Optimal. Leaf size=404 \[ \frac{1}{6} a^2 e x^6 \left (a C e^2+3 c \left (e (A e+3 B d)+3 C d^2\right )\right )+\frac{1}{3} a^2 d x^3 \left (3 A \left (a e^2+c d^2\right )+a d (3 B e+C d)\right )+\frac{1}{4} a^3 e x^4 \left (e (A e+3 B d)+3 C d^2\right )+a^3 A d^3 x+\frac{1}{10} c^2 e x^{10} \left (3 a C e^2+c \left (e (A e+3 B d)+3 C d^2\right )\right )+\frac{1}{9} c^2 x^9 \left (3 a e^2 (B e+3 C d)+c d \left (3 e (A e+B d)+C d^2\right )\right )+\frac{3}{8} a c e x^8 \left (a C e^2+c \left (e (A e+3 B d)+3 C d^2\right )\right )+\frac{1}{7} c x^7 \left (A c d \left (9 a e^2+c d^2\right )+3 a \left (a e^2 (B e+3 C d)+c d^2 (3 B e+C d)\right )\right )+\frac{1}{5} a x^5 \left (3 A c d \left (3 a e^2+c d^2\right )+a \left (a e^2 (B e+3 C d)+3 c d^2 (3 B e+C d)\right )\right )+\frac{d^2 \left (a+c x^2\right )^4 (3 A e+B d)}{8 c}+\frac{1}{11} c^3 e^2 x^{11} (B e+3 C d)+\frac{1}{12} c^3 C e^3 x^{12} \]
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Rubi [A] time = 0.691394, antiderivative size = 400, normalized size of antiderivative = 0.99, 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} \[ \frac{1}{6} a^2 e x^6 \left (a C e^2+3 c e (A e+3 B d)+9 c C d^2\right )+\frac{1}{3} a^2 d x^3 \left (3 A \left (a e^2+c d^2\right )+a d (3 B e+C d)\right )+\frac{1}{4} a^3 e x^4 \left (e (A e+3 B d)+3 C d^2\right )+a^3 A d^3 x+\frac{1}{10} c^2 e x^{10} \left (3 a C e^2+c e (A e+3 B d)+3 c C d^2\right )+\frac{1}{9} c^2 x^9 \left (3 a e^2 (B e+3 C d)+3 c d e (A e+B d)+c C d^3\right )+\frac{3}{8} a c e x^8 \left (a C e^2+c e (A e+3 B d)+3 c C d^2\right )+\frac{1}{7} c x^7 \left (A c d \left (9 a e^2+c d^2\right )+3 a \left (a e^2 (B e+3 C d)+c d^2 (3 B e+C d)\right )\right )+\frac{1}{5} a x^5 \left (3 A c d \left (3 a e^2+c d^2\right )+a \left (a e^2 (B e+3 C d)+3 c d^2 (3 B e+C d)\right )\right )+\frac{d^2 \left (a+c x^2\right )^4 (3 A e+B d)}{8 c}+\frac{1}{11} c^3 e^2 x^{11} (B e+3 C d)+\frac{1}{12} c^3 C e^3 x^{12} \]
Antiderivative was successfully verified.
