time = 1.44165, size = 108, normalized size = 1.1 \[ -\frac{2}{15} \, c^{-\frac{5}{2} \, n - 1}{\left (\frac{15 \, \sqrt{b x^{n} + a} b^{2} x^{-\frac{1}{2} \, n}}{a^{3} n} - \frac{10 \,{\left (b x^{n} + a\right )}^{\frac{3}{2}} b x^{-\frac{3}{2} \, n}}{a^{3} n} + \frac{3 \,{\left (b x^{n} + a\right )}^{\frac{5}{2}} x^{-\frac{5}{2} \, n}}{a^{3} n}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/15*c^(-5/2*n - 1)*(15*sqrt(b*x^n + a)*b^2*x^(-1/2*n)/(a^3*n) - 10*(b*x^n + a)
^(3/2)*b*x^(-3/2*n)/(a^3*n) + 3*(b*x^n + a)^(5/2)*x^(-5/2*n)/(a^3*n))

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Fricas [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: TypeError

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Sympy [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x)**(-1-5/2*n)/(a+b*x**n)**(1/2),x)

[Out]

Exception raised: TypeError

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (c x\right )^{-\frac{5}{2} \, n - 1}}{\sqrt{b x^{n} + a}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((c*x)^(-5/2*n - 1)/sqrt(b*x^n + a), x)