Optimal. Leaf size=96 \[ a^2 A x+a A b x^2+\frac {1}{3} \left (A \left (b^2+2 a c\right )+a^2 C\right ) x^3+\frac {1}{2} b (A c+a C) x^4+\frac {1}{5} \left (A c^2+\left (b^2+2 a c\right ) C\right ) x^5+\frac {1}{3} b c C x^6+\frac {1}{7} c^2 C x^7 \]
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Rubi [A]
time = 0.07, antiderivative size = 96, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 1, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {1671}
\begin {gather*} \frac {1}{3} x^3 \left (a^2 C+A \left (2 a c+b^2\right )\right )+a^2 A x+\frac {1}{5} x^5 \left (C \left (2 a c+b^2\right )+A c^2\right )+\frac {1}{2} b x^4 (a C+A c)+a A b x^2+\frac {1}{3} b c C x^6+\frac {1}{7} c^2 C x^7 \end {gather*}
Antiderivative was successfully verified.
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Rule 1671
Rubi steps
\begin {align*} \int \left (a+b x+c x^2\right )^2 \left (A+C x^2\right ) \, dx &=\int \left (a^2 A+2 a A b x+\left (A \left (b^2+2 a c\right )+a^2 C\right ) x^2+2 b (A c+a C) x^3+\left (A c^2+\left (b^2+2 a c\right ) C\right ) x^4+2 b c C x^5+c^2 C x^6\right ) \, dx\\ &=a^2 A x+a A b x^2+\frac {1}{3} \left (A \left (b^2+2 a c\right )+a^2 C\right ) x^3+\frac {1}{2} b (A c+a C) x^4+\frac {1}{5} \left (A c^2+\left (b^2+2 a c\right ) C\right ) x^5+\frac {1}{3} b c C x^6+\frac {1}{7} c^2 C x^7\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 96, normalized size = 1.00 \begin {gather*} a^2 A x+a A b x^2+\frac {1}{3} \left (A b^2+2 a A c+a^2 C\right ) x^3+\frac {1}{2} b (A c+a C) x^4+\frac {1}{5} \left (A c^2+b^2 C+2 a c C\right ) x^5+\frac {1}{3} b c C x^6+\frac {1}{7} c^2 C x^7 \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.11, size = 90, normalized size = 0.94
method | result | size |
default | \(\frac {c^{2} C \,x^{7}}{7}+\frac {b c C \,x^{6}}{3}+\frac {\left (A \,c^{2}+\left (2 a c +b^{2}\right ) C \right ) x^{5}}{5}+\frac {\left (2 b c A +2 a b C \right ) x^{4}}{4}+\frac {\left (A \left (2 a c +b^{2}\right )+a^{2} C \right ) x^{3}}{3}+a A b \,x^{2}+a^{2} A x\) | \(90\) |
norman | \(\frac {c^{2} C \,x^{7}}{7}+\frac {b c C \,x^{6}}{3}+\left (\frac {1}{5} A \,c^{2}+\frac {2}{5} a c C +\frac {1}{5} C \,b^{2}\right ) x^{5}+\left (\frac {1}{2} b c A +\frac {1}{2} a b C \right ) x^{4}+\left (\frac {2}{3} a c A +\frac {1}{3} A \,b^{2}+\frac {1}{3} a^{2} C \right ) x^{3}+a A b \,x^{2}+a^{2} A x\) | \(91\) |
gosper | \(\frac {1}{7} c^{2} C \,x^{7}+\frac {1}{3} b c C \,x^{6}+\frac {1}{5} x^{5} A \,c^{2}+\frac {2}{5} x^{5} a c C +\frac {1}{5} x^{5} C \,b^{2}+\frac {1}{2} x^{4} b c A +\frac {1}{2} x^{4} a b C +\frac {2}{3} x^{3} a c A +\frac {1}{3} x^{3} A \,b^{2}+\frac {1}{3} x^{3} a^{2} C +a A b \,x^{2}+a^{2} A x\) | \(100\) |
risch | \(\frac {1}{7} c^{2} C \,x^{7}+\frac {1}{3} b c C \,x^{6}+\frac {1}{5} x^{5} A \,c^{2}+\frac {2}{5} x^{5} a c C +\frac {1}{5} x^{5} C \,b^{2}+\frac {1}{2} x^{4} b c A +\frac {1}{2} x^{4} a b C +\frac {2}{3} x^{3} a c A +\frac {1}{3} x^{3} A \,b^{2}+\frac {1}{3} x^{3} a^{2} C +a A b \,x^{2}+a^{2} A x\) | \(100\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 87, normalized size = 0.91 \begin {gather*} \frac {1}{7} \, C c^{2} x^{7} + \frac {1}{3} \, C b c x^{6} + \frac {1}{5} \, {\left (C b^{2} + 2 \, C a c + A c^{2}\right )} x^{5} + A a b x^{2} + \frac {1}{2} \, {\left (C a b + A b c\right )} x^{4} + A a^{2} x + \frac {1}{3} \, {\left (C a^{2} + A b^{2} + 2 \, A a c\right )} x^{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.34, size = 87, normalized size = 0.91 \begin {gather*} \frac {1}{7} \, C c^{2} x^{7} + \frac {1}{3} \, C b c x^{6} + \frac {1}{5} \, {\left (C b^{2} + 2 \, C a c + A c^{2}\right )} x^{5} + A a b x^{2} + \frac {1}{2} \, {\left (C a b + A b c\right )} x^{4} + A a^{2} x + \frac {1}{3} \, {\left (C a^{2} + A b^{2} + 2 \, A a c\right )} x^{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.02, size = 102, normalized size = 1.06 \begin {gather*} A a^{2} x + A a b x^{2} + \frac {C b c x^{6}}{3} + \frac {C c^{2} x^{7}}{7} + x^{5} \left (\frac {A c^{2}}{5} + \frac {2 C a c}{5} + \frac {C b^{2}}{5}\right ) + x^{4} \left (\frac {A b c}{2} + \frac {C a b}{2}\right ) + x^{3} \cdot \left (\frac {2 A a c}{3} + \frac {A b^{2}}{3} + \frac {C a^{2}}{3}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 3.76, size = 99, normalized size = 1.03 \begin {gather*} \frac {1}{7} \, C c^{2} x^{7} + \frac {1}{3} \, C b c x^{6} + \frac {1}{5} \, C b^{2} x^{5} + \frac {2}{5} \, C a c x^{5} + \frac {1}{5} \, A c^{2} x^{5} + \frac {1}{2} \, C a b x^{4} + \frac {1}{2} \, A b c x^{4} + \frac {1}{3} \, C a^{2} x^{3} + \frac {1}{3} \, A b^{2} x^{3} + \frac {2}{3} \, A a c x^{3} + A a b x^{2} + A a^{2} x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 4.09, size = 88, normalized size = 0.92 \begin {gather*} x^3\,\left (\frac {C\,a^2}{3}+\frac {2\,A\,c\,a}{3}+\frac {A\,b^2}{3}\right )+x^5\,\left (\frac {C\,b^2}{5}+\frac {A\,c^2}{5}+\frac {2\,C\,a\,c}{5}\right )+\frac {C\,c^2\,x^7}{7}+A\,a^2\,x+\frac {b\,x^4\,\left (A\,c+C\,a\right )}{2}+A\,a\,b\,x^2+\frac {C\,b\,c\,x^6}{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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