3.19.51 \(\int (d+e x)^2 (a d e+(c d^2+a e^2) x+c d e x^2)^3 \, dx\) [1851]

Optimal. Leaf size=111 \[ -\frac {\left (c d^2-a e^2\right )^3 (d+e x)^6}{6 e^4}+\frac {3 c d \left (c d^2-a e^2\right )^2 (d+e x)^7}{7 e^4}-\frac {3 c^2 d^2 \left (c d^2-a e^2\right ) (d+e x)^8}{8 e^4}+\frac {c^3 d^3 (d+e x)^9}{9 e^4} \]

[Out]

-1/6*(-a*e^2+c*d^2)^3*(e*x+d)^6/e^4+3/7*c*d*(-a*e^2+c*d^2)^2*(e*x+d)^7/e^4-3/8*c^2*d^2*(-a*e^2+c*d^2)*(e*x+d)^
8/e^4+1/9*c^3*d^3*(e*x+d)^9/e^4

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Rubi [A]
time = 0.17, antiderivative size = 111, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.057, Rules used = {640, 45} \begin {gather*} -\frac {3 c^2 d^2 (d+e x)^8 \left (c d^2-a e^2\right )}{8 e^4}+\frac {3 c d (d+e x)^7 \left (c d^2-a e^2\right )^2}{7 e^4}-\frac {(d+e x)^6 \left (c d^2-a e^2\right )^3}{6 e^4}+\frac {c^3 d^3 (d+e x)^9}{9 e^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(d + e*x)^2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^3,x]

[Out]

-1/6*((c*d^2 - a*e^2)^3*(d + e*x)^6)/e^4 + (3*c*d*(c*d^2 - a*e^2)^2*(d + e*x)^7)/(7*e^4) - (3*c^2*d^2*(c*d^2 -
a*e^2)*(d + e*x)^8)/(8*e^4) + (c^3*d^3*(d + e*x)^9)/(9*e^4)

Rule 45

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 640

Int[((d_) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[(d + e*x)^(m + p)*(a
/d + (c/e)*x)^p, x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a*e^2, 0] &&
IntegerQ[p]

Rubi steps

\begin {align*} \int (d+e x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3 \, dx &=\int (a e+c d x)^3 (d+e x)^5 \, dx\\ &=\int \left (\frac {\left (-c d^2+a e^2\right )^3 (d+e x)^5}{e^3}+\frac {3 c d \left (c d^2-a e^2\right )^2 (d+e x)^6}{e^3}-\frac {3 c^2 d^2 \left (c d^2-a e^2\right ) (d+e x)^7}{e^3}+\frac {c^3 d^3 (d+e x)^8}{e^3}\right ) \, dx\\ &=-\frac {\left (c d^2-a e^2\right )^3 (d+e x)^6}{6 e^4}+\frac {3 c d \left (c d^2-a e^2\right )^2 (d+e x)^7}{7 e^4}-\frac {3 c^2 d^2 \left (c d^2-a e^2\right ) (d+e x)^8}{8 e^4}+\frac {c^3 d^3 (d+e x)^9}{9 e^4}\\ \end {align*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(255\) vs. \(2(111)=222\).
time = 0.05, size = 255, normalized size = 2.30 \begin {gather*} \frac {1}{504} x \left (84 a^3 e^3 \left (6 d^5+15 d^4 e x+20 d^3 e^2 x^2+15 d^2 e^3 x^3+6 d e^4 x^4+e^5 x^5\right )+36 a^2 c d e^2 x \left (21 d^5+70 d^4 e x+105 d^3 e^2 x^2+84 d^2 e^3 x^3+35 d e^4 x^4+6 e^5 x^5\right )+9 a c^2 d^2 e x^2 \left (56 d^5+210 d^4 e x+336 d^3 e^2 x^2+280 d^2 e^3 x^3+120 d e^4 x^4+21 e^5 x^5\right )+c^3 d^3 x^3 \left (126 d^5+504 d^4 e x+840 d^3 e^2 x^2+720 d^2 e^3 x^3+315 d e^4 x^4+56 e^5 x^5\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x)^2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^3,x]

