Optimal. Leaf size=277 \[ \frac{A (e x)^{m+1} \left (\frac{2 c x}{b-\sqrt{b^2-4 a c}}+1\right )^{-p} \left (\frac{2 c x}{\sqrt{b^2-4 a c}+b}+1\right )^{-p} \left (a+b x+c x^2\right )^p F_1\left (m+1;-p,-p;m+2;-\frac{2 c x}{b-\sqrt{b^2-4 a c}},-\frac{2 c x}{b+\sqrt{b^2-4 a c}}\right )}{e (m+1)}+\frac{B (e x)^{m+2} \left (\frac{2 c x}{b-\sqrt{b^2-4 a c}}+1\right )^{-p} \left (\frac{2 c x}{\sqrt{b^2-4 a c}+b}+1\right )^{-p} \left (a+b x+c x^2\right )^p F_1\left (m+2;-p,-p;m+3;-\frac{2 c x}{b-\sqrt{b^2-4 a c}},-\frac{2 c x}{b+\sqrt{b^2-4 a c}}\right )}{e^2 (m+2)} \]
[Out]
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Rubi [A] time = 0.881264, antiderivative size = 277, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.13 \[ \frac{A (e x)^{m+1} \left (\frac{2 c x}{b-\sqrt{b^2-4 a c}}+1\right )^{-p} \left (\frac{2 c x}{\sqrt{b^2-4 a c}+b}+1\right )^{-p} \left (a+b x+c x^2\right )^p F_1\left (m+1;-p,-p;m+2;-\frac{2 c x}{b-\sqrt{b^2-4 a c}},-\frac{2 c x}{b+\sqrt{b^2-4 a c}}\right )}{e (m+1)}+\frac{B (e x)^{m+2} \left (\frac{2 c x}{b-\sqrt{b^2-4 a c}}+1\right )^{-p} \left (\frac{2 c x}{\sqrt{b^2-4 a c}+b}+1\right )^{-p} \left (a+b x+c x^2\right )^p F_1\left (m+2;-p,-p;m+3;-\frac{2 c x}{b-\sqrt{b^2-4 a c}},-\frac{2 c x}{b+\sqrt{b^2-4 a c}}\right )}{e^2 (m+2)} \]
Antiderivative was successfully verified.
[In] Int[(e*x)^m*(A + B*x)*(a + b*x + c*x^2)^p,x]
[Out]
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Rubi in Sympy [A] time = 86.0343, size = 235, normalized size = 0.85 \[ \frac{A \left (e x\right )^{m + 1} \left (\frac{2 c x}{b - \sqrt{- 4 a c + b^{2}}} + 1\right )^{- p} \left (\frac{2 c x}{b + \sqrt{- 4 a c + b^{2}}} + 1\right )^{- p} \left (a + b x + c x^{2}\right )^{p} \operatorname{appellf_{1}}{\left (m + 1,- p,- p,m + 2,- \frac{2 c x}{b - \sqrt{- 4 a c + b^{2}}},- \frac{2 c x}{b + \sqrt{- 4 a c + b^{2}}} \right )}}{e \left (m + 1\right )} + \frac{B \left (e x\right )^{m + 2} \left (\frac{2 c x}{b - \sqrt{- 4 a c + b^{2}}} + 1\right )^{- p} \left (\frac{2 c x}{b + \sqrt{- 4 a c + b^{2}}} + 1\right )^{- p} \left (a + b x + c x^{2}\right )^{p} \operatorname{appellf_{1}}{\left (m + 2,- p,- p,m + 3,- \frac{2 c x}{b - \sqrt{- 4 a c + b^{2}}},- \frac{2 c x}{b + \sqrt{- 4 a c + b^{2}}} \right )}}{e^{2} \left (m + 2\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((e*x)**m*(B*x+A)*(c*x**2+b*x+a)**p,x)
[Out]
