3.2642 \(\int x^{-1+4 n} \sqrt{a+b x^n} \, dx\)

Optimal. Leaf size=92 \[ -\frac{2 a^3 \left (a+b x^n\right )^{3/2}}{3 b^4 n}+\frac{6 a^2 \left (a+b x^n\right )^{5/2}}{5 b^4 n}+\frac{2 \left (a+b x^n\right )^{9/2}}{9 b^4 n}-\frac{6 a \left (a+b x^n\right )^{7/2}}{7 b^4 n} \]

[Out]

(-2*a^3*(a + b*x^n)^(3/2))/(3*b^4*n) + (6*a^2*(a + b*x^n)^(5/2))/(5*b^4*n) - (6*
a*(a + b*x^n)^(7/2))/(7*b^4*n) + (2*(a + b*x^n)^(9/2))/(9*b^4*n)

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Rubi [A]  time = 0.116044, antiderivative size = 92, 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 \[ -\frac{2 a^3 \left (a+b x^n\right )^{3/2}}{3 b^4 n}+\frac{6 a^2 \left (a+b x^n\right )^{5/2}}{5 b^4 n}+\frac{2 \left (a+b x^n\right )^{9/2}}{9 b^4 n}-\frac{6 a \left (a+b x^n\right )^{7/2}}{7 b^4 n} \]

Antiderivative was successfully verified.

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

[Out]

(-2*a^3*(a + b*x^n)^(3/2))/(3*b^4*n) + (6*a^2*(a + b*x^n)^(5/2))/(5*b^4*n) - (6*
a*(a + b*x^n)^(7/2))/(7*b^4*n) + (2*(a + b*x^n)^(9/2))/(9*b^4*n)

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Rubi in Sympy [A]  time = 16.837, size = 82, normalized size = 0.89 \[ - \frac{2 a^{3} \left (a + b x^{n}\right )^{\frac{3}{2}}}{3 b^{4} n} + \frac{6 a^{2} \left (a + b x^{n}\right )^{\frac{5}{2}}}{5 b^{4} n} - \frac{6 a \left (a + b x^{n}\right )^{\frac{7}{2}}}{7 b^{4} n} + \frac{2 \left (a + b x^{n}\right )^{\frac{9}{2}}}{9 b^{4} n} \]

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*(a + b*x**n)**(3/2)/(3*b**4*n) + 6*a**2*(a + b*x**n)**(5/2)/(5*b**4*n) -
6*a*(a + b*x**n)**(7/2)/(7*b**4*n) + 2*(a + b*x**n)**(9/2)/(9*b**4*n)

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Mathematica [A]  time = 0.0519288, size = 70, normalized size = 0.76 \[ \frac{2 \sqrt{a+b x^n} \left (-16 a^4+8 a^3 b x^n-6 a^2 b^2 x^{2 n}+5 a b^3 x^{3 n}+35 b^4 x^{4 n}\right )}{315 b^4 n} \]

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^4 + 8*a^3*b*x^n - 6*a^2*b^2*x^(2*n) + 5*a*b^3*x^(3*n)
+ 35*b^4*x^(4*n)))/(315*b^4*n)

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Maple [A]  time = 0.035, size = 67, normalized size = 0.7 \[ -{\frac{-70\, \left ({x}^{n} \right ) ^{4}{b}^{4}-10\,a \left ({x}^{n} \right ) ^{3}{b}^{3}+12\,{a}^{2} \left ({x}^{n} \right ) ^{2}{b}^{2}-16\,{a}^{3}{x}^{n}b+32\,{a}^{4}}{315\,{b}^{4}n}\sqrt{a+b{x}^{n}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/315*(-35*(x^n)^4*b^4-5*a*(x^n)^3*b^3+6*a^2*(x^n)^2*b^2-8*a^3*x^n*b+16*a^4)*(a
+b*x^n)^(1/2)/b^4/n

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Maxima [A]  time = 1.46528, size = 89, normalized size = 0.97 \[ \frac{2 \,{\left (35 \, b^{4} x^{4 \, n} + 5 \, a b^{3} x^{3 \, n} - 6 \, a^{2} b^{2} x^{2 \, n} + 8 \, a^{3} b x^{n} - 16 \, a^{4}\right )} \sqrt{b x^{n} + a}}{315 \, b^{4} n} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/315*(35*b^4*x^(4*n) + 5*a*b^3*x^(3*n) - 6*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 = 0.2203, size = 89, normalized size = 0.97 \[ \frac{2 \,{\left (35 \, b^{4} x^{4 \, n} + 5 \, a b^{3} x^{3 \, n} - 6 \, a^{2} b^{2} x^{2 \, n} + 8 \, a^{3} b x^{n} - 16 \, a^{4}\right )} \sqrt{b x^{n} + a}}{315 \, b^{4} n} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/315*(35*b^4*x^(4*n) + 5*a*b^3*x^(3*n) - 6*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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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. \[ \int \sqrt{b x^{n} + a} x^{4 \, n - 1}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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