3.2643 \(\int x^{-1+3 n} \sqrt{a+b x^n} \, dx\)

Optimal. Leaf size=68 \[ \frac{2 a^2 \left (a+b x^n\right )^{3/2}}{3 b^3 n}+\frac{2 \left (a+b x^n\right )^{7/2}}{7 b^3 n}-\frac{4 a \left (a+b x^n\right )^{5/2}}{5 b^3 n} \]

[Out]

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

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Rubi [A]  time = 0.0919977, antiderivative size = 68, 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 \[ \frac{2 a^2 \left (a+b x^n\right )^{3/2}}{3 b^3 n}+\frac{2 \left (a+b x^n\right )^{7/2}}{7 b^3 n}-\frac{4 a \left (a+b x^n\right )^{5/2}}{5 b^3 n} \]

Antiderivative was successfully verified.

[In]  Int[x^(-1 + 3*n)*Sqrt[a + b*x^n],x]

[Out]

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

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Rubi in Sympy [A]  time = 12.3256, size = 60, normalized size = 0.88 \[ \frac{2 a^{2} \left (a + b x^{n}\right )^{\frac{3}{2}}}{3 b^{3} n} - \frac{4 a \left (a + b x^{n}\right )^{\frac{5}{2}}}{5 b^{3} n} + \frac{2 \left (a + b x^{n}\right )^{\frac{7}{2}}}{7 b^{3} n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**(-1+3*n)*(a+b*x**n)**(1/2),x)

[Out]

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

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Mathematica [A]  time = 0.0415162, size = 57, normalized size = 0.84 \[ \frac{2 \sqrt{a+b x^n} \left (8 a^3-4 a^2 b x^n+3 a b^2 x^{2 n}+15 b^3 x^{3 n}\right )}{105 b^3 n} \]

Antiderivative was successfully verified.

[In]  Integrate[x^(-1 + 3*n)*Sqrt[a + b*x^n],x]

[Out]

(2*Sqrt[a + b*x^n]*(8*a^3 - 4*a^2*b*x^n + 3*a*b^2*x^(2*n) + 15*b^3*x^(3*n)))/(10
5*b^3*n)

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Maple [A]  time = 0.032, size = 54, normalized size = 0.8 \[{\frac{30\, \left ({x}^{n} \right ) ^{3}{b}^{3}+6\,a \left ({x}^{n} \right ) ^{2}{b}^{2}-8\,{a}^{2}{x}^{n}b+16\,{a}^{3}}{105\,{b}^{3}n}\sqrt{a+b{x}^{n}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(-1+3*n)*(a+b*x^n)^(1/2),x)

[Out]

2/105*(15*(x^n)^3*b^3+3*a*(x^n)^2*b^2-4*a^2*x^n*b+8*a^3)*(a+b*x^n)^(1/2)/b^3/n

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Maxima [A]  time = 1.46713, size = 72, normalized size = 1.06 \[ \frac{2 \,{\left (15 \, b^{3} x^{3 \, n} + 3 \, a b^{2} x^{2 \, n} - 4 \, a^{2} b x^{n} + 8 \, a^{3}\right )} \sqrt{b x^{n} + a}}{105 \, b^{3} n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(b*x^n + a)*x^(3*n - 1),x, algorithm="maxima")

[Out]

2/105*(15*b^3*x^(3*n) + 3*a*b^2*x^(2*n) - 4*a^2*b*x^n + 8*a^3)*sqrt(b*x^n + a)/(
b^3*n)

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Fricas [A]  time = 0.221839, size = 72, normalized size = 1.06 \[ \frac{2 \,{\left (15 \, b^{3} x^{3 \, n} + 3 \, a b^{2} x^{2 \, n} - 4 \, a^{2} b x^{n} + 8 \, a^{3}\right )} \sqrt{b x^{n} + a}}{105 \, b^{3} n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(b*x^n + a)*x^(3*n - 1),x, algorithm="fricas")

[Out]

2/105*(15*b^3*x^(3*n) + 3*a*b^2*x^(2*n) - 4*a^2*b*x^n + 8*a^3)*sqrt(b*x^n + a)/(
b^3*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+3*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. \[ \int \sqrt{b x^{n} + a} x^{3 \, n - 1}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(b*x^n + a)*x^(3*n - 1),x, algorithm="giac")

[Out]

integrate(sqrt(b*x^n + a)*x^(3*n - 1), x)