3.19.39 \(\int (d+e x)^2 (a d e+(c d^2+a e^2) x+c d e x^2)^2 \, dx\) [1839]

Optimal. Leaf size=77 \[ \frac {\left (c d^2-a e^2\right )^2 (d+e x)^5}{5 e^3}-\frac {c d \left (c d^2-a e^2\right ) (d+e x)^6}{3 e^3}+\frac {c^2 d^2 (d+e x)^7}{7 e^3} \]

[Out]

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

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Rubi [A]
time = 0.10, antiderivative size = 77, 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 {c d (d+e x)^6 \left (c d^2-a e^2\right )}{3 e^3}+\frac {(d+e x)^5 \left (c d^2-a e^2\right )^2}{5 e^3}+\frac {c^2 d^2 (d+e x)^7}{7 e^3} \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)^2,x]

[Out]

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

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

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(160\) vs. \(2(77)=154\).
time = 0.02, size = 160, normalized size = 2.08 \begin {gather*} \frac {1}{15} a c d e x^2 \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 )+\frac {1}{105} c^2 d^2 x^3 \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 )+a^2 \left (d^4 e^2 x+2 d^3 e^3 x^2+2 d^2 e^4 x^3+d e^5 x^4+\frac {e^6 x^5}{5}\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)^2,x]

[Out]

(a*c*d*e*x^2*(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))/15 + (c^2*d^2*x^3*(35*d^4 + 10
5*d^3*e*x + 126*d^2*e^2*x^2 + 70*d*e^3*x^3 + 15*e^4*x^4))/105 + a^2*(d^4*e^2*x + 2*d^3*e^3*x^2 + 2*d^2*e^4*x^3
+ d*e^5*x^4 + (e^6*x^5)/5)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(294\) vs. \(2(71)=142\).
time = 0.69, size = 295, normalized size = 3.83

method	result	size
norman	\(\frac {c^{2} d^{2} e^{4} x^{7}}{7}+\left (\frac {1}{3} a c d \,e^{5}+\frac {2}{3} c^{2} d^{3} e^{3}\right ) x^{6}+\left (\frac {1}{5} a^{2} e^{6}+\frac {8}{5} a c \,d^{2} e^{4}+\frac {6}{5} c^{2} d^{4} e^{2}\right ) x^{5}+\left (a^{2} e^{5} d +3 a c \,d^{3} e^{3}+c^{2} d^{5} e \right ) x^{4}+\left (2 a^{2} d^{2} e^{4}+\frac {8}{3} a \,d^{4} e^{2} c +\frac {1}{3} d^{6} c^{2}\right ) x^{3}+\left (2 a^{2} e^{3} d^{3}+a c \,d^{5} e \right ) x^{2}+a^{2} e^{2} d^{4} x\)	\(173\)
risch	\(\frac {1}{7} c^{2} d^{2} e^{4} x^{7}+\frac {1}{3} x^{6} a c d \,e^{5}+\frac {2}{3} x^{6} c^{2} d^{3} e^{3}+\frac {1}{5} x^{5} a^{2} e^{6}+\frac {8}{5} x^{5} a c \,d^{2} e^{4}+\frac {6}{5} x^{5} c^{2} d^{4} e^{2}+a^{2} d \,e^{5} x^{4}+3 a c \,d^{3} e^{3} x^{4}+c^{2} d^{5} e \,x^{4}+2 x^{3} a^{2} d^{2} e^{4}+\frac {8}{3} x^{3} a \,d^{4} e^{2} c +\frac {1}{3} x^{3} d^{6} c^{2}+2 a^{2} d^{3} e^{3} x^{2}+a c \,d^{5} e \,x^{2}+a^{2} e^{2} d^{4} x\)	\(187\)
gosper	\(\frac {x \left (15 c^{2} d^{2} e^{4} x^{6}+35 x^{5} a c d \,e^{5}+70 x^{5} c^{2} d^{3} e^{3}+21 x^{4} a^{2} e^{6}+168 x^{4} a c \,d^{2} e^{4}+126 x^{4} c^{2} d^{4} e^{2}+105 a^{2} d \,e^{5} x^{3}+315 a c \,d^{3} e^{3} x^{3}+105 c^{2} d^{5} e \,x^{3}+210 x^{2} a^{2} d^{2} e^{4}+280 x^{2} a \,d^{4} e^{2} c +35 x^{2} d^{6} c^{2}+210 a^{2} d^{3} e^{3} x +105 a c \,d^{5} e x +105 a^{2} e^{2} d^{4}\right )}{105}\)	\(189\)
default	\(\frac {c^{2} d^{2} e^{4} x^{7}}{7}+\frac {\left (2 c^{2} d^{3} e^{3}+2 e^{3} c d \left (e^{2} a +c \,d^{2}\right )\right ) x^{6}}{6}+\frac {\left (c^{2} d^{4} e^{2}+4 d^{2} e^{2} c \left (e^{2} a +c \,d^{2}\right )+e^{2} \left (2 a c \,d^{2} e^{2}+\left (e^{2} a +c \,d^{2}\right )^{2}\right )\right ) x^{5}}{5}+\frac {\left (2 d^{3} c e \left (e^{2} a +c \,d^{2}\right )+2 d e \left (2 a c \,d^{2} e^{2}+\left (e^{2} a +c \,d^{2}\right )^{2}\right )+2 e^{3} a d \left (e^{2} a +c \,d^{2}\right )\right ) x^{4}}{4}+\frac {\left (d^{2} \left (2 a c \,d^{2} e^{2}+\left (e^{2} a +c \,d^{2}\right )^{2}\right )+4 d^{2} e^{2} a \left (e^{2} a +c \,d^{2}\right )+a^{2} d^{2} e^{4}\right ) x^{3}}{3}+\frac {\left (2 d^{3} a e \left (e^{2} a +c \,d^{2}\right )+2 a^{2} e^{3} d^{3}\right ) x^{2}}{2}+a^{2} e^{2} d^{4} x\)	\(295\)

