Optimal. Leaf size=300 \[ -\frac {a e \left (c d^2-5 a e^2\right )-3 c d x \left (3 a e^2+c d^2\right )}{8 a^2 \left (a+c x^2\right ) (d+e x) \left (a e^2+c d^2\right )^2}+\frac {3 e \left (c d^2-a e^2\right ) \left (5 a e^2+c d^2\right )}{8 a^2 (d+e x) \left (a e^2+c d^2\right )^3}+\frac {3 \sqrt {c} \left (-5 a^3 e^6+15 a^2 c d^2 e^4+5 a c^2 d^4 e^2+c^3 d^6\right ) \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{8 a^{5/2} \left (a e^2+c d^2\right )^4}+\frac {a e+c d x}{4 a \left (a+c x^2\right )^2 (d+e x) \left (a e^2+c d^2\right )}-\frac {3 c d e^5 \log \left (a+c x^2\right )}{\left (a e^2+c d^2\right )^4}+\frac {6 c d e^5 \log (d+e x)}{\left (a e^2+c d^2\right )^4} \]
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Rubi [A] time = 0.36, antiderivative size = 300, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.353, Rules used = {741, 823, 801, 635, 205, 260} \begin {gather*} \frac {3 \sqrt {c} \left (15 a^2 c d^2 e^4-5 a^3 e^6+5 a c^2 d^4 e^2+c^3 d^6\right ) \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{8 a^{5/2} \left (a e^2+c d^2\right )^4}-\frac {a e \left (c d^2-5 a e^2\right )-3 c d x \left (3 a e^2+c d^2\right )}{8 a^2 \left (a+c x^2\right ) (d+e x) \left (a e^2+c d^2\right )^2}+\frac {3 e \left (c d^2-a e^2\right ) \left (5 a e^2+c d^2\right )}{8 a^2 (d+e x) \left (a e^2+c d^2\right )^3}+\frac {a e+c d x}{4 a \left (a+c x^2\right )^2 (d+e x) \left (a e^2+c d^2\right )}-\frac {3 c d e^5 \log \left (a+c x^2\right )}{\left (a e^2+c d^2\right )^4}+\frac {6 c d e^5 \log (d+e x)}{\left (a e^2+c d^2\right )^4} \end {gather*}
Antiderivative was successfully verified.
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Rule 205
Rule 260
Rule 635
Rule 741
Rule 801
Rule 823
Rubi steps
\begin {align*} \int \frac {1}{(d+e x)^2 \left (a+c x^2\right )^3} \, dx &=\frac {a e+c d x}{4 a \left (c d^2+a e^2\right ) (d+e x) \left (a+c x^2\right )^2}-\frac {\int \frac {-3 c d^2-5 a e^2-4 c d e x}{(d+e x)^2 \left (a+c x^2\right )^2} \, dx}{4 a \left (c d^2+a e^2\right )}\\ &=\frac {a e+c d x}{4 a \left (c d^2+a e^2\right ) (d+e x) \left (a+c x^2\right )^2}-\frac {a e \left (c d^2-5 a e^2\right )-3 c d \left (c d^2+3 a e^2\right ) x}{8 a^2 \left (c d^2+a e^2\right )^2 (d+e x) \left (a+c x^2\right )}+\frac {\int \frac {3 c \left (c^2 d^4+2 a c d^2 e^2+5 a^2 e^4\right )+6 c^2 d e \left (c d^2+3 a e^2\right ) x}{(d+e x)^2 \left (a+c x^2\right )} \, dx}{8 a^2 c \left (c d^2+a e^2\right )^2}\\ &=\frac {a e+c d x}{4 a \left (c d^2+a