Optimal. Leaf size=153 \[ -\frac {\left (a e^2+c d^2\right ) \left (A e^2-B d e+C d^2\right )}{e^5 (d+e x)}-\frac {\log (d+e x) \left (a e^2 (2 C d-B e)+c d \left (4 C d^2-e (3 B d-2 A e)\right )\right )}{e^5}+\frac {x \left (a C e^2+c \left (3 C d^2-e (2 B d-A e)\right )\right )}{e^4}-\frac {c x^2 (2 C d-B e)}{2 e^3}+\frac {c C x^3}{3 e^2} \]
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Rubi [A] time = 0.20, antiderivative size = 151, normalized size of antiderivative = 0.99, number of steps used = 2, number of rules used = 1, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.040, Rules used = {1628} \[ \frac {x \left (a C e^2-c e (2 B d-A e)+3 c C d^2\right )}{e^4}-\frac {\left (a e^2+c d^2\right ) \left (A e^2-B d e+C d^2\right )}{e^5 (d+e x)}-\frac {\log (d+e x) \left (a e^2 (2 C d-B e)-c d e (3 B d-2 A e)+4 c C d^3\right )}{e^5}-\frac {c x^2 (2 C d-B e)}{2 e^3}+\frac {c C x^3}{3 e^2} \]
Antiderivative was successfully verified.
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Rule 1628
Rubi steps
\begin {align*} \int \frac {\left (a+c x^2\right ) \left (A+B x+C x^2\right )}{(d+e x)^2} \, dx &=\int \left (\frac {3 c C d^2+a C e^2-c e (2 B d-A e)}{e^4}+\frac {c (-2 C d+B e) x}{e^3}+\frac {c C x^2}{e^2}+\frac {\left (c d^2+a e^2\right ) \left (C d^2-B d e+A e^2\right )}{e^4 (d+e x)^2}+\frac {-4 c C d^3+c d e (3 B d-2 A e)-a e^2 (2 C d-B e)}{e^4 (d+e x)}\right ) \, dx\\ &=\frac {\left (3 c C d^2+a C e^2-c e (2 B d-A e)\right ) x}{e^4}-\frac {c (2 C d-B e) x^2}{2 e^3}+\frac {c C x^3}{3 e^2}-\frac {\left (c d^2+a e^2\right ) \left (C d^2-B d e+A e^2\right )}{e^5 (d+e x)}-\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 ) \log (d+e x)}{e^5}\\ \end {align*}
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Mathematica [A] time = 0.15, size = 142, normalized size = 0.93 \[ \frac {6 \log (d+e x) \left (a e^2 (B e-2 C d)+c d e (3 B d-2 A e)-4 c C d^3\right )+6 e x \left (a C e^2+c e (A e-2 B d)+3 c C d^2\right )-\frac {6 \left (a e^2+c d^2\right ) \left (e (A e-B d)+C d^2\right )}{d+e x}+3 c e^2 x^2 (B e-2 C d)+2 c C e^3 x^3}{6 e^5} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.87, size = 250, normalized size = 1.63 \[ \frac {2 \, C c e^{4} x^{4} - 6 \, C c d^{4} + 6 \, B c d^{3} e + 6 \, B a d e^{3} - 6 \, A a e^{4} - 6 \, {\left (C a + A c\right )} d^{2} e^{2} - {\left (4 \, C c d e^{3} - 3 \, B c e^{4}\right )} x^{3} + 3 \, {\left (4 \, C c d^{2} e^{2} - 3 \, B c d e^{3} + 2 \, {\left (C a + A c\right )} e^{4}\right )} x^{2} + 6 \, {\left (3 \, C c d^{3} e - 2 \, B c d^{2} e^{2} + {\left (C a + A c\right )} d e^{3}\right )} x - 6 \, {\left (4 \, C c d^{4} - 3 \, B c d^{3} e - B a d e^{3} + 2 \, {\left (C a + A c\right )} d^{2} e^{2} + {\left (4 \, C c d^{3} e - 3 \, B c d^{2} e^{2} - B a e^{4} + 2 \, {\left (C a + A c\right )} d e^{3}\right )} x\right )} \log \left (e x + d\right )}{6 \, {\left (e^{6} x + d e^{5}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 240, normalized size = 1.57 \[ \frac {1}{6} \, {\left (2 \, C c - \frac {3 \, {\left (4 \, C c d e - B c e^{2}\right )} e^{\left (-1\right )}}{x e + d} + \frac {6 \, {\left (6 \, C c d^{2} e^{2} - 3 \, B c d e^{3} + C a