3.1840 \(\int (d+e x) (a d e+(c d^2+a e^2) x+c d e x^2)^2 \, dx\)

Optimal. Leaf size=77 \[ -\frac {2 c d (d+e x)^5 \left (c d^2-a e^2\right )}{5 e^3}+\frac {(d+e x)^4 \left (c d^2-a e^2\right )^2}{4 e^3}+\frac {c^2 d^2 (d+e x)^6}{6 e^3} \]

[Out]

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

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Rubi [A]  time = 0.08, antiderivative size = 77, 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} \[ -\frac {2 c d (d+e x)^5 \left (c d^2-a e^2\right )}{5 e^3}+\frac {(d+e x)^4 \left (c d^2-a e^2\right )^2}{4 e^3}+\frac {c^2 d^2 (d+e x)^6}{6 e^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

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Mathematica [A]  time = 0.03, size = 120, normalized size = 1.56 \[ \frac {1}{60} x \left (15 a^2 e^2 \left (4 d^3+6 d^2 e x+4 d e^2 x^2+e^3 x^3\right )+6 a c d e x \left (10 d^3+20 d^2 e x+15 d e^2 x^2+4 e^3 x^3\right )+c^2 d^2 x^2 \left (20 d^3+45 d^2 e x+36 d e^2 x^2+10 e^3 x^3\right )\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(x*(15*a^2*e^2*(4*d^3 + 6*d^2*e*x + 4*d*e^2*x^2 + e^3*x^3) + 6*a*c*d*e*x*(10*d^3 + 20*d^2*e*x + 15*d*e^2*x^2 +
4*e^3*x^3) + c^2*d^2*x^2*(20*d^3 + 45*d^2*e*x + 36*d*e^2*x^2 + 10*e^3*x^3)))/60

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fricas [B]  time = 0.88, size = 146, normalized size = 1.90 \[ \frac {1}{6} x^{6} e^{3} d^{2} c^{2} + \frac {3}{5} x^{5} e^{2} d^{3} c^{2} + \frac {2}{5} x^{5} e^{4} d c a + \frac {3}{4} x^{4} e d^{4} c^{2} + \frac {3}{2} x^{4} e^{3} d^{2} c a + \frac {1}{4} x^{4} e^{5} a^{2} + \frac {1}{3} x^{3} d^{5} c^{2} + 2 x^{3} e^{2} d^{3} c a + x^{3} e^{4} d a^{2} + x^{2} e d^{4} c a + \frac {3}{2} x^{2} e^{3} d^{2} a^{2} + x e^{2} d^{3} a^{2} \]

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)^2,x, algorithm="fricas")

[Out]

1/6*x^6*e^3*d^2*c^2 + 3/5*x^5*e^2*d^3*c^2 + 2/5*x^5*e^4*d*c*a + 3/4*x^4*e*d^4*c^2 + 3/2*x^4*e^3*d^2*c*a + 1/4*
x^4*e^5*a^2 + 1/3*x^3*d^5*c^2 + 2*x^3*e^2*d^3*c*a + x^3*e^4*d*a^2 + x^2*e*d^4*c*a + 3/2*x^2*e^3*d^2*a^2 + x*e^
2*d^3*a^2

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giac [A]  time = 0.16, size = 139, normalized size = 1.81 \[ \frac {1}{6} \, c^{2} d^{2} x^{6} e^{3} + \frac {3}{5} \, c^{2} d^{3} x^{5} e^{2} + \frac {3}{4} \, c^{2} d^{4} x^{4} e + \frac {1}{3} \, c^{2} d^{5} x^{3} + \frac {2}{5} \, a c d x^{5} e^{4} + \frac {3}{2} \, a c d^{2} x^{4} e^{3} + 2 \, a c d^{3} x^{3} e^{2} + a c d^{4} x^{2} e + \frac {1}{4} \, a^{2} x^{4} e^{5} + a^{2} d x^{3} e^{4} + \frac {3}{2} \, a^{2} d^{2} x^{2} e^{3} + a^{2} d^{3} x e^{2} \]

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)^2,x, algorithm="giac")

[Out]

1/6*c^2*d^2*x^6*e^3 + 3/5*c^2*d^3*x^5*e^2 + 3/4*c^2*d^4*x^4*e + 1/3*c^2*d^5*x^3 + 2/5*a*c*d*x^5*e^4 + 3/2*a*c*
d^2*x^4*e^3 + 2*a*c*d^3*x^3*e^2 + a*c*d^4*x^2*e + 1/4*a^2*x^4*e^5 + a^2*d*x^3*e^4 + 3/2*a^2*d^2*x^2*e^3 + a^2*
d^3*x*e^2

