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

Optimal. Leaf size=56 \[ -\frac{b \left (b^2-3 a c\right ) (b+c x)^4}{2 c^3}+\frac{b^2 x \left (b^2-3 a c\right )^2}{c^2}+\frac{(b+c x)^7}{7 c^3} \]

[Out]

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

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Rubi [A]  time = 0.0678793, antiderivative size = 56, 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} \[ -\frac{b \left (b^2-3 a c\right ) (b+c x)^4}{2 c^3}+\frac{b^2 x \left (b^2-3 a c\right )^2}{c^2}+\frac{(b+c x)^7}{7 c^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 )^2 \, dx &=\frac{1}{9} \operatorname{Subst}\left (\int \left (3 b \left (3 a-\frac{b^2}{c}\right )+3 c^2 x^3\right )^2 \, dx,x,\frac{b}{c}+x\right )\\ &=\frac{1}{9} \operatorname{Subst}\left (\int \left (\frac{9 \left (-b^3+3 a b c\right )^2}{c^2}-18 b c \left (b^2-3 a c\right ) x^3+9 c^4 x^6\right ) \, dx,x,\frac{b}{c}+x\right )\\ &=\frac{b^2 \left (b^2-3 a c\right )^2 x}{c^2}-\frac{b \left (b^2-3 a c\right ) (b+c x)^4}{2 c^3}+\frac{(b+c x)^7}{7 c^3}\\ \end{align*}

Mathematica [A]  time = 0.0080767, size = 82, normalized size = 1.46 \[ 9 a^2 b^2 x+\frac{3}{2} b c x^4 \left (a c+3 b^2\right )+3 b^2 x^3 \left (2 a c+b^2\right )+9 a b^3 x^2+3 b^2 c^2 x^5+b c^3 x^6+\frac{c^4 x^7}{7} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.001, size = 84, normalized size = 1.5 \begin{align*}{\frac{{c}^{4}{x}^{7}}{7}}+b{c}^{3}{x}^{6}+3\,{b}^{2}{c}^{2}{x}^{5}+{\frac{ \left ( 6\,ab{c}^{2}+18\,{b}^{3}c \right ){x}^{4}}{4}}+{\frac{ \left ( 18\,a{b}^{2}c+9\,{b}^{4} \right ){x}^{3}}{3}}+9\,a{b}^{3}{x}^{2}+9\,{b}^{2}{a}^{2}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)^2,x)

[Out]

1/7*c^4*x^7+b*c^3*x^6+3*b^2*c^2*x^5+1/4*(6*a*b*c^2+18*b^3*c)*x^4+1/3*(18*a*b^2*c+9*b^4)*x^3+9*a*b^3*x^2+9*b^2*
a^2*x

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Maxima [A]  time = 1.25047, size = 126, normalized size = 2.25 \begin{align*} \frac{1}{7} \, c^{4} x^{7} + b c^{3} x^{6} + \frac{9}{5} \, b^{2} c^{2} x^{5} + 3 \, b^{4} x^{3} + 9 \, a^{2} b^{2} x + \frac{3}{2} \,{\left (c^{2} x^{4} + 4 \, b c x^{3} + 6 \, b^{2} x^{2}\right )} a b + \frac{3}{10} \,{\left (4 \, c^{2} x^{5} + 15 \, b c x^{4}\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)^2,x, algorithm="maxima")

[Out]

1/7*c^4*x^7 + b*c^3*x^6 + 9/5*b^2*c^2*x^5 + 3*b^4*x^3 + 9*a^2*b^2*x + 3/2*(c^2*x^4 + 4*b*c*x^3 + 6*b^2*x^2)*a*
b + 3/10*(4*c^2*x^5 + 15*b*c*x^4)*b^2

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Fricas [A]  time = 1.14051, size = 177, normalized size = 3.16 \begin{align*} \frac{1}{7} x^{7} c^{4} + x^{6} c^{3} b + 3 x^{5} c^{2} b^{2} + \frac{9}{2} x^{4} c b^{3} + \frac{3}{2} x^{4} c^{2} b a + 3 x^{3} b^{4} + 6 x^{3} c b^{2} a + 9 x^{2} b^{3} a + 9 x b^{2} a^{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)^2,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.078979, size = 87, normalized size = 1.55 \begin{align*} 9 a^{2} b^{2} x + 9 a b^{3} x^{2} + 3 b^{2} c^{2} x^{5} + b c^{3} x^{6} + \frac{c^{4} x^{7}}{7} + x^{4} \left (\frac{3 a b c^{2}}{2} + \frac{9 b^{3} c}{2}\right ) + x^{3} \left (6 a b^{2} c + 3 b^{4}\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)**2,x)

[Out]

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

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Giac [A]  time = 1.07039, size = 112, normalized size = 2. \begin{align*} \frac{1}{7} \, c^{4} x^{7} + b c^{3} x^{6} + 3 \, b^{2} c^{2} x^{5} + \frac{9}{2} \, b^{3} c x^{4} + \frac{3}{2} \, a b c^{2} x^{4} + 3 \, b^{4} x^{3} + 6 \, a b^{2} c x^{3} + 9 \, a b^{3} x^{2} + 9 \, a^{2} b^{2} 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)^2,x, algorithm="giac")

[Out]

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