e^{4} + A c e^{4}\right )} e^{\left (-2\right )}}{{\left (x e + d\right )}^{2}}\right )} {\left (x e + d\right )}^{3} e^{\left (-5\right )} + {\left (4 \, C c d^{3} - 3 \, B c d^{2} e + 2 \, C a d e^{2} + 2 \, A c d e^{2} - B a e^{3}\right )} e^{\left (-5\right )} \log \left (\frac {{\left | x e + d \right |} e^{\left (-1\right )}}{{\left (x e + d\right )}^{2}}\right ) - {\left (\frac {C c d^{4} e^{3}}{x e + d} - \frac {B c d^{3} e^{4}}{x e + d} + \frac {C a d^{2} e^{5}}{x e + d} + \frac {A c d^{2} e^{5}}{x e + d} - \frac {B a d e^{6}}{x e + d} + \frac {A a e^{7}}{x e + d}\right )} e^{\left (-8\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 234, normalized size = 1.53 \[ \frac {C c \,x^{3}}{3 e^{2}}+\frac {B c \,x^{2}}{2 e^{2}}-\frac {C c d \,x^{2}}{e^{3}}-\frac {A a}{\left (e x +d \right ) e}-\frac {A c \,d^{2}}{\left (e x +d \right ) e^{3}}-\frac {2 A c d \ln \left (e x +d \right )}{e^{3}}+\frac {A c x}{e^{2}}+\frac {B a d}{\left (e x +d \right ) e^{2}}+\frac {B a \ln \left (e x +d \right )}{e^{2}}+\frac {B c \,d^{3}}{\left (e x +d \right ) e^{4}}+\frac {3 B c \,d^{2} \ln \left (e x +d \right )}{e^{4}}-\frac {2 B c d x}{e^{3}}-\frac {C a \,d^{2}}{\left (e x +d \right ) e^{3}}-\frac {2 C a d \ln \left (e x +d \right )}{e^{3}}+\frac {C a x}{e^{2}}-\frac {C c \,d^{4}}{\left (e x +d \right ) e^{5}}-\frac {4 C c \,d^{3} \ln \left (e x +d \right )}{e^{5}}+\frac {3 C c \,d^{2} x}{e^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.45, size = 169, normalized size = 1.10 \[ -\frac {C c d^{4} - B c d^{3} e - B a d e^{3} + A a e^{4} + {\left (C a + A c\right )} d^{2} e^{2}}{e^{6} x + d e^{5}} + \frac {2 \, C c e^{2} x^{3} - 3 \, {\left (2 \, C c d e - B c e^{2}\right )} x^{2} + 6 \, {\left (3 \, C c d^{2} - 2 \, B c d e + {\left (C a + A c\right )} e^{2}\right )} x}{6 \, e^{4}} - \frac {{\left (4 \, C c d^{3} - 3 \, B c d^{2} e - B a e^{3} + 2 \, {\left (C a + A c\right )} d e^{2}\right )} \log \left (e x + d\right )}{e^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.09, size = 192, normalized size = 1.25 \[ x^2\,\left (\frac {B\,c}{2\,e^2}-\frac {C\,c\,d}{e^3}\right )-x\,\left (\frac {2\,d\,\left (\frac {B\,c}{e^2}-\frac {2\,C\,c\,d}{e^3}\right )}{e}-\frac {A\,c+C\,a}{e^2}+\frac {C\,c\,d^2}{e^4}\right )-\frac {\ln \left (d+e\,x\right )\,\left (4\,C\,c\,d^3-B\,a\,e^3+2\,A\,c\,d\,e^2+2\,C\,a\,d\,e^2-3\,B\,c\,d^2\,e\right )}{e^5}-\frac {A\,a\,e^4+C\,c\,d^4-B\,a\,d\,e^3-B\,c\,d^3\,e+A\,c\,d^2\,e^2+C\,a\,d^2\,e^2}{e\,\left (x\,e^5+d\,e^4\right )}+\frac {C\,c\,x^3}{3\,e^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.26, size = 185, normalized size = 1.21 \[ \frac {C c x^{3}}{3 e^{2}} + x^{2} \left (\frac {B c}{2 e^{2}} - \frac {C c d}{e^{3}}\right ) + x \left (\frac {A c}{e^{2}} - \frac {2 B c d}{e^{3}} + \frac {C a}{e^{2}} + \frac {3 C c d^{2}}{e^{4}}\right ) + \frac {- A a e^{4} - A c d^{2} e^{2} + B a d e^{3} + B c d^{3} e - C a d^{2} e^{2} - C c d^{4}}{d e^{5} + e^{6} x} - \frac {\left (2 A c d e^{2} - B a e^{3} - 3 B c d^{2} e + 2 C a d e^{2} + 4 C c d^{3}\right ) \log {\left (d + e x \right )}}{e^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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