Optimal. Leaf size=194 \[ -\frac {(c d-b e) (2 B d (2 c d-b e)-A e (3 c d-b e)) x}{e^5}+\frac {(c d-b e) (3 B c d-b B e-2 A c e) x^2}{2 e^4}-\frac {c (2 B c d-2 b B e-A c e) x^3}{3 e^3}+\frac {B c^2 x^4}{4 e^2}+\frac {d^2 (B d-A e) (c d-b e)^2}{e^6 (d+e x)}+\frac {d (c d-b e) (B d (5 c d-3 b e)-2 A e (2 c d-b e)) \log (d+e x)}{e^6} \]
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Rubi [A]
time = 0.19, antiderivative size = 194, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 1, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.042, Rules used = {785}
\begin {gather*} \frac {d^2 (B d-A e) (c d-b e)^2}{e^6 (d+e x)}+\frac {d (c d-b e) \log (d+e x) (B d (5 c d-3 b e)-2 A e (2 c d-b e))}{e^6}-\frac {x (c d-b e) (2 B d (2 c d-b e)-A e (3 c d-b e))}{e^5}+\frac {x^2 (c d-b e) (-2 A c e-b B e+3 B c d)}{2 e^4}-\frac {c x^3 (-A c e-2 b B e+2 B c d)}{3 e^3}+\frac {B c^2 x^4}{4 e^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 785
Rubi steps
\begin {align*} \int \frac {(A+B x) \left (b x+c x^2\right )^2}{(d+e x)^2} \, dx &=\int \left (\frac {(c d-b e) (-2 B d (2 c d-b e)+A e (3 c d-b e))}{e^5}+\frac {(-c d+b e) (-3 B c d+b B e+2 A c e) x}{e^4}+\frac {c (-2 B c d+2 b B e+A c e) x^2}{e^3}+\frac {B c^2 x^3}{e^2}-\frac {d^2 (B d-A e) (c d-b e)^2}{e^5 (d+e x)^2}+\frac {d (c d-b e) (B d (5 c d-3 b e)-2 A e (2 c d-b e))}{e^5 (d+e x)}\right ) \, dx\\ &=-\frac {(c d-b e) (2 B d (2 c d-b e)-A e (3 c d-b e)) x}{e^5}+\frac {(c d-b e) (3 B c d-b B e-2 A c e) x^2}{2 e^4}-\frac {c (2 B c d-2 b B e-A c e) x^3}{3 e^3}+\frac {B c^2 x^4}{4 e^2}+\frac {d^2 (B d-A e) (c d-b e)^2}{e^6 (d+e x)}+\frac {d (c d-b e) (B d (5 c d-3 b e)-2 A e (2 c d-b e)) \log (d+e x)}{e^6}\\ \end {align*}
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Mathematica [A]
time = 0.05, size = 184, normalized size = 0.95 \begin {gather*} \frac {12 e (-c d+b e) (2 B d (2 c d-b e)+A e (-3 c d+b e)) x+6 e^2 (-c d+b e) (-3 B c d+b B e+2 A c e) x^2+4 c e^3 (-2 B c d+2 b B e+A c e) x^3+3 B c^2 e^4 x^4+\frac {12 d^2 (B d-A e) (c d-b e)^2}{d+e x}+12 d (c d-b e) (B d (5 c d-3 b e)+2 A e (-2 c d+b e)) \log (d+e x)}{12 e^6} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.70, size = 315, normalized size = 1.62
method | result | size |
norman | \(\frac {\frac {\left (2 A \,b^{2} d^{2} e^{3}-6 A b c \,d^{3} e^{2}+4 A \,c^{2} d^{4} e -3 B \,b^{2} d^{3} e^{2}+8 B b c \,d^{4} e -5 B \,c^{2} d^{5}\right ) x}{d \,e^{5}}+\frac {\left (2 A \,b^{2} e^{3}-6 A b c d \,e^{2}+4 A \,c^{2} d^{2} e -3 B \,b^{2} d \,e^{2}+8 B b c \,d^{2} e -5 B \,c^{2} d^{3}\right ) x^{2}}{2 e^{4}}+\frac {\left (6 A b c \,e^{2}-4 A \,c^{2} d e +3 B \,e^{2} b^{2}-8 B b c d e +5 B \,c^{2} d^{2}\right ) x^{3}}{6 e^{3}}+\frac {B \,c^{2} x^{5}}{4 e}+\frac {c \left (4 A c e +8 b B e -5 B c d \right ) x^{4}}{12 e^{2}}}{e x +d}-\frac {d \left (2 A \,b^{2} e^{3}-6 A b c d \,e^{2}+4 A \,c^{2} d^{2} e -3 B \,b^{2} d \,e^{2}+8 B b c \,d^{2} e -5 B \,c^{2} d^{3}\right ) \ln \left (e x +d \right )}{e^{6}}\) | \(304\) |
