Optimal. Leaf size=209 \[ -\frac {(a B-(A c-a C) x) (d+e x)^3}{4 a c \left (a+c x^2\right )^2}-\frac {(d+e x) \left (a e (3 A c d+5 a C d+3 a B e)-\left (3 A c^2 d^2-a \left (4 a C e^2-c d (C d+3 B e)\right )\right ) x\right )}{8 a^2 c^2 \left (a+c x^2\right )}+\frac {\left (3 a e^2 (A c d+3 a C d+a B e)+c d^2 (3 A c d+a C d+3 a B e)\right ) \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{8 a^{5/2} c^{5/2}}+\frac {C e^3 \log \left (a+c x^2\right )}{2 c^3} \]
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Rubi [A]
time = 0.18, antiderivative size = 209, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {1659, 833, 649,
211, 266} \begin {gather*} \frac {\text {ArcTan}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right ) \left (c d^2 (3 a B e+a C d+3 A c d)+3 a e^2 (a B e+3 a C d+A c d)\right )}{8 a^{5/2} c^{5/2}}-\frac {(d+e x) \left (a e (3 a B e+5 a C d+3 A c d)-x \left (3 A c^2 d^2-a \left (4 a C e^2-c d (3 B e+C d)\right )\right )\right )}{8 a^2 c^2 \left (a+c x^2\right )}-\frac {(d+e x)^3 (a B-x (A c-a C))}{4 a c \left (a+c x^2\right )^2}+\frac {C e^3 \log \left (a+c x^2\right )}{2 c^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 211
Rule 266
Rule 649
Rule 833
Rule 1659
Rubi steps
\begin {align*} \int \frac {(d+e x)^3 \left (A+B x+C x^2\right )}{\left (a+c x^2\right )^3} \, dx &=-\frac {(a B-(A c-a C) x) (d+e x)^3}{4 a c \left (a+c x^2\right )^2}-\frac {\int \frac {(d+e x)^2 (-3 A c d-a C d-3 a B e-4 a C e x)}{\left (a+c x^2\right )^2} \, dx}{4 a c}\\ &=-\frac {(a B-(A c-a C) x) (d+e x)^3}{4 a c \left (a+c x^2\right )^2}-\frac {(d+e x) \left (a e (3 A c d+5 a C d+3 a B e)-\left (3 A c^2 d^2-a \left (4 a C e^2-c d (C d+3 B e)\right )\right ) x\right )}{8 a^2 c^2 \left (a+c x^2\right )}-\frac {\int \frac {-3 a e^2 (A c d+3 a C d+a B e)-c d^2 (3 A c d+a C d+3 a B e)-8 a^2 C e^3 x}{a+c x^2} \, dx}{8 a^2 c^2}\\ &=-\frac {(a B-(A c-a C) x) (d+e x)^3}{4 a c \left (a+c x^2\right )^2}-\frac {(d+e x) \left (a e (3 A c d+5 a C d+3 a B e)-\left (3 A c^2 d^2-a \left (4 a C e^2-c d (C d+3 B e)\right )\right ) x\right )}{8 a^2 c^2 \left (a+c x^2\right )}+\frac {\left (C e^3\right ) \int \frac {x}{a+c x^2} \, dx}{c^2}+\frac {\left (3 a e^2 (A c d+3 a C d+a B e)+c d^2 (3 A c d+a C d+3 a B e)\right ) \int \frac {1}{a+c x^2} \, dx}{8 a^2 c^2}\\ &=-\frac {(a B-(A c-a C) x) (d+e x)^3}{4 a c \left (a+c x^2\right )^2}-\frac {(d+e x) \left (a e (3 A c d+5 a C d+3 a B e)-\left (3 A c^2 d^2-a \left (4 a C e^2-c d (C d+3 B e)\right )\right ) x\right )}{8 a^2 c^2 \left (a+c x^2\right )}+\frac {\left (3 a e^2 (A c d+3 a C d+a B e)+c d^2 (3 A c d+a C d+3 a B e)\right ) \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{8 a^{5/2} c^{5/2}}+\frac {C e^3 \log \left (a+c x^2\right )}{2 c^3}\\ \end {align*}
