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 = 0.0632473, antiderivative size = 57, normalized size of antiderivative = 1., 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*}
Mathematica [A] time = 0.0161587, size = 101, normalized size = 1.77 \[ \frac{1}{5} e^2 x^5 \left (a e^2+6 c d^2\right )+d e x^4 \left (a e^2+c d^2\right )+\frac{1}{3} d^2 x^3 \left (6 a e^2+c d^2\right )+2 a d^3 e x^2+a d^4 x+\frac{2}{3} c d e^3 x^6+\frac{1}{7} c e^4 x^7 \]
Antiderivative was successfully verified.
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Maple [A] time = 0.042, size = 97, normalized size = 1.7 \begin{align*}{\frac{{e}^{4}c{x}^{7}}{7}}+{\frac{2\,d{e}^{3}c{x}^{6}}{3}}+{\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}ec \right ){x}^{4}}{4}}+{\frac{ \left ( 6\,{d}^{2}{e}^{2}a+{d}^{4}c \right ){x}^{3}}{3}}+2\,{d}^{3}ea{x}^{2}+{d}^{4}ax \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.1672, size = 126, normalized size = 2.21 \begin{align*} \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.64037, size = 212, normalized size = 3.72 \begin{align*} \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 \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.127347, size = 100, normalized size = 1.75 \begin{align*} 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 ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.17834, size = 124, normalized size = 2.18 \begin{align*} \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 \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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