3.9 \(\int (3 a b+3 b^2 x+3 b c x^2+c^2 x^3)^3 \, dx\)

Optimal. Leaf size=84 \[ -\frac{3 b \left (b^2-3 a c\right ) (b+c x)^7}{7 c^4}+\frac{3 b^2 \left (b^2-3 a c\right )^2 (b+c x)^4}{4 c^4}-\frac{b^3 x \left (b^2-3 a c\right )^3}{c^3}+\frac{(b+c x)^{10}}{10 c^4} \]

[Out]

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

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Rubi [A]  time = 0.124573, antiderivative size = 84, normalized size of antiderivative = 1., 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 = {2060, 194} \[ -\frac{3 b \left (b^2-3 a c\right ) (b+c x)^7}{7 c^4}+\frac{3 b^2 \left (b^2-3 a c\right )^2 (b+c x)^4}{4 c^4}-\frac{b^3 x \left (b^2-3 a c\right )^3}{c^3}+\frac{(b+c x)^{10}}{10 c^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 2060

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2 + (d_.)*(x_)^3)^(p_), x_Symbol] :> Dist[1/3^p, Subst[Int[Simp[(3*a*c -
b^2)/c + (c^2*x^3)/b, x]^p, x], x, c/(3*d) + x], x] /; FreeQ[{a, b, c, d}, x] && IGtQ[p, 0] && EqQ[c^2 - 3*b*d
, 0]

Rule 194

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

Rubi steps

\begin{align*} \int \left (3 a b+3 b^2 x+3 b c x^2+c^2 x^3\right )^3 \, dx &=\frac{1}{27} \operatorname{Subst}\left (\int \left (3 b \left (3 a-\frac{b^2}{c}\right )+3 c^2 x^3\right )^3 \, dx,x,\frac{b}{c}+x\right )\\ &=\frac{1}{27} \operatorname{Subst}\left (\int \left (\frac{27 \left (-b^3+3 a b c\right )^3}{c^3}+81 \left (b^3-3 a b c\right )^2 x^3-81 b c^3 \left (b^2-3 a c\right ) x^6+27 c^6 x^9\right ) \, dx,x,\frac{b}{c}+x\right )\\ &=-\frac{b^3 \left (b^2-3 a c\right )^3 x}{c^3}+\frac{3 b^2 \left (b^2-3 a c\right )^2 (b+c x)^4}{4 c^4}-\frac{3 b \left (b^2-3 a c\right ) (b+c x)^7}{7 c^4}+\frac{(b+c x)^{10}}{10 c^4}\\ \end{align*}

Mathematica [A]  time = 0.0187318, size = 159, normalized size = 1.89 \[ \frac{27}{4} b^2 x^4 \left (a^2 c^2+6 a b^2 c+b^4\right )+\frac{81}{2} a^2 b^4 x^2+27 a^3 b^3 x+\frac{9}{7} b c^3 x^7 \left (a c+9 b^2\right )+9 b^2 c^2 x^6 \left (a c+2 b^2\right )+\frac{27}{5} b^3 c x^5 \left (5 a c+3 b^2\right )+27 a b^3 x^3 \left (a c+b^2\right )+\frac{9}{2} b^2 c^4 x^8+b c^5 x^9+\frac{c^6 x^{10}}{10} \]

Antiderivative was successfully verified.

[In]

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

[Out]

27*a^3*b^3*x + (81*a^2*b^4*x^2)/2 + 27*a*b^3*(b^2 + a*c)*x^3 + (27*b^2*(b^4 + 6*a*b^2*c + a^2*c^2)*x^4)/4 + (2
7*b^3*c*(3*b^2 + 5*a*c)*x^5)/5 + 9*b^2*c^2*(2*b^2 + a*c)*x^6 + (9*b*c^3*(9*b^2 + a*c)*x^7)/7 + (9*b^2*c^4*x^8)
/2 + b*c^5*x^9 + (c^6*x^10)/10

