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

Optimal. Leaf size=131 \[ \frac{b x \left (c+d x^n\right )^{2-\frac{1}{n}}}{2 a n (b c-a d) \left (a+b x^n\right )^2}-\frac{c x \left (c+d x^n\right )^{-1/n} (2 a d n+b c (1-2 n)) \, _2F_1\left (2,\frac{1}{n};1+\frac{1}{n};-\frac{(b c-a d) x^n}{a \left (d x^n+c\right )}\right )}{2 a^3 n (b c-a d)} \]

[Out]

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

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Rubi [A]  time = 0.16193, antiderivative size = 131, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.08 \[ \frac{b x \left (c+d x^n\right )^{2-\frac{1}{n}}}{2 a n (b c-a d) \left (a+b x^n\right )^2}-\frac{c x \left (c+d x^n\right )^{-1/n} (2 a d n+b (c-2 c n)) \, _2F_1\left (2,\frac{1}{n};1+\frac{1}{n};-\frac{(b c-a d) x^n}{a \left (d x^n+c\right )}\right )}{2 a^3 n (b c-a d)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 19.6972, size = 105, normalized size = 0.8 \[ - \frac{b x \left (c + d x^{n}\right )^{2 - \frac{1}{n}}}{2 a n \left (a + b x^{n}\right )^{2} \left (a d - b c\right )} + \frac{c x \left (c + d x^{n}\right )^{- \frac{1}{n}} \left (2 a d n - 2 b c n + b c\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{n}, 2 \\ 1 + \frac{1}{n} \end{matrix}\middle |{- \frac{x^{n} \left (- a d + b c\right )}{a \left (c + d x^{n}\right )}} \right )}}{2 a^{3} n \left (a d - b c\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-b*x*(c + d*x**n)**(2 - 1/n)/(2*a*n*(a + b*x**n)**2*(a*d - b*c)) + c*x*(c + d*x*
*n)**(-1/n)*(2*a*d*n - 2*b*c*n + b*c)*hyper((1/n, 2), (1 + 1/n,), -x**n*(-a*d +
b*c)/(a*(c + d*x**n)))/(2*a**3*n*(a*d - b*c))

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Mathematica [B]  time = 43.7676, size = 1241, normalized size = 9.47 \[ \text{result too large to display} \]

Warning: Unable to verify antiderivative.

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

[Out]

