3.3029 \(\int (a+b x)^p (A+B x) (d+e x)^{-2-p} \, dx\)

Optimal. Leaf size=125 \[ -\frac{(a+b x)^{p+1} (B d-A e) (d+e x)^{-p-1}}{e (p+1) (b d-a e)}-\frac{B (a+b x)^p (d+e x)^{-p} \left (-\frac{e (a+b x)}{b d-a e}\right )^{-p} \, _2F_1\left (-p,-p;1-p;\frac{b (d+e x)}{b d-a e}\right )}{e^2 p} \]

[Out]

-(((B*d - A*e)*(a + b*x)^(1 + p)*(d + e*x)^(-1 - p))/(e*(b*d - a*e)*(1 + p))) -
(B*(a + b*x)^p*Hypergeometric2F1[-p, -p, 1 - p, (b*(d + e*x))/(b*d - a*e)])/(e^2
*p*(-((e*(a + b*x))/(b*d - a*e)))^p*(d + e*x)^p)

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Rubi [A]  time = 0.206315, antiderivative size = 125, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125 \[ -\frac{(a+b x)^{p+1} (B d-A e) (d+e x)^{-p-1}}{e (p+1) (b d-a e)}-\frac{B (a+b x)^p (d+e x)^{-p} \left (-\frac{e (a+b x)}{b d-a e}\right )^{-p} \, _2F_1\left (-p,-p;1-p;\frac{b (d+e x)}{b d-a e}\right )}{e^2 p} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x)^p*(A + B*x)*(d + e*x)^(-2 - p),x]

[Out]

-(((B*d - A*e)*(a + b*x)^(1 + p)*(d + e*x)^(-1 - p))/(e*(b*d - a*e)*(1 + p))) -
(B*(a + b*x)^p*Hypergeometric2F1[-p, -p, 1 - p, (b*(d + e*x))/(b*d - a*e)])/(e^2
*p*(-((e*(a + b*x))/(b*d - a*e)))^p*(d + e*x)^p)

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Rubi in Sympy [A]  time = 22.3429, size = 95, normalized size = 0.76 \[ - \frac{B \left (\frac{e \left (a + b x\right )}{a e - b d}\right )^{- p} \left (a + b x\right )^{p} \left (d + e x\right )^{- p}{{}_{2}F_{1}\left (\begin{matrix} - p, - p \\ - p + 1 \end{matrix}\middle |{\frac{b \left (- d - e x\right )}{a e - b d}} \right )}}{e^{2} p} - \frac{\left (a + b x\right )^{p + 1} \left (d + e x\right )^{- p - 1} \left (A e - B d\right )}{e \left (p + 1\right ) \left (a e - b d\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**p*(B*x+A)*(e*x+d)**(-2-p),x)

[Out]

-B*(e*(a + b*x)/(a*e - b*d))**(-p)*(a + b*x)**p*(d + e*x)**(-p)*hyper((-p, -p),
(-p + 1,), b*(-d - e*x)/(a*e - b*d))/(e**2*p) - (a + b*x)**(p + 1)*(d + e*x)**(-
p - 1)*(A*e - B*d)/(e*(p + 1)*(a*e - b*d))

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Mathematica [A]  time = 0.355286, size = 132, normalized size = 1.06 \[ \frac{(a+b x)^p (d+e x)^{-p-1} \left (A e^2 p (a+b x)-B (p+1) (d+e x) (b d-a e) \left (\frac{e (a+b x)}{a e-b d}\right )^{-p} \, _2F_1\left (-p,-p;1-p;\frac{b (d+e x)}{b d-a e}\right )-B d e p (a+b x)\right )}{e^2 p (p+1) (b d-a e)} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x)^p*(A + B*x)*(d + e*x)^(-2 - p),x]

[Out]

((a + b*x)^p*(d + e*x)^(-1 - p)*(-(B*d*e*p*(a + b*x)) + A*e^2*p*(a + b*x) - (B*(
b*d - a*e)*(1 + p)*(d + e*x)*Hypergeometric2F1[-p, -p, 1 - p, (b*(d + e*x))/(b*d
- a*e)])/((e*(a + b*x))/(-(b*d) + a*e))^p))/(e^2*(b*d - a*e)*p*(1 + p))

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Maple [F]  time = 0.077, size = 0, normalized size = 0. \[ \int \left ( bx+a \right ) ^{p} \left ( Bx+A \right ) \left ( ex+d \right ) ^{-2-p}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^p*(B*x+A)*(e*x+d)^(-2-p),x)

[Out]

int((b*x+a)^p*(B*x+A)*(e*x+d)^(-2-p),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (B x + A\right )}{\left (b x + a\right )}^{p}{\left (e x + d\right )}^{-p - 2}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)^p*(e*x + d)^(-p - 2),x, algorithm="maxima")

[Out]

integrate((B*x + A)*(b*x + a)^p*(e*x + d)^(-p - 2), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left ({\left (B x + A\right )}{\left (b x + a\right )}^{p}{\left (e x + d\right )}^{-p - 2}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)^p*(e*x + d)^(-p - 2),x, algorithm="fricas")

[Out]

integral((B*x + A)*(b*x + a)^p*(e*x + d)^(-p - 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((b*x+a)**p*(B*x+A)*(e*x+d)**(-2-p),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (B x + A\right )}{\left (b x + a\right )}^{p}{\left (e x + d\right )}^{-p - 2}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)^p*(e*x + d)^(-p - 2),x, algorithm="giac")

[Out]

integrate((B*x + A)*(b*x + a)^p*(e*x + d)^(-p - 2), x)