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

Optimal. Leaf size=139 \[ \frac{d (a+b x)^{n+1} (a d-b c (1-n)) \, _2F_1\left (1,n+1;n+2;-\frac{d (a+b x)}{b c-a d}\right )}{c^2 (n+1) (b c-a d)^2}-\frac{(a+b x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac{b x}{a}+1\right )}{a c^2 (n+1)}-\frac{d (a+b x)^{n+1}}{c (c+d x) (b c-a d)} \]

[Out]

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

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Rubi [A]  time = 0.243495, antiderivative size = 139, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222 \[ \frac{d (a+b x)^{n+1} (a d-b c (1-n)) \, _2F_1\left (1,n+1;n+2;-\frac{d (a+b x)}{b c-a d}\right )}{c^2 (n+1) (b c-a d)^2}-\frac{(a+b x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac{b x}{a}+1\right )}{a c^2 (n+1)}-\frac{d (a+b x)^{n+1}}{c (c+d x) (b c-a d)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 41.4574, size = 107, normalized size = 0.77 \[ \frac{d \left (a + b x\right )^{n + 1}}{c \left (c + d x\right ) \left (a d - b c\right )} + \frac{d \left (a + b x\right )^{n + 1} \left (a d + b c n - b c\right ){{}_{2}F_{1}\left (\begin{matrix} 1, n + 1 \\ n + 2 \end{matrix}\middle |{\frac{d \left (a + b x\right )}{a d - b c}} \right )}}{c^{2} \left (n + 1\right ) \left (a d - b c\right )^{2}} - \frac{\left (a + b x\right )^{n + 1}{{}_{2}F_{1}\left (\begin{matrix} 1, n + 1 \\ n + 2 \end{matrix}\middle |{1 + \frac{b x}{a}} \right )}}{a c^{2} \left (n + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0774103, size = 0, normalized size = 0. \[ \int \frac{(a+b x)^n}{x (c+d x)^2} \, dx \]

Verification is Not applicable to the result.

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

[Out]

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

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Maple [F]  time = 0.066, size = 0, normalized size = 0. \[ \int{\frac{ \left ( bx+a \right ) ^{n}}{x \left ( dx+c \right ) ^{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^n/x/(d*x+c)^2,x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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