3.222 \(\int \left (a+b x^n\right )^3 \left (c+d x^n\right )^{-4-\frac{1}{n}} \, dx\)

Optimal. Leaf size=178 \[ \frac{6 a^3 n^3 x \left (c+d x^n\right )^{-1/n}}{c^4 (n+1) (2 n+1) (3 n+1)}+\frac{6 a^2 n^2 x \left (a+b x^n\right ) \left (c+d x^n\right )^{-\frac{1}{n}-1}}{c^3 (n+1) (2 n+1) (3 n+1)}+\frac{3 a n x \left (a+b x^n\right )^2 \left (c+d x^n\right )^{-\frac{1}{n}-2}}{c^2 \left (6 n^2+5 n+1\right )}+\frac{x \left (a+b x^n\right )^3 \left (c+d x^n\right )^{-\frac{1}{n}-3}}{c (3 n+1)} \]

[Out]

(x*(a + b*x^n)^3*(c + d*x^n)^(-3 - n^(-1)))/(c*(1 + 3*n)) + (3*a*n*x*(a + b*x^n)
^2*(c + d*x^n)^(-2 - n^(-1)))/(c^2*(1 + 5*n + 6*n^2)) + (6*a^2*n^2*x*(a + b*x^n)
*(c + d*x^n)^(-1 - n^(-1)))/(c^3*(1 + n)*(1 + 2*n)*(1 + 3*n)) + (6*a^3*n^3*x)/(c
^4*(1 + n)*(1 + 2*n)*(1 + 3*n)*(c + d*x^n)^n^(-1))

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Rubi [A]  time = 0.240627, antiderivative size = 178, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.08 \[ \frac{6 a^3 n^3 x \left (c+d x^n\right )^{-1/n}}{c^4 (n+1) (2 n+1) (3 n+1)}+\frac{6 a^2 n^2 x \left (a+b x^n\right ) \left (c+d x^n\right )^{-\frac{1}{n}-1}}{c^3 (n+1) (2 n+1) (3 n+1)}+\frac{3 a n x \left (a+b x^n\right )^2 \left (c+d x^n\right )^{-\frac{1}{n}-2}}{c^2 \left (6 n^2+5 n+1\right )}+\frac{x \left (a+b x^n\right )^3 \left (c+d x^n\right )^{-\frac{1}{n}-3}}{c (3 n+1)} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^n)^3*(c + d*x^n)^(-4 - n^(-1)),x]

[Out]

(x*(a + b*x^n)^3*(c + d*x^n)^(-3 - n^(-1)))/(c*(1 + 3*n)) + (3*a*n*x*(a + b*x^n)
^2*(c + d*x^n)^(-2 - n^(-1)))/(c^2*(1 + 5*n + 6*n^2)) + (6*a^2*n^2*x*(a + b*x^n)
*(c + d*x^n)^(-1 - n^(-1)))/(c^3*(1 + n)*(1 + 2*n)*(1 + 3*n)) + (6*a^3*n^3*x)/(c
^4*(1 + n)*(1 + 2*n)*(1 + 3*n)*(c + d*x^n)^n^(-1))

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Rubi in Sympy [A]  time = 35.5955, size = 156, normalized size = 0.88 \[ \frac{6 a^{3} n^{3} x \left (c + d x^{n}\right )^{- \frac{1}{n}}}{c^{4} \left (n + 1\right ) \left (2 n + 1\right ) \left (3 n + 1\right )} + \frac{6 a^{2} n^{2} x \left (a + b x^{n}\right ) \left (c + d x^{n}\right )^{-1 - \frac{1}{n}}}{c^{3} \left (n + 1\right ) \left (2 n + 1\right ) \left (3 n + 1\right )} + \frac{3 a n x \left (a + b x^{n}\right )^{2} \left (c + d x^{n}\right )^{-2 - \frac{1}{n}}}{c^{2} \left (2 n + 1\right ) \left (3 n + 1\right )} + \frac{x \left (a + b x^{n}\right )^{3} \left (c + d x^{n}\right )^{-3 - \frac{1}{n}}}{c \left (3 n + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((a+b*x**n)**3*(c+d*x**n)**(-4-1/n),x)

