3.2651 \(\int \frac{x^{-1+4 n}}{\sqrt{a+b x^n}} \, dx\)

Optimal. Leaf size=88 \[ -\frac{2 a^3 \sqrt{a+b x^n}}{b^4 n}+\frac{2 a^2 \left (a+b x^n\right )^{3/2}}{b^4 n}+\frac{2 \left (a+b x^n\right )^{7/2}}{7 b^4 n}-\frac{6 a \left (a+b x^n\right )^{5/2}}{5 b^4 n} \]

[Out]

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

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Rubi [A]  time = 0.115258, antiderivative size = 88, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105 \[ -\frac{2 a^3 \sqrt{a+b x^n}}{b^4 n}+\frac{2 a^2 \left (a+b x^n\right )^{3/2}}{b^4 n}+\frac{2 \left (a+b x^n\right )^{7/2}}{7 b^4 n}-\frac{6 a \left (a+b x^n\right )^{5/2}}{5 b^4 n} \]

Antiderivative was successfully verified.

[In]  Int[x^(-1 + 4*n)/Sqrt[a + b*x^n],x]

[Out]

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

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Rubi in Sympy [A]  time = 16.5084, size = 78, normalized size = 0.89 \[ - \frac{2 a^{3} \sqrt{a + b x^{n}}}{b^{4} n} + \frac{2 a^{2} \left (a + b x^{n}\right )^{\frac{3}{2}}}{b^{4} n} - \frac{6 a \left (a + b x^{n}\right )^{\frac{5}{2}}}{5 b^{4} n} + \frac{2 \left (a + b x^{n}\right )^{\frac{7}{2}}}{7 b^{4} n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**(-1+4*n)/(a+b*x**n)**(1/2),x)

[Out]

-2*a**3*sqrt(a + b*x**n)/(b**4*n) + 2*a**2*(a + b*x**n)**(3/2)/(b**4*n) - 6*a*(a
+ b*x**n)**(5/2)/(5*b**4*n) + 2*(a + b*x**n)**(7/2)/(7*b**4*n)

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Mathematica [A]  time = 0.051858, size = 57, normalized size = 0.65 \[ \frac{2 \sqrt{a+b x^n} \left (-16 a^3+8 a^2 b x^n-6 a b^2 x^{2 n}+5 b^3 x^{3 n}\right )}{35 b^4 n} \]

Antiderivative was successfully verified.

[In]  Integrate[x^(-1 + 4*n)/Sqrt[a + b*x^n],x]

[Out]

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

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Maple [A]  time = 0.035, size = 54, normalized size = 0.6 \[ -{\frac{-10\, \left ({x}^{n} \right ) ^{3}{b}^{3}+12\,a \left ({x}^{n} \right ) ^{2}{b}^{2}-16\,{a}^{2}{x}^{n}b+32\,{a}^{3}}{35\,{b}^{4}n}\sqrt{a+b{x}^{n}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(-1+4*n)/(a+b*x^n)^(1/2),x)

[Out]

-2/35*(-5*(x^n)^3*b^3+6*a*(x^n)^2*b^2-8*a^2*x^n*b+16*a^3)*(a+b*x^n)^(1/2)/b^4/n

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Maxima [A]  time = 1.46873, size = 89, normalized size = 1.01 \[ \frac{2 \,{\left (5 \, b^{4} x^{4 \, n} - a b^{3} x^{3 \, n} + 2 \, a^{2} b^{2} x^{2 \, n} - 8 \, a^{3} b x^{n} - 16 \, a^{4}\right )}}{35 \, \sqrt{b x^{n} + a} b^{4} n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^(4*n - 1)/sqrt(b*x^n + a),x, algorithm="maxima")

[Out]

2/35*(5*b^4*x^(4*n) - a*b^3*x^(3*n) + 2*a^2*b^2*x^(2*n) - 8*a^3*b*x^n - 16*a^4)/
(sqrt(b*x^n + a)*b^4*n)

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Fricas [A]  time = 0.229767, size = 72, normalized size = 0.82 \[ \frac{2 \,{\left (5 \, b^{3} x^{3 \, n} - 6 \, a b^{2} x^{2 \, n} + 8 \, a^{2} b x^{n} - 16 \, a^{3}\right )} \sqrt{b x^{n} + a}}{35 \, b^{4} n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^(4*n - 1)/sqrt(b*x^n + a),x, algorithm="fricas")

[Out]

2/35*(5*b^3*x^(3*n) - 6*a*b^2*x^(2*n) + 8*a^2*b*x^n - 16*a^3)*sqrt(b*x^n + a)/(b
^4*n)

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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**(-1+4*n)/(a+b*x**n)**(1/2),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{4 \, n - 1}}{\sqrt{b x^{n} + a}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^(4*n - 1)/sqrt(b*x^n + a),x, algorithm="giac")

[Out]

integrate(x^(4*n - 1)/sqrt(b*x^n + a), x)