Optimal. Leaf size=374 \[ -\frac {e \left (C d^2-B d e+A e^2\right )}{\left (c d^2+a e^2\right )^2 (d+e x)}-\frac {a \left (B c d^2-2 A c d e+2 a C d e-a B e^2\right )-\left (A c \left (c d^2-a e^2\right )+a \left (a C e^2-c d (C d-2 B e)\right )\right ) x}{2 a \left (c d^2+a e^2\right )^2 \left (a+c x^2\right )}+\frac {\left (A c \left (c^2 d^4+6 a c d^2 e^2-3 a^2 e^4\right )+a \left (a^2 C e^4+c^2 d^3 (C d-2 B e)-6 a c d e^2 (C d-B e)\right )\right ) \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{2 a^{3/2} \sqrt {c} \left (c d^2+a e^2\right )^3}-\frac {e \left (a e^2 (2 C d-B e)-c d \left (2 C d^2-e (3 B d-4 A e)\right )\right ) \log (d+e x)}{\left (c d^2+a e^2\right )^3}+\frac {e \left (a e^2 (2 C d-B e)-c d \left (2 C d^2-e (3 B d-4 A e)\right )\right ) \log \left (a+c x^2\right )}{2 \left (c d^2+a e^2\right )^3} \]
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Rubi [A]
time = 0.59, antiderivative size = 371, normalized size of antiderivative = 0.99, number of steps
used = 6, number of rules used = 5, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {1661, 1643,
649, 211, 266} \begin {gather*} \frac {\text {ArcTan}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right ) \left (A c \left (-3 a^2 e^4+6 a c d^2 e^2+c^2 d^4\right )+a \left (a^2 C e^4-6 a c d e^2 (C d-B e)+c^2 d^3 (C d-2 B e)\right )\right )}{2 a^{3/2} \sqrt {c} \left (a e^2+c d^2\right )^3}-\frac {a \left (-a B e^2+2 a C d e-2 A c d e+B c d^2\right )-x \left (A c \left (c d^2-a e^2\right )+a \left (a C e^2-c d (C d-2 B e)\right )\right )}{2 a \left (a+c x^2\right ) \left (a e^2+c d^2\right )^2}-\frac {e \left (A e^2-B d e+C d^2\right )}{(d+e x) \left (a e^2+c d^2\right )^2}-\frac {e \log \left (a+c x^2\right ) \left (-a e^2 (2 C d-B e)-c d e (3 B d-4 A e)+2 c C d^3\right )}{2 \left (a e^2+c d^2\right )^3}+\frac {e \log (d+e x) \left (-a e^2 (2 C d-B e)-c d e (3 B d-4 A e)+2 c C d^3\right )}{\left (a e^2+c d^2\right )^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 211
Rule 266
Rule 649
Rule 1643
Rule 1661
Rubi steps
\begin {align*} \int \frac {A+B x+C x^2}{(d+e x)^2 \left (a+c x^2\right )^2} \, dx &=-\frac {a \left (B c d^2-2 A c d e+2 a C d e-a B e^2\right )-\left (A c \left (c d^2-a e^2\right )+a \left (a C e^2-c d (C d-2 B e)\right )\right ) x}{2 a \left (c d^2+a e^2\right )^2 \left (a+c x^2\right )}-\frac {\int \frac {-\frac {c \left (A \left (c^2 d^4+5 a c d^2 e^2+2 a^2 