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Mathematica [A]
time = 0.17, size = 281, normalized size = 1.34 \begin {gather*} \frac {\frac {-2 a^3 C e^3+2 A c^3 d^3 x-2 a c^2 d \left (C d^2 x+3 A e (d+e x)+B d (d+3 e x)\right )+2 a^2 c e (3 C d (d+e x)+e (3 B d+A e+B e x))}{a \left (a+c x^2\right )^2}+\frac {8 a^3 C e^3+3 A c^3 d^3 x+a c^2 d \left (C d^2+3 e (B d+A e)\right ) x-a^2 c e (3 C d (4 d+5 e x)+e (12 B d+4 A e+5 B e x))}{a^2 \left (a+c x^2\right )}+\frac {\sqrt {c} \left (3 A c d \left (c d^2+a e^2\right )+a \left (3 a e^2 (3 C d+B e)+c d^2 (C d+3 B e)\right )\right ) \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{a^{5/2}}+4 C e^3 \log \left (a+c x^2\right )}{8 c^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.15, size = 333, normalized size = 1.59
method | result | size |
default | \(\frac {\frac {\left (3 A a c d \,e^{2}+3 A \,c^{2} d^{3}-5 B \,a^{2} e^{3}+3 B a c \,d^{2} e -15 C \,a^{2} d \,e^{2}+C a c \,d^{3}\right ) x^{3}}{8 a^{2} c}-\frac {e \left (A c \,e^{2}+3 B c d e -2 a C \,e^{2}+3 C c \,d^{2}\right ) x^{2}}{2 c^{2}}-\frac {\left (3 A a c d \,e^{2}-5 A \,c^{2} d^{3}+3 B \,a^{2} e^{3}+3 B a c \,d^{2} e +9 C \,a^{2} d \,e^{2}+C a c \,d^{3}\right ) x}{8 c^{2} a}-\frac {A a c \,e^{3}+3 A \,c^{2} d^{2} e +3 B a c d \,e^{2}+B \,c^{2} d^{3}-3 C \,a^{2} e^{3}+3 C a c \,d^{2} e}{4 c^{3}}}{\left (c \,x^{2}+a \right )^{2}}+\frac {\frac {4 C \,a^{2} e^{3} \ln \left (c \,x^{2}+a \right )}{c}+\frac {\left (3 A a c d \,e^{2}+3 A \,c^{2} d^{3}+3 B \,a^{2} e^{3}+3 B a c \,d^{2} e +9 C \,a^{2} d \,e^{2}+C a c \,d^{3}\right ) \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{\sqrt {a c}}}{8 a^{2} c^{2}}\) | \(333\) |
risch | \(\frac {\frac {\left (3 A a c d \,e^{2}+3 A \,c^{2} d^{3}-5 B \,a^{2} e^{3}+3 B a c \,d^{2} e -15 C \,a^{2} d \,e^{2}+C a c \,d^{3}\right ) x^{3}}{8 a^{2} c}-\frac {e \left (A c \,e^{2}+3 B c d e -2 a C \,e^{2}+3 C c \,d^{2}\right ) x^{2}}{2 c^{2}}-\frac {\left (3 A a c d \,e^{2}-5 A \,c^{2} d^{3}+3 B \,a^{2} e^{3}+3 B a c \,d^{2} e +9 C \,a^{2} d \,e^{2}+C a c \,d^{3}\right ) x}{8 c^{2} a}-\frac {A a c \,e^{3}+3 A \,c^{2} d^{2} e +3 B a c d \,e^{2}+B \,c^{2} d^{3}-3 C \,a^{2} e^{3}+3 C a c \,d^{2} e}{4 c^{3}}}{\left (c \,x^{2}+a \right )^{2}}+\frac {\ln \left (3 A \,a^{2} c d \,e^{2}+3 A a \,c^{2} d^{3}+3 B \,a^{3} e^{3}+3 B \,a^{2} c \,d^{2} e +9 C \,a^{3} d \,e^{2}+C \,a^{2} c \,d^{3}-\sqrt {-a c \left (3 A a c d \,e^{2}+3 A \,c^{2} d^{3}+3 B \,a^{2} e^{3}+3 