3.2857 \(\int (c e+d e x)^3 (a+b (c+d x)^3)^2 \, dx\)

Optimal. Leaf size=60 \[ \frac{a^2 e^3 (c+d x)^4}{4 d}+\frac{2 a b e^3 (c+d x)^7}{7 d}+\frac{b^2 e^3 (c+d x)^{10}}{10 d} \]

[Out]

(a^2*e^3*(c + d*x)^4)/(4*d) + (2*a*b*e^3*(c + d*x)^7)/(7*d) + (b^2*e^3*(c + d*x)^10)/(10*d)

________________________________________________________________________________________

Rubi [A]  time = 0.0623818, antiderivative size = 60, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083, Rules used = {372, 270} \[ \frac{a^2 e^3 (c+d x)^4}{4 d}+\frac{2 a b e^3 (c+d x)^7}{7 d}+\frac{b^2 e^3 (c+d x)^{10}}{10 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a^2*e^3*(c + d*x)^4)/(4*d) + (2*a*b*e^3*(c + d*x)^7)/(7*d) + (b^2*e^3*(c + d*x)^10)/(10*d)

Rule 372

Int[(u_)^(m_.)*((a_) + (b_.)*(v_)^(n_))^(p_.), x_Symbol] :> Dist[u^m/(Coefficient[v, x, 1]*v^m), Subst[Int[x^m
*(a + b*x^n)^p, x], x, v], x] /; FreeQ[{a, b, m, n, p}, x] && LinearPairQ[u, v, x]

Rule 270

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*(a + b*x^n)^p,
x], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0]

Rubi steps

\begin{align*} \int (c e+d e x)^3 \left (a+b (c+d x)^3\right )^2 \, dx &=\frac{e^3 \operatorname{Subst}\left (\int x^3 \left (a+b x^3\right )^2 \, dx,x,c+d x\right )}{d}\\ &=\frac{e^3 \operatorname{Subst}\left (\int \left (a^2 x^3+2 a b x^6+b^2 x^9\right ) \, dx,x,c+d x\right )}{d}\\ &=\frac{a^2 e^3 (c+d x)^4}{4 d}+\frac{2 a b e^3 (c+d x)^7}{7 d}+\frac{b^2 e^3 (c+d x)^{10}}{10 d}\\ \end{align*}

Mathematica [B]  time = 0.0103369, size = 207, normalized size = 3.45 \[ e^3 \left (\frac{1}{4} d^3 x^4 \left (a^2+40 a b c^3+84 b^2 c^6\right )+c d^2 x^3 \left (a^2+10 a b c^3+12 b^2 c^6\right )+\frac{3}{2} c^2 d x^2 \left (a^2+4 a b c^3+3 b^2 c^6\right )+\frac{2}{7} b d^6 x^7 \left (a+42 b c^3\right )+b c d^5 x^6 \left (2 a+21 b c^3\right )+\frac{6}{5} b c^2 d^4 x^5 \left (5 a+21 b c^3\right )+c^3 x \left (a+b c^3\right )^2+\frac{9}{2} b^2 c^2 d^7 x^8+b^2 c d^8 x^9+\frac{1}{10} b^2 d^9 x^{10}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

e^3*(c^3*(a + b*c^3)^2*x + (3*c^2*(a^2 + 4*a*b*c^3 + 3*b^2*c^6)*d*x^2)/2 + c*(a^2 + 10*a*b*c^3 + 12*b^2*c^6)*d
^2*x^3 + ((a^2 + 40*a*b*c^3 + 84*b^2*c^6)*d^3*x^4)/4 + (6*b*c^2*(5*a + 21*b*c^3)*d^4*x^5)/5 + b*c*(2*a + 21*b*
c^3)*d^5*x^6 + (2*b*(a + 42*b*c^3)*d^6*x^7)/7 + (9*b^2*c^2*d^7*x^8)/2 + b^2*c*d^8*x^9 + (b^2*d^9*x^10)/10)

