Optimal. Leaf size=188 \[ \frac {(d e-c f) (a+b x)^{1+m} (c+d x)^{-3-m}}{d (b c-a d) (3+m)}-\frac {(a d f (3+m)-b (2 d e+c f (1+m))) (a+b x)^{1+m} (c+d x)^{-2-m}}{d (b c-a d)^2 (2+m) (3+m)}-\frac {b (a d f (3+m)-b (2 d e+c f (1+m))) (a+b x)^{1+m} (c+d x)^{-1-m}}{d (b c-a d)^3 (1+m) (2+m) (3+m)} \]
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Rubi [A]
time = 0.06, antiderivative size = 184, normalized size of antiderivative = 0.98, number of steps
used = 3, number of rules used = 3, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {80, 47, 37}
\begin {gather*} \frac {(a+b x)^{m+1} (d e-c f) (c+d x)^{-m-3}}{d (m+3) (b c-a d)}+\frac {(a+b x)^{m+1} (c+d x)^{-m-2} (-a d f (m+3)+b c f (m+1)+2 b d e)}{d (m+2) (m+3) (b c-a d)^2}+\frac {b (a+b x)^{m+1} (c+d x)^{-m-1} (-a d f (m+3)+b c f (m+1)+2 b d e)}{d (m+1) (m+2) (m+3) (b c-a d)^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 37
Rule 47
Rule 80
Rubi steps
\begin {align*} \int (a+b x)^m (c+d x)^{-4-m} (e+f x) \, dx &=\frac {(d e-c f) (a+b x)^{1+m} (c+d x)^{-3-m}}{d (b c-a d) (3+m)}+\frac {(2 b d e+b c f (1+m)-a d f (3+m)) \int (a+b x)^m (c+d x)^{-3-m} \, dx}{d (b c-a d) (3+m)}\\ &=\frac {(d e-c f) (a+b x)^{1+m} (c+d x)^{-3-m}}{d (b c-a d) (3+m)}+\frac {(2 b d e+b c f (1+m)-a d f (3+m)) (a+b x)^{1+m} (c+d x)^{-2-m}}{d (b c-a d)^2 (2+m) (3+m)}+\frac {(b (2 b d e+b c f (1+m)-a d f (3+m))) \int (a+b x)^m (c+d x)^{-2-m} \, dx}{d (b c-a d)^2 (2+m) (3+m)}\\ &=\frac {(d e-c f) (a+b x)^{1+m} (c+d x)^{-3-m}}{d (b c-a d) (3+m)}+\frac {(2 b d e+b c f (1+m)-a d f (3+m)) (a+b x)^{1+m} (c+d x)^{-2-m}}{d (b c-a d)^2 (2+m) (3+m)}+\frac {b (2 b d e+b c f (1+m)-a d f (3+m)) (a+b x)^{1+m} (c+d x)^{-1-m}}{d (b c-a d)^3 (1+m) (2+m) (3+m)}\\ \end {align*}
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Mathematica [A]
time = 0.16, size = 179, normalized size = 0.95 \begin {gather*} \frac {(a+b x)^{1+m} (c+d x)^{-3-m} \left (a^2 d (1+m) (c f+d e (2+m)+d f (3+m) x)+b^2 \left (2 d^2 e x^2+c^2 (3+m) (e (2+m)+f (1+m) x)+c d x (2 e (3+m)+f (1+m) x)\right )-a b \left (c^2 f (3+m)+d^2 x (2 e (1+m)+f (3+m) x)+2 c d \left (e \left (3+4 m+m^2\right )+f \left (5+4 m+m^2\right ) x\right )\right )\right )}{(b c-a d)^3 (1+m) (2+m) (3+m)} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(502\) vs.
\(2(188)=376\).
