Optimal. Leaf size=131 \[ \frac{f (a+b x)^{m+1} (c+d x)^{n+1}}{b d (m+n+2)}-\frac{(a+b x)^{m+1} (c+d x)^{n+1} (b d e (m+n+2)-f (a d (n+1)+b c (m+1))) \, _2F_1\left (1,m+n+2;n+2;\frac{b (c+d x)}{b c-a d}\right )}{b d (n+1) (m+n+2) (b c-a d)} \]
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Rubi [A] time = 0.234711, antiderivative size = 141, normalized size of antiderivative = 1.08, number of steps used = 3, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15 \[ \frac{(a+b x)^{m+1} (c+d x)^n \left (\frac{b (c+d x)}{b c-a d}\right )^{-n} (b d e (m+n+2)-f (a d (n+1)+b c (m+1))) \, _2F_1\left (m+1,-n;m+2;-\frac{d (a+b x)}{b c-a d}\right )}{b^2 d (m+1) (m+n+2)}+\frac{f (a+b x)^{m+1} (c+d x)^{n+1}}{b d (m+n+2)} \]
Antiderivative was successfully verified.
[In] Int[(a + b*x)^m*(c + d*x)^n*(e + f*x),x]
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Rubi in Sympy [A] time = 27.7638, size = 117, normalized size = 0.89 \[ \frac{f \left (a + b x\right )^{m + 1} \left (c + d x\right )^{n + 1}}{b d \left (m + n + 2\right )} - \frac{\left (\frac{b \left (- c - d x\right )}{a d - b c}\right )^{- n} \left (a + b x\right )^{m + 1} \left (c + d x\right )^{n} \left (- b d e \left (m + n + 2\right ) + f \left (a d \left (n + 1\right ) + b c \left (m + 1\right )\right )\right ){{}_{2}F_{1}\left (\begin{matrix} - n, m + 1 \\ m + 2 \end{matrix}\middle |{\frac{d \left (a + b x\right )}{a d - b c}} \right )}}{b^{2} d \left (m + 1\right ) \left (m + n + 2\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*x+a)**m*(d*x+c)**n*(f*x+e),x)
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Mathematica [C] time = 0.408829, size = 202, normalized size = 1.54 \[ (a+b x)^m (c+d x)^n \left (\frac{3 a c f x^2 F_1\left (2;-m,-n;3;-\frac{b x}{a},-\frac{d x}{c}\right )}{6 a c F_1\left (2;-m,-n;3;-\frac{b x}{a},-\frac{d x}{c}\right )+2 b c m x F_1\left (3;1-m,-n;4;-\frac{b x}{a},-\frac{d x}{c}\right )+2 a d n x F_1\left (3;-m,1-n;4;-\frac{b x}{a},-\frac{d x}{c}\right )}+\frac{e (c+d x) \left (\frac{d (a+b x)}{a d-b c}\right )^{-m} \, _2F_1\left (-m,n+1;n+2;\frac{b (c+d x)}{b c-a d}\right )}{d (n+1)}\right ) \]
Warning: Unable to verify antiderivative.
[In] Integrate[(a + b*x)^m*(c + d*x)^n*(e + f*x),x]
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Maple [F] time = 0.085, size = 0, normalized size = 0. \[ \int \left ( bx+a \right ) ^{m} \left ( dx+c \right ) ^{n} \left ( fx+e \right ) \, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x+a)^m*(d*x+c)^n*(f*x+e),x)
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int{\left (f x + e\right )}{\left (b x + a\right )}^{m}{\left (d x + c\right )}^{n}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((f*x + e)*(b*x + a)^m*(d*x + c)^n,x, algorithm="maxima")
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left ({\left (f x + e\right )}{\left (b x + a\right )}^{m}{\left (d x + c\right )}^{n}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((f*x + e)*(b*x + a)^m*(d*x + c)^n,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+a)**m*(d*x+c)**n*(f*x+e),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int{\left (f x + e\right )}{\left (b x + a\right )}^{m}{\left (d x + c\right )}^{n}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((f*x + e)*(b*x + a)^m*(d*x + c)^n,x, algorithm="giac")
[Out]