________________________________________________________________________________________

Maple [B]  time = 0.001, size = 536, normalized size = 8.9 \begin{align*}{\frac{{d}^{9}{e}^{3}{b}^{2}{x}^{10}}{10}}+c{e}^{3}{d}^{8}{b}^{2}{x}^{9}+{\frac{9\,{c}^{2}{e}^{3}{d}^{7}{b}^{2}{x}^{8}}{2}}+{\frac{ \left ( 64\,{c}^{3}{e}^{3}{b}^{2}{d}^{6}+{d}^{3}{e}^{3} \left ( 2\, \left ( b{c}^{3}+a \right ) b{d}^{3}+18\,{b}^{2}{c}^{3}{d}^{3} \right ) \right ){x}^{7}}{7}}+{\frac{ \left ( 51\,{c}^{4}{e}^{3}{b}^{2}{d}^{5}+3\,c{e}^{3}{d}^{2} \left ( 2\, \left ( b{c}^{3}+a \right ) b{d}^{3}+18\,{b}^{2}{c}^{3}{d}^{3} \right ) +{d}^{3}{e}^{3} \left ( 6\, \left ( b{c}^{3}+a \right ) bc{d}^{2}+9\,{b}^{2}{c}^{4}{d}^{2} \right ) \right ){x}^{6}}{6}}+{\frac{ \left ( 15\,{c}^{5}{e}^{3}{b}^{2}{d}^{4}+3\,{c}^{2}{e}^{3}d \left ( 2\, \left ( b{c}^{3}+a \right ) b{d}^{3}+18\,{b}^{2}{c}^{3}{d}^{3} \right ) +3\,c{e}^{3}{d}^{2} \left ( 6\, \left ( b{c}^{3}+a \right ) bc{d}^{2}+9\,{b}^{2}{c}^{4}{d}^{2} \right ) +6\,{d}^{4}{e}^{3} \left ( b{c}^{3}+a \right ) b{c}^{2} \right ){x}^{5}}{5}}+{\frac{ \left ({c}^{3}{e}^{3} \left ( 2\, \left ( b{c}^{3}+a \right ) b{d}^{3}+18\,{b}^{2}{c}^{3}{d}^{3} \right ) +3\,{c}^{2}{e}^{3}d \left ( 6\, \left ( b{c}^{3}+a \right ) bc{d}^{2}+9\,{b}^{2}{c}^{4}{d}^{2} \right ) +18\,{c}^{3}{e}^{3}{d}^{3} \left ( b{c}^{3}+a \right ) b+{d}^{3}{e}^{3} \left ( b{c}^{3}+a \right ) ^{2} \right ){x}^{4}}{4}}+{\frac{ \left ({c}^{3}{e}^{3} \left ( 6\, \left ( b{c}^{3}+a \right ) bc{d}^{2}+9\,{b}^{2}{c}^{4}{d}^{2} \right ) +18\,{c}^{4}{e}^{3}{d}^{2} \left ( b{c}^{3}+a \right ) b+3\,c{e}^{3}{d}^{2} \left ( b{c}^{3}+a \right ) ^{2} \right ){x}^{3}}{3}}+{\frac{ \left ( 6\,{c}^{5}{e}^{3} \left ( b{c}^{3}+a \right ) bd+3\,{c}^{2}{e}^{3}d \left ( b{c}^{3}+a \right ) ^{2} \right ){x}^{2}}{2}}+{c}^{3}{e}^{3} \left ( b{c}^{3}+a \right ) ^{2}x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/10*d^9*e^3*b^2*x^10+c*e^3*d^8*b^2*x^9+9/2*c^2*e^3*d^7*b^2*x^8+1/7*(64*c^3*e^3*b^2*d^6+d^3*e^3*(2*(b*c^3+a)*b
*d^3+18*b^2*c^3*d^3))*x^7+1/6*(51*c^4*e^3*b^2*d^5+3*c*e^3*d^2*(2*(b*c^3+a)*b*d^3+18*b^2*c^3*d^3)+d^3*e^3*(6*(b
*c^3+a)*b*c*d^2+9*b^2*c^4*d^2))*x^6+1/5*(15*c^5*e^3*b^2*d^4+3*c^2*e^3*d*(2*(b*c^3+a)*b*d^3+18*b^2*c^3*d^3)+3*c
*e^3*d^2*(6*(b*c^3+a)*b*c*d^2+9*b^2*c^4*d^2)+6*d^4*e^3*(b*c^3+a)*b*c^2)*x^5+1/4*(c^3*e^3*(2*(b*c^3+a)*b*d^3+18
*b^2*c^3*d^3)+3*c^2*e^3*d*(6*(b*c^3+a)*b*c*d^2+9*b^2*c^4*d^2)+18*c^3*e^3*d^3*(b*c^3+a)*b+d^3*e^3*(b*c^3+a)^2)*
x^4+1/3*(c^3*e^3*(6*(b*c^3+a)*b*c*d^2+9*b^2*c^4*d^2)+18*c^4*e^3*d^2*(b*c^3+a)*b+3*c*e^3*d^2*(b*c^3+a)^2)*x^3+1
/2*(6*c^5*e^3*(b*c^3+a)*b*d+3*c^2*e^3*d*(b*c^3+a)^2)*x^2+c^3*e^3*(b*c^3+a)^2*x

