Optimal. Leaf size=144 \[ \frac {\left (c d^2-b d e+a e^2\right ) (e f-d g) (d+e x)^{1+m}}{e^4 (1+m)}-\frac {(c d (2 e f-3 d g)-e (b e f-2 b d g+a e g)) (d+e x)^{2+m}}{e^4 (2+m)}+\frac {(c e f-3 c d g+b e g) (d+e x)^{3+m}}{e^4 (3+m)}+\frac {c g (d+e x)^{4+m}}{e^4 (4+m)} \]
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Rubi [A]
time = 0.07, antiderivative size = 144, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 1, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.043, Rules used = {785}
\begin {gather*} \frac {(e f-d g) (d+e x)^{m+1} \left (a e^2-b d e+c d^2\right )}{e^4 (m+1)}-\frac {(d+e x)^{m+2} (c d (2 e f-3 d g)-e (a e g-2 b d g+b e f))}{e^4 (m+2)}+\frac {(d+e x)^{m+3} (b e g-3 c d g+c e f)}{e^4 (m+3)}+\frac {c g (d+e x)^{m+4}}{e^4 (m+4)} \end {gather*}
Antiderivative was successfully verified.
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Rule 785
Rubi steps
\begin {align*} \int (d+e x)^m (f+g x) \left (a+b x+c x^2\right ) \, dx &=\int \left (\frac {\left (c d^2-b d e+a e^2\right ) (e f-d g) (d+e x)^m}{e^3}+\frac {(-c d (2 e f-3 d g)+e (b e f-2 b d g+a e g)) (d+e x)^{1+m}}{e^3}+\frac {(c e f-3 c d g+b e g) (d+e x)^{2+m}}{e^3}+\frac {c g (d+e x)^{3+m}}{e^3}\right ) \, dx\\ &=\frac {\left (c d^2-b d e+a e^2\right ) (e f-d g) (d+e x)^{1+m}}{e^4 (1+m)}-\frac {(c d (2 e f-3 d g)-e (b e f-2 b d g+a e g)) (d+e x)^{2+m}}{e^4 (2+m)}+\frac {(c e f-3 c d g+b e g) (d+e x)^{3+m}}{e^4 (3+m)}+\frac {c g (d+e x)^{4+m}}{e^4 (4+m)}\\ \end {align*}
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Mathematica [A]
time = 0.31, size = 180, normalized size = 1.25 \begin {gather*} \frac {(d+e x)^{1+m} \left (-\frac {\left (c d^2+e (-b d+a e)\right ) (6 c d g+b e g (1+m)-2 c e f (4+m))}{e^2 (1+m)}+\frac {\left (-b^2 e^2 g (2+m)+2 c^2 d (3 d g-e f (4+m))+c e (b d g (-2+m)+2 a e g (3+m)+b e f (4+m))\right ) (d+e x)}{e^2 (2+m)}+(a+x (b+c x)) (b e g+c (-3 d g+e f (4+m)+e g (3+m) x))\right )}{c e^2 (3+m) (4+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.
\(2(144)=288\).