[Out]

(x*(84*a^3*e^3*(6*d^5 + 15*d^4*e*x + 20*d^3*e^2*x^2 + 15*d^2*e^3*x^3 + 6*d*e^4*x^4 + e^5*x^5) + 36*a^2*c*d*e^2
*x*(21*d^5 + 70*d^4*e*x + 105*d^3*e^2*x^2 + 84*d^2*e^3*x^3 + 35*d*e^4*x^4 + 6*e^5*x^5) + 9*a*c^2*d^2*e*x^2*(56
*d^5 + 210*d^4*e*x + 336*d^3*e^2*x^2 + 280*d^2*e^3*x^3 + 120*d*e^4*x^4 + 21*e^5*x^5) + c^3*d^3*x^3*(126*d^5 +
504*d^4*e*x + 840*d^3*e^2*x^2 + 720*d^2*e^3*x^3 + 315*d*e^4*x^4 + 56*e^5*x^5)))/504

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(800\) vs. \(2(103)=206\).
time = 0.71, size = 801, normalized size = 7.22

method	result	size
norman	\(\frac {c^{3} d^{3} e^{5} x^{9}}{9}+\left (\frac {3}{8} c^{2} d^{2} a \,e^{6}+\frac {5}{8} c^{3} d^{4} e^{4}\right ) x^{8}+\left (\frac {3}{7} d \,e^{7} a^{2} c +\frac {15}{7} c^{2} d^{3} a \,e^{5}+\frac {10}{7} c^{3} d^{5} e^{3}\right ) x^{7}+\left (\frac {1}{6} e^{8} a^{3}+\frac {5}{2} d^{2} e^{6} a^{2} c +5 c^{2} d^{4} a \,e^{4}+\frac {5}{3} c^{3} d^{6} e^{2}\right ) x^{6}+\left (e^{7} a^{3} d +6 d^{3} e^{5} a^{2} c +6 c^{2} d^{5} a \,e^{3}+c^{3} d^{7} e \right ) x^{5}+\left (\frac {5}{2} e^{6} a^{3} d^{2}+\frac {15}{2} d^{4} e^{4} a^{2} c +\frac {15}{4} c^{2} d^{6} a \,e^{2}+\frac {1}{4} c^{3} d^{8}\right ) x^{4}+\left (\frac {10}{3} e^{5} a^{3} d^{3}+5 d^{5} e^{3} a^{2} c +c^{2} d^{7} a e \right ) x^{3}+\left (\frac {5}{2} e^{4} a^{3} d^{4}+\frac {3}{2} d^{6} e^{2} a^{2} c \right ) x^{2}+e^{3} a^{3} d^{5} x\)	\(299\)
risch	\(\frac {1}{9} c^{3} d^{3} e^{5} x^{9}+\frac {3}{8} x^{8} c^{2} d^{2} a \,e^{6}+\frac {5}{8} x^{8} c^{3} d^{4} e^{4}+\frac {3}{7} x^{7} d \,e^{7} a^{2} c +\frac {15}{7} x^{7} c^{2} d^{3} a \,e^{5}+\frac {10}{7} x^{7} c^{3} d^{5} e^{3}+\frac {1}{6} x^{6} e^{8} a^{3}+\frac {5}{2} x^{6} d^{2} e^{6} a^{2} c +5 x^{6} c^{2} d^{4} a \,e^{4}+\frac {5}{3} x^{6} c^{3} d^{6} e^{2}+a^{3} d \,e^{7} x^{5}+6 a^{2} c \,d^{3} e^{5} x^{5}+6 a \,c^{2} d^{5} e^{3} x^{5}+c^{3} d^{7} e \,x^{5}+\frac {5}{2} x^{4} e^{6} a^{3} d^{2}+\frac {15}{2} x^{4} d^{4} e^{4} a^{2} c +\frac {15}{4} x^{4} c^{2} d^{6} a \,e^{2}+\frac {1}{4} x^{4} c^{3} d^{8}+\frac {10}{3} x^{3} e^{5} a^{3} d^{3}+5 x^{3} d^{5} e^{3} a^{2} c +x^{3} c^{2} d^{7} a e +\frac {5}{2} x^{2} e^{4} a^{3} d^{4}+\frac {3}{2} x^{2} d^{6} e^{2} a^{2} c +e^{3} a^{3} d^{5} x\)	\(330\)
gosper	\(\frac {x \left (56 c^{3} d^{3} e^{5} x^{8}+189 x^{7} c^{2} d^{2} a \,e^{6}+315 x^{7} c^{3} d^{4} e^{4}+216 x^{6} d \,e^{7} a^{2} c +1080 x^{6} c^{2} d^{3} a \,e^{5}+720 x^{6} c^{3} d^{5} e^{3}+84 x^{5} e^{8} a^{3}+1260 x^{5} d^{2} e^{6} a^{2} c +2520 x^{5} c^{2} d^{4} a \,e^{4}+840 x^{5} c^{3} d^{6} e^{2}+504 a^{3} d \,e^{7} x^{4}+3024 a^{2} c \,d^{3} e^{5} x^{4}+3024 a \,c^{2} d^{5} e^{3} x^{4}+504 c^{3} d^{7} e \,x^{4}+1260 x^{3} e^{6} a^{3} d^{2}+3780 x^{3} d^{4} e^{4} a^{2} c +1890 x^{3} c^{2} d^{6} a \,e^{2}+126 x^{3} c^{3} d^{8}+1680 x^{2} e^{5} a^{3} d^{3}+2520 x^{2} d^{5} e^{3} a^{2} c +504 x^{2} c^{2} d^{7} a e +1260 x \,e^{4} a^{3} d^{4}+756 x \,d^{6} e^{2} a^{2} c +504 e^{3} a^{3} d^{5}\right )}{504}\)	\(332\)
