3.19.52 \(\int (d+e x) (a d e+(c d^2+a e^2) x+c d e x^2)^3 \, dx\) [1852]

Optimal. Leaf size=111 \[ -\frac {\left (c d^2-a e^2\right )^3 (d+e x)^5}{5 e^4}+\frac {c d \left (c d^2-a e^2\right )^2 (d+e x)^6}{2 e^4}-\frac {3 c^2 d^2 \left (c d^2-a e^2\right ) (d+e x)^7}{7 e^4}+\frac {c^3 d^3 (d+e x)^8}{8 e^4} \]

[Out]

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

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Rubi [A]
time = 0.11, antiderivative size = 111, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.061, Rules used = {640, 45} \begin {gather*} -\frac {3 c^2 d^2 (d+e x)^7 \left (c d^2-a e^2\right )}{7 e^4}+\frac {c d (d+e x)^6 \left (c d^2-a e^2\right )^2}{2 e^4}-\frac {(d+e x)^5 \left (c d^2-a e^2\right )^3}{5 e^4}+\frac {c^3 d^3 (d+e x)^8}{8 e^4} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/5*((c*d^2 - a*e^2)^3*(d + e*x)^5)/e^4 + (c*d*(c*d^2 - a*e^2)^2*(d + e*x)^6)/(2*e^4) - (3*c^2*d^2*(c*d^2 - a
*e^2)*(d + e*x)^7)/(7*e^4) + (c^3*d^3*(d + e*x)^8)/(8*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) \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)^4 \, dx\\ &=\int \left (\frac {\left (-c d^2+a e^2\right )^3 (d+e x)^4}{e^3}+\frac {3 c d \left (c d^2-a e^2\right )^2 (d+e x)^5}{e^3}-\frac {3 c^2 d^2 \left (c d^2-a e^2\right ) (d+e x)^6}{e^3}+\frac {c^3 d^3 (d+e x)^7}{e^3}\right ) \, dx\\ &=-\frac {\left (c d^2-a e^2\right )^3 (d+e x)^5}{5 e^4}+\frac {c d \left (c d^2-a e^2\right )^2 (d+e x)^6}{2 e^4}-\frac {3 c^2 d^2 \left (c d^2-a e^2\right ) (d+e x)^7}{7 e^4}+\frac {c^3 d^3 (d+e x)^8}{8 e^4}\\ \end {align*}

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Mathematica [A]
time = 0.04, size = 211, normalized size = 1.90 \begin {gather*} \frac {1}{280} x \left (56 a^3 e^3 \left (5 d^4+10 d^3 e x+10 d^2 e^2 x^2+5 d e^3 x^3+e^4 x^4\right )+28 a^2 c d e^2 x \left (15 d^4+40 d^3 e x+45 d^2 e^2 x^2+24 d e^3 x^3+5 e^4 x^4\right )+8 a c^2 d^2 e x^2 \left (35 d^4+105 d^3 e x+126 d^2 e^2 x^2+70 d e^3 x^3+15 e^4 x^4\right )+c^3 d^3 x^3 \left (70 d^4+224 d^3 e x+280 d^2 e^2 x^2+160 d e^3 x^3+35 e^4 x^4\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(x*(56*a^3*e^3*(5*d^4 + 10*d^3*e*x + 10*d^2*e^2*x^2 + 5*d*e^3*x^3 + e^4*x^4) + 28*a^2*c*d*e^2*x*(15*d^4 + 40*d
^3*e*x + 45*d^2*e^2*x^2 + 24*d*e^3*x^3 + 5*e^4*x^4) + 8*a*c^2*d^2*e*x^2*(35*d^4 + 105*d^3*e*x + 126*d^2*e^2*x^
2 + 70*d*e^3*x^3 + 15*e^4*x^4) + c^3*d^3*x^3*(70*d^4 + 224*d^3*e*x + 280*d^2*e^2*x^2 + 160*d*e^3*x^3 + 35*e^4*
x^4)))/280

