3.24.50 \(\int x (d+e x)^m (a+b x+c x^2) \, dx\) [2350]

Optimal. Leaf size=121 \[ -\frac {d \left (c d^2-b d e+a e^2\right ) (d+e x)^{1+m}}{e^4 (1+m)}+\frac {\left (3 c d^2-e (2 b d-a e)\right ) (d+e x)^{2+m}}{e^4 (2+m)}-\frac {(3 c d-b e) (d+e x)^{3+m}}{e^4 (3+m)}+\frac {c (d+e x)^{4+m}}{e^4 (4+m)} \]

[Out]

-d*(a*e^2-b*d*e+c*d^2)*(e*x+d)^(1+m)/e^4/(1+m)+(3*c*d^2-e*(-a*e+2*b*d))*(e*x+d)^(2+m)/e^4/(2+m)-(-b*e+3*c*d)*(
e*x+d)^(3+m)/e^4/(3+m)+c*(e*x+d)^(4+m)/e^4/(4+m)

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Rubi [A]
time = 0.06, antiderivative size = 121, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {785} \begin {gather*} \frac {(d+e x)^{m+2} \left (3 c d^2-e (2 b d-a e)\right )}{e^4 (m+2)}-\frac {d (d+e x)^{m+1} \left (a e^2-b d e+c d^2\right )}{e^4 (m+1)}-\frac {(3 c d-b e) (d+e x)^{m+3}}{e^4 (m+3)}+\frac {c (d+e x)^{m+4}}{e^4 (m+4)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x*(d + e*x)^m*(a + b*x + c*x^2),x]

[Out]

-((d*(c*d^2 - b*d*e + a*e^2)*(d + e*x)^(1 + m))/(e^4*(1 + m))) + ((3*c*d^2 - e*(2*b*d - a*e))*(d + e*x)^(2 + m
))/(e^4*(2 + m)) - ((3*c*d - b*e)*(d + e*x)^(3 + m))/(e^4*(3 + m)) + (c*(d + e*x)^(4 + m))/(e^4*(4 + m))

Rule 785

Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> In
t[ExpandIntegrand[(d + e*x)^m*(f + g*x)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && N
eQ[b^2 - 4*a*c, 0] && IntegerQ[p] && (GtQ[p, 0] || (EqQ[a, 0] && IntegerQ[m]))

Rubi steps

\begin {align*} \int x (d+e x)^m \left (a+b x+c x^2\right ) \, dx &=\int \left (-\frac {d \left (c d^2-b d e+a e^2\right ) (d+e x)^m}{e^3}+\frac {\left (3 c d^2-e (2 b d-a e)\right ) (d+e x)^{1+m}}{e^3}+\frac {(-3 c d+b e) (d+e x)^{2+m}}{e^3}+\frac {c (d+e x)^{3+m}}{e^3}\right ) \, dx\\ &=-\frac {d \left (c d^2-b d e+a e^2\right ) (d+e x)^{1+m}}{e^4 (1+m)}+\frac {\left (3 c d^2-e (2 b d-a e)\right ) (d+e x)^{2+m}}{e^4 (2+m)}-\frac {(3 c d-b e) (d+e x)^{3+m}}{e^4 (3+m)}+\frac {c (d+e x)^{4+m}}{e^4 (4+m)}\\ \end {align*}

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Mathematica [A]
time = 0.23, size = 142, normalized size = 1.17 \begin {gather*} \frac {(d+e x)^{1+m} \left (c \left (-6 d^3+6 d^2 e (1+m) x-3 d e^2 \left (2+3 m+m^2\right ) x^2+e^3 \left (6+11 m+6 m^2+m^3\right ) x^3\right )+e (4+m) \left (a e (3+m) (-d+e (1+m) x)+b \left (2 d^2-2 d e (1+m) x+e^2 \left (2+3 m+m^2\right ) x^2\right )\right )\right )}{e^4 (1+m) (2+m) (3+m) (4+m)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x*(d + e*x)^m*(a + b*x + c*x^2),x]

[Out]