default	\(\frac {c^{3} d^{3} e^{5} x^{9}}{9}+\frac {\left (2 c^{3} d^{4} e^{4}+3 e^{4} \left (e^{2} a +c \,d^{2}\right ) c^{2} d^{2}\right ) x^{8}}{8}+\frac {\left (c^{3} d^{5} e^{3}+6 d^{3} e^{3} \left (e^{2} a +c \,d^{2}\right ) c^{2}+e^{2} \left (d^{3} e^{3} c^{2} a +2 \left (e^{2} a +c \,d^{2}\right )^{2} c d e +c d e \left (2 a c \,d^{2} e^{2}+\left (e^{2} a +c \,d^{2}\right )^{2}\right )\right )\right ) x^{7}}{7}+\frac {\left (3 d^{4} \left (e^{2} a +c \,d^{2}\right ) c^{2} e^{2}+2 d e \left (d^{3} e^{3} c^{2} a +2 \left (e^{2} a +c \,d^{2}\right )^{2} c d e +c d e \left (2 a c \,d^{2} e^{2}+\left (e^{2} a +c \,d^{2}\right )^{2}\right )\right )+e^{2} \left (4 a \,d^{2} e^{2} c \left (e^{2} a +c \,d^{2}\right )+\left (e^{2} a +c \,d^{2}\right ) \left (2 a c \,d^{2} e^{2}+\left (e^{2} a +c \,d^{2}\right )^{2}\right )\right )\right ) x^{6}}{6}+\frac {\left (d^{2} \left (d^{3} e^{3} c^{2} a +2 \left (e^{2} a +c \,d^{2}\right )^{2} c d e +c d e \left (2 a c \,d^{2} e^{2}+\left (e^{2} a +c \,d^{2}\right )^{2}\right )\right )+2 d e \left (4 a \,d^{2} e^{2} c \left (e^{2} a +c \,d^{2}\right )+\left (e^{2} a +c \,d^{2}\right ) \left (2 a c \,d^{2} e^{2}+\left (e^{2} a +c \,d^{2}\right )^{2}\right )\right )+e^{2} \left (a d e \left (2 a c \,d^{2} e^{2}+\left (e^{2} a +c \,d^{2}\right )^{2}\right )+2 \left (e^{2} a +c \,d^{2}\right )^{2} a d e +c \,d^{3} e^{3} a^{2}\right )\right ) x^{5}}{5}+\frac {\left (d^{2} \left (4 a \,d^{2} e^{2} c \left (e^{2} a +c \,d^{2}\right )+\left (e^{2} a +c \,d^{2}\right ) \left (2 a c \,d^{2} e^{2}+\left (e^{2} a +c \,d^{2}\right )^{2}\right )\right )+2 d e \left (a d e \left (2 a c \,d^{2} e^{2}+\left (e^{2} a +c \,d^{2}\right )^{2}\right )+2 \left (e^{2} a +c \,d^{2}\right )^{2} a d e +c \,d^{3} e^{3} a^{2}\right )+3 e^{4} a^{2} d^{2} \left (e^{2} a +c \,d^{2}\right )\right ) x^{4}}{4}+\frac {\left (d^{2} \left (a d e \left (2 a c \,d^{2} e^{2}+\left (e^{2} a +c \,d^{2}\right )^{2}\right )+2 \left (e^{2} a +c \,d^{2}\right )^{2} a d e +c \,d^{3} e^{3} a^{2}\right )+6 d^{3} e^{3} a^{2} \left (e^{2} a +c \,d^{2}\right )+e^{5} a^{3} d^{3}\right ) x^{3}}{3}+\frac {\left (3 d^{4} a^{2} e^{2} \left (e^{2} a +c \,d^{2}\right )+2 e^{4} a^{3} d^{4}\right ) x^{2}}{2}+e^{3} a^{3} d^{5} x\)	\(801\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^3,x,method=_RETURNVERBOSE)

