3.10.0 ∫ (d + ex)m(f + gx)(a + bx + cx2) dx [921]

Integrand size = 23, antiderivative size = 144 \[ \int (d+e x)^m (f+g x) \left (a+b x+c x^2\right ) \, dx=\frac {\left (c d^2-b d e+a e^2\right ) (e f-d g) (d+e x)^{1+m}}{e^4 (1+m)}-\frac {(c d (2 e f-3 d g)-e (b e f-2 b d g+a e g)) (d+e x)^{2+m}}{e^4 (2+m)}+\frac {(c e f-3 c d g+b e g) (d+e x)^{3+m}}{e^4 (3+m)}+\frac {c g (d+e x)^{4+m}}{e^4 (4+m)} \]

Rubi [A] (veriﬁed)

Time = 0.07 (sec) , antiderivative size = 144, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.043, Rules used = {785} \[ \int (d+e x)^m (f+g x) \left (a+b x+c x^2\right ) \, dx=\frac {(e f-d g) (d+e x)^{m+1} \left (a e^2-b d e+c d^2\right )}{e^4 (m+1)}-\frac {(d+e x)^{m+2} (c d (2 e f-3 d g)-e (a e g-2 b d g+b e f))}{e^4 (m+2)}+\frac {(d+e x)^{m+3} (b e g-3 c d g+c e f)}{e^4 (m+3)}+\frac {c g (d+e x)^{m+4}}{e^4 (m+4)} \]

Rubi steps \begin{align*} \text {integral}& = \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*}

Mathematica [A] (veriﬁed)

Time = 0.32 (sec) , antiderivative size = 180, normalized size of antiderivative = 1.25 \[ \int (d+e x)^m (f+g x) \left (a+b x+c x^2\right ) \, dx=\frac {(d+e x)^{1+m} \left (-\frac {\left (c d^2+e (-b d+a e)\right ) (6 c d g+b e g (1+m)-2 c e f (4+m))}{e^2 (1+m)}+\frac {\left (-b^2 e^2 g (2+m)+2 c^2 d (3 d g-e f (4+m))+c e (b d g (-2+m)+2 a e g (3+m)+b e f (4+m))\right ) (d+e x)}{e^2 (2+m)}+(a+x (b+c x)) (b e g+c (-3 d g+e f (4+m)+e g (3+m) x))\right )}{c e^2 (3+m) (4+m)} \]

Maple [B] (veriﬁed)

Leaf count of result is larger than twice the leaf count of optimal. \(502\) vs. \(2(144)=288\).

Time = 0.49 (sec) , antiderivative size = 503, normalized size of antiderivative = 3.49