time = 0.10, size = 503, normalized size = 2.68
method | result | size |
gosper | \(-\frac {\left (d x +c \right )^{-3-m} \left (b x +a \right )^{1+m} \left (a^{2} d^{2} f \,m^{2} x -2 a b c d f \,m^{2} x -a b \,d^{2} f m \,x^{2}+b^{2} c^{2} f \,m^{2} x +b^{2} c d f m \,x^{2}+a^{2} d^{2} e \,m^{2}+4 a^{2} d^{2} f m x -2 a b c d e \,m^{2}-8 a b c d f m x -2 a b \,d^{2} e m x -3 a b \,d^{2} f \,x^{2}+b^{2} c^{2} e \,m^{2}+4 b^{2} c^{2} f m x +2 b^{2} c d e m x +b^{2} c d f \,x^{2}+2 b^{2} d^{2} e \,x^{2}+a^{2} c d f m +3 a^{2} d^{2} e m +3 a^{2} d^{2} f x -a b \,c^{2} f m -8 a b c d e m -10 a b c d f x -2 a b \,d^{2} e x +5 b^{2} c^{2} e m +3 b^{2} c^{2} f x +6 b^{2} c d e x +a^{2} c d f +2 a^{2} d^{2} e -3 a b \,c^{2} f -6 a b c d e +6 b^{2} c^{2} e \right )}{a^{3} d^{3} m^{3}-3 a^{2} b c \,d^{2} m^{3}+3 a \,b^{2} c^{2} d \,m^{3}-b^{3} c^{3} m^{3}+6 a^{3} d^{3} m^{2}-18 a^{2} b c \,d^{2} m^{2}+18 a \,b^{2} c^{2} d \,m^{2}-6 b^{3} c^{3} m^{2}+11 a^{3} d^{3} m -33 a^{2} b c \,d^{2} m +33 a \,b^{2} c^{2} d m -11 b^{3} c^{3} m +6 a^{3} d^{3}-18 a^{2} b c \,d^{2}+18 a \,b^{2} c^{2} d -6 b^{3} c^{3}}\) | \(503\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 904 vs.
\(2 (192) = 384\).
time = 1.05, size = 904, normalized size = 4.81 \begin {gather*} \frac {{\left ({\left ({\left (b^{3} c d^{2} - a b^{2} d^{3}\right )} f m + {\left (b^{3} c d^{2} - 3 \, a b^{2} d^{3}\right )} f\right )} x^{4} + {\left ({\left (b^{3} c^{2} d - 2 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} f m^{2} + {\left (5 \, b^{3} c^{2} d - 8 \, a b^{2} c d^{2} + 3 \, a^{2} b d^{3}\right )} f m + 4 \, {\left (b^{3} c^{2} d - 3 \, a b^{2} c d^{2}\right )} f\right )} x^{3} - {\left (a^{2} b c^{3} - a^{3} c^{2} d\right )} f m + {\left ({\left (b^{3} c^{3} - a b^{2} c^{2} d - a^{2} b c d^{2} + a^{3} d^{3}\right )} f m^{2} + 4 \, {\left (b^{3} c^{3} - a b^{2} c^{2} d - a^{2} b c d^{2} + a^{3} d^{3}\right )} f m + 3 \, {\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d - 3 \, a^{2} b c d^{2} + a^{3} d^{3}\right )} f\right )} x^{2} - {\left (3 \, a^{2} b c^{3} - a^{3} c^{2} d\right )} f + {\left ({\left (a b^{2} c^{3} - 2 \, a^{2} b c^{2} d + a^{3} c d^{2}\right )} f m^{2} + {\left (3 \, a b^{2} c^{3} - 8 \, a^{2} b c^{2} d + 5 \, a^{3} c d^{2}\right )} f m - 4 \, {\left (3 \, a^{2} b c^{2} d - a^{3} c d^{2}\right )} f\right )} x + {\left (2 \, b^{3} d^{3} x^{4} + 6 \, a b^{2} c^{3} - 6 \, a^{2} b c^{2} d + 2 \, a^{3} c d^{2} + 2 \, {\left (4 \, b^{3} c d^{2} + {\left (b^{3} c d^{2} - a b^{2} d^{3}\right )} m\right )} x^{3} + {\left (a b^{2} c^{3} - 2 \, a^{2} b c^{2} d + a^{3} c d^{2}\right )} m^{2} + {\left (12 \, b^{3} c^{2} d + {\left (b^{3} c^{2} d - 2 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} m^{2} + {\left (7 \, b^{3} c^{2} d - 8 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} m\right )} x^{2} + {\left (5 \, a b^{2} c^{3} - 8 \, a^{2} b c^{2} d + 3 \, a^{3} c d^{2}\right )} m + {\left (6 \, b^{3} c^{3} + 6 \, a b^{2} c^{2} d - 6 \, a^{2} b c d^{2} + 2 \, a^{3} d^{3} + {\left (b^{3} c^{3} - a b^{2} c^{2} d - a^{2} b c d^{2} + a^{3} d^{3}\right )} m^{2} + {\left (5 \, b^{3} c^{3} - a b^{2} c^{2} d - 7 \, a^{2} b c d^{2} + 3 \, a^{3} d^{3}\right )} m\right )} x\right )} e\right )} {\left (b x + a\right )}^{m} {\left (d x + c\right )}^{-m - 4}}{6 \, b^{3} c^{3} - 18 \, a b^{2} c^{2} d + 18 \, a^{2} b c d^{2} - 6 \, a^{3} d^{3} + {\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} m^{3} + 6 \, {\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} m^{2} + 11 \, {\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} m} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: HeuristicGCDFailed} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 3.44, size = 874, normalized size = 4.65 \begin {gather*} -\frac {x\,{\left (a+b\,x\right )}^m\,\left (f\,a^3\,c\,d^2\,m^2+5\,f\,a^3\,c\,d^2\,m+4\,f\,a^3\,c\,d^2+e\,a^3\,d^3\,m^2+3\,e\,a^3\,d^3\,m+2\,e\,a^3\,d^3-2\,f\,a^2\,b\,c^2\,d\,m^2-8\,f\,a^2\,b\,c^2\,d\,m-12\,f\,a^2\,b\,c^2\,d-e\,a^2\,b\,c\,d^2\,m^2-7\,e\,a^2\,b\,c\,d^2\,m-6\,e\,a^2\,b\,c\,d^2+f\,a\,b^2\,c^3\,m^2+3\,f\,a\,b^2\,c^3\,m-e\,a\,b^2\,c^2\,d\,m^2-e\,a\,b^2\,c^2\,d\,m+6\,e\,a\,b^2\,c^2\,d+e\,b^3\,c^3\,m^2+5\,e\,b^3\,c^3\,m+6\,e\,b^3\,c^3\right )}{{\left (a\,d-b\,c\right )}^3\,{\left (c+d\,x\right )}^{m+4}\,\left (m^3+6\,m^2+11\,m+6\right )}-\frac {x^2\,{\left (a+b\,x\right )}^m\,\left (f\,a^3\,d^3\,m^2+4\,f\,a^3\,d^3\,m+3\,f\,a^3\,d^3-f\,a^2\,b\,c\,d^2\,m^2-4\,f\,a^2\,b\,c\,d^2\,m-9\,f\,a^2\,b\,c\,d^2+e\,a^2\,b\,d^3\,m^2+e\,a^2\,b\,d^3\,m-f\,a\,b^2\,c^2\,d\,m^2-4\,f\,a\,b^2\,c^2\,d\,m-9\,f\,a\,b^2\,c^2\,d-2\,e\,a\,b^2\,c\,d^2\,m^2-8\,e\,a\,b^2\,c\,d^2\,m+f\,b^3\,c^3\,m^2+4\,f\,b^3\,c^3\,m+3\,f\,b^3\,c^3+e\,b^3\,c^2\,d\,m^2+7\,e\,b^3\,c^2\,d\,m+12\,e\,b^3\,c^2\,d\right )}{{\left (a\,d-b\,c\right )}^3\,{\left (c+d\,x\right )}^{m+4}\,\left (m^3+6\,m^2+11\,m+6\right )}-\frac {a\,c\,{\left (a+b\,x\right )}^m\,\left (f\,a^2\,c\,d\,m+f\,a^2\,c\,d+e\,a^2\,d^2\,m^2+3\,e\,a^2\,d^2\,m+2\,e\,a^2\,d^2-f\,a\,b\,c^2\,m-3\,f\,a\,b\,c^2-2\,e\,a\,b\,c\,d\,m^2-8\,e\,a\,b\,c\,d\,m-6\,e\,a\,b\,c\,d+e\,b^2\,c^2\,m^2+5\,e\,b^2\,c^2\,m+6\,e\,b^2\,c^2\right )}{{\left (a\,d-b\,c\right )}^3\,{\left (c+d\,x\right )}^{m+4}\,\left (m^3+6\,m^2+11\,m+6\right )}-\frac {b^2\,d^2\,x^4\,{\left (a+b\,x\right )}^m\,\left (b\,c\,f-3\,a\,d\,f+2\,b\,d\,e-a\,d\,f\,m+b\,c\,f\,m\right )}{{\left (a\,d-b\,c\right )}^3\,{\left (c+d\,x\right )}^{m+4}\,\left (m^3+6\,m^2+11\,m+6\right )}-\frac {b\,d\,x^3\,{\left (a+b\,x\right )}^m\,\left (4\,b\,c-a\,d\,m+b\,c\,m\right )\,\left (b\,c\,f-3\,a\,d\,f+2\,b\,d\,e-a\,d\,f\,m+b\,c\,f\,m\right )}{{\left (a\,d-b\,c\right )}^3\,{\left (c+d\,x\right )}^{m+4}\,\left (m^3+6\,m^2+11\,m+6\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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