[Out]

1/9*c^3*d^3*e^5*x^9+1/8*(2*c^3*d^4*e^4+3*e^4*(a*e^2+c*d^2)*c^2*d^2)*x^8+1/7*(c^3*d^5*e^3+6*d^3*e^3*(a*e^2+c*d^
2)*c^2+e^2*(d^3*e^3*c^2*a+2*(a*e^2+c*d^2)^2*c*d*e+c*d*e*(2*a*c*d^2*e^2+(a*e^2+c*d^2)^2)))*x^7+1/6*(3*d^4*(a*e^
2+c*d^2)*c^2*e^2+2*d*e*(d^3*e^3*c^2*a+2*(a*e^2+c*d^2)^2*c*d*e+c*d*e*(2*a*c*d^2*e^2+(a*e^2+c*d^2)^2))+e^2*(4*a*
d^2*e^2*c*(a*e^2+c*d^2)+(a*e^2+c*d^2)*(2*a*c*d^2*e^2+(a*e^2+c*d^2)^2)))*x^6+1/5*(d^2*(d^3*e^3*c^2*a+2*(a*e^2+c
*d^2)^2*c*d*e+c*d*e*(2*a*c*d^2*e^2+(a*e^2+c*d^2)^2))+2*d*e*(4*a*d^2*e^2*c*(a*e^2+c*d^2)+(a*e^2+c*d^2)*(2*a*c*d
^2*e^2+(a*e^2+c*d^2)^2))+e^2*(a*d*e*(2*a*c*d^2*e^2+(a*e^2+c*d^2)^2)+2*(a*e^2+c*d^2)^2*a*d*e+c*d^3*e^3*a^2))*x^
5+1/4*(d^2*(4*a*d^2*e^2*c*(a*e^2+c*d^2)+(a*e^2+c*d^2)*(2*a*c*d^2*e^2+(a*e^2+c*d^2)^2))+2*d*e*(a*d*e*(2*a*c*d^2
*e^2+(a*e^2+c*d^2)^2)+2*(a*e^2+c*d^2)^2*a*d*e+c*d^3*e^3*a^2)+3*e^4*a^2*d^2*(a*e^2+c*d^2))*x^4+1/3*(d^2*(a*d*e*
(2*a*c*d^2*e^2+(a*e^2+c*d^2)^2)+2*(a*e^2+c*d^2)^2*a*d*e+c*d^3*e^3*a^2)+6*d^3*e^3*a^2*(a*e^2+c*d^2)+e^5*a^3*d^3
)*x^3+1/2*(3*d^4*a^2*e^2*(a*e^2+c*d^2)+2*e^4*a^3*d^4)*x^2+e^3*a^3*d^5*x

