Optimal. Leaf size=205 \[ \frac {\sqrt {c} \left (-3 a^2 e^4+6 a c d^2 e^2+c^2 d^4\right ) \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{2 a^{3/2} \left (a e^2+c d^2\right )^3}+\frac {a e+c d x}{2 a \left (a+c x^2\right ) (d+e x) \left (a e^2+c d^2\right )}+\frac {e \left (c d^2-3 a e^2\right )}{2 a (d+e x) \left (a e^2+c d^2\right )^2}-\frac {2 c d e^3 \log \left (a+c x^2\right )}{\left (a e^2+c d^2\right )^3}+\frac {4 c d e^3 \log (d+e x)}{\left (a e^2+c d^2\right )^3} \]
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Rubi [A] time = 0.19, antiderivative size = 205, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.294, Rules used = {741, 801, 635, 205, 260} \begin {gather*} \frac {\sqrt {c} \left (-3 a^2 e^4+6 a c d^2 e^2+c^2 d^4\right ) \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{2 a^{3/2} \left (a e^2+c d^2\right )^3}+\frac {a e+c d x}{2 a \left (a+c x^2\right ) (d+e x) \left (a e^2+c d^2\right )}-\frac {2 c d e^3 \log \left (a+c x^2\right )}{\left (a e^2+c d^2\right )^3}+\frac {e \left (c d^2-3 a e^2\right )}{2 a (d+e x) \left (a e^2+c d^2\right )^2}+\frac {4 c d e^3 \log (d+e x)}{\left (a e^2+c d^2\right )^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 205
Rule 260
Rule 635
Rule 741
Rule 801
Rubi steps
\begin {align*} \int \frac {1}{(d+e x)^2 \left (a+c x^2\right )^2} \, dx &=\frac {a e+c d x}{2 a \left (c d^2+a e^2\right ) (d+e x) \left (a+c x^2\right )}-\frac {\int \frac {-c d^2-3 a e^2-2 c d e x}{(d+e x)^2 \left (a+c x^2\right )} \, dx}{2 a \left (c d^2+a e^2\right )}\\ &=\frac {a e+c d x}{2 a \left (c d^2+a e^2\right ) (d+e x) \left (a+c x^2\right )}-\frac {\int \left (\frac {c d^2 e^2-3 a e^4}{\left (c d^2+a e^2\right ) (d+e x)^2}-\frac {8 a c d e^4}{\left (c d^2+a e^2\right )^2 (d+e x)}-\frac {c \left (c^2 d^4+6 a c d^2 e^2-3 a^2 e^4-8 a c d e^3 x\right )}{\left (c d^2+a e^2\right )^2 \left (a+c x^2\right )}\right ) \, dx}{2 a \left (c d^2+a e^2\right )}\\ &=\frac {e \left (c d^2-3 a e^2\right )}{2 a \left (c d^2+a e^2\right )^2 (d+e x)}+\frac {a e+c d x}{2 a \left (c d^2+a e^2\right ) (d+e x) \left (a+c x^2\right )}+\frac {4 c d e^3 \log (d+e x)}{\left (c d^2+a e^2\right )^3}+\frac {c \int \frac {c^2 d^4+6 a c d^2 e^2-3 a^2 e^4-8 a c d e^3 x}{a+c x^2} \, dx}{2 a \left (c d^2+a e^2\right )^3}\\ &=\frac {e \left (c d^2-3 a e^2\right )}{2 a \left (c d^2+a e^2\right )^2 (d+e x)}+\frac {a e+c d x}{2 a \left (c d^2+a e^2\right ) (d+e x) \left (a+c x^2\right )}+\frac {4 c d e^3 \log (d+e x)}{\left (c d^2+a e^2\right )^3}-\frac {\left (4 c^2 d e^3\right ) \int \frac {x}{a+c x^2} \, dx}{\left (c d^2+a e^2\right )^3}+\frac {\left (c \left (c^2 d^4+6 a c d^2 e^2-3 a^2 e^4\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^2-3 a