3.19.53 \(\int (a d e+(c d^2+a e^2) x+c d e x^2)^3 \, dx\) [1853]

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

[Out]

-1/4*(-a*e^2+c*d^2)^3*(e*x+d)^4/e^4+3/5*c*d*(-a*e^2+c*d^2)^2*(e*x+d)^5/e^4-1/2*c^2*d^2*(-a*e^2+c*d^2)*(e*x+d)^
6/e^4+1/7*c^3*d^3*(e*x+d)^7/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 = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.074, Rules used = {624, 45} \begin {gather*} -\frac {c^2 d^2 (d+e x)^6 \left (c d^2-a e^2\right )}{2 e^4}+\frac {3 c d (d+e x)^5 \left (c d^2-a e^2\right )^2}{5 e^4}-\frac {(d+e x)^4 \left (c d^2-a e^2\right )^3}{4 e^4}+\frac {c^3 d^3 (d+e x)^7}{7 e^4} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/4*((c*d^2 - a*e^2)^3*(d + e*x)^4)/e^4 + (3*c*d*(c*d^2 - a*e^2)^2*(d + e*x)^5)/(5*e^4) - (c^2*d^2*(c*d^2 - a
*e^2)*(d + e*x)^6)/(2*e^4) + (c^3*d^3*(d + e*x)^7)/(7*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 624

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Dist[1/c^p, Int[Simp[
b/2 - q/2 + c*x, x]^p*Simp[b/2 + q/2 + c*x, x]^p, x], x]] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && IGt
Q[p, 0] && PerfectSquareQ[b^2 - 4*a*c]

Rubi steps

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

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Mathematica [A]
time = 0.03, size = 167, normalized size = 1.50 \begin {gather*} \frac {1}{140} x \left (35 a^3 e^3 \left (4 d^3+6 d^2 e x+4 d e^2 x^2+e^3 x^3\right )+21 a^2 c d e^2 x \left (10 d^3+20 d^2 e x+15 d e^2 x^2+4 e^3 x^3\right )+7 a c^2 d^2 e x^2 \left (20 d^3+45 d^2 e x+36 d e^2 x^2+10 e^3 x^3\right )+c^3 d^3 x^3 \left (35 d^3+84 d^2 e x+70 d e^2 x^2+20 e^3 x^3\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(x*(35*a^3*e^3*(4*d^3 + 6*d^2*e*x + 4*d*e^2*x^2 + e^3*x^3) + 21*a^2*c*d*e^2*x*(10*d^3 + 20*d^2*e*x + 15*d*e^2*
x^2 + 4*e^3*x^3) + 7*a*c^2*d^2*e*x^2*(20*d^3 + 45*d^2*e*x + 36*d*e^2*x^2 + 10*e^3*x^3) + c^3*d^3*x^3*(35*d^3 +
84*d^2*e*x + 70*d*e^2*x^2 + 20*e^3*x^3)))/140

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(265\) vs. \(2(103)=206\).
time = 0.61, size = 266, normalized size = 2.40

