Optimal. Leaf size=251 \[ \frac{(2 c d-b e) \left (-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right )}{e^7 (d+e x)}-\frac{3 \left (a e^2-b d e+c d^2\right ) \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{2 e^7 (d+e x)^2}+\frac{3 c \log (d+e x) \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{e^7}+\frac{(2 c d-b e) \left (a e^2-b d e+c d^2\right )^2}{e^7 (d+e x)^3}-\frac{\left (a e^2-b d e+c d^2\right )^3}{4 e^7 (d+e x)^4}-\frac{c^2 x (5 c d-3 b e)}{e^6}+\frac{c^3 x^2}{2 e^5} \]
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Rubi [A] time = 0.274186, antiderivative size = 251, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05, Rules used = {698} \[ \frac{(2 c d-b e) \left (-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right )}{e^7 (d+e x)}-\frac{3 \left (a e^2-b d e+c d^2\right ) \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{2 e^7 (d+e x)^2}+\frac{3 c \log (d+e x) \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{e^7}+\frac{(2 c d-b e) \left (a e^2-b d e+c d^2\right )^2}{e^7 (d+e x)^3}-\frac{\left (a e^2-b d e+c d^2\right )^3}{4 e^7 (d+e x)^4}-\frac{c^2 x (5 c d-3 b e)}{e^6}+\frac{c^3 x^2}{2 e^5} \]
Antiderivative was successfully verified.
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Rule 698
Rubi steps
\begin{align*} \int \frac{\left (a+b x+c x^2\right )^3}{(d+e x)^5} \, dx &=\int \left (-\frac{c^2 (5 c d-3 b e)}{e^6}+\frac{c^3 x}{e^5}+\frac{\left (c d^2-b d e+a e^2\right )^3}{e^6 (d+e x)^5}+\frac{3 (-2 c d+b e) \left (c d^2-b d e+a e^2\right )^2}{e^6 (d+e x)^4}+\frac{3 \left (c d^2-b d e+a e^2\right ) \left (5 c^2 d^2-5 b c d e+b^2 e^2+a c e^2\right )}{e^6 (d+e x)^3}+\frac{(2 c d-b e) \left (-10 c^2 d^2-b^2 e^2+2 c e (5 b d-3 a e)\right )}{e^6 (d+e x)^2}+\frac{3 c \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right )}{e^6 (d+e x)}\right ) \, dx\\ &=-\frac{c^2 (5 c d-3 b e) x}{e^6}+\frac{c^3 x^2}{2 e^5}-\frac{\left (c d^2-b d e+a e^2\right )^3}{4 e^7 (d+e x)^4}+\frac{(2 c d-b e) \left (c d^2-b d e+a e^2\right )^2}{e^7 (d+e x)^3}-\frac{3 \left (c d^2-b d e+a e^2\right ) \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right )}{2 e^7 (d+e x)^2}+\frac{(2 c d-b e) \left (10 c^2 d^2+b^2 e^2-2 c e (5 b d-3 a e)\right )}{e^7 (d+e x)}+\frac{3 c \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right ) \log (d+e x)}{e^7}\\ \end{align*}
