Optimal. Leaf size=147 \[ \frac{f (a+b x)^{m+1} (c+d x)^{3-m}}{4 b d}-\frac{(b c-a d)^2 (a+b x)^{m+1} (c+d x)^{-m} \left (\frac{b (c+d x)}{b c-a d}\right )^m (a d f (3-m)-b (4 d e-c f (m+1))) \, _2F_1\left (m-2,m+1;m+2;-\frac{d (a+b x)}{b c-a d}\right )}{4 b^4 d (m+1)} \]
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Rubi [A] time = 0.24438, antiderivative size = 146, normalized size of antiderivative = 0.99, number of steps used = 3, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125 \[ \frac{(b c-a d)^2 (a+b x)^{m+1} (c+d x)^{-m} \left (\frac{b (c+d x)}{b c-a d}\right )^m (-a d f (3-m)-b c f (m+1)+4 b d e) \, _2F_1\left (m-2,m+1;m+2;-\frac{d (a+b x)}{b c-a d}\right )}{4 b^4 d (m+1)}+\frac{f (a+b x)^{m+1} (c+d x)^{3-m}}{4 b d} \]
Antiderivative was successfully verified.
[In] Int[(a + b*x)^m*(c + d*x)^(2 - m)*(e + f*x),x]
[Out]
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Rubi in Sympy [A] time = 27.7612, size = 116, normalized size = 0.79 \[ \frac{f \left (a + b x\right )^{m + 1} \left (c + d x\right )^{- m + 3}}{4 b d} - \frac{\left (\frac{b \left (- c - d x\right )}{a d - b c}\right )^{m} \left (a + b x\right )^{m + 1} \left (c + d x\right )^{- m} \left (a d - b c\right )^{2} \left (- 4 b d e + f \left (a d \left (- m + 3\right ) + b c \left (m + 1\right )\right )\right ){{}_{2}F_{1}\left (\begin{matrix} m - 2, m + 1 \\ m + 2 \end{matrix}\middle |{\frac{d \left (a + b x\right )}{a d - b c}} \right )}}{4 b^{4} d \left (m + 1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*x+a)**m*(d*x+c)**(2-m)*(f*x+e),x)
[Out]
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Mathematica [C] time = 5.43263, size = 509, normalized size = 3.46 \[ c (a+b x)^m (c+d x)^{-m} \left (\frac{5 a d^2 f x^4 F_1\left (4;-m,m;5;-\frac{b x}{a},-\frac{d x}{c}\right )}{20 a c F_1\left (4;-m,m;5;-\frac{b x}{a},-\frac{d x}{c}\right )+4 b c m x F_1\left (5;1-m,m;6;-\frac{b x}{a},-\frac{d x}{c}\right )-4 a d m x F_1\left (5;-m,m+1;6;-\frac{b x}{a},-\frac{d x}{c}\right )}+\frac{4 a d x^3 (2 c f+d e) F_1\left (3;-m,m;4;-\frac{b x}{a},-\frac{d x}{c}\right )}{3 \left (4 a c F_1\left (3;-m,m;4;-\frac{b x}{a},-\frac{d x}{c}\right )+b c m x F_1\left (4;1-m,m;5;-\frac{b x}{a},-\frac{d x}{c}\right )-a d m x F_1\left (4;-m,m+1;5;-\frac{b x}{a},-\frac{d x}{c}\right )\right )}+\frac{3 a c x^2 (c f+2 d e) F_1\left (2;-m,m;3;-\frac{b x}{a},-\frac{d x}{c}\right )}{6 a c F_1\left (2;-m,m;3;-\frac{b x}{a},-\frac{d x}{c}\right )+2 m x \left (b c F_1\left (3;1-m,m;4;-\frac{b x}{a},-\frac{d x}{c}\right )-a d F_1\left (3;-m,m+1;4;-\frac{b x}{a},-\frac{d x}{c}\right )\right )}-\frac{c^2 e \left (\frac{d (a+b x)}{a d-b c}\right )^{-m} \, _2F_1\left (1-m,-m;2-m;\frac{b (c+d x)}{b c-a d}\right )}{d (m-1)}-\frac{c e x \left (\frac{d (a+b x)}{a d-b c}\right )^{-m} \, _2F_1\left (1-m,-m;2-m;\frac{b (c+d x)}{b c-a d}\right )}{m-1}\right ) \]
Warning: Unable to verify antiderivative.
[In] Integrate[(a + b*x)^m*(c + d*x)^(2 - m)*(e + f*x),x]
[Out]
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Maple [F] time = 0.088, size = 0, normalized size = 0. \[ \int \left ( bx+a \right ) ^{m} \left ( dx+c \right ) ^{2-m} \left ( fx+e \right ) \, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x+a)^m*(d*x+c)^(2-m)*(f*x+e),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int{\left (f x + e\right )}{\left (b x + a\right )}^{m}{\left (d x + c\right )}^{-m + 2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((f*x + e)*(b*x + a)^m*(d*x + c)^(-m + 2),x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left ({\left (f x + e\right )}{\left (b x + a\right )}^{m}{\left (d x + c\right )}^{-m + 2}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((f*x + e)*(b*x + a)^m*(d*x + c)^(-m + 2),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)**(2-m)*(f*x+e),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int{\left (f x + e\right )}{\left (b x + a\right )}^{m}{\left (d x + c\right )}^{-m + 2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((f*x + e)*(b*x + a)^m*(d*x + c)^(-m + 2),x, algorithm="giac")
[Out]