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Mathematica [B] time = 4.76446, size = 725, normalized size = 2.62 \[ \frac{c 2^{-p-1} x \left (\sqrt{b^2-4 a c}+b\right ) (e x)^m \left (x \left (b-\sqrt{b^2-4 a c}\right )+2 a\right )^2 \left (\frac{b-\sqrt{b^2-4 a c}}{2 c}+x\right )^{-p} \left (\frac{-\sqrt{b^2-4 a c}+b+2 c x}{c}\right )^{p+1} (a+x (b+c x))^{p-1} \left (\frac{B (m+3) x F_1\left (m+2;-p,-p;m+3;-\frac{2 c x}{b+\sqrt{b^2-4 a c}},\frac{2 c x}{\sqrt{b^2-4 a c}-b}\right )}{p x \left (\left (\sqrt{b^2-4 a c}-b\right ) F_1\left (m+3;1-p,-p;m+4;-\frac{2 c x}{b+\sqrt{b^2-4 a c}},\frac{2 c x}{\sqrt{b^2-4 a c}-b}\right )-\left (\sqrt{b^2-4 a c}+b\right ) F_1\left (m+3;-p,1-p;m+4;-\frac{2 c x}{b+\sqrt{b^2-4 a c}},\frac{2 c x}{\sqrt{b^2-4 a c}-b}\right )\right )-2 a (m+3) F_1\left (m+2;-p,-p;m+3;-\frac{2 c x}{b+\sqrt{b^2-4 a c}},\frac{2 c x}{\sqrt{b^2-4 a c}-b}\right )}-\frac{A (m+2)^2 F_1\left (m+1;-p,-p;m+2;-\frac{2 c x}{b+\sqrt{b^2-4 a c}},\frac{2 c x}{\sqrt{b^2-4 a c}-b}\right )}{(m+1) \left (2 a (m+2) F_1\left (m+1;-p,-p;m+2;-\frac{2 c x}{b+\sqrt{b^2-4 a c}},\frac{2 c x}{\sqrt{b^2-4 a c}-b}\right )+p x \left (\left (b-\sqrt{b^2-4 a c}\right ) F_1\left (m+2;1-p,-p;m+3;-\frac{2 c x}{b+\sqrt{b^2-4 a c}},\frac{2 c x}{\sqrt{b^2-4 a c}-b}\right )+\left (\sqrt{b^2-4 a c}+b\right ) F_1\left (m+2;-p,1-p;m+3;-\frac{2 c x}{b+\sqrt{b^2-4 a c}},\frac{2 c x}{\sqrt{b^2-4 a c}-b}\right )\right )\right )}\right )}{(m+2) \left (\sqrt{b^2-4 a c}-b\right ) \left (\sqrt{b^2-4 a c}+b+2 c x\right )} \]
Warning: Unable to verify antiderivative.
[In] Integrate[(e*x)^m*(A + B*x)*(a + b*x + c*x^2)^p,x]
[Out]
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Maple [F] time = 0.16, size = 0, normalized size = 0. \[ \int \left ( ex \right ) ^{m} \left ( Bx+A \right ) \left ( c{x}^{2}+bx+a \right ) ^{p}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((e*x)^m*(B*x+A)*(c*x^2+b*x+a)^p,x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int{\left (B x + A\right )}{\left (c x^{2} + b x + a\right )}^{p} \left (e x\right )^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)*(c*x^2 + b*x + a)^p*(e*x)^m,x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left ({\left (B x + A\right )}{\left (c x^{2} + b x + a\right )}^{p} \left (e x\right )^{m}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)*(c*x^2 + b*x + a)^p*(e*x)^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)**m*(B*x+A)*(c*x**2+b*x+a)**p,x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int{\left (B x + A\right )}{\left (c x^{2} + b x + a\right )}^{p} \left (e x\right )^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)*(c*x^2 + b*x + a)^p*(e*x)^m,x, algorithm="giac")
[Out]