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(530\) vs. \(2(103)=206\).
time = 0.78, size = 531, normalized size = 4.78

method	result	size
norman	\(\frac {c^{3} d^{3} e^{4} x^{8}}{8}+\left (\frac {3}{7} c^{2} d^{2} a \,e^{5}+\frac {4}{7} c^{3} d^{4} e^{3}\right ) x^{7}+\left (\frac {1}{2} d \,e^{6} a^{2} c +2 c^{2} d^{3} a \,e^{4}+c^{3} d^{5} e^{2}\right ) x^{6}+\left (\frac {1}{5} e^{7} a^{3}+\frac {12}{5} d^{2} e^{5} a^{2} c +\frac {18}{5} c^{2} d^{4} a \,e^{3}+\frac {4}{5} c^{3} d^{6} e \right ) x^{5}+\left (e^{6} a^{3} d +\frac {9}{2} d^{3} e^{4} a^{2} c +3 c^{2} d^{5} a \,e^{2}+\frac {1}{4} c^{3} d^{7}\right ) x^{4}+\left (2 e^{5} a^{3} d^{2}+4 d^{4} e^{3} a^{2} c +c^{2} d^{6} a e \right ) x^{3}+\left (2 e^{4} a^{3} d^{3}+\frac {3}{2} d^{5} e^{2} a^{2} c \right ) x^{2}+e^{3} a^{3} d^{4} x\)	\(248\)
risch	\(\frac {1}{8} c^{3} d^{3} e^{4} x^{8}+\frac {3}{7} x^{7} c^{2} d^{2} a \,e^{5}+\frac {4}{7} x^{7} c^{3} d^{4} e^{3}+\frac {1}{2} x^{6} d \,e^{6} a^{2} c +2 x^{6} c^{2} d^{3} a \,e^{4}+x^{6} c^{3} d^{5} e^{2}+\frac {1}{5} x^{5} e^{7} a^{3}+\frac {12}{5} x^{5} d^{2} e^{5} a^{2} c +\frac {18}{5} x^{5} c^{2} d^{4} a \,e^{3}+\frac {4}{5} x^{5} c^{3} d^{6} e +x^{4} e^{6} a^{3} d +\frac {9}{2} x^{4} d^{3} e^{4} a^{2} c +3 x^{4} c^{2} d^{5} a \,e^{2}+\frac {1}{4} x^{4} c^{3} d^{7}+2 a^{3} d^{2} e^{5} x^{3}+4 a^{2} c \,d^{4} e^{3} x^{3}+a \,c^{2} d^{6} e \,x^{3}+2 x^{2} e^{4} a^{3} d^{3}+\frac {3}{2} x^{2} d^{5} e^{2} a^{2} c +e^{3} a^{3} d^{4} x\)	\(272\)
gosper	\(\frac {x \left (35 c^{3} d^{3} e^{4} x^{7}+120 x^{6} c^{2} d^{2} a \,e^{5}+160 x^{6} c^{3} d^{4} e^{3}+140 x^{5} d \,e^{6} a^{2} c +560 x^{5} c^{2} d^{3} a \,e^{4}+280 x^{5} c^{3} d^{5} e^{2}+56 x^{4} e^{7} a^{3}+672 x^{4} d^{2} e^{5} a^{2} c +1008 x^{4} c^{2} d^{4} a \,e^{3}+224 x^{4} c^{3} d^{6} e +280 x^{3} e^{6} a^{3} d +1260 x^{3} d^{3} e^{4} a^{2} c +840 x^{3} c^{2} d^{5} a \,e^{2}+70 x^{3} c^{3} d^{7}+560 a^{3} d^{2} e^{5} x^{2}+1120 a^{2} c \,d^{4} e^{3} x^{2}+280 a \,c^{2} d^{6} e \,x^{2}+560 x \,e^{4} a^{3} d^{3}+420 x \,d^{5} e^{2} a^{2} c +280 e^{3} a^{3} d^{4}\right )}{280}\)	\(274\)
default	\(\frac {c^{3} d^{3} e^{4} x^{8}}{8}+\frac {\left (c^{3} d^{4} e^{3}+3 e^{3} \left (e^{2} a +c \,d^{2}\right ) c^{2} d^{2}\right ) x^{7}}{7}+\frac {\left (3 d^{3} \left (e^{2} a +c \,d^{2}\right ) c^{2} e^{2}+e \left (d^{3} e^{3} c^{2} a +2 \left (e^{2} a +c \,d^{2}\right )^{2} c d e +c d e \left (2 a c \,d^{2} e^{2}+\left (e^{2} a +c \,d^{2}\right )^{2}\right )\right )\right ) x^{6}}{6}+\frac {\left (d \left (d^{3} e^{3} c^{2} a +2 \left (e^{2} a +c \,d^{2}\right )^{2} c d e +c d e \left (2 a c \,d^{2} e^{2}+\left (e^{2} a +c \,d^{2}\right )^{2}\right )\right )+e \left (4 a \,d^{2} e^{2} c \left (e^{2} a +c \,d^{2}\right )+\left (e^{2} a +c \,d^{2}\right ) \left (2 a c \,d^{2} e^{2}+\left (e^{2} a +c \,d^{2}\right )^{2}\right )\right )\right ) x^{5}}{5}+\frac {\left (d \left (4 a \,d^{2} e^{2} c \left (e^{2} a +c \,d^{2}\right )+\left (e^{2} a +c \,d^{2}\right ) \left (2 a c \,d^{2} e^{2}+\left (e^{2} a +c \,d^{2}\right )^{2}\right )\right )+e \left (a d e \left (2 a c \,d^{2} e^{2}+\left (e^{2} a +c \,d^{2}\right )^{2}\right )+2 \left (e^{2} a +c \,d^{2}\right )^{2} a d e +c \,d^{3} e^{3} a^{2}\right )\right ) x^{4}}{4}+\frac {\left (d \left (a d e \left (2 a c \,d^{2} e^{2}+\left (e^{2} a +c \,d^{2}\right )^{2}\right )+2 \left (e^{2} a +c \,d^{2}\right )^{2} a d e +c \,d^{3} e^{3} a^{2}\right )+3 e^{3} a^{2} d^{2} \left (e^{2} a +c \,d^{2}\right )\right ) x^{3}}{3}+\frac {\left (3 d^{3} a^{2} e^{2} \left (e^{2} a +c \,d^{2}\right )+e^{4} a^{3} d^{3}\right ) x^{2}}{2}+e^{3} a^{3} d^{4} x\)	\(531\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 236 vs. \(2 (100) = 200\).
time = 0.29, size = 236, normalized size = 2.13 \begin {gather*} \frac {1}{8} \, c^{3} d^{3} x^{8} e^{4} + a^{3} d^{4} x e^{3} + \frac {1}{7} \, {\left (4 \, c^{3} d^{4} e^{3} + 3 \, a c^{2} d^{2} e^{5}\right )} x^{7} + \frac {1}{2} \, {\left (2 \, c^{3} d^{5} e^{2} + 4 \, a c^{2} d^{3} e^{4} + a^{2} c d e^{6}\right )} x^{6} + \frac {1}{5} \, {\left (4 \, c^{3} d^{6} e + 18 \, a c^{2} d^{4} e^{3} + 12 \, a^{2} c d^{2} e^{5} + a^{3} e^{7}\right )} x^{5} + \frac {1}{4} \, {\left (c^{3} d^{7} + 12 \, a c^{2} d^{5} e^{2} + 18 \, a^{2} c d^{3} e^{4} + 4 \, a^{3} d e^{6}\right )} x^{4} + {\left (a c^{2} d^{6} e + 4 \, a^{2} c d^{4} e^{3} + 2 \, a^{3} d^{2} e^{5}\right )} x^{3} + \frac {1}{2} \, {\left (3 \, a^{2} c d^{5} e^{2} + 4 \, a^{3} d^{3} e^{4}\right )} x^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*c^3*d^3*x^8*e^4 + a^3*d^4*x*e^3 + 1/7*(4*c^3*d^4*e^3 + 3*a*c^2*d^2*e^5)*x^7 + 1/2*(2*c^3*d^5*e^2 + 4*a*c^2
*d^3*e^4 + a^2*c*d*e^6)*x^6 + 1/5*(4*c^3*d^6*e + 18*a*c^2*d^4*e^3 + 12*a^2*c*d^2*e^5 + a^3*e^7)*x^5 + 1/4*(c^3
*d^7 + 12*a*c^2*d^5*e^2 + 18*a^2*c*d^3*e^4 + 4*a^3*d*e^6)*x^4 + (a*c^2*d^6*e + 4*a^2*c*d^4*e^3 + 2*a^3*d^2*e^5
)*x^3 + 1/2*(3*a^2*c*d^5*e^2 + 4*a^3*d^3*e^4)*x^2