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Rule 1582
Rule 1810
Rubi steps
\begin{align*} \int (d+e x)^3 \left (a+c x^2\right )^3 \left (A+B x+C x^2\right ) \, dx &=\frac{d^2 (B d+3 A e) \left (a+c x^2\right )^4}{8 c}+\int \left (a+c x^2\right )^3 \left (-\left (B d^3+3 A d^2 e\right ) x+(d+e x)^3 \left (A+B x+C x^2\right )\right ) \, dx\\ &=\frac{d^2 (B d+3 A e) \left (a+c x^2\right )^4}{8 c}+\int \left (a^3 A d^3+a^2 d \left (a d (C d+3 B e)+3 A \left (c d^2+a e^2\right )\right ) x^2+a^3 e \left (3 C d^2+e (3 B d+A e)\right ) x^3+a \left (3 A c d \left (c d^2+3 a e^2\right )+a \left (a e^2 (3 C d+B e)+3 c d^2 (C d+3 B e)\right )\right ) x^4+a^2 e \left (9 c C d^2+a C e^2+3 c e (3 B d+A e)\right ) x^5+c \left (A c d \left (c d^2+9 a e^2\right )+3 a \left (a e^2 (3 C d+B e)+c d^2 (C d+3 B e)\right )\right ) x^6+3 a c e \left (3 c C d^2+a C e^2+c e (3 B d+A e)\right ) x^7+c^2 \left (c C d^3+3 c d e (B d+A e)+3 a e^2 (3 C d+B e)\right ) x^8+c^2 e \left (3 c C d^2+3 a C e^2+c e (3 B d+A e)\right ) x^9+c^3 e^2 (3 C d+B e) x^{10}+c^3 C e^3 x^{11}\right ) \, dx\\ &=a^3 A d^3 x+\frac{1}{3} a^2 d \left (a d (C d+3 B e)+3 A \left (c d^2+a e^2\right )\right ) x^3+\frac{1}{4} a^3 e \left (3 C d^2+e (3 B d+A e)\right ) x^4+\frac{1}{5} a \left (3 A c d \left (c d^2+3 a e^2\right )+a \left (a e^2 (3 C d+B e)+3 c d^2 (C d+3 B e)\right )\right ) x^5+\frac{1}{6} a^2 e \left (9 c C d^2+a C e^2+3 c e (3 B d+A e)\right ) x^6+\frac{1}{7} c \left (A c d \left (c d^2+9 a e^2\right )+3 a \left (a e^2 (3 C d+B e)+c d^2 (C d+3 B e)\right )\right ) x^7+\frac{3}{8} a c e \left (3 c C d^2+a C e^2+c e (3 B d+A e)\right ) x^8+\frac{1}{9} c^2 \left (c C d^3+3 c d e (B d+A e)+3 a e^2 (3 C d+B e)\right ) x^9+\frac{1}{10} c^2 e \left (3 c C d^2+3 a C e^2+c e (3 B d+A e)\right ) x^{10}+\frac{1}{11} c^3 e^2 (3 C d+B e) x^{11}+\frac{1}{12} c^3 C e^3 x^{12}+\frac{d^2 (B d+3 A e) \left (a+c x^2\right )^4}{8 c}\\ \end{align*}
Mathematica [A] time = 0.204747, size = 459, normalized size = 1.14 \[ \frac{1}{4} a^2 x^4 \left (a A e^3+3 a B d e^2+3 a C d^2 e+9 A c d^2 e+3 B c d^3\right )+\frac{1}{3} a^2 d x^3 \left (3 A \left (a e^2+c d^2\right )+a d (3 B e+C d)\right )+\frac{1}{2} a^3 d^2 x^2 (3 A e+B d)+a^3 A d^3 x+\frac{1}{10} c^2 e x^{10} \left (3 a C e^2+c e (A e+3 B d)+3 c C d^2\right )+\frac{1}{9} c^2 x^9 \left (3 a e^2 (B e+3 C d)+3 c d e (A e+B d)+c C