________________________________________________________________________________________

Maxima [B]  time = 1.41778, size = 324, normalized size = 5.4 \begin{align*} \frac{1}{10} \, b^{2} d^{9} e^{3} x^{10} + b^{2} c d^{8} e^{3} x^{9} + \frac{9}{2} \, b^{2} c^{2} d^{7} e^{3} x^{8} + \frac{2}{7} \,{\left (42 \, b^{2} c^{3} + a b\right )} d^{6} e^{3} x^{7} +{\left (21 \, b^{2} c^{4} + 2 \, a b c\right )} d^{5} e^{3} x^{6} + \frac{6}{5} \,{\left (21 \, b^{2} c^{5} + 5 \, a b c^{2}\right )} d^{4} e^{3} x^{5} + \frac{1}{4} \,{\left (84 \, b^{2} c^{6} + 40 \, a b c^{3} + a^{2}\right )} d^{3} e^{3} x^{4} +{\left (12 \, b^{2} c^{7} + 10 \, a b c^{4} + a^{2} c\right )} d^{2} e^{3} x^{3} + \frac{3}{2} \,{\left (3 \, b^{2} c^{8} + 4 \, a b c^{5} + a^{2} c^{2}\right )} d e^{3} x^{2} +{\left (b^{2} c^{9} + 2 \, a b c^{6} + a^{2} c^{3}\right )} e^{3} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/10*b^2*d^9*e^3*x^10 + b^2*c*d^8*e^3*x^9 + 9/2*b^2*c^2*d^7*e^3*x^8 + 2/7*(42*b^2*c^3 + a*b)*d^6*e^3*x^7 + (21
*b^2*c^4 + 2*a*b*c)*d^5*e^3*x^6 + 6/5*(21*b^2*c^5 + 5*a*b*c^2)*d^4*e^3*x^5 + 1/4*(84*b^2*c^6 + 40*a*b*c^3 + a^
2)*d^3*e^3*x^4 + (12*b^2*c^7 + 10*a*b*c^4 + a^2*c)*d^2*e^3*x^3 + 3/2*(3*b^2*c^8 + 4*a*b*c^5 + a^2*c^2)*d*e^3*x
^2 + (b^2*c^9 + 2*a*b*c^6 + a^2*c^3)*e^3*x