e^2\right ) (d+e x) \left (a+c x^2\right )^2}-\frac {a e \left (c d^2-5 a e^2\right )-3 c d \left (c d^2+3 a e^2\right ) x}{8 a^2 \left (c d^2+a e^2\right )^2 (d+e x) \left (a+c x^2\right )}+\frac {\int \left (\frac {3 c e^2 \left (-c d^2-5 a e^2\right ) \left (c d^2-a e^2\right )}{\left (c d^2+a e^2\right ) (d+e x)^2}+\frac {48 a^2 c^2 d e^6}{\left (c d^2+a e^2\right )^2 (d+e x)}+\frac {3 c^2 \left (c^3 d^6+5 a c^2 d^4 e^2+15 a^2 c d^2 e^4-5 a^3 e^6-16 a^2 c d e^5 x\right )}{\left (c d^2+a e^2\right )^2 \left (a+c x^2\right )}\right ) \, dx}{8 a^2 c \left (c d^2+a e^2\right )^2}\\ &=\frac {3 e \left (c d^2-a e^2\right ) \left (c d^2+5 a e^2\right )}{8 a^2 \left (c d^2+a e^2\right )^3 (d+e x)}+\frac {a e+c d x}{4 a \left (c d^2+a e^2\right ) (d+e x) \left (a+c x^2\right )^2}-\frac {a e \left (c d^2-5 a e^2\right )-3 c d \left (c d^2+3 a e^2\right ) x}{8 a^2 \left (c d^2+a e^2\right )^2 (d+e x) \left (a+c x^2\right )}+\frac {6 c d e^5 \log (d+e x)}{\left (c d^2+a e^2\right )^4}+\frac {(3 c) \int \frac {c^3 d^6+5 a c^2 d^4 e^2+15 a^2 c d^2 e^4-5 a^3 e^6-16 a^2 c d e^5 x}{a+c x^2} \, dx}{8 a^2 \left (c d^2+a e^2\right )^4}\\ &=\frac {3 e \left (c d^2-a e^2\right ) \left (c d^2+5 a e^2\right )}{8 a^2 \left (c d^2+a e^2\right )^3 (d+e x)}+\frac {a e+c d x}{4 a \left (c d^2+a e^2\right ) (d+e x) \left (a+c x^2\right )^2}-\frac {a e \left (c d^2-5 a e^2\right )-3 c d \left (c d^2+3 a e^2\right ) x}{8 a^2 \left (c d^2+a e^2\right )^2 (d+e x) \left (a+c x^2\right )}+\frac {6 c d e^5 \log (d+e x)}{\left (c d^2+a e^2\right )^4}-\frac {\left (6 c^2 d e^5\right ) \int \frac {x}{a+c x^2} \, dx}{\left (c d^2+a e^2\right )^4}+\frac {\left (3 c \left (c^3 d^6+5 a c^2 d^4 e^2+15 a^2 c d^2 e^4-5 a^3 e^6\right )\right ) \int \frac {1}{a+c x^2} \, dx}{8 a^2 \left (c d^2+a e^2\right )^4}\\ &=\frac {3 e \left (c d^2-a e^2\right ) \left (c d^2+5 a e^2\right )}{8 a^2 \left (c d^2+a e^2\right )^3 (d+e x)}+\frac {a e+c d x}{4 a \left (c d^2+a e^2\right ) (d+e x) \left (a+c x^2\right )^2}-\frac {a e \left (c d^2-5 a e^2\right )-3 c d \left (c d^2+3 a e^2\right ) x}{8 a^2 \left (c d^2+a e^2\right )^2 (d+e x) \left (a+c x^2\right )}+\frac {3 \sqrt {c} \left (c^3 d^6+5 a c^2 d^4 e^2+15 a^2 c d^2 e^4-5 a^3 e^6\right ) \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{8 a^{5/2} \left (c d^2+a e^2\right )^4}+\frac {6 c d e^5 \log (d+e x)}{\left (c d^2+a e^2\right )^4}-\frac {3 c d e^5 \log \left (a+c x^2\right )}{\left (c d^2+a e^2\right )^4}\\ \end {align*}