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Maple [B]  time = 0.001, size = 295, normalized size = 3.5 \begin{align*}{\frac{{c}^{6}{x}^{10}}{10}}+b{c}^{5}{x}^{9}+{\frac{9\,{b}^{2}{c}^{4}{x}^{8}}{2}}+{\frac{ \left ( 3\,ab{c}^{4}+63\,{b}^{3}{c}^{3}+{c}^{2} \left ( 6\,ab{c}^{2}+18\,{b}^{3}c \right ) \right ){x}^{7}}{7}}+{\frac{ \left ( 18\,a{b}^{2}{c}^{3}+45\,{b}^{4}{c}^{2}+3\,bc \left ( 6\,ab{c}^{2}+18\,{b}^{3}c \right ) +{c}^{2} \left ( 18\,a{b}^{2}c+9\,{b}^{4} \right ) \right ){x}^{6}}{6}}+{\frac{ \left ( 63\,a{b}^{3}{c}^{2}+3\,{b}^{2} \left ( 6\,ab{c}^{2}+18\,{b}^{3}c \right ) +3\,bc \left ( 18\,a{b}^{2}c+9\,{b}^{4} \right ) \right ){x}^{5}}{5}}+{\frac{ \left ( 3\,ab \left ( 6\,ab{c}^{2}+18\,{b}^{3}c \right ) +3\,{b}^{2} \left ( 18\,a{b}^{2}c+9\,{b}^{4} \right ) +54\,{b}^{4}ca+9\,{a}^{2}{b}^{2}{c}^{2} \right ){x}^{4}}{4}}+{\frac{ \left ( 3\,ab \left ( 18\,a{b}^{2}c+9\,{b}^{4} \right ) +54\,{b}^{5}a+27\,{b}^{3}c{a}^{2} \right ){x}^{3}}{3}}+{\frac{81\,{a}^{2}{b}^{4}{x}^{2}}{2}}+27\,{a}^{3}{b}^{3}x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/10*c^6*x^10+b*c^5*x^9+9/2*b^2*c^4*x^8+1/7*(3*a*b*c^4+63*b^3*c^3+c^2*(6*a*b*c^2+18*b^3*c))*x^7+1/6*(18*a*b^2*
c^3+45*b^4*c^2+3*b*c*(6*a*b*c^2+18*b^3*c)+c^2*(18*a*b^2*c+9*b^4))*x^6+1/5*(63*a*b^3*c^2+3*b^2*(6*a*b*c^2+18*b^
3*c)+3*b*c*(18*a*b^2*c+9*b^4))*x^5+1/4*(3*a*b*(6*a*b*c^2+18*b^3*c)+3*b^2*(18*a*b^2*c+9*b^4)+54*b^4*c*a+9*a^2*b
^2*c^2)*x^4+1/3*(3*a*b*(18*a*b^2*c+9*b^4)+54*b^5*a+27*b^3*c*a^2)*x^3+81/2*a^2*b^4*x^2+27*a^3*b^3*x

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Maxima [B]  time = 1.71501, size = 275, normalized size = 3.27 \begin{align*} \frac{1}{10} \, c^{6} x^{10} + b c^{5} x^{9} + \frac{27}{8} \, b^{2} c^{4} x^{8} + \frac{27}{7} \, b^{3} c^{3} x^{7} + \frac{27}{4} \, b^{6} x^{4} + 27 \, a^{3} b^{3} x + \frac{27}{4} \,{\left (c^{2} x^{4} + 4 \, b c x^{3} + 6 \, b^{2} x^{2}\right )} a^{2} b^{2} + \frac{9}{10} \,{\left (5 \, c^{2} x^{6} + 18 \, b c x^{5}\right )} b^{4} + \frac{9}{70} \,{\left (10 \, c^{4} x^{7} + 70 \, b c^{3} x^{6} + 126 \, b^{2} c^{2} x^{5} + 210 \, b^{4} x^{3} + 21 \,{\left (4 \, c^{2} x^{5} + 15 \, b c x^{4}\right )} b^{2}\right )} a b + \frac{9}{56} \,{\left (7 \, c^{4} x^{8} + 48 \, b c^{3} x^{7} + 84 \, b^{2} c^{2} x^{6}\right )} b^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/10*c^6*x^10 + b*c^5*x^9 + 27/8*b^2*c^4*x^8 + 27/7*b^3*c^3*x^7 + 27/4*b^6*x^4 + 27*a^3*b^3*x + 27/4*(c^2*x^4
+ 4*b*c*x^3 + 6*b^2*x^2)*a^2*b^2 + 9/10*(5*c^2*x^6 + 18*b*c*x^5)*b^4 + 9/70*(10*c^4*x^7 + 70*b*c^3*x^6 + 126*b
^2*c^2*x^5 + 210*b^4*x^3 + 21*(4*c^2*x^5 + 15*b*c*x^4)*b^2)*a*b + 9/56*(7*c^4*x^8 + 48*b*c^3*x^7 + 84*b^2*c^2*
x^6)*b^2