d^3\right )+\frac{1}{8} c x^8 \left (3 e \left (A c \left (a e^2+c d^2\right )+a C \left (a e^2+3 c d^2\right )\right )+B c d \left (9 a e^2+c d^2\right )\right )+\frac{1}{7} c x^7 \left (A c d \left (9 a e^2+c d^2\right )+3 a \left (a e^2 (B e+3 C d)+c d^2 (3 B e+C d)\right )\right )+\frac{1}{6} a x^6 \left (3 A c e \left (a e^2+3 c d^2\right )+3 B c d \left (3 a e^2+c d^2\right )+a C e \left (a e^2+9 c d^2\right )\right )+\frac{1}{5} a x^5 \left (3 A c d \left (3 a e^2+c d^2\right )+a \left (a e^2 (B e+3 C d)+3 c d^2 (3 B e+C d)\right )\right )+\frac{1}{11} c^3 e^2 x^{11} (B e+3 C d)+\frac{1}{12} c^3 C e^3 x^{12} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.046, size = 553, normalized size = 1.4 \begin{align*}{\frac{{c}^{3}C{e}^{3}{x}^{12}}{12}}+{\frac{ \left ({e}^{3}{c}^{3}B+3\,d{e}^{2}{c}^{3}C \right ){x}^{11}}{11}}+{\frac{ \left ( \left ( 3\,{e}^{3}a{c}^{2}+3\,{d}^{2}e{c}^{3} \right ) C+3\,d{e}^{2}{c}^{3}B+{e}^{3}{c}^{3}A \right ){x}^{10}}{10}}+{\frac{ \left ( \left ( 9\,{c}^{2}ad{e}^{2}+{c}^{3}{d}^{3} \right ) C+ \left ( 3\,{e}^{3}a{c}^{2}+3\,{d}^{2}e{c}^{3} \right ) B+3\,d{e}^{2}{c}^{3}A \right ){x}^{9}}{9}}+{\frac{ \left ( \left ( 3\,{e}^{3}{a}^{2}c+9\,a{c}^{2}{d}^{2}e \right ) C+ \left ( 9\,{c}^{2}ad{e}^{2}+{c}^{3}{d}^{3} \right ) B+ \left ( 3\,{e}^{3}a{c}^{2}+3\,{d}^{2}e{c}^{3} \right ) A \right ){x}^{8}}{8}}+{\frac{ \left ( \left ( 9\,{a}^{2}cd{e}^{2}+3\,{d}^{3}a{c}^{2} \right ) C+ \left ( 3\,{e}^{3}{a}^{2}c+9\,a{c}^{2}{d}^{2}e \right ) B+ \left ( 9\,{c}^{2}ad{e}^{2}+{c}^{3}{d}^{3} \right ) A \right ){x}^{7}}{7}}+{\frac{ \left ( \left ({e}^{3}{a}^{3}+9\,{d}^{2}e{a}^{2}c \right ) C+ \left ( 9\,{a}^{2}cd{e}^{2}+3\,{d}^{3}a{c}^{2} \right ) B+ \left ( 3\,{e}^{3}{a}^{2}c+9\,a{c}^{2}{d}^{2}e \right ) A \right ){x}^{6}}{6}}+{\frac{ \left ( \left ( 3\,d{e}^{2}{a}^{3}+3\,{d}^{3}{a}^{2}c \right ) C+ \left ({e}^{3}{a}^{3}+9\,{d}^{2}e{a}^{2}c \right ) B+ \left ( 9\,{a}^{2}cd{e}^{2}+3\,{d}^{3}a{c}^{2} \right ) A \right ){x}^{5}}{5}}+{\frac{ \left ( 3\,{d}^{2}e{a}^{3}C+ \left ( 3\,d{e}^{2}{a}^{3}+3\,{d}^{3}{a}^{2}c \right ) B+ \left ({e}^{3}{a}^{3}+9\,{d}^{2}e{a}^{2}c \right ) A \right ){x}^{4}}{4}}+{\frac{ \left ({a}^{3}{d}^{3}C+3\,{d}^{2}e{a}^{3}B+ \left ( 3\,d{e}^{2}{a}^{3}+3\,{d}^{3}{a}^{2}c \right ) A \right ){x}^{3}}{3}}+{\frac{ \left ( 3\,{d}^{2}e{a}^{3}A+{a}^{3}{d}^{3}B \right ){x}^{2}}{2}}+{a}^{3}A{d}^{3}x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.0111, size = 691, normalized size = 1.71 \begin{align*} \frac{1}{12} \, C c^{3} e^{3} x^{12} + \frac{1}{11} \,{\left (3 \, C c^{3} d e^{2} + B c^{3} e^{3}\right )} x^{11} + \frac{1}{10} \,{\left (3 \, C c^{3} d^{2} e + 3 \, B c^{3} d e^{2} +{\left (3 \, C a c^{2} + A c^{3}\right )} e^{3}\right )} x^{10} + \frac{1}{9} \,{\left (C c^{3} d^{3} + 3 \, B c^{3} d^{2} e + 3 \, B a c^{2} e^{3} + 3 \,{\left (3 \, C a c^{2} + A c^{3}\right )} d e^{2}\right )} x^{9} + \frac{1}{8} \,{\left (B c^{3} d^{3} + 9 \, B a c^{2} d e^{2} + 3 \,{\left (3 \, C a c^{2} + A c^{3}\right )} d^{2} e + 3 \,{\left (C a^{2} c + A a c^{2}\right )} e^{3}\right )} x^{8} + A a^{3} d^{3} x + \frac{1}{7} \,{\left (9 \, B a c^{2} d^{2} e + 3 \, B a^{2} c e^{3} +{\left (3 \, C a c^{2} + A c^{3}\right )} d^{3} + 9 \,{\left (C a^{2} c + A a c^{2}\right )} d e^{2}\right )} x^{7} + \frac{1}{6} \,{\left (3 \, B a c^{2} d^{3} + 9 \, B a^{2} c d e^{2} + 9 \,{\left (C a^{2} c + A a c^{2}\right )} d^{2} e +{\left (C a^{3} + 3 \, A a^{2} c\right )} e^{3}\right )} x^{6} + \frac{1}{5} \,{\left (9 \, B a^{2} c d^{2} e + B a^{3} e^{3} + 3 \,{\left (C a^{2} c + A a c^{2}\right )} d^{3} + 3 \,{\left (C a^{3} + 3 \, A a^{2} c\right )} d e^{2}\right )} x^{5} + \frac{1}{4} \,{\left (3 \, B a^{2} c d^{3} + 3 \, B a^{3} d e^{2} + A a^{3} e^{3} + 3 \,{\left (C a^{3} + 3 \, A a^{2} c\right )} d^{2} e\right )} x^{4} + \frac{1}{3} \,{\left (3 \, B a^{3} d^{2} e + 3 \, A a^{3} d e^{2} +{\left (C a^{3} + 3 \, A a^{2} c\right )} d^{3}\right )} x^{3} + \frac{1}{2} \,{\left (B a^{3} d^{3} + 3 \, A a^{3} 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 = 1.43126, size = 1413, normalized size = 3.5 \begin{align*} \frac{1}{12} x^{12} e^{3} c^{3} C + \frac{3}{11} x^{11} e^{2} d c^{3} C + \frac{1}{11} x^{11} e^{3} c^{3} B + \frac{3}{10} x^{10} e d^{2} c^{3} C + \frac{3}{10} x^{10} e^{3} c^{2} a C + \frac{3}{10} x^{10} e^{2} d c^{3} B + \frac{1}{10} x^{10} e^{3} c^{3} A + \frac{1}{9} x^{9} d^{3} c^{3} C + x^{9} e^{2} d c^{2} a C + \frac{1}{3} x^{9} e d^{2} c^{3} B + \frac{1}{3} x^{9} e^{3} c^{2} a B + \frac{1}{3} x^{9} e^{2} d c^{3} A + \frac{9}{8} x^{8} e d^{2} c^{2} a C + \frac{3}{8} x^{8} e^{3} c a^{2} C + \frac{1}{8} x^{8} d^{3} c^{3} B + \frac{9}{8} x^{8} e^{2} d c^{2} a B + \frac{3}{8} x^{8} e d^{2} c^{3} A + \frac{3}{8} x^{8} e^{3} c^{2} a