((d + e*x)^(1 + m)*(c*(-6*d^3 + 6*d^2*e*(1 + m)*x - 3*d*e^2*(2 + 3*m + m^2)*x^2 + e^3*(6 + 11*m + 6*m^2 + m^3)
*x^3) + e*(4 + m)*(a*e*(3 + m)*(-d + e*(1 + m)*x) + b*(2*d^2 - 2*d*e*(1 + m)*x + e^2*(2 + 3*m + m^2)*x^2))))/(
e^4*(1 + m)*(2 + m)*(3 + m)*(4 + m))

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(280\) vs. \(2(121)=242\).
time = 0.03, size = 281, normalized size = 2.32

method	result	size
gosper	\(-\frac {\left (e x +d \right )^{1+m} \left (-c \,e^{3} m^{3} x^{3}-b \,e^{3} m^{3} x^{2}-6 c \,e^{3} m^{2} x^{3}-a \,e^{3} m^{3} x -7 b \,e^{3} m^{2} x^{2}+3 c d \,e^{2} m^{2} x^{2}-11 c \,e^{3} m \,x^{3}-8 a \,e^{3} m^{2} x +2 b d \,e^{2} m^{2} x -14 b \,e^{3} m \,x^{2}+9 c d \,e^{2} m \,x^{2}-6 c \,x^{3} e^{3}+a d \,e^{2} m^{2}-19 a \,e^{3} m x +10 b d \,e^{2} m x -8 b \,e^{3} x^{2}-6 c \,d^{2} e m x +6 c d \,e^{2} x^{2}+7 a d \,e^{2} m -12 a \,e^{3} x -2 b \,d^{2} e m +8 b d \,e^{2} x -6 c \,d^{2} e x +12 a d \,e^{2}-8 b \,d^{2} e +6 c \,d^{3}\right )}{e^{4} \left (m^{4}+10 m^{3}+35 m^{2}+50 m +24\right )}\)	\(281\)
norman	\(\frac {c \,x^{4} {\mathrm e}^{m \ln \left (e x +d \right )}}{4+m}+\frac {\left (b e m +c d m +4 b e \right ) x^{3} {\mathrm e}^{m \ln \left (e x +d \right )}}{e \left (m^{2}+7 m +12\right )}+\frac {\left (a \,e^{2} m^{2}+b d e \,m^{2}+7 a \,e^{2} m +4 b d e m -3 c \,d^{2} m +12 a \,e^{2}\right ) x^{2} {\mathrm e}^{m \ln \left (e x +d \right )}}{e^{2} \left (m^{3}+9 m^{2}+26 m +24\right )}+\frac {m d \left (a \,e^{2} m^{2}+7 a \,e^{2} m -2 b d e m +12 a \,e^{2}-8 b d e +6 c \,d^{2}\right ) x \,{\mathrm e}^{m \ln \left (e x +d \right )}}{e^{3} \left (m^{4}+10 m^{3}+35 m^{2}+50 m +24\right )}-\frac {d^{2} \left (a \,e^{2} m^{2}+7 a \,e^{2} m -2 b d e m +12 a \,e^{2}-8 b d e +6 c \,d^{2}\right ) {\mathrm e}^{m \ln \left (e x +d \right )}}{e^{4} \left (m^{4}+10 m^{3}+35 m^{2}+50 m +24\right )}\)	\(284\)
risch	\(-\frac {\left (-c \,e^{4} m^{3} x^{4}-b \,e^{4} m^{3} x^{3}-c d \,e^{3} m^{3} x^{3}-6 c \,e^{4} m^{2} x^{4}-a \,e^{4} m^{3} x^{2}-b d \,e^{3} m^{3} x^{2}-7 b \,e^{4} m^{2} x^{3}-3 c d \,e^{3} m^{2} x^{3}-11 c \,e^{4} m \,x^{4}-a d \,e^{3} m^{3} x -8 a \,e^{4} m^{2} x^{2}-5 b d \,e^{3} m^{2} x^{2}-14 b \,e^{4} m \,x^{3}+3 c \,d^{2} e^{2} m^{2} x^{2}-2 c d \,e^{3} m \,x^{3}-6 c \,x^{4} e^{4}-7 a d \,e^{3} m^{2} x -19 a \,e^{4} m \,x^{2}+2 b \,d^{2} e^{2} m^{2} x -4 b d \,e^{3} m \,x^{2}-8 b \,e^{4} x^{3}+3 c \,d^{2} e^{2} m \,x^{2}+a \,d^{2} e^{2} m^{2}-12 a d \,e^{3} m x -12 a \,e^{4} x^{2}+8 b \,d^{2} e^{2} m x -6 c \,d^{3} e m x +7 a \,d^{2} e^{2} m -2 b \,d^{3} e m +12 a \,d^{2} e^{2}-8 b \,d^{3} e +6 c \,d^{4}\right ) \left (e x +d \right )^{m}}{\left (3+m \right ) \left (4+m \right ) \left (2+m \right ) \left (1+m \right ) e^{4}}\)	\(380\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*(e*x+d)^m*(c*x^2+b*x+a),x,method=_RETURNVERBOSE)