method	result	size
norman	\(\frac {c^{3} d^{3} e^{3} x^{7}}{7}+\left (\frac {1}{2} e^{4} d^{2} c^{2} a +\frac {1}{2} d^{4} e^{2} c^{3}\right ) x^{6}+\left (\frac {3}{5} d \,e^{5} a^{2} c +\frac {9}{5} d^{3} e^{3} c^{2} a +\frac {3}{5} d^{5} e \,c^{3}\right ) x^{5}+\left (\frac {1}{4} e^{6} a^{3}+\frac {9}{4} e^{4} d^{2} a^{2} c +\frac {9}{4} d^{4} e^{2} c^{2} a +\frac {1}{4} d^{6} c^{3}\right ) x^{4}+\left (e^{5} a^{3} d +3 c \,d^{3} e^{3} a^{2}+c^{2} d^{5} a e \right ) x^{3}+\left (\frac {3}{2} e^{4} a^{3} d^{2}+\frac {3}{2} d^{4} e^{2} a^{2} c \right ) x^{2}+e^{3} a^{3} d^{3} x\)	\(198\)
risch	\(\frac {1}{7} c^{3} d^{3} e^{3} x^{7}+\frac {1}{2} a \,c^{2} d^{2} e^{4} x^{6}+\frac {1}{2} c^{3} d^{4} e^{2} x^{6}+\frac {3}{5} x^{5} d \,e^{5} a^{2} c +\frac {9}{5} x^{5} d^{3} e^{3} c^{2} a +\frac {3}{5} x^{5} d^{5} e \,c^{3}+\frac {1}{4} x^{4} e^{6} a^{3}+\frac {9}{4} x^{4} e^{4} d^{2} a^{2} c +\frac {9}{4} x^{4} d^{4} e^{2} c^{2} a +\frac {1}{4} x^{4} d^{6} c^{3}+a^{3} d \,e^{5} x^{3}+3 a^{2} c \,d^{3} e^{3} x^{3}+a \,c^{2} d^{5} e \,x^{3}+\frac {3}{2} a^{3} d^{2} e^{4} x^{2}+\frac {3}{2} a^{2} c \,d^{4} e^{2} x^{2}+e^{3} a^{3} d^{3} x\)	\(215\)
gosper	\(\frac {x \left (20 c^{3} d^{3} e^{3} x^{6}+70 x^{5} e^{4} d^{2} c^{2} a +70 x^{5} d^{4} e^{2} c^{3}+84 x^{4} d \,e^{5} a^{2} c +252 x^{4} d^{3} e^{3} c^{2} a +84 x^{4} d^{5} e \,c^{3}+35 x^{3} e^{6} a^{3}+315 x^{3} e^{4} d^{2} a^{2} c +315 x^{3} d^{4} e^{2} c^{2} a +35 x^{3} d^{6} c^{3}+140 a^{3} d \,e^{5} x^{2}+420 a^{2} c \,d^{3} e^{3} x^{2}+140 a \,c^{2} d^{5} e \,x^{2}+210 x \,e^{4} a^{3} d^{2}+210 x \,d^{4} e^{2} a^{2} c +140 e^{3} a^{3} d^{3}\right )}{140}\)	\(216\)
default	\(\frac {c^{3} d^{3} e^{3} x^{7}}{7}+\frac {\left (e^{2} a +c \,d^{2}\right ) c^{2} d^{2} e^{2} x^{6}}{2}+\frac {\left (d^{3} e^{3} c^{2} a +2 \left (e^{2} a +c \,d^{2}\right )^{2} c d e +c d e \left (2 a c \,d^{2} e^{2}+\left (e^{2} a +c \,d^{2}\right )^{2}\right )\right ) x^{5}}{5}+\frac {\left (4 a \,d^{2} e^{2} c \left (e^{2} a +c \,d^{2}\right )+\left (e^{2} a +c \,d^{2}\right ) \left (2 a c \,d^{2} e^{2}+\left (e^{2} a +c \,d^{2}\right )^{2}\right )\right ) x^{4}}{4}+\frac {\left (a d e \left (2 a c \,d^{2} e^{2}+\left (e^{2} a +c \,d^{2}\right )^{2}\right )+2 \left (e^{2} a +c \,d^{2}\right )^{2} a d e +c \,d^{3} e^{3} a^{2}\right ) x^{3}}{3}+\frac {3 a^{2} d^{2} e^{2} \left (e^{2} a +c \,d^{2}\right ) x^{2}}{2}+e^{3} a^{3} d^{3} x\)	\(266\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.28, size = 176, normalized size = 1.59 \begin {gather*} \frac {1}{7} \, c^{3} d^{3} x^{7} e^{3} + \frac {1}{2} \, {\left (c d^{2} + a e^{2}\right )} c^{2} d^{2} x^{6} e^{2} + \frac {3}{5} \, {\left (c d^{2} + a e^{2}\right )}^{2} c d x^{5} e + a^{3} d^{3} x e^{3} + \frac {1}{4} \, {\left (c d^{2} + a e^{2}\right )}^{3} x^{4} + \frac {1}{2} \, {\left (2 \, c d x^{3} e + 3 \, {\left (c d^{2} + a e^{2}\right )} x^{2}\right )} a^{2} d^{2} e^{2} + \frac {1}{10} \, {\left (6 \, c^{2} d^{2} x^{5} e^{2} + 15 \, {\left (c d^{2} + a e^{2}\right )} c d x^{4} e + 10 \, {\left (c d^{2} + a e^{2}\right )}^{2} x^{3}\right )} a d e \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 203 vs. \(2 (100) = 200\).
time = 3.80, size = 203, normalized size = 1.83 \begin {gather*} \frac {1}{4} \, c^{3} d^{6} x^{4} + \frac {1}{4} \, a^{3} x^{4} e^{6} + \frac {1}{5} \, {\left (3 \, a^{2} c d x^{5} + 5 \, a^{3} d x^{3}\right )} e^{5} + \frac {1}{4} \, {\left (2 \, a c^{2} d^{2} x^{6} + 9 \, a^{2} c d^{2} x^{4} + 6 \, a^{3} d^{2} x^{2}\right )} e^{4} + \frac {1}{35} \, {\left (5 \, c^{3} d^{3} x^{7} + 63 \, a c^{2} d^{3} x^{5} + 105 \, a^{2} c d^{3} x^{3} + 35 \, a^{3} d^{3} x\right )} e^{3} + \frac {1}{4} \, {\left (2 \, c^{3} d^{4} x^{6} + 9 \, a c^{2} d^{4} x^{4} + 6 \, a^{2} c d^{4} x^{2}\right )} e^{2} + \frac {1}{5} \, {\left (3 \, c^{3} d^{5} x^{5} + 5 \, a c^{2} d^{5} x^{3}\right )} e \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((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^6*x^4 + 1/4*a^3*x^4*e^6 + 1/5*(3*a^2*c*d*x^5 + 5*a^3*d*x^3)*e^5 + 1/4*(2*a*c^2*d^2*x^6 + 9*a^2*c*d^2
*x^4 + 6*a^3*d^2*x^2)*e^4 + 1/35*(5*c^3*d^3*x^7 + 63*a*c^2*d^3*x^5 + 105*a^2*c*d^3*x^3 + 35*a^3*d^3*x)*e^3 + 1
/4*(2*c^3*d^4*x^6 + 9*a*c^2*d^4*x^4 + 6*a^2*c*d^4*x^2)*e^2 + 1/5*(3*c^3*d^5*x^5 + 5*a*c^2*d^5*x^3)*e

