Optimal. Leaf size=103 \[ \frac{\left (b x+c x^2\right )^p (d+e x)^{m+1} \left (-\frac{e x}{d}\right )^{-p} \left (1-\frac{c (d+e x)}{c d-b e}\right )^{-p} F_1\left (m+1;-p,-p;m+2;\frac{d+e x}{d},\frac{c (d+e x)}{c d-b e}\right )}{e (m+1)} \]
[Out]
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Rubi [A] time = 0.147408, antiderivative size = 103, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105 \[ \frac{\left (b x+c x^2\right )^p (d+e x)^{m+1} \left (-\frac{e x}{d}\right )^{-p} \left (1-\frac{c (d+e x)}{c d-b e}\right )^{-p} F_1\left (m+1;-p,-p;m+2;\frac{d+e x}{d},\frac{c (d+e x)}{c d-b e}\right )}{e (m+1)} \]
Antiderivative was successfully verified.
[In] Int[(d + e*x)^m*(b*x + c*x^2)^p,x]
[Out]
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Rubi in Sympy [A] time = 21.5791, size = 76, normalized size = 0.74 \[ \frac{\left (- \frac{e x}{d}\right )^{- p} \left (d + e x\right )^{m + 1} \left (b x + c x^{2}\right )^{p} \left (\frac{c \left (d + e x\right )}{b e - c d} + 1\right )^{- p} \operatorname{appellf_{1}}{\left (m + 1,- p,- p,m + 2,\frac{d + e x}{d},\frac{c \left (- d - e x\right )}{b e - c d} \right )}}{e \left (m + 1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((e*x+d)**m*(c*x**2+b*x)**p,x)
[Out]
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Mathematica [A] time = 0.538833, size = 160, normalized size = 1.55 \[ \frac{b d (p+2) x (x (b+c x))^p (d+e x)^m F_1\left (p+1;-p,-m;p+2;-\frac{c x}{b},-\frac{e x}{d}\right )}{(p+1) \left (b d (p+2) F_1\left (p+1;-p,-m;p+2;-\frac{c x}{b},-\frac{e x}{d}\right )+x \left (c d p F_1\left (p+2;1-p,-m;p+3;-\frac{c x}{b},-\frac{e x}{d}\right )+b e m F_1\left (p+2;-p,1-m;p+3;-\frac{c x}{b},-\frac{e x}{d}\right )\right )\right )} \]
Warning: Unable to verify antiderivative.
[In] Integrate[(d + e*x)^m*(b*x + c*x^2)^p,x]
[Out]
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Maple [F] time = 0.128, size = 0, normalized size = 0. \[ \int \left ( ex+d \right ) ^{m} \left ( c{x}^{2}+bx \right ) ^{p}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((e*x+d)^m*(c*x^2+b*x)^p,x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int{\left (c x^{2} + b x\right )}^{p}{\left (e x + d\right )}^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x)^p*(e*x + d)^m,x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left ({\left (c x^{2} + b x\right )}^{p}{\left (e x + d\right )}^{m}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x)^p*(e*x + d)^m,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((e*x+d)**m*(c*x**2+b*x)**p,x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int{\left (c x^{2} + b x\right )}^{p}{\left (e x + d\right )}^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x)^p*(e*x + d)^m,x, algorithm="giac")
[Out]