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