3.1852 \(\int (d+e x) (a d e+(c d^2+a e^2) x+c d e x^2)^3 \, dx\)

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

[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 = 0.16, 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 = {626, 43} \[ -\frac {3 c^2 d^2 (d+e x)^7 \left (c d^2-a e^2\right )}{7 e^4}+\frac {c d (d+e x)^6 \left (c d^2-a e^2\right )^2}{2 e^4}-\frac {(d+e x)^5 \left (c d^2-a e^2\right )^3}{5 e^4}+\frac {c^3 d^3 (d+e x)^8}{8 e^4} \]

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]

-((c*d^2 - a*e^2)^3*(d + e*x)^5)/(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 43

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 626

Int[((d_) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[(d + e*x)^(m + p)*(a
/d + (c*x)/e)^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 = 0.06, size = 211, normalized size = 1.90 \[ \frac {1}{280} x \left (56 a^3 e^3 \left (5 d^4+10 d^3 e x+10 d^2 e^2 x^2+5 d e^3 x^3+e^4 x^4\right )+28 a^2 c d e^2 x \left (15 d^4+40 d^3 e x+45 d^2 e^2 x^2+24 d e^3 x^3+5 e^4 x^4\right )+8 a c^2 d^2 e x^2 \left (35 d^4+105 d^3 e x+126 d^2 e^2 x^2+70 d e^3 x^3+15 e^4 x^4\right )+c^3 d^3 x^3 \left (70 d^4+224 d^3 e x+280 d^2 e^2 x^2+160 d e^3 x^3+35 e^4 x^4\right )\right ) \]

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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fricas [B]  time = 0.77, size = 271, normalized size = 2.44 \[ \frac {1}{8} x^{8} e^{4} d^{3} c^{3} + \frac {4}{7} x^{7} e^{3} d^{4} c^{3} + \frac {3}{7} x^{7} e^{5} d^{2} c^{2} a + x^{6} e^{2} d^{5} c^{3} + 2 x^{6} e^{4} d^{3} c^{2} a + \frac {1}{2} x^{6} e^{6} d c a^{2} + \frac {4}{5} x^{5} e d^{6} c^{3} + \frac {18}{5} x^{5} e^{3} d^{4} c^{2} a + \frac {12}{5} x^{5} e^{5} d^{2} c a^{2} + \frac {1}{5} x^{5} e^{7} a^{3} + \frac {1}{4} x^{4} d^{7} c^{3} + 3 x^{4} e^{2} d^{5} c^{2} a + \frac {9}{2} x^{4} e^{4} d^{3} c a^{2} + x^{4} e^{6} d a^{3} + x^{3} e d^{6} c^{2} a + 4 x^{3} e^{3} d^{4} c a^{2} + 2 x^{3} e^{5} d^{2} a^{3} + \frac {3}{2} x^{2} e^{2} d^{5} c a^{2} + 2 x^{2} e^{4} d^{3} a^{3} + x e^{3} d^{4} a^{3} \]

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]

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

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giac [B]  time = 0.19, size = 256, normalized size = 2.31 \[ \frac {1}{8} \, c^{3} d^{3} x^{8} e^{4} + \frac {4}{7} \, c^{3} d^{4} x^{7} e^{3} + c^{3} d^{5} x^{6} e^{2} + \frac {4}{5} \, c^{3} d^{6} x^{5} e + \frac {1}{4} \, c^{3} d^{7} x^{4} + \frac {3}{7} \, a c^{2} d^{2} x^{7} e^{5} + 2 \, a c^{2} d^{3} x^{6} e^{4} + \frac {18}{5} \, a c^{2} d^{4} x^{5} e^{3} + 3 \, a c^{2} d^{5} x^{4} e^{2} + a c^{2} d^{6} x^{3} e + \frac {1}{2} \, a^{2} c d x^{6} e^{6} + \frac {12}{5} \, a^{2} c d^{2} x^{5} e^{5} + \frac {9}{2} \, a^{2} c d^{3} x^{4} e^{4} + 4 \, a^{2} c d^{4} x^{3} e^{3} + \frac {3}{2} \, a^{2} c d^{5} x^{2} e^{2} + \frac {1}{5} \, a^{3} x^{5} e^{7} + a^{3} d x^{4} e^{6} + 2 \, a^{3} d^{2} x^{3} e^{5} + 2 \, a^{3} d^{3} x^{2} e^{4} + a^{3} d^{4} x e^{3} \]