method	result	size

gosper	\(-\frac {\left (e x +d \right )^{1+m} \left (-c \,e^{3} g \,m^{3} x^{3}-b \,e^{3} g \,m^{3} x^{2}-c \,e^{3} f \,m^{3} x^{2}-6 c \,e^{3} g \,m^{2} x^{3}-a \,e^{3} g \,m^{3} x -b \,e^{3} f \,m^{3} x -7 b \,e^{3} g \,m^{2} x^{2}+3 c d \,e^{2} g \,m^{2} x^{2}-7 c \,e^{3} f \,m^{2} x^{2}-11 c \,e^{3} g m \,x^{3}-a \,e^{3} f \,m^{3}-8 a \,e^{3} g \,m^{2} x +2 b d \,e^{2} g \,m^{2} x -8 b \,e^{3} f \,m^{2} x -14 b \,e^{3} g m \,x^{2}+2 c d \,e^{2} f \,m^{2} x +9 c d \,e^{2} g m \,x^{2}-14 c \,e^{3} f m \,x^{2}-6 c \,e^{3} g \,x^{3}+a d \,e^{2} g \,m^{2}-9 a \,e^{3} f \,m^{2}-19 a \,e^{3} g m x +b d \,e^{2} f \,m^{2}+10 b d \,e^{2} g m x -19 b \,e^{3} f m x -8 b \,e^{3} g \,x^{2}-6 c \,d^{2} e g m x +10 c d \,e^{2} f m x +6 c d \,e^{2} g \,x^{2}-8 c \,e^{3} f \,x^{2}+7 a d \,e^{2} g m -26 a \,e^{3} f m -12 a \,e^{3} g x -2 b \,d^{2} e g m +7 b d \,e^{2} f m +8 b d \,e^{2} g x -12 b \,e^{3} f x -2 c \,d^{2} e f m -6 c \,d^{2} e g x +8 c d \,e^{2} f x +12 a \,e^{2} g d -24 a \,e^{3} f -8 b \,d^{2} e g +12 b \,e^{2} f d +6 c \,d^{3} g -8 c \,d^{2} e f \right )}{e^{4} \left (m^{4}+10 m^{3}+35 m^{2}+50 m +24\right )}\)	\(503\)
norman	\(\frac {c g \,x^{4} {\mathrm e}^{m \ln \left (e x +d \right )}}{4+m}+\frac {\left (b e g m +c d g m +c e f m +4 b e g +4 c e f \right ) x^{3} {\mathrm e}^{m \ln \left (e x +d \right )}}{e \left (m^{2}+7 m +12\right )}+\frac {\left (a \,e^{2} g \,m^{2}+b d e g \,m^{2}+b \,e^{2} f \,m^{2}+c d e f \,m^{2}+7 a \,e^{2} g m +4 b d e g m +7 b \,e^{2} f m -3 c \,d^{2} g m +4 c d e f m +12 a \,e^{2} g +12 b \,e^{2} f \right ) x^{2} {\mathrm e}^{m \ln \left (e x +d \right )}}{e^{2} \left (m^{3}+9 m^{2}+26 m +24\right )}+\frac {\left (a d \,e^{2} g \,m^{3}+a \,e^{3} f \,m^{3}+b d \,e^{2} f \,m^{3}+7 a d \,e^{2} g \,m^{2}+9 a \,e^{3} f \,m^{2}-2 b \,d^{2} e g \,m^{2}+7 b d \,e^{2} f \,m^{2}-2 c \,d^{2} e f \,m^{2}+12 a d \,e^{2} g m +26 a \,e^{3} f m -8 b \,d^{2} e g m +12 b d \,e^{2} f m +6 c \,d^{3} g m -8 c \,d^{2} e f m +24 a \,e^{3} f \right ) x \,{\mathrm e}^{m \ln \left (e x +d \right )}}{e^{3} \left (m^{4}+10 m^{3}+35 m^{2}+50 m +24\right )}-\frac {d \left (-a \,e^{3} f \,m^{3}+a d \,e^{2} g \,m^{2}-9 a \,e^{3} f \,m^{2}+b d \,e^{2} f \,m^{2}+7 a d \,e^{2} g m -26 a \,e^{3} f m -2 b \,d^{2} e g m +7 b d \,e^{2} f m -2 c \,d^{2} e f m +12 a \,e^{2} g d -24 a \,e^{3} f -8 b \,d^{2} e g +12 b \,e^{2} f d +6 c \,d^{3} g -8 c \,d^{2} e f \right ) {\mathrm e}^{m \ln \left (e x +d \right )}}{e^{4} \left (m^{4}+10 m^{3}+35 m^{2}+50 m +24\right )}\)	\(535\)
risch	\(-\frac {\left (-c \,e^{4} g \,m^{3} x^{4}-b \,e^{4} g \,m^{3} x^{3}-c d \,e^{3} g \,m^{3} x^{3}-c \,e^{4} f \,m^{3} x^{3}-6 c \,e^{4} g \,m^{2} x^{4}-a \,e^{4} g \,m^{3} x^{2}-b d \,e^{3} g \,m^{3} x^{2}-b \,e^{4} f \,m^{3} x^{2}-7 b \,e^{4} g \,m^{2} x^{3}-c d \,e^{3} f \,m^{3} x^{2}-3 c d \,e^{3} g \,m^{2} x^{3}-7 c \,e^{4} f \,m^{2} x^{3}-11 c \,e^{4} g m \,x^{4}-a d \,e^{3} g \,m^{3} x -a \,e^{4} f \,m^{3} x -8 a \,e^{4} g \,m^{2} x^{2}-b d \,e^{3} f \,m^{3} x -5 b d \,e^{3} g \,m^{2} x^{2}-8 b \,e^{4} f \,m^{2} x^{2}-14 b \,e^{4} g m \,x^{3}+3 c \,d^{2} e^{2} g \,m^{2} x^{2}-5 c d \,e^{3} f \,m^{2} x^{2}-2 c d \,e^{3} g m \,x^{3}-14 c \,e^{4} f m \,x^{3}-6 c g \,x^{4} e^{4}-a d \,e^{3} f \,m^{3}-7 a d \,e^{3} g \,m^{2} x -9 a \,e^{4} f \,m^{2} x -19 a \,e^{4} g m \,x^{2}+2 b \,d^{2} e^{2} g \,m^{2} x -7 b d \,e^{3} f \,m^{2} x -4 b d \,e^{3} g m \,x^{2}-19 b \,e^{4} f m \,x^{2}-8 b \,e^{4} g \,x^{3}+2 c \,d^{2} e^{2} f \,m^{2} x +3 c \,d^{2} e^{2} g m \,x^{2}-4 c d \,e^{3} f m \,x^{2}-8 c \,e^{4} f \,x^{3}+a \,d^{2} e^{2} g \,m^{2}-9 a d \,e^{3} f \,m^{2}-12 a d \,e^{3} g m x -26 a \,e^{4} f m x -12 a \,e^{4} g \,x^{2}+b \,d^{2} e^{2} f \,m^{2}+8 b \,d^{2} e^{2} g m x -12 b d \,e^{3} f m x -12 b \,e^{4} f \,x^{2}-6 c \,d^{3} e g m x +8 c \,d^{2} e^{2} f m x +7 a \,d^{2} e^{2} g m -26 a d \,e^{3} f m -24 a \,e^{4} f x -2 b \,d^{3} e g m +7 b \,d^{2} e^{2} f m -2 c \,d^{3} e f m +12 a \,d^{2} e^{2} g -24 a d \,e^{3} f -8 b \,d^{3} e g +12 b \,d^{2} e^{2} f +6 c \,d^{4} g -8 c \,d^{3} e f \right ) \left (e x +d \right )^{m}}{\left (3+m \right ) \left (4+m \right ) \left (2+m \right ) \left (1+m \right ) e^{4}}\)	\(734\)
parallelrisch	\(\text {Expression too large to display}\)	\(1143\)

