Optimal. Leaf size=187 \[ \frac{(d+e x)^{m+1} \left (a+b x+c x^2\right )^p \left (1-\frac{2 c (d+e x)}{2 c d-e \left (b-\sqrt{b^2-4 a c}\right )}\right )^{-p} \left (1-\frac{2 c (d+e x)}{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}\right )^{-p} F_1\left (m+1;-p,-p;m+2;\frac{2 c (d+e x)}{2 c d-\left (b-\sqrt{b^2-4 a c}\right ) e},\frac{2 c (d+e x)}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{e (m+1)} \]
[Out]
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Rubi [A] time = 0.421005, antiderivative size = 187, 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 \[ \frac{(d+e x)^{m+1} \left (a+b x+c x^2\right )^p \left (1-\frac{2 c (d+e x)}{2 c d-e \left (b-\sqrt{b^2-4 a c}\right )}\right )^{-p} \left (1-\frac{2 c (d+e x)}{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}\right )^{-p} F_1\left (m+1;-p,-p;m+2;\frac{2 c (d+e x)}{2 c d-\left (b-\sqrt{b^2-4 a c}\right ) e},\frac{2 c (d+e x)}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{e (m+1)} \]
Antiderivative was successfully verified.
[In] Int[(d + e*x)^m*(a + b*x + c*x^2)^p,x]
[Out]
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Rubi in Sympy [A] time = 37.6143, size = 167, normalized size = 0.89 \[ \frac{\left (d + e x\right )^{m + 1} \left (\frac{c \left (- 2 d - 2 e x\right )}{2 c d - e \left (b + \sqrt{- 4 a c + b^{2}}\right )} + 1\right )^{- p} \left (\frac{c \left (2 d + 2 e x\right )}{b e - 2 c d - e \sqrt{- 4 a c + b^{2}}} + 1\right )^{- p} \left (a + b x + c x^{2}\right )^{p} \operatorname{appellf_{1}}{\left (m + 1,- p,- p,m + 2,\frac{c \left (- 2 d - 2 e x\right )}{b e - 2 c d - e \sqrt{- 4 a c + b^{2}}},\frac{c \left (2 d + 2 e x\right )}{2 c d - e \left (b + \sqrt{- 4 a c + b^{2}}\right )} \right )}}{e \left (m + 1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((e*x+d)**m*(c*x**2+b*x+a)**p,x)
[Out]
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Mathematica [A] time = 0.755637, size = 205, normalized size = 1.1 \[ \frac{(d+e x)^{m+1} (a+x (b+c x))^p \left (\frac{e \left (\sqrt{b^2-4 a c}-b-2 c x\right )}{e \left (\sqrt{b^2-4 a c}-b\right )+2 c d}\right )^{-p} \left (\frac{e \left (\sqrt{b^2-4 a c}+b+2 c x\right )}{e \left (\sqrt{b^2-4 a c}+b\right )-2 c d}\right )^{-p} F_1\left (m+1;-p,-p;m+2;\frac{2 c (d+e x)}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e},\frac{2 c (d+e x)}{2 c d+\left (\sqrt{b^2-4 a c}-b\right ) e}\right )}{e (m+1)} \]
Warning: Unable to verify antiderivative.
[In] Integrate[(d + e*x)^m*(a + b*x + c*x^2)^p,x]
[Out]
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Maple [F] time = 0.234, size = 0, normalized size = 0. \[ \int \left ( ex+d \right ) ^{m} \left ( c{x}^{2}+bx+a \right ) ^{p}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((e*x+d)^m*(c*x^2+b*x+a)^p,x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int{\left (c x^{2} + b x + a\right )}^{p}{\left (e x + d\right )}^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x + a)^p*(e*x + d)^m,x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left ({\left (c x^{2} + b x + a\right )}^{p}{\left (e x + d\right )}^{m}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x + a)^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+a)**p,x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int{\left (c x^{2} + b x + a\right )}^{p}{\left (e x + d\right )}^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x + a)^p*(e*x + d)^m,x, algorithm="giac")
[Out]