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

Optimal. Leaf size=124 \[ \frac{d (e+f x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac{d (e+f x)}{d e-c f}\right )}{(n+1) (b c-a d) (d e-c f)}-\frac{b (e+f x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac{b (e+f x)}{b e-a f}\right )}{(n+1) (b c-a d) (b e-a f)} \]

[Out]

-((b*(e + f*x)^(1 + n)*Hypergeometric2F1[1, 1 + n, 2 + n, (b*(e + f*x))/(b*e - a
*f)])/((b*c - a*d)*(b*e - a*f)*(1 + n))) + (d*(e + f*x)^(1 + n)*Hypergeometric2F
1[1, 1 + n, 2 + n, (d*(e + f*x))/(d*e - c*f)])/((b*c - a*d)*(d*e - c*f)*(1 + n))

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Rubi [A]  time = 0.131416, antiderivative size = 124, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091 \[ \frac{d (e+f x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac{d (e+f x)}{d e-c f}\right )}{(n+1) (b c-a d) (d e-c f)}-\frac{b (e+f x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac{b (e+f x)}{b e-a f}\right )}{(n+1) (b c-a d) (b e-a f)} \]

Antiderivative was successfully verified.

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

[Out]

-((b*(e + f*x)^(1 + n)*Hypergeometric2F1[1, 1 + n, 2 + n, (b*(e + f*x))/(b*e - a
*f)])/((b*c - a*d)*(b*e - a*f)*(1 + n))) + (d*(e + f*x)^(1 + n)*Hypergeometric2F
1[1, 1 + n, 2 + n, (d*(e + f*x))/(d*e - c*f)])/((b*c - a*d)*(d*e - c*f)*(1 + n))

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Rubi in Sympy [A]  time = 20.3654, size = 94, normalized size = 0.76 \[ - \frac{b \left (e + f x\right )^{n + 1}{{}_{2}F_{1}\left (\begin{matrix} 1, n + 1 \\ n + 2 \end{matrix}\middle |{\frac{b \left (- e - f x\right )}{a f - b e}} \right )}}{\left (n + 1\right ) \left (a d - b c\right ) \left (a f - b e\right )} + \frac{d \left (e + f x\right )^{n + 1}{{}_{2}F_{1}\left (\begin{matrix} 1, n + 1 \\ n + 2 \end{matrix}\middle |{\frac{d \left (- e - f x\right )}{c f - d e}} \right )}}{\left (n + 1\right ) \left (a d - b c\right ) \left (c f - d e\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.121052, size = 116, normalized size = 0.94 \[ \frac{(e+f x)^{n+1} \left (b (d e-c f) \, _2F_1\left (1,n+1;n+2;\frac{b (e+f x)}{b e-a f}\right )+d (a f-b e) \, _2F_1\left (1,n+1;n+2;\frac{d (e+f x)}{d e-c f}\right )\right )}{(n+1) (b c-a d) (b e-a f) (c f-d e)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [F]  time = 0.082, size = 0, normalized size = 0. \[ \int{\frac{ \left ( fx+e \right ) ^{n}}{ \left ( bx+a \right ) \left ( dx+c \right ) }}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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