Optimal. Leaf size=297 \[ -\frac {\left (a^2 e^4 (C d-B e)+c^2 d^3 \left (C d^2-e (B d-A e)\right )+2 a c d e^2 \left (C d^2-e (B d-A e)\right )\right ) x}{e^6}+\frac {\left (a^2 C e^4+c^2 d^2 \left (C d^2-e (B d-A e)\right )+2 a c e^2 \left (C d^2-e (B d-A e)\right )\right ) x^2}{2 e^5}-\frac {c \left (2 a e^2 (C d-B e)+c d \left (C d^2-e (B d-A e)\right )\right ) x^3}{3 e^4}+\frac {c \left (2 a C e^2+c \left (C d^2-e (B d-A e)\right )\right ) x^4}{4 e^3}-\frac {c^2 (C d-B e) x^5}{5 e^2}+\frac {c^2 C x^6}{6 e}+\frac {\left (c d^2+a e^2\right )^2 \left (C d^2-B d e+A e^2\right ) \log (d+e x)}{e^7} \]
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Rubi [A]
time = 0.40, antiderivative size = 295, normalized size of antiderivative = 0.99, number of steps
used = 2, number of rules used = 1, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.037, Rules used = {1642}
\begin {gather*} \frac {x^2 \left (a^2 C e^4+2 a c e^2 \left (C d^2-e (B d-A e)\right )+c^2 \left (C d^4-d^2 e (B d-A e)\right )\right )}{2 e^5}-\frac {x \left (a^2 e^4 (C d-B e)+2 a c d e^2 \left (C d^2-e (B d-A e)\right )+c^2 \left (C d^5-d^3 e (B d-A e)\right )\right )}{e^6}-\frac {c x^3 \left (2 a e^2 (C d-B e)-c d e (B d-A e)+c C d^3\right )}{3 e^4}+\frac {\left (a e^2+c d^2\right )^2 \log (d+e x) \left (A e^2-B d e+C d^2\right )}{e^7}+\frac {c x^4 \left (2 a C e^2-c e (B d-A e)+c C d^2\right )}{4 e^3}-\frac {c^2 x^5 (C d-B e)}{5 e^2}+\frac {c^2 C x^6}{6 e} \end {gather*}
Antiderivative was successfully verified.
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Rule 1642
Rubi steps
\begin {align*} \int \frac {\left (a+c x^2\right )^2 \left (A+B x+C x^2\right )}{d+e x} \, dx &=\int \left (\frac {-a^2 e^4 (C d-B e)-2 a c d e^2 \left (C d^2-e (B d-A e)\right )-c^2 \left (C d^5-d^3 e (B d-A e)\right )}{e^6}+\frac {\left (a^2 C e^4+2 a c e^2 \left (C d^2-e (B d-A e)\right )+c^2 \left (C d^4-d^2 e (B d-A e)\right )\right ) x}{e^5}+\frac {c \left (-c C d^3+c d e (B d-A e)-2 a e^2 (C d-B e)\right ) x^2}{e^4}+\frac {c \left (c C d^2+2 a C e^2-c e (B d-A e)\right ) x^3}{e^3}+\frac {c^2 (-C d+B e) x^4}{e^2}+\frac {c^2 C x^5}{e}+\frac {\left (c d^2+a e^2\right )^2 \left (C d^2-B d e+A e^2\right )}{e^6 (d+e x)}\right ) \, dx\\ &=-\frac {\left (a^2 e^4 (C d-B e)+2 a c d e^2 \left (C d^2-e (B d-A e)\right )+c^2 \left (C d^5-d^3 e (B d-A e)\right )\right ) x}{e^6}+\frac {\left (a^2 C e^4+2 a c e^2 \left (C d^2-e (B d-A e)\right )+c^2 \left (C d^4-d^2 e (B d-A e)\right )\right ) x^2}{2 e^5}-\frac {c \left (c C d^3-c d e (B d-A e)+2 a e^2 (C d-B e)\right ) x^3}{3 e^4}+\frac {c \left (c C d^2+2 a C e^2-c e (B d-A e)\right ) x^4}{4 e^3}-\frac {c^2 (C d-B e) x^5}{5 e^2}+\frac {c^2 C x^6}{6 e}+\frac {\left (c d^2+a e^2\right )^2 \left (C d^2-B d e+A e^2\right ) \log (d+e x)}{e^7}\\ \end {align*}
