Optimal. Leaf size=175 \[ \frac{(d+e x)^6 \left (a C e^2+c \left (6 C d^2-e (3 B d-A e)\right )\right )}{6 e^5}-\frac{(d+e x)^5 \left (a e^2 (2 C d-B e)+c d \left (4 C d^2-e (3 B d-2 A e)\right )\right )}{5 e^5}+\frac{(d+e x)^4 \left (a e^2+c d^2\right ) \left (A e^2-B d e+C d^2\right )}{4 e^5}-\frac{c (d+e x)^7 (4 C d-B e)}{7 e^5}+\frac{c C (d+e x)^8}{8 e^5} \]
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Rubi [A] time = 0.313178, antiderivative size = 173, normalized size of antiderivative = 0.99, number of steps used = 2, number of rules used = 1, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.04, Rules used = {1628} \[ \frac{(d+e x)^6 \left (a C e^2-c e (3 B d-A e)+6 c C d^2\right )}{6 e^5}-\frac{(d+e x)^5 \left (a e^2 (2 C d-B e)-c d e (3 B d-2 A e)+4 c C d^3\right )}{5 e^5}+\frac{(d+e x)^4 \left (a e^2+c d^2\right ) \left (A e^2-B d e+C d^2\right )}{4 e^5}-\frac{c (d+e x)^7 (4 C d-B e)}{7 e^5}+\frac{c C (d+e x)^8}{8 e^5} \]
Antiderivative was successfully verified.
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Rule 1628
Rubi steps
\begin{align*} \int (d+e x)^3 \left (a+c x^2\right ) \left (A+B x+C x^2\right ) \, dx &=\int \left (\frac{\left (c d^2+a e^2\right ) \left (C d^2-B d e+A e^2\right ) (d+e x)^3}{e^4}+\frac{\left (-4 c C d^3+c d e (3 B d-2 A e)-a e^2 (2 C d-B e)\right ) (d+e x)^4}{e^4}+\frac{\left (6 c C d^2+a C e^2-c e (3 B d-A e)\right ) (d+e x)^5}{e^4}+\frac{c (-4 C d+B e) (d+e x)^6}{e^4}+\frac{c C (d+e x)^7}{e^4}\right ) \, dx\\ &=\frac{\left (c d^2+a e^2\right ) \left (C d^2-B d e+A e^2\right ) (d+e x)^4}{4 e^5}-\frac{\left (4 c C d^3-c d e (3 B d-2 A e)+a e^2 (2 C d-B e)\right ) (d+e x)^5}{5 e^5}+\frac{\left (6 c C d^2+a C e^2-c e (3 B d-A e)\right ) (d+e x)^6}{6 e^5}-\frac{c (4 C d-B e) (d+e x)^7}{7 e^5}+\frac{c C (d+e x)^8}{8 e^5}\\ \end{align*}
Mathematica [A] time = 0.0895023, size = 208, normalized size = 1.19 \[ \frac{1}{6} e x^6 \left (a C e^2+c e (A e+3 B d)+3 c C d^2\right )+\frac{1}{5} x^5 \left (a e^2 (B e+3 C d)+3 c d e (A e+B d)+c C d^3\right )+\frac{1}{4} x^4 \left (a A e^3+3 a B d e^2+3 a C d^2 e+3 A c d^2 e+B c d^3\right )+\frac{1}{3} d x^3 \left (A \left (3 a e^2+c d^2\right )+a d (3 B e+C d)\right )+\frac{1}{2} a d^2 x^2 (3 A e+B d)+a A d^3 x+\frac{1}{7} c e^2 x^7 (B e+3 C d)+\frac{1}{8} c C e^3 x^8 \]
Antiderivative was successfully verified.