[Out]

-(e*x+d)^(1+m)*(-c*e^3*m^3*x^3-b*e^3*m^3*x^2-6*c*e^3*m^2*x^3-a*e^3*m^3*x-7*b*e^3*m^2*x^2+3*c*d*e^2*m^2*x^2-11*
c*e^3*m*x^3-8*a*e^3*m^2*x+2*b*d*e^2*m^2*x-14*b*e^3*m*x^2+9*c*d*e^2*m*x^2-6*c*e^3*x^3+a*d*e^2*m^2-19*a*e^3*m*x+
10*b*d*e^2*m*x-8*b*e^3*x^2-6*c*d^2*e*m*x+6*c*d*e^2*x^2+7*a*d*e^2*m-12*a*e^3*x-2*b*d^2*e*m+8*b*d*e^2*x-6*c*d^2*
e*x+12*a*d*e^2-8*b*d^2*e+6*c*d^3)/e^4/(m^4+10*m^3+35*m^2+50*m+24)

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Maxima [A]
time = 0.29, size = 218, normalized size = 1.80 \begin {gather*} \frac {{\left ({\left (m + 1\right )} x^{2} e^{2} + d m x e - d^{2}\right )} a 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 )} b 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 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.

[In]

integrate(x*(e*x+d)^m*(c*x^2+b*x+a),x, algorithm="maxima")

[Out]

((m + 1)*x^2*e^2 + d*m*x*e - d^2)*a*e^(m*log(x*e + d) - 2)/(m^2 + 3*m + 2) + ((m^2 + 3*m + 2)*x^3*e^3 + (m^2 +
m)*d*x^2*e^2 - 2*d^2*m*x*e + 2*d^3)*b*e^(m*log(x*e + d) - 3)/(m^3 + 6*m^2 + 11*m + 6) + ((m^3 + 6*m^2 + 11*m
+ 6)*x^4*e^4 + (m^3 + 3*m^2 + 2*m)*d*x^3*e^3 - 3*(m^2 + m)*d^2*x^2*e^2 + 6*d^3*m*x*e - 6*d^4)*c*e^(m*log(x*e +
d) - 4)/(m^4 + 10*m^3 + 35*m^2 + 50*m + 24)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 274 vs. \(2 (124) = 248\).
time = 4.24, size = 274, normalized size = 2.26 \begin {gather*} -\frac {{\left (6 \, c d^{4} - {\left ({\left (c m^{3} + 6 \, c m^{2} + 11 \, c m + 6 \, c\right )} x^{4} + {\left (b m^{3} + 7 \, b m^{2} + 14 \, b m + 8 \, b\right )} x^{3} + {\left (a m^{3} + 8 \, a m^{2} + 19 \, a m + 12 \, a\right )} x^{2}\right )} e^{4} - {\left ({\left (c d m^{3} + 3 \, c d m^{2} + 2 \, c d m\right )} x^{3} + {\left (b d m^{3} + 5 \, b d m^{2} + 4 \, b d m\right )} x^{2} + {\left (a d m^{3} + 7 \, a d m^{2} + 12 \, a d m\right )} x\right )} e^{3} + {\left (a d^{2} m^{2} + 7 \, a d^{2} m + 12 \, a d^{2} + 3 \, {\left (c d^{2} m^{2} + c d^{2} m\right )} x^{2} + 2 \, {\left (b d^{2} m^{2} + 4 \, b d^{2} m\right )} x\right )} e^{2} - 2 \, {\left (3 \, c d^{3} m x + b d^{3} m + 4 \, b d^{3}\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.