[Out]

6*a**3*n**3*x*(c + d*x**n)**(-1/n)/(c**4*(n + 1)*(2*n + 1)*(3*n + 1)) + 6*a**2*n
**2*x*(a + b*x**n)*(c + d*x**n)**(-1 - 1/n)/(c**3*(n + 1)*(2*n + 1)*(3*n + 1)) +
3*a*n*x*(a + b*x**n)**2*(c + d*x**n)**(-2 - 1/n)/(c**2*(2*n + 1)*(3*n + 1)) + x
*(a + b*x**n)**3*(c + d*x**n)**(-3 - 1/n)/(c*(3*n + 1))

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Mathematica [C]  time = 0.607878, size = 198, normalized size = 1.11 \[ \frac{x \left (c+d x^n\right )^{-1/n} \left (a^3 \left (\frac{d x^n}{c}+1\right )^{\frac{1}{n}} \, _2F_1\left (4+\frac{1}{n},\frac{1}{n};1+\frac{1}{n};-\frac{d x^n}{c}\right )+\frac{3 a^2 b x^n \left (\frac{d x^n}{c}+1\right )^{\frac{1}{n}} \, _2F_1\left (1+\frac{1}{n},4+\frac{1}{n};2+\frac{1}{n};-\frac{d x^n}{c}\right )}{n+1}+\frac{3 a b^2 x^{2 n} \left (\frac{d x^n}{c}+1\right )^{\frac{1}{n}} \, _2F_1\left (2+\frac{1}{n},4+\frac{1}{n};3+\frac{1}{n};-\frac{d x^n}{c}\right )}{2 n+1}+\frac{b^3 c^3 x^{3 n}}{(3 n+1) \left (c+d x^n\right )^3}\right )}{c^4} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x^n)^3*(c + d*x^n)^(-4 - n^(-1)),x]

[Out]

(x*((b^3*c^3*x^(3*n))/((1 + 3*n)*(c + d*x^n)^3) + (3*a^2*b*x^n*(1 + (d*x^n)/c)^n
^(-1)*Hypergeometric2F1[1 + n^(-1), 4 + n^(-1), 2 + n^(-1), -((d*x^n)/c)])/(1 +
n) + (3*a*b^2*x^(2*n)*(1 + (d*x^n)/c)^n^(-1)*Hypergeometric2F1[2 + n^(-1), 4 + n
^(-1), 3 + n^(-1), -((d*x^n)/c)])/(1 + 2*n) + a^3*(1 + (d*x^n)/c)^n^(-1)*Hyperge
ometric2F1[4 + n^(-1), n^(-1), 1 + n^(-1), -((d*x^n)/c)]))/(c^4*(c + d*x^n)^n^(-
1))

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Maple [F]  time = 0.203, size = 0, normalized size = 0. \[ \int \left ( a+b{x}^{n} \right ) ^{3} \left ( c+d{x}^{n} \right ) ^{-4-{n}^{-1}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b*x^n)^3*(c+d*x^n)^(-4-1/n),x)

[Out]

int((a+b*x^n)^3*(c+d*x^n)^(-4-1/n),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (b x^{n} + a\right )}^{3}{\left (d x^{n} + c\right )}^{-\frac{1}{n} - 4}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^n + a)^3*(d*x^n + c)^(-1/n - 4),x, algorithm="maxima")

[Out]

integrate((b*x^n + a)^3*(d*x^n + c)^(-1/n - 4), x)

