\(\int (d+e x)^2 (c d^2+2 c d e x+c e^2 x^2)^2 \, dx\) [987]

Optimal result

Integrand size = 30, antiderivative size = 17 \[ \int (d+e x)^2 \left (c d^2+2 c d e x+c e^2 x^2\right )^2 \, dx=\frac {c^2 (d+e x)^7}{7 e} \]

[Out]

1/7*c^2*(e*x+d)^7/e

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {27, 12, 32} \[ \int (d+e x)^2 \left (c d^2+2 c d e x+c e^2 x^2\right )^2 \, dx=\frac {c^2 (d+e x)^7}{7 e} \]

[In]

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

[Out]

(c^2*(d + e*x)^7)/(7*e)

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 27

Int[(u_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[u*Cancel[(b/2 + c*x)^(2*p)/c^p], x] /; Fr
eeQ[{a, b, c}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]

Rule 32

Int[((a_.) + (b_.)*(x_))^(m_), x_Symbol] :> Simp[(a + b*x)^(m + 1)/(b*(m + 1)), x] /; FreeQ[{a, b, m}, x] && N
eQ[m, -1]

Rubi steps \begin{align*} \text {integral}& = \int c^2 (d+e x)^6 \, dx \\ & = c^2 \int (d+e x)^6 \, dx \\ & = \frac {c^2 (d+e x)^7}{7 e} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.00 \[ \int (d+e x)^2 \left (c d^2+2 c d e x+c e^2 x^2\right )^2 \, dx=\frac {c^2 (d+e x)^7}{7 e} \]

[In]

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

[Out]

(c^2*(d + e*x)^7)/(7*e)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(68\) vs. \(2(15)=30\).

Time = 2.47 (sec) , antiderivative size = 69, normalized size of antiderivative = 4.06

method	result	size
gosper	\(\frac {x \left (e^{6} x^{6}+7 d \,e^{5} x^{5}+21 d^{2} e^{4} x^{4}+35 x^{3} d^{3} e^{3}+35 d^{4} e^{2} x^{2}+21 d^{5} e x +7 d^{6}\right ) c^{2}}{7}\)	\(69\)
default	\(\frac {1}{7} e^{6} c^{2} x^{7}+d \,e^{5} c^{2} x^{6}+3 d^{2} e^{4} c^{2} x^{5}+5 d^{3} c^{2} e^{3} x^{4}+5 d^{4} c^{2} e^{2} x^{3}+3 d^{5} e \,c^{2} x^{2}+d^{6} c^{2} x\)	\(86\)
norman	\(\frac {1}{7} e^{6} c^{2} x^{7}+d \,e^{5} c^{2} x^{6}+3 d^{2} e^{4} c^{2} x^{5}+5 d^{3} c^{2} e^{3} x^{4}+5 d^{4} c^{2} e^{2} x^{3}+3 d^{5} e \,c^{2} x^{2}+d^{6} c^{2} x\)	\(86\)
risch	\(\frac {1}{7} e^{6} c^{2} x^{7}+d \,e^{5} c^{2} x^{6}+3 d^{2} e^{4} c^{2} x^{5}+5 d^{3} c^{2} e^{3} x^{4}+5 d^{4} c^{2} e^{2} x^{3}+3 d^{5} e \,c^{2} x^{2}+d^{6} c^{2} x\)	\(86\)
parallelrisch	\(\frac {1}{7} e^{6} c^{2} x^{7}+d \,e^{5} c^{2} x^{6}+3 d^{2} e^{4} c^{2} x^{5}+5 d^{3} c^{2} e^{3} x^{4}+5 d^{4} c^{2} e^{2} x^{3}+3 d^{5} e \,c^{2} x^{2}+d^{6} c^{2} x\)	\(86\)

[In]

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

[Out]

1/7*x*(e^6*x^6+7*d*e^5*x^5+21*d^2*e^4*x^4+35*d^3*e^3*x^3+35*d^4*e^2*x^2+21*d^5*e*x+7*d^6)*c^2

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 85 vs. \(2 (15) = 30\).

Time = 0.27 (sec) , antiderivative size = 85, normalized size of antiderivative = 5.00 \[ \int (d+e x)^2 \left (c d^2+2 c d e x+c e^2 x^2\right )^2 \, dx=\frac {1}{7} \, c^{2} e^{6} x^{7} + c^{2} d e^{5} x^{6} + 3 \, c^{2} d^{2} e^{4} x^{5} + 5 \, c^{2} d^{3} e^{3} x^{4} + 5 \, c^{2} d^{4} e^{2} x^{3} + 3 \, c^{2} d^{5} e x^{2} + c^{2} d^{6} x \]

[In]

integrate((e*x+d)^2*(c*e^2*x^2+2*c*d*e*x+c*d^2)^2,x, algorithm="fricas")

[Out]

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

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 90 vs. \(2 (12) = 24\).

Time = 0.02 (sec) , antiderivative size = 90, normalized size of antiderivative = 5.29 \[ \int (d+e x)^2 \left (c d^2+2 c d e x+c e^2 x^2\right )^2 \, dx=c^{2} d^{6} x + 3 c^{2} d^{5} e x^{2} + 5 c^{2} d^{4} e^{2} x^{3} + 5 c^{2} d^{3} e^{3} x^{4} + 3 c^{2} d^{2} e^{4} x^{5} + c^{2} d e^{5} x^{6} + \frac {c^{2} e^{6} x^{7}}{7} \]

[In]

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

[Out]

c**2*d**6*x + 3*c**2*d**5*e*x**2 + 5*c**2*d**4*e**2*x**3 + 5*c**2*d**3*e**3*x**4 + 3*c**2*d**2*e**4*x**5 + c**
2*d*e**5*x**6 + c**2*e**6*x**7/7

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 85 vs. \(2 (15) = 30\).

Time = 0.19 (sec) , antiderivative size = 85, normalized size of antiderivative = 5.00 \[ \int (d+e x)^2 \left (c d^2+2 c d e x+c e^2 x^2\right )^2 \, dx=\frac {1}{7} \, c^{2} e^{6} x^{7} + c^{2} d e^{5} x^{6} + 3 \, c^{2} d^{2} e^{4} x^{5} + 5 \, c^{2} d^{3} e^{3} x^{4} + 5 \, c^{2} d^{4} e^{2} x^{3} + 3 \, c^{2} d^{5} e x^{2} + c^{2} d^{6} x \]

[In]

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

[Out]

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 85 vs. \(2 (15) = 30\).

Time = 0.27 (sec) , antiderivative size = 85, normalized size of antiderivative = 5.00 \[ \int (d+e x)^2 \left (c d^2+2 c d e x+c e^2 x^2\right )^2 \, dx=\frac {1}{7} \, c^{2} e^{6} x^{7} + c^{2} d e^{5} x^{6} + 3 \, c^{2} d^{2} e^{4} x^{5} + 5 \, c^{2} d^{3} e^{3} x^{4} + 5 \, c^{2} d^{4} e^{2} x^{3} + 3 \, c^{2} d^{5} e x^{2} + c^{2} d^{6} x \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 85, normalized size of antiderivative = 5.00 \[ \int (d+e x)^2 \left (c d^2+2 c d e x+c e^2 x^2\right )^2 \, dx=c^2\,d^6\,x+3\,c^2\,d^5\,e\,x^2+5\,c^2\,d^4\,e^2\,x^3+5\,c^2\,d^3\,e^3\,x^4+3\,c^2\,d^2\,e^4\,x^5+c^2\,d\,e^5\,x^6+\frac {c^2\,e^6\,x^7}{7} \]

[In]

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

[Out]

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