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 252 vs. \(2 (100) = 200\).
time = 3.03, size = 252, normalized size = 2.27 \begin {gather*} \frac {1}{4} \, c^{3} d^{7} x^{4} + \frac {1}{5} \, a^{3} x^{5} e^{7} + \frac {1}{2} \, {\left (a^{2} c d x^{6} + 2 \, a^{3} d x^{4}\right )} e^{6} + \frac {1}{35} \, {\left (15 \, a c^{2} d^{2} x^{7} + 84 \, a^{2} c d^{2} x^{5} + 70 \, a^{3} d^{2} x^{3}\right )} e^{5} + \frac {1}{8} \, {\left (c^{3} d^{3} x^{8} + 16 \, a c^{2} d^{3} x^{6} + 36 \, a^{2} c d^{3} x^{4} + 16 \, a^{3} d^{3} x^{2}\right )} e^{4} + \frac {1}{35} \, {\left (20 \, c^{3} d^{4} x^{7} + 126 \, a c^{2} d^{4} x^{5} + 140 \, a^{2} c d^{4} x^{3} + 35 \, a^{3} d^{4} x\right )} e^{3} + \frac {1}{2} \, {\left (2 \, c^{3} d^{5} x^{6} + 6 \, a c^{2} d^{5} x^{4} + 3 \, a^{2} c d^{5} x^{2}\right )} e^{2} + \frac {1}{5} \, {\left (4 \, c^{3} d^{6} x^{5} + 5 \, a c^{2} d^{6} x^{3}\right )} e \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)*(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^7*x^4 + 1/5*a^3*x^5*e^7 + 1/2*(a^2*c*d*x^6 + 2*a^3*d*x^4)*e^6 + 1/35*(15*a*c^2*d^2*x^7 + 84*a^2*c*d^
2*x^5 + 70*a^3*d^2*x^3)*e^5 + 1/8*(c^3*d^3*x^8 + 16*a*c^2*d^3*x^6 + 36*a^2*c*d^3*x^4 + 16*a^3*d^3*x^2)*e^4 + 1
/35*(20*c^3*d^4*x^7 + 126*a*c^2*d^4*x^5 + 140*a^2*c*d^4*x^3 + 35*a^3*d^4*x)*e^3 + 1/2*(2*c^3*d^5*x^6 + 6*a*c^2
*d^5*x^4 + 3*a^2*c*d^5*x^2)*e^2 + 1/5*(4*c^3*d^6*x^5 + 5*a*c^2*d^6*x^3)*e