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Fricas [B]  time = 1.22675, size = 375, normalized size = 4.46 \begin{align*} \frac{1}{10} x^{10} c^{6} + x^{9} c^{5} b + \frac{9}{2} x^{8} c^{4} b^{2} + \frac{81}{7} x^{7} c^{3} b^{3} + \frac{9}{7} x^{7} c^{4} b a + 18 x^{6} c^{2} b^{4} + 9 x^{6} c^{3} b^{2} a + \frac{81}{5} x^{5} c b^{5} + 27 x^{5} c^{2} b^{3} a + \frac{27}{4} x^{4} b^{6} + \frac{81}{2} x^{4} c b^{4} a + \frac{27}{4} x^{4} c^{2} b^{2} a^{2} + 27 x^{3} b^{5} a + 27 x^{3} c b^{3} a^{2} + \frac{81}{2} x^{2} b^{4} a^{2} + 27 x b^{3} a^{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/10*x^10*c^6 + x^9*c^5*b + 9/2*x^8*c^4*b^2 + 81/7*x^7*c^3*b^3 + 9/7*x^7*c^4*b*a + 18*x^6*c^2*b^4 + 9*x^6*c^3*
b^2*a + 81/5*x^5*c*b^5 + 27*x^5*c^2*b^3*a + 27/4*x^4*b^6 + 81/2*x^4*c*b^4*a + 27/4*x^4*c^2*b^2*a^2 + 27*x^3*b^
5*a + 27*x^3*c*b^3*a^2 + 81/2*x^2*b^4*a^2 + 27*x*b^3*a^3

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Sympy [B]  time = 0.092548, size = 175, normalized size = 2.08 \begin{align*} 27 a^{3} b^{3} x + \frac{81 a^{2} b^{4} x^{2}}{2} + \frac{9 b^{2} c^{4} x^{8}}{2} + b c^{5} x^{9} + \frac{c^{6} x^{10}}{10} + x^{7} \left (\frac{9 a b c^{4}}{7} + \frac{81 b^{3} c^{3}}{7}\right ) + x^{6} \left (9 a b^{2} c^{3} + 18 b^{4} c^{2}\right ) + x^{5} \left (27 a b^{3} c^{2} + \frac{81 b^{5} c}{5}\right ) + x^{4} \left (\frac{27 a^{2} b^{2} c^{2}}{4} + \frac{81 a b^{4} c}{2} + \frac{27 b^{6}}{4}\right ) + x^{3} \left (27 a^{2} b^{3} c + 27 a b^{5}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

27*a**3*b**3*x + 81*a**2*b**4*x**2/2 + 9*b**2*c**4*x**8/2 + b*c**5*x**9 + c**6*x**10/10 + x**7*(9*a*b*c**4/7 +
81*b**3*c**3/7) + x**6*(9*a*b**2*c**3 + 18*b**4*c**2) + x**5*(27*a*b**3*c**2 + 81*b**5*c/5) + x**4*(27*a**2*b
**2*c**2/4 + 81*a*b**4*c/2 + 27*b**6/4) + x**3*(27*a**2*b**3*c + 27*a*b**5)

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Giac [B]  time = 1.07557, size = 224, normalized size = 2.67 \begin{align*} \frac{1}{10} \, c^{6} x^{10} + b c^{5} x^{9} + \frac{9}{2} \, b^{2} c^{4} x^{8} + \frac{81}{7} \, b^{3} c^{3} x^{7} + \frac{9}{7} \, a b c^{4} x^{7} + 18 \, b^{4} c^{2} x^{6} + 9 \, a b^{2} c^{3} x^{6} + \frac{81}{5} \, b^{5} c x^{5} + 27 \, a b^{3} c^{2} x^{5} + \frac{27}{4} \, b^{6} x^{4} + \frac{81}{2} \, a b^{4} c x^{4} + \frac{27}{4} \, a^{2} b^{2} c^{2} x^{4} + 27 \, a b^{5} x^{3} + 27 \, a^{2} b^{3} c x^{3} + \frac{81}{2} \, a^{2} b^{4} x^{2} + 27 \, a^{3} b^{3} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/10*c^6*x^10 + b*c^5*x^9 + 9/2*b^2*c^4*x^8 + 81/7*b^3*c^3*x^7 + 9/7*a*b*c^4*x^7 + 18*b^4*c^2*x^6 + 9*a*b^2*c^
3*x^6 + 81/5*b^5*c*x^5 + 27*a*b^3*c^2*x^5 + 27/4*b^6*x^4 + 81/2*a*b^4*c*x^4 + 27/4*a^2*b^2*c^2*x^4 + 27*a*b^5*
x^3 + 27*a^2*b^3*c*x^3 + 81/2*a^2*b^4*x^2 + 27*a^3*b^3*x