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

Optimal. Leaf size=160 \[ -\frac{x \left (-a^2 d^2 \left (2 n^2-3 n+1\right )+2 a b c d (1-n)-b^2 c^2 (n+1)\right ) \, _2F_1\left (1,\frac{1}{n};1+\frac{1}{n};-\frac{d x^n}{c}\right )}{2 c^3 d^2 n^2}+\frac{x (b c-a d) (a d (1-2 n)-b c (n+1))}{2 c^2 d^2 n^2 \left (c+d x^n\right )}-\frac{x (b c-a d) \left (a+b x^n\right )}{2 c d n \left (c+d x^n\right )^2} \]

[Out]

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

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Rubi [A]  time = 0.359242, antiderivative size = 160, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158 \[ -\frac{x \left (-a^2 d^2 \left (2 n^2-3 n+1\right )+2 a b c d (1-n)-b^2 c^2 (n+1)\right ) \, _2F_1\left (1,\frac{1}{n};1+\frac{1}{n};-\frac{d x^n}{c}\right )}{2 c^3 d^2 n^2}+\frac{x (b c-a d) (a d (1-2 n)-b c (n+1))}{2 c^2 d^2 n^2 \left (c+d x^n\right )}-\frac{x (b c-a d) \left (a+b x^n\right )}{2 c d n \left (c+d x^n\right )^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 27.9249, size = 141, normalized size = 0.88 \[ \frac{x \left (a + b x^{n}\right ) \left (a d - b c\right )}{2 c d n \left (c + d x^{n}\right )^{2}} - \frac{x \left (a d - b c\right ) \left (- 2 a d n + a d - b c n - b c\right )}{2 c^{2} d^{2} n^{2} \left (c + d x^{n}\right )} - \frac{x \left (- a d \left (- n + 1\right ) \left (- 2 a d n + a d - b c\right ) + b c \left (a d \left (- n + 1\right ) - b c \left (n + 1\right )\right )\right ){{}_{2}F_{1}\left (\begin{matrix} 1, \frac{1}{n} \\ 1 + \frac{1}{n} \end{matrix}\middle |{- \frac{d x^{n}}{c}} \right )}}{2 c^{3} d^{2} n^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.173422, size = 133, normalized size = 0.83 \[ \frac{x \left (\left (a^2 d^2 \left (2 n^2-3 n+1\right )+2 a b c d (n-1)+b^2 c^2 (n+1)\right ) \, _2F_1\left (1,\frac{1}{n};1+\frac{1}{n};-\frac{d x^n}{c}\right )+\frac{c^2 n (b c-a d)^2}{\left (c+d x^n\right )^2}-\frac{c (b c-a d) (a d (2 n-1)+b (2 c n+c))}{c+d x^n}\right )}{2 c^3 d^2 n^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [F]  time = 0.079, size = 0, normalized size = 0. \[ \int{\frac{ \left ( a+b{x}^{n} \right ) ^{2}}{ \left ( c+d{x}^{n} \right ) ^{3}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b*x^n)^2/(c+d*x^n)^3,x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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