3.2687 \(\int \frac{x^{-1-\frac{7 n}{2}}}{\sqrt{a+b x^n}} \, dx\)

Optimal. Leaf size=120 \[ \frac{32 b^3 x^{-n/2} \sqrt{a+b x^n}}{35 a^4 n}-\frac{16 b^2 x^{-3 n/2} \sqrt{a+b x^n}}{35 a^3 n}+\frac{12 b x^{-5 n/2} \sqrt{a+b x^n}}{35 a^2 n}-\frac{2 x^{-7 n/2} \sqrt{a+b x^n}}{7 a n} \]

[Out]

(-2*Sqrt[a + b*x^n])/(7*a*n*x^((7*n)/2)) + (12*b*Sqrt[a + b*x^n])/(35*a^2*n*x^((
5*n)/2)) - (16*b^2*Sqrt[a + b*x^n])/(35*a^3*n*x^((3*n)/2)) + (32*b^3*Sqrt[a + b*
x^n])/(35*a^4*n*x^(n/2))

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Rubi [A]  time = 0.126124, antiderivative size = 120, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095 \[ \frac{32 b^3 x^{-n/2} \sqrt{a+b x^n}}{35 a^4 n}-\frac{16 b^2 x^{-3 n/2} \sqrt{a+b x^n}}{35 a^3 n}+\frac{12 b x^{-5 n/2} \sqrt{a+b x^n}}{35 a^2 n}-\frac{2 x^{-7 n/2} \sqrt{a+b x^n}}{7 a n} \]

Antiderivative was successfully verified.

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

[Out]

(-2*Sqrt[a + b*x^n])/(7*a*n*x^((7*n)/2)) + (12*b*Sqrt[a + b*x^n])/(35*a^2*n*x^((
5*n)/2)) - (16*b^2*Sqrt[a + b*x^n])/(35*a^3*n*x^((3*n)/2)) + (32*b^3*Sqrt[a + b*
x^n])/(35*a^4*n*x^(n/2))

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Rubi in Sympy [A]  time = 13.4536, size = 105, normalized size = 0.88 \[ - \frac{2 x^{- \frac{7 n}{2}} \sqrt{a + b x^{n}}}{7 a n} + \frac{12 b x^{- \frac{5 n}{2}} \sqrt{a + b x^{n}}}{35 a^{2} n} - \frac{16 b^{2} x^{- \frac{3 n}{2}} \sqrt{a + b x^{n}}}{35 a^{3} n} + \frac{32 b^{3} x^{- \frac{n}{2}} \sqrt{a + b x^{n}}}{35 a^{4} n} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*x**(-7*n/2)*sqrt(a + b*x**n)/(7*a*n) + 12*b*x**(-5*n/2)*sqrt(a + b*x**n)/(35*
a**2*n) - 16*b**2*x**(-3*n/2)*sqrt(a + b*x**n)/(35*a**3*n) + 32*b**3*x**(-n/2)*s
qrt(a + b*x**n)/(35*a**4*n)

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Mathematica [A]  time = 0.0671014, size = 64, normalized size = 0.53 \[ \frac{2 x^{-7 n/2} \sqrt{a+b x^n} \left (-5 a^3+6 a^2 b x^n-8 a b^2 x^{2 n}+16 b^3 x^{3 n}\right )}{35 a^4 n} \]

Antiderivative was successfully verified.

[In]  Integrate[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*n*x^((7*n)/2))

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Maple [F]  time = 0.085, size = 0, normalized size = 0. \[ \int{1{x}^{-1-{\frac{7\,n}{2}}}{\frac{1}{\sqrt{a+b{x}^{n}}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 1.5498, size = 130, normalized size = 1.08 \[ \frac{2 \, \sqrt{b x^{n} + a} b^{3} x^{-\frac{1}{2} \, n}}{a^{4} n} - \frac{2 \,{\left (b x^{n} + a\right )}^{\frac{3}{2}} b^{2} x^{-\frac{3}{2} \, n}}{a^{4} n} + \frac{6 \,{\left (b x^{n} + a\right )}^{\frac{5}{2}} b x^{-\frac{5}{2} \, n}}{5 \, a^{4} n} - \frac{2 \,{\left (b x^{n} + a\right )}^{\frac{7}{2}} x^{-\frac{7}{2} \, n}}{7 \, a^{4} n} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*sqrt(b*x^n + a)*b^3*x^(-1/2*n)/(a^4*n) - 2*(b*x^n + a)^(3/2)*b^2*x^(-3/2*n)/(a
^4*n) + 6/5*(b*x^n + a)^(5/2)*b*x^(-5/2*n)/(a^4*n) - 2/7*(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(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(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. \[ \int \frac{x^{-\frac{7}{2} \, n - 1}}{\sqrt{b x^{n} + a}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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