Optimal. Leaf size=466 \[ \frac{c x^2 \left (3 a^2 C e^4+3 a c e^2 \left (6 C d^2-e (3 B d-A e)\right )+c^2 d^2 \left (15 C d^2-2 e (5 B d-3 A e)\right )\right )}{2 e^7}-\frac{c x \left (3 a^2 e^4 (3 C d-B e)+3 a c d e^2 \left (10 C d^2-3 e (2 B d-A e)\right )+c^2 d^3 \left (21 C d^2-5 e (3 B d-2 A e)\right )\right )}{e^8}+\frac{\left (a e^2+c d^2\right ) \log (d+e x) \left (a^2 C e^4+a c e^2 \left (17 C d^2-3 e (3 B d-A e)\right )+c^2 d^2 \left (28 C d^2-3 e (7 B d-5 A e)\right )\right )}{e^9}+\frac{c^2 x^4 \left (3 a C e^2+c \left (6 C d^2-e (3 B d-A e)\right )\right )}{4 e^5}-\frac{c^2 x^3 \left (3 a e^2 (3 C d-B e)+c d \left (10 C d^2-3 e (2 B d-A e)\right )\right )}{3 e^6}+\frac{\left (a e^2+c d^2\right )^2 \left (a e^2 (2 C d-B e)+c d \left (8 C d^2-e (7 B d-6 A e)\right )\right )}{e^9 (d+e x)}-\frac{\left (a e^2+c d^2\right )^3 \left (A e^2-B d e+C d^2\right )}{2 e^9 (d+e x)^2}-\frac{c^3 x^5 (3 C d-B e)}{5 e^4}+\frac{c^3 C x^6}{6 e^3} \]
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Rubi [A] time = 0.966645, antiderivative size = 463, normalized size of antiderivative = 0.99, number of steps used = 2, number of rules used = 1, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.037, Rules used = {1628} \[ \frac{c x^2 \left (3 a^2 C e^4+3 a c e^2 \left (6 C d^2-e (3 B d-A e)\right )+c^2 \left (15 C d^4-2 d^2 e (5 B d-3 A e)\right )\right )}{2 e^7}-\frac{c x \left (3 a^2 e^4 (3 C d-B e)+3 a c d e^2 \left (10 C d^2-3 e (2 B d-A e)\right )+c^2 \left (21 C d^5-5 d^3 e (3 B d-2 A e)\right )\right )}{e^8}+\frac{\left (a e^2+c d^2\right ) \log (d+e x) \left (a^2 C e^4+a c e^2 \left (17 C d^2-3 e (3 B d-A e)\right )+c^2 \left (28 C d^4-3 d^2 e (7 B d-5 A e)\right )\right )}{e^9}+\frac{c^2 x^4 \left (3 a C e^2-c e (3 B d-A e)+6 c C d^2\right )}{4 e^5}-\frac{c^2 x^3 \left (3 a e^2 (3 C d-B e)-3 c d e (2 B d-A e)+10 c C d^3\right )}{3 e^6}+\frac{\left (a e^2+c d^2\right )^2 \left (a e^2 (2 C d-B e)-c d e (7 B d-6 A e)+8 c C d^3\right )}{e^9 (d+e x)}-\frac{\left (a e^2+c d^2\right )^3 \left (A e^2-B d e+C d^2\right )}{2 e^9 (d+e x)^2}-\frac{c^3 x^5 (3 C d-B e)}{5 e^4}+\frac{c^3 C x^6}{6 e^3} \]
Antiderivative was successfully verified.