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 284 vs. \(2 (100) = 200\).
time = 0.32, size = 284, normalized size = 2.56 \begin {gather*} \frac {1}{9} \, c^{3} d^{3} x^{9} e^{5} + a^{3} d^{5} x e^{3} + \frac {1}{8} \, {\left (5 \, c^{3} d^{4} e^{4} + 3 \, a c^{2} d^{2} e^{6}\right )} x^{8} + \frac {1}{7} \, {\left (10 \, c^{3} d^{5} e^{3} + 15 \, a c^{2} d^{3} e^{5} + 3 \, a^{2} c d e^{7}\right )} x^{7} + \frac {1}{6} \, {\left (10 \, c^{3} d^{6} e^{2} + 30 \, a c^{2} d^{4} e^{4} + 15 \, a^{2} c d^{2} e^{6} + a^{3} e^{8}\right )} x^{6} + {\left (c^{3} d^{7} e + 6 \, a c^{2} d^{5} e^{3} + 6 \, a^{2} c d^{3} e^{5} + a^{3} d e^{7}\right )} x^{5} + \frac {1}{4} \, {\left (c^{3} d^{8} + 15 \, a c^{2} d^{6} e^{2} + 30 \, a^{2} c d^{4} e^{4} + 10 \, a^{3} d^{2} e^{6}\right )} x^{4} + \frac {1}{3} \, {\left (3 \, a c^{2} d^{7} e + 15 \, a^{2} c d^{5} e^{3} + 10 \, a^{3} d^{3} e^{5}\right )} x^{3} + \frac {1}{2} \, {\left (3 \, a^{2} c d^{6} e^{2} + 5 \, a^{3} d^{4} e^{4}\right )} x^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^3,x, algorithm="maxima")

[Out]

1/9*c^3*d^3*x^9*e^5 + a^3*d^5*x*e^3 + 1/8*(5*c^3*d^4*e^4 + 3*a*c^2*d^2*e^6)*x^8 + 1/7*(10*c^3*d^5*e^3 + 15*a*c
^2*d^3*e^5 + 3*a^2*c*d*e^7)*x^7 + 1/6*(10*c^3*d^6*e^2 + 30*a*c^2*d^4*e^4 + 15*a^2*c*d^2*e^6 + a^3*e^8)*x^6 + (
c^3*d^7*e + 6*a*c^2*d^5*e^3 + 6*a^2*c*d^3*e^5 + a^3*d*e^7)*x^5 + 1/4*(c^3*d^8 + 15*a*c^2*d^6*e^2 + 30*a^2*c*d^
4*e^4 + 10*a^3*d^2*e^6)*x^4 + 1/3*(3*a*c^2*d^7*e + 15*a^2*c*d^5*e^3 + 10*a^3*d^3*e^5)*x^3 + 1/2*(3*a^2*c*d^6*e
^2 + 5*a^3*d^4*e^4)*x^2

