Optimal. Leaf size=406 \[ \frac{2 b^2 (a+b x)^{m+1} (d e-c f)^3 (c+d x)^{-m-1}}{d^3 (m+1) (m+2) (m+3) (b c-a d)^3}-\frac{f^3 (a+b x)^m (c+d x)^{-m} \left (-\frac{d (a+b x)}{b c-a d}\right )^{-m} \, _2F_1\left (-m,-m;1-m;\frac{b (c+d x)}{b c-a d}\right )}{d^4 m}+\frac{3 f^2 (a+b x)^{m+1} (d e-c f) (c+d x)^{-m-1}}{d^3 (m+1) (b c-a d)}+\frac{(a+b x)^{m+1} (d e-c f)^3 (c+d x)^{-m-3}}{d^3 (m+3) (b c-a d)}+\frac{3 f (a+b x)^{m+1} (d e-c f)^2 (c+d x)^{-m-2}}{d^3 (m+2) (b c-a d)}+\frac{2 b (a+b x)^{m+1} (d e-c f)^3 (c+d x)^{-m-2}}{d^3 (m+2) (m+3) (b c-a d)^2}+\frac{3 b f (a+b x)^{m+1} (d e-c f)^2 (c+d x)^{-m-1}}{d^3 (m+1) (m+2) (b c-a d)^2} \]
[Out]
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Rubi [A] time = 0.741184, antiderivative size = 406, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 5, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.192 \[ \frac{2 b^2 (a+b x)^{m+1} (d e-c f)^3 (c+d x)^{-m-1}}{d^3 (m+1) (m+2) (m+3) (b c-a d)^3}-\frac{f^3 (a+b x)^m (c+d x)^{-m} \left (-\frac{d (a+b x)}{b c-a d}\right )^{-m} \, _2F_1\left (-m,-m;1-m;\frac{b (c+d x)}{b c-a d}\right )}{d^4 m}+\frac{3 f^2 (a+b x)^{m+1} (d e-c f) (c+d x)^{-m-1}}{d^3 (m+1) (b c-a d)}+\frac{(a+b x)^{m+1} (d e-c f)^3 (c+d x)^{-m-3}}{d^3 (m+3) (b c-a d)}+\frac{3 f (a+b x)^{m+1} (d e-c f)^2 (c+d x)^{-m-2}}{d^3 (m+2) (b c-a d)}+\frac{2 b (a+b x)^{m+1} (d e-c f)^3 (c+d x)^{-m-2}}{d^3 (m+2) (m+3) (b c-a d)^2}+\frac{3 b f (a+b x)^{m+1} (d e-c f)^2 (c+d x)^{-m-1}}{d^3 (m+1) (m+2) (b c-a d)^2} \]
Antiderivative was successfully verified.
[In] Int[(a + b*x)^m*(c + d*x)^(-4 - m)*(e + f*x)^3,x]
[Out]
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Rubi in Sympy [A] time = 141.596, size = 342, normalized size = 0.84 \[ \frac{2 b^{2} \left (a + b x\right )^{m + 1} \left (c + d x\right )^{- m - 1} \left (c f - d e\right )^{3}}{d^{3} \left (m + 1\right ) \left (m + 2\right ) \left (m + 3\right ) \left (a d - b c\right )^{3}} + \frac{3 b f \left (a + b x\right )^{m + 1} \left (c + d x\right )^{- m - 1} \left (c f - d e\right )^{2}}{d^{3} \left (m + 1\right ) \left (m + 2\right ) \left (a d - b c\right )^{2}} - \frac{2 b \left (a + b x\right )^{m + 1} \left (c + d x\right )^{- m - 2} \left (c f - d e\right )^{3}}{d^{3} \left (m + 2\right ) \left (m + 3\right ) \left (a d - b c\right )^{2}} + \frac{3 f^{2} \left (a + b x\right )^{m + 1} \left (c + d x\right )^{- m - 1} \left (c f - d e\right )}{d^{3} \left (m + 1\right ) \left (a d - b c\right )} - \frac{3 f \left (a + b x\right )^{m + 1} \left (c + d x\right )^{- m - 2} \left (c f - d e\right )^{2}}{d^{3} \left (m + 2\right ) \left (a d - b c\right )} + \frac{\left (a + b x\right )^{m + 1} \left (c + d x\right )^{- m - 3} \left (c f - d e\right )^{3}}{d^{3} \left (m + 3\right ) \left (a d - b c\right )} - \frac{f^{3} \left (\frac{d \left (a + b x\right )}{a d - b c}\right )^{- m} \left (a + b x\right )^{m} \left (c + d x\right )^{- m}{{}_{2}F_{1}\left (\begin{matrix} - m, - m \\ - m + 1 \end{matrix}\middle |{\frac{b \left (- c - d x\right )}{a d - b c}} \right )}}{d^{4} m} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*x+a)**m*(d*x+c)**(-4-m)*(f*x+e)**3,x)
[Out]
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Mathematica [C] time = 57.4284, size = 1833, normalized size = 4.51 \[ \text{result too large to display} \]
Warning: Unable to verify antiderivative.
[In] Integrate[(a + b*x)^m*(c + d*x)^(-4 - m)*(e + f*x)^3,x]
[Out]
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Maple [F] time = 0.085, size = 0, normalized size = 0. \[ \int \left ( bx+a \right ) ^{m} \left ( dx+c \right ) ^{-4-m} \left ( fx+e \right ) ^{3}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x+a)^m*(d*x+c)^(-4-m)*(f*x+e)^3,x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int{\left (f x + e\right )}^{3}{\left (b x + a\right )}^{m}{\left (d x + c\right )}^{-m - 4}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((f*x + e)^3*(b*x + a)^m*(d*x + c)^(-m - 4),x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left ({\left (f^{3} x^{3} + 3 \, e f^{2} x^{2} + 3 \, e^{2} f x + e^{3}\right )}{\left (b x + a\right )}^{m}{\left (d x + c\right )}^{-m - 4}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((f*x + e)^3*(b*x + a)^m*(d*x + c)^(-m - 4),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)**(-4-m)*(f*x+e)**3,x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int{\left (f x + e\right )}^{3}{\left (b x + a\right )}^{m}{\left (d x + c\right )}^{-m - 4}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((f*x + e)^3*(b*x + a)^m*(d*x + c)^(-m - 4),x, algorithm="giac")
[Out]