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Rule 1628
Rubi steps
\begin{align*} \int \frac{\left (a+c x^2\right )^3 \left (A+B x+C x^2\right )}{(d+e x)^3} \, dx &=\int \left (\frac{c \left (-3 a^2 e^4 (3 C d-B e)-c^2 \left (21 C d^5-5 d^3 e (3 B d-2 A e)\right )-3 a c d e^2 \left (10 C d^2-3 e (2 B d-A e)\right )\right )}{e^8}+\frac{c \left (3 a^2 C e^4+c^2 \left (15 C d^4-2 d^2 e (5 B d-3 A e)\right )+3 a c e^2 \left (6 C d^2-e (3 B d-A e)\right )\right ) x}{e^7}+\frac{c^2 \left (-10 c C d^3+3 c d e (2 B d-A e)-3 a e^2 (3 C d-B e)\right ) x^2}{e^6}+\frac{c^2 \left (6 c C d^2+3 a C e^2-c e (3 B d-A e)\right ) x^3}{e^5}+\frac{c^3 (-3 C d+B e) x^4}{e^4}+\frac{c^3 C x^5}{e^3}+\frac{\left (c d^2+a e^2\right )^3 \left (C d^2-B d e+A e^2\right )}{e^8 (d+e x)^3}+\frac{\left (c d^2+a e^2\right )^2 \left (-8 c C d^3+c d e (7 B d-6 A e)-a e^2 (2 C d-B e)\right )}{e^8 (d+e x)^2}+\frac{\left (c d^2+a e^2\right ) \left (a^2 C e^4+c^2 \left (28 C d^4-3 d^2 e (7 B d-5 A e)\right )+a c e^2 \left (17 C d^2-3 e (3 B d-A e)\right )\right )}{e^8 (d+e x)}\right ) \, dx\\ &=-\frac{c \left (3 a^2 e^4 (3 C d-B e)+c^2 \left (21 C d^5-5 d^3 e (3 B d-2 A e)\right )+3 a c d e^2 \left (10 C d^2-3 e (2 B d-A e)\right )\right ) x}{e^8}+\frac{c \left (3 a^2 C e^4+c^2 \left (15 C d^4-2 d^2 e (5 B d-3 A e)\right )+3 a c e^2 \left (6 C d^2-e (3 B d-A e)\right )\right ) x^2}{2 e^7}-\frac{c^2 \left (10 c C d^3-3 c d e (2 B d-A e)+3 a e^2 (3 C d-B e)\right ) x^3}{3 e^6}+\frac{c^2 \left (6 c C d^2+3 a C e^2-c e (3 B d-A e)\right ) x^4}{4 e^5}-\frac{c^3 (3 C d-B e) x^5}{5 e^4}+\frac{c^3 C x^6}{6 e^3}-\frac{\left (c d^2+a e^2\right )^3 \left (C d^2-B d e+A e^2\right )}{2 e^9 (d+e x)^2}+\frac{\left (c d^2+a e^2\right )^2 \left (8 c C d^3-c d e (7 B d-6 A e)+a e^2 (2 C d-B e)\right )}{e^9 (d+e x)}+\frac{\left (c d^2+a e^2\right ) \left (a^2 C e^4+c^2 \left (28 C d^4-3 d^2 e (7 B d-5 A e)\right )+a c e^2 \left (17 C d^2-3 e (3 B d-A e)\right )\right ) \log (d+e x)}{e^9}\\ \end{align*}
Mathematica [A] time = 0.235061, size = 438, normalized size = 0.94 \[ \frac{30 c e^2 x^2 \left (3 a^2 C e^4+3 a c e^2 \left (e (A e-3 B d)+6 C d^2\right )+c^2 \left (2 d^2 e (3 A e-5 B d)+15 C d^4\right )\right )-60 c e x \left (-3 a^2 e^4 (B e-3 C d)+3 a c d e^2 \left (3 e (A e-2 B d)+10 C d^2\right )+c^2 \left (5 d^3 e (2 A e-3 B d)+21 C d^5\right )\right )+60 \left (a e^2+c d^2\right ) \log (d+e x) \left (a^2 C e^4+a c e^2 \left (3 e (A e-3 B d)+17 C d^2\right )+c^2 \left (3 d^2 e (5 A e-7 B d)+28 C d^4\right )\right )+15 c^2 e^4 x^4 \left (3 a C e^2+c e (A e-3 B d)+6 c C d^2\right )-20 c^2 e^3 x^3 \left (-3 a e^2 (B e-3 C d)+3 c d e (A e-2 B d)+10 c C d^3\right )+\frac{60 \left (a e^2+c d^2\right )^2 \left (a e^2 (2 C d-B e)+c d e (6 A e-7 B d)+8 c C d^3\right )}{d+e x}-\frac{30 \left (a e^2+c d^2\right )^3 \left (e (A e-B d)+C d^2\right )}{(d+e x)^2}+12 c^3 e^5 x^5 (B e-3 C d)+10 c^3 C e^6 x^6}{60 e^9} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.061, size = 978, normalized size = 2.1 