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)^2,x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 162 vs. \(2 (69) = 138\).
time = 0.28, size = 162, normalized size = 2.10 \begin {gather*} \frac {1}{7} \, c^{2} d^{2} x^{7} e^{4} + a^{2} d^{4} x e^{2} + \frac {1}{3} \, {\left (2 \, c^{2} d^{3} e^{3} + a c d e^{5}\right )} x^{6} + \frac {1}{5} \, {\left (6 \, c^{2} d^{4} e^{2} + 8 \, a c d^{2} e^{4} + a^{2} e^{6}\right )} x^{5} + {\left (c^{2} d^{5} e + 3 \, a c d^{3} e^{3} + a^{2} d e^{5}\right )} x^{4} + \frac {1}{3} \, {\left (c^{2} d^{6} + 8 \, a c d^{4} e^{2} + 6 \, a^{2} d^{2} e^{4}\right )} x^{3} + {\left (a c d^{5} e + 2 \, a^{2} d^{3} e^{3}\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)^2,x, algorithm="maxima")

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 175 vs. \(2 (69) = 138\).
time = 2.89, size = 175, normalized size = 2.27 \begin {gather*} \frac {1}{3} \, c^{2} d^{6} x^{3} + \frac {1}{5} \, a^{2} x^{5} e^{6} + \frac {1}{3} \, {\left (a c d x^{6} + 3 \, a^{2} d x^{4}\right )} e^{5} + \frac {1}{35} \, {\left (5 \, c^{2} d^{2} x^{7} + 56 \, a c d^{2} x^{5} + 70 \, a^{2} d^{2} x^{3}\right )} e^{4} + \frac {1}{3} \, {\left (2 \, c^{2} d^{3} x^{6} + 9 \, a c d^{3} x^{4} + 6 \, a^{2} d^{3} x^{2}\right )} e^{3} + \frac {1}{15} \, {\left (18 \, c^{2} d^{4} x^{5} + 40 \, a c d^{4} x^{3} + 15 \, a^{2} d^{4} x\right )} e^{2} + {\left (c^{2} d^{5} x^{4} + a c d^{5} x^{2}\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)^2,x, algorithm="fricas")