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 301 vs. \(2 (100) = 200\).
time = 3.64, size = 301, normalized size = 2.71 \begin {gather*} \frac {1}{4} \, c^{3} d^{8} x^{4} + \frac {1}{6} \, a^{3} x^{6} e^{8} + \frac {1}{7} \, {\left (3 \, a^{2} c d x^{7} + 7 \, a^{3} d x^{5}\right )} e^{7} + \frac {1}{8} \, {\left (3 \, a c^{2} d^{2} x^{8} + 20 \, a^{2} c d^{2} x^{6} + 20 \, a^{3} d^{2} x^{4}\right )} e^{6} + \frac {1}{63} \, {\left (7 \, c^{3} d^{3} x^{9} + 135 \, a c^{2} d^{3} x^{7} + 378 \, a^{2} c d^{3} x^{5} + 210 \, a^{3} d^{3} x^{3}\right )} e^{5} + \frac {5}{8} \, {\left (c^{3} d^{4} x^{8} + 8 \, a c^{2} d^{4} x^{6} + 12 \, a^{2} c d^{4} x^{4} + 4 \, a^{3} d^{4} x^{2}\right )} e^{4} + \frac {1}{7} \, {\left (10 \, c^{3} d^{5} x^{7} + 42 \, a c^{2} d^{5} x^{5} + 35 \, a^{2} c d^{5} x^{3} + 7 \, a^{3} d^{5} x\right )} e^{3} + \frac {1}{12} \, {\left (20 \, c^{3} d^{6} x^{6} + 45 \, a c^{2} d^{6} x^{4} + 18 \, a^{2} c d^{6} x^{2}\right )} e^{2} + {\left (c^{3} d^{7} x^{5} + a c^{2} d^{7} x^{3}\right )} e \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^3,x, algorithm="fricas")

[Out]

1/4*c^3*d^8*x^4 + 1/6*a^3*x^6*e^8 + 1/7*(3*a^2*c*d*x^7 + 7*a^3*d*x^5)*e^7 + 1/8*(3*a*c^2*d^2*x^8 + 20*a^2*c*d^
2*x^6 + 20*a^3*d^2*x^4)*e^6 + 1/63*(7*c^3*d^3*x^9 + 135*a*c^2*d^3*x^7 + 378*a^2*c*d^3*x^5 + 210*a^3*d^3*x^3)*e
^5 + 5/8*(c^3*d^4*x^8 + 8*a*c^2*d^4*x^6 + 12*a^2*c*d^4*x^4 + 4*a^3*d^4*x^2)*e^4 + 1/7*(10*c^3*d^5*x^7 + 42*a*c
^2*d^5*x^5 + 35*a^2*c*d^5*x^3 + 7*a^3*d^5*x)*e^3 + 1/12*(20*c^3*d^6*x^6 + 45*a*c^2*d^6*x^4 + 18*a^2*c*d^6*x^2)
*e^2 + (c^3*d^7*x^5 + a*c^2*d^7*x^3)*e

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 335 vs. \(2 (100) = 200\).
time = 0.04, size = 335, normalized size = 3.02 \begin {gather*} a^{3} d^{5} e^{3} x + \frac {c^{3} d^{3} e^{5} x^{9}}{9} + x^{8} \cdot \left (\frac {3 a c^{2} d^{2} e^{6}}{8} + \frac {5 c^{3} d^{4} e^{4}}{8}\right ) + x^{7} \cdot \left (\frac {3 a^{2} c d e^{7}}{7} + \frac {15 a c^{2} d^{3} e^{5}}{7} + \frac {10 c^{3} d^{5} e^{3}}{7}\right ) + x^{6} \left (\frac {a^{3} e^{8}}{6} + \frac {5 a^{2} c d^{2} e^{6}}{2} + 5 a c^{2} d^{4} e^{4} + \frac {5 c^{3} d^{6} e^{2}}{3}\right ) + x^{5} \left (a^{3} d e^{7} + 6 a^{2} c d^{3} e^{5} + 6 a c^{2} d^{5} e^{3} + c^{3} d^{7} e\right ) + x^{4} \cdot \left (\frac {5 a^{3} d^{2} e^{6}}{2} + \frac {15 a^{2} c d^{4} e^{4}}{2} + \frac {15 a c^{2} d^{6} e^{2}}{4} + \frac {c^{3} d^{8}}{4}\right ) + x^{3} \cdot \left (\frac {10 a^{3} d^{3} e^{5}}{3} + 5 a^{2} c d^{5} e^{3} + a c^{2} d^{7} e\right ) + x^{2} \cdot \left (\frac {5 a^{3} d^{4} e^{4}}{2} + \frac {3 a^{2} c d^{6} e^{2}}{2}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)**2*(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**3,x)

