Optimal. Leaf size=189 \[ -\frac {2 c \left (a B e^2-2 A c d e+5 B c d^2\right )}{e^6 (d+e x)}-\frac {\left (a e^2+c d^2\right ) \left (a B e^2-4 A c d e+5 B c d^2\right )}{3 e^6 (d+e x)^3}+\frac {\left (a e^2+c d^2\right )^2 (B d-A e)}{4 e^6 (d+e x)^4}+\frac {c \left (-a A e^3+3 a B d e^2-3 A c d^2 e+5 B c d^3\right )}{e^6 (d+e x)^2}-\frac {c^2 (5 B d-A e) \log (d+e x)}{e^6}+\frac {B c^2 x}{e^5} \]
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Rubi [A] time = 0.17, antiderivative size = 189, 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} \begin {gather*} -\frac {2 c \left (a B e^2-2 A c d e+5 B c d^2\right )}{e^6 (d+e x)}+\frac {c \left (-a A e^3+3 a B d e^2-3 A c d^2 e+5 B c d^3\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 )}{3 e^6 (d+e x)^3}+\frac {\left (a e^2+c d^2\right )^2 (B d-A e)}{4 e^6 (d+e x)^4}-\frac {c^2 (5 B d-A e) \log (d+e x)}{e^6}+\frac {B c^2 x}{e^5} \end {gather*}
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)^5} \, dx &=\int \left (\frac {B c^2}{e^5}+\frac {(-B d+A e) \left (c d^2+a e^2\right )^2}{e^5 (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 )}{e^5 (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 )}{e^5 (d+e x)^3}-\frac {2 c \left (-5 B c d^2+2 A c d e-a B e^2\right )}{e^5 (d+e x)^2}+\frac {c^2 (-5 B d+A e)}{e^5 (d+e x)}\right ) \, dx\\ &=\frac {B c^2 x}{e^5}+\frac {(B d-A e) \left (c d^2+a e^2\right )^2}{4 e^6 (d+e x)^4}-\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 )}{3 e^6 (d+e x)^3}+\frac {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^6 (d+e x)^2}-\frac {2 c \left (5 B c d^2-2 A c d e+a B e^2\right )}{e^6 (d+e x)}-\frac {c^2 (5 B d-A e) \log (d+e x)}{e^6}\\ \end {align*}
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Mathematica [A] time = 0.11, size = 221, normalized size = 1.17 \begin {gather*} \frac {A e \left (-3 a^2 e^4-2 a c e^2 \left (d^2+4 d e x+6 e^2 x^2\right )+c^2 d \left (25 d^3+88 d^2 e x+108 d e^2 x^2+48 e^3 x^3\right )\right )-B \left (a^2 e^4 (d+4 e x)+6 a c e^2 \left (d^3+4 d^2 e x+6 d e^2 x^2+4 e^3 x^3\right )+c^2 \left (77 d^5+248 d^4 e x+252 d^3 e^2 x^2+48 d^2 e^3 x^3-48 d e^4 x^4-12 e^5 x^5\right )\right )-12 c^2 (d+e x)^4 (5 B d-A e) \log (d+e x)}{12 e^6 (d+e x)^4} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(A+B x) \left (a+c x^2\right )^2}{(d+e x)^5} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [B] time = 0.39, size = 405, normalized size = 2.14 \begin {gather*} \frac {12 \, B c^{2} e^{5} x^{5} + 48 \, B c^{2} d e^{4} x^{4} - 77 \, B c^{2} d^{5} + 25 \, A c^{2} d^{4} e - 6 \, B a c d^{3} e^{2} - 2 \, A a c d^{2} e^{3} - B a^{2} d e^{4} - 3 \, A a^{2} e^{5} - 24 \, {\left (2 \, B c^{2} d^{2} e^{3} - 2 \, A c^{2} d e^{4} + B a c e^{5}\right )} x^{3} - 12 \, {\left (21 \, B c^{2} d^{3} e^{2} - 9 \, A c^{2} d^{2} e^{3} + 3 \, B a c d e^{4} + A a c e^{5}\right )} x^{2} - 4 \, {\left (62 \, B c^{2} d^{4} e - 22 \, A c^{2} d^{3} e^{2} + 6 \, B a c d^{2} e^{3} + 2 \, A a c d e^{4} + B a^{2} e^{5}\right )} x - 12 \, {\left (5 \, B c^{2} d^{5} - A c^{2} d^{4} e + {\left (5 \, B c^{2} d e^{4} - A c^{2} e^{5}\right )} x^{4} + 4 \, {\left (5 \, B c^{2} d^{2} e^{3} - A c^{2} d e^{4}\right )} x^{3} + 6 \, {\left (5 \, B c^{2} d^{3} e^{2} - A c^{2} d^{2} e^{3}\right )} x^{2} + 4 \, {\left (5 \, B c^{2} d^{4} e - A c^{2} d^{3} e^{2}\right )} x\right )} \log \left (e x + d\right )}{12 \, {\left (e^{10} x^{4} + 4 \, d e^{9} x^{3} + 6 \, d^{2} e^{8} x^{2} + 4 \, d^{3} e^{7} x + d^{4} e^{6}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.17, size = 372, normalized size = 1.97 \begin {gather*} {\left (x e + d\right )} B c^{2} e^{\left (-6\right )} + {\left (5 \, B c^{2} d - A c^{2} e\right )} e^{\left (-6\right )} \log \left (\frac {{\left | x e + d \right |} e^{\left (-1\right )}}{{\left (x e + d\right )}^{2}}\right ) - \frac {1}{12} \, {\left (\frac {120 \, B c^{2} d^{2} e^{22}}{x e + d} - \frac {60 \, B c^{2} d^{3} e^{22}}{{\left (x e + d\right )}^{2}} + \frac {20 \, B c^{2} d^{4} e^{22}}{{\left (x e + d\right )}^{3}} - \frac {3 \, B c^{2} d^{5} e^{22}}{{\left (x e + d\right )}^{4}} - \frac {48 \, A c^{2} d e^{23}}{x e + d} + \frac {36 \, A c^{2} d^{2} e^{23}}{{\left (x e + d\right )}^{2}} - \frac {16 \, A c^{2} d^{3} e^{23}}{{\left (x e + d\right )}^{3}} + \frac {3 \, A c^{2} d^{4} e^{23}}{{\left (x e + d\right )}^{4}} + \frac {24 \, B a c e^{24}}{x e + d} - \frac {36 \, B a c d e^{24}}{{\left (x e + d\right )}^{2}} + \frac {24 \, B a c d^{2} e^{24}}{{\left (x e + d\right )}^{3}} - \frac {6 \, B a c d^{3} e^{24}}{{\left (x e + d\right )}^{4}} + \frac {12 \, A a c e^{25}}{{\left (x e + d\right )}^{2}} - \frac {16 \, A a c d e^{25}}{{\left (x e + d\right )}^{3}} + \frac {6 \, A a c d^{2} e^{25}}{{\left (x e + d\right )}^{4}} + \frac {4 \, B a^{2} e^{26}}{{\left (x e + d\right )}^{3}} - \frac {3 \, B a^{2} d e^{26}}{{\left (x e + d\right )}^{4}} + \frac {3 \, A a^{2} e^{27}}{{\left (x e + d\right )}^{4}}\right )} e^{\left (-28\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 356, normalized size = 1.88 \begin {gather*} -\frac {A \,a^{2}}{4 \left (e x +d \right )^{4} e}-\frac {A a c \,d^{2}}{2 \left (e x +d \right )^{4} e^{3}}-\frac {A \,c^{2} d^{4}}{4 \left (e x +d \right )^{4} e^{5}}+\frac {B \,a^{2} d}{4 \left (e x +d \right )^{4} e^{2}}+\frac {B a c \,d^{3}}{2 \left (e x +d \right )^{4} e^{4}}+\frac {B \,c^{2} d^{5}}{4 \left (e x +d \right )^{4} e^{6}}+\frac {4 A a c d}{3 \left (e x +d \right )^{3} e^{3}}+\frac {4 A \,c^{2} d^{3}}{3 \left (e x +d \right )^{3} e^{5}}-\frac {B \,a^{2}}{3 \left (e x +d \right )^{3} e^{2}}-\frac {2 B a c \,d^{2}}{\left (e x +d \right )^{3} e^{4}}-\frac {5 B \,c^{2} d^{4}}{3 \left (e x +d \right )^{3} e^{6}}-\frac {A a c}{\left (e x +d \right )^{2} e^{3}}-\frac {3 