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Mathematica [A] time = 0.36, size = 241, normalized size = 0.80 \begin {gather*} \frac {\frac {c \left (a e^2+c d^2\right ) \left (a^2 e^3 (16 d-7 e x)+12 a c d^2 e^2 x+3 c^2 d^4 x\right )}{a^2 \left (a+c x^2\right )}+\frac {3 \sqrt {c} \left (-5 a^3 e^6+15 a^2 c d^2 e^4+5 a c^2 d^4 e^2+c^3 d^6\right ) \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{a^{5/2}}+\frac {2 c \left (a e^2+c d^2\right )^2 \left (a e (2 d-e x)+c d^2 x\right )}{a \left (a+c x^2\right )^2}-\frac {8 e^5 \left (a e^2+c d^2\right )}{d+e x}-24 c d e^5 \log \left (a+c x^2\right )+48 c d e^5 \log (d+e x)}{8 \left (a e^2+c d^2\right )^4} \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 {1}{(d+e x)^2 \left (a+c x^2\right )^3} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [B] time = 9.39, size = 2322, normalized size = 7.74
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.19, size = 569, normalized size = 1.90 \begin {gather*} -\frac {3 \, c d e^{5} \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 )}{c^{4} d^{8} + 4 \, a c^{3} d^{6} e^{2} + 6 \, a^{2} c^{2} d^{4} e^{4} + 4 \, a^{3} c d^{2} e^{6} + a^{4} e^{8}} + \frac {3 \, {\left (c^{4} d^{6} e^{2} + 5 \, a c^{3} d^{4} e^{4} + 15 \, a^{2} c^{2} d^{2} e^{6} - 5 \, a^{3} c e^{8}\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 )}}{8 \, {\left (a^{2} c^{4} d^{8} + 4 \, a^{3} c^{3} d^{6} e^{2} + 6 \, a^{4} c^{2} d^{4} e^{4} + 4 \, a^{5} c d^{2} e^{6} + a^{6} e^{8}\right )} \sqrt {a c}} - \frac {e^{11}}{{\left (c^{3} d^{6} e^{6} + 3 \, a c^{2} d^{4} e^{8} + 3 \, a^{2} c d^{2} e^{10} + a^{3} e^{12}\right )} {\left (x e + d\right )}} + \frac {3 \, c^{5} d^{5} e + 14 \, a c^{4} d^{3} e^{3} - 29 \, a^{2} c^{3} d e^{5} - \frac {{\left (9 \, c^{5} d^{6} e^{2} + 41 \, a c^{4} d^{4} e^{4} - 121 \, a^{2} c^{3} d^{2} e^{6} + 7 \, a^{3} c^{2} e^{8}\right )} e^{\left (-1\right )}}{x e + d} + \frac {{\left (9 \, c^{5} d^{7} e^{3} + 45 \, a c^{4} d^{5} e^{5} - 145 \, a^{2} c^{3} d^{3} e^{7} - 21 \, a^{3} c^{2} d e^{9}\right )} e^{\left (-2\right )}}{{\left (x e + d\right )}^{2}} - \frac {3 \, {\left (c^{5} d^{8} e^{4} + 6 \, a c^{4} d^{6} e^{6} - 20 \, a^{2} c^{3} d^{4} e^{8} - 22 \, a^{3} c^{2} d^{2} e^{10} + 3 \, a^{4} c e^{12}\right )} e^{\left (-3\right )}}{{\left (x e + d\right )}^{3}}}{8 \, {\left (c d^{2} + a e^{2}\right )}^{4} a^{2} {\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}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.07, size = 680, normalized