[Out]

a**3*d**5*e**3*x + c**3*d**3*e**5*x**9/9 + x**8*(3*a*c**2*d**2*e**6/8 + 5*c**3*d**4*e**4/8) + x**7*(3*a**2*c*d
*e**7/7 + 15*a*c**2*d**3*e**5/7 + 10*c**3*d**5*e**3/7) + x**6*(a**3*e**8/6 + 5*a**2*c*d**2*e**6/2 + 5*a*c**2*d
**4*e**4 + 5*c**3*d**6*e**2/3) + x**5*(a**3*d*e**7 + 6*a**2*c*d**3*e**5 + 6*a*c**2*d**5*e**3 + c**3*d**7*e) +
x**4*(5*a**3*d**2*e**6/2 + 15*a**2*c*d**4*e**4/2 + 15*a*c**2*d**6*e**2/4 + c**3*d**8/4) + x**3*(10*a**3*d**3*e
**5/3 + 5*a**2*c*d**5*e**3 + a*c**2*d**7*e) + x**2*(5*a**3*d**4*e**4/2 + 3*a**2*c*d**6*e**2/2)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 310 vs. \(2 (100) = 200\).
time = 1.34, size = 310, normalized size = 2.79 \begin {gather*} \frac {1}{9} \, c^{3} d^{3} x^{9} e^{5} + \frac {5}{8} \, c^{3} d^{4} x^{8} e^{4} + \frac {10}{7} \, c^{3} d^{5} x^{7} e^{3} + \frac {5}{3} \, c^{3} d^{6} x^{6} e^{2} + c^{3} d^{7} x^{5} e + \frac {1}{4} \, c^{3} d^{8} x^{4} + \frac {3}{8} \, a c^{2} d^{2} x^{8} e^{6} + \frac {15}{7} \, a c^{2} d^{3} x^{7} e^{5} + 5 \, a c^{2} d^{4} x^{6} e^{4} + 6 \, a c^{2} d^{5} x^{5} e^{3} + \frac {15}{4} \, a c^{2} d^{6} x^{4} e^{2} + a c^{2} d^{7} x^{3} e + \frac {3}{7} \, a^{2} c d x^{7} e^{7} + \frac {5}{2} \, a^{2} c d^{2} x^{6} e^{6} + 6 \, a^{2} c d^{3} x^{5} e^{5} + \frac {15}{2} \, a^{2} c d^{4} x^{4} e^{4} + 5 \, a^{2} c d^{5} x^{3} e^{3} + \frac {3}{2} \, a^{2} c d^{6} x^{2} e^{2} + \frac {1}{6} \, a^{3} x^{6} e^{8} + a^{3} d x^{5} e^{7} + \frac {5}{2} \, a^{3} d^{2} x^{4} e^{6} + \frac {10}{3} \, a^{3} d^{3} x^{3} e^{5} + \frac {5}{2} \, a^{3} d^{4} x^{2} e^{4} + a^{3} d^{5} x e^{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^3,x, algorithm="giac")

[Out]