[In]

integrate(x*(e*x+d)^m*(c*x^2+b*x+a),x, algorithm="fricas")

[Out]

-(6*c*d^4 - ((c*m^3 + 6*c*m^2 + 11*c*m + 6*c)*x^4 + (b*m^3 + 7*b*m^2 + 14*b*m + 8*b)*x^3 + (a*m^3 + 8*a*m^2 +
19*a*m + 12*a)*x^2)*e^4 - ((c*d*m^3 + 3*c*d*m^2 + 2*c*d*m)*x^3 + (b*d*m^3 + 5*b*d*m^2 + 4*b*d*m)*x^2 + (a*d*m^
3 + 7*a*d*m^2 + 12*a*d*m)*x)*e^3 + (a*d^2*m^2 + 7*a*d^2*m + 12*a*d^2 + 3*(c*d^2*m^2 + c*d^2*m)*x^2 + 2*(b*d^2*
m^2 + 4*b*d^2*m)*x)*e^2 - 2*(3*c*d^3*m*x + b*d^3*m + 4*b*d^3)*e)*(x*e + d)^m*e^(-4)/(m^4 + 10*m^3 + 35*m^2 + 5
0*m + 24)

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 3267 vs. \(2 (104) = 208\).
time = 0.95, size = 3267, normalized size = 27.00 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(e*x+d)**m*(c*x**2+b*x+a),x)

[Out]

