3.2781 \(\int \frac{(c x)^{-1-\frac{7 n}{2}}}{\sqrt{a+b x^n}} \, dx\)

Optimal. Leaf size=129 \[ \frac{32 (c x)^{-7 n/2} \left (a+b x^n\right )^{7/2}}{35 a^4 c n}-\frac{16 (c x)^{-7 n/2} \left (a+b x^n\right )^{5/2}}{5 a^3 c n}+\frac{4 (c x)^{-7 n/2} \left (a+b x^n\right )^{3/2}}{a^2 c n}-\frac{2 (c x)^{-7 n/2} \sqrt{a+b x^n}}{a c n} \]

[Out]

(-2*Sqrt[a + b*x^n])/(a*c*n*(c*x)^((7*n)/2)) + (4*(a + b*x^n)^(3/2))/(a^2*c*n*(c
*x)^((7*n)/2)) - (16*(a + b*x^n)^(5/2))/(5*a^3*c*n*(c*x)^((7*n)/2)) + (32*(a + b
*x^n)^(7/2))/(35*a^4*c*n*(c*x)^((7*n)/2))

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Rubi [A]  time = 0.148921, antiderivative size = 129, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.087 \[ \frac{32 (c x)^{-7 n/2} \left (a+b x^n\right )^{7/2}}{35 a^4 c n}-\frac{16 (c x)^{-7 n/2} \left (a+b x^n\right )^{5/2}}{5 a^3 c n}+\frac{4 (c x)^{-7 n/2} \left (a+b x^n\right )^{3/2}}{a^2 c n}-\frac{2 (c x)^{-7 n/2} \sqrt{a+b x^n}}{a c n} \]

Antiderivative was successfully verified.

[In]  Int[(c*x)^(-1 - (7*n)/2)/Sqrt[a + b*x^n],x]

[Out]

(-2*Sqrt[a + b*x^n])/(a*c*n*(c*x)^((7*n)/2)) + (4*(a + b*x^n)^(3/2))/(a^2*c*n*(c
*x)^((7*n)/2)) - (16*(a + b*x^n)^(5/2))/(5*a^3*c*n*(c*x)^((7*n)/2)) + (32*(a + b
*x^n)^(7/2))/(35*a^4*c*n*(c*x)^((7*n)/2))

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Rubi in Sympy [A]  time = 17.9944, size = 109, normalized size = 0.84 \[ - \frac{2 \left (c x\right )^{- \frac{7 n}{2}} \sqrt{a + b x^{n}}}{a c n} + \frac{4 \left (c x\right )^{- \frac{7 n}{2}} \left (a + b x^{n}\right )^{\frac{3}{2}}}{a^{2} c n} - \frac{16 \left (c x\right )^{- \frac{7 n}{2}} \left (a + b x^{n}\right )^{\frac{5}{2}}}{5 a^{3} c n} + \frac{32 \left (c x\right )^{- \frac{7 n}{2}} \left (a + b x^{n}\right )^{\frac{7}{2}}}{35 a^{4} c n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((c*x)**(-1-7/2*n)/(a+b*x**n)**(1/2),x)

[Out]

-2*(c*x)**(-7*n/2)*sqrt(a + b*x**n)/(a*c*n) + 4*(c*x)**(-7*n/2)*(a + b*x**n)**(3
/2)/(a**2*c*n) - 16*(c*x)**(-7*n/2)*(a + b*x**n)**(5/2)/(5*a**3*c*n) + 32*(c*x)*
*(-7*n/2)*(a + b*x**n)**(7/2)/(35*a**4*c*n)

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Mathematica [A]  time = 0.0718752, size = 69, normalized size = 0.53 \[ \frac{2 (c x)^{-7 n/2} \sqrt{a+b x^n} \left (-5 a^3+6 a^2 b x^n-8 a b^2 x^{2 n}+16 b^3 x^{3 n}\right )}{35 a^4 c n} \]

Antiderivative was successfully verified.

[In]  Integrate[(c*x)^(-1 - (7*n)/2)/Sqrt[a + b*x^n],x]

[Out]

(2*Sqrt[a + b*x^n]*(-5*a^3 + 6*a^2*b*x^n - 8*a*b^2*x^(2*n) + 16*b^3*x^(3*n)))/(3
5*a^4*c*n*(c*x)^((7*n)/2))

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Maple [F]  time = 0.074, size = 0, normalized size = 0. \[ \int{1 \left ( cx \right ) ^{-1-{\frac{7\,n}{2}}}{\frac{1}{\sqrt{a+b{x}^{n}}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x)^(-1-7/2*n)/(a+b*x^n)^(1/2),x)

[Out]

int((c*x)^(-1-7/2*n)/(a+b*x^n)^(1/2),x)

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Maxima [A]  time = 1.4411, size = 142, normalized size = 1.1 \[ \frac{2}{35} \, c^{-\frac{7}{2} \, n - 1}{\left (\frac{35 \, \sqrt{b x^{n} + a} b^{3} x^{-\frac{1}{2} \, n}}{a^{4} n} - \frac{35 \,{\left (b x^{n} + a\right )}^{\frac{3}{2}} b^{2} x^{-\frac{3}{2} \, n}}{a^{4} n} + \frac{21 \,{\left (b x^{n} + a\right )}^{\frac{5}{2}} b x^{-\frac{5}{2} \, n}}{a^{4} n} - \frac{5 \,{\left (b x^{n} + a\right )}^{\frac{7}{2}} x^{-\frac{7}{2} \, n}}{a^{4} n}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x)^(-7/2*n - 1)/sqrt(b*x^n + a),x, algorithm="maxima")

[Out]

2/35*c^(-7/2*n - 1)*(35*sqrt(b*x^n + a)*b^3*x^(-1/2*n)/(a^4*n) - 35*(b*x^n + a)^
(3/2)*b^2*x^(-3/2*n)/(a^4*n) + 21*(b*x^n + a)^(5/2)*b*x^(-5/2*n)/(a^4*n) - 5*(b*
x^n + a)^(7/2)*x^(-7/2*n)/(a^4*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)^(-7/2*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)**(-1-7/2*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. \[ \int \frac{\left (c x\right )^{-\frac{7}{2} \, n - 1}}{\sqrt{b x^{n} + a}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x)^(-7/2*n - 1)/sqrt(b*x^n + a),x, algorithm="giac")

[Out]

integrate((c*x)^(-7/2*n - 1)/sqrt(b*x^n + a), x)