________________________________________________________________________________________

Fricas [B]  time = 1.30916, size = 644, normalized size = 10.73 \begin{align*} \frac{1}{10} x^{10} e^{3} d^{9} b^{2} + x^{9} e^{3} d^{8} c b^{2} + \frac{9}{2} x^{8} e^{3} d^{7} c^{2} b^{2} + 12 x^{7} e^{3} d^{6} c^{3} b^{2} + 21 x^{6} e^{3} d^{5} c^{4} b^{2} + \frac{126}{5} x^{5} e^{3} d^{4} c^{5} b^{2} + 21 x^{4} e^{3} d^{3} c^{6} b^{2} + \frac{2}{7} x^{7} e^{3} d^{6} b a + 12 x^{3} e^{3} d^{2} c^{7} b^{2} + 2 x^{6} e^{3} d^{5} c b a + \frac{9}{2} x^{2} e^{3} d c^{8} b^{2} + 6 x^{5} e^{3} d^{4} c^{2} b a + x e^{3} c^{9} b^{2} + 10 x^{4} e^{3} d^{3} c^{3} b a + 10 x^{3} e^{3} d^{2} c^{4} b a + 6 x^{2} e^{3} d c^{5} b a + 2 x e^{3} c^{6} b a + \frac{1}{4} x^{4} e^{3} d^{3} a^{2} + x^{3} e^{3} d^{2} c a^{2} + \frac{3}{2} x^{2} e^{3} d c^{2} a^{2} + x e^{3} c^{3} a^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/10*x^10*e^3*d^9*b^2 + x^9*e^3*d^8*c*b^2 + 9/2*x^8*e^3*d^7*c^2*b^2 + 12*x^7*e^3*d^6*c^3*b^2 + 21*x^6*e^3*d^5*
c^4*b^2 + 126/5*x^5*e^3*d^4*c^5*b^2 + 21*x^4*e^3*d^3*c^6*b^2 + 2/7*x^7*e^3*d^6*b*a + 12*x^3*e^3*d^2*c^7*b^2 +
2*x^6*e^3*d^5*c*b*a + 9/2*x^2*e^3*d*c^8*b^2 + 6*x^5*e^3*d^4*c^2*b*a + x*e^3*c^9*b^2 + 10*x^4*e^3*d^3*c^3*b*a +
10*x^3*e^3*d^2*c^4*b*a + 6*x^2*e^3*d*c^5*b*a + 2*x*e^3*c^6*b*a + 1/4*x^4*e^3*d^3*a^2 + x^3*e^3*d^2*c*a^2 + 3/
2*x^2*e^3*d*c^2*a^2 + x*e^3*c^3*a^2

________________________________________________________________________________________

Sympy [B]  time = 0.12715, size = 323, normalized size = 5.38 \begin{align*} \frac{9 b^{2} c^{2} d^{7} e^{3} x^{8}}{2} + b^{2} c d^{8} e^{3} x^{9} + \frac{b^{2} d^{9} e^{3} x^{10}}{10} + x^{7} \left (\frac{2 a b d^{6} e^{3}}{7} + 12 b^{2} c^{3} d^{6} e^{3}\right ) + x^{6} \left (2 a b c d^{5} e^{3} + 21 b^{2} c^{4} d^{5} e^{3}\right ) + x^{5} \left (6 a b c^{2} d^{4} e^{3} + \frac{126 b^{2} c^{5} d^{4} e^{3}}{5}\right ) + x^{4} \left (\frac{a^{2} d^{3} e^{3}}{4} + 10 a b c^{3} d^{3} e^{3} + 21 b^{2} c^{6} d^{3} e^{3}\right ) + x^{3} \left (a^{2} c d^{2} e^{3} + 10 a b c^{4} d^{2} e^{3} + 12 b^{2} c^{7} d^{2} e^{3}\right ) + x^{2} \left (\frac{3 a^{2} c^{2} d e^{3}}{2} + 6 a b c^{5} d e^{3} + \frac{9 b^{2} c^{8} d e^{3}}{2}\right ) + x \left (a^{2} c^{3} e^{3} + 2 a b c^{6} e^{3} + b^{2} c^{9} e^{3}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