Piecewise((d**m*(a*x**2/2 + b*x**3/3 + c*x**4/4), Eq(e, 0)), (-a*d*e**2/(6*d**3*e**4 + 18*d**2*e**5*x + 18*d*e
**6*x**2 + 6*e**7*x**3) - 3*a*e**3*x/(6*d**3*e**4 + 18*d**2*e**5*x + 18*d*e**6*x**2 + 6*e**7*x**3) - 2*b*d**2*
e/(6*d**3*e**4 + 18*d**2*e**5*x + 18*d*e**6*x**2 + 6*e**7*x**3) - 6*b*d*e**2*x/(6*d**3*e**4 + 18*d**2*e**5*x +
18*d*e**6*x**2 + 6*e**7*x**3) - 6*b*e**3*x**2/(6*d**3*e**4 + 18*d**2*e**5*x + 18*d*e**6*x**2 + 6*e**7*x**3) +
6*c*d**3*log(d/e + x)/(6*d**3*e**4 + 18*d**2*e**5*x + 18*d*e**6*x**2 + 6*e**7*x**3) + 11*c*d**3/(6*d**3*e**4
+ 18*d**2*e**5*x + 18*d*e**6*x**2 + 6*e**7*x**3) + 18*c*d**2*e*x*log(d/e + x)/(6*d**3*e**4 + 18*d**2*e**5*x +
18*d*e**6*x**2 + 6*e**7*x**3) + 27*c*d**2*e*x/(6*d**3*e**4 + 18*d**2*e**5*x + 18*d*e**6*x**2 + 6*e**7*x**3) +
18*c*d*e**2*x**2*log(d/e + x)/(6*d**3*e**4 + 18*d**2*e**5*x + 18*d*e**6*x**2 + 6*e**7*x**3) + 18*c*d*e**2*x**2
/(6*d**3*e**4 + 18*d**2*e**5*x + 18*d*e**6*x**2 + 6*e**7*x**3) + 6*c*e**3*x**3*log(d/e + x)/(6*d**3*e**4 + 18*
d**2*e**5*x + 18*d*e**6*x**2 + 6*e**7*x**3), Eq(m, -4)), (-a*d*e**2/(2*d**2*e**4 + 4*d*e**5*x + 2*e**6*x**2) -
2*a*e**3*x/(2*d**2*e**4 + 4*d*e**5*x + 2*e**6*x**2) + 2*b*d**2*e*log(d/e + x)/(2*d**2*e**4 + 4*d*e**5*x + 2*e
**6*x**2) + 3*b*d**2*e/(2*d**2*e**4 + 4*d*e**5*x + 2*e**6*x**2) + 4*b*d*e**2*x*log(d/e + x)/(2*d**2*e**4 + 4*d
*e**5*x + 2*e**6*x**2) + 4*b*d*e**2*x/(2*d**2*e**4 + 4*d*e**5*x + 2*e**6*x**2) + 2*b*e**3*x**2*log(d/e + x)/(2
*d**2*e**4 + 4*d*e**5*x + 2*e**6*x**2) - 6*c*d**3*log(d/e + x)/(2*d**2*e**4 + 4*d*e**5*x + 2*e**6*x**2) - 9*c*
d**3/(2*d**2*e**4 + 4*d*e**5*x + 2*e**6*x**2) - 12*c*d**2*e*x*log(d/e + x)/(2*d**2*e**4 + 4*d*e**5*x + 2*e**6*
x**2) - 12*c*d**2*e*x/(2*d**2*e**4 + 4*d*e**5*x + 2*e**6*x**2) - 6*c*d*e**2*x**2*log(d/e + x)/(2*d**2*e**4 + 4
*d*e**5*x + 2*e**6*x**2) + 2*c*e**3*x**3/(2*d**2*e**4 + 4*d*e**5*x + 2*e**6*x**2), Eq(m, -3)), (2*a*d*e**2*log
(d/e + x)/(2*d*e**4 + 2*e**5*x) + 2*a*d*e**2/(2*d*e**4 + 2*e**5*x) + 2*a*e**3*x*log(d/e + x)/(2*d*e**4 + 2*e**
5*x) - 4*b*d**2*e*log(d/e + x)/(2*d*e**4 + 2*e**5*x) - 4*b*d**2*e/(2*d*e**4 + 2*e**5*x) - 4*b*d*e**2*x*log(d/e
+ x)/(2*d*e**4 + 2*e**5*x) + 2*b*e**3*x**2/(2*d*e**4 + 2*e**5*x) + 6*c*d**3*log(d/e + x)/(2*d*e**4 + 2*e**5*x
) + 6*c*d**3/(2*d*e**4 + 2*e**5*x) + 6*c*d**2*e*x*log(d/e + x)/(2*d*e**4 + 2*e**5*x) - 3*c*d*e**2*x**2/(2*d*e*
*4 + 2*e**5*x) + c*e**3*x**3/(2*d*e**4 + 2*e**5*x), Eq(m, -2)), (-a*d*log(d/e + x)/e**2 + a*x/e + b*d**2*log(d
/e + x)/e**3 - b*d*x/e**2 + b*x**2/(2*e) - c*d**3*log(d/e + x)/e**4 + c*d**2*x/e**3 - c*d*x**2/(2*e**2) + c*x*
