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]
________________________________________________________________________________________
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]
[Out]
Rule 2060
Rule 194
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]
[Out]
________________________________________________________________________________________
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]
[Out]
________________________________________________________________________________________
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]
[Out]
________________________________________________________________________________________
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]
[Out]
________________________________________________________________________________________
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]
[Out]
________________________________________________________________________________________
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]
[Out]