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Maple [A] time = 0.044, size = 217, normalized size = 1.2 \begin{align*}{\frac{{e}^{3}cC{x}^{8}}{8}}+{\frac{ \left ({e}^{3}cB+3\,d{e}^{2}cC \right ){x}^{7}}{7}}+{\frac{ \left ( \left ( a{e}^{3}+3\,{d}^{2}ec \right ) C+3\,d{e}^{2}cB+{e}^{3}cA \right ){x}^{6}}{6}}+{\frac{ \left ( \left ( 3\,ad{e}^{2}+c{d}^{3} \right ) C+ \left ( a{e}^{3}+3\,{d}^{2}ec \right ) B+3\,Acd{e}^{2} \right ){x}^{5}}{5}}+{\frac{ \left ( 3\,{d}^{2}eaC+ \left ( 3\,ad{e}^{2}+c{d}^{3} \right ) B+ \left ( a{e}^{3}+3\,{d}^{2}ec \right ) A \right ){x}^{4}}{4}}+{\frac{ \left ({d}^{3}aC+3\,Ba{d}^{2}e+ \left ( 3\,ad{e}^{2}+c{d}^{3} \right ) A \right ){x}^{3}}{3}}+{\frac{ \left ( 3\,{d}^{2}eaA+{d}^{3}aB \right ){x}^{2}}{2}}+{d}^{3}aAx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.02217, size = 273, normalized size = 1.56 \begin{align*} \frac{1}{8} \, C c e^{3} x^{8} + \frac{1}{7} \,{\left (3 \, C c d e^{2} + B c e^{3}\right )} x^{7} + \frac{1}{6} \,{\left (3 \, C c d^{2} e + 3 \, B c d e^{2} +{\left (C a + A c\right )} e^{3}\right )} x^{6} + A a d^{3} x + \frac{1}{5} \,{\left (C c d^{3} + 3 \, B c d^{2} e + B a e^{3} + 3 \,{\left (C a + A c\right )} d e^{2}\right )} x^{5} + \frac{1}{4} \,{\left (B c d^{3} + 3 \, B a d e^{2} + A a e^{3} + 3 \,{\left (C a + A c\right )} d^{2} e\right )} x^{4} + \frac{1}{3} \,{\left (3 \, B a d^{2} e + 3 \, A a d e^{2} +{\left (C a + A c\right )} d^{3}\right )} x^{3} + \frac{1}{2} \,{\left (B a d^{3} + 3 \, A a d^{2} e\right )} x^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.53428, size = 595, normalized size = 3.4 \begin{align*} \frac{1}{8} x^{8} e^{3} c C + \frac{3}{7} x^{7} e^{2} d c C + \frac{1}{7} x^{7} e^{3} c B + \frac{1}{2} x^{6} e d^{2} c C + \frac{1}{6} x^{6} e^{3} a C + \frac{1}{2} x^{6} e^{2} d c B + \frac{1}{6} x^{6} e^{3} c A + \frac{1}{5} x^{5} d^{3} c C + \frac{3}{5} x^{5} e^{2} d a C + \frac{3}{5} x^{5} e d^{2} c B + \frac{1}{5} x^{5} e^{3} a B + \frac{3}{5} x^{5} e^{2} d c A + \frac{3}{4} x^{4} e d^{2} a C + \frac{1}{4} x^{4} d^{3} c B + \frac{3}{4} x^{4} e^{2} d a B + \frac{3}{4} x^{4} e d^{2} c A + \frac{1}{4} x^{4} e^{3} a A + \frac{1}{3} x^{3} d^{3} a C + x^{3} e d^{2} a B + \frac{1}{3} x^{3} d^{3} c A + x^{3} e^{2} d a A + \frac{1}{2} x^{2} d^{3} a B + \frac{3}{2} x^{2} e d^{2} a A + x d^{3} a A \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.092206, size = 257, normalized size = 1.47 \begin{align*} A a d^{3} x + \frac{C c e^{3} x^{8}}{8} + x^{7} \left (\frac{B c e^{3}}{7} + \frac{3 C c d e^{2}}{7}\right ) + x^{6} \left (\frac{A c e^{3}}{6} + \frac{B c d e^{2}}{2} + \frac{C a e^{3}}{6} + \frac{C c d^{2} e}{2}\right ) + x^{5} \left (\frac{3 A c d e^{2}}{5} + \frac{B a e^{3}}{5} + \frac{3 B c d^{2} e}{5} + \frac{3 C a d e^{2}}{5} + \frac{C c d^{3}}{5}\right ) + x^{4} \left (\frac{A a e^{3}}{4} + \frac{3 A c d^{2} e}{4} + \frac{3 B a d e^{2}}{4} + \frac{B c d^{3}}{4} + \frac{3 C a d^{2} e}{4}\right ) + x^{3} \left (A a d e^{2} + \frac{A c d^{3}}{3} + B a d^{2} e + \frac{C a d^{3}}{3}\right ) + x^{2} \left (\frac{3 A a d^{2} e}{2} + \frac{B a d^{3}}{2}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.14464, size = 327, normalized size = 1.87 \begin{align*} \frac{1}{8} \, C c x^{8} e^{3} + \frac{3}{7} \, C c d x^{7} e^{2} + \frac{1}{2} \, C c d^{2} x^{6} e + \frac{1}{5} \, C c d^{3} x^{5} + \frac{1}{7} \, B c x^{7} e^{3} + \frac{1}{2} \, B c d x^{6} e^{2} + \frac{3}{5} \, B c d^{2} x^{5} e + \frac{1}{4} \, B c d^{3} x^{4} + \frac{1}{6} \, C a x^{6} e^{3} + \frac{1}{6} \, A c x^{6} e^{3} + \frac{3}{5} \, C a d x^{5} e^{2} + \frac{3}{5} \, A c d x^{5} e^{2} + \frac{3}{4} \, C a d^{2} x^{4} e + \frac{3}{4} \, A c d^{2} x^{4} e + \frac{1}{3} \, C a d^{3} x^{3} + \frac{1}{3} \, A c d^{3} x^{3} + \frac{1}{5} \, B a x^{5} e^{3} + \frac{3}{4} \, B a d x^{4} e^{2} + B a d^{2} x^{3} e + \frac{1}{2} \, B a d^{3} x^{2} + \frac{1}{4} \, A a x^{4} e^{3} + A a d x^{3} e^{2} + \frac{3}{2} \, A a d^{2} x^{2} e + A a d^{3} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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