3.207 \(\int \frac{\left (c+d x^n\right )^3}{\left (a+b x^n\right )^2} \, dx\)

Optimal. Leaf size=200 \[ -\frac{x (b c-a d)^2 (b c (1-n)-a d (2 n+1)) \, _2F_1\left (1,\frac{1}{n};1+\frac{1}{n};-\frac{b x^n}{a}\right )}{a^2 b^3 n}-\frac{d x \left (a^2 d^2 \left (2 n^2+3 n+1\right )-a b c d \left (3 n^2+4 n+2\right )+b^2 c^2 (n+1)\right )}{a b^3 n (n+1)}-\frac{d x \left (c+d x^n\right ) (b c (n+1)-a d (2 n+1))}{a b^2 n (n+1)}+\frac{x (b c-a d) \left (c+d x^n\right )^2}{a b n \left (a+b x^n\right )} \]

[Out]

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

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Rubi [A]  time = 0.609973, antiderivative size = 200, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21 \[ -\frac{x (b c-a d)^2 (b c (1-n)-a d (2 n+1)) \, _2F_1\left (1,\frac{1}{n};1+\frac{1}{n};-\frac{b x^n}{a}\right )}{a^2 b^3 n}-\frac{d x \left (a^2 d^2 \left (2 n^2+3 n+1\right )-a b c d \left (3 n^2+4 n+2\right )+b^2 c^2 (n+1)\right )}{a b^3 n (n+1)}-\frac{d x \left (c+d x^n\right ) (b c (n+1)-a d (2 n+1))}{a b^2 n (n+1)}+\frac{x (b c-a d) \left (c+d x^n\right )^2}{a b n \left (a+b x^n\right )} \]

Antiderivative was successfully verified.

[In]  Int[(c + d*x^n)^3/(a + b*x^n)^2,x]

[Out]

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

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Rubi in Sympy [A]  time = 65.6107, size = 172, normalized size = 0.86 \[ - \frac{x \left (c + d x^{n}\right )^{2} \left (a d - b c\right )}{a b n \left (a + b x^{n}\right )} + \frac{d x \left (c \left (a d - b c \left (- n + 1\right )\right ) + d x^{n} \left (a d \left (2 n + 1\right ) - b c \left (n + 1\right )\right )\right )}{a b^{2} n \left (n + 1\right )} - \frac{d x \left (a d n + a d - b c\right ) \left (2 a d n + a d - 3 b c n - b c\right )}{a b^{3} n \left (n + 1\right )} + \frac{x \left (a d - b c\right )^{2} \left (2 a d n + a d + b c n - b c\right ){{}_{2}F_{1}\left (\begin{matrix} 1, \frac{1}{n} \\ 1 + \frac{1}{n} \end{matrix}\middle |{- \frac{b x^{n}}{a}} \right )}}{a^{2} b^{3} n} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.457563, size = 167, normalized size = 0.84 \[ \frac{x \left (\frac{a \left (-a^3 d^3 \left (2 n^2+3 n+1\right )+a^2 b d^2 \left (3 c (n+1)^2-d n (2 n+1) x^n\right )+a b^2 d \left (-3 c^2 (n+1)+3 c d n (n+1) x^n+d^2 n x^{2 n}\right )+b^3 c^3 (n+1)\right )}{(n+1) \left (a+b x^n\right )}+(b c-a d)^2 (a d (2 n+1)+b c (n-1)) \, _2F_1\left (1,\frac{1}{n};1+\frac{1}{n};-\frac{b x^n}{a}\right )\right )}{a^2 b^3 n} \]

Antiderivative was successfully verified.

[In]  Integrate[(c + d*x^n)^3/(a + b*x^n)^2,x]

[Out]

(x*((a*(b^3*c^3*(1 + n) - a^3*d^3*(1 + 3*n + 2*n^2) + a^2*b*d^2*(3*c*(1 + n)^2 -
d*n*(1 + 2*n)*x^n) + a*b^2*d*(-3*c^2*(1 + n) + 3*c*d*n*(1 + n)*x^n + d^2*n*x^(2
*n))))/((1 + n)*(a + b*x^n)) + (b*c - a*d)^2*(b*c*(-1 + n) + a*d*(1 + 2*n))*Hype
rgeometric2F1[1, n^(-1), 1 + n^(-1), -((b*x^n)/a)]))/(a^2*b^3*n)

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Maple [F]  time = 0.073, size = 0, normalized size = 0. \[ \int{\frac{ \left ( c+d{x}^{n} \right ) ^{3}}{ \left ( a+b{x}^{n} \right ) ^{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c+d*x^n)^3/(a+b*x^n)^2,x)

[Out]

int((c+d*x^n)^3/(a+b*x^n)^2,x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[{\left (a^{3} d^{3}{\left (2 \, n + 1\right )} - 3 \, a^{2} b c d^{2}{\left (n + 1\right )} + b^{3} c^{3}{\left (n - 1\right )} + 3 \, a b^{2} c^{2} d\right )} \int \frac{1}{a b^{4} n x^{n} + a^{2} b^{3} n}\,{d x} + \frac{a b^{2} d^{3} n x x^{2 \, n} +{\left (3 \,{\left (n^{2} + n\right )} a b^{2} c d^{2} -{\left (2 \, n^{2} + n\right )} a^{2} b d^{3}\right )} x x^{n} +{\left (3 \,{\left (n^{2} + 2 \, n + 1\right )} a^{2} b c d^{2} -{\left (2 \, n^{2} + 3 \, n + 1\right )} a^{3} d^{3} + b^{3} c^{3}{\left (n + 1\right )} - 3 \, a b^{2} c^{2} d{\left (n + 1\right )}\right )} x}{{\left (n^{2} + n\right )} a b^{4} x^{n} +{\left (n^{2} + n\right )} a^{2} b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x^n + c)^3/(b*x^n + a)^2,x, algorithm="maxima")

[Out]

(a^3*d^3*(2*n + 1) - 3*a^2*b*c*d^2*(n + 1) + b^3*c^3*(n - 1) + 3*a*b^2*c^2*d)*in
tegrate(1/(a*b^4*n*x^n + a^2*b^3*n), x) + (a*b^2*d^3*n*x*x^(2*n) + (3*(n^2 + n)*
a*b^2*c*d^2 - (2*n^2 + n)*a^2*b*d^3)*x*x^n + (3*(n^2 + 2*n + 1)*a^2*b*c*d^2 - (2
*n^2 + 3*n + 1)*a^3*d^3 + b^3*c^3*(n + 1) - 3*a*b^2*c^2*d*(n + 1))*x)/((n^2 + n)
*a*b^4*x^n + (n^2 + n)*a^2*b^3)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x^n + c)^3/(b*x^n + a)^2,x, algorithm="fricas")

[Out]

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

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (d x^{n} + c\right )}^{3}}{{\left (b x^{n} + a\right )}^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x^n + c)^3/(b*x^n + a)^2,x, algorithm="giac")

[Out]

integrate((d*x^n + c)^3/(b*x^n + a)^2, x)