-((c^3*(1 + n)*(1 + 2*n)*(1 + 3*n)*x*(c + d*x^n)^(2 - n^(-1))*Gamma[2 + n^(-1)]*
(Hypergeometric2F1[1, 3, 1 + n^(-1), ((b*c - a*d)*x^n)/(c*(a + b*x^n))] + (d*n*x
^n*((c*Hypergeometric2F1[1, 3, 2 + n^(-1), ((b*c - a*d)*x^n)/(c*(a + b*x^n))])/(
1 + n) + (3*(b*c - a*d)*x^n*Gamma[1 + n^(-1)]*Hypergeometric2F1[2, 4, 3 + n^(-1)
, ((b*c - a*d)*x^n)/(c*(a + b*x^n))])/((1 + 2*n)*(a + b*x^n)*Gamma[2 + n^(-1)]))
)/c^2))/(c*d*(1 - 2*n)*(1 + 3*n)*x^n*(a + b*x^n)^2*(c^2*(1 + n)*(1 + 2*n)*(a + b
*x^n)*Gamma[2 + n^(-1)]*Hypergeometric2F1[1, 3, 1 + n^(-1), ((b*c - a*d)*x^n)/(c
*(a + b*x^n))] + d*n*x^n*(c*(1 + 2*n)*(a + b*x^n)*Gamma[2 + n^(-1)]*Hypergeometr
ic2F1[1, 3, 2 + n^(-1), ((b*c - a*d)*x^n)/(c*(a + b*x^n))] + 3*(b*c - a*d)*(1 +
n)*x^n*Gamma[1 + n^(-1)]*Hypergeometric2F1[2, 4, 3 + n^(-1), ((b*c - a*d)*x^n)/(
c*(a + b*x^n))])) + 3*b*c*n*(1 + 3*n)*x^n*(a + b*x^n)*(c + d*x^n)*(c^2*(1 + n)*(
1 + 2*n)*(a + b*x^n)*Gamma[2 + n^(-1)]*Hypergeometric2F1[1, 3, 1 + n^(-1), ((b*c
- a*d)*x^n)/(c*(a + b*x^n))] + d*n*x^n*(c*(1 + 2*n)*(a + b*x^n)*Gamma[2 + n^(-1
)]*Hypergeometric2F1[1, 3, 2 + n^(-1), ((b*c - a*d)*x^n)/(c*(a + b*x^n))] + 3*(b
*c - a*d)*(1 + n)*x^n*Gamma[1 + n^(-1)]*Hypergeometric2F1[2, 4, 3 + n^(-1), ((b*
c - a*d)*x^n)/(c*(a + b*x^n))])) - c*(1 + 3*n)*(a + b*x^n)^2*(c + d*x^n)*(c^2*(1
+ n)*(1 + 2*n)*(a + b*x^n)*Gamma[2 + n^(-1)]*Hypergeometric2F1[1, 3, 1 + n^(-1)
, ((b*c - a*d)*x^n)/(c*(a + b*x^n))] + d*n*x^n*(c*(1 + 2*n)*(a + b*x^n)*Gamma[2
+ n^(-1)]*Hypergeometric2F1[1, 3, 2 + n^(-1), ((b*c - a*d)*x^n)/(c*(a + b*x^n))]
+ 3*(b*c - a*d)*(1 + n)*x^n*Gamma[1 + n^(-1)]*Hypergeometric2F1[2, 4, 3 + n^(-1
), ((b*c - a*d)*x^n)/(c*(a + b*x^n))])) + n^2*x^n*(c + d*x^n)*(3*a*c^2*(-(b*c) +
a*d)*(1 + 2*n)*(1 + 3*n)*(a + b*x^n)*Gamma[2 + n^(-1)]*Hypergeometric2F1[2, 4,
2 + n^(-1), ((b*c - a*d)*x^n)/(c*(a + b*x^n))] - c*d*(1 + 3*n)*(a + b*x^n)^2*(c*
(1 + 2*n)*(a + b*x^n)*Gamma[2 + n^(-1)]*Hypergeometric2F1[1, 3, 2 + n^(-1), ((b*
c - a*d)*x^n)/(c*(a + b*x^n))] + 3*(b*c - a*d)*(1 + n)*x^n*Gamma[1 + n^(-1)]*Hyp
ergeometric2F1[2, 4, 3 + n^(-1), ((b*c - a*d)*x^n)/(c*(a + b*x^n))]) + 3*d*(b*c
- a*d)*x^n*(b*c*(1 + n)*(1 + 3*n)*x^n*(a + b*x^n)*Gamma[1 + n^(-1)]*Hypergeometr
ic2F1[2, 4, 3 + n^(-1), ((b*c - a*d)*x^n)/(c*(a + b*x^n))] - c*(1 + n)*(1 + 3*n)
*(a + b*x^n)^2*Gamma[1 + n^(-1)]*Hypergeometric2F1[2, 4, 3 + n^(-1), ((b*c - a*d
)*x^n)/(c*(a + b*x^n))] - a*c*n*(1 + 3*n)*(a + b*x^n)*Gamma[2 + n^(-1)]*Hypergeo
metric2F1[2, 4, 3 + n^(-1), ((b*c - a*d)*x^n)/(c*(a + b*x^n))] + 8*a*(-(b*c) + a
*d)*n*(1 + n)*x^n*Gamma[1 + n^(-1)]*Hypergeometric2F1[3, 5, 4 + n^(-1), ((b*c -
a*d)*x^n)/(c*(a + b*x^n))]))))

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Maple [F]  time = 0.113, size = 0, normalized size = 0. \[ \int{\frac{1}{ \left ( a+b{x}^{n} \right ) ^{3}} \left ( c+d{x}^{n} \right ) ^{1-{n}^{-1}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (d x^{n} + c\right )}^{\frac{n - 1}{n}}}{b^{3} x^{3 \, n} + 3 \, a b^{2} x^{2 \, n} + 3 \, a^{2} b x^{n} + a^{3}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((d*x^n + c)^((n - 1)/n)/(b^3*x^(3*n) + 3*a*b^2*x^(2*n) + 3*a^2*b*x^n +
a^3), x)

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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