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maple [B]  time = 0.05, size = 195, normalized size = 2.53 \[ \frac {c^{2} d^{2} e^{3} x^{6}}{6}+a^{2} d^{3} e^{2} x +\frac {\left (c^{2} d^{3} e^{2}+2 \left (a \,e^{2}+c \,d^{2}\right ) c d \,e^{2}\right ) x^{5}}{5}+\frac {\left (2 \left (a \,e^{2}+c \,d^{2}\right ) c \,d^{2} e +\left (2 a c \,d^{2} e^{2}+\left (a \,e^{2}+c \,d^{2}\right )^{2}\right ) e \right ) x^{4}}{4}+\frac {\left (2 \left (a \,e^{2}+c \,d^{2}\right ) a d \,e^{2}+\left (2 a c \,d^{2} e^{2}+\left (a \,e^{2}+c \,d^{2}\right )^{2}\right ) d \right ) x^{3}}{3}+\frac {\left (a^{2} d^{2} e^{3}+2 \left (a \,e^{2}+c \,d^{2}\right ) a \,d^{2} e \right ) x^{2}}{2} \]

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)^2,x)

[Out]

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

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maxima [A]  time = 1.09, size = 140, normalized size = 1.82 \[ \frac {1}{6} \, c^{2} d^{2} e^{3} x^{6} + a^{2} d^{3} e^{2} x + \frac {1}{5} \, {\left (3 \, c^{2} d^{3} e^{2} + 2 \, a c d e^{4}\right )} x^{5} + \frac {1}{4} \, {\left (3 \, c^{2} d^{4} e + 6 \, a c d^{2} e^{3} + a^{2} e^{5}\right )} x^{4} + \frac {1}{3} \, {\left (c^{2} d^{5} + 6 \, a c d^{3} e^{2} + 3 \, a^{2} d e^{4}\right )} x^{3} + \frac {1}{2} \, {\left (2 \, a c d^{4} e + 3 \, a^{2} d^{2} e^{3}\right )} x^{2} \]

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)^2,x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 0.05, size = 135, normalized size = 1.75 \[ x^3\,\left (a^2\,d\,e^4+2\,a\,c\,d^3\,e^2+\frac {c^2\,d^5}{3}\right )+x^4\,\left (\frac {a^2\,e^5}{4}+\frac {3\,a\,c\,d^2\,e^3}{2}+\frac {3\,c^2\,d^4\,e}{4}\right )+a^2\,d^3\,e^2\,x+\frac {c^2\,d^2\,e^3\,x^6}{6}+\frac {a\,d^2\,e\,x^2\,\left (2\,c\,d^2+3\,a\,e^2\right )}{2}+\frac {c\,d\,e^2\,x^5\,\left (3\,c\,d^2+2\,a\,e^2\right )}{5} \]

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)^2,x)

[Out]

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

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sympy [B]  time = 0.11, size = 150, normalized size = 1.95 \[ a^{2} d^{3} e^{2} x + \frac {c^{2} d^{2} e^{3} x^{6}}{6} + x^{5} \left (\frac {2 a c d e^{4}}{5} + \frac {3 c^{2} d^{3} e^{2}}{5}\right ) + x^{4} \left (\frac {a^{2} e^{5}}{4} + \frac {3 a c d^{2} e^{3}}{2} + \frac {3 c^{2} d^{4} e}{4}\right ) + x^{3} \left (a^{2} d e^{4} + 2 a c d^{3} e^{2} + \frac {c^{2} d^{5}}{3}\right ) + x^{2} \left (\frac {3 a^{2} d^{2} e^{3}}{2} + a c d^{4} e\right ) \]

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)**2,x)

[Out]

a**2*d**3*e**2*x + c**2*d**2*e**3*x**6/6 + x**5*(2*a*c*d*e**4/5 + 3*c**2*d**3*e**2/5) + x**4*(a**2*e**5/4 + 3*
a*c*d**2*e**3/2 + 3*c**2*d**4*e/4) + x**3*(a**2*d*e**4 + 2*a*c*d**3*e**2 + c**2*d**5/3) + x**2*(3*a**2*d**2*e*
*3/2 + a*c*d**4*e)

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