Optimal. Leaf size=200 \[ -\frac{2^{2 p-1} \left (a+b x+c x^2\right )^p \left (\frac{e \left (-\sqrt{b^2-4 a c}+b+2 c x\right )}{c (d+e x)}\right )^{-p} \left (\frac{e \left (\sqrt{b^2-4 a c}+b+2 c x\right )}{c (d+e x)}\right )^{-p} F_1\left (2 (1-p);-p,-p;3-2 p;\frac{2 c d-\left (b-\sqrt{b^2-4 a c}\right ) e}{2 c (d+e x)},\frac{2 d-\frac{\left (b+\sqrt{b^2-4 a c}\right ) e}{c}}{2 (d+e x)}\right )}{e (1-p) (d+e x)^2} \]
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Rubi [A] time = 0.475663, antiderivative size = 200, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1 \[ -\frac{2^{2 p-1} \left (a+b x+c x^2\right )^p \left (\frac{e \left (-\sqrt{b^2-4 a c}+b+2 c x\right )}{c (d+e x)}\right )^{-p} \left (\frac{e \left (\sqrt{b^2-4 a c}+b+2 c x\right )}{c (d+e x)}\right )^{-p} F_1\left (2 (1-p);-p,-p;3-2 p;\frac{2 c d-\left (b-\sqrt{b^2-4 a c}\right ) e}{2 c (d+e x)},\frac{2 d-\frac{\left (b+\sqrt{b^2-4 a c}\right ) e}{c}}{2 (d+e x)}\right )}{e (1-p) (d+e x)^2} \]
Antiderivative was successfully verified.
[In] Int[(a + b*x + c*x^2)^p/(d + e*x)^3,x]
[Out]
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Rubi in Sympy [A] time = 31.7882, size = 167, normalized size = 0.84 \[ - \frac{\left (\frac{e \left (b + 2 c x - \sqrt{- 4 a c + b^{2}}\right )}{2 c \left (d + e x\right )}\right )^{- p} \left (\frac{e \left (b + 2 c x + \sqrt{- 4 a c + b^{2}}\right )}{2 c \left (d + e x\right )}\right )^{- p} \left (a + b x + c x^{2}\right )^{p} \left (\frac{1}{d + e x}\right )^{2 p} \left (\frac{1}{d + e x}\right )^{- 2 p + 2} \operatorname{appellf_{1}}{\left (- 2 p + 2,- p,- p,- 2 p + 3,\frac{c d - \frac{e \left (b - \sqrt{- 4 a c + b^{2}}\right )}{2}}{c \left (d + e x\right )},\frac{c d - \frac{e \left (b + \sqrt{- 4 a c + b^{2}}\right )}{2}}{c \left (d + e x\right )} \right )}}{2 e \left (- p + 1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((c*x**2+b*x+a)**p/(e*x+d)**3,x)
[Out]
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Mathematica [A] time = 0.814973, size = 193, normalized size = 0.96 \[ \frac{2^{2 p-1} (a+x (b+c x))^p \left (\frac{e \left (-\sqrt{b^2-4 a c}+b+2 c x\right )}{c (d+e x)}\right )^{-p} \left (\frac{e \left (\sqrt{b^2-4 a c}+b+2 c x\right )}{c (d+e x)}\right )^{-p} F_1\left (2-2 p;-p,-p;3-2 p;\frac{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}{2 c (d+e x)},\frac{2 c d-b e+\sqrt{b^2-4 a c} e}{2 c d+2 c e x}\right )}{e (p-1) (d+e x)^2} \]
Warning: Unable to verify antiderivative.
[In] Integrate[(a + b*x + c*x^2)^p/(d + e*x)^3,x]
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Maple [F] time = 0.195, size = 0, normalized size = 0. \[ \int{\frac{ \left ( c{x}^{2}+bx+a \right ) ^{p}}{ \left ( ex+d \right ) ^{3}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((c*x^2+b*x+a)^p/(e*x+d)^3,x)
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (c x^{2} + b x + a\right )}^{p}}{{\left (e x + d\right )}^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x + a)^p/(e*x + d)^3,x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (c x^{2} + b x + a\right )}^{p}}{e^{3} x^{3} + 3 \, d e^{2} x^{2} + 3 \, d^{2} e x + d^{3}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x + a)^p/(e*x + d)^3,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((c*x**2+b*x+a)**p/(e*x+d)**3,x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (c x^{2} + b x + a\right )}^{p}}{{\left (e x + d\right )}^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x + a)^p/(e*x + d)^3,x, algorithm="giac")
[Out]