Optimal. Leaf size=57 \[ \frac {(d+e x)^4 \left (a e^2+c d^2\right )}{4 e^3}+\frac {c (d+e x)^6}{6 e^3}-\frac {2 c d (d+e x)^5}{5 e^3} \]
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Rubi [A] time = 0.05, antiderivative size = 57, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {697} \[ \frac {(d+e x)^4 \left (a e^2+c d^2\right )}{4 e^3}+\frac {c (d+e x)^6}{6 e^3}-\frac {2 c d (d+e x)^5}{5 e^3} \]
Antiderivative was successfully verified.
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Rule 697
Rubi steps
\begin {align*} \int (d+e x)^3 \left (a+c x^2\right ) \, dx &=\int \left (\frac {\left (c d^2+a e^2\right ) (d+e x)^3}{e^2}-\frac {2 c d (d+e x)^4}{e^2}+\frac {c (d+e x)^5}{e^2}\right ) \, dx\\ &=\frac {\left (c d^2+a e^2\right ) (d+e x)^4}{4 e^3}-\frac {2 c d (d+e x)^5}{5 e^3}+\frac {c (d+e x)^6}{6 e^3}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 74, normalized size = 1.30 \[ \frac {1}{4} a x \left (4 d^3+6 d^2 e x+4 d e^2 x^2+e^3 x^3\right )+\frac {1}{60} c x^3 \left (20 d^3+45 d^2 e x+36 d e^2 x^2+10 e^3 x^3\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.99, size = 73, normalized size = 1.28 \[ \frac {1}{6} x^{6} e^{3} c + \frac {3}{5} x^{5} e^{2} d c + \frac {3}{4} x^{4} e d^{2} c + \frac {1}{4} x^{4} e^{3} a + \frac {1}{3} x^{3} d^{3} c + x^{3} e^{2} d a + \frac {3}{2} x^{2} e d^{2} a + x d^{3} a \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 71, normalized size = 1.25 \[ \frac {1}{6} \, c x^{6} e^{3} + \frac {3}{5} \, c d x^{5} e^{2} + \frac {3}{4} \, c d^{2} x^{4} e + \frac {1}{3} \, c d^{3} x^{3} + \frac {1}{4} \, a x^{4} e^{3} + a d x^{3} e^{2} + \frac {3}{2} \, a d^{2} x^{2} e + a d^{3} x \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 73, normalized size = 1.28 \[ \frac {c \,e^{3} x^{6}}{6}+\frac {3 c d \,e^{2} x^{5}}{5}+\frac {3 a \,d^{2} e \,x^{2}}{2}+a \,d^{3} x +\frac {\left (e^{3} a +3 d^{2} e c \right ) x^{4}}{4}+\frac {\left (3 d \,e^{2} a +d^{3} c \right ) x^{3}}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.33, size = 72, normalized size = 1.26 \[ \frac {1}{6} \, c e^{3} x^{6} + \frac {3}{5} \, c d e^{2} x^{5} + \frac {3}{2} \, a d^{2} e x^{2} + a d^{3} x + \frac {1}{4} \, {\left (3 \, c d^{2} e + a e^{3}\right )} x^{4} + \frac {1}{3} \, {\left (c d^{3} + 3 \, a d e^{2}\right )} x^{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.03, size = 71, normalized size = 1.25 \[ x^3\,\left (\frac {c\,d^3}{3}+a\,d\,e^2\right )+x^4\,\left (\frac {3\,c\,d^2\,e}{4}+\frac {a\,e^3}{4}\right )+\frac {c\,e^3\,x^6}{6}+a\,d^3\,x+\frac {3\,a\,d^2\,e\,x^2}{2}+\frac {3\,c\,d\,e^2\,x^5}{5} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.08, size = 80, normalized size = 1.40 \[ a d^{3} x + \frac {3 a d^{2} e x^{2}}{2} + \frac {3 c d e^{2} x^{5}}{5} + \frac {c e^{3} x^{6}}{6} + x^{4} \left (\frac {a e^{3}}{4} + \frac {3 c d^{2} e}{4}\right ) + x^{3} \left (a d e^{2} + \frac {c d^{3}}{3}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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