*3/(3*e), Eq(m, -1)), (-a*d**2*e**2*m**2*(d + e*x)**m/(e**4*m**4 + 10*e**4*m**3 + 35*e**4*m**2 + 50*e**4*m + 2
4*e**4) - 7*a*d**2*e**2*m*(d + e*x)**m/(e**4*m**4 + 10*e**4*m**3 + 35*e**4*m**2 + 50*e**4*m + 24*e**4) - 12*a*
d**2*e**2*(d + e*x)**m/(e**4*m**4 + 10*e**4*m**3 + 35*e**4*m**2 + 50*e**4*m + 24*e**4) + a*d*e**3*m**3*x*(d +
e*x)**m/(e**4*m**4 + 10*e**4*m**3 + 35*e**4*m**2 + 50*e**4*m + 24*e**4) + 7*a*d*e**3*m**2*x*(d + e*x)**m/(e**4
*m**4 + 10*e**4*m**3 + 35*e**4*m**2 + 50*e**4*m + 24*e**4) + 12*a*d*e**3*m*x*(d + e*x)**m/(e**4*m**4 + 10*e**4
*m**3 + 35*e**4*m**2 + 50*e**4*m + 24*e**4) + a*e**4*m**3*x**2*(d + e*x)**m/(e**4*m**4 + 10*e**4*m**3 + 35*e**
4*m**2 + 50*e**4*m + 24*e**4) + 8*a*e**4*m**2*x**2*(d + e*x)**m/(e**4*m**4 + 10*e**4*m**3 + 35*e**4*m**2 + 50*
e**4*m + 24*e**4) + 19*a*e**4*m*x**2*(d + e*x)**m/(e**4*m**4 + 10*e**4*m**3 + 35*e**4*m**2 + 50*e**4*m + 24*e*
*4) + 12*a*e**4*x**2*(d + e*x)**m/(e**4*m**4 + 10*e**4*m**3 + 35*e**4*m**2 + 50*e**4*m + 24*e**4) + 2*b*d**3*e
*m*(d + e*x)**m/(e**4*m**4 + 10*e**4*m**3 + 35*e**4*m**2 + 50*e**4*m + 24*e**4) + 8*b*d**3*e*(d + e*x)**m/(e**
4*m**4 + 10*e**4*m**3 + 35*e**4*m**2 + 50*e**4*m + 24*e**4) - 2*b*d**2*e**2*m**2*x*(d + e*x)**m/(e**4*m**4 + 1
0*e**4*m**3 + 35*e**4*m**2 + 50*e**4*m + 24*e**4) - 8*b*d**2*e**2*m*x*(d + e*x)**m/(e**4*m**4 + 10*e**4*m**3 +
35*e**4*m**2 + 50*e**4*m + 24*e**4) + b*d*e**3*m**3*x**2*(d + e*x)**m/(e**4*m**4 + 10*e**4*m**3 + 35*e**4*m**
2 + 50*e**4*m + 24*e**4) + 5*b*d*e**3*m**2*x**2*(d + e*x)**m/(e**4*m**4 + 10*e**4*m**3 + 35*e**4*m**2 + 50*e**
4*m + 24*e**4) + 4*b*d*e**3*m*x**2*(d + e*x)**m/(e**4*m**4 + 10*e**4*m**3 + 35*e**4*m**2 + 50*e**4*m + 24*e**4
) + b*e**4*m**3*x**3*(d + e*x)**m/(e**4*m**4 + 10*e**4*m**3 + 35*e**4*m**2 + 50*e**4*m + 24*e**4) + 7*b*e**4*m
**2*x**3*(d + e*x)**m/(e**4*m**4 + 10*e**4*m**3 + 35*e**4*m**2 + 50*e**4*m + 24*e**4) + 14*b*e**4*m*x**3*(d +
e*x)**m/(e**4*m**4 + 10*e**4*m**3 + 35*e**4*m**2 + 50*e**4*m + 24*e**4) + 8*b*e**4*x**3*(d + e*x)**m/(e**4*m**
4 + 10*e**4*m**3 + 35*e**4*m**2 + 50*e**4*m + 24*e**4) - 6*c*d**4*(d + e*x)**m/(e**4*m**4 + 10*e**4*m**3 + 35*
e**4*m**2 + 50*e**4*m + 24*e**4) + 6*c*d**3*e*m*x*(d + e*x)**m/(e**4*m**4 + 10*e**4*m**3 + 35*e**4*m**2 + 50*e
**4*m + 24*e**4) - 3*c*d**2*e**2*m**2*x**2*(d + e*x)**m/(e**4*m**4 + 10*e**4*m**3 + 35*e**4*m**2 + 50*e**4*m +
24*e**4) - 3*c*d**2*e**2*m*x**2*(d + e*x)**m/(e**4*m**4 + 10*e**4*m**3 + 35*e**4*m**2 + 50*e**4*m + 24*e**4)
+ c*d*e**3*m**3*x**3*(d + e*x)**m/(e**4*m**4 + ...