time = 0.09, size = 503, normalized size = 3.49
method | result | size |
gosper | \(-\frac {\left (e x +d \right )^{1+m} \left (-c \,e^{3} g \,m^{3} x^{3}-b \,e^{3} g \,m^{3} x^{2}-c \,e^{3} f \,m^{3} x^{2}-6 c \,e^{3} g \,m^{2} x^{3}-a \,e^{3} g \,m^{3} x -b \,e^{3} f \,m^{3} x -7 b \,e^{3} g \,m^{2} x^{2}+3 c d \,e^{2} g \,m^{2} x^{2}-7 c \,e^{3} f \,m^{2} x^{2}-11 c \,e^{3} g m \,x^{3}-a \,e^{3} f \,m^{3}-8 a \,e^{3} g \,m^{2} x +2 b d \,e^{2} g \,m^{2} x -8 b \,e^{3} f \,m^{2} x -14 b \,e^{3} g m \,x^{2}+2 c d \,e^{2} f \,m^{2} x +9 c d \,e^{2} g m \,x^{2}-14 c \,e^{3} f m \,x^{2}-6 c g \,x^{3} e^{3}+a d \,e^{2} g \,m^{2}-9 a \,e^{3} f \,m^{2}-19 a \,e^{3} g m x +b d \,e^{2} f \,m^{2}+10 b d \,e^{2} g m x -19 b \,e^{3} f m x -8 b \,e^{3} g \,x^{2}-6 c \,d^{2} e g m x +10 c d \,e^{2} f m x +6 c d \,e^{2} g \,x^{2}-8 c \,e^{3} f \,x^{2}+7 a d \,e^{2} g m -26 a \,e^{3} f m -12 a \,e^{3} g x -2 b \,d^{2} e g m +7 b d \,e^{2} f m +8 b d \,e^{2} g x -12 b \,e^{3} f x -2 c \,d^{2} e f m -6 c \,d^{2} e g x +8 c d \,e^{2} f x +12 a d \,e^{2} g -24 a \,e^{3} f -8 b \,d^{2} e g +12 b d \,e^{2} f +6 c \,d^{3} g -8 c \,d^{2} e f \right )}{e^{4} \left (m^{4}+10 m^{3}+35 m^{2}+50 m +24\right )}\) | \(503\) |
norman | \(\frac {c g \,x^{4} {\mathrm e}^{m \ln \left (e x +d \right )}}{4+m}+\frac {\left (b e g m +c d g m +c e f m +4 b e g +4 c e f \right ) x^{3} {\mathrm e}^{m \ln \left (e x +d \right )}}{e \left (m^{2}+7 m +12\right )}+\frac {\left (a \,e^{2} g \,m^{2}+b d e g \,m^{2}+b \,e^{2} f \,m^{2}+c d e f \,m^{2}+7 a \,e^{2} g m +4 b d e g m +7 b \,e^{2} f m -3 c \,d^{2} g m +4 c d e f m +12 a \,e^{2} g +12 b \,e^{2} f \right ) x^{2} {\mathrm e}^{m \ln \left (e x +d \right )}}{e^{2} \left (m^{3}+9 m^{2}+26 m +24\right )}+\frac {\left (a d \,e^{2} g \,m^{3}+a \,e^{3} f \,m^{3}+b d \,e^{2} f \,m^{3}+7 a d \,e^{2} g \,m^{2}+9 a \,e^{3} f \,m^{2}-2 b \,d^{2} e g \,m^{2}+7 b d \,e^{2} f \,m^{2}-2 c \,d^{2} e f \,m^{2}+12 a d \,e^{2} g m +26 a \,e^{3} f m -8 b \,d^{2} e g m +12 b d \,e^{2} f m +6 c \,d^{3} g m -8 c \,d^{2} e f m +24 a \,e^{3} f \right ) x \,{\mathrm e}^{m \ln \left (e x +d \right )}}{e^{3} \left (m^{4}+10 m^{3}+35 m^{2}+50 m +24\right )}-\frac {d \left (-a \,e^{3} f \,m^{3}+a d \,e^{2} g \,m^{2}-9 a \,e^{3} f \,m^{2}+b d \,e^{2} f \,m^{2}+7 a d \,e^{2} g m -26 a \,e^{3} f m -2 b \,d^{2} e g m +7 b d \,e^{2} f m -2 c \,d^{2} e f m +12 a d \,e^{2} g -24 a \,e^{3} f -8 b \,d^{2} e g +12 b d \,e^{2} f +6 c \,d^{3} g -8 c \,d^{2} e f \right ) {\mathrm e}^{m \ln \left (e x +d \right )}}{e^{4} \left (m^{4}+10 m^{3}+35 m^{2}+50 m +24\right )}\) | \(535\) |
