Optimal. Leaf size=286 \[ -\frac {\log (d+e x) \left (2 A c e (2 c d-b e)-B \left (-2 c e (4 b d-a e)+b^2 e^2+10 c^2 d^2\right )\right )}{e^6}+\frac {B \left (-6 c d e (2 b d-a e)+b e^2 (3 b d-2 a e)+10 c^2 d^3\right )-A e \left (-2 c e (3 b d-a e)+b^2 e^2+6 c^2 d^2\right )}{e^6 (d+e x)}+\frac {(B d-A e) \left (a e^2-b d e+c d^2\right )^2}{3 e^6 (d+e x)^3}+\frac {\left (a e^2-b d e+c d^2\right ) \left (2 A e (2 c d-b e)-B \left (5 c d^2-e (3 b d-a e)\right )\right )}{2 e^6 (d+e x)^2}-\frac {c x (-A c e-2 b B e+4 B c d)}{e^5}+\frac {B c^2 x^2}{2 e^4} \]
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Rubi [A] time = 0.41, antiderivative size = 284, 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 = {771} \begin {gather*} \frac {B \left (-6 c d e (2 b d-a e)+b e^2 (3 b d-2 a e)+10 c^2 d^3\right )-A e \left (-2 c e (3 b d-a e)+b^2 e^2+6 c^2 d^2\right )}{e^6 (d+e x)}-\frac {\log (d+e x) \left (2 A c e (2 c d-b e)-B \left (-2 c e (4 b d-a e)+b^2 e^2+10 c^2 d^2\right )\right )}{e^6}+\frac {(B d-A e) \left (a e^2-b d e+c d^2\right )^2}{3 e^6 (d+e x)^3}-\frac {\left (a e^2-b d e+c d^2\right ) \left (-B e (3 b d-a e)-2 A e (2 c d-b e)+5 B c d^2\right )}{2 e^6 (d+e x)^2}-\frac {c x (-A c e-2 b B e+4 B c d)}{e^5}+\frac {B c^2 x^2}{2 e^4} \end {gather*}
Antiderivative was successfully verified.
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Rule 771
Rubi steps
\begin {align*} \int \frac {(A+B x) \left (a+b x+c x^2\right )^2}{(d+e x)^4} \, dx &=\int \left (\frac {c (-4 B c d+2 b B e+A c e)}{e^5}+\frac {B c^2 x}{e^4}+\frac {(-B d+A e) \left (c d^2-b d e+a e^2\right )^2}{e^5 (d+e x)^4}+\frac {\left (c d^2-b d e+a e^2\right ) \left (5 B c d^2-B e (3 b d-a e)-2 A e (2 c d-b e)\right )}{e^5 (d+e x)^3}+\frac {-B \left (10 c^2 d^3+b e^2 (3 b d-2 a e)-6 c d e (2 b d-a e)\right )+A e \left (6 c^2 d^2+b^2 e^2-2 c e (3 b d-a e)\right )}{e^5 (d+e x)^2}+\frac {-2 A c e (2 c d-b e)+B \left (10 c^2 d^2+b^2 e^2-2 c e (4 b d-a e)\right )}{e^5 (d+e x)}\right ) \, dx\\ &=-\frac {c (4 B c d-2 b B e-A c e) x}{e^5}+\frac {B c^2 x^2}{2 e^4}+\frac {(B d-A e) \left (c d^2-b d e+a e^2\right )^2}{3 e^6 (d+e x)^3}-\frac {\left (c d^2-b d e+a e^2\right ) \left (5 B c d^2-B e (3 b d-a e)-2 A e (2 c d-b e)\right )}{2 e^6 (d+e x)^2}+\frac {B \left (10 c^2 d^3+b e^2 (3 b d-2 a e)-6 c d e (2 b d-a e)\right )-A e \left (6 c^2 d^2+b^2 e^2-2 c e (3 b d-a e)\right )}{e^6 (d+e x)}-\frac {\left (2 A c e (2 c d-b e)-B \left (10 c^2 d^2+b^2 e^2-2 c e (4 b d-a e)\right )\right ) \log (d+e x)}{e^6}\\ \end {align*}