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.08593, size = 946, normalized size = 2.03 \begin{align*} \frac{15 \, C c^{3} d^{8} - 13 \, B c^{3} d^{7} e - 27 \, B a c^{2} d^{5} e^{3} - 15 \, B a^{2} c d^{3} e^{5} - B a^{3} d e^{7} - A a^{3} e^{8} + 11 \,{\left (3 \, C a c^{2} + A c^{3}\right )} d^{6} e^{2} + 21 \,{\left (C a^{2} c + A a c^{2}\right )} d^{4} e^{4} + 3 \,{\left (C a^{3} + 3 \, A a^{2} c\right )} d^{2} e^{6} + 2 \,{\left (8 \, C c^{3} d^{7} e - 7 \, B c^{3} d^{6} e^{2} - 15 \, B a c^{2} d^{4} e^{4} - 9 \, B a^{2} c d^{2} e^{6} - B a^{3} e^{8} + 6 \,{\left (3 \, C a c^{2} + A c^{3}\right )} d^{5} e^{3} + 12 \,{\left (C a^{2} c + A a c^{2}\right )} d^{3} e^{5} + 2 \,{\left (C a^{3} + 3 \, A a^{2} c\right )} d e^{7}\right )} x}{2 \,{\left (e^{11} x^{2} + 2 \, d e^{10} x + d^{2} e^{9}\right )}} + \frac{10 \, C c^{3} e^{5} x^{6} - 12 \,{\left (3 \, C c^{3} d e^{4} - B c^{3} e^{5}\right )} x^{5} + 15 \,{\left (6 \, C c^{3} d^{2} e^{3} - 3 \, B c^{3} d e^{4} +{\left (3 \, C a c^{2} + A c^{3}\right )} e^{5}\right )} x^{4} - 20 \,{\left (10 \, C c^{3} d^{3} e^{2} - 6 \, B c^{3} d^{2} e^{3} - 3 \, B a c^{2} e^{5} + 3 \,{\left (3 \, C a c^{2} + A c^{3}\right )} d e^{4}\right )} x^{3} + 30 \,{\left (15 \, C c^{3} d^{4} e - 10 \, B c^{3} d^{3} e^{2} - 9 \, B a c^{2} d e^{4} + 6 \,{\left (3 \, C a c^{2} + A c^{3}\right )} d^{2} e^{3} + 3 \,{\left (C a^{2} c + A a c^{2}\right )} e^{5}\right )} x^{2} - 60 \,{\left (21 \, C c^{3} d^{5} - 15 \, B c^{3} d^{4} e - 18 \, B a c^{2} d^{2} e^{3} - 3 \, B a^{2} c e^{5} + 10 \,{\left (3 \, C a c^{2} + A c^{3}\right )} d^{3} e^{2} + 9 \,{\left (C a^{2} c + A a c^{2}\right )} d e^{4}\right )} x}{60 \, e^{8}} + \frac{{\left (28 \, C c^{3} d^{6} - 21 \, B c^{3} d^{5} e - 30 \, B a c^{2} d^{3} e^{3} - 9 \, B a^{2} c d e^{5} + 15 \,{\left (3 \, C a c^{2} + A c^{3}\right )} d^{4} e^{2} + 18 \,{\left (C a^{2} c + A a c^{2}\right )} d^{2} e^{4} +{\left (C a^{3} + 3 \, A a^{2} c\right )} e^{6}\right )} \log \left (e x + d\right )}{e^{9}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.82771, size = 2198, normalized size = 4.72 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 36.9157, size = 799, normalized size = 1.71 \begin{align*} \frac{C c^{3} x^{6}}{6 e^{3}} + \frac{- A a^{3} e^{8} + 9 A a^{2} c d^{2} e^{6} + 21 A a c^{2} d^{4} e^{4} + 11 A c^{3} d^{6} e^{2} - B a^{3} d e^{7} - 15 B a^{2} c d^{3} e^{5} - 27 B a c^{2} d^{5} e^{3} - 13 B c^{3} d^{7} e + 3 C a^{3} d^{2} e^{6} + 21 C a^{2} c d^{4} e^{4} + 33 C a c^{2} d^{6} e^{2} + 15 C c^{3} d^{8} + x \left (12 A a^{2} c d e^{7} + 24 A a c^{2} d^{3} e^{5} + 12 A c^{3} d^{5} e^{3} - 2 B a^{3} e^{8} - 18 B a^{2} c d^{2} e^{6} - 30 B a c^{2} d^{4} e^{4} - 14 B c^{3} d^{6} e^{2} + 4 C a^{3} d e^{7} + 24 C a^{2} c d^{3} e^{5} + 36 C a c^{2} d^{5} e^{3} + 16 C c^{3} d^{7} e\right )}{2 d^{2} e^{9} + 4 d e^{10} x + 2 e^{11} x^{2}} - \frac{x^{5} \left (- B c^{3} e + 3 C c^{3} d\right )}{5 e^{4}} + \frac{x^{4} \left (A c^{3} e^{2} - 3 B c^{3} d e + 3 C a c^{2} e^{2} + 