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 270 vs. \(2 (99) = 198\).
time = 0.04, size = 270, normalized size = 2.43 \begin {gather*} a^{3} d^{4} e^{3} x + \frac {c^{3} d^{3} e^{4} x^{8}}{8} + x^{7} \cdot \left (\frac {3 a c^{2} d^{2} e^{5}}{7} + \frac {4 c^{3} d^{4} e^{3}}{7}\right ) + x^{6} \left (\frac {a^{2} c d e^{6}}{2} + 2 a c^{2} d^{3} e^{4} + c^{3} d^{5} e^{2}\right ) + x^{5} \left (\frac {a^{3} e^{7}}{5} + \frac {12 a^{2} c d^{2} e^{5}}{5} + \frac {18 a c^{2} d^{4} e^{3}}{5} + \frac {4 c^{3} d^{6} e}{5}\right ) + x^{4} \left (a^{3} d e^{6} + \frac {9 a^{2} c d^{3} e^{4}}{2} + 3 a c^{2} d^{5} e^{2} + \frac {c^{3} d^{7}}{4}\right ) + x^{3} \cdot \left (2 a^{3} d^{2} e^{5} + 4 a^{2} c d^{4} e^{3} + a c^{2} d^{6} e\right ) + x^{2} \cdot \left (2 a^{3} d^{3} e^{4} + \frac {3 a^{2} c d^{5} e^{2}}{2}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a**3*d**4*e**3*x + c**3*d**3*e**4*x**8/8 + x**7*(3*a*c**2*d**2*e**5/7 + 4*c**3*d**4*e**3/7) + x**6*(a**2*c*d*e
**6/2 + 2*a*c**2*d**3*e**4 + c**3*d**5*e**2) + x**5*(a**3*e**7/5 + 12*a**2*c*d**2*e**5/5 + 18*a*c**2*d**4*e**3
/5 + 4*c**3*d**6*e/5) + x**4*(a**3*d*e**6 + 9*a**2*c*d**3*e**4/2 + 3*a*c**2*d**5*e**2 + c**3*d**7/4) + x**3*(2
*a**3*d**2*e**5 + 4*a**2*c*d**4*e**3 + a*c**2*d**6*e) + x**2*(2*a**3*d**3*e**4 + 3*a**2*c*d**5*e**2/2)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 256 vs. \(2 (100) = 200\).
time = 0.66, size = 256, normalized size = 2.31 \begin {gather*} \frac {1}{8} \, c^{3} d^{3} x^{8} e^{4} + \frac {4}{7} \, c^{3} d^{4} x^{7} e^{3} + c^{3} d^{5} x^{6} e^{2} + \frac {4}{5} \, c^{3} d^{6} x^{5} e + \frac {1}{4} \, c^{3} d^{7} x^{4} + \frac {3}{7} \, a c^{2} d^{2} x^{7} e^{5} + 2 \, a c^{2} d^{3} x^{6} e^{4} + \frac {18}{5} \, a c^{2} d^{4} x^{5} e^{3} + 3 \, a c^{2} d^{5} x^{4} e^{2} + a c^{2} d^{6} x^{3} e + \frac {1}{2} \, a^{2} c d x^{6} e^{6} + \frac {12}{5} \, a^{2} c d^{2} x^{5} e^{5} + \frac {9}{2} \, a^{2} c d^{3} x^{4} e^{4} + 4 \, a^{2} c d^{4} x^{3} e^{3} + \frac {3}{2} \, a^{2} c d^{5} x^{2} e^{2} + \frac {1}{5} \, a^{3} x^{5} e^{7} + a^{3} d x^{4} e^{6} + 2 \, a^{3} d^{2} x^{3} e^{5} + 2 \, a^{3} d^{3} x^{2} e^{4} + a^{3} d^{4} x e^{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*c^3*d^3*x^8*e^4 + 4/7*c^3*d^4*x^7*e^3 + c^3*d^5*x^6*e^2 + 4/5*c^3*d^6*x^5*e + 1/4*c^3*d^7*x^4 + 3/7*a*c^2*
d^2*x^7*e^5 + 2*a*c^2*d^3*x^6*e^4 + 18/5*a*c^2*d^4*x^5*e^3 + 3*a*c^2*d^5*x^4*e^2 + a*c^2*d^6*x^3*e + 1/2*a^2*c
*d*x^6*e^6 + 12/5*a^2*c*d^2*x^5*e^5 + 9/2*a^2*c*d^3*x^4*e^4 + 4*a^2*c*d^4*x^3*e^3 + 3/2*a^2*c*d^5*x^2*e^2 + 1/
5*a^3*x^5*e^7 + a^3*d*x^4*e^6 + 2*a^3*d^2*x^3*e^5 + 2*a^3*d^3*x^2*e^4 + a^3*d^4*x*e^3