[Out]

1/3*c^2*d^6*x^3 + 1/5*a^2*x^5*e^6 + 1/3*(a*c*d*x^6 + 3*a^2*d*x^4)*e^5 + 1/35*(5*c^2*d^2*x^7 + 56*a*c*d^2*x^5 +
70*a^2*d^2*x^3)*e^4 + 1/3*(2*c^2*d^3*x^6 + 9*a*c*d^3*x^4 + 6*a^2*d^3*x^2)*e^3 + 1/15*(18*c^2*d^4*x^5 + 40*a*c
*d^4*x^3 + 15*a^2*d^4*x)*e^2 + (c^2*d^5*x^4 + a*c*d^5*x^2)*e

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 185 vs. \(2 (66) = 132\).
time = 0.03, size = 185, normalized size = 2.40 \begin {gather*} a^{2} d^{4} e^{2} x + \frac {c^{2} d^{2} e^{4} x^{7}}{7} + x^{6} \left (\frac {a c d e^{5}}{3} + \frac {2 c^{2} d^{3} e^{3}}{3}\right ) + x^{5} \left (\frac {a^{2} e^{6}}{5} + \frac {8 a c d^{2} e^{4}}{5} + \frac {6 c^{2} d^{4} e^{2}}{5}\right ) + x^{4} \left (a^{2} d e^{5} + 3 a c d^{3} e^{3} + c^{2} d^{5} e\right ) + x^{3} \cdot \left (2 a^{2} d^{2} e^{4} + \frac {8 a c d^{4} e^{2}}{3} + \frac {c^{2} d^{6}}{3}\right ) + x^{2} \cdot \left (2 a^{2} d^{3} e^{3} + a c d^{5} e\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)**2,x)

[Out]

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

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 176 vs. \(2 (69) = 138\).
time = 0.95, size = 176, normalized size = 2.29 \begin {gather*} \frac {1}{7} \, c^{2} d^{2} x^{7} e^{4} + \frac {2}{3} \, c^{2} d^{3} x^{6} e^{3} + \frac {6}{5} \, c^{2} d^{4} x^{5} e^{2} + c^{2} d^{5} x^{4} e + \frac {1}{3} \, c^{2} d^{6} x^{3} + \frac {1}{3} \, a c d x^{6} e^{5} + \frac {8}{5} \, a c d^{2} x^{5} e^{4} + 3 \, a c d^{3} x^{4} e^{3} + \frac {8}{3} \, a c d^{4} x^{3} e^{2} + a c d^{5} x^{2} e + \frac {1}{5} \, a^{2} x^{5} e^{6} + a^{2} d x^{4} e^{5} + 2 \, a^{2} d^{2} x^{3} e^{4} + 2 \, a^{2} d^{3} x^{2} e^{3} + a^{2} d^{4} x e^{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)^2,x, algorithm="giac")

[Out]

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

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Mupad [B]
time = 0.60, size = 168, normalized size = 2.18 \begin {gather*} x^3\,\left (2\,a^2\,d^2\,e^4+\frac {8\,a\,c\,d^4\,e^2}{3}+\frac {c^2\,d^6}{3}\right )+x^5\,\left (\frac {a^2\,e^6}{5}+\frac {8\,a\,c\,d^2\,e^4}{5}+\frac {6\,c^2\,d^4\,e^2}{5}\right )+x^4\,\left (a^2\,d\,e^5+3\,a\,c\,d^3\,e^3+c^2\,d^5\,e\right )+a^2\,d^4\,e^2\,x+\frac {c^2\,d^2\,e^4\,x^7}{7}+a\,d^3\,e\,x^2\,\left (c\,d^2+2\,a\,e^2\right )+\frac {c\,d\,e^3\,x^6\,\left (2\,c\,d^2+a\,e^2\right )}{3} \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)^2,x)

[Out]

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

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