e^4\right )-a d^2 \left (a C e^2-c d (C d-2 B e)\right )\right )}{\left (c d^2+a e^2\right )^2}-\frac {2 c e (A c d-a C d+a B e) x}{c d^2+a e^2}-\frac {c e^2 \left (A c \left (c d^2-a e^2\right )+a \left (a C e^2-c d (C d-2 B e)\right )\right ) x^2}{\left (c d^2+a e^2\right )^2}}{(d+e x)^2 \left (a+c x^2\right )} \, dx}{2 a c}\\ &=-\frac {a \left (B c d^2-2 A c d e+2 a C d e-a B e^2\right )-\left (A c \left (c d^2-a e^2\right )+a \left (a C e^2-c d (C d-2 B e)\right )\right ) x}{2 a \left (c d^2+a e^2\right )^2 \left (a+c x^2\right )}-\frac {\int \left (-\frac {2 a c e^2 \left (C d^2-B d e+A e^2\right )}{\left (c d^2+a e^2\right )^2 (d+e x)^2}+\frac {2 a c e^2 \left (-2 c C d^3+c d e (3 B d-4 A e)+a e^2 (2 C d-B e)\right )}{\left (c d^2+a e^2\right )^3 (d+e x)}+\frac {c \left (-A c \left (c^2 d^4+6 a c d^2 e^2-3 a^2 e^4\right )-a \left (a^2 C e^4+c^2 d^3 (C d-2 B e)-6 a c d e^2 (C d-B e)\right )+2 a c e \left (2 c C d^3-c d e (3 B d-4 A e)-a e^2 (2 C d-B e)\right ) x\right )}{\left (c d^2+a e^2\right )^3 \left (a+c x^2\right )}\right ) \, dx}{2 a c}\\ &=-\frac {e \left (C d^2-B d e+A e^2\right )}{\left (c d^2+a e^2\right )^2 (d+e x)}-\frac {a \left (B c d^2-2 A c d e+2 a C d e-a B e^2\right )-\left (A c \left (c d^2-a e^2\right )+a \left (a C e^2-c d (C d-2 B e)\right )\right ) x}{2 a \left (c d^2+a e^2\right )^2 \left (a+c x^2\right )}+\frac {e \left (2 c C d^3-c d e (3 B d-4 A e)-a e^2 (2 C d-B e)\right ) \log (d+e x)}{\left (c d^2+a e^2\right )^3}-\frac {\int \frac {-A c \left (c^2 d^4+6 a c d^2 e^2-3 a^2 e^4\right )-a \left (a^2 C e^4+c^2 d^3 (C d-2 B e)-6 a c d e^2 (C d-B e)\right )+2 a c e \left (2 c C d^3-c d e (3 B d-4 A e)-a e^2 (2 C d-B e)\right ) x}{a+c x^2} \, dx}{2 a \left (c d^2+a e^2\right )^3}\\ &=-\frac {e \left (C d^2-B d e+A e^2\right )}{\left (c d^2+a e^2\right )^2 (d+e x)}-\frac {a \left (B c d^2-2 A c d e+2 a C d e-a B e^2\right )-\left (A c \left (c d^2-a e^2\right )+a \left (a C e^2-c d (C d-2 B e)\right )\right ) x}{2 a \left (c d^2+a e^2\right )^2 \left (a+c x^2\right )}+\frac {e \left (2 c C d^3-c d e (3 B d-4 A e)-a e^2 (2 C d-B e)\right ) \log (d+e x)}{\left (c d^2+a e^2\right )^3}-\frac {\left (c e \left (2 c C d^3-c d e (3 B d-4 A e)-a e^2 (2 C d-B e)\right )\right ) \int \frac {x}{a+c x^2} \, dx}{\left (c d^2+a e^2\right )^3}+\frac {\left (A c \left (c^2 d^4+6 a c d^2 e^2-3 a^2 e^4\right )+a \left (a^2 C e^4+c^2 d^3 (C d-2 B e)-6 a c d e^2 (C d-B e)\right )\right ) \int \frac {1}{a+c x^2} \, dx}{2 a \left (c d^2+a e^2\right )^3}\\ &=-\frac {e \left (C d^2-B d e+A e^2\right )}{\left (c d^2+a e^2\right )^2 (d+e x)}-\frac {a \left (B c d^2-2 A c d e+2 a C d e-a B e^2\right )-\left (A c \left (c d^2-a e^2\right )+a \left (a C e^2-c d (C d-2 B e)\right )\right ) x}{2 a \left (c d^2+a e^2\right )^2 \left (a+c x^2\right )}+\frac {\left (A c \left (c^2 d^4+6 a c d^2 e^2-3 a^2 e^4\right )+a \left (a^2 C e^4+c^2 d^3 (C d-2 B e)-6 a c d e^2 (C d-B e)\right )\right ) \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{2 a^{3/2} \sqrt {c} \left (c d^2+a e^2\right )^3}+\frac {e \left (2 c C d^3-c d e (3 B d-4 A e)-a e^2 (2 C d-B e)\right ) \log (d+e x)}{\left (c d^2+a e^2\right )^3}-\frac {e \left (2 c C d^3-c d e (3 B d-4 A e)-a e^2 (2 C d-B e)\right ) \log \left (a+c x^2\right )}{2 \left (c d^2+a e^2\right )^3}\\ \end {align*}
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Mathematica [A]
time = 0.28, size = 320, normalized size = 0.86 \begin {gather*} \frac {-\frac {2 e \left (c d^2+a e^2\right ) \left (C d^2+e (-B d+A e)\right )}{d+e x}+\frac {\left (c d^2+a e^2\right ) \left (A c^2 d^2 x+a^2 e (-2 C d+B e+C e x)-a c \left (C d^2 x+B d (d-2 e x)+A e (-2 d+e x)\right )\right )}{a \left (a+c x^2\right )}+\frac {\left (A c \left (c^2 d^4+6 a c d^2 e^2-3 a^2 e^4\right )+a \left (a^2 C e^4+c^2 d^3 (C d-2 B e)+6 a c d e^2 (-C d+B e)\right )\right ) \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{a^{3/2} \sqrt {c}}+2 e \left (2 c C d^3+c d e (-3 B d+4 A e)+a e^2 (-2 C d+B e)\right ) \log (d+e x)-e \left (2 c C d^3+c d e (-3 B d+4 A e)+a e^2 (-2 C d+B e)\right ) \log \left (a+c x^2\right )}{2 \left (c d^2+a e^2\right )^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.14, size = 424, normalized size = 1.13
method | result | size |
default | \(\frac {e \left (4 d \,e^{2} c A +B a \,e^{3}-3 B c \,d^{2} e -2 C a d \,e^{2}+2 C c \,d^{3}\right ) \ln \left (e x +d \right )}{\left (a \,e^{2}+c \,d^{2}\right )^{3}}-\frac {e \left (A \,e^{2}-B d e +C \,d^{2}\right )}{\left (a \,e^{2}+c \,d^{2}\right )^{2} \left (e x +d \right )}-\frac {\frac {\frac {\left (A \,a^{2} c \,e^{4}-A \,c^{3} d^{4}-2 B \,a^{2} c d \,e^{3}-2 B a \,c^{2} d^{3} e -C \,a^{3} e^{4}+C a \,c^{2} d^{4}\right ) x}{2 a}-A a c d \,e^{3}-A \,c^{2} d^{3} e -\frac {B \,a^{2} e^{4}}{2}+\frac {B \,c^{2} d^{4}}{2}+C \,a^{2} d \,e^{3}+C a c \,d^{3} e}{c \,x^{2}+a}+\frac {\frac {\left (8 A a \,c^{2} d \,e^{3}+2 B \,a^{2} c \,e^{4}-6 B a \,c^{2} d^{2} e^{2}-4 C \,a^{2} c d \,e^{3}+4 C a \,c^{2} d^{3} e \right ) \ln \left (c \,x^{2}+a \right )}{2 c}+\frac {\left (3 A \,a^{2} c \,e^{4}-6 A a \,c^{2} d^{2} e^{2}-A \,c^{3} d^{4}-6 B \,a^{2} c d \,e^{3}+2 B a \,c^{2} d^{3} e -C \,a^{3} e^{4}+6 C \,a^{2} c \,d^{2} e^{2}-C a \,c^{2} d^{4}\right ) \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{\sqrt {a c}}}{2 a}}{\left (a \,e^{2}+c \,d^{2}\right )^{3}}\) | \(424\) |
risch | \(\text {Expression too large to display}\) | \(1606\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.51, size = 593, normalized size = 1.59 \begin {gather*} -\frac {{\left (2 \, C c d^{3} e - 3 \, B c d^{2} e^{2} + B a e^{4} - 2 \, {\left (C a e^{3} - 2 \, A c e^{3}\right )} d\right )} \log \left (c x^{2} + a\right )}{2 \, {\left (c^{3} d^{6} + 3 \, a c^{2} d^{4} e^{2} + 3 \, a^{2} c d^{2} e^{4} + a^{3} e^{6}\right )}} + \frac {{\left (2 \, C c d^{3} e - 3 \, B c d^{2} e^{2} + B a e^{4} - 2 \, {\left (C a e^{3} - 2 \, A c e^{3}\right )} d\right )} \log \left (x e + d\right )}{c^{3} d^{6} + 3 \, a c^{2} d^{4} e^{2} + 3 \, a^{2} c d^{2} e^{4} + a^{3} e^{6}} - \frac {{\left (2 \, B a c^{2} d^{3} e - 6 \, B a^{2} c d e^{3} - {\left (C a c^{2} + A c^{3}\right )} d^{4} - C a^{3} e^{4} + 3 \, A a^{2} c e^{4} + 6 \, {\left (C a^{2} c e^{2} - A a c^{2} e^{2}\right )} d^{2}\right )} \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{2 \, {\left (a c^{3} d^{6} + 3 \, a^{2} c^{2} d^{4} e^{2} + 3 \, a^{3} c d^{2} e^{4} + a^{4} e^{6}\right )} \sqrt {a c}} - \frac {B a c d^{3} - 3 \, B a^{2} d e^{2} + 2 \, A a^{2} e^{3} + 2 \, {\left (2 \, C a^{2} e - A a c e\right )} d^{2} - {\left (4 \, B a c d e^{2} + C a^{2} e^{3} - 3 \, A a c e^{3} - {\left (3 \, C a c e - A c^{2} e\right )} d^{2}\right )} x^{2} - {\left (B a c d^{2} e - {\left (C a c - A c^{2}\right )} d^{3} + B a^{2} e^{3} - {\left (C a^{2} e^{2} - A a c e^{2}\right )} d\right )} x}{2 \, {\left (a^{2} c^{2} d^{5} + 2 \, a^{3} c d^{3} e^{2} + a^{4} d e^{4} + {\left (a c^{3} d^{4} e + 2 \, a^{2} c^{2} d^{2} e^{3} + a^{3} c e^{5}\right )} x^{3} + {\left (a c^{3} d^{5} + 2 \, a^{2} c^{2} d^{3} e^{2} + a^{3} c d e^{4}\right )} x^{2} + {\left (a^{2} c^{2} d^{4} e + 2 \, a^{3} c d^{2} e^{3} + a^{4} e^{5}\right )} x\right )}} \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. 1415 vs.
\(2 (362) = 724\).