risch | \(-\frac {\left (-c \,e^{4} g \,m^{3} x^{4}-b \,e^{4} g \,m^{3} x^{3}-c d \,e^{3} g \,m^{3} x^{3}-c \,e^{4} f \,m^{3} x^{3}-6 c \,e^{4} g \,m^{2} x^{4}-a \,e^{4} g \,m^{3} x^{2}-b d \,e^{3} g \,m^{3} x^{2}-b \,e^{4} f \,m^{3} x^{2}-7 b \,e^{4} g \,m^{2} x^{3}-c d \,e^{3} f \,m^{3} x^{2}-3 c d \,e^{3} g \,m^{2} x^{3}-7 c \,e^{4} f \,m^{2} x^{3}-11 c \,e^{4} g m \,x^{4}-a d \,e^{3} g \,m^{3} x -a \,e^{4} f \,m^{3} x -8 a \,e^{4} g \,m^{2} x^{2}-b d \,e^{3} f \,m^{3} x -5 b d \,e^{3} g \,m^{2} x^{2}-8 b \,e^{4} f \,m^{2} x^{2}-14 b \,e^{4} g m \,x^{3}+3 c \,d^{2} e^{2} g \,m^{2} x^{2}-5 c d \,e^{3} f \,m^{2} x^{2}-2 c d \,e^{3} g m \,x^{3}-14 c \,e^{4} f m \,x^{3}-6 c g \,x^{4} e^{4}-a d \,e^{3} f \,m^{3}-7 a d \,e^{3} g \,m^{2} x -9 a \,e^{4} f \,m^{2} x -19 a \,e^{4} g m \,x^{2}+2 b \,d^{2} e^{2} g \,m^{2} x -7 b d \,e^{3} f \,m^{2} x -4 b d \,e^{3} g m \,x^{2}-19 b \,e^{4} f m \,x^{2}-8 b \,e^{4} g \,x^{3}+2 c \,d^{2} e^{2} f \,m^{2} x +3 c \,d^{2} e^{2} g m \,x^{2}-4 c d \,e^{3} f m \,x^{2}-8 c \,e^{4} f \,x^{3}+a \,d^{2} e^{2} g \,m^{2}-9 a d \,e^{3} f \,m^{2}-12 a d \,e^{3} g m x -26 a \,e^{4} f m x -12 a \,e^{4} g \,x^{2}+b \,d^{2} e^{2} f \,m^{2}+8 b \,d^{2} e^{2} g m x -12 b d \,e^{3} f m x -12 b \,e^{4} f \,x^{2}-6 c \,d^{3} e g m x +8 c \,d^{2} e^{2} f m x +7 a \,d^{2} e^{2} g m -26 a d \,e^{3} f m -24 a \,e^{4} f x -2 b \,d^{3} e g m +7 b \,d^{2} e^{2} f m -2 c \,d^{3} e f m +12 a \,d^{2} e^{2} g -24 a d \,e^{3} f -8 b \,d^{3} e g +12 b \,d^{2} e^{2} f +6 c \,d^{4} g -8 c \,d^{3} e f \right ) \left (e x +d \right )^{m}}{\left (3+m \right ) \left (4+m \right ) \left (2+m \right ) \left (1+m \right ) e^{4}}\) | \(734\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 358 vs.
\(2 (156) = 312\).
time = 0.33, size = 358, normalized size = 2.49 \begin {gather*} \frac {{\left (x e + d\right )}^{m + 1} a f e^{\left (-1\right )}}{m + 1} + \frac {{\left ({\left (m + 1\right )} x^{2} e^{2} + d m x e - d^{2}\right )} b f e^{\left (m \log \left (x e + d\right ) - 2\right )}}{m^{2} + 3 \, m + 2} + \frac {{\left ({\left (m + 1\right )} x^{2} e^{2} + d m x e - d^{2}\right )} a g e^{\left (m \log \left (x e + d\right ) - 2\right )}}{m^{2} + 3 \, m + 2} + \frac {{\left ({\left (m^{2} + 3 \, m + 2\right )} x^{3} e^{3} + {\left (m^{2} + m\right )} d x^{2} e^{2} - 2 \, d^{2} m x e + 2 \, d^{3}\right )} c f e^{\left (m \log \left (x e + d\right ) - 3\right )}}{m^{3} + 6 \, m^{2} + 11 \, m + 6} + \frac {{\left ({\left (m^{2} + 3 \, m + 2\right )} x^{3} e^{3} + {\left (m^{2} + m\right )} d x^{2} e^{2} - 2 \, d^{2} m x e + 2 \, d^{3}\right )} b g e^{\left (m \log \left (x e + d\right ) - 3\right )}}{m^{3} + 6 \, m^{2} + 11 \, m + 6} + \frac {{\left ({\left (m^{3} + 6 \, m^{2} + 11 \, m + 6\right )} x^{4} e^{4} + {\left (m^{3} + 3 \, m^{2} + 2 \, m\right )} d x^{3} e^{3} - 3 \, {\left (m^{2} + m\right )} d^{2} x^{2} e^{2} + 6 \, d^{3} m x e - 6 \, d^{4}\right )} c g e^{\left (m \log \left (x e + d\right ) - 4\right )}}{m^{4} + 10 \, m^{3} + 35 \, m^{2} + 50 \, m + 24} \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. 485 vs.
