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 = 0.07, antiderivative size = 56, normalized size of antiderivative = 1.00, 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 194
Rule 2060
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 = 0.01, size = 82, normalized size = 1.46 \[ 9 a^2 b^2 x+9 a b^3 x^2+\frac {3}{2} b c x^4 \left (a c+3 b^2\right )+3 b^2 x^3 \left (2 a c+b^2\right )+3 b^2 c^2 x^5+b c^3 x^6+\frac {c^4 x^7}{7} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.69, size = 83, normalized size = 1.48 \[ \frac {1}{7} x^{7} c^{4} + x^{6} c^{3} b + 3 x^{5} c^{2} b^{2} + \frac {9}{2} x^{4} c b^{3} + \frac {3}{2} x^{4} c^{2} b a + 3 x^{3} b^{4} + 6 x^{3} c b^{2} a + 9 x^{2} b^{3} a + 9 x b^{2} a^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.34, size = 83, normalized size = 1.48 \[ \frac {1}{7} \, c^{4} x^{7} + b c^{3} x^{6} + 3 \, b^{2} c^{2} x^{5} + \frac {9}{2} \, b^{3} c x^{4} + \frac {3}{2} \, a b c^{2} x^{4} + 3 \, b^{4} x^{3} + 6 \, a b^{2} c x^{3} + 9 \, a b^{3} x^{2} + 9 \, a^{2} b^{2} x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.00, size = 84, normalized size = 1.50 \[ \frac {c^{4} x^{7}}{7}+b \,c^{3} x^{6}+3 b^{2} c^{2} x^{5}+9 a \,b^{3} x^{2}+9 a^{2} b^{2} x +\frac {\left (6 a b \,c^{2}+18 b^{3} c \right ) x^{4}}{4}+\frac {\left (18 a \,b^{2} c +9 b^{4}\right ) x^{3}}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.64, size = 93, normalized size = 1.66 \[ \frac {1}{7} \, c^{4} x^{7} + b c^{3} x^{6} + \frac {9}{5} \, b^{2} c^{2} x^{5} + 3 \, b^{4} x^{3} + 9 \, a^{2} b^{2} x + \frac {3}{2} \, {\left (c^{2} x^{4} + 4 \, b c x^{3} + 6 \, b^{2} x^{2}\right )} a b + \frac {3}{10} \, {\left (4 \, c^{2} x^{5} + 15 \, b c x^{4}\right )} b^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.04, size = 79, normalized size = 1.41 \[ x^3\,\left (3\,b^4+6\,a\,c\,b^2\right )+\frac {c^4\,x^7}{7}+9\,a^2\,b^2\,x+9\,a\,b^3\,x^2+b\,c^3\,x^6+3\,b^2\,c^2\,x^5+\frac {3\,b\,c\,x^4\,\left (3\,b^2+a\,c\right )}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 0.08, size = 87, normalized size = 1.55 \[ 9 a^{2} b^{2} x + 9 a b^{3} x^{2} + 3 b^{2} c^{2} x^{5} + b c^{3} x^{6} + \frac {c^{4} x^{7}}{7} + x^{4} \left (\frac {3 a b c^{2}}{2} + \frac {9 b^{3} c}{2}\right ) + x^{3} \left (6 a b^{2} c + 3 b^{4}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________