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Mathematica [A] time = 0.11, size = 263, normalized size = 0.92 \begin {gather*} \frac {6 \log (d+e x) \left (B \left (2 c e (a e-4 b d)+b^2 e^2+10 c^2 d^2\right )+2 A c e (b e-2 c d)\right )-\frac {6 \left (A e \left (2 c e (a e-3 b d)+b^2 e^2+6 c^2 d^2\right )+B \left (6 c d e (2 b d-a e)+b e^2 (2 a e-3 b d)-10 c^2 d^3\right )\right )}{d+e x}+\frac {2 (B d-A e) \left (e (a e-b d)+c d^2\right )^2}{(d+e x)^3}-\frac {3 \left (e (a e-b d)+c d^2\right ) \left (B e (a e-3 b d)+2 A e (b e-2 c d)+5 B c d^2\right )}{(d+e x)^2}+6 c e x (A c e+2 b B e-4 B c d)+3 B c^2 e^2 x^2}{6 e^6} \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+b x+c x^2\right )^2}{(d+e x)^4} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [B] time = 0.40, size = 618, normalized size = 2.16 \begin {gather*} \frac {3 \, B c^{2} e^{5} x^{5} + 47 \, B c^{2} d^{5} - 2 \, A a^{2} e^{5} - 26 \, {\left (2 \, B b c + A c^{2}\right )} d^{4} e + 11 \, {\left (B b^{2} + 2 \, {\left (B a + A b\right )} c\right )} d^{3} e^{2} - 2 \, {\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} d^{2} e^{3} - {\left (B a^{2} + 2 \, A a b\right )} d e^{4} - 3 \, {\left (5 \, B c^{2} d e^{4} - 2 \, {\left (2 \, B b c + A c^{2}\right )} e^{5}\right )} x^{4} - 9 \, {\left (7 \, B c^{2} d^{2} e^{3} - 2 \, {\left (2 \, B b c + A c^{2}\right )} d e^{4}\right )} x^{3} - 3 \, {\left (3 \, B c^{2} d^{3} e^{2} + 6 \, {\left (2 \, B b c + A c^{2}\right )} d^{2} e^{3} - 6 \, {\left (B b^{2} + 2 \, {\left (B a + A b\right )} c\right )} d e^{4} + 2 \, {\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} e^{5}\right )} x^{2} + 3 \, {\left (27 \, B c^{2} d^{4} e - 18 \, {\left (2 \, B b c + A c^{2}\right )} d^{3} e^{2} + 9 \, {\left (B b^{2} + 2 \, {\left (B a + A b\right )} c\right )} d^{2} e^{3} - 2 \, {\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} d e^{4} - {\left (B a^{2} + 2 \, A a b\right )} e^{5}\right )} x + 6 \, {\left (10 \, B c^{2} d^{5} - 4 \, {\left (2 \, B b c + A c^{2}\right )} d^{4} e + {\left (B b^{2} + 2 \, {\left (B a + A b\right )} c\right )} d^{3} e^{2} + {\left (10 \, B c^{2} d^{2} e^{3} - 4 \, {\left (2 \, B b c + A c^{2}\right )} d e^{4} + {\left (B b^{2} + 2 \, {\left (B a + A b\right )} c\right )} e^{5}\right )} x^{3} + 3 \, {\left (10 \, B c^{2} d^{3} e^{2} - 4 \, {\left (2 \, B b c + A c^{2}\right )} d^{2} e^{3} + {\left (B b^{2} + 2 \, {\left (B a + A b\right )} c\right )} d e^{4}\right )} x^{2} + 3 \, {\left (10 \, B c^{2} d^{4} e - 4 \, {\left (2 \, B b c + A