A + \frac{3}{7} x^{7} d^{3} c^{2} a C + \frac{9}{7} x^{7} e^{2} d c a^{2} C + \frac{9}{7} x^{7} e d^{2} c^{2} a B + \frac{3}{7} x^{7} e^{3} c a^{2} B + \frac{1}{7} x^{7} d^{3} c^{3} A + \frac{9}{7} x^{7} e^{2} d c^{2} a A + \frac{3}{2} x^{6} e d^{2} c a^{2} C + \frac{1}{6} x^{6} e^{3} a^{3} C + \frac{1}{2} x^{6} d^{3} c^{2} a B + \frac{3}{2} x^{6} e^{2} d c a^{2} B + \frac{3}{2} x^{6} e d^{2} c^{2} a A + \frac{1}{2} x^{6} e^{3} c a^{2} A + \frac{3}{5} x^{5} d^{3} c a^{2} C + \frac{3}{5} x^{5} e^{2} d a^{3} C + \frac{9}{5} x^{5} e d^{2} c a^{2} B + \frac{1}{5} x^{5} e^{3} a^{3} B + \frac{3}{5} x^{5} d^{3} c^{2} a A + \frac{9}{5} x^{5} e^{2} d c a^{2} A + \frac{3}{4} x^{4} e d^{2} a^{3} C + \frac{3}{4} x^{4} d^{3} c a^{2} B + \frac{3}{4} x^{4} e^{2} d a^{3} B + \frac{9}{4} x^{4} e d^{2} c a^{2} A + \frac{1}{4} x^{4} e^{3} a^{3} A + \frac{1}{3} x^{3} d^{3} a^{3} C + x^{3} e d^{2} a^{3} B + x^{3} d^{3} c a^{2} A + x^{3} e^{2} d a^{3} A + \frac{1}{2} x^{2} d^{3} a^{3} B + \frac{3}{2} x^{2} e d^{2} a^{3} A + x d^{3} a^{3} A \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.136789, size = 646, normalized size = 1.6 \begin{align*} A a^{3} d^{3} x + \frac{C c^{3} e^{3} x^{12}}{12} + x^{11} \left (\frac{B c^{3} e^{3}}{11} + \frac{3 C c^{3} d e^{2}}{11}\right ) + x^{10} \left (\frac{A c^{3} e^{3}}{10} + \frac{3 B c^{3} d e^{2}}{10} + \frac{3 C a c^{2} e^{3}}{10} + \frac{3 C c^{3} d^{2} e}{10}\right ) + x^{9} \left (\frac{A c^{3} d e^{2}}{3} + \frac{B a c^{2} e^{3}}{3} + \frac{B c^{3} d^{2} e}{3} + C a c^{2} d e^{2} + \frac{C c^{3} d^{3}}{9}\right ) + x^{8} \left (\frac{3 A a c^{2} e^{3}}{8} + \frac{3 A c^{3} d^{2} e}{8} + \frac{9 B a c^{2} d e^{2}}{8} + \frac{B c^{3} d^{3}}{8} + \frac{3 C a^{2} c e^{3}}{8} + \frac{9 C a c^{2} d^{2} e}{8}\right ) + x^{7} \left (\frac{9 A a c^{2} d e^{2}}{7} + \frac{A c^{3} d^{3}}{7} + \frac{3 B a^{2} c e^{3}}{7} + \frac{9 B a c^{2} d^{2} e}{7} + \frac{9 C a^{2} c d e^{2}}{7} + \frac{3 C a c^{2} d^{3}}{7}\right ) + x^{6} \left (\frac{A a^{2} c e^{3}}{2} + \frac{3 A a c^{2} d^{2} e}{2} + \frac{3 B a^{2} c d e^{2}}{2} + \frac{B a c^{2} d^{3}}{2} + \frac{C a^{3} e^{3}}{6} + \frac{3 C a^{2} c d^{2} e}{2}\right ) + x^{5} \left (\frac{9 A a^{2} c d e^{2}}{5} + \frac{3 A a c^{2} d^{3}}{5} + \frac{B a^{3} e^{3}}{5} + \frac{9 B a^{2} c d^{2} e}{5} + \frac{3 C a^{3} d e^{2}}{5} + \frac{3 C a^{2} c d^{3}}{5}\right ) + x^{4} \left (\frac{A a^{3} e^{3}}{4} + \frac{9 A a^{2} c d^{2} e}{4} + \frac{3 B a^{3} d e^{2}}{4} + \frac{3 B a^{2} c d^{3}}{4} + \frac{3 C a^{3} d^{2} e}{4}\right ) + x^{3} \left (A a^{3} d e^{2} + A a^{2} c d^{3} + B a^{3} d^{2} e + \frac{C a^{3} d^{3}}{3}\right ) + x^{2} \left (\frac{3 A a^{3} d^{2} e}{2} + \frac{B a^{3} d^{3}}{2}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.14244, size = 818, normalized size = 2.02 \begin{align*} \frac{1}{12} \, C c^{3} x^{12} e^{3} + \frac{3}{11} \, C c^{3} d x^{11} e^{2} + \frac{3}{10} \, C c^{3} d^{2} x^{10} e + \frac{1}{9} \, C c^{3} d^{3} x^{9} + \frac{1}{11} \, B c^{3} x^{11} e^{3} + \frac{3}{10} \, B c^{3} d x^{10} e^{2} + \frac{1}{3} \, B c^{3} d^{2} x^{9} e + \frac{1}{8} \, B c^{3} d^{3} x^{8} + \frac{3}{10} \, C a c^{2} x^{10} e^{3} + \frac{1}{10} \, A c^{3} x^{10} e^{3} + C a c^{2} d x^{9} e^{2} + \frac{1}{3} \, A c^{3} d x^{9} e^{2} + \frac{9}{8} \, C a c^{2} d^{2} x^{8} e + \frac{3}{8} \, A c^{3} d^{2} x^{8} e + \frac{3}{7} \, C a c^{2} d^{3} x^{7} + \frac{1}{7} \, A c^{3} d^{3} x^{7} + \frac{1}{3} \, B a c^{2} x^{9} e^{3} + \frac{9}{8} \, B a c^{2} d x^{8} e^{2} + \frac{9}{7} \, B a c^{2} d^{2} x^{7} e + \frac{1}{2} \, B a c^{2} d^{3} x^{6} + \frac{3}{8} \, C a^{2} c x^{8} e^{3} + \frac{3}{8} \, A a c^{2} x^{8} e^{3} + \frac{9}{7} \, C a^{2} c d x^{7} e^{2} + \frac{9}{7} \, A a c^{2} d x^{7} e^{2} + \frac{3}{2} \, C a^{2} c d^{2} x^{6} e + \frac{3}{2} \, A a c^{2} d^{2} x^{6} e + \frac{3}{5} \, C a^{2} c d^{3} x^{5} + \frac{3}{5} \, A a c^{2} d^{3} x^{5} + \frac{3}{7} \, B a^{2} c x^{7} e^{3} + \frac{3}{2} \, B a^{2} c d x^{6} e^{2} + \frac{9}{5} \, B a^{2} c d^{2} x^{5} e + \frac{3}{4} \, B a^{2} c d^{3} x^{4} + \frac{1}{6} \, C a^{3} x^{6} e^{3} + \frac{1}{2} \, A a^{2} c x^{6} e^{3} + \frac{3}{5} \, C a^{3} d x^{5} e^{2} + \frac{9}{5} \, A a^{2} c d x^{5} e^{2} + \frac{3}{4} \, C a^{3} d^{2} x^{4} e + \frac{9}{4} \, A a^{2} c d^{2} x^{4} e + \frac{1}{3} \, C a^{3} d^{3} x^{3} + A a^{2} c d^{3} x^{3} + \frac{1}{5} \, B a^{3} x^{5} e^{3} + \frac{3}{4} \, B a^{3} d x^{4} e^{2} + B a^{3} d^{2} x^{3} e + \frac{1}{2} \, B a^{3} d^{3} x^{2} + \frac{1}{4} \, A a^{3} x^{4} e^{3} + A a^{3} d x^{3} e^{2} + \frac{3}{2} \, A a^{3} d^{2} x^{2} e + A a^{3} d^{3} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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