Optimal. Leaf size=289 \[ \frac{1}{3} a^2 x^3 \left (A \left (a e^2+3 c d^2\right )+a d (2 B e+C d)\right )+a^3 A d^2 x+\frac{1}{2} a^2 c e x^6 (B e+2 C d)+\frac{1}{4} a^3 e x^4 (B e+2 C d)+\frac{1}{9} c^2 x^9 \left (3 a C e^2+c \left (e (A e+2 B d)+C d^2\right )\right )+\frac{1}{7} c x^7 \left (A c \left (3 a e^2+c d^2\right )+3 a \left (a C e^2+c d (2 B e+C d)\right )\right )+\frac{1}{5} a x^5 \left (3 A c \left (a e^2+c d^2\right )+a \left (a C e^2+3 c d (2 B e+C d)\right )\right )+\frac{d \left (a+c x^2\right )^4 (2 A e+B d)}{8 c}+\frac{3}{8} a c^2 e x^8 (B e+2 C d)+\frac{1}{10} c^3 e x^{10} (B e+2 C d)+\frac{1}{11} c^3 C e^2 x^{11} \]
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Rubi [A] time = 0.424141, antiderivative size = 288, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.074, Rules used = {1582, 1810} \[ \frac{1}{3} a^2 x^3 \left (A \left (a e^2+3 c d^2\right )+a d (2 B e+C d)\right )+a^3 A d^2 x+\frac{1}{2} a^2 c e x^6 (B e+2 C d)+\frac{1}{4} a^3 e x^4 (B e+2 C d)+\frac{1}{9} c^2 x^9 \left (3 a C e^2+c e (A e+2 B d)+c C d^2\right )+\frac{1}{7} c x^7 \left (A c \left (3 a e^2+c d^2\right )+3 a \left (a C e^2+c d (2 B e+C d)\right )\right )+\frac{1}{5} a x^5 \left (3 A c \left (a e^2+c d^2\right )+a \left (a C e^2+3 c d (2 B e+C d)\right )\right )+\frac{d \left (a+c x^2\right )^4 (2 A e+B d)}{8 c}+\frac{3}{8} a c^2 e x^8 (B e+2 C d)+\frac{1}{10} c^3 e x^{10} (B e+2 C d)+\frac{1}{11} c^3 C e^2 x^{11} \]
Antiderivative was successfully verified.
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Rule 1582
Rule 1810
Rubi steps
\begin{align*} \int (d+e x)^2 \left (a+c x^2\right )^3 \left (A+B x+C x^2\right ) \, dx &=\frac{d (B d+2 A e) \left (a+c x^2\right )^4}{8 c}+\int \left (a+c x^2\right )^3 \left (-\left (B d^2+2 A d e\right ) x+(d+e x)^2 \left (A+B x+C x^2\right )\right ) \, dx\\ &=\frac{d (B d+2 A e) \left (a+c x^2\right )^4}{8 c}+\int \left (a^3 A d^2+a^2 \left (a d (C d+2 B e)+A \left (3 c d^2+a e^2\right )\right ) x^2+a^3 e (2 C d+B e) x^3+a \left (3 A c \left (c d^2+a e^2\right )+a \left (a C e^2+3 c d (C d+2 B e)\right )\right ) x^4+3 a^2 c e (2 C d+B e) x^5+c \left (A c \left (c d^2+3 a e^2\right )+3 a \left (a C e^2+c d (C d+2 B e)\right )\right ) x^6+3 a c^2 e (2 C d+B e) x^7+c^2 \left (c C d^2+3 a C e^2+c e (2 B d+A e)\right ) x^8+c^3 e (2 C d+B e) x^9+c^3 C e^2 x^{10}\right ) \, dx\\ &=a^3 A d^2 x+\frac{1}{3} a^2 \left (a d (C d+2 B e)+A \left (3 c d^2+a e^2\right )\right ) x^3+\frac{1}{4} a^3 e (2 C d+B e) x^4+\frac{1}{5} a \left (3 A c \left (c d^2+a e^2\right )+a \left (a C e^2+3 c d (C d+2 B e)\right )\right ) x^5+\frac{1}{2} a^2 c e (2 C d+B e) x^6+\frac{1}{7} c \left (A c \left (c d^2+3 a e^2\right )+3 a \left (a C e^2+c d (C d+2 B e)\right )\right ) x^7+\frac{3}{8} a c^2 e (2 C d+B e) x^8+\frac{1}{9} c^2 \left (c C d^2+3 a C e^2+c e (2 B d+A e)\right ) x^9+\frac{1}{10} c^3 e (2 C d+B e) x^{10}+\frac{1}{11} c^3 C e^2 x^{11}+\frac{d (B d+2 A e) \left (a+c x^2\right )^4}{8 c}\\ \end{align*}
Mathematica [A] time = 0.131723, size = 329, normalized size = 1.14 \[ \frac{1}{4} a^2 x^4 \left (a B e^2+2 a C d e+6 A c d e+3 B c d^2\right )+\frac{1}{3} a^2 x^3 \left (A \left (a e^2+3 c d^2\right )+a d (2 B e+C d)\right )+\frac{1}{2} a^3 d x^2 (2 A e+B d)+a^3 A d^2 x+\frac{1}{9} c^2 x^9 \left (3 a C e^2+c e (A e+2 B d)+c C d^2\right )+\frac{1}{8} c^2 x^8 \left (3 a B e^2+6 a C d e+2 A c d e+B c d^2\right )+\frac{1}{7} c x^7 \left (A c \left (3 a e^2+c d^2\right )+3 a \left (a C e^2+c d (2 B e+C d)\right )\right )+\frac{1}{2} a c x^6 \left (2 d e (a C+A c)+B \left (a e^2+c d^2\right )\right )+\frac{1}{5} a x^5 \left (3 A c \left (a e^2+c d^2\right )+a \left (a C e^2+3 c d (2 B e+C d)\right )\right )+\frac{1}{10} c^3 e x^{10} (B e+2 C d)+\frac{1}{11} c^3 C e^2 x^{11} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.044, size = 388, normalized size = 1.3 \begin{align*}{\frac{{c}^{3}C{e}^{2}{x}^{11}}{11}}+{\frac{ \left ({e}^{2}{c}^{3}B+2\,de{c}^{3}C \right ){x}^{10}}{10}}+{\frac{ \left ( \left ( 3\,a{c}^{2}{e}^{2}+{c}^{3}{d}^{2} \right ) C+2\,de{c}^{3}B+{e}^{2}{c}^{3}A \right ){x}^{9}}{9}}+{\frac{ \left ( 6\,a{c}^{2}deC+ \left ( 3\,a{c}^{2}{e}^{2}+{c}^{3}{d}^{2} \right ) B+2\,de{c}^{3}A \right ){x}^{8}}{8}}+{\frac{ \left ( \left ( 3\,{a}^{2}c{e}^{2}+3\,{d}^{2}a{c}^{2} \right ) C+6\,a{c}^{2}deB+ \left ( 3\,a{c}^{2}{e}^{2}+{c}^{3}{d}^{2} \right ) A \right ){x}^{7}}{7}}+{\frac{ \left ( 6\,de{a}^{2}cC+ \left ( 3\,{a}^{2}c{e}^{2}+3\,{d}^{2}a{c}^{2} \right ) B+6\,a{c}^{2}deA \right ){x}^{6}}{6}}+{\frac{ \left ( \left ({e}^{2}{a}^{3}+3\,{a}^{2}c{d}^{2} \right ) C+6\,de{a}^{2}cB+ \left ( 3\,{a}^{2}c{e}^{2}+3\,{d}^{2}a{c}^{2} \right ) A \right ){x}^{5}}{5}}+{\frac{ \left ( 2\,de{a}^{3}C+ \left ({e}^{2}{a}^{3}+3\,{a}^{2}c{d}^{2} \right ) B+6\,de{a}^{2}cA \right ){x}^{4}}{4}}+{\frac{ \left ({a}^{3}{d}^{2}C+2\,de{a}^{3}B+ \left ({e}^{2}{a}^{3}+3\,{a}^{2}c{d}^{2} \right ) A \right ){x}^{3}}{3}}+{\frac{ \left ( 2\,de{a}^{3}A+{a}^{3}{d}^{2}B \right ){x}^{2}}{2}}+{a}^{3}A{d}^{2}x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.00264, size = 495, normalized size = 1.71 \begin{align*} \frac{1}{11} \, C