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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maple [B]  time = 0.05, size = 531, normalized size = 4.78 \[ \frac {c^{3} d^{3} e^{4} x^{8}}{8}+a^{3} d^{4} e^{3} x +\frac {\left (c^{3} d^{4} e^{3}+3 \left (a \,e^{2}+c \,d^{2}\right ) c^{2} d^{2} e^{3}\right ) x^{7}}{7}+\frac {\left (3 \left (a \,e^{2}+c \,d^{2}\right ) c^{2} d^{3} e^{2}+\left (a \,c^{2} d^{3} e^{3}+2 \left (a \,e^{2}+c \,d^{2}\right )^{2} c d e +\left (2 a c \,d^{2} e^{2}+\left (a \,e^{2}+c \,d^{2}\right )^{2}\right ) c d e \right ) e \right ) x^{6}}{6}+\frac {\left (\left (a \,c^{2} d^{3} e^{3}+2 \left (a \,e^{2}+c \,d^{2}\right )^{2} c d e +\left (2 a c \,d^{2} e^{2}+\left (a \,e^{2}+c \,d^{2}\right )^{2}\right ) c d e \right ) d +\left (4 \left (a \,e^{2}+c \,d^{2}\right ) a c \,d^{2} e^{2}+\left (a \,e^{2}+c \,d^{2}\right ) \left (2 a c \,d^{2} e^{2}+\left (a \,e^{2}+c \,d^{2}\right )^{2}\right )\right ) e \right ) x^{5}}{5}+\frac {\left (\left (4 \left (a \,e^{2}+c \,d^{2}\right ) a c \,d^{2} e^{2}+\left (a \,e^{2}+c \,d^{2}\right ) \left (2 a c \,d^{2} e^{2}+\left (a \,e^{2}+c \,d^{2}\right )^{2}\right )\right ) d +\left (a^{2} c \,d^{3} e^{3}+\left (2 a c \,d^{2} e^{2}+\left (a \,e^{2}+c \,d^{2}\right )^{2}\right ) a d e +2 \left (a \,e^{2}+c \,d^{2}\right )^{2} a d e \right ) e \right ) x^{4}}{4}+\frac {\left (3 \left (a \,e^{2}+c \,d^{2}\right ) a^{2} d^{2} e^{3}+\left (a^{2} c \,d^{3} e^{3}+\left (2 a c \,d^{2} e^{2}+\left (a \,e^{2}+c \,d^{2}\right )^{2}\right ) a d e +2 \left (a \,e^{2}+c \,d^{2}\right )^{2} a d e \right ) d \right ) x^{3}}{3}+\frac {\left (a^{3} d^{3} e^{4}+3 \left (a \,e^{2}+c \,d^{2}\right ) a^{2} d^{3} e^{2}\right ) x^{2}}{2} \]

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)

[Out]

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

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maxima [B]  time = 1.11, size = 251, normalized size = 2.26 \[ \frac {1}{8} \, c^{3} d^{3} e^{4} x^{8} + a^{3} d^{4} e^{3} x + \frac {1}{7} \, {\left (4 \, c^{3} d^{4} e^{3} + 3 \, a c^{2} d^{2} e^{5}\right )} x^{7} + \frac {1}{2} \, {\left (2 \, c^{3} d^{5} e^{2} + 4 \, a c^{2} d^{3} e^{4} + a^{2} c d e^{6}\right )} x^{6} + \frac {1}{5} \, {\left (4 \, c^{3} d^{6} e + 18 \, a c^{2} d^{4} e^{3} + 12 \, a^{2} c d^{2} e^{5} + a^{3} e^{7}\right )} x^{5} + \frac {1}{4} \, {\left (c^{3} d^{7} + 12 \, a c^{2} d^{5} e^{2} + 18 \, a^{2} c d^{3} e^{4} + 4 \, a^{3} d e^{6}\right )} x^{4} + {\left (a c^{2} d^{6} e + 4 \, a^{2} c d^{4} e^{3} + 2 \, a^{3} d^{2} e^{5}\right )} x^{3} + \frac {1}{2} \, {\left (3 \, a^{2} c d^{5} e^{2} + 4 \, a^{3} d^{3} e^{4}\right )} x^{2} \]

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

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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sympy [B]  time = 0.14, size = 270, normalized size = 2.43 \[ a^{3} d^{4} e^{3} x + \frac {c^{3} d^{3} e^{4} x^{8}}{8} + x^{7} \left (\frac {3 a c^{2} d^{2} e^{5}}{7} + \frac {4 c^{3} d^{4} e^{3}}{7}\right ) + x^{6} \left (\frac {a^{2} c d e^{6}}{2} + 2 a c^{2} d^{3} e^{4} + c^{3} d^{5} e^{2}\right ) + x^{5} \left (\frac {a^{3} e^{7}}{5} + \frac {12 a^{2} c d^{2} e^{5}}{5} + \frac {18 a c^{2} d^{4} e^{3}}{5} + \frac {4 c^{3} d^{6} e}{5}\right ) + x^{4} \left (a^{3} d e^{6} + \frac {9 a^{2} c d^{3} e^{4}}{2} + 3 a c^{2} d^{5} e^{2} + \frac {c^{3} d^{7}}{4}\right ) + x^{3} \left (2 a^{3} d^{2} e^{5} + 4 a^{2} c d^{4} e^{3} + a c^{2} d^{6} e\right ) + x^{2} \left (2 a^{3} d^{3} e^{4} + \frac {3 a^{2} c d^{5} e^{2}}{2}\right ) \]

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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