B a c \,d^{2} e +9 C \,a^{2} d \,e^{2}+C a c \,d^{3}\right )^{2}}\, x \right ) C \,e^{3}}{2 c^{3}}+\frac {\ln \left (3 A \,a^{2} c d \,e^{2}+3 A a \,c^{2} d^{3}+3 B \,a^{3} e^{3}+3 B \,a^{2} c \,d^{2} e +9 C \,a^{3} d \,e^{2}+C \,a^{2} c \,d^{3}-\sqrt {-a c \left (3 A a c d \,e^{2}+3 A \,c^{2} d^{3}+3 B \,a^{2} e^{3}+3 B a c \,d^{2} e +9 C \,a^{2} d \,e^{2}+C a c \,d^{3}\right )^{2}}\, x \right ) \sqrt {-a c \left (3 A a c d \,e^{2}+3 A \,c^{2} d^{3}+3 B \,a^{2} e^{3}+3 B a c \,d^{2} e +9 C \,a^{2} d \,e^{2}+C a c \,d^{3}\right )^{2}}}{16 a^{3} c^{3}}+\frac {\ln \left (3 A \,a^{2} c d \,e^{2}+3 A a \,c^{2} d^{3}+3 B \,a^{3} e^{3}+3 B \,a^{2} c \,d^{2} e +9 C \,a^{3} d \,e^{2}+C \,a^{2} c \,d^{3}+\sqrt {-a c \left (3 A a c d \,e^{2}+3 A \,c^{2} d^{3}+3 B \,a^{2} e^{3}+3 B a c \,d^{2} e +9 C \,a^{2} d \,e^{2}+C a c \,d^{3}\right )^{2}}\, x \right ) C \,e^{3}}{2 c^{3}}-\frac {\ln \left (3 A \,a^{2} c d \,e^{2}+3 A a \,c^{2} d^{3}+3 B \,a^{3} e^{3}+3 B \,a^{2} c \,d^{2} e +9 C \,a^{3} d \,e^{2}+C \,a^{2} c \,d^{3}+\sqrt {-a c \left (3 A a c d \,e^{2}+3 A \,c^{2} d^{3}+3 B \,a^{2} e^{3}+3 B a c \,d^{2} e +9 C \,a^{2} d \,e^{2}+C a c \,d^{3}\right )^{2}}\, x \right ) \sqrt {-a c \left (3 A a c d \,e^{2}+3 A \,c^{2} d^{3}+3 B \,a^{2} e^{3}+3 B a c \,d^{2} e +9 C \,a^{2} d \,e^{2}+C a c \,d^{3}\right )^{2}}}{16 a^{3} c^{3}}\) | \(899\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.52, size = 378, normalized size = 1.81 \begin {gather*} -\frac {2 \, B a^{2} c^{2} d^{3} + 6 \, B a^{3} c d e^{2} - 6 \, C a^{4} e^{3} + 2 \, A a^{3} c e^{3} - {\left (3 \, B a c^{3} d^{2} e - 5 \, B a^{2} c^{2} e^{3} + {\left (C a c^{3} + 3 \, A c^{4}\right )} d^{3} - 3 \, {\left (5 \, C a^{2} c^{2} e^{2} - A a c^{3} e^{2}\right )} d\right )} x^{3} + 6 \, {\left (C a^{3} c e + A a^{2} c^{2} e\right )} d^{2} + 4 \, {\left (3 \, C a^{2} c^{2} d^{2} e + 3 \, B a^{2} c^{2} d e^{2} - 2 \, C a^{3} c e^{3} + A a^{2} c^{2} e^{3}\right )} x^{2} + {\left (3 \, B a^{2} c^{2} d^{2} e + 3 \, B a^{3} c e^{3} + {\left (C a^{2} c^{2} - 5 \, A a c^{3}\right )} d^{3} + 3 \, {\left (3 \, C a^{3} c e^{2} + A a^{2} c^{2} e^{2}\right )} d\right )} x}{8 \, {\left (a^{2} c^{5} x^{4} + 2 \, a^{3} c^{4} x^{2} + a^{4} c^{3}\right )}} + \frac {C e^{3} \log \left (c x^{2} + a\right )}{2 \, c^{3}} + \frac {{\left (3 \, B a c d^{2} e + {\left (C a c + 3 \, A c^{2}\right )} d^{3} + 3 \, B a^{2} e^{3} + 3 \, {\left (3 \, C a^{2} e^{2} + A a c e^{2}\right )} d\right )} \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{8 \, \sqrt {a c} a^{2} c^{2}} \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. 545 vs.
