3.3030 \(\int (a+b x) (c+d x)^n (e+f x)^{-n} \, dx\)

Optimal. Leaf size=134 \[ \frac{(c+d x)^{n+1} (e+f x)^{-n} \left (\frac{d (e+f x)}{d e-c f}\right )^n (2 a d f-b (c f (1-n)+d e (n+1))) \, _2F_1\left (n,n+1;n+2;-\frac{f (c+d x)}{d e-c f}\right )}{2 d^2 f (n+1)}+\frac{b (c+d x)^{n+1} (e+f x)^{1-n}}{2 d f} \]

[Out]

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

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Rubi [A]  time = 0.2355, antiderivative size = 134, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136 \[ \frac{(c+d x)^{n+1} (e+f x)^{-n} \left (\frac{d (e+f x)}{d e-c f}\right )^n (2 a d f-b c f (1-n)-b d e (n+1)) \, _2F_1\left (n,n+1;n+2;-\frac{f (c+d x)}{d e-c f}\right )}{2 d^2 f (n+1)}+\frac{b (c+d x)^{n+1} (e+f x)^{1-n}}{2 d f} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 22.8304, size = 105, normalized size = 0.78 \[ \frac{b \left (c + d x\right )^{n + 1} \left (e + f x\right )^{- n + 1}}{2 d f} - \frac{\left (\frac{d \left (- e - f x\right )}{c f - d e}\right )^{n} \left (c + d x\right )^{n + 1} \left (e + f x\right )^{- n} \left (- 2 a d f + b \left (c f \left (- n + 1\right ) + d e \left (n + 1\right )\right )\right ){{}_{2}F_{1}\left (\begin{matrix} n, n + 1 \\ n + 2 \end{matrix}\middle |{\frac{f \left (c + d x\right )}{c f - d e}} \right )}}{2 d^{2} f \left (n + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [C]  time = 0.688552, size = 201, normalized size = 1.5 \[ (c+d x)^n (e+f x)^{-n} \left (\frac{3 b c e x^2 F_1\left (2;-n,n;3;-\frac{d x}{c},-\frac{f x}{e}\right )}{6 c e F_1\left (2;-n,n;3;-\frac{d x}{c},-\frac{f x}{e}\right )+2 n x \left (d e F_1\left (3;1-n,n;4;-\frac{d x}{c},-\frac{f x}{e}\right )-c f F_1\left (3;-n,n+1;4;-\frac{d x}{c},-\frac{f x}{e}\right )\right )}-\frac{a (e+f x) \left (\frac{f (c+d x)}{c f-d e}\right )^{-n} \, _2F_1\left (1-n,-n;2-n;\frac{d (e+f x)}{d e-c f}\right )}{f (n-1)}\right ) \]

Warning: Unable to verify antiderivative.

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

[Out]

((c + d*x)^n*((3*b*c*e*x^2*AppellF1[2, -n, n, 3, -((d*x)/c), -((f*x)/e)])/(6*c*e
*AppellF1[2, -n, n, 3, -((d*x)/c), -((f*x)/e)] + 2*n*x*(d*e*AppellF1[3, 1 - n, n
, 4, -((d*x)/c), -((f*x)/e)] - c*f*AppellF1[3, -n, 1 + n, 4, -((d*x)/c), -((f*x)
/e)])) - (a*(e + f*x)*Hypergeometric2F1[1 - n, -n, 2 - n, (d*(e + f*x))/(d*e - c
*f)])/(f*(-1 + n)*((f*(c + d*x))/(-(d*e) + c*f))^n)))/(e + f*x)^n

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Maple [F]  time = 0.086, size = 0, normalized size = 0. \[ \int{\frac{ \left ( bx+a \right ) \left ( dx+c \right ) ^{n}}{ \left ( fx+e \right ) ^{n}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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