A \,c^{2} d^{2}}{\left (e x +d \right )^{2} e^{5}}+\frac {3 B a c d}{\left (e x +d \right )^{2} e^{4}}+\frac {5 B \,c^{2} d^{3}}{\left (e x +d \right )^{2} e^{6}}+\frac {4 A \,c^{2} d}{\left (e x +d \right ) e^{5}}+\frac {A \,c^{2} \ln \left (e x +d \right )}{e^{5}}-\frac {2 B a c}{\left (e x +d \right ) e^{4}}-\frac {10 B \,c^{2} d^{2}}{\left (e x +d \right ) e^{6}}-\frac {5 B \,c^{2} d \ln \left (e x +d \right )}{e^{6}}+\frac {B \,c^{2} x}{e^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.58, size = 279, normalized size = 1.48 \begin {gather*} -\frac {77 \, B c^{2} d^{5} - 25 \, A c^{2} d^{4} e + 6 \, B a c d^{3} e^{2} + 2 \, A a c d^{2} e^{3} + B a^{2} d e^{4} + 3 \, A a^{2} e^{5} + 24 \, {\left (5 \, B c^{2} d^{2} e^{3} - 2 \, A c^{2} d e^{4} + B a c e^{5}\right )} x^{3} + 12 \, {\left (25 \, B c^{2} d^{3} e^{2} - 9 \, A c^{2} d^{2} e^{3} + 3 \, B a c d e^{4} + A a c e^{5}\right )} x^{2} + 4 \, {\left (65 \, B c^{2} d^{4} e - 22 \, A c^{2} d^{3} e^{2} + 6 \, B a c d^{2} e^{3} + 2 \, A a c d e^{4} + B a^{2} e^{5}\right )} x}{12 \, {\left (e^{10} x^{4} + 4 \, d e^{9} x^{3} + 6 \, d^{2} e^{8} x^{2} + 4 \, d^{3} e^{7} x + d^{4} e^{6}\right )}} + \frac {B c^{2} x}{e^{5}} - \frac {{\left (5 \, B c^{2} d - A c^{2} e\right )} \log \left (e x + d\right )}{e^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.84, size = 277, normalized size = 1.47 \begin {gather*} \frac {\ln \left (d+e\,x\right )\,\left (A\,c^2\,e-5\,B\,c^2\,d\right )}{e^6}-\frac {x^3\,\left (10\,B\,c^2\,d^2\,e^2-4\,A\,c^2\,d\,e^3+2\,B\,a\,c\,e^4\right )+x\,\left (\frac {B\,a^2\,e^4}{3}+2\,B\,a\,c\,d^2\,e^2+\frac {2\,A\,a\,c\,d\,e^3}{3}+\frac {65\,B\,c^2\,d^4}{3}-\frac {22\,A\,c^2\,d^3\,e}{3}\right )+\frac {B\,a^2\,d\,e^4+3\,A\,a^2\,e^5+6\,B\,a\,c\,d^3\,e^2+2\,A\,a\,c\,d^2\,e^3+77\,B\,c^2\,d^5-25\,A\,c^2\,d^4\,e}{12\,e}+x^2\,\left (25\,B\,c^2\,d^3\,e-9\,A\,c^2\,d^2\,e^2+3\,B\,a\,c\,d\,e^3+A\,a\,c\,e^4\right )}{d^4\,e^5+4\,d^3\,e^6\,x+6\,d^2\,e^7\,x^2+4\,d\,e^8\,x^3+e^9\,x^4}+\frac {B\,c^2\,x}{e^5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 21.57, size = 304, normalized size = 1.61 \begin {gather*} \frac {B c^{2} x}{e^{5}} - \frac {c^{2} \left (- A e + 5 B d\right ) \log {\left (d + e x \right )}}{e^{6}} + \frac {- 3 A a^{2} e^{5} - 2 A a c d^{2} e^{3} + 25 A c^{2} d^{4} e - B a^{2} d e^{4} - 6 B a c d^{3} e^{2} - 77 B c^{2} d^{5} + x^{3} \left (48 A c^{2} d e^{4} - 24 B a c e^{5} - 120 B c^{2} d^{2} e^{3}\right ) + x^{2} \left (- 12 A a c e^{5} + 108 A c^{2} d^{2} e^{3} - 36 B a c d e^{4} - 300 B c^{2} d^{3} e^{2}\right ) + x \left (- 8 A a c d e^{4} + 88 A c^{2} d^{3} e^{2} - 4 B a^{2} e^{5} - 24 B a c d^{2} e^{3} - 260 B c^{2} d^{4} e\right )}{12 d^{4} e^{6} + 48 d^{3} e^{7} x + 72 d^{2} e^{8} x^{2} + 48 d e^{9} x^{3} + 12 e^{10} x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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