Mathematica [A] time = 0.175801, size = 402, normalized size = 1.6 \[ \frac{-c e^2 \left (a^2 e^2 \left (d^2+4 d e x+6 e^2 x^2\right )+6 a b e \left (4 d^2 e x+d^3+6 d e^2 x^2+4 e^3 x^3\right )+b^2 (-d) \left (88 d^2 e x+25 d^3+108 d e^2 x^2+48 e^3 x^3\right )\right )-e^3 \left (a^2 b e^2 (d+4 e x)+a^3 e^3+a b^2 e \left (d^2+4 d e x+6 e^2 x^2\right )+b^3 \left (4 d^2 e x+d^3+6 d e^2 x^2+4 e^3 x^3\right )\right )+12 c (d+e x)^4 \log (d+e x) \left (c e (a e-5 b d)+b^2 e^2+5 c^2 d^2\right )+c^2 e \left (a d e \left (88 d^2 e x+25 d^3+108 d e^2 x^2+48 e^3 x^3\right )-b \left (252 d^3 e^2 x^2+48 d^2 e^3 x^3+248 d^4 e x+77 d^5-48 d e^4 x^4-12 e^5 x^5\right )\right )+c^3 \left (132 d^4 e^2 x^2-32 d^3 e^3 x^3-68 d^2 e^4 x^4+168 d^5 e x+57 d^6-12 d e^5 x^5+2 e^6 x^6\right )}{4 e^7 (d+e x)^4} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.051, size = 678, normalized size = 2.7 \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 = 1.04103, size = 595, normalized size = 2.37 \begin{align*} \frac{57 \, c^{3} d^{6} - 77 \, b c^{2} d^{5} e - a^{2} b d e^{5} - a^{3} e^{6} + 25 \,{\left (b^{2} c + a c^{2}\right )} d^{4} e^{2} -{\left (b^{3} + 6 \, a b c\right )} d^{3} e^{3} -{\left (a b^{2} + a^{2} c\right )} d^{2} e^{4} + 4 \,{\left (20 \, c^{3} d^{3} e^{3} - 30 \, b c^{2} d^{2} e^{4} + 12 \,{\left (b^{2} c + a c^{2}\right )} d e^{5} -{\left (b^{3} + 6 \, a b c\right )} e^{6}\right )} x^{3} + 6 \,{\left (35 \, c^{3} d^{4} e^{2} - 50 \, b c^{2} d^{3} e^{3} + 18 \,{\left (b^{2} c + a c^{2}\right )} d^{2} e^{4} -{\left (b^{3} + 6 \, a b c\right )} d e^{5} -{\left (a b^{2} + a^{2} c\right )} e^{6}\right )} x^{2} + 4 \,{\left (47 \, c^{3} d^{5} e - 65 \, b c^{2} d^{4} e^{2} - a^{2} b e^{6} + 22 \,{\left (b^{2} c + a c^{2}\right )} d^{3} e^{3} -{\left (b^{3} + 6 \, a b c\right )} d^{2} e^{4} -{\left (a b^{2} + a^{2} c\right )} d e^{5}\right )} x}{4 \,{\left (e^{11} x^{4} + 4 \, d e^{10} x^{3} + 6 \, d^{2} e^{9} x^{2} + 4 \, d^{3} e^{8} x + d^{4} e^{7}\right )}} + \frac{c^{3} e x^{2} - 2 \,{\left (5 \, c^{3} d - 3 \, b c^{2} e\right )} x}{2 \, e^{6}} + \frac{3 \,{\left (5 \, c^{3} d^{2} - 5 \, b c^{2} d e +{\left (b^{2} c + a c^{2}\right )} e^{2}\right )} \log \left (e x + d\right )}{e^{7}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.98217, size = 1308, normalized size = 5.21 \begin{align*} \frac{2 \, c^{3} e^{6} x^{6} + 57 \, c^{3} d^{6} - 77 \, b c^{2} d^{5} e - a^{2} b d e^{5} - a^{3} e^{6} + 25 \,{\left (b^{2} c + a c^{2}\right )} d^{4} e^{2} -{\left (b^{3} + 6 \, a b c\right )} d^{3} e^{3} -{\left (a b^{2} + a^{2} c\right )} d^{2} e^{4} - 12 \,{\left (c^{3} d e^{5} - b c^{2} e^{6}\right )} x^{5} - 4 \,{\left (17 \, c^{3} d^{2} e^{4} - 12 \, b c^{2} d e^{5}\right )} x^{4} - 4 \,{\left (8 \, c^{3} d^{3} e^{3} + 12 \, b c^{2} d^{2} e^{4} - 12 \,{\left (b^{2} c + a c^{2}\right )} d e^{5} +{\left (b^{3} + 6 \, a b c\right )} e^{6}\right )} x^{3} + 6 \,{\left (22 \, c^{3} d^{4} e^{2} - 42 \, b c^{2} d^{3} e^{3} + 18 \,{\left (b^{2} c + a c^{2}\right )} d^{2} e^{4} -{\left (b^{3} + 6 \, a b c\right )} d e^{5} -{\left (a b^{2} + a^{2} c\right )} e^{6}\right )} x^{2} + 4 \,{\left (42 \, c^{3} d^{5} e - 62 \, b c^{2} d^{4} e^{2} - a^{2} b e^{6} + 22 \,{\left (b^{2} c + a c^{2}\right )} d^{3} e^{3} -{\left (b^{3} + 6 \, a b c\right )} d^{2} e^{4} -{\left (a b^{2} + a^{2} c\right )} d e^{5}\right )} x + 12 \,{\left (5 \, c^{3} d^{6} - 5 \, b c^{2} d^{5} e +{\left (b^{2} c + a c^{2}\right )} d^{4} e^{2} +{\left (5 \, c^{3} d^{2} e^{4} - 5 \, b c^{2} d e^{5} +{\left (b^{2} c + a c^{2}\right )} e^{6}\right )} x^{4} + 4 \,{\left (5 \, c^{3} d^{3} e^{3} - 5 \, b c^{2} d^{2} e^{4} +{\left (b^{2} c + a c^{2}\right )} d e^{5}\right )} x^{3} + 6 \,{\left (5 \, c^{3} d^{4} e^{2} - 5 \, b c^{2} d^{3} e^{3} +{\left (b^{2} c + a c^{2}\right )} d^{2} e^{4}\right )} x^{2} + 4 \,{\left (5 \, c^{3} d^{5} e - 5 \, b c^{2} d^{4} e^{2} +{\left (b^{2} c + a c^{2}\right )} d^{3} e^{3}\right )} x\right )} \log \left (e x + d\right )}{4 \,{\left (e^{11} x^{4} + 4 \, d e^{10} x^{3} + 6 \, d^{2} e^{9} x^{2} + 4 \, d^{3} e^{8} x + d^{4} e^{7}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 141.101, size = 518, normalized size = 2.06 \begin{align*} \frac{c^{3} x^{2}}{2 e^{5}} + \frac{3 c \left (a c e^{2} + b^{2} e^{2} - 5 b c d e + 5 c^{2} d^{2}\right ) \log{\left (d + e x \right )}}{e^{7}} - \frac{a^{3} e^{6} + a^{2} b d e^{5} + a^{2} c d^{2} e^{4} + a b^{2} d^{2} e^{4} + 6 a b c d^{3} e^{3} - 25 a c^{2} d^{4} e^{2} + b^{3} d^{3} e^{3} - 25 b^{2} c d^{4} e^{2} + 77 b c^{2} d^{5} e - 57 c^{3} d^{6} + x^{3} \left (24 a b c e^{6} - 48 a c^{2} d e^{5} + 4 b^{3} e^{6} - 48 b^{2} c d e^{5} + 120 b c^{2} d^{2} e^{4} - 80 c^{3} d^{3} e^{3}\right ) + x^{2} \left (6 a^{2} c e^{6} + 6 a b^{2} e^{6} + 36 a b c d e^{5} - 108 a c^{2} d^{2} e^{4} + 6 b^{3} d e^{5} - 108 b^{2} c d^{2} e^{4} + 300 b c^{2} d^{3} e^{3} - 210 c^{3} d^{4} e^{2}\right ) + x \left (4 a^{2} b e^{6} + 4 a^{2} c d e^{5} + 4 a b^{2} d e^{5} + 24 a b c d^{2} e^{4} - 88 a c^{2} d^{3} e^{3} + 4 b^{3} d^{2} e^{4} - 88 b^{2} c d^{3} e^{3} + 260 b c^{2} d^{4} e^{2} - 188 c^{3} d^{5} e\right )}{4 d^{4} e^{7} + 16 d^{3} e^{8} x + 24 d^{2} e^{9} x^{2} + 16 d e^{10} x^{3} + 4 e^{11} x^{4}} + \frac{x \left (3 b c^{2} e - 5 c^{3} d\right )}{e^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.16593, size = 927, normalized size = 3.69 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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