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

Optimal. Leaf size=190 \[ \frac{(a+b x)^{n+1} (2 a d-b c n) \, _2F_1\left (1,n+1;n+2;\frac{b x}{a}+1\right )}{a^2 c^3 (n+1)}-\frac{d^2 (a+b x)^{n+1} (2 a d-b c (2-n)) \, _2F_1\left (1,n+1;n+2;-\frac{d (a+b x)}{b c-a d}\right )}{c^3 (n+1) (b c-a d)^2}-\frac{d (b c-2 a d) (a+b x)^{n+1}}{a c^2 (c+d x) (b c-a d)}-\frac{(a+b x)^{n+1}}{a c x (c+d x)} \]

[Out]

-((d*(b*c - 2*a*d)*(a + b*x)^(1 + n))/(a*c^2*(b*c - a*d)*(c + d*x))) - (a + b*x)
^(1 + n)/(a*c*x*(c + d*x)) - (d^2*(2*a*d - b*c*(2 - n))*(a + b*x)^(1 + n)*Hyperg
eometric2F1[1, 1 + n, 2 + n, -((d*(a + b*x))/(b*c - a*d))])/(c^3*(b*c - a*d)^2*(
1 + n)) + ((2*a*d - b*c*n)*(a + b*x)^(1 + n)*Hypergeometric2F1[1, 1 + n, 2 + n,
1 + (b*x)/a])/(a^2*c^3*(1 + n))

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Rubi [A]  time = 0.567042, antiderivative size = 190, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.278 \[ \frac{(a+b x)^{n+1} (2 a d-b c n) \, _2F_1\left (1,n+1;n+2;\frac{b x}{a}+1\right )}{a^2 c^3 (n+1)}-\frac{d^2 (a+b x)^{n+1} (2 a d-b c (2-n)) \, _2F_1\left (1,n+1;n+2;-\frac{d (a+b x)}{b c-a d}\right )}{c^3 (n+1) (b c-a d)^2}-\frac{d (b c-2 a d) (a+b x)^{n+1}}{a c^2 (c+d x) (b c-a d)}-\frac{(a+b x)^{n+1}}{a c x (c+d x)} \]

Antiderivative was successfully verified.

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

[Out]

-((d*(b*c - 2*a*d)*(a + b*x)^(1 + n))/(a*c^2*(b*c - a*d)*(c + d*x))) - (a + b*x)
^(1 + n)/(a*c*x*(c + d*x)) - (d^2*(2*a*d - b*c*(2 - n))*(a + b*x)^(1 + n)*Hyperg
eometric2F1[1, 1 + n, 2 + n, -((d*(a + b*x))/(b*c - a*d))])/(c^3*(b*c - a*d)^2*(
1 + n)) + ((2*a*d - b*c*n)*(a + b*x)^(1 + n)*Hypergeometric2F1[1, 1 + n, 2 + n,
1 + (b*x)/a])/(a^2*c^3*(1 + n))

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Rubi in Sympy [A]  time = 75.7803, size = 156, normalized size = 0.82 \[ \frac{d \left (a + b x\right )^{n + 1}}{c x \left (c + d x\right ) \left (a d - b c\right )} - \frac{d^{2} \left (a + b x\right )^{n + 1} \left (2 a d + b c n - 2 b c\right ){{}_{2}F_{1}\left (\begin{matrix} 1, n + 1 \\ n + 2 \end{matrix}\middle |{\frac{d \left (a + b x\right )}{a d - b c}} \right )}}{c^{3} \left (n + 1\right ) \left (a d - b c\right )^{2}} - \frac{\left (a + b x\right )^{n + 1} \left (2 a d - b c\right )}{a c^{2} x \left (a d - b c\right )} + \frac{\left (a + b x\right )^{n + 1} \left (2 a d - b c n\right ){{}_{2}F_{1}\left (\begin{matrix} 1, n + 1 \\ n + 2 \end{matrix}\middle |{1 + \frac{b x}{a}} \right )}}{a^{2} c^{3} \left (n + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0891338, size = 0, normalized size = 0. \[ \int \frac{(a+b x)^n}{x^2 (c+d x)^2} \, dx \]

Verification is Not applicable to the result.

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

[Out]

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

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Maple [F]  time = 0.093, size = 0, normalized size = 0. \[ \int{\frac{ \left ( bx+a \right ) ^{n}}{{x}^{2} \left ( dx+c \right ) ^{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^n/x^2/(d*x+c)^2,x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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