c^{3} e^{2} x^{11} + \frac{1}{10} \,{\left (2 \, C c^{3} d e + B c^{3} e^{2}\right )} x^{10} + \frac{1}{9} \,{\left (C c^{3} d^{2} + 2 \, B c^{3} d e +{\left (3 \, C a c^{2} + A c^{3}\right )} e^{2}\right )} x^{9} + \frac{1}{8} \,{\left (B c^{3} d^{2} + 3 \, B a c^{2} e^{2} + 2 \,{\left (3 \, C a c^{2} + A c^{3}\right )} d e\right )} x^{8} + \frac{1}{7} \,{\left (6 \, B a c^{2} d e +{\left (3 \, C a c^{2} + A c^{3}\right )} d^{2} + 3 \,{\left (C a^{2} c + A a c^{2}\right )} e^{2}\right )} x^{7} + A a^{3} d^{2} x + \frac{1}{2} \,{\left (B a c^{2} d^{2} + B a^{2} c e^{2} + 2 \,{\left (C a^{2} c + A a c^{2}\right )} d e\right )} x^{6} + \frac{1}{5} \,{\left (6 \, B a^{2} c d e + 3 \,{\left (C a^{2} c + A a c^{2}\right )} d^{2} +{\left (C a^{3} + 3 \, A a^{2} c\right )} e^{2}\right )} x^{5} + \frac{1}{4} \,{\left (3 \, B a^{2} c d^{2} + B a^{3} e^{2} + 2 \,{\left (C a^{3} + 3 \, A a^{2} c\right )} d e\right )} x^{4} + \frac{1}{3} \,{\left (2 \, B a^{3} d e + A a^{3} e^{2} +{\left (C a^{3} + 3 \, A a^{2} c\right )} d^{2}\right )} x^{3} + \frac{1}{2} \,{\left (B a^{3} d^{2} + 2 \, A a^{3} d e\right )} x^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.39606, size = 996, normalized size = 3.45 \begin{align*} \frac{1}{11} x^{11} e^{2} c^{3} C + \frac{1}{5} x^{10} e d c^{3} C + \frac{1}{10} x^{10} e^{2} c^{3} B + \frac{1}{9} x^{9} d^{2} c^{3} C + \frac{1}{3} x^{9} e^{2} c^{2} a C + \frac{2}{9} x^{9} e d c^{3} B + \frac{1}{9} x^{9} e^{2} c^{3} A + \frac{3}{4} x^{8} e d c^{2} a C + \frac{1}{8} x^{8} d^{2} c^{3} B + \frac{3}{8} x^{8} e^{2} c^{2} a B + \frac{1}{4} x^{8} e d c^{3} A + \frac{3}{7} x^{7} d^{2} c^{2} a C + \frac{3}{7} x^{7} e^{2} c a^{2} C + \frac{6}{7} x^{7} e d c^{2} a B + \frac{1}{7} x^{7} d^{2} c^{3} A + \frac{3}{7} x^{7} e^{2} c^{2} a A + x^{6} e d c a^{2} C + \frac{1}{2} x^{6} d^{2} c^{2} a B + \frac{1}{2} x^{6} e^{2} c a^{2} B + x^{6} e d c^{2} a A + \frac{3}{5} x^{5} d^{2} c a^{2} C + \frac{1}{5} x^{5} e^{2} a^{3} C + \frac{6}{5} x^{5} e d c a^{2} B + \frac{3}{5} x^{5} d^{2} c^{2} a A + \frac{3}{5} x^{5} e^{2} c a^{2} A + \frac{1}{2} x^{4} e d a^{3} C + \frac{3}{4} x^{4} d^{2} c a^{2} B + \frac{1}{4} x^{4} e^{2} a^{3} B + \frac{3}{2} x^{4} e d c a^{2} A + \frac{1}{3} x^{3} d^{2} a^{3} C + \frac{2}{3} x^{3} e d a^{3} B + x^{3} d^{2} c a^{2} A + \frac{1}{3} x^{3} e^{2} a^{3} A + \frac{1}{2} x^{2} d^{2} a^{3} B + x^{2} e d a^{3} A + x d^{2} a^{3} A \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.121491, size = 447, normalized size = 1.55 \begin{align*} A a^{3} d^{2} x + \frac{C