\(2 (156) = 312\).
time = 3.42, size = 485, normalized size = 3.37 \begin {gather*} -\frac {{\left (6 \, c d^{4} g - {\left ({\left (c g m^{3} + 6 \, c g m^{2} + 11 \, c g m + 6 \, c g\right )} x^{4} + {\left ({\left (c f + b g\right )} m^{3} + 7 \, {\left (c f + b g\right )} m^{2} + 8 \, c f + 8 \, b g + 14 \, {\left (c f + b g\right )} m\right )} x^{3} + {\left ({\left (b f + a g\right )} m^{3} + 8 \, {\left (b f + a g\right )} m^{2} + 12 \, b f + 12 \, a g + 19 \, {\left (b f + a g\right )} m\right )} x^{2} + {\left (a f m^{3} + 9 \, a f m^{2} + 26 \, a f m + 24 \, a f\right )} x\right )} e^{4} - {\left (a d f m^{3} + 9 \, a d f m^{2} + 26 \, a d f m + {\left (c d g m^{3} + 3 \, c d g m^{2} + 2 \, c d g m\right )} x^{3} + 24 \, a d f + {\left ({\left (c d f + b d g\right )} m^{3} + 5 \, {\left (c d f + b d g\right )} m^{2} + 4 \, {\left (c d f + b d g\right )} m\right )} x^{2} + {\left ({\left (b d f + a d g\right )} m^{3} + 7 \, {\left (b d f + a d g\right )} m^{2} + 12 \, {\left (b d f + a d g\right )} m\right )} x\right )} e^{3} + {\left (12 \, b d^{2} f + 12 \, a d^{2} g + {\left (b d^{2} f + a d^{2} g\right )} m^{2} + 3 \, {\left (c d^{2} g m^{2} + c d^{2} g m\right )} x^{2} + 7 \, {\left (b d^{2} f + a d^{2} g\right )} m + 2 \, {\left ({\left (c d^{2} f + b d^{2} g\right )} m^{2} + 4 \, {\left (c d^{2} f + b d^{2} g\right )} m\right )} x\right )} e^{2} - 2 \, {\left (3 \, c d^{3} g m x + 4 \, c d^{3} f + 4 \, b d^{3} g + {\left (c d^{3} f + b d^{3} g\right )} m\right )} e\right )} {\left (x e + d\right )}^{m} e^{\left (-4\right )}}{m^{4} + 10 \, m^{3} + 35 \, m^{2} + 50 \, m + 24} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 5930 vs.
\(2 (134) = 268\).
time = 1.38, size = 5930, normalized size = 41.18 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 1162 vs.
