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

Optimal. Leaf size=73 \[ \frac{x (b c-a d (1-n)) \, _2F_1\left (1,\frac{1}{n};1+\frac{1}{n};-\frac{d x^n}{c}\right )}{c^2 d n}-\frac{x (b c-a d)}{c d n \left (c+d x^n\right )} \]

[Out]

-(((b*c - a*d)*x)/(c*d*n*(c + d*x^n))) + ((b*c - a*d*(1 - n))*x*Hypergeometric2F
1[1, n^(-1), 1 + n^(-1), -((d*x^n)/c)])/(c^2*d*n)

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Rubi [A]  time = 0.0910764, antiderivative size = 73, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118 \[ \frac{x (b c-a d (1-n)) \, _2F_1\left (1,\frac{1}{n};1+\frac{1}{n};-\frac{d x^n}{c}\right )}{c^2 d n}-\frac{x (b c-a d)}{c d n \left (c+d x^n\right )} \]

Antiderivative was successfully verified.

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

[Out]

-(((b*c - a*d)*x)/(c*d*n*(c + d*x^n))) + ((b*c - a*d*(1 - n))*x*Hypergeometric2F
1[1, n^(-1), 1 + n^(-1), -((d*x^n)/c)])/(c^2*d*n)

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Rubi in Sympy [A]  time = 9.66263, size = 53, normalized size = 0.73 \[ \frac{x \left (a d - b c\right )}{c d n \left (c + d x^{n}\right )} + \frac{x \left (- a d \left (- n + 1\right ) + b c\right ){{}_{2}F_{1}\left (\begin{matrix} 1, \frac{1}{n} \\ 1 + \frac{1}{n} \end{matrix}\middle |{- \frac{d x^{n}}{c}} \right )}}{c^{2} d n} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0790748, size = 68, normalized size = 0.93 \[ \frac{x \left (\left (c+d x^n\right ) (a d (n-1)+b c) \, _2F_1\left (1,\frac{1}{n};1+\frac{1}{n};-\frac{d x^n}{c}\right )+c (a d-b c)\right )}{c^2 d n \left (c+d x^n\right )} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [F]  time = 0.064, size = 0, normalized size = 0. \[ \int{\frac{a+b{x}^{n}}{ \left ( c+d{x}^{n} \right ) ^{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b*x^n)/(c+d*x^n)^2,x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [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((a+b*x**n)/(c+d*x**n)**2,x)

[Out]

Exception raised: TypeError

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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