1/9*c^3*d^3*x^9*e^5 + 5/8*c^3*d^4*x^8*e^4 + 10/7*c^3*d^5*x^7*e^3 + 5/3*c^3*d^6*x^6*e^2 + c^3*d^7*x^5*e + 1/4*c
^3*d^8*x^4 + 3/8*a*c^2*d^2*x^8*e^6 + 15/7*a*c^2*d^3*x^7*e^5 + 5*a*c^2*d^4*x^6*e^4 + 6*a*c^2*d^5*x^5*e^3 + 15/4
*a*c^2*d^6*x^4*e^2 + a*c^2*d^7*x^3*e + 3/7*a^2*c*d*x^7*e^7 + 5/2*a^2*c*d^2*x^6*e^6 + 6*a^2*c*d^3*x^5*e^5 + 15/
2*a^2*c*d^4*x^4*e^4 + 5*a^2*c*d^5*x^3*e^3 + 3/2*a^2*c*d^6*x^2*e^2 + 1/6*a^3*x^6*e^8 + a^3*d*x^5*e^7 + 5/2*a^3*
d^2*x^4*e^6 + 10/3*a^3*d^3*x^3*e^5 + 5/2*a^3*d^4*x^2*e^4 + a^3*d^5*x*e^3

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Mupad [B]
time = 0.11, size = 295, normalized size = 2.66 \begin {gather*} x^4\,\left (\frac {5\,a^3\,d^2\,e^6}{2}+\frac {15\,a^2\,c\,d^4\,e^4}{2}+\frac {15\,a\,c^2\,d^6\,e^2}{4}+\frac {c^3\,d^8}{4}\right )+x^6\,\left (\frac {a^3\,e^8}{6}+\frac {5\,a^2\,c\,d^2\,e^6}{2}+5\,a\,c^2\,d^4\,e^4+\frac {5\,c^3\,d^6\,e^2}{3}\right )+x^5\,\left (a^3\,d\,e^7+6\,a^2\,c\,d^3\,e^5+6\,a\,c^2\,d^5\,e^3+c^3\,d^7\,e\right )+a^3\,d^5\,e^3\,x+\frac {c^3\,d^3\,e^5\,x^9}{9}+\frac {a\,d^3\,e\,x^3\,\left (10\,a^2\,e^4+15\,a\,c\,d^2\,e^2+3\,c^2\,d^4\right )}{3}+\frac {c\,d\,e^3\,x^7\,\left (3\,a^2\,e^4+15\,a\,c\,d^2\,e^2+10\,c^2\,d^4\right )}{7}+\frac {a^2\,d^4\,e^2\,x^2\,\left (3\,c\,d^2+5\,a\,e^2\right )}{2}+\frac {c^2\,d^2\,e^4\,x^8\,\left (5\,c\,d^2+3\,a\,e^2\right )}{8} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d + e*x)^2*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^3,x)

[Out]

x^4*((c^3*d^8)/4 + (5*a^3*d^2*e^6)/2 + (15*a*c^2*d^6*e^2)/4 + (15*a^2*c*d^4*e^4)/2) + x^6*((a^3*e^8)/6 + (5*c^
3*d^6*e^2)/3 + 5*a*c^2*d^4*e^4 + (5*a^2*c*d^2*e^6)/2) + x^5*(a^3*d*e^7 + c^3*d^7*e + 6*a*c^2*d^5*e^3 + 6*a^2*c
*d^3*e^5) + a^3*d^5*e^3*x + (c^3*d^3*e^5*x^9)/9 + (a*d^3*e*x^3*(10*a^2*e^4 + 3*c^2*d^4 + 15*a*c*d^2*e^2))/3 +
(c*d*e^3*x^7*(3*a^2*e^4 + 10*c^2*d^4 + 15*a*c*d^2*e^2))/7 + (a^2*d^4*e^2*x^2*(5*a*e^2 + 3*c*d^2))/2 + (c^2*d^2
*e^4*x^8*(3*a*e^2 + 5*c*d^2))/8

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