Fricas [B] (veriﬁcation not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 613 vs. \(2 (144) = 288\).

Time = 0.30 (sec) , antiderivative size = 613, normalized size of antiderivative = 4.26 \[ \int (d+e x)^m (f+g x) \left (a+b x+c x^2\right ) \, dx=\frac {{\left (a d e^{3} f m^{3} + {\left (c e^{4} g m^{3} + 6 \, c e^{4} g m^{2} + 11 \, c e^{4} g m + 6 \, c e^{4} g\right )} x^{4} + {\left (8 \, c e^{4} f + 8 \, b e^{4} g + {\left (c e^{4} f + {\left (c d e^{3} + b e^{4}\right )} g\right )} m^{3} + {\left (7 \, c e^{4} f + {\left (3 \, c d e^{3} + 7 \, b e^{4}\right )} g\right )} m^{2} + 2 \, {\left (7 \, c e^{4} f + {\left (c d e^{3} + 7 \, b e^{4}\right )} g\right )} m\right )} x^{3} - {\left (a d^{2} e^{2} g + {\left (b d^{2} e^{2} - 9 \, a d e^{3}\right )} f\right )} m^{2} + {\left (12 \, b e^{4} f + 12 \, a e^{4} g + {\left ({\left (c d e^{3} + b e^{4}\right )} f + {\left (b d e^{3} + a e^{4}\right )} g\right )} m^{3} + {\left ({\left (5 \, c d e^{3} + 8 \, b e^{4}\right )} f - {\left (3 \, c d^{2} e^{2} - 5 \, b d e^{3} - 8 \, a e^{4}\right )} g\right )} m^{2} + {\left ({\left (4 \, c d e^{3} + 19 \, b e^{4}\right )} f - {\left (3 \, c d^{2} e^{2} - 4 \, b d e^{3} - 19 \, a e^{4}\right )} g\right )} m\right )} x^{2} + 4 \, {\left (2 \, c d^{3} e - 3 \, b d^{2} e^{2} + 6 \, a d e^{3}\right )} f - 2 \, {\left (3 \, c d^{4} - 4 \, b d^{3} e + 6 \, a d^{2} e^{2}\right )} g + {\left ({\left (2 \, c d^{3} e - 7 \, b d^{2} e^{2} + 26 \, a d e^{3}\right )} f + {\left (2 \, b d^{3} e - 7 \, a d^{2} e^{2}\right )} g\right )} m + {\left (24 \, a e^{4} f + {\left (a d e^{3} g + {\left (b d e^{3} + a e^{4}\right )} f\right )} m^{3} - {\left ({\left (2 \, c d^{2} e^{2} - 7 \, b d e^{3} - 9 \, a e^{4}\right )} f + {\left (2 \, b d^{2} e^{2} - 7 \, a d e^{3}\right )} g\right )} m^{2} - 2 \, {\left ({\left (4 \, c d^{2} e^{2} - 6 \, b d e^{3} - 13 \, a e^{4}\right )} f - {\left (3 \, c d^{3} e - 4 \, b d^{2} e^{2} + 6 \, a d e^{3}\right )} g\right )} m\right )} x\right )} {\left (e x + d\right )}^{m}}{e^{4} m^{4} + 10 \, e^{4} m^{3} + 35 \, e^{4} m^{2} + 50 \, e^{4} m + 24 \, e^{4}} \]

Sympy [B] (veriﬁcation not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 5930 vs. \(2 (134) = 268\).

Time = 1.27 (sec) , antiderivative size = 5930, normalized size of antiderivative = 41.18 \[ \int (d+e x)^m (f+g x) \left (a+b x+c x^2\right ) \, dx=\text {Too large to display} \]

Maxima [B] (veriﬁcation not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 352 vs. \(2 (144) = 288\).

Time = 0.23 (sec) , antiderivative size = 352, normalized size of antiderivative = 2.44 \[ \int (d+e x)^m (f+g x) \left (a+b x+c x^2\right ) \, dx=\frac {{\left (e^{2} {\left (m + 1\right )} x^{2} + d e m x - d^{2}\right )} {\left (e x + d\right )}^{m} b f}{{\left (m^{2} + 3 \, m + 2\right )} e^{2}} + \frac {{\left (e^{2} {\left (m + 1\right )} x^{2} + d e m x - d^{2}\right )} {\left (e x + d\right )}^{m} a g}{{\left (m^{2} + 3 \, m + 2\right )} e^{2}} + \frac {{\left (e x + d\right )}^{m + 1} a f}{e {\left (m + 1\right )}} + \frac {{\left ({\left (m^{2} + 3 \, m + 2\right )} e^{3} x^{3} + {\left (m^{2} + m\right )} d e^{2} x^{2} - 2 \, d^{2} e m x + 2 \, d^{3}\right )} {\left (e x + d\right )}^{m} c f}{{\left (m^{3} + 6 \, m^{2} + 11 \, m + 6\right )} e^{3}} + \frac {{\left ({\left (m^{2} + 3 \, m + 2\right )} e^{3} x^{3} + {\left (m^{2} + m\right )} d e^{2} x^{2} - 2 \, d^{2} e m x + 2 \, d^{3}\right )} {\left (e x + d\right )}^{m} b g}{{\left (m^{3} + 6 \, m^{2} + 11 \, m + 6\right )} e^{3}} + \frac {{\left ({\left (m^{3} + 6 \, m^{2} + 11 \, m + 6\right )} e^{4} x^{4} + {\left (m^{3} + 3 \, m^{2} + 2 \, m\right )} d e^{3} x^{3} - 3 \, {\left (m^{2} + m\right )} d^{2} e^{2} x^{2} + 6 \, d^{3} e m x - 6 \, d^{4}\right )} {\left (e x + d\right )}^{m} c g}{{\left (m^{4} + 10 \, m^{3} + 35 \, m^{2} + 50 \, m + 24\right )} e^{4}} \]

Giac [B] (veriﬁcation not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1156 vs. \(2 (144) = 288\).