c^{2}\right )} d^{3} e^{2} + {\left (B b^{2} + 2 \, {\left (B a + A b\right )} c\right )} d^{2} e^{3}\right )} x\right )} \log \left (e x + d\right )}{6 \, {\left (e^{9} x^{3} + 3 \, d e^{8} x^{2} + 3 \, d^{2} e^{7} x + d^{3} e^{6}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 424, normalized size = 1.48 \begin {gather*} {\left (10 \, B c^{2} d^{2} - 8 \, B b c d e - 4 \, A c^{2} d e + B b^{2} e^{2} + 2 \, B a c e^{2} + 2 \, A b c e^{2}\right )} e^{\left (-6\right )} \log \left ({\left | x e + d \right |}\right ) + \frac {1}{2} \, {\left (B c^{2} x^{2} e^{4} - 8 \, B c^{2} d x e^{3} + 4 \, B b c x e^{4} + 2 \, A c^{2} x e^{4}\right )} e^{\left (-8\right )} + \frac {{\left (47 \, B c^{2} d^{5} - 52 \, B b c d^{4} e - 26 \, A c^{2} d^{4} e + 11 \, B b^{2} d^{3} e^{2} + 22 \, B a c d^{3} e^{2} + 22 \, A b c d^{3} e^{2} - 4 \, B a b d^{2} e^{3} - 2 \, A b^{2} d^{2} e^{3} - 4 \, A a c d^{2} e^{3} - B a^{2} d e^{4} - 2 \, A a b d e^{4} - 2 \, A a^{2} e^{5} + 6 \, {\left (10 \, B c^{2} d^{3} e^{2} - 12 \, B b c d^{2} e^{3} - 6 \, A c^{2} d^{2} e^{3} + 3 \, B b^{2} d e^{4} + 6 \, B a c d e^{4} + 6 \, A b c d e^{4} - 2 \, B a b e^{5} - A b^{2} e^{5} - 2 \, A a c e^{5}\right )} x^{2} + 3 \, {\left (35 \, B c^{2} d^{4} e - 40 \, B b c d^{3} e^{2} - 20 \, A c^{2} d^{3} e^{2} + 9 \, B b^{2} d^{2} e^{3} + 18 \, B a c d^{2} e^{3} + 18 \, A b c d^{2} e^{3} - 4 \, B a b d e^{4} - 2 \, A b^{2} d e^{4} - 4 \, A a c d e^{4} - B a^{2} e^{5} - 2 \, A a b e^{5}\right )} x\right )} e^{\left (-6\right )}}{6 \, {\left (x e + d\right )}^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.06, size = 690, normalized size = 2.41 \begin {gather*} -\frac {A \,a^{2}}{3 \left (e x +d \right )^{3} e}+\frac {2 A a b d}{3 \left (e x +d \right )^{3} e^{2}}-\frac {2 A a c \,d^{2}}{3 \left (e x +d \right )^{3} e^{3}}-\frac {A \,b^{2} d^{2}}{3 \left (e x +d \right )^{3} e^{3}}+\frac {2 A b c \,d^{3}}{3 \left (e x +d \right )^{3} e^{4}}-\frac {A \,c^{2} d^{4}}{3 \left (e x +d \right )^{3} e^{5}}+\frac {B \,a^{2} d}{3 \left (e x +d \right )^{3} e^{2}}-\frac {2 B a b \,d^{2}}{3 \left (e x +d \right )^{3} e^{3}}+\frac {2 B a c \,d^{3}}{3 \left (e x +d \right )^{3} e^{4}}+\frac {B \,b^{2} d^{3}}{3 \left (e x +d \right )^{3} e^{4}}-\frac {2 B b c \,d^{4}}{3 \left (e x +d \right )^{3} e^{5}}+\frac {B \,c^{2} d^{5}}{3 \left (e x +d \right )^{3} e^{6}}-\frac {A a b}{\left (e x +d \right )^{2} e^{2}}+\frac {2 A a c d}{\left (e x +d \right )^{2} e^{3}}+\frac {A \,b^{2} d}{\left (e x +d \right )^{2} e^{3}}-\frac {3 A b c \,d^{2}}{\left (e x +d \right )^{2} e^{4}}+\frac {2 A \,c^{2} d^{3}}{\left (e x +d \right )^{2} e^{5}}-\frac {B \,a^{2}}{2 \left (e x +d \right )^{2} e^{2}}+\frac {2 B a b d}{\left (e x +d \right )^{2} e^{3}}-\frac {3 B a c \,d^{2}}{\left (e x +d \right )^{2} e^{4}}-\frac {3 B \,b^{2} d^{2}}{2 \left (e x +d \right )^{2} e^{4}}+\frac {4 B b c \,d^{3}}{\left (e x +d \right )^{2} e^{5}}-\frac {5 B \,c^{2} d^{4}}{2 \left (e x +d \right )^{2} e^{6}}+\frac {B \,c^{2} x^{2}}{2 e^{4}}-\frac {2 A a c}{\left (e x +d \right ) e^{3}}-\frac {A \,b^{2}}{\left (e x +d \right ) e^{3}}+\frac {6 A b c d}{\left (e x +d \right ) e^{4}}+\frac {2 A b c \ln \left (e x +d \right )}{e^{4}}-\frac {6 A \,c^{2} d^{2}}{\left (e x +d \right ) e^{5}}-\frac {4 A \,c^{2} d \ln \left (e x +d \right )}{e^{5}}+\frac {A \,c^{2} x}{e^{4}}-\frac {2 B a b}{\left (e x +d \right ) e^{3}}+\frac {6 B a c d}{\left (e x +d \right ) e^{4}}+\frac {2 B a c \ln \left (e x +d \right )}{e^{4}}+\frac {3 B \,b^{2} d}{\left (e x +d \right ) e^{4}}+\frac {B \,b^{2} \ln \left (e x +d \right )}{e^{4}}-\frac {12 B b c \,d^{2}}{\left (e x +d \right ) e^{5}}-\frac {8 B b c d \ln \left (e x +d \right )}{e^{5}}+\frac {2 B b c x}{e^{4}}+\frac {10 B \,c^{2} d^{3}}{\left (e x +d \right ) e^{6}}+\frac {10 B \,c^{2} d^{2} \ln \left (e x +d \right )}{e^{6}}-\frac {4 B \,c^{2} d x}{e^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.66, size = 409, normalized size = 1.43 \begin {gather*} \frac {47 \, B c^{2} d^{5} - 2 \, A a^{2} e^{5} - 26 \, {\left (2 \, B b c + A c^{2}\right )} d^{4} e + 11 \, {\left (B b^{2} + 2 \, {\left (B a + A b\right )} c\right )} d^{3} e^{2} - 2 \, {\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} d^{2} e^{3} - {\left (B a^{2} + 2 \, A a b\right )} d e^{4} + 6 \, {\left (10 \, B c^{2} d^{3} e^{2} - 6 \, {\left (2 \, B b c + A c^{2}\right )} d^{2} e^{3} + 3 \, {\left (B b^{2} + 2 \, {\left (B a + A b\right )} c\right )} d e^{4} - {\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} e^{5}\right )} x^{2} + 3 \, {\left (35 \, B c^{2} d^{4} e - 20 \, {\left (2 \, B b c + A c^{2}\right )} d^{3} e^{2} + 9 \, {\left (B b^{2} + 2 \, {\left (B a + A b\right )} c\right )} d^{2} e^{3} - 2 \, {\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} d e^{4} - {\left (B a^{2} + 2 \, A a b\right )} e^{5}\right )} x}{6 \, {\left (e^{9} x^{3} + 3 \, d e^{8} x^{2} + 3 \, d^{2} e^{7} x + d^{3} e^{6}\right )}} + \frac {B c^{2} e x^{2} - 2 \, {\left (4 \, B c^{2} d - {\left (2 \, B b c + A c^{2}\right )} e\right )} x}{2 \, e^{5}} + \frac {{\left (10 \, B c^{2} d^{2} - 4 \, {\left (2 \, B b c + A c^{2}\right )} d e + {\left (B b^{2} + 2 \, {\left (B a + A b\right )} c\right )} e^{2}\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 = 2.37, size = 465, normalized size = 1.63 \begin {gather*} x\,\left (\frac {A\,c^2+2\,B\,b\,c}{e^4}-\frac {4\,B\,c^2\,d}{e^5}\right )-\frac {x^2\,\left (-3\,B\,b^2\,d\,e^3+A\,b^2\,e^4+12\,B\,b\,c\,d^2\,e^2-6\,A\,b\,c\,d\,e^3+2\,B\,a\,b\,e^4-10\,B\,c^2\,d^3\,e+6\,A\,c^2\,d^2\,e^2-6\,B\,a\,c\,d\,e^3+2\,A\,a\,c\,e^4\right )+x\,\left (\frac {B\,a^2\,e^4}{2}+2\,B\,a\,b\,d\,e^3+A\,a\,b\,e^4-9\,B\,a\,c\,d^2\,e^2+2\,A\,a\,c\,d\,e^3-\frac {9\,B\,b^2\,d^2\,e^2}{2}+A\,b^2\,d\,e^3+20\,B\,b\,c\,d^3\,e-9\,A\,b\,c\,d^2\,e^2-\frac {35\,B\,c^2\,d^4}{2}+10\,A\,c^2\,d^3\,e\right )+\frac {B\,a^2\,d\,e^4+2\,A\,a^2\,e^5+4\,B\,a\,b\,d^2\,e^3+2\,A\,a\,b\,d\,e^4-22\,B\,a\,c\,d^3\,e^2+4\,A\,a\,c\,d^2\,e^3-11\,B\,b^2\,d^3\,e^2+2\,A\,b^2\,d^2\,e^3+52\,B\,b\,c\,d^4\,e-22\,A\,b\,c\,d^3\,e^2-47\,B\,c^2\,d^5+26\,A\,c^2\,d^4\,e}{6\,e}}{d^3\,e^5+3\,d^2\,e^6\,x+3\,d\,e^7\,x^2+e^8\,x^3}+\frac {\ln \left (d+e\,x\right )\,\left (B\,b^2\,e^2-8\,B\,b\,c\,d\,e+2\,A\,b\,c\,e^2+10\,B\,c^2\,d^2-4\,A\,c^2\,d\,e+2\,B\,a\,c\,e^2\right )}{e^6}+\frac {B\,c^2\,x^2}{2\,e^4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 152.98, size = 547, normalized size = 1.91 \begin {gather*} \frac {B c^{2} x^{2}}{2 e^{4}} + x \left (\frac {A c^{2}}{e^{4}} + \frac {2 B b c}{e^{4}} - \frac {4 B c^{2} d}{e^{5}}\right ) + \frac {- 2 A a^{2} e^{5} - 2 A a b d e^{4} - 4 A a c d^{2} e^{3} - 2 A b^{2} d^{2} e^{3} + 22 A b c d^{3} e^{2} - 26 A c^{2} d^{4} e - B a^{2} d e^{4} - 4 B a b d^{2} e^{3} + 22 B a c d^{3} e^{2} + 11 B b^{2} d^{3} e^{2} - 52 B b c d^{4} e + 47 B c^{2} d^{5} + x^{2} \left (- 12 A a c e^{5} - 6 A b^{2} e^{5} + 36 A b c d e^{4} - 36 A c^{2} d^{2} e^{3} - 12 B a b e^{5} + 36 B a c d e^{4} + 18 B b^{2} d e^{4} - 72 B b c d^{2} e^{3} + 60 B c^{2} d^{3} e^{2}\right ) + x \left (- 6 A a b e^{5} - 12 A a c d e^{4} - 6 A b^{2} d e^{4} + 54 A b c d^{2} e^{3} - 60 A c^{2} d^{3} e^{2} - 3 B a^{2} e^{5} - 12 B a b d e^{4} + 54 B a c d^{2} e^{3} + 27 B b^{2} d^{2} e^{3} - 120 B b c d^{3} e^{2} + 105 B c^{2} d^{4} e\right )}{6 d^{3} e^{6} + 18 d^{2} e^{7} x + 18 d e^{8} x^{2} + 6 e^{9} x^{3}} + \frac {\left (2 A b c e^{2} - 4 A c^{2} d e + 2 B a c e^{2} + B b^{2} e^{2} - 8 B b c d e + 10 B c^{2} d^{2}\right ) \log {\left (d + e x \right )}}{e^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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