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Fricas [A]  time = 0.255308, size = 645, normalized size = 3.62 \[ \frac{{\left (6 \, a^{3} d^{4} n^{3} + b^{3} c^{3} d +{\left (2 \, b^{3} c^{3} d + 3 \, a b^{2} c^{2} d^{2} + 6 \, a^{2} b c d^{3}\right )} n^{2} + 3 \,{\left (b^{3} c^{3} d + a b^{2} c^{2} d^{2}\right )} n\right )} x x^{4 \, n} +{\left (24 \, a^{3} c d^{3} n^{3} + b^{3} c^{4} + 3 \, a b^{2} c^{3} d + 2 \,{\left (b^{3} c^{4} + 6 \, a b^{2} c^{3} d + 12 \, a^{2} b c^{2} d^{2} + 3 \, a^{3} c d^{3}\right )} n^{2} + 3 \,{\left (b^{3} c^{4} + 5 \, a b^{2} c^{3} d + 2 \, a^{2} b c^{2} d^{2}\right )} n\right )} x x^{3 \, n} + 3 \,{\left (12 \, a^{3} c^{2} d^{2} n^{3} + a b^{2} c^{4} + a^{2} b c^{3} d +{\left (3 \, a b^{2} c^{4} + 12 \, a^{2} b c^{3} d + 7 \, a^{3} c^{2} d^{2}\right )} n^{2} +{\left (4 \, a b^{2} c^{4} + 7 \, a^{2} b c^{3} d + a^{3} c^{2} d^{2}\right )} n\right )} x x^{2 \, n} +{\left (24 \, a^{3} c^{3} d n^{3} + 3 \, a^{2} b c^{4} + a^{3} c^{3} d + 2 \,{\left (9 \, a^{2} b c^{4} + 13 \, a^{3} c^{3} d\right )} n^{2} + 3 \,{\left (5 \, a^{2} b c^{4} + 3 \, a^{3} c^{3} d\right )} n\right )} x x^{n} +{\left (6 \, a^{3} c^{4} n^{3} + 11 \, a^{3} c^{4} n^{2} + 6 \, a^{3} c^{4} n + a^{3} c^{4}\right )} x}{{\left (6 \, c^{4} n^{3} + 11 \, c^{4} n^{2} + 6 \, c^{4} n + c^{4}\right )}{\left (d x^{n} + c\right )}^{\frac{4 \, n + 1}{n}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^n + a)^3*(d*x^n + c)^(-1/n - 4),x, algorithm="fricas")

[Out]

((6*a^3*d^4*n^3 + b^3*c^3*d + (2*b^3*c^3*d + 3*a*b^2*c^2*d^2 + 6*a^2*b*c*d^3)*n^
2 + 3*(b^3*c^3*d + a*b^2*c^2*d^2)*n)*x*x^(4*n) + (24*a^3*c*d^3*n^3 + b^3*c^4 + 3
*a*b^2*c^3*d + 2*(b^3*c^4 + 6*a*b^2*c^3*d + 12*a^2*b*c^2*d^2 + 3*a^3*c*d^3)*n^2
+ 3*(b^3*c^4 + 5*a*b^2*c^3*d + 2*a^2*b*c^2*d^2)*n)*x*x^(3*n) + 3*(12*a^3*c^2*d^2
*n^3 + a*b^2*c^4 + a^2*b*c^3*d + (3*a*b^2*c^4 + 12*a^2*b*c^3*d + 7*a^3*c^2*d^2)*
n^2 + (4*a*b^2*c^4 + 7*a^2*b*c^3*d + a^3*c^2*d^2)*n)*x*x^(2*n) + (24*a^3*c^3*d*n
^3 + 3*a^2*b*c^4 + a^3*c^3*d + 2*(9*a^2*b*c^4 + 13*a^3*c^3*d)*n^2 + 3*(5*a^2*b*c
^4 + 3*a^3*c^3*d)*n)*x*x^n + (6*a^3*c^4*n^3 + 11*a^3*c^4*n^2 + 6*a^3*c^4*n + a^3
*c^4)*x)/((6*c^4*n^3 + 11*c^4*n^2 + 6*c^4*n + c^4)*(d*x^n + c)^((4*n + 1)/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((a+b*x**n)**3*(c+d*x**n)**(-4-1/n),x)

[Out]

Timed out

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GIAC/XCAS [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((b*x^n + a)^3*(d*x^n + c)^(-1/n - 4),x, algorithm="giac")

[Out]

Exception raised: TypeError