\(2 (156) = 312\).
time = 2.18, size = 1162, normalized size = 8.07 \begin {gather*} \frac {{\left (x e + d\right )}^{m} c g m^{3} x^{4} e^{4} + {\left (x e + d\right )}^{m} c d g m^{3} x^{3} e^{3} + {\left (x e + d\right )}^{m} c f m^{3} x^{3} e^{4} + {\left (x e + d\right )}^{m} b g m^{3} x^{3} e^{4} + 6 \, {\left (x e + d\right )}^{m} c g m^{2} x^{4} e^{4} + {\left (x e + d\right )}^{m} c d f m^{3} x^{2} e^{3} + {\left (x e + d\right )}^{m} b d g m^{3} x^{2} e^{3} + 3 \, {\left (x e + d\right )}^{m} c d g m^{2} x^{3} e^{3} - 3 \, {\left (x e + d\right )}^{m} c d^{2} g m^{2} x^{2} e^{2} + {\left (x e + d\right )}^{m} b f m^{3} x^{2} e^{4} + {\left (x e + d\right )}^{m} a g m^{3} x^{2} e^{4} + 7 \, {\left (x e + d\right )}^{m} c f m^{2} x^{3} e^{4} + 7 \, {\left (x e + d\right )}^{m} b g m^{2} x^{3} e^{4} + 11 \, {\left (x e + d\right )}^{m} c g m x^{4} e^{4} + {\left (x e + d\right )}^{m} b d f m^{3} x e^{3} + {\left (x e + d\right )}^{m} a d g m^{3} x e^{3} + 5 \, {\left (x e + d\right )}^{m} c d f m^{2} x^{2} e^{3} + 5 \, {\left (x e + d\right )}^{m} b d g m^{2} x^{2} e^{3} + 2 \, {\left (x e + d\right )}^{m} c d g m x^{3} e^{3} - 2 \, {\left (x e + d\right )}^{m} c d^{2} f m^{2} x e^{2} - 2 \, {\left (x e + d\right )}^{m} b d^{2} g m^{2} x e^{2} - 3 \, {\left (x e + d\right )}^{m} c d^{2} g m x^{2} e^{2} + 6 \, {\left (x e + d\right )}^{m} c d^{3} g m x e + {\left (x e + d\right )}^{m} a f m^{3} x e^{4} + 8 \, {\left (x e + d\right )}^{m} b f m^{2} x^{2} e^{4} + 8 \, {\left (x e + d\right )}^{m} a g m^{2} x^{2} e^{4} + 14 \, {\left (x e + d\right )}^{m} c f m x^{3} e^{4} + 14 \, {\left (x e + d\right )}^{m} b g m x^{3} e^{4} + 6 \, {\left (x e + d\right )}^{m} c g x^{4} e^{4} + {\left (x e + d\right )}^{m} a d f m^{3} e^{3} + 7 \, {\left (x e + d\right )}^{m} b d f m^{2} x e^{3} + 7 \, {\left (x e + d\right )}^{m} a d g m^{2} x e^{3} + 4 \, {\left (x e + d\right )}^{m} c d f m x^{2} e^{3} + 4 \, {\left (x e + d\right )}^{m} b d g m x^{2} e^{3} - {\left (x e + d\right )}^{m} b d^{2} f m^{2} e^{2} - {\left (x e + d\right )}^{m} a d^{2} g m^{2} e^{2} - 8 \, {\left (x e + d\right )}^{m} c d^{2} f m x e^{2} - 8 \, {\left (x e + d\right )}^{m} b d^{2} g m x e^{2} + 2 \, {\left (x e + d\right )}^{m} c d^{3} f m e + 2 \, {\left (x e + d\right )}^{m} b d^{3} g m e - 6 \, {\left (x e + d\right )}^{m} c d^{4} g + 9 \, {\left (x e + d\right )}^{m} a f m^{2} x e^{4} + 19 \, {\left (x e + d\right )}^{m} b f m x^{2} e^{4} + 19 \, {\left (x e + d\right )}^{m} a g m x^{2} e^{4} + 8 \, {\left (x e + d\right )}^{m} c f x^{3} e^{4} + 8 \, {\left (x e + d\right )}^{m} b g x^{3} e^{4} + 9 \, {\left (x e + d\right )}^{m} a d f m^{2} e^{3} + 12 \, {\left (x e + d\right )}^{m} b d f m x e^{3} + 12 \, {\left (x e + d\right )}^{m} a d g m x e^{3} - 7 \, {\left (x e + d\right )}^{m} b d^{2} f m e^{2} - 7 \, {\left (x e + d\right )}^{m} a d^{2} g m e^{2} + 8 \, {\left (x e + d\right )}^{m} c d^{3} f e + 8 \, {\left (x e + d\right )}^{m} b d^{3} g e + 26 \, {\left (x e + d\right )}^{m} a f m x e^{4} + 12 \, {\left (x e + d\right )}^{m} b f x^{2} e^{4} + 12 \, {\left (x e + d\right )}^{m} a g x^{2} e^{4} + 26 \, {\left (x e + d\right )}^{m} a d f m e^{3} - 12 \, {\left (x e + d\right )}^{m} b d^{2} f e^{2} - 12 \, {\left (x e + d\right )}^{m} a d^{2} g e^{2} + 24 \, {\left (x e + d\right )}^{m} a f x e^{4} + 24 \, {\left (x e + d\right )}^{m} a d f e^{3}}{m^{4} e^{4} + 10 \, m^{3} e^{4} + 35 \, m^{2} e^{4} + 50 \, m e^{4} + 24 \, e^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 3.59, size = 602, normalized size = 4.18 \begin {gather*} \frac {{\left (d+e\,x\right )}^m\,\left (24\,a\,d\,e^3\,f-6\,c\,d^4\,g+8\,b\,d^3\,e\,g+8\,c\,d^3\,e\,f-12\,a\,d^2\,e^2\,g-12\,b\,d^2\,e^2\,f+9\,a\,d\,e^3\,f\,m^2+a\,d\,e^3\,f\,m^3-7\,a\,d^2\,e^2\,g\,m-7\,b\,d^2\,e^2\,f\,m-a\,d^2\,e^2\,g\,m^2-b\,d^2\,e^2\,f\,m^2+26\,a\,d\,e^3\,f\,m+2\,b\,d^3\,e\,g\,m+2\,c\,d^3\,e\,f\,m\right )}{e^4\,\left (m^4+10\,m^3+35\,m^2+50\,m+24\right )}+\frac {x\,{\left (d+e\,x\right )}^m\,\left (24\,a\,e^4\,f+26\,a\,e^4\,f\,m+9\,a\,e^4\,f\,m^2+a\,e^4\,f\,m^3+7\,a\,d\,e^3\,g\,m^2+7\,b\,d\,e^3\,f\,m^2+a\,d\,e^3\,g\,m^3+b\,d\,e^3\,f\,m^3-8\,b\,d^2\,e^2\,g\,m-8\,c\,d^2\,e^2\,f\,m-2\,b\,d^2\,e^2\,g\,m^2-2\,c\,d^2\,e^2\,f\,m^2+12\,a\,d\,e^3\,g\,m+12\,b\,d\,e^3\,f\,m+6\,c\,d^3\,e\,g\,m\right )}{e^4\,\left (m^4+10\,m^3+35\,m^2+50\,m+24\right )}+\frac {x^2\,\left (m+1\right )\,{\left (d+e\,x\right )}^m\,\left (12\,a\,e^2\,g+12\,b\,e^2\,f+7\,a\,e^2\,g\,m+7\,b\,e^2\,f\,m-3\,c\,d^2\,g\,m+a\,e^2\,g\,m^2+b\,e^2\,f\,m^2+4\,b\,d\,e\,g\,m+4\,c\,d\,e\,f\,m+b\,d\,e\,g\,m^2+c\,d\,e\,f\,m^2\right )}{e^2\,\left (m^4+10\,m^3+35\,m^2+50\,m+24\right )}+\frac {c\,g\,x^4\,{\left (d+e\,x\right )}^m\,\left (m^3+6\,m^2+11\,m+6\right )}{m^4+10\,m^3+35\,m^2+50\,m+24}+\frac {x^3\,{\left (d+e\,x\right )}^m\,\left (m^2+3\,m+2\right )\,\left (4\,b\,e\,g+4\,c\,e\,f+b\,e\,g\,m+c\,d\,g\,m+c\,e\,f\,m\right )}{e\,\left (m^4+10\,m^3+35\,m^2+50\,m+24\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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