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Mathematica [A]
time = 0.11, size = 285, normalized size = 0.96 \begin {gather*} \frac {e x \left (30 a^2 e^4 (-2 C d+2 B e+C e x)+10 a c e^2 \left (C \left (-12 d^3+6 d^2 e x-4 d e^2 x^2+3 e^3 x^3\right )+2 e \left (3 A e (-2 d+e x)+B \left (6 d^2-3 d e x+2 e^2 x^2\right )\right )\right )+c^2 \left (C \left (-60 d^5+30 d^4 e x-20 d^3 e^2 x^2+15 d^2 e^3 x^3-12 d e^4 x^4+10 e^5 x^5\right )+e \left (5 A e \left (-12 d^3+6 d^2 e x-4 d e^2 x^2+3 e^3 x^3\right )+B \left (60 d^4-30 d^3 e x+20 d^2 e^2 x^2-15 d e^3 x^3+12 e^4 x^4\right )\right )\right )\right )+60 \left (c d^2+a e^2\right )^2 \left (C d^2+e (-B d+A e)\right ) \log (d+e x)}{60 e^7} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.09, size = 441, normalized size = 1.48
method | result | size |
norman | \(\frac {\left (2 A a c \,e^{4}+A \,c^{2} d^{2} e^{2}-2 B a c d \,e^{3}-B \,c^{2} d^{3} e +a^{2} C \,e^{4}+2 C a c \,d^{2} e^{2}+C \,c^{2} d^{4}\right ) x^{2}}{2 e^{5}}-\frac {\left (2 A a c d \,e^{4}+A \,c^{2} d^{3} e^{2}-B \,a^{2} e^{5}-2 B a c \,d^{2} e^{3}-B \,c^{2} d^{4} e +C \,a^{2} d \,e^{4}+2 C a c \,d^{3} e^{2}+C \,c^{2} d^{5}\right ) x}{e^{6}}-\frac {c \left (d \,e^{2} c A -2 B a \,e^{3}-B c \,d^{2} e +2 C a d \,e^{2}+C c \,d^{3}\right ) x^{3}}{3 e^{4}}+\frac {c \left (A c \,e^{2}-B c d e +2 a C \,e^{2}+C c \,d^{2}\right ) x^{4}}{4 e^{3}}+\frac {c^{2} C \,x^{6}}{6 e}+\frac {c^{2} \left (B e -C d \right ) x^{5}}{5 e^{2}}+\frac {\left (A \,a^{2} e^{6}+2 A a c \,d^{2} e^{4}+A \,c^{2} d^{4} e^{2}-B \,a^{2} d \,e^{5}-2 B a c \,d^{3} e^{3}-B \,c^{2} d^{5} e +C \,a^{2} d^{2} e^{4}+2 C a c \,d^{4} e^{2}+C \,c^{2} d^{6}\right ) \ln \left (e x +d \right )}{e^{7}}\) | \(375\) |
default | \(\frac {\left (A \,a^{2} e^{6}+2 A a c \,d^{2} e^{4}+A \,c^{2} d^{4} e^{2}-B \,a^{2} d \,e^{5}-2 B a c \,d^{3} e^{3}-B \,c^{2} d^{5} e +C \,a^{2} d^{2} e^{4}+2 C a c \,d^{4} e^{2}+C \,c^{2} d^{6}\right ) \ln \left (e x +d \right )}{e^{7}}-\frac {B a c d \,e^{4} x^{2}+\frac {1}{3} A \,c^{2} d \,e^{4} x^{3}-\frac {2}{3} B a c \,e^{5} x^{3}-\frac {1}{3} B \,c^{2} d^{2} e^{3} x^{3}+\frac {1}{3} C \,c^{2} d^{3} e^{2} x^{3}-A a c \,e^{5} x^{2}-\frac {1}{2} A \,c^{2} d^{2} e^{3} x^{2}+\frac {1}{2} B \,c^{2} d^{3} e^{2} x^{2}-\frac {1}{2} C \,c^{2} d^{4} e \,x^{2}+\frac {1}{4} B \,c^{2} d \,e^{4} x^{4}-\frac {1}{2} C a c \,e^{5} x^{4}-\frac {1}{4} C \,c^{2} d^{2} e^{3} x^{4}+\frac {1}{5} C \,c^{2} d \,e^{4} x^{5}-B \,c^{2} d^{4} e x +C \,a^{2} d \,e^{4} x +A \,c^{2} d^{3} e^{2} x -\frac {1}{6} c^{2} C \,x^{6} e^{5}-\frac {1}{5} B \,c^{2} e^{5} x^{5}-\frac {1}{4} A \,c^{2} e^{5} x^{4}-\frac {1}{2} C \,a^{2} e^{5} x^{2}-B \,a^{2} e^{5} x +C \,c^{2} d^{5} x +\frac {2}{3} C a c d \,e^{4} x^{3}-C a c \,d^{2} e^{3} x^{2}+2 A a c d \,e^{4} x -2 B a c \,d^{2} e^{3} x +2 C a c \,d^{3} e^{2} x}{e^{6}}\) | \(441\) |
risch | \(\frac {B \,c^{2} x^{5}}{5 e}+\frac {A \,c^{2} x^{4}}{4 e}+\frac {C \,a^{2} x^{2}}{2 e}+\frac {B \,a^{2} x}{e}+\frac {\ln \left (e x +d \right ) A \,a^{2}}{e}-\frac {2 \ln \left (e x +d \right ) B a c \,d^{3}}{e^{4}}+\frac {2 \ln \left (e x +d \right ) C a c \,d^{4}}{e^{5}}-\frac {2 C a c d \,x^{3}}{3 e^{2}}+\frac {C a c \,d^{2} x^{2}}{e^{3}}-\frac {2 A a c d x}{e^{2}}+\frac {2 B a c \,d^{2} x}{e^{3}}-\frac {2 C a c \,d^{3} x}{e^{4}}+\frac {2 \ln \left (e x +d \right ) A a c \,d^{2}}{e^{3}}-\frac {B a c d \,x^{2}}{e^{2}}+\frac {c^{2} C \,x^{6}}{6 e}+\frac {B \,c^{2} d^{4} x}{e^{5}}-\frac {C \,a^{2} d x}{e^{2}}-\frac {A \,c^{2} d^{3} x}{e^{4}}-\frac {C \,c^{2} d^{5} x}{e^{6}}+\frac {\ln \left (e x +d \right ) A \,c^{2} d^{4}}{e^{5}}-\frac {\ln \left (e x +d \right ) B \,a^{2} d}{e^{2}}-\frac {\ln \left (e x +d \right ) B \,c^{2} d^{5}}{e^{6}}+\frac {\ln \left (e x +d \right ) C \,a^{2} d^{2}}{e^{3}}+\frac {\ln \left (e x +d \right ) C \,c^{2} d^{6}}{e^{7}}-\frac {A \,c^{2} d \,x^{3}}{3 e^{2}}+\frac {2 B a c \,x^{3}}{3 e}+\frac {B \,c^{2} d^{2} x^{3}}{3 e^{3}}-\frac {C \,c^{2} d^{3} x^{3}}{3 e^{4}}+\frac {A a c \,x^{2}}{e}+\frac {A \,c^{2} d^{2} x^{2}}{2 e^{3}}-\frac {B \,c^{2} d^{3} x^{2}}{2 e^{4}}+\frac {C \,c^{2} d^{4} x^{2}}{2 e^{5}}-\frac {B \,c^{2} d \,x^{4}}{4 e^{2}}+\frac {C a c \,x^{4}}{2 e}+\frac {C \,c^{2} d^{2} x^{4}}{4 e^{3}}-\frac {C \,c^{2} d \,x^{5}}{5 e^{2}}\) | \(490\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.33, size = 368, normalized size = 1.24 \begin {gather*} {\left (C c^{2} d^{6} - B c^{2} d^{5} e - 2 \, B a c d^{3} e^{3} + {\left (2 \, C a c e^{2} + A c^{2} e^{2}\right )} d^{4} - B a^{2} d e^{5} + A a^{2} e^{6} + {\left (C a^{2} e^{4} + 2 \, A a c e^{4}\right )} d^{2}\right )} e^{\left (-7\right )} \log \left (x e + d\right ) + \frac {1}{60} \, {\left (10 \, C c^{2} x^{6} e^{5} - 12 \, {\left (C c^{2} d e^{4} - B c^{2} e^{5}\right )} x^{5} + 15 \, {\left (C c^{2} d^{2} e^{3} - B c^{2} d e^{4} + 2 \, C a c e^{5} + A c^{2} e^{5}\right )} x^{4} - 20 \, {\left (C c^{2} d^{3} e^{2} - B c^{2} d^{2} e^{3} - 2 \, B a c e^{5} + {\left (2 \, C a c e^{4} + A c^{2} e^{4}\right )} d\right )} x^{3} + 30 \, {\left (C c^{2} d^{4} e - B c^{2} d^{3} e^{2} - 2 \, B a c d e^{4} + C a^{2} e^{5} + 2 \, A a c e^{5} + {\left (2 \, C a c e^{3} + A c^{2} e^{3}\right )} d^{2}\right )} x^{2} - 60 \, {\left (C c^{2} d^{5} - B c^{2} d^{4} e - 2 \, B a c d^{2} e^{3} + {\left (2 \, C a c e^{2} + A c^{2} e^{2}\right )} d^{3} - B a^{2} e^{5} + {\left (C a^{2} e^{4} + 2 \, A a c e^{4}\right )} d\right )} x\right )} e^{\left (-6\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.34, size = 370, normalized size = 1.25 \begin {gather*} -\frac {1}{60} \, {\left (60 \, C c^{2} d^{5} x e - {\left (10 \, C c^{2} x^{6} + 12 \, B c^{2} x^{5} + 40 \, B a c x^{3} + 15 \, {\left (2 \, C a c + A c^{2}\right )} x^{4} + 60 \, B a^{2} x + 30 \, {\left (C a^{2} + 2 \, A a c\right )} x^{2}\right )} e^{6} + {\left (12 \, C c^{2} d x^{5} + 15 \, B c^{2} d x^{4} + 60 \, B a c d x^{2} + 20 \, {\left (2 \, C a c + A c^{2}\right )} d x^{3} + 60 \, {\left (C a^{2} + 2 \, A a c\right )} d x\right )} e^{5} - 5 \, {\left (3 \, C c^{2} d^{2} x^{4} + 4 \, B c^{2} d^{2} x^{3} + 24 \, B a c d^{2} x + 6 \, {\left (2 \, C a c + A c^{2}\right )} d^{2} x^{2}\right )} e^{4} + 10 \, {\left (2 \, C c^{2} d^{3} x^{3} + 3 \, B c^{2} d^{3} x^{2} + 6 \, {\left (2 \, C a c + A c^{2}\right )} d^{3} x\right )} e^{3} - 30 \, {\left (C c^{2} d^{4} x^{2} + 2 \, B c^{2} d^{4} x\right )} e^{2} - 60 \, {\left (C c^{2} d^{6} - B c^{2} d^{5} e - 2 \, B a c d^{3} e^{3} + {\left (2 \, C a c + A c^{2}\right )} d^{4} e^{2} - B a^{2} d e^{5} + A a^{2} e^{6} + {\left (C a^{2} + 2 \, A a c\right )} d^{2} e^{4}\right )} \log \left (x e + d\right )\right )} e^{\left (-7\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.47, size = 359, normalized size = 1.21 \begin {gather*} \frac {C c^{2} x^{6}}{6 e} + x^{5} \left (\frac {B c^{2}}{5 e} - \frac {C c^{2} d}{5 e^{2}}\right ) + x^{4} \left (\frac {A c^{2}}{4 e} - \frac {B c^{2} d}{4 e^{2}} + \frac {C a c}{2 e} + \frac {C c^{2} d^{2}}{4 e^{3}}\right ) + x^{3} \left (- \frac {A c^{2} d}{3 e^{2}} + \frac {2 B a c}{3 e} + \frac {B c^{2} d^{2}}{3 e^{3}} - \frac {2 C a c d}{3 e^{2}} - \frac {C c^{2} d^{3}}{3 e^{4}}\right ) + x^{2} \left (\frac {A a c}{e} + \frac {A c^{2} d^{2}}{2 e^{3}} - \frac {B a c d}{e^{2}} - \frac {B c^{2} d^{3}}{2 e^{4}} + \frac {C a^{2}}{2 e} + \frac {C a c d^{2}}{e^{3}} + \frac {C c^{2} d^{4}}{2 e^{5}}\right ) + x \left (- \frac {2 A a c d}{e^{2}} - \frac {A c^{2} d^{3}}{e^{4}} + \frac {B a^{2}}{e} + \frac {2 B a c d^{2}}{e^{3}} + \frac {B c^{2} d^{4}}{e^{5}} - \frac {C a^{2} d}{e^{2}} - \frac {2 C a c d^{3}}{e^{4}} - \frac {C c^{2} d^{5}}{e^{6}}\right ) + \frac {\left (a e^{2} + c d^{2}\right )^{2} \left (A e^{2} - B d e + C d^{2}\right ) \log {\left (d + e x \right )}}{e^{7}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 4.54, size = 416, normalized size = 1.40 \begin {gather*} {\left (C c^{2} d^{6} - B c^{2} d^{5} e + 2 \, C a c d^{4} e^{2} + A c^{2} d^{4} e^{2} - 2 \, B a c d^{3} e^{3} + C a^{2} d^{2} e^{4} + 2 \, A a c d^{2} e^{4} - B a^{2} d e^{5} + A a^{2} e^{6}\right )} e^{\left (-7\right )} \log \left ({\left | x e + d \right |}\right ) + \frac {1}{60} \, {\left (10 \, C c^{2} x^{6} e^{5} - 12 \, C c^{2} d x^{5} e^{4} + 15 \, C c^{2} d^{2} x^{4} e^{3} - 20 \, C c^{2} d^{3} x^{3} e^{2} + 30 \, C c^{2} d^{4} x^{2} e - 60 \, C c^{2} d^{5} x + 12 \, B c^{2} x^{5} e^{5} - 15 \, B c^{2} d x^{4} e^{4} + 20 \, B c^{2} d^{2} x^{3} e^{3} - 30 \, B c^{2} d^{3} x^{2} e^{2} + 60 \, B c^{2} d^{4} x e + 30 \, C a c x^{4} e^{5} + 15 \, A c^{2} x^{4} e^{5} - 40 \, C a c d x^{3} e^{4} - 20 \, A c^{2} d x^{3} e^{4} + 60 \, C a c d^{2} x^{2} e^{3} + 30 \, A c^{2} d^{2} x^{2} e^{3} - 120 \, C a c d^{3} x e^{2} - 60 \, A c^{2} d^{3} x e^{2} + 40 \, B a c x^{3} e^{5} - 60 \, B a c d x^{2} e^{4} + 120 \, B a c d^{2} x e^{3} + 30 \, C a^{2} x^{2} e^{5} + 60 \, A a c x^{2} e^{5} - 60 \, C a^{2} d x e^{4} - 120 \, A a c d x e^{4} + 60 \, B a^{2} x e^{5}\right )} e^{\left (-6\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 3.68, size = 422, normalized size = 1.42 \begin {gather*} x^5\,\left (\frac {B\,c^2}{5\,e}-\frac {C\,c^2\,d}{5\,e^2}\right )-x\,\left (\frac {d\,\left (\frac {C\,a^2+2\,A\,c\,a}{e}+\frac {d\,\left (\frac {d\,\left (\frac {A\,c^2+2\,C\,a\,c}{e}-\frac {d\,\left (\frac {B\,c^2}{e}-\frac {C\,c^2\,d}{e^2}\right )}{e}\right )}{e}-\frac {2\,B\,a\,c}{e}\right )}{e}\right )}{e}-\frac {B\,a^2}{e}\right )+x^4\,\left (\frac {A\,c^2+2\,C\,a\,c}{4\,e}-\frac {d\,\left (\frac {B\,c^2}{e}-\frac {C\,c^2\,d}{e^2}\right )}{4\,e}\right )-x^3\,\left (\frac {d\,\left (\frac {A\,c^2+2\,C\,a\,c}{e}-\frac {d\,\left (\frac {B\,c^2}{e}-\frac {C\,c^2\,d}{e^2}\right )}{e}\right )}{3\,e}-\frac {2\,B\,a\,c}{3\,e}\right )+x^2\,\left (\frac {C\,a^2+2\,A\,c\,a}{2\,e}+\frac {d\,\left (\frac {d\,\left (\frac {A\,c^2+2\,C\,a\,c}{e}-\frac {d\,\left (\frac {B\,c^2}{e}-\frac {C\,c^2\,d}{e^2}\right )}{e}\right )}{e}-\frac {2\,B\,a\,c}{e}\right )}{2\,e}\right )+\frac {\ln \left (d+e\,x\right )\,\left (C\,a^2\,d^2\,e^4-B\,a^2\,d\,e^5+A\,a^2\,e^6+2\,C\,a\,c\,d^4\,e^2-2\,B\,a\,c\,d^3\,e^3+2\,A\,a\,c\,d^2\,e^4+C\,c^2\,d^6-B\,c^2\,d^5\,e+A\,c^2\,d^4\,e^2\right )}{e^7}+\frac {C\,c^2\,x^6}{6\,e} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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