Optimal. Leaf size=57 \[ \frac {(d+e x)^5 \left (a e^2+c d^2\right )}{5 e^3}+\frac {c (d+e x)^7}{7 e^3}-\frac {c d (d+e x)^6}{3 e^3} \]
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Rubi [A] time = 0.06, 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} \[ \frac {(d+e x)^5 \left (a e^2+c d^2\right )}{5 e^3}+\frac {c (d+e x)^7}{7 e^3}-\frac {c d (d+e x)^6}{3 e^3} \]
Antiderivative was successfully verified.
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Rule 697
Rubi steps
\begin {align*} \int (d+e x)^4 \left (a+c x^2\right ) \, dx &=\int \left (\frac {\left (c d^2+a e^2\right ) (d+e x)^4}{e^2}-\frac {2 c d (d+e x)^5}{e^2}+\frac {c (d+e x)^6}{e^2}\right ) \, dx\\ &=\frac {\left (c d^2+a e^2\right ) (d+e x)^5}{5 e^3}-\frac {c d (d+e x)^6}{3 e^3}+\frac {c (d+e x)^7}{7 e^3}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 101, normalized size = 1.77 \[ \frac {1}{5} e^2 x^5 \left (a e^2+6 c d^2\right )+d e x^4 \left (a e^2+c d^2\right )+\frac {1}{3} d^2 x^3 \left (6 a e^2+c d^2\right )+a d^4 x+2 a d^3 e x^2+\frac {2}{3} c d e^3 x^6+\frac {1}{7} c e^4 x^7 \]
Antiderivative was successfully verified.
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fricas [A] time = 0.98, size = 96, normalized size = 1.68 \[ \frac {1}{7} x^{7} e^{4} c + \frac {2}{3} x^{6} e^{3} d c + \frac {6}{5} x^{5} e^{2} d^{2} c + \frac {1}{5} x^{5} e^{4} a + x^{4} e d^{3} c + x^{4} e^{3} d a + \frac {1}{3} x^{3} d^{4} c + 2 x^{3} e^{2} d^{2} a + 2 x^{2} e d^{3} a + x d^{4} a \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 92, normalized size = 1.61 \[ \frac {1}{7} \, c x^{7} e^{4} + \frac {2}{3} \, c d x^{6} e^{3} + \frac {6}{5} \, c d^{2} x^{5} e^{2} + c d^{3} x^{4} e + \frac {1}{3} \, c d^{4} x^{3} + \frac {1}{5} \, a x^{5} e^{4} + a d x^{4} e^{3} + 2 \, a d^{2} x^{3} e^{2} + 2 \, a d^{3} x^{2} e + a d^{4} x \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 97, normalized size = 1.70 \[ \frac {c \,e^{4} x^{7}}{7}+\frac {2 c d \,e^{3} x^{6}}{3}+2 a \,d^{3} e \,x^{2}+a \,d^{4} x +\frac {\left (e^{4} a +6 d^{2} e^{2} c \right ) x^{5}}{5}+\frac {\left (4 d \,e^{3} a +4 d^{3} e c \right ) x^{4}}{4}+\frac {\left (6 d^{2} e^{2} a +d^{4} c \right ) x^{3}}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.32, size = 93, normalized size = 1.63 \[ \frac {1}{7} \, c e^{4} x^{7} + \frac {2}{3} \, c d e^{3} x^{6} + 2 \, a d^{3} e x^{2} + a d^{4} x + \frac {1}{5} \, {\left (6 \, c d^{2} e^{2} + a e^{4}\right )} x^{5} + {\left (c d^{3} e + a d e^{3}\right )} x^{4} + \frac {1}{3} \, {\left (c d^{4} + 6 \, a d^{2} e^{2}\right )} x^{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.27, size = 93, normalized size = 1.63 \[ x^3\,\left (\frac {c\,d^4}{3}+2\,a\,d^2\,e^2\right )+x^5\,\left (\frac {6\,c\,d^2\,e^2}{5}+\frac {a\,e^4}{5}\right )+x^4\,\left (c\,d^3\,e+a\,d\,e^3\right )+\frac {c\,e^4\,x^7}{7}+a\,d^4\,x+2\,a\,d^3\,e\,x^2+\frac {2\,c\,d\,e^3\,x^6}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.09, size = 100, normalized size = 1.75 \[ a d^{4} x + 2 a d^{3} e x^{2} + \frac {2 c d e^{3} x^{6}}{3} + \frac {c e^{4} x^{7}}{7} + x^{5} \left (\frac {a e^{4}}{5} + \frac {6 c d^{2} e^{2}}{5}\right ) + x^{4} \left (a d e^{3} + c d^{3} e\right ) + x^{3} \left (2 a d^{2} e^{2} + \frac {c d^{4}}{3}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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