default | \(\frac {\frac {1}{4} B \,c^{2} x^{4} e^{3}+\frac {1}{3} A \,c^{2} e^{3} x^{3}+\frac {2}{3} B b c \,e^{3} x^{3}-\frac {2}{3} B \,c^{2} d \,e^{2} x^{3}+A b c \,e^{3} x^{2}-A \,c^{2} d \,e^{2} x^{2}+\frac {1}{2} B \,b^{2} e^{3} x^{2}-2 B b c d \,e^{2} x^{2}+\frac {3}{2} B \,c^{2} d^{2} e \,x^{2}+A \,b^{2} e^{3} x -4 A b c d \,e^{2} x +3 A \,c^{2} d^{2} e x -2 B \,b^{2} d \,e^{2} x +6 B b c \,d^{2} e x -4 B \,c^{2} d^{3} x}{e^{5}}-\frac {d^{2} \left (A \,b^{2} e^{3}-2 A b c d \,e^{2}+A \,c^{2} d^{2} e -B \,b^{2} d \,e^{2}+2 B b c \,d^{2} e -B \,c^{2} d^{3}\right )}{e^{6} \left (e x +d \right )}-\frac {d \left (2 A \,b^{2} e^{3}-6 A b c d \,e^{2}+4 A \,c^{2} d^{2} e -3 B \,b^{2} d \,e^{2}+8 B b c \,d^{2} e -5 B \,c^{2} d^{3}\right ) \ln \left (e x +d \right )}{e^{6}}\) | \(315\) |
risch | \(\frac {2 B b c \,x^{3}}{3 e^{2}}-\frac {2 B \,c^{2} d \,x^{3}}{3 e^{3}}-\frac {d^{2} A \,b^{2}}{e^{3} \left (e x +d \right )}-\frac {d^{4} A \,c^{2}}{e^{5} \left (e x +d \right )}+\frac {d^{3} B \,b^{2}}{e^{4} \left (e x +d \right )}+\frac {6 d^{2} \ln \left (e x +d \right ) A b c}{e^{4}}-\frac {8 d^{3} \ln \left (e x +d \right ) B b c}{e^{5}}+\frac {A \,c^{2} x^{3}}{3 e^{2}}+\frac {A \,b^{2} x}{e^{2}}-\frac {2 d \ln \left (e x +d \right ) A \,b^{2}}{e^{3}}-\frac {4 d^{3} \ln \left (e x +d \right ) A \,c^{2}}{e^{5}}+\frac {3 d^{2} \ln \left (e x +d \right ) B \,b^{2}}{e^{4}}+\frac {5 d^{4} \ln \left (e x +d \right ) B \,c^{2}}{e^{6}}-\frac {2 B b c d \,x^{2}}{e^{3}}-\frac {4 A b c d x}{e^{3}}+\frac {6 B b c \,d^{2} x}{e^{4}}+\frac {2 d^{3} A b c}{e^{4} \left (e x +d \right )}-\frac {2 d^{4} B b c}{e^{5} \left (e x +d \right )}+\frac {d^{5} B \,c^{2}}{e^{6} \left (e x +d \right )}+\frac {A b c \,x^{2}}{e^{2}}-\frac {A \,c^{2} d \,x^{2}}{e^{3}}+\frac {3 B \,c^{2} d^{2} x^{2}}{2 e^{4}}+\frac {3 A \,c^{2} d^{2} x}{e^{4}}-\frac {2 B \,b^{2} d x}{e^{3}}-\frac {4 B \,c^{2} d^{3} x}{e^{5}}+\frac {B \,c^{2} x^{4}}{4 e^{2}}+\frac {b^{2} B \,x^{2}}{2 e^{2}}\) | \(394\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 295, normalized size = 1.52 \begin {gather*} {\left (5 \, B c^{2} d^{4} - 2 \, A b^{2} d e^{3} - 4 \, {\left (2 \, B b c e + A c^{2} e\right )} d^{3} + 3 \, {\left (B b^{2} e^{2} + 2 \, A b c e^{2}\right )} d^{2}\right )} e^{\left (-6\right )} \log \left (x e + d\right ) + \frac {1}{12} \, {\left (3 \, B c^{2} x^{4} e^{3} - 4 \, {\left (2 \, B c^{2} d e^{2} - 2 \, B b c e^{3} - A c^{2} e^{3}\right )} x^{3} + 6 \, {\left (3 \, B c^{2} d^{2} e + B b^{2} e^{3} + 2 \, A b c e^{3} - 2 \, {\left (2 \, B b c e^{2} + A c^{2} e^{2}\right )} d\right )} x^{2} - 12 \, {\left (4 \, B c^{2} d^{3} - A b^{2} e^{3} - 3 \, {\left (2 \, B b c e + A c^{2} e\right )} d^{2} + 2 \, {\left (B b^{2} e^{2} + 2 \, A b c e^{2}\right )} d\right )} x\right )} e^{\left (-5\right )} + \frac {B c^{2} d^{5} - A b^{2} d^{2} e^{3} - {\left (2 \, B b c e + A c^{2} e\right )} d^{4} + {\left (B b^{2} e^{2} + 2 \, A b c e^{2}\right )} d^{3}}{x e^{7} + d e^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 404 vs.
