Optimal. Leaf size=87 \[ \frac {1}{5} x^5 \left (b^2 e^2+4 b c d e+c^2 d^2\right )+\frac {1}{3} b^2 d^2 x^3+\frac {1}{3} c e x^6 (b e+c d)+\frac {1}{2} b d x^4 (b e+c d)+\frac {1}{7} c^2 e^2 x^7 \]
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Rubi [A] time = 0.08, antiderivative size = 87, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {698} \[ \frac {1}{5} x^5 \left (b^2 e^2+4 b c d e+c^2 d^2\right )+\frac {1}{3} b^2 d^2 x^3+\frac {1}{3} c e x^6 (b e+c d)+\frac {1}{2} b d x^4 (b e+c d)+\frac {1}{7} c^2 e^2 x^7 \]
Antiderivative was successfully verified.
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Rule 698
Rubi steps
\begin {align*} \int (d+e x)^2 \left (b x+c x^2\right )^2 \, dx &=\int \left (b^2 d^2 x^2+2 b d (c d+b e) x^3+\left (c^2 d^2+4 b c d e+b^2 e^2\right ) x^4+2 c e (c d+b e) x^5+c^2 e^2 x^6\right ) \, dx\\ &=\frac {1}{3} b^2 d^2 x^3+\frac {1}{2} b d (c d+b e) x^4+\frac {1}{5} \left (c^2 d^2+4 b c d e+b^2 e^2\right ) x^5+\frac {1}{3} c e (c d+b e) x^6+\frac {1}{7} c^2 e^2 x^7\\ \end {align*}
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Mathematica [A] time = 0.01, size = 87, normalized size = 1.00 \[ \frac {1}{5} x^5 \left (b^2 e^2+4 b c d e+c^2 d^2\right )+\frac {1}{3} b^2 d^2 x^3+\frac {1}{3} c e x^6 (b e+c d)+\frac {1}{2} b d x^4 (b e+c d)+\frac {1}{7} c^2 e^2 x^7 \]
Antiderivative was successfully verified.
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fricas [A] time = 0.94, size = 94, normalized size = 1.08 \[ \frac {1}{7} x^{7} e^{2} c^{2} + \frac {1}{3} x^{6} e d c^{2} + \frac {1}{3} x^{6} e^{2} c b + \frac {1}{5} x^{5} d^{2} c^{2} + \frac {4}{5} x^{5} e d c b + \frac {1}{5} x^{5} e^{2} b^{2} + \frac {1}{2} x^{4} d^{2} c b + \frac {1}{2} x^{4} e d b^{2} + \frac {1}{3} x^{3} d^{2} b^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 94, normalized size = 1.08 \[ \frac {1}{7} \, c^{2} x^{7} e^{2} + \frac {1}{3} \, c^{2} d x^{6} e + \frac {1}{5} \, c^{2} d^{2} x^{5} + \frac {1}{3} \, b c x^{6} e^{2} + \frac {4}{5} \, b c d x^{5} e + \frac {1}{2} \, b c d^{2} x^{4} + \frac {1}{5} \, b^{2} x^{5} e^{2} + \frac {1}{2} \, b^{2} d x^{4} e + \frac {1}{3} \, b^{2} d^{2} x^{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 90, normalized size = 1.03 \[ \frac {c^{2} e^{2} x^{7}}{7}+\frac {b^{2} d^{2} x^{3}}{3}+\frac {\left (2 e^{2} b c +2 d e \,c^{2}\right ) x^{6}}{6}+\frac {\left (b^{2} e^{2}+4 b c d e +c^{2} d^{2}\right ) x^{5}}{5}+\frac {\left (2 b^{2} d e +2 b c \,d^{2}\right ) x^{4}}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.39, size = 85, normalized size = 0.98 \[ \frac {1}{7} \, c^{2} e^{2} x^{7} + \frac {1}{3} \, b^{2} d^{2} x^{3} + \frac {1}{3} \, {\left (c^{2} d e + b c e^{2}\right )} x^{6} + \frac {1}{5} \, {\left (c^{2} d^{2} + 4 \, b c d e + b^{2} e^{2}\right )} x^{5} + \frac {1}{2} \, {\left (b c d^{2} + b^{2} d e\right )} x^{4} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.04, size = 78, normalized size = 0.90 \[ x^5\,\left (\frac {b^2\,e^2}{5}+\frac {4\,b\,c\,d\,e}{5}+\frac {c^2\,d^2}{5}\right )+\frac {b^2\,d^2\,x^3}{3}+\frac {c^2\,e^2\,x^7}{7}+\frac {b\,d\,x^4\,\left (b\,e+c\,d\right )}{2}+\frac {c\,e\,x^6\,\left (b\,e+c\,d\right )}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.09, size = 94, normalized size = 1.08 \[ \frac {b^{2} d^{2} x^{3}}{3} + \frac {c^{2} e^{2} x^{7}}{7} + x^{6} \left (\frac {b c e^{2}}{3} + \frac {c^{2} d e}{3}\right ) + x^{5} \left (\frac {b^{2} e^{2}}{5} + \frac {4 b c d e}{5} + \frac {c^{2} d^{2}}{5}\right ) + x^{4} \left (\frac {b^{2} d e}{2} + \frac {b c d^{2}}{2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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