Optimal. Leaf size=197 \[ -\frac {c \left (a B e^2-2 A c d e+5 B c d^2\right )}{e^6 (d+e x)^2}-\frac {\left (a e^2+c d^2\right ) \left (a B e^2-4 A c d e+5 B c d^2\right )}{4 e^6 (d+e x)^4}+\frac {\left (a e^2+c d^2\right )^2 (B d-A e)}{5 e^6 (d+e x)^5}+\frac {2 c \left (-a A e^3+3 a B d e^2-3 A c d^2 e+5 B c d^3\right )}{3 e^6 (d+e x)^3}+\frac {c^2 (5 B d-A e)}{e^6 (d+e x)}+\frac {B c^2 \log (d+e x)}{e^6} \]
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Rubi [A] time = 0.15, antiderivative size = 197, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {772} \[ -\frac {c \left (a B e^2-2 A c d e+5 B c d^2\right )}{e^6 (d+e x)^2}+\frac {2 c \left (-a A e^3+3 a B d e^2-3 A c d^2 e+5 B c d^3\right )}{3 e^6 (d+e x)^3}-\frac {\left (a e^2+c d^2\right ) \left (a B e^2-4 A c d e+5 B c d^2\right )}{4 e^6 (d+e x)^4}+\frac {\left (a e^2+c d^2\right )^2 (B d-A e)}{5 e^6 (d+e x)^5}+\frac {c^2 (5 B d-A e)}{e^6 (d+e x)}+\frac {B c^2 \log (d+e x)}{e^6} \]
Antiderivative was successfully verified.
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Rule 772
Rubi steps
\begin {align*} \int \frac {(A+B x) \left (a+c x^2\right )^2}{(d+e x)^6} \, dx &=\int \left (\frac {(-B d+A e) \left (c d^2+a e^2\right )^2}{e^5 (d+e x)^6}+\frac {\left (c d^2+a e^2\right ) \left (5 B c d^2-4 A c d e+a B e^2\right )}{e^5 (d+e x)^5}+\frac {2 c \left (-5 B c d^3+3 A c d^2 e-3 a B d e^2+a A e^3\right )}{e^5 (d+e x)^4}-\frac {2 c \left (-5 B c d^2+2 A c d e-a B e^2\right )}{e^5 (d+e x)^3}+\frac {c^2 (-5 B d+A e)}{e^5 (d+e x)^2}+\frac {B c^2}{e^5 (d+e x)}\right ) \, dx\\ &=\frac {(B d-A e) \left (c d^2+a e^2\right )^2}{5 e^6 (d+e x)^5}-\frac {\left (c d^2+a e^2\right ) \left (5 B c d^2-4 A c d e+a B e^2\right )}{4 e^6 (d+e x)^4}+\frac {2 c \left (5 B c d^3-3 A c d^2 e+3 a B d e^2-a A e^3\right )}{3 e^6 (d+e x)^3}-\frac {c \left (5 B c d^2-2 A c d e+a B e^2\right )}{e^6 (d+e x)^2}+\frac {c^2 (5 B d-A e)}{e^6 (d+e x)}+\frac {B c^2 \log (d+e x)}{e^6}\\ \end {align*}
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Mathematica [A] time = 0.11, size = 212, normalized size = 1.08 \[ \frac {-4 A e \left (3 a^2 e^4+a c e^2 \left (d^2+5 d e x+10 e^2 x^2\right )+3 c^2 \left (d^4+5 d^3 e x+10 d^2 e^2 x^2+10 d e^3 x^3+5 e^4 x^4\right )\right )+B \left (-3 a^2 e^4 (d+5 e x)-6 a c e^2 \left (d^3+5 d^2 e x+10 d e^2 x^2+10 e^3 x^3\right )+c^2 d \left (137 d^4+625 d^3 e x+1100 d^2 e^2 x^2+900 d e^3 x^3+300 e^4 x^4\right )\right )+60 B c^2 (d+e x)^5 \log (d+e x)}{60 e^6 (d+e x)^5} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.57, size = 365, normalized size = 1.85 \[ \frac {137 \, B c^{2} d^{5} - 12 \, A c^{2} d^{4} e - 6 \, B a c d^{3} e^{2} - 4 \, A a c d^{2} e^{3} - 3 \, B a^{2} d e^{4} - 