6 C c^{3} d^{2}\right )}{4 e^{5}} - \frac{x^{3} \left (3 A c^{3} d e^{2} - 3 B a c^{2} e^{3} - 6 B c^{3} d^{2} e + 9 C a c^{2} d e^{2} + 10 C c^{3} d^{3}\right )}{3 e^{6}} + \frac{x^{2} \left (3 A a c^{2} e^{4} + 6 A c^{3} d^{2} e^{2} - 9 B a c^{2} d e^{3} - 10 B c^{3} d^{3} e + 3 C a^{2} c e^{4} + 18 C a c^{2} d^{2} e^{2} + 15 C c^{3} d^{4}\right )}{2 e^{7}} - \frac{x \left (9 A a c^{2} d e^{4} + 10 A c^{3} d^{3} e^{2} - 3 B a^{2} c e^{5} - 18 B a c^{2} d^{2} e^{3} - 15 B c^{3} d^{4} e + 9 C a^{2} c d e^{4} + 30 C a c^{2} d^{3} e^{2} + 21 C c^{3} d^{5}\right )}{e^{8}} + \frac{\left (a e^{2} + c d^{2}\right ) \left (3 A a c e^{4} + 15 A c^{2} d^{2} e^{2} - 9 B a c d e^{3} - 21 B c^{2} d^{3} e + C a^{2} e^{4} + 17 C a c d^{2} e^{2} + 28 C c^{2} d^{4}\right ) \log{\left (d + e x \right )}}{e^{9}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.15108, size = 981, normalized size = 2.11 \begin{align*}{\left (28 \, C c^{3} d^{6} - 21 \, B c^{3} d^{5} e + 45 \, C a c^{2} d^{4} e^{2} + 15 \, A c^{3} d^{4} e^{2} - 30 \, B a c^{2} d^{3} e^{3} + 18 \, C a^{2} c d^{2} e^{4} + 18 \, A a c^{2} d^{2} e^{4} - 9 \, B a^{2} c d e^{5} + C a^{3} e^{6} + 3 \, A a^{2} c e^{6}\right )} e^{\left (-9\right )} \log \left ({\left | x e + d \right |}\right ) + \frac{1}{60} \,{\left (10 \, C c^{3} x^{6} e^{15} - 36 \, C c^{3} d x^{5} e^{14} + 90 \, C c^{3} d^{2} x^{4} e^{13} - 200 \, C c^{3} d^{3} x^{3} e^{12} + 450 \, C c^{3} d^{4} x^{2} e^{11} - 1260 \, C c^{3} d^{5} x e^{10} + 12 \, B c^{3} x^{5} e^{15} - 45 \, B c^{3} d x^{4} e^{14} + 120 \, B c^{3} d^{2} x^{3} e^{13} - 300 \, B c^{3} d^{3} x^{2} e^{12} + 900 \, B c^{3} d^{4} x e^{11} + 45 \, C a c^{2} x^{4} e^{15} + 15 \, A c^{3} x^{4} e^{15} - 180 \, C a c^{2} d x^{3} e^{14} - 60 \, A c^{3} d x^{3} e^{14} + 540 \, C a c^{2} d^{2} x^{2} e^{13} + 180 \, A c^{3} d^{2} x^{2} e^{13} - 1800 \, C a c^{2} d^{3} x e^{12} - 600 \, A c^{3} d^{3} x e^{12} + 60 \, B a c^{2} x^{3} e^{15} - 270 \, B a c^{2} d x^{2} e^{14} + 1080 \, B a c^{2} d^{2} x e^{13} + 90 \, C a^{2} c x^{2} e^{15} + 90 \, A a c^{2} x^{2} e^{15} - 540 \, C a^{2} c d x e^{14} - 540 \, A a c^{2} d x e^{14} + 180 \, B a^{2} c x e^{15}\right )} e^{\left (-18\right )} + \frac{{\left (15 \, C c^{3} d^{8} - 13 \, B c^{3} d^{7} e + 33 \, C a c^{2} d^{6} e^{2} + 11 \, A c^{3} d^{6} e^{2} - 27 \, B a c^{2} d^{5} e^{3} + 21 \, C a^{2} c d^{4} e^{4} + 21 \, A a c^{2} d^{4} e^{4} - 15 \, B a^{2} c d^{3} e^{5} + 3 \, C a^{3} d^{2} e^{6} + 9 \, A a^{2} c d^{2} e^{6} - B a^{3} d e^{7} - A a^{3} e^{8} + 2 \,{\left (8 \, C c^{3} d^{7} e - 7 \, B c^{3} d^{6} e^{2} + 18 \, C a c^{2} d^{5} e^{3} + 6 \, A c^{3} d^{5} e^{3} - 15 \, B a c^{2} d^{4} e^{4} + 12 \, C a^{2} c d^{3} e^{5} + 12 \, A a c^{2} d^{3} e^{5} - 9 \, B a^{2} c d^{2} e^{6} + 2 \, C a^{3} d e^{7} + 6 \, A a^{2} c d e^{7} - B a^{3} e^{8}\right )} x\right )} e^{\left (-9\right )}}{2 \,{\left (x e + d\right )}^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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