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 606 vs. \(2 (124) = 248\).
time = 1.12, size = 606, normalized size = 5.01 \begin {gather*} \frac {{\left (x e + d\right )}^{m} c m^{3} x^{4} e^{4} + {\left (x e + d\right )}^{m} c d m^{3} x^{3} e^{3} + {\left (x e + d\right )}^{m} b m^{3} x^{3} e^{4} + 6 \, {\left (x e + d\right )}^{m} c m^{2} x^{4} e^{4} + {\left (x e + d\right )}^{m} b d m^{3} x^{2} e^{3} + 3 \, {\left (x e + d\right )}^{m} c d m^{2} x^{3} e^{3} - 3 \, {\left (x e + d\right )}^{m} c d^{2} m^{2} x^{2} e^{2} + {\left (x e + d\right )}^{m} a m^{3} x^{2} e^{4} + 7 \, {\left (x e + d\right )}^{m} b m^{2} x^{3} e^{4} + 11 \, {\left (x e + d\right )}^{m} c m x^{4} e^{4} + {\left (x e + d\right )}^{m} a d m^{3} x e^{3} + 5 \, {\left (x e + d\right )}^{m} b d m^{2} x^{2} e^{3} + 2 \, {\left (x e + d\right )}^{m} c d m x^{3} e^{3} - 2 \, {\left (x e + d\right )}^{m} b d^{2} m^{2} x e^{2} - 3 \, {\left (x e + d\right )}^{m} c d^{2} m x^{2} e^{2} + 6 \, {\left (x e + d\right )}^{m} c d^{3} m x e + 8 \, {\left (x e + d\right )}^{m} a m^{2} x^{2} e^{4} + 14 \, {\left (x e + d\right )}^{m} b m x^{3} e^{4} + 6 \, {\left (x e + d\right )}^{m} c x^{4} e^{4} + 7 \, {\left (x e + d\right )}^{m} a d m^{2} x e^{3} + 4 \, {\left (x e + d\right )}^{m} b d m x^{2} e^{3} - {\left (x e + d\right )}^{m} a d^{2} m^{2} e^{2} - 8 \, {\left (x e + d\right )}^{m} b d^{2} m x e^{2} + 2 \, {\left (x e + d\right )}^{m} b d^{3} m e - 6 \, {\left (x e + d\right )}^{m} c d^{4} + 19 \, {\left (x e + d\right )}^{m} a m x^{2} e^{4} + 8 \, {\left (x e + d\right )}^{m} b x^{3} e^{4} + 12 \, {\left (x e + d\right )}^{m} a d m x e^{3} - 7 \, {\left (x e + d\right )}^{m} a d^{2} m e^{2} + 8 \, {\left (x e + d\right )}^{m} b d^{3} e + 12 \, {\left (x e + d\right )}^{m} a x^{2} e^{4} - 12 \, {\left (x e + d\right )}^{m} a d^{2} e^{2}}{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.