12 \, A a^{2} e^{5} + 60 \, {\left (5 \, B c^{2} d e^{4} - A c^{2} e^{5}\right )} x^{4} + 60 \, {\left (15 \, B c^{2} d^{2} e^{3} - 2 \, A c^{2} d e^{4} - B a c e^{5}\right )} x^{3} + 20 \, {\left (55 \, B c^{2} d^{3} e^{2} - 6 \, A c^{2} d^{2} e^{3} - 3 \, B a c d e^{4} - 2 \, A a c e^{5}\right )} x^{2} + 5 \, {\left (125 \, B c^{2} d^{4} e - 12 \, A c^{2} d^{3} e^{2} - 6 \, B a c d^{2} e^{3} - 4 \, A a c d e^{4} - 3 \, B a^{2} e^{5}\right )} x + 60 \, {\left (B c^{2} e^{5} x^{5} + 5 \, B c^{2} d e^{4} x^{4} + 10 \, B c^{2} d^{2} e^{3} x^{3} + 10 \, B c^{2} d^{3} e^{2} x^{2} + 5 \, B c^{2} d^{4} e x + B c^{2} d^{5}\right )} \log \left (e x + d\right )}{60 \, {\left (e^{11} x^{5} + 5 \, d e^{10} x^{4} + 10 \, d^{2} e^{9} x^{3} + 10 \, d^{3} e^{8} x^{2} + 5 \, d^{4} e^{7} x + d^{5} e^{6}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.20, size = 239, normalized size = 1.21 \[ B c^{2} e^{\left (-6\right )} \log \left ({\left | x e + d \right |}\right ) + \frac {{\left (60 \, {\left (5 \, B c^{2} d e^{3} - A c^{2} e^{4}\right )} x^{4} + 60 \, {\left (15 \, B c^{2} d^{2} e^{2} - 2 \, A c^{2} d e^{3} - B a c e^{4}\right )} x^{3} + 20 \, {\left (55 \, B c^{2} d^{3} e - 6 \, A c^{2} d^{2} e^{2} - 3 \, B a c d e^{3} - 2 \, A a c e^{4}\right )} x^{2} + 5 \, {\left (125 \, B c^{2} d^{4} - 12 \, A c^{2} d^{3} e - 6 \, B a c d^{2} e^{2} - 4 \, A a c d e^{3} - 3 \, B a^{2} e^{4}\right )} x + {\left (137 \, B c^{2} d^{5} - 12 \, A c^{2} d^{4} e - 6 \, B a c d^{3} e^{2} - 4 \, A a c d^{2} e^{3} - 3 \, B a^{2} d e^{4} - 12 \, A a^{2} e^{5}\right )} e^{\left (-1\right )}\right )} e^{\left (-5\right )}}{60 \, {\left (x e + d\right )}^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 362, normalized size = 1.84 \[ -\frac {A \,a^{2}}{5 \left (e x +d \right )^{5} e}-\frac {2 A a c \,d^{2}}{5 \left (e x +d \right )^{5} e^{3}}-\frac {A \,c^{2} d^{4}}{5 \left (e x +d \right )^{5} e^{5}}+\frac {B \,a^{2} d}{5 \left (e x +d \right )^{5} e^{2}}+\frac {2 B a c \,d^{3}}{5 \left (e x +d \right )^{5} e^{4}}+\frac {B \,c^{2} d^{5}}{5 \left (e x +d \right )^{5} e^{6}}+\frac {A a c d}{\left (e x +d \right )^{4} e^{3}}+\frac {A \,c^{2} d^{3}}{\left (e x +d \right )^{4} e^{5}}-\frac {B \,a^{2}}{4 \left (e x +d \right )^{4} e^{2}}-\frac {3 B a c \,d^{2}}{2 \left (e x +d \right )^{4} e^{4}}-\frac {5 B \,c^{2} d^{4}}{4 \left (e x +d \right )^{4} e^{6}}-\frac {2 A a c}{3 \left (e x +d \right )^{3} e^{3}}-\frac {2 A \,c^{2} d^{2}}{\left (e x +d \right )^{3} e^{5}}+\frac {2 B a c d}{\left (e x +d \right )^{3} e^{4}}+\frac {10 B \,c^{2} d^{3}}{3 \left (e x +d \right )^{3} e^{6}}+\frac {2 A \,c^{2} d}{\left (e x +d \right )^{2} e^{5}}-\frac {B a c}{\left (e x +d \right )^{2} e^{4}}-\frac {5 B \,c^{2} d^{2}}{\left (e x +d \right )^{2} e^{6}}-\frac {A \,c^{2}}{\left (e x +d \right ) e^{5}}+\frac {5 B \,c^{2} d}{\left (e x +d \right ) e^{6}}+\frac {B \,c^{2} \ln \left (e x +d \right )}{e^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.64, size = 298, normalized size = 1.51 \[ \frac {137 \, B c^{2} d^{5} - 12 \, A c^{2} d^{4} e - 6 \, B a c d^{3} e^{2} - 4 \, A a c d^{2} e^{3} - 3 \, B a^{2} d e^{4} - 12 \, A a^{2} e^{5} + 60 \, {\left (5 \, B c^{2} d e^{4} - A c^{2} e^{5}\right )} x^{4} + 60 \, {\left (15 \, B c^{2} d^{2} e^{3} - 2 \, A c^{2} d e^{4} - B a c e^{5}\right )} x^{3} + 20 \, {\left (55 \, B c^{2} d^{3} e^{2} - 6 \, A c^{2} d^{2} e^{3} - 3 \, B a c d e^{4} - 2 \, A a c e^{5}\right )} x^{2} + 5 \, {\left (125 \, B c^{2} d^{4} e - 12 \, A c^{2} d^{3} e^{2} - 6 \, B a c d^{2} e^{3} - 4 \, A a c d e^{4} - 3 \, B a^{2} e^{5}\right )} x}{60 \, {\left (e^{11} x^{5} + 5 \, d e^{10} x^{4} + 10 \, d^{2} e^{9} x^{3} + 10 \, d^{3} e^{8} x^{2} + 5 \, d^{4} e^{7} x + d^{5} e^{6}\right )}} + \frac {B c^{2} \log \left (e x + d\right )}{e^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.78, size = 243, normalized size = 1.23 \[ \frac {B\,c^2\,\ln \left (d+e\,x\right )}{e^6}-\frac {x^2\,\left (-\frac {55\,B\,c^2\,d^3\,e^2}{3}+2\,A\,c^2\,d^2\,e^3+B\,a\,c\,d\,e^4+\frac {2\,A\,a\,c\,e^5}{3}\right )+x^3\,\left (-15\,B\,c^2\,d^2\,e^3+2\,A\,c^2\,d\,e^4+B\,a\,c\,e^5\right )+x^4\,\left (A\,c^2\,e^5-5\,B\,c^2\,d\,e^4\right )+x\,\left (\frac {B\,a^2\,e^5}{4}+\frac {B\,a\,c\,d^2\,e^3}{2}+\frac {A\,a\,c\,d\,e^4}{3}-\frac {125\,B\,c^2\,d^4\,e}{12}+A\,c^2\,d^3\,e^2\right )+\frac {A\,a^2\,e^5}{5}-\frac {137\,B\,c^2\,d^5}{60}+\frac {B\,a^2\,d\,e^4}{20}+\frac {A\,c^2\,d^4\,e}{5}+\frac {A\,a\,c\,d^2\,e^3}{15}+\frac {B\,a\,c\,d^3\,e^2}{10}}{e^6\,{\left (d+e\,x\right )}^5} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 54.06, size = 326, normalized size = 1.65 \[ \frac {B c^{2} \log {\left (d + e x \right )}}{e^{6}} + \frac {- 12 A a^{2} e^{5} - 4 A a c d^{2} e^{3} - 12 A c^{2} d^{4} e - 3 B a^{2} d e^{4} - 6 B a c d^{3} e^{2} + 137 B c^{2} d^{5} + x^{4} \left (- 60 A c^{2} e^{5} + 300 B c^{2} d e^{4}\right ) + x^{3} \left (- 120 A c^{2} d e^{4} - 60 B a c e^{5} + 900 B c^{2} d^{2} e^{3}\right ) + x^{2} \left (- 40 A a c e^{5} - 120 A c^{2} d^{2} e^{3} - 60 B a c d e^{4} + 1100 B c^{2} d^{3} e^{2}\right ) + x \left (- 20 A a c d e^{4} - 60 A c^{2} d^{3} e^{2} - 15 B a^{2} e^{5} - 30 B a c d^{2} e^{3} + 625 B c^{2} d^{4} e\right )}{60 d^{5} e^{6} + 300 d^{4} e^{7} x + 600 d^{3} e^{8} x^{2} + 600 d^{2} e^{9} x^{3} + 300 d e^{10} x^{4} + 60 e^{11} x^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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