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Mupad [B]
time = 0.59, size = 242, normalized size = 2.18 \begin {gather*} x^4\,\left (a^3\,d\,e^6+\frac {9\,a^2\,c\,d^3\,e^4}{2}+3\,a\,c^2\,d^5\,e^2+\frac {c^3\,d^7}{4}\right )+x^5\,\left (\frac {a^3\,e^7}{5}+\frac {12\,a^2\,c\,d^2\,e^5}{5}+\frac {18\,a\,c^2\,d^4\,e^3}{5}+\frac {4\,c^3\,d^6\,e}{5}\right )+a^3\,d^4\,e^3\,x+\frac {c^3\,d^3\,e^4\,x^8}{8}+a\,d^2\,e\,x^3\,\left (2\,a^2\,e^4+4\,a\,c\,d^2\,e^2+c^2\,d^4\right )+\frac {c\,d\,e^2\,x^6\,\left (a^2\,e^4+4\,a\,c\,d^2\,e^2+2\,c^2\,d^4\right )}{2}+\frac {a^2\,d^3\,e^2\,x^2\,\left (3\,c\,d^2+4\,a\,e^2\right )}{2}+\frac {c^2\,d^2\,e^3\,x^7\,\left (4\,c\,d^2+3\,a\,e^2\right )}{7} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x^4*((c^3*d^7)/4 + a^3*d*e^6 + 3*a*c^2*d^5*e^2 + (9*a^2*c*d^3*e^4)/2) + x^5*((a^3*e^7)/5 + (4*c^3*d^6*e)/5 + (
18*a*c^2*d^4*e^3)/5 + (12*a^2*c*d^2*e^5)/5) + a^3*d^4*e^3*x + (c^3*d^3*e^4*x^8)/8 + a*d^2*e*x^3*(2*a^2*e^4 + c
^2*d^4 + 4*a*c*d^2*e^2) + (c*d*e^2*x^6*(a^2*e^4 + 2*c^2*d^4 + 4*a*c*d^2*e^2))/2 + (a^2*d^3*e^2*x^2*(4*a*e^2 +
3*c*d^2))/2 + (c^2*d^2*e^3*x^7*(3*a*e^2 + 4*c*d^2))/7

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