\(2 (199) = 398\).
time = 0.36, size = 1111, normalized size = 5.32 \begin {gather*} \left [-\frac {4 \, B a^{3} c^{2} d^{3} - 2 \, {\left (C a^{2} c^{3} + 3 \, A a c^{4}\right )} d^{3} x^{3} + 2 \, {\left (C a^{3} c^{2} - 5 \, A a^{2} c^{3}\right )} d^{3} x - 8 \, {\left (C a^{3} c^{2} x^{4} + 2 \, C a^{4} c x^{2} + C a^{5}\right )} e^{3} \log \left (c x^{2} + a\right ) + {\left ({\left (C a c^{3} + 3 \, A c^{4}\right )} d^{3} x^{4} + 2 \, {\left (C a^{2} c^{2} + 3 \, A a c^{3}\right )} d^{3} x^{2} + {\left (C a^{3} c + 3 \, A a^{2} c^{2}\right )} d^{3} + 3 \, {\left (B a^{2} c^{2} x^{4} + 2 \, B a^{3} c x^{2} + B a^{4}\right )} e^{3} + 3 \, {\left ({\left (3 \, C a^{2} c^{2} + A a c^{3}\right )} d x^{4} + 2 \, {\left (3 \, C a^{3} c + A a^{2} c^{2}\right )} d x^{2} + {\left (3 \, C a^{4} + A a^{3} c\right )} d\right )} e^{2} + 3 \, {\left (B a c^{3} d^{2} x^{4} + 2 \, B a^{2} c^{2} d^{2} x^{2} + B a^{3} c d^{2}\right )} e\right )} \sqrt {-a c} \log \left (\frac {c x^{2} - 2 \, \sqrt {-a c} x - a}{c x^{2} + a}\right ) + 2 \, {\left (5 \, B a^{3} c^{2} x^{3} + 3 \, B a^{4} c x - 6 \, C a^{5} + 2 \, A a^{4} c - 4 \, {\left (2 \, C a^{4} c - A a^{3} c^{2}\right )} x^{2}\right )} e^{3} + 6 \, {\left (4 \, B a^{3} c^{2} d x^{2} + 2 \, B a^{4} c d + {\left (5 \, C a^{3} c^{2} - A a^{2} c^{3}\right )} d x^{3} + {\left (3 \, C a^{4} c + A a^{3} c^{2}\right )} d x\right )} e^{2} - 6 \, {\left (B a^{2} c^{3} d^{2} x^{3} - 4 \, C a^{3} c^{2} d^{2} x^{2} - B a^{3} c^{2} d^{2} x - 2 \, {\left (C a^{4} c + A a^{3} c^{2}\right )} d^{2}\right )} e}{16 \, {\left (a^{3} c^{5} x^{4} + 2 \, a^{4} c^{4} x^{2} + a^{5} c^{3}\right )}}, -\frac {2 \, B a^{3} c^{2} d^{3} - {\left (C a^{2} c^{3} + 3 \, A a c^{4}\right )} d^{3} x^{3} + {\left (C a^{3} c^{2} - 5 \, A a^{2} c^{3}\right )} d^{3} x - 4 \, {\left (C a^{3} c^{2} x^{4} + 2 \, C a^{4} c x^{2} + C a^{5}\right )} e^{3} \log \left (c x^{2} + a\right ) - {\left ({\left (C a c^{3} + 3 \, A c^{4}\right )} d^{3} x^{4} + 2 \, {\left (C a^{2} c^{2} + 3 \, A a c^{3}\right )} d^{3} x^{2} + {\left (C a^{3} c + 3 \, A a^{2} c^{2}\right )} d^{3} + 3 \, {\left (B a^{2} c^{2} x^{4} + 2 \, B a^{3} c x^{2} + B a^{4}\right )} e^{3} + 3 \, {\left ({\left (3 \, C a^{2} c^{2} + A a