[In]

integrate(x*(e*x+d)^m*(c*x^2+b*x+a),x, algorithm="giac")

[Out]

((x*e + d)^m*c*m^3*x^4*e^4 + (x*e + d)^m*c*d*m^3*x^3*e^3 + (x*e + d)^m*b*m^3*x^3*e^4 + 6*(x*e + d)^m*c*m^2*x^4
*e^4 + (x*e + d)^m*b*d*m^3*x^2*e^3 + 3*(x*e + d)^m*c*d*m^2*x^3*e^3 - 3*(x*e + d)^m*c*d^2*m^2*x^2*e^2 + (x*e +
d)^m*a*m^3*x^2*e^4 + 7*(x*e + d)^m*b*m^2*x^3*e^4 + 11*(x*e + d)^m*c*m*x^4*e^4 + (x*e + d)^m*a*d*m^3*x*e^3 + 5*
(x*e + d)^m*b*d*m^2*x^2*e^3 + 2*(x*e + d)^m*c*d*m*x^3*e^3 - 2*(x*e + d)^m*b*d^2*m^2*x*e^2 - 3*(x*e + d)^m*c*d^
2*m*x^2*e^2 + 6*(x*e + d)^m*c*d^3*m*x*e + 8*(x*e + d)^m*a*m^2*x^2*e^4 + 14*(x*e + d)^m*b*m*x^3*e^4 + 6*(x*e +
d)^m*c*x^4*e^4 + 7*(x*e + d)^m*a*d*m^2*x*e^3 + 4*(x*e + d)^m*b*d*m*x^2*e^3 - (x*e + d)^m*a*d^2*m^2*e^2 - 8*(x*
e + d)^m*b*d^2*m*x*e^2 + 2*(x*e + d)^m*b*d^3*m*e - 6*(x*e + d)^m*c*d^4 + 19*(x*e + d)^m*a*m*x^2*e^4 + 8*(x*e +
d)^m*b*x^3*e^4 + 12*(x*e + d)^m*a*d*m*x*e^3 - 7*(x*e + d)^m*a*d^2*m*e^2 + 8*(x*e + d)^m*b*d^3*e + 12*(x*e + d
)^m*a*x^2*e^4 - 12*(x*e + d)^m*a*d^2*e^2)/(m^4*e^4 + 10*m^3*e^4 + 35*m^2*e^4 + 50*m*e^4 + 24*e^4)

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Mupad [B]
time = 2.55, size = 300, normalized size = 2.48 \begin {gather*} {\left (d+e\,x\right )}^m\,\left (\frac {c\,x^4\,\left (m^3+6\,m^2+11\,m+6\right )}{m^4+10\,m^3+35\,m^2+50\,m+24}-\frac {d^2\,\left (6\,c\,d^2-2\,b\,d\,e\,m-8\,b\,d\,e+a\,e^2\,m^2+7\,a\,e^2\,m+12\,a\,e^2\right )}{e^4\,\left (m^4+10\,m^3+35\,m^2+50\,m+24\right )}+\frac {x^3\,\left (4\,b\,e+b\,e\,m+c\,d\,m\right )\,\left (m^2+3\,m+2\right )}{e\,\left (m^4+10\,m^3+35\,m^2+50\,m+24\right )}+\frac {x^2\,\left (m+1\right )\,\left (-3\,c\,d^2\,m+b\,d\,e\,m^2+4\,b\,d\,e\,m+a\,e^2\,m^2+7\,a\,e^2\,m+12\,a\,e^2\right )}{e^2\,\left (m^4+10\,m^3+35\,m^2+50\,m+24\right )}+\frac {d\,m\,x\,\left (6\,c\,d^2-2\,b\,d\,e\,m-8\,b\,d\,e+a\,e^2\,m^2+7\,a\,e^2\,m+12\,a\,e^2\right )}{e^3\,\left (m^4+10\,m^3+35\,m^2+50\,m+24\right )}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*(d + e*x)^m*(a + b*x + c*x^2),x)

[Out]

(d + e*x)^m*((c*x^4*(11*m + 6*m^2 + m^3 + 6))/(50*m + 35*m^2 + 10*m^3 + m^4 + 24) - (d^2*(12*a*e^2 + 6*c*d^2 +
a*e^2*m^2 - 8*b*d*e + 7*a*e^2*m - 2*b*d*e*m))/(e^4*(50*m + 35*m^2 + 10*m^3 + m^4 + 24)) + (x^3*(4*b*e + b*e*m
+ c*d*m)*(3*m + m^2 + 2))/(e*(50*m + 35*m^2 + 10*m^3 + m^4 + 24)) + (x^2*(m + 1)*(12*a*e^2 + a*e^2*m^2 + 7*a*
e^2*m - 3*c*d^2*m + b*d*e*m^2 + 4*b*d*e*m))/(e^2*(50*m + 35*m^2 + 10*m^3 + m^4 + 24)) + (d*m*x*(12*a*e^2 + 6*c
*d^2 + a*e^2*m^2 - 8*b*d*e + 7*a*e^2*m - 2*b*d*e*m))/(e^3*(50*m + 35*m^2 + 10*m^3 + m^4 + 24)))

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