Time = 0.28 (sec) , antiderivative size = 1156, normalized size of antiderivative = 8.03 \[ \int (d+e x)^m (f+g x) \left (a+b x+c x^2\right ) \, dx=\text {Too large to display} \]

Mupad [B] (veriﬁcation not implemented)

Time = 13.54 (sec) , antiderivative size = 602, normalized size of antiderivative = 4.18 \[ \int (d+e x)^m (f+g x) \left (a+b x+c x^2\right ) \, dx=\frac {{\left (d+e\,x\right )}^m\,\left (24\,a\,d\,e^3\,f-6\,c\,d^4\,g+8\,b\,d^3\,e\,g+8\,c\,d^3\,e\,f-12\,a\,d^2\,e^2\,g-12\,b\,d^2\,e^2\,f+9\,a\,d\,e^3\,f\,m^2+a\,d\,e^3\,f\,m^3-7\,a\,d^2\,e^2\,g\,m-7\,b\,d^2\,e^2\,f\,m-a\,d^2\,e^2\,g\,m^2-b\,d^2\,e^2\,f\,m^2+26\,a\,d\,e^3\,f\,m+2\,b\,d^3\,e\,g\,m+2\,c\,d^3\,e\,f\,m\right )}{e^4\,\left (m^4+10\,m^3+35\,m^2+50\,m+24\right )}+\frac {x\,{\left (d+e\,x\right )}^m\,\left (24\,a\,e^4\,f+26\,a\,e^4\,f\,m+9\,a\,e^4\,f\,m^2+a\,e^4\,f\,m^3+7\,a\,d\,e^3\,g\,m^2+7\,b\,d\,e^3\,f\,m^2+a\,d\,e^3\,g\,m^3+b\,d\,e^3\,f\,m^3-8\,b\,d^2\,e^2\,g\,m-8\,c\,d^2\,e^2\,f\,m-2\,b\,d^2\,e^2\,g\,m^2-2\,c\,d^2\,e^2\,f\,m^2+12\,a\,d\,e^3\,g\,m+12\,b\,d\,e^3\,f\,m+6\,c\,d^3\,e\,g\,m\right )}{e^4\,\left (m^4+10\,m^3+35\,m^2+50\,m+24\right )}+\frac {x^2\,\left (m+1\right )\,{\left (d+e\,x\right )}^m\,\left (12\,a\,e^2\,g+12\,b\,e^2\,f+7\,a\,e^2\,g\,m+7\,b\,e^2\,f\,m-3\,c\,d^2\,g\,m+a\,e^2\,g\,m^2+b\,e^2\,f\,m^2+4\,b\,d\,e\,g\,m+4\,c\,d\,e\,f\,m+b\,d\,e\,g\,m^2+c\,d\,e\,f\,m^2\right )}{e^2\,\left (m^4+10\,m^3+35\,m^2+50\,m+24\right )}+\frac {c\,g\,x^4\,{\left (d+e\,x\right )}^m\,\left (m^3+6\,m^2+11\,m+6\right )}{m^4+10\,m^3+35\,m^2+50\,m+24}+\frac {x^3\,{\left (d+e\,x\right )}^m\,\left (m^2+3\,m+2\right )\,\left (4\,b\,e\,g+4\,c\,e\,f+b\,e\,g\,m+c\,d\,g\,m+c\,e\,f\,m\right )}{e\,\left (m^4+10\,m^3+35\,m^2+50\,m+24\right )} \]

\(\int (d+e x)^m (f+g x) (a+b x+c x^2) \, dx\) [921]

Optimal result