size = 2.27 \begin {gather*} -\frac {7 a \,c^{2} e^{6} x^{3}}{8 \left (a \,e^{2}+c \,d^{2}\right )^{4} \left (c \,x^{2}+a \right )^{2}}+\frac {15 c^{4} d^{4} e^{2} x^{3}}{8 \left (a \,e^{2}+c \,d^{2}\right )^{4} \left (c \,x^{2}+a \right )^{2} a}+\frac {3 c^{5} d^{6} x^{3}}{8 \left (a \,e^{2}+c \,d^{2}\right )^{4} \left (c \,x^{2}+a \right )^{2} a^{2}}+\frac {5 c^{3} d^{2} e^{4} x^{3}}{8 \left (a \,e^{2}+c \,d^{2}\right )^{4} \left (c \,x^{2}+a \right )^{2}}+\frac {2 a \,c^{2} d \,e^{5} x^{2}}{\left (a \,e^{2}+c \,d^{2}\right )^{4} \left (c \,x^{2}+a \right )^{2}}+\frac {2 c^{3} d^{3} e^{3} x^{2}}{\left (a \,e^{2}+c \,d^{2}\right )^{4} \left (c \,x^{2}+a \right )^{2}}-\frac {9 a^{2} c \,e^{6} x}{8 \left (a \,e^{2}+c \,d^{2}\right )^{4} \left (c \,x^{2}+a \right )^{2}}+\frac {3 a \,c^{2} d^{2} e^{4} x}{8 \left (a \,e^{2}+c \,d^{2}\right )^{4} \left (c \,x^{2}+a \right )^{2}}+\frac {5 c^{4} d^{6} x}{8 \left (a \,e^{2}+c \,d^{2}\right )^{4} \left (c \,x^{2}+a \right )^{2} a}+\frac {17 c^{3} d^{4} e^{2} x}{8 \left (a \,e^{2}+c \,d^{2}\right )^{4} \left (c \,x^{2}+a \right )^{2}}+\frac {5 a^{2} c d \,e^{5}}{2 \left (a \,e^{2}+c \,d^{2}\right )^{4} \left (c \,x^{2}+a \right )^{2}}+\frac {3 a \,c^{2} d^{3} e^{3}}{\left (a \,e^{2}+c \,d^{2}\right )^{4} \left (c \,x^{2}+a \right )^{2}}-\frac {15 a c \,e^{6} \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{8 \left (a \,e^{2}+c \,d^{2}\right )^{4} \sqrt {a c}}+\frac {15 c^{3} d^{4} e^{2} \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{8 \left (a \,e^{2}+c \,d^{2}\right )^{4} \sqrt {a c}\, a}+\frac {3 c^{4} d^{6} \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{8 \left (a \,e^{2}+c \,d^{2}\right )^{4} \sqrt {a c}\, a^{2}}+\frac {c^{3} d^{5} e}{2 \left (a \,e^{2}+c \,d^{2}\right )^{4} \left (c \,x^{2}+a \right )^{2}}+\frac {45 c^{2} d^{2} e^{4} \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{8 \left (a \,e^{2}+c \,d^{2}\right )^{4} \sqrt {a c}}-\frac {3 c d \,e^{5} \ln \left (c \,x^{2}+a \right )}{\left (a \,e^{2}+c \,d^{2}\right )^{4}}+\frac {6 c d \,e^{5} \ln \left (e x +d \right )}{\left (a \,e^{2}+c \,d^{2}\right )^{4}}-\frac {e^{5}}{\left (a \,e^{2}+c \,d^{2}\right )^{3} \left (e x +d \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 3.35, size = 749, normalized size = 2.50 \begin {gather*} -\frac {3 \, c d e^{5} \log \left (c x^{2} + a\right )}{c^{4} d^{8} + 4 \, a c^{3} d^{6} e^{2} + 6 \, a^{2} c^{2} d^{4} e^{4} + 4 \, a^{3} c d^{2} e^{6} + a^{4} e^{8}} + \frac {6 \, c d