\(2 (199) = 398\).
time = 2.40, size = 404, normalized size = 2.08 \begin {gather*} \frac {12 \, B c^{2} d^{5} + {\left (3 \, B c^{2} x^{5} + 12 \, A b^{2} x^{2} + 4 \, {\left (2 \, B b c + A c^{2}\right )} x^{4} + 6 \, {\left (B b^{2} + 2 \, A b c\right )} x^{3}\right )} e^{5} - {\left (5 \, B c^{2} d x^{4} - 12 \, A b^{2} d x + 8 \, {\left (2 \, B b c + A c^{2}\right )} d x^{3} + 18 \, {\left (B b^{2} + 2 \, A b c\right )} d x^{2}\right )} e^{4} + 2 \, {\left (5 \, B c^{2} d^{2} x^{3} - 6 \, A b^{2} d^{2} + 12 \, {\left (2 \, B b c + A c^{2}\right )} d^{2} x^{2} - 12 \, {\left (B b^{2} + 2 \, A b c\right )} d^{2} x\right )} e^{3} - 6 \, {\left (5 \, B c^{2} d^{3} x^{2} - 6 \, {\left (2 \, B b c + A c^{2}\right )} d^{3} x - 2 \, {\left (B b^{2} + 2 \, A b c\right )} d^{3}\right )} e^{2} - 12 \, {\left (4 \, B c^{2} d^{4} x + {\left (2 \, B b c + A c^{2}\right )} d^{4}\right )} e + 12 \, {\left (5 \, B c^{2} d^{5} - 2 \, A b^{2} d x e^{4} - {\left (2 \, A b^{2} d^{2} - 3 \, {\left (B b^{2} + 2 \, A b c\right )} d^{2} x\right )} e^{3} - {\left (4 \, {\left (2 \, B b c + A c^{2}\right )} d^{3} x - 3 \, {\left (B b^{2} + 2 \, A b c\right )} d^{3}\right )} e^{2} + {\left (5 \, B c^{2} d^{4} x - 4 \, {\left (2 \, B b c + A c^{2}\right )} d^{4}\right )} e\right )} \log \left (x e + d\right )}{12 \, {\left (x e^{7} + d e^{6}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.74, size = 316, normalized size = 1.63 \begin {gather*} \frac {B c^{2} x^{4}}{4 e^{2}} + \frac {d \left (b e - c d\right ) \left (- 2 A b e^{2} + 4 A c d e + 3 B b d e - 5 B c d^{2}\right ) \log {\left (d + e x \right )}}{e^{6}} + x^{3} \left (\frac {A c^{2}}{3 e^{2}} + \frac {2 B b c}{3 e^{2}} - \frac {2 B c^{2} d}{3 e^{3}}\right ) + x^{2} \left (\frac {A b c}{e^{2}} - \frac {A c^{2} d}{e^{3}} + \frac {B b^{2}}{2 e^{2}} - \frac {2 B b c d}{e^{3}} + \frac {3 B c^{2} d^{2}}{2 e^{4}}\right ) + x \left (\frac {A b^{2}}{e^{2}} - \frac {4 A b c d}{e^{3}} + \frac {3 A c^{2} d^{2}}{e^{4}} - \frac {2 B b^{2} d}{e^{3}} + \frac {6 B b c d^{2}}{e^{4}} - \frac {4 B c^{2} d^{3}}{e^{5}}\right ) + \frac {- A b^{2} d^{2} e^{3} + 2 A b c d^{3} e^{2} - A c^{2} d^{4} e + B b^{2} d^{3} e^{2} - 2 B b c d^{4} e + B c^{2} d^{5}}{d e^{6} + e^{7} x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.34, size = 380, normalized size = 1.96 \begin {gather*} \frac {1}{12} \, {\left (3 \, B c^{2} - \frac {4 \, {\left (5 \, B c^{2} d e - 2 \, B b c e^{2} - A c^{2} e^{2}\right )} e^{\left (-1\right )}}{x e + d} + \frac {6 \, {\left (10 \, B c^{2} d^{2} e^{2} - 8 \, B b c d e^{3} - 4 \, A c^{2} d e^{3} + B b^{2} e^{4} + 2 \, A b c e^{4}\right )} e^{\left (-2\right )}}{{\left (x e + d\right )}^{2}} - \frac {12 \, {\left (10 \, B c^{2} d^{3} e^{3} - 12 \, B b c d^{2} e^{4} - 6 \, A c^{2} d^{2} e^{4} + 3 \, B b^{2} d e^{5} + 6 \, A b c d e^{5} - A b^{2} e^{6}\right )} e^{\left (-3\right )}}{{\left (x e + d\right )}^{3}}\right )} {\left (x e + d\right )}^{4} e^{\left (-6\right )} - {\left (5 \, B c^{2} d^{4} - 8 \, B b c d^{3} e - 4 \, A c^{2} d^{3} e + 3 \, B b^{2} d^{2} e^{2} + 6 \, A b c d^{2} e^{2} - 2 \, A b^{2} d e^{3}\right )} e^{\left (-6\right )} \log \left (\frac {{\left | x e + d \right |} e^{\left (-1\right )}}{{\left (x e + d\right )}^{2}}\right ) + {\left (\frac {B c^{2} d^{5} e^{4}}{x e + d} - \frac {2 \, B b c d^{4} e^{5}}{x e + d} - \frac {A c^{2} d^{4} e^{5}}{x e + d} + \frac {B b^{2} d^{3} e^{6}}{x e + d} + \frac {2 \, A b c d^{3} e^{6}}{x e + d} - \frac {A b^{2} d^{2} e^{7}}{x e + d}\right )} e^{\left (-10\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.43, size = 371, normalized size = 1.91 \begin {gather*} x\,\left (\frac {A\,b^2}{e^2}-\frac {d^2\,\left (\frac {A\,c^2+2\,B\,b\,c}{e^2}-\frac {2\,B\,c^2\,d}{e^3}\right )}{e^2}+\frac {2\,d\,\left (\frac {2\,d\,\left (\frac {A\,c^2+2\,B\,b\,c}{e^2}-\frac {2\,B\,c^2\,d}{e^3}\right )}{e}-\frac {B\,b^2+2\,A\,c\,b}{e^2}+\frac {B\,c^2\,d^2}{e^4}\right )}{e}\right )+x^3\,\left (\frac {A\,c^2+2\,B\,b\,c}{3\,e^2}-\frac {2\,B\,c^2\,d}{3\,e^3}\right )-x^2\,\left (\frac {d\,\left (\frac {A\,c^2+2\,B\,b\,c}{e^2}-\frac {2\,B\,c^2\,d}{e^3}\right )}{e}-\frac {B\,b^2+2\,A\,c\,b}{2\,e^2}+\frac {B\,c^2\,d^2}{2\,e^4}\right )+\frac {B\,b^2\,d^3\,e^2-A\,b^2\,d^2\,e^3-2\,B\,b\,c\,d^4\,e+2\,A\,b\,c\,d^3\,e^2+B\,c^2\,d^5-A\,c^2\,d^4\,e}{e\,\left (x\,e^6+d\,e^5\right )}+\frac {\ln \left (d+e\,x\right )\,\left (3\,B\,b^2\,d^2\,e^2-2\,A\,b^2\,d\,e^3-8\,B\,b\,c\,d^3\,e+6\,A\,b\,c\,d^2\,e^2+5\,B\,c^2\,d^4-4\,A\,c^2\,d^3\,e\right )}{e^6}+\frac {B\,c^2\,x^4}{4\,e^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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