c^{3} e^{2} x^{11}}{11} + x^{10} \left (\frac{B c^{3} e^{2}}{10} + \frac{C c^{3} d e}{5}\right ) + x^{9} \left (\frac{A c^{3} e^{2}}{9} + \frac{2 B c^{3} d e}{9} + \frac{C a c^{2} e^{2}}{3} + \frac{C c^{3} d^{2}}{9}\right ) + x^{8} \left (\frac{A c^{3} d e}{4} + \frac{3 B a c^{2} e^{2}}{8} + \frac{B c^{3} d^{2}}{8} + \frac{3 C a c^{2} d e}{4}\right ) + x^{7} \left (\frac{3 A a c^{2} e^{2}}{7} + \frac{A c^{3} d^{2}}{7} + \frac{6 B a c^{2} d e}{7} + \frac{3 C a^{2} c e^{2}}{7} + \frac{3 C a c^{2} d^{2}}{7}\right ) + x^{6} \left (A a c^{2} d e + \frac{B a^{2} c e^{2}}{2} + \frac{B a c^{2} d^{2}}{2} + C a^{2} c d e\right ) + x^{5} \left (\frac{3 A a^{2} c e^{2}}{5} + \frac{3 A a c^{2} d^{2}}{5} + \frac{6 B a^{2} c d e}{5} + \frac{C a^{3} e^{2}}{5} + \frac{3 C a^{2} c d^{2}}{5}\right ) + x^{4} \left (\frac{3 A a^{2} c d e}{2} + \frac{B a^{3} e^{2}}{4} + \frac{3 B a^{2} c d^{2}}{4} + \frac{C a^{3} d e}{2}\right ) + x^{3} \left (\frac{A a^{3} e^{2}}{3} + A a^{2} c d^{2} + \frac{2 B a^{3} d e}{3} + \frac{C a^{3} d^{2}}{3}\right ) + x^{2} \left (A a^{3} d e + \frac{B a^{3} d^{2}}{2}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.16747, size = 583, normalized size = 2.02 \begin{align*} \frac{1}{11} \, C c^{3} x^{11} e^{2} + \frac{1}{5} \, C c^{3} d x^{10} e + \frac{1}{9} \, C c^{3} d^{2} x^{9} + \frac{1}{10} \, B c^{3} x^{10} e^{2} + \frac{2}{9} \, B c^{3} d x^{9} e + \frac{1}{8} \, B c^{3} d^{2} x^{8} + \frac{1}{3} \, C a c^{2} x^{9} e^{2} + \frac{1}{9} \, A c^{3} x^{9} e^{2} + \frac{3}{4} \, C a c^{2} d x^{8} e + \frac{1}{4} \, A c^{3} d x^{8} e + \frac{3}{7} \, C a c^{2} d^{2} x^{7} + \frac{1}{7} \, A c^{3} d^{2} x^{7} + \frac{3}{8} \, B a c^{2} x^{8} e^{2} + \frac{6}{7} \, B a c^{2} d x^{7} e + \frac{1}{2} \, B a c^{2} d^{2} x^{6} + \frac{3}{7} \, C a^{2} c x^{7} e^{2} + \frac{3}{7} \, A a c^{2} x^{7} e^{2} + C a^{2} c d x^{6} e + A a c^{2} d x^{6} e + \frac{3}{5} \, C a^{2} c d^{2} x^{5} + \frac{3}{5} \, A a c^{2} d^{2} x^{5} + \frac{1}{2} \, B a^{2} c x^{6} e^{2} + \frac{6}{5} \, B a^{2} c d x^{5} e + \frac{3}{4} \, B a^{2} c d^{2} x^{4} + \frac{1}{5} \, C a^{3} x^{5} e^{2} + \frac{3}{5} \, A a^{2} c x^{5} e^{2} + \frac{1}{2} \, C a^{3} d x^{4} e + \frac{3}{2} \, A a^{2} c d x^{4} e + \frac{1}{3} \, C a^{3} d^{2} x^{3} + A a^{2} c d^{2} x^{3} + \frac{1}{4} \, B a^{3} x^{4} e^{2} + \frac{2}{3} \, B a^{3} d x^{3} e + \frac{1}{2} \, B a^{3} d^{2} x^{2} + \frac{1}{3} \, A a^{3} x^{3} e^{2} + A a^{3} d x^{2} e + A a^{3} d^{2} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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