Optimal. Leaf size=81 \[ \frac{(b x)^{m+1} (c+d x)^n \left (\frac{d x}{c}+1\right )^{-n} (e+f x)^p \left (\frac{f x}{e}+1\right )^{-p} F_1\left (m+1;-n,-p;m+2;-\frac{d x}{c},-\frac{f x}{e}\right )}{b (m+1)} \]
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Rubi [A] time = 0.139547, antiderivative size = 81, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1 \[ \frac{(b x)^{m+1} (c+d x)^n \left (\frac{d x}{c}+1\right )^{-n} (e+f x)^p \left (\frac{f x}{e}+1\right )^{-p} F_1\left (m+1;-n,-p;m+2;-\frac{d x}{c},-\frac{f x}{e}\right )}{b (m+1)} \]
Antiderivative was successfully verified.
[In] Int[(b*x)^m*(c + d*x)^n*(e + f*x)^p,x]
[Out]
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Rubi in Sympy [A] time = 19.9772, size = 61, normalized size = 0.75 \[ \frac{\left (b x\right )^{m + 1} \left (1 + \frac{d x}{c}\right )^{- n} \left (1 + \frac{f x}{e}\right )^{- p} \left (c + d x\right )^{n} \left (e + f x\right )^{p} \operatorname{appellf_{1}}{\left (m + 1,- n,- p,m + 2,- \frac{d x}{c},- \frac{f x}{e} \right )}}{b \left (m + 1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*x)**m*(d*x+c)**n*(f*x+e)**p,x)
[Out]
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Mathematica [B] time = 0.516383, size = 163, normalized size = 2.01 \[ \frac{c e (m+2) x (b x)^m (c+d x)^n (e+f x)^p F_1\left (m+1;-n,-p;m+2;-\frac{d x}{c},-\frac{f x}{e}\right )}{(m+1) \left (c e (m+2) F_1\left (m+1;-n,-p;m+2;-\frac{d x}{c},-\frac{f x}{e}\right )+x \left (d e n F_1\left (m+2;1-n,-p;m+3;-\frac{d x}{c},-\frac{f x}{e}\right )+c f p F_1\left (m+2;-n,1-p;m+3;-\frac{d x}{c},-\frac{f x}{e}\right )\right )\right )} \]
Warning: Unable to verify antiderivative.
[In] Integrate[(b*x)^m*(c + d*x)^n*(e + f*x)^p,x]
[Out]
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Maple [F] time = 0.181, size = 0, normalized size = 0. \[ \int \left ( bx \right ) ^{m} \left ( dx+c \right ) ^{n} \left ( fx+e \right ) ^{p}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x)^m*(d*x+c)^n*(f*x+e)^p,x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \left (b x\right )^{m}{\left (d x + c\right )}^{n}{\left (f x + e\right )}^{p}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x)^m*(d*x + c)^n*(f*x + e)^p,x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\left (b x\right )^{m}{\left (d x + c\right )}^{n}{\left (f x + e\right )}^{p}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x)^m*(d*x + c)^n*(f*x + e)^p,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)**m*(d*x+c)**n*(f*x+e)**p,x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \left (b x\right )^{m}{\left (d x + c\right )}^{n}{\left (f x + e\right )}^{p}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x)^m*(d*x + c)^n*(f*x + e)^p,x, algorithm="giac")
[Out]