e^{5} \log \left (e x + d\right )}{c^{4} d^{8} + 4 \, a c^{3} d^{6} e^{2} + 6 \, a^{2} c^{2} d^{4} e^{4} + 4 \, a^{3} c d^{2} e^{6} + a^{4} e^{8}} + \frac {3 \, {\left (c^{4} d^{6} + 5 \, a c^{3} d^{4} e^{2} + 15 \, a^{2} c^{2} d^{2} e^{4} - 5 \, a^{3} c e^{6}\right )} \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{8 \, {\left (a^{2} c^{4} d^{8} + 4 \, a^{3} c^{3} d^{6} e^{2} + 6 \, a^{4} c^{2} d^{4} e^{4} + 4 \, a^{5} c d^{2} e^{6} + a^{6} e^{8}\right )} \sqrt {a c}} + \frac {4 \, a^{2} c^{2} d^{4} e + 20 \, a^{3} c d^{2} e^{3} - 8 \, a^{4} e^{5} + 3 \, {\left (c^{4} d^{4} e + 4 \, a c^{3} d^{2} e^{3} - 5 \, a^{2} c^{2} e^{5}\right )} x^{4} + 3 \, {\left (c^{4} d^{5} + 4 \, a c^{3} d^{3} e^{2} + 3 \, a^{2} c^{2} d e^{4}\right )} x^{3} + {\left (5 \, a c^{3} d^{4} e + 28 \, a^{2} c^{2} d^{2} e^{3} - 25 \, a^{3} c e^{5}\right )} x^{2} + {\left (5 \, a c^{3} d^{5} + 16 \, a^{2} c^{2} d^{3} e^{2} + 11 \, a^{3} c d e^{4}\right )} x}{8 \, {\left (a^{4} c^{3} d^{7} + 3 \, a^{5} c^{2} d^{5} e^{2} + 3 \, a^{6} c d^{3} e^{4} + a^{7} d e^{6} + {\left (a^{2} c^{5} d^{6} e + 3 \, a^{3} c^{4} d^{4} e^{3} + 3 \, a^{4} c^{3} d^{2} e^{5} + a^{5} c^{2} e^{7}\right )} x^{5} + {\left (a^{2} c^{5} d^{7} + 3 \, a^{3} c^{4} d^{5} e^{2} + 3 \, a^{4} c^{3} d^{3} e^{4} + a^{5} c^{2} d e^{6}\right )} x^{4} + 2 \, {\left (a^{3} c^{4} d^{6} e + 3 \, a^{4} c^{3} d^{4} e^{3} + 3 \, a^{5} c^{2} d^{2} e^{5} + a^{6} c e^{7}\right )} x^{3} + 2 \, {\left (a^{3} c^{4} d^{7} + 3 \, a^{4} c^{3} d^{5} e^{2} + 3 \, a^{5} c^{2} d^{3} e^{4} + a^{6} c d e^{6}\right )} x^{2} + {\left (a^{4} c^{3} d^{6} e + 3 \, a^{5} c^{2} d^{4} e^{3} + 3 \, a^{6} c d^{2} e^{5} + a^{7} e^{7}\right )} x\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.84, size = 1296, normalized size = 4.32 \begin {gather*} \frac {\frac {-2\,a^2\,e^5+5\,a\,c\,d^2\,e^3+c^2\,d^4\,e}{2\,\left (c\,d^2+a\,e^2\right )\,\left (a^2\,e^4+2\,a\,c\,d^2\,e^2+c^2\,d^4\right )}+\frac {3\,x^3\,\left (c^3\,d^3+3\,a\,c^2\,d\,e^2\right )}{8\,a^2\,\left (a^2\,e^4+2\,a\,c\,d^2\,e^2+c^2\,d^4\right )}+\frac {x\,\left (5\,c^2\,d^3+11\,a\,c\,d\,e^2\right )}{8\,a\,\left (a^2\,e^4+2\,a\,c\,d^2\,e^2+c^2\,d^4\right )}+\frac {3\,x^4\,\left (-5\,a^2\,c^2\,e^5+4\,a\,c^3\,d^2\,e^3+c^4\,d^4\,e\right )}{8\,a^2\,\left (a^3\,e^6+3\,a^2\,c\,d^2\,e^4+3\,a\,c^2\,d^4\,e^2+c^3\,d^6\right )}+\frac {x^2\,\left (-25\,a^2\,c\,e^5+28\,a\,c^2\,d^2\,e^3+5\,c^3\,d^4\,e\right )}{8\,a\,\left (c\,d^2+a\,e^2\right )\,\left (a^2\,e^4+2\,a\,c\,d^2\,e^2+c^2\,d^4\right )}}{e\,a^2\,x+d\,a^2+2\,e\,a\,c\,x^3+2\,d\,a\,c\,x^2+e\,c^2\,x^5+d\,c^2\,x^4}-\frac {\ln \left (c^7\,d^{16}\,{\left (-a^5\,c\right )}^{3/2}-25\,a^{13}\,e^{16}\,\sqrt {-a^5\,c}+a^7\,c^9\,d^{16}\,x-4508\,a\,d^4\,e^{12}\,{\left (-a^5\,c\right )}^{5/2}-2644\,c\,d^6\,e^{10}\,{\left (-a^5\,c\right )}^{5/2}+2204\,a^7\,d^2\,e^{14}\,{\left (-a^5\,c\right )}^{3/2}+25\,a^{15}\,c\,e^{16}\,x+76\,a^2\,c^5\,d^{12}\,e^4\,{\left (-a^5\,c\right )}^{3/2}+260\,a^3\,c^4\,d^{10}\,e^6\,{\left (-a^5\,c\right )}^{3/2}+510\,a^4\,c^3\,d^8\,e^8\,{\left (-a^5\,c\right )}^{3/2}+12\,a^8\,c^8\,d^{14}\,e^2\,x+76\,a^9\,c^7\,d^{12}\,e^4\,x+260\,a^{10}\,c^6\,d^{10}\,e^6\,x+510\,a^{11}\,c^5\,d^8\,e^8\,x+2644\,a^{12}\,c^4\,d^6\,e^{10}\,x+4508\,a^{13}\,c^3\,d^4\,e^{12}\,x+2204\,a^{14}\,c^2\,d^2\,e^{14}\,x+12\,a\,c^6\,d^{14}\,e^2\,{\left (-a^5\,c\right )}^{3/2}\right )\,\left (c\,\left (3\,a^5\,d\,e^5+\frac {45\,a^2\,d^2\,e^4\,\sqrt {-a^5\,c}}{16}\right )-\frac {15\,a^3\,e^6\,\sqrt {-a^5\,c}}{16}+\frac {3\,c^3\,d^6\,\sqrt {-a^5\,c}}{16}+\frac {15\,a\,c^2\,d^4\,e^2\,\sqrt {-a^5\,c}}{16}\right )}{a^9\,e^8+4\,a^8\,c\,d^2\,e^6+6\,a^7\,c^2\,d^4\,e^4+4\,a^6\,c^3\,d^6\,e^2+a^5\,c^4\,d^8}-\frac {\ln \left (25\,a^{13}\,e^{16}\,\sqrt {-a^5\,c}-c^7\,d^{16}\,{\left (-a^5\,c\right )}^{3/2}+a^7\,c^9\,d^{16}\,x+4508\,a\,d^4\,e^{12}\,{\left (-a^5\,c\right )}^{5/2}+2644\,c\,d^6\,e^{10}\,{\left (-a^5\,c\right )}^{5/2}-2204\,a^7\,d^2\,e^{14}\,{\left (-a^5\,c\right )}^{3/2}+25\,a^{15}\,c\,e^{16}\,x-76\,a^2\,c^5\,d^{12}\,e^4\,{\left (-a^5\,c\right )}^{3/2}-260\,a^3\,c^4\,d^{10}\,e^6\,{\left (-a^5\,c\right )}^{3/2}-510\,a^4\,c^3\,d^8\,e^8\,{\left (-a^5\,c\right )}^{3/2}+12\,a^8\,c^8\,d^{14}\,e^2\,x+76\,a^9\,c^7\,d^{12}\,e^4\,x+260\,a^{10}\,c^6\,d^{10}\,e^6\,x+510\,a^{11}\,c^5\,d^8\,e^8\,x+2644\,a^{12}\,c^4\,d^6\,e^{10}\,x+4508\,a^{13}\,c^3\,d^4\,e^{12}\,x+2204\,a^{14}\,c^2\,d^2\,e^{14}\,x-12\,a\,c^6\,d^{14}\,e^2\,{\left (-a^5\,c\right )}^{3/2}\right )\,\left (c\,\left (3\,a^5\,d\,e^5-\frac {45\,a^2\,d^2\,e^4\,\sqrt {-a^5\,c}}{16}\right )+\frac {15\,a^3\,e^6\,\sqrt {-a^5\,c}}{16}-\frac {3\,c^3\,d^6\,\sqrt {-a^5\,c}}{16}-\frac {15\,a\,c^2\,d^4\,e^2\,\sqrt {-a^5\,c}}{16}\right )}{a^9\,e^8+4\,a^8\,c\,d^2\,e^6+6\,a^7\,c^2\,d^4\,e^4+4\,a^6\,c^3\,d^6\,e^2+a^5\,c^4\,d^8}+\frac {6\,c\,d\,e^5\,\ln \left (d+e\,x\right )}{{\left (c\,d^2+a\,e^2\right )}^4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] 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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