time = 126.70, size = 2853, normalized size = 7.63 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 3.64, size = 608, normalized size = 1.63 \begin {gather*} \frac {{\left (C a c^{2} d^{4} e^{2} + A c^{3} d^{4} e^{2} - 2 \, B a c^{2} d^{3} e^{3} - 6 \, C a^{2} c d^{2} e^{4} + 6 \, A a c^{2} d^{2} e^{4} + 6 \, B a^{2} c d e^{5} + C a^{3} e^{6} - 3 \, A a^{2} c e^{6}\right )} \arctan \left (\frac {{\left (c d - \frac {c d^{2}}{x e + d} - \frac {a e^{2}}{x e + d}\right )} e^{\left (-1\right )}}{\sqrt {a c}}\right ) e^{\left (-2\right )}}{2 \, {\left (a c^{3} d^{6} + 3 \, a^{2} c^{2} d^{4} e^{2} + 3 \, a^{3} c d^{2} e^{4} + a^{4} e^{6}\right )} \sqrt {a c}} - \frac {{\left (2 \, C c d^{3} e - 3 \, B c d^{2} e^{2} - 2 \, C a d e^{3} + 4 \, A c d e^{3} + B a e^{4}\right )} \log \left (c - \frac {2 \, c d}{x e + d} + \frac {c d^{2}}{{\left (x e + d\right )}^{2}} + \frac {a e^{2}}{{\left (x e + d\right )}^{2}}\right )}{2 \, {\left (c^{3} d^{6} + 3 \, a c^{2} d^{4} e^{2} + 3 \, a^{2} c d^{2} e^{4} + a^{3} e^{6}\right )}} - \frac {\frac {C d^{2} e^{5}}{x e + d} - \frac {B d e^{6}}{x e + d} + \frac {A e^{7}}{x e + d}}{c^{2} d^{4} e^{4} + 2 \, a c d^{2} e^{6} + a^{2} e^{8}} - \frac {\frac {C a c^{2} d^{3} e - A c^{3} d^{3} e - 3 \, B a c^{2} d^{2} e^{2} - 3 \, C a^{2} c d e^{3} + 3 \, A a c^{2} d e^{3} + B a^{2} c e^{4}}{c d^{2} + a e^{2}} - \frac {{\left (C a c^{2} d^{4} e^{2} - A c^{3} d^{4} e^{2} - 4 \, B a c^{2} d^{3} e^{3} - 6 \, C a^{2} c d^{2} e^{4} + 6 \, A a c^{2} d^{2} e^{4} + 4 \, B a^{2} c d e^{5} + C a^{3} e^{6} - A a^{2} c e^{6}\right )} e^{\left (-1\right )}}{{\left (c d^{2} + a e^{2}\right )} {\left (x e + d\right )}}}{2 \, {\left (c d^{2} + a e^{2}\right )}^{2} a {\left (c - \frac {2 \, c d}{x e + d} + \frac {c d^{2}}{{\left (x e + d\right )}^{2}} + \frac {a e^{2}}{{\left (x e + d\right )}^{2}}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 9.91, size = 2094, normalized size = 5.60 \begin {gather*} \frac {\frac {x^2\,\left (C\,a^2\,e^3-3\,C\,a\,c\,d^2\,e+4\,B\,a\,c\,d\,e^2-3\,A\,a\,c\,e^3+A\,c^2\,d^2\,e\right )}{2\,a\,\left (a^2\,e^4+2\,a\,c\,d^2\,e^2+c^2\,d^4\right )}-\frac {2\,A\,a\,e^3+B\,c\,d^3-3\,B\,a\,d\,e^2-2\,A\,c\,d^2\,e+4\,C\,a\,d^2\,e}{2\,{\left (c\,d^2+a\,e^2\right )}^2}+\frac {x\,\left (A\,c\,d+B\,a\,e-C\,a\,d\right )}{2\,a\,\left (c\,d^2+a\,e^2\right )}}{c\,e\,x^3+c\,d\,x^2+a\,e\,x+a\,d}-\frac {\ln \left (3\,A\,e^6\,{\left (-a^3\,c\right )}^{3/2}-A\,c^4\,d^6\,\sqrt {-a^3\,c}+C\,a^4\,e^6\,\sqrt {-a^3\,c}+31\,C\,d^2\,e^4\,{\left (-a^3\,c\right )}^{3/2}+6\,B\,a^5\,c\,e^6-18\,B\,d\,e^5\,{\left (-a^3\,c\right )}^{3/2}-6\,B\,e^6\,x\,{\left (-a^3\,c\right )}^{3/2}-C\,a^5\,c\,e^6\,x+14\,C\,d\,e^5\,x\,{\left (-a^3\,c\right )}^{3/2}-2\,A\,a^2\,c^4\,d^5\,e+30\,A\,a^4\,c^2\,d\,e^5-14\,C\,a^3\,c^3\,d^5\,e+3\,A\,a^4\,c^2\,e^6\,x+C\,a^2\,c^4\,d^6\,x-C\,a\,c^3\,d^6\,\sqrt {-a^3\,c}-36\,A\,a^3\,c^3\,d^3\,e^3+22\,B\,a^3\,c^3\,d^4\,e^2-36\,B\,a^4\,c^2\,d^2\,e^4+36\,C\,a^4\,c^2\,d^3\,e^3-14\,C\,a^5\,c\,d\,e^5+A\,a\,c^5\,d^6\,x+5\,A\,a^2\,c^4\,d^4\,e^2\,x-57\,A\,a^3\,c^3\,d^2\,e^4\,x+44\,B\,a^3\,c^3\,d^3\,e^3\,x-31\,C\,a^3\,c^3\,d^4\,e^2\,x+31\,C\,a^4\,c^2\,d^2\,e^4\,x-5\,A\,a\,c^3\,d^4\,e^2\,\sqrt {-a^3\,c}+57\,A\,a^2\,c^2\,d^2\,e^4\,\sqrt {-a^3\,c}-44\,B\,a^2\,c^2\,d^3\,e^3\,\sqrt {-a^3\,c}+31\,C\,a^2\,c^2\,d^4\,e^2\,\sqrt {-a^3\,c}-2\,B\,a^2\,c^4\,d^5\,e\,x-18\,B\,a^4\,c^2\,d\,e^5\,x+2\,B\,a\,c^3\,d^5\,e\,\sqrt {-a^3\,c}-2\,A\,c^4\,d^5\,e\,x\,\sqrt {-a^3\,c}-36\,B\,a^2\,c^2\,d^2\,e^4\,x\,\sqrt {-a^3\,c}+36\,C\,a^2\,c^2\,d^3\,e^3\,x\,\sqrt {-a^3\,c}-14\,C\,a\,c^3\,d^5\,e\,x\,\sqrt {-a^3\,c}-36\,A\,a\,c^3\,d^3\,e^3\,x\,\sqrt {-a^3\,c}+30\,A\,a^2\,c^2\,d\,e^5\,x\,\sqrt {-a^3\,c}+22\,B\,a\,c^3\,d^4\,e^2\,x\,\sqrt {-a^3\,c}\right )\,\left (c^2\,\left (a\,\left (\frac {C\,d^4\,\sqrt {-a^3\,c}}{4}+\frac {3\,A\,d^2\,e^2\,\sqrt {-a^3\,c}}{2}-\frac {B\,d^3\,e\,\sqrt {-a^3\,c}}{2}\right )+a^3\,\left (C\,d^3\,e-\frac {3\,B\,d^2\,e^2}{2}+2\,A\,d\,e^3\right )\right )-c\,\left (a^2\,\left (\frac {3\,A\,e^4\,\sqrt {-a^3\,c}}{4}+\frac {3\,C\,d^2\,e^2\,\sqrt {-a^3\,c}}{2}-\frac {3\,B\,d\,e^3\,\sqrt {-a^3\,c}}{2}\right )-a^4\,\left (\frac {B\,e^4}{2}-C\,d\,e^3\right )\right )+\frac {A\,c^3\,d^4\,\sqrt {-a^3\,c}}{4}+\frac {C\,a^3\,e^4\,\sqrt {-a^3\,c}}{4}\right )}{a^6\,c\,e^6+3\,a^5\,c^2\,d^2\,e^4+3\,a^4\,c^3\,d^4\,e^2+a^3\,c^4\,d^6}+\frac {\ln \left (3\,A\,e^6\,{\left (-a^3\,c\right )}^{3/2}-A\,c^4\,d^6\,\sqrt {-a^3\,c}+C\,a^4\,e^6\,\sqrt {-a^3\,c}+31\,C\,d^2\,e^4\,{\left (-a^3\,c\right )}^{3/2}-6\,B\,a^5\,c\,e^6-18\,B\,d\,e^5\,{\left (-a^3\,c\right )}^{3/2}-6\,B\,e^6\,x\,{\left (-a^3\,c\right )}^{3/2}+C\,a^5\,c\,e^6\,x+14\,C\,d\,e^5\,x\,{\left (-a^3\,c\right )}^{3/2}+2\,A\,a^2\,c^4\,d^5\,e-30\,A\,a^4\,c^2\,d\,e^5+14\,C\,a^3\,c^3\,d^5\,e-3\,A\,a^4\,c^2\,e^6\,x-C\,a^2\,c^4\,d^6\,x-C\,a\,c^3\,d^6\,\sqrt {-a^3\,c}+36\,A\,a^3\,c^3\,d^3\,e^3-22\,B\,a^3\,c^3\,d^4\,e^2+36\,B\,a^4\,c^2\,d^2\,e^4-36\,C\,a^4\,c^2\,d^3\,e^3+14\,C\,a^5\,c\,d\,e^5-A\,a\,c^5\,d^6\,x-5\,A\,a^2\,c^4\,d^4\,e^2\,x+57\,A\,a^3\,c^3\,d^2\,e^4\,x-44\,B\,a^3\,c^3\,d^3\,e^3\,x+31\,C\,a^3\,c^3\,d^4\,e^2\,x-31\,C\,a^4\,c^2\,d^2\,e^4\,x-5\,A\,a\,c^3\,d^4\,e^2\,\sqrt {-a^3\,c}+57\,A\,a^2\,c^2\,d^2\,e^4\,\sqrt {-a^3\,c}-44\,B\,a^2\,c^2\,d^3\,e^3\,\sqrt {-a^3\,c}+31\,C\,a^2\,c^2\,d^4\,e^2\,\sqrt {-a^3\,c}+2\,B\,a^2\,c^4\,d^5\,e\,x+18\,B\,a^4\,c^2\,d\,e^5\,x+2\,B\,a\,c^3\,d^5\,e\,\sqrt {-a^3\,c}-2\,A\,c^4\,d^5\,e\,x\,\sqrt {-a^3\,c}-36\,B\,a^2\,c^2\,d^2\,e^4\,x\,\sqrt {-a^3\,c}+36\,C\,a^2\,c^2\,d^3\,e^3\,x\,\sqrt {-a^3\,c}-14\,C\,a\,c^3\,d^5\,e\,x\,\sqrt {-a^3\,c}-36\,A\,a\,c^3\,d^3\,e^3\,x\,\sqrt {-a^3\,c}+30\,A\,a^2\,c^2\,d\,e^5\,x\,\sqrt {-a^3\,c}+22\,B\,a\,c^3\,d^4\,e^2\,x\,\sqrt {-a^3\,c}\right )\,\left (c^2\,\left (a\,\left (\frac {C\,d^4\,\sqrt {-a^3\,c}}{4}+\frac {3\,A\,d^2\,e^2\,\sqrt {-a^3\,c}}{2}-\frac {B\,d^3\,e\,\sqrt {-a^3\,c}}{2}\right )-a^3\,\left (C\,d^3\,e-\frac {3\,B\,d^2\,e^2}{2}+2\,A\,d\,e^3\right )\right )-c\,\left (a^2\,\left (\frac {3\,A\,e^4\,\sqrt {-a^3\,c}}{4}+\frac {3\,C\,d^2\,e^2\,\sqrt {-a^3\,c}}{2}-\frac {3\,B\,d\,e^3\,\sqrt {-a^3\,c}}{2}\right )+a^4\,\left (\frac {B\,e^4}{2}-C\,d\,e^3\right )\right )+\frac {A\,c^3\,d^4\,\sqrt {-a^3\,c}}{4}+\frac {C\,a^3\,e^4\,\sqrt {-a^3\,c}}{4}\right )}{a^6\,c\,e^6+3\,a^5\,c^2\,d^2\,e^4+3\,a^4\,c^3\,d^4\,e^2+a^3\,c^4\,d^6}+\frac {\ln \left (d+e\,x\right )\,\left (a\,\left (B\,e^4-2\,C\,d\,e^3\right )+c\,\left (2\,C\,d^3\,e-3\,B\,d^2\,e^2+4\,A\,d\,e^3\right )\right )}{a^3\,e^6+3\,a^2\,c\,d^2\,e^4+3\,a\,c^2\,d^4\,e^2+c^3\,d^6} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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