3.117 \(\int \frac{(e+f x)^n}{x (a+b x) (c+d x)} \, dx\)

Optimal. Leaf size=175 \[ \frac{b^2 (e+f x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac{b (e+f x)}{b e-a f}\right )}{a (n+1) (b c-a d) (b e-a f)}-\frac{d^2 (e+f x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac{d (e+f x)}{d e-c f}\right )}{c (n+1) (b c-a d) (d e-c f)}-\frac{(e+f x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac{f x}{e}+1\right )}{a c e (n+1)} \]

[Out]

(b^2*(e + f*x)^(1 + n)*Hypergeometric2F1[1, 1 + n, 2 + n, (b*(e + f*x))/(b*e - a
*f)])/(a*(b*c - a*d)*(b*e - a*f)*(1 + n)) - (d^2*(e + f*x)^(1 + n)*Hypergeometri
c2F1[1, 1 + n, 2 + n, (d*(e + f*x))/(d*e - c*f)])/(c*(b*c - a*d)*(d*e - c*f)*(1
+ n)) - ((e + f*x)^(1 + n)*Hypergeometric2F1[1, 1 + n, 2 + n, 1 + (f*x)/e])/(a*c
*e*(1 + n))

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Rubi [A]  time = 0.35225, antiderivative size = 175, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.12 \[ \frac{b^2 (e+f x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac{b (e+f x)}{b e-a f}\right )}{a (n+1) (b c-a d) (b e-a f)}-\frac{d^2 (e+f x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac{d (e+f x)}{d e-c f}\right )}{c (n+1) (b c-a d) (d e-c f)}-\frac{(e+f x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac{f x}{e}+1\right )}{a c e (n+1)} \]

Antiderivative was successfully verified.

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

[Out]

(b^2*(e + f*x)^(1 + n)*Hypergeometric2F1[1, 1 + n, 2 + n, (b*(e + f*x))/(b*e - a
*f)])/(a*(b*c - a*d)*(b*e - a*f)*(1 + n)) - (d^2*(e + f*x)^(1 + n)*Hypergeometri
c2F1[1, 1 + n, 2 + n, (d*(e + f*x))/(d*e - c*f)])/(c*(b*c - a*d)*(d*e - c*f)*(1
+ n)) - ((e + f*x)^(1 + n)*Hypergeometric2F1[1, 1 + n, 2 + n, 1 + (f*x)/e])/(a*c
*e*(1 + n))

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Rubi in Sympy [A]  time = 49.5462, size = 129, normalized size = 0.74 \[ - \frac{d^{2} \left (e + f x\right )^{n + 1}{{}_{2}F_{1}\left (\begin{matrix} 1, n + 1 \\ n + 2 \end{matrix}\middle |{\frac{d \left (- e - f x\right )}{c f - d e}} \right )}}{c \left (n + 1\right ) \left (a d - b c\right ) \left (c f - d e\right )} + \frac{b^{2} \left (e + f x\right )^{n + 1}{{}_{2}F_{1}\left (\begin{matrix} 1, n + 1 \\ n + 2 \end{matrix}\middle |{\frac{b \left (- e - f x\right )}{a f - b e}} \right )}}{a \left (n + 1\right ) \left (a d - b c\right ) \left (a f - b e\right )} - \frac{\left (e + f x\right )^{n + 1}{{}_{2}F_{1}\left (\begin{matrix} 1, n + 1 \\ n + 2 \end{matrix}\middle |{1 + \frac{f x}{e}} \right )}}{a c e \left (n + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.629879, size = 177, normalized size = 1.01 \[ (e+f x)^n \left (\frac{b^2 (e+f x) \, _2F_1\left (1,n+1;n+2;\frac{b (e+f x)}{b e-a f}\right )}{a (n+1) (a d-b c) (a f-b e)}+\frac{\frac{d^2 (e+f x) \, _2F_1\left (1,n+1;n+2;\frac{d (e+f x)}{d e-c f}\right )}{(n+1) (b c-a d) (c f-d e)}+\frac{\left (\frac{e}{f x}+1\right )^{-n} \, _2F_1\left (-n,-n;1-n;-\frac{e}{f x}\right )}{a n}}{c}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(e + f*x)^n*((b^2*(e + f*x)*Hypergeometric2F1[1, 1 + n, 2 + n, (b*(e + f*x))/(b*
e - a*f)])/(a*(-(b*c) + a*d)*(-(b*e) + a*f)*(1 + n)) + ((d^2*(e + f*x)*Hypergeom
etric2F1[1, 1 + n, 2 + n, (d*(e + f*x))/(d*e - c*f)])/((b*c - a*d)*(-(d*e) + c*f
)*(1 + n)) + Hypergeometric2F1[-n, -n, 1 - n, -(e/(f*x))]/(a*n*(1 + e/(f*x))^n))
/c)

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Maple [F]  time = 0.068, size = 0, normalized size = 0. \[ \int{\frac{ \left ( fx+e \right ) ^{n}}{x \left ( bx+a \right ) \left ( dx+c \right ) }}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((f*x+e)^n/x/(b*x+a)/(d*x+c),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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