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 218 vs. \(2 (99) = 198\).
time = 0.03, size = 218, normalized size = 1.96 \begin {gather*} a^{3} d^{3} e^{3} x + \frac {c^{3} d^{3} e^{3} x^{7}}{7} + x^{6} \left (\frac {a c^{2} d^{2} e^{4}}{2} + \frac {c^{3} d^{4} e^{2}}{2}\right ) + x^{5} \cdot \left (\frac {3 a^{2} c d e^{5}}{5} + \frac {9 a c^{2} d^{3} e^{3}}{5} + \frac {3 c^{3} d^{5} e}{5}\right ) + x^{4} \left (\frac {a^{3} e^{6}}{4} + \frac {9 a^{2} c d^{2} e^{4}}{4} + \frac {9 a c^{2} d^{4} e^{2}}{4} + \frac {c^{3} d^{6}}{4}\right ) + x^{3} \left (a^{3} d e^{5} + 3 a^{2} c d^{3} e^{3} + a c^{2} d^{5} e\right ) + x^{2} \cdot \left (\frac {3 a^{3} d^{2} e^{4}}{2} + \frac {3 a^{2} c d^{4} e^{2}}{2}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 203 vs. \(2 (100) = 200\).
time = 0.98, size = 203, normalized size = 1.83 \begin {gather*} \frac {1}{7} \, c^{3} d^{3} x^{7} e^{3} + \frac {1}{2} \, c^{3} d^{4} x^{6} e^{2} + \frac {3}{5} \, c^{3} d^{5} x^{5} e + \frac {1}{4} \, c^{3} d^{6} x^{4} + \frac {1}{2} \, a c^{2} d^{2} x^{6} e^{4} + \frac {9}{5} \, a c^{2} d^{3} x^{5} e^{3} + \frac {9}{4} \, a c^{2} d^{4} x^{4} e^{2} + a c^{2} d^{5} x^{3} e + \frac {3}{5} \, a^{2} c d x^{5} e^{5} + \frac {9}{4} \, a^{2} c d^{2} x^{4} e^{4} + 3 \, a^{2} c d^{3} x^{3} e^{3} + \frac {3}{2} \, a^{2} c d^{4} x^{2} e^{2} + \frac {1}{4} \, a^{3} x^{4} e^{6} + a^{3} d x^{3} e^{5} + \frac {3}{2} \, a^{3} d^{2} x^{2} e^{4} + a^{3} d^{3} x e^{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [B]
time = 0.60, size = 186, normalized size = 1.68 \begin {gather*} x^4\,\left (\frac {a^3\,e^6}{4}+\frac {9\,a^2\,c\,d^2\,e^4}{4}+\frac {9\,a\,c^2\,d^4\,e^2}{4}+\frac {c^3\,d^6}{4}\right )+a^3\,d^3\,e^3\,x+\frac {c^3\,d^3\,e^3\,x^7}{7}+a\,d\,e\,x^3\,\left (a^2\,e^4+3\,a\,c\,d^2\,e^2+c^2\,d^4\right )+\frac {3\,c\,d\,e\,x^5\,\left (a^2\,e^4+3\,a\,c\,d^2\,e^2+c^2\,d^4\right )}{5}+\frac {3\,a^2\,d^2\,e^2\,x^2\,\left (c\,d^2+a\,e^2\right )}{2}+\frac {c^2\,d^2\,e^2\,x^6\,\left (c\,d^2+a\,e^2\right )}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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