9*b**2*c**2*d**7*e**3*x**8/2 + b**2*c*d**8*e**3*x**9 + b**2*d**9*e**3*x**10/10 + x**7*(2*a*b*d**6*e**3/7 + 12*
b**2*c**3*d**6*e**3) + x**6*(2*a*b*c*d**5*e**3 + 21*b**2*c**4*d**5*e**3) + x**5*(6*a*b*c**2*d**4*e**3 + 126*b*
*2*c**5*d**4*e**3/5) + x**4*(a**2*d**3*e**3/4 + 10*a*b*c**3*d**3*e**3 + 21*b**2*c**6*d**3*e**3) + x**3*(a**2*c
*d**2*e**3 + 10*a*b*c**4*d**2*e**3 + 12*b**2*c**7*d**2*e**3) + x**2*(3*a**2*c**2*d*e**3/2 + 6*a*b*c**5*d*e**3
+ 9*b**2*c**8*d*e**3/2) + x*(a**2*c**3*e**3 + 2*a*b*c**6*e**3 + b**2*c**9*e**3)

________________________________________________________________________________________

Giac [B]  time = 1.08665, size = 392, normalized size = 6.53 \begin{align*} \frac{1}{10} \, b^{2} d^{9} x^{10} e^{3} + b^{2} c d^{8} x^{9} e^{3} + \frac{9}{2} \, b^{2} c^{2} d^{7} x^{8} e^{3} + 12 \, b^{2} c^{3} d^{6} x^{7} e^{3} + 21 \, b^{2} c^{4} d^{5} x^{6} e^{3} + \frac{126}{5} \, b^{2} c^{5} d^{4} x^{5} e^{3} + 21 \, b^{2} c^{6} d^{3} x^{4} e^{3} + \frac{2}{7} \, a b d^{6} x^{7} e^{3} + 12 \, b^{2} c^{7} d^{2} x^{3} e^{3} + 2 \, a b c d^{5} x^{6} e^{3} + \frac{9}{2} \, b^{2} c^{8} d x^{2} e^{3} + 6 \, a b c^{2} d^{4} x^{5} e^{3} + b^{2} c^{9} x e^{3} + 10 \, a b c^{3} d^{3} x^{4} e^{3} + 10 \, a b c^{4} d^{2} x^{3} e^{3} + 6 \, a b c^{5} d x^{2} e^{3} + 2 \, a b c^{6} x e^{3} + \frac{1}{4} \, a^{2} d^{3} x^{4} e^{3} + a^{2} c d^{2} x^{3} e^{3} + \frac{3}{2} \, a^{2} c^{2} d x^{2} e^{3} + a^{2} c^{3} x e^{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/10*b^2*d^9*x^10*e^3 + b^2*c*d^8*x^9*e^3 + 9/2*b^2*c^2*d^7*x^8*e^3 + 12*b^2*c^3*d^6*x^7*e^3 + 21*b^2*c^4*d^5*
x^6*e^3 + 126/5*b^2*c^5*d^4*x^5*e^3 + 21*b^2*c^6*d^3*x^4*e^3 + 2/7*a*b*d^6*x^7*e^3 + 12*b^2*c^7*d^2*x^3*e^3 +
2*a*b*c*d^5*x^6*e^3 + 9/2*b^2*c^8*d*x^2*e^3 + 6*a*b*c^2*d^4*x^5*e^3 + b^2*c^9*x*e^3 + 10*a*b*c^3*d^3*x^4*e^3 +
10*a*b*c^4*d^2*x^3*e^3 + 6*a*b*c^5*d*x^2*e^3 + 2*a*b*c^6*x*e^3 + 1/4*a^2*d^3*x^4*e^3 + a^2*c*d^2*x^3*e^3 + 3/
2*a^2*c^2*d*x^2*e^3 + a^2*c^3*x*e^3