c^{3}\right )} d x^{4} + 2 \, {\left (3 \, C a^{3} c + A a^{2} c^{2}\right )} d x^{2} + {\left (3 \, C a^{4} + A a^{3} c\right )} d\right )} e^{2} + 3 \, {\left (B a c^{3} d^{2} x^{4} + 2 \, B a^{2} c^{2} d^{2} x^{2} + B a^{3} c d^{2}\right )} e\right )} \sqrt {a c} \arctan \left (\frac {\sqrt {a c} x}{a}\right ) + {\left (5 \, B a^{3} c^{2} x^{3} + 3 \, B a^{4} c x - 6 \, C a^{5} + 2 \, A a^{4} c - 4 \, {\left (2 \, C a^{4} c - A a^{3} c^{2}\right )} x^{2}\right )} e^{3} + 3 \, {\left (4 \, B a^{3} c^{2} d x^{2} + 2 \, B a^{4} c d + {\left (5 \, C a^{3} c^{2} - A a^{2} c^{3}\right )} d x^{3} + {\left (3 \, C a^{4} c + A a^{3} c^{2}\right )} d x\right )} e^{2} - 3 \, {\left (B a^{2} c^{3} d^{2} x^{3} - 4 \, C a^{3} c^{2} d^{2} x^{2} - B a^{3} c^{2} d^{2} x - 2 \, {\left (C a^{4} c + A a^{3} c^{2}\right )} d^{2}\right )} e}{8 \, {\left (a^{3} c^{5} x^{4} + 2 \, a^{4} c^{4} x^{2} + a^{5} c^{3}\right )}}\right ] \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 = 4.42, size = 348, normalized size = 1.67 \begin {gather*} \frac {C e^{3} \log \left (c x^{2} + a\right )}{2 \, c^{3}} + \frac {{\left (C a c d^{3} + 3 \, A c^{2} d^{3} + 3 \, B a c d^{2} e + 9 \, C a^{2} d e^{2} + 3 \, A a c d e^{2} + 3 \, B a^{2} e^{3}\right )} \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{8 \, \sqrt {a c} a^{2} c^{2}} + \frac {{\left (C a c^{2} d^{3} + 3 \, A c^{3} d^{3} + 3 \, B a c^{2} d^{2} e - 15 \, C a^{2} c d e^{2} + 3 \, A a c^{2} d e^{2} - 5 \, B a^{2} c e^{3}\right )} x^{3} - 4 \, {\left (3 \, C a^{2} c d^{2} e + 3 \, B a^{2} c d e^{2} - 2 \, C a^{3} e^{3} + A a^{2} c e^{3}\right )} x^{2} - {\left (C a^{2} c d^{3} - 5 \, A a c^{2} d^{3} + 3 \, B a^{2} c d^{2} e + 9 \, C a^{3} d e^{2} + 3 \, A a^{2} c d e^{2} + 3 \, B a^{3} e^{3}\right )} x - \frac {2 \, {\left (B a^{2} c^{2} d^{3} + 3 \, C a^{3} c d^{2} e + 3 \, A a^{2} c^{2} d^{2} e + 3 \, B a^{3} c d e^{2} - 3 \, C a^{4} e^{3} + A a^{3} c e^{3}\right )}}{c}}{8 \, {\left (c x^{2} + a\right )}^{2} a^{2} c^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.77, size = 920, normalized size = 4.40 \begin {gather*} \frac {5\,A\,d^3\,x}{8\,\left (a^3+2\,a^2\,c\,x^2+a\,c^2\,x^4\right )}-\frac {B\,d^3}{4\,\left (a^2\,c+2\,a\,c^2\,x^2+c^3\,x^4\right )}+\frac {3\,C\,a^2\,e^3}{4\,\left (a^2\,c^3+2\,a\,c^4\,x^2+c^5\,x^4\right )}-\frac {3\,A\,d^2\,e}{4\,\left (a^2\,c+2\,a\,c^2\,x^2+c^3\,x^4\right )}+\frac {C\,d^3\,x^3}{8\,\left (a^3+2\,a^2\,c\,x^2+a\,c^2\,x^4\right )}-\frac {C\,d^3\,x}{8\,\left (a^2\,c+2\,a\,c^2\,x^2+c^3\,x^4\right )}-\frac {A\,a\,e^3}{4\,\left (a^2\,c^2+2\,a\,c^3\,x^2+c^4\,x^4\right )}-\frac {A\,e^3\,x^2}{2\,\left (a^2\,c+2\,a\,c^2\,x^2+c^3\,x^4\right )}-\frac {5\,B\,e^3\,x^3}{8\,\left (a^2\,c+2\,a\,c^2\,x^2+c^3\,x^4\right )}+\frac {C\,e^3\,\ln \left (c\,x^2+a\right )}{2\,c^3}-\frac {3\,B\,a\,d\,e^2}{4\,\left (a^2\,c^2+2\,a\,c^3\,x^2+c^4\,x^4\right )}-\frac {3\,C\,a\,d^2\,e}{4\,\left (a^2\,c^2+2\,a\,c^3\,x^2+c^4\,x^4\right )}+\frac {3\,A\,c\,d^3\,x^3}{8\,\left (a^4+2\,a^3\,c\,x^2+a^2\,c^2\,x^4\right )}-\frac {3\,B\,a\,e^3\,x}{8\,\left (a^2\,c^2+2\,a\,c^3\,x^2+c^4\,x^4\right )}-\frac {3\,B\,d\,e^2\,x^2}{2\,\left (a^2\,c+2\,a\,c^2\,x^2+c^3\,x^4\right )}-\frac {3\,C\,d^2\,e\,x^2}{2\,\left (a^2\,c+2\,a\,c^2\,x^2+c^3\,x^4\right )}-\frac {15\,C\,d\,e^2\,x^3}{8\,\left (a^2\,c+2\,a\,c^2\,x^2+c^3\,x^4\right )}+\frac {C\,a\,e^3\,x^2}{a^2\,c^2+2\,a\,c^3\,x^2+c^4\,x^4}+\frac {3\,A\,d^3\,\mathrm {atan}\left (\frac {\sqrt {c}\,x}{\sqrt {a}}\right )}{8\,a^{5/2}\,\sqrt {c}}+\frac {3\,B\,e^3\,\mathrm {atan}\left (\frac {\sqrt {c}\,x}{\sqrt {a}}\right )}{8\,\sqrt {a}\,c^{5/2}}+\frac {C\,d^3\,\mathrm {atan}\left (\frac {\sqrt {c}\,x}{\sqrt {a}}\right )}{8\,a^{3/2}\,c^{3/2}}+\frac {3\,A\,d\,e^2\,x^3}{8\,\left (a^3+2\,a^2\,c\,x^2+a\,c^2\,x^4\right )}+\frac {3\,B\,d^2\,e\,x^3}{8\,\left (a^3+2\,a^2\,c\,x^2+a\,c^2\,x^4\right )}-\frac {3\,A\,d\,e^2\,x}{8\,\left (a^2\,c+2\,a\,c^2\,x^2+c^3\,x^4\right )}-\frac {3\,B\,d^2\,e\,x}{8\,\left (a^2\,c+2\,a\,c^2\,x^2+c^3\,x^4\right )}+\frac {3\,A\,d\,e^2\,\mathrm {atan}\left (\frac {\sqrt {c}\,x}{\sqrt {a}}\right )}{8\,a^{3/2}\,c^{3/2}}+\frac {3\,B\,d^2\,e\,\mathrm {atan}\left (\frac {\sqrt {c}\,x}{\sqrt {a}}\right )}{8\,a^{3/2}\,c^{3/2}}+\frac {9\,C\,d\,e^2\,\mathrm {atan}\left (\frac {\sqrt {c}\,x}{\sqrt {a}}\right )}{8\,\sqrt {a}\,c^{5/2}}-\frac {9\,C\,a\,d\,e^2\,x}{8\,\left (a^2\,c^2+2\,a\,c^3\,x^2+c^4\,x^4\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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