e^2\right )}{2 a \left (c d^2+a e^2\right )^2 (d+e x)}+\frac {a e+c d x}{2 a \left (c d^2+a e^2\right ) (d+e x) \left (a+c x^2\right )}+\frac {\sqrt {c} \left (c^2 d^4+6 a c d^2 e^2-3 a^2 e^4\right ) \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{2 a^{3/2} \left (c d^2+a e^2\right )^3}+\frac {4 c d e^3 \log (d+e x)}{\left (c d^2+a e^2\right )^3}-\frac {2 c d e^3 \log \left (a+c x^2\right )}{\left (c d^2+a e^2\right )^3}\\ \end {align*}
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Mathematica [A] time = 0.20, size = 162, normalized size = 0.79 \begin {gather*} \frac {\frac {\sqrt {c} \left (-3 a^2 e^4+6 a c d^2 e^2+c^2 d^4\right ) \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{a^{3/2}}+\frac {c \left (a e^2+c d^2\right ) \left (a e (2 d-e x)+c d^2 x\right )}{a \left (a+c x^2\right )}-\frac {2 e^3 \left (a e^2+c d^2\right )}{d+e x}-4 c d e^3 \log \left (a+c x^2\right )+8 c d e^3 \log (d+e x)}{2 \left (a e^2+c d^2\right )^3} \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 )^2} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [B] time = 2.38, size = 1111, normalized size = 5.42 \begin {gather*} \left [\frac {4 \, a c^{2} d^{4} e - 4 \, a^{3} e^{5} + 2 \, {\left (c^{3} d^{4} e - 2 \, a c^{2} d^{2} e^{3} - 3 \, a^{2} c e^{5}\right )} x^{2} - {\left (a c^{2} d^{5} + 6 \, a^{2} c d^{3} e^{2} - 3 \, a^{3} d e^{4} + {\left (c^{3} d^{4} e + 6 \, a c^{2} d^{2} e^{3} - 3 \, a^{2} c e^{5}\right )} x^{3} + {\left (c^{3} d^{5} + 6 \, a c^{2} d^{3} e^{2} - 3 \, a^{2} c d e^{4}\right )} x^{2} + {\left (a c^{2} d^{4} e + 6 \, a^{2} c d^{2} e^{3} - 3 \, a^{3} e^{5}\right )} x\right )} \sqrt {-\frac {c}{a}} \log \left (\frac {c x^{2} - 2 \, a x \sqrt {-\frac {c}{a}} - a}{c x^{2} + a}\right ) + 2 \, {\left (c^{3} d^{5} + 2 \, a c^{2} d^{3} e^{2} + a^{2} c d e^{4}\right )} x - 8 \, {\left (a c^{2} d e^{4} x^{3} + a c^{2} d^{2} e^{3} x^{2} + a^{2} c d e^{4} x + a^{2} c d^{2} e^{3}\right )} \log \left (c x^{2} + a\right ) + 16 \, {\left (a c^{2} d e^{4} x^{3} + a c^{2} d^{2} e^{3} x^{2} + a^{2} c d e^{4} x + a^{2} c d^{2} e^{3}\right )} \log \left (e x + d\right )}{4 \, {\left (a^{2} c^{3} d^{7} + 3 \, a^{3} c^{2} d^{5} e^{2} + 3 \, a^{4} c d^{3} e^{4} + a^{5} d e^{6} + {\left (a c^{4} d^{6} e + 3 \, a^{2} c^{3} d^{4} e^{3} + 3 \, a^{3} c^{2} d^{2} e^{5} + a^{4} c e^{7}\right )} x^{3} + {\left (a c^{4} d^{7} + 3 \, a^{2} c^{3} d^{5} e^{2} + 3 \, a^{3} c^{2} d^{3} e^{4} + a^{4} c d e^{6}\right )} x^{2} + {\left (a^{2} c^{3} d^{6} e + 3 \, a^{3} c^{2} d^{4} e^{3} + 3 \, a^{4} c d^{2} e^{5} + a^{5} e^{7}\right )} x\right )}}, \frac {2 \, a c^{2} d^{4} e - 2 \, a^{3} e^{5} + {\left (c^{3} d^{4} e - 2 \, a c^{2} d^{2} e^{3} - 3 \, a^{2} c e^{5}\right )} x^{2} + {\left (a c^{2} d^{5} + 6 \, a^{2} c d^{3} e^{2} - 3 \, a^{3} d e^{4} + {\left (c^{3} d^{4} e + 6 \, a c^{2} d^{2} e^{3} - 3 \, a^{2} c e^{5}\right )} x^{3} + {\left (c^{3} d^{5} + 6 \, a c^{2} d^{3} e^{2} - 3 \, a^{2} c d e^{4}\right )} x^{2} + {\left (a c^{2} d^{4} e + 6 \, a^{2} c d^{2} e^{3} - 3 \, a^{3} e^{5}\right )} x\right )} \sqrt {\frac {c}{a}} \arctan \left (x \sqrt {\frac {c}{a}}\right ) + {\left (c^{3} d^{5} + 2 \, a c^{2} d^{3} e^{2} + a^{2} c d e^{4}\right )} x - 4 \, {\left (a c^{2} d e^{4} x^{3} + a c^{2} d^{2} e^{3} x^{2} + a^{2} c d e^{4} x + a^{2} c d^{2} e^{3}\right )} \log \left (c x^{2} + a\right ) + 8 \, {\left (a c^{2} d e^{4} x^{3} + a c^{2} d^{2} e^{3} x^{2} + a^{2} c d e^{4} x + a^{2} c d^{2} e^{3}\right )} \log \left (e x + d\right )}{2 \, {\left (a^{2} c^{3} d^{7} + 3 \, a^{3} c^{2} d^{5} e^{2} + 3 \, a^{4} c d^{3} e^{4} + a^{5} d e^{6} + {\left (a c^{4} d^{6} e + 3 \, a^{2} c^{3} d^{4} e^{3} + 3 \, a^{3} c^{2} d^{2} e^{5} + a^{4} c e^{7}\right )} x^{3} + {\left (a c^{4} d^{7} + 3 \, a^{2} c^{3} d^{5} e^{2} + 3 \, a^{3} c^{2} d^{3} e^{4} + a^{4} c d e^{6}\right )} x^{2} + {\left (a^{2} c^{3} d^{6} e + 3 \, a^{3} c^{2} d^{4} e^{3} + 3 \, a^{4} c d^{2} e^{5} + a^{5} e^{7}\right )} x\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.17, size = 386, normalized size = 1.88 \begin {gather*} -\frac {2 \, c d e^{3} \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^{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 (c^{3} d^{4} e^{2} + 6 \, a c^{2} d^{2} e^{4} - 3 \, 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 {e^{7}}{{\left (c^{2} d^{4} e^{4} + 2 \, a c d^{2} e^{6} + a^{2} e^{8}\right )} {\left (x e + d\right )}} + \frac {\frac {c^{3} d^{3} e - 3 \, a c^{2} d e^{3}}{c d^{2} + a e^{2}} - \frac {{\left (c^{3} d^{4} e^{2} - 6 \, a c^{2} d^{2} e^{4} + 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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maple [A] time = 0.07, size = 314, normalized size = 1.53 \begin {gather*} -\frac {a c \,e^{4} x}{2 \left (a \,e^{2}+c \,d^{2}\right )^{3} \left (c \,x^{2}+a \right )}-\frac {3 a c \,e^{4} \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{2 \left (a \,e^{2}+c \,d^{2}\right )^{3} \sqrt {a c}}+\frac {c^{3} d^{4} x}{2 \left (a \,e^{2}+c \,d^{2}\right )^{3} \left (c \,x^{2}+a \right ) a}+\frac {c^{3} d^{4} \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{2 \left (a \,e^{2}+c \,d^{2}\right )^{3} \sqrt {a c}\, a}+\frac {3 c^{2} d^{2} e^{2} \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{\left (a \,e^{2}+c \,d^{2}\right )^{3} \sqrt {a c}}+\frac {a c d \,e^{3}}{\left (a \,e^{2}+c \,d^{2}\right )^{3} \left (c \,x^{2}+a \right )}+\frac {c^{2} d^{3} e}{\left (a \,e^{2}+c \,d^{2}\right )^{3} \left (c \,x^{2}+a \right )}-\frac {2 c d \,e^{3} \ln \left (c \,x^{2}+a \right )}{\left (a \,e^{2}+c \,d^{2}\right )^{3}}+\frac {4 c d \,e^{3} \ln \left (e x +d \right )}{\left (a \,e^{2}+c \,d^{2}\right )^{3}}-\frac {e^{3}}{\left (a \,e^{2}+c \,d^{2}\right )^{2} \left (e x +d \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 3.06, size = 394, normalized size = 1.92 \begin {gather*} -\frac {2 \, c d e^{3} \log \left (c x^{2} + a\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 {4 \, c d e^{3} \log \left (e x + 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 (c^{3} d^{4} + 6 \, a c^{2} d^{2} e^{2} - 3 \, a^{2} c e^{4}\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 {2 \, a c d^{2} e - 2 \, a^{2} e^{3} + {\left (c^{2} d^{2} e - 3 \, a c e^{3}\right )} x^{2} + {\left (c^{2} d^{3} + a c d e^{2}\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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mupad [B] time = 1.31, size = 819, normalized size = 4.00 \begin {gather*} \frac {4\,c\,d\,e^3\,\ln \left (d+e\,x\right )}{{\left (c\,d^2+a\,e^2\right )}^3}-\frac {\ln \left (c^5\,d^{12}\,{\left (-a^3\,c\right )}^{3/2}-9\,a^9\,e^{12}\,\sqrt {-a^3\,c}+a^4\,c^7\,d^{12}\,x-1119\,a\,d^4\,e^8\,{\left (-a^3\,c\right )}^{5/2}-612\,c\,d^6\,e^6\,{\left (-a^3\,c\right )}^{5/2}+558\,a^5\,d^2\,e^{10}\,{\left (-a^3\,c\right )}^{3/2}+9\,a^{10}\,c\,e^{12}\,x+55\,a^2\,c^3\,d^8\,e^4\,{\left (-a^3\,c\right )}^{3/2}+14\,a^5\,c^6\,d^{10}\,e^2\,x+55\,a^6\,c^5\,d^8\,e^4\,x+612\,a^7\,c^4\,d^6\,e^6\,x+1119\,a^8\,c^3\,d^4\,e^8\,x+558\,a^9\,c^2\,d^2\,e^{10}\,x+14\,a\,c^4\,d^{10}\,e^2\,{\left (-a^3\,c\right )}^{3/2}\right )\,\left (c\,\left (2\,a^3\,d\,e^3+\frac {3\,a\,d^2\,e^2\,\sqrt {-a^3\,c}}{2}\right )-\frac {3\,a^2\,e^4\,\sqrt {-a^3\,c}}{4}+\frac {c^2\,d^4\,\sqrt {-a^3\,c}}{4}\right )}{a^6\,e^6+3\,a^5\,c\,d^2\,e^4+3\,a^4\,c^2\,d^4\,e^2+a^3\,c^3\,d^6}-\frac {\ln \left (9\,a^9\,e^{12}\,\sqrt {-a^3\,c}-c^5\,d^{12}\,{\left (-a^3\,c\right )}^{3/2}+a^4\,c^7\,d^{12}\,x+1119\,a\,d^4\,e^8\,{\left (-a^3\,c\right )}^{5/2}+612\,c\,d^6\,e^6\,{\left (-a^3\,c\right )}^{5/2}-558\,a^5\,d^2\,e^{10}\,{\left (-a^3\,c\right )}^{3/2}+9\,a^{10}\,c\,e^{12}\,x-55\,a^2\,c^3\,d^8\,e^4\,{\left (-a^3\,c\right )}^{3/2}+14\,a^5\,c^6\,d^{10}\,e^2\,x+55\,a^6\,c^5\,d^8\,e^4\,x+612\,a^7\,c^4\,d^6\,e^6\,x+1119\,a^8\,c^3\,d^4\,e^8\,x+558\,a^9\,c^2\,d^2\,e^{10}\,x-14\,a\,c^4\,d^{10}\,e^2\,{\left (-a^3\,c\right )}^{3/2}\right )\,\left (c\,\left (2\,a^3\,d\,e^3-\frac {3\,a\,d^2\,e^2\,\sqrt {-a^3\,c}}{2}\right )+\frac {3\,a^2\,e^4\,\sqrt {-a^3\,c}}{4}-\frac {c^2\,d^4\,\sqrt {-a^3\,c}}{4}\right )}{a^6\,e^6+3\,a^5\,c\,d^2\,e^4+3\,a^4\,c^2\,d^4\,e^2+a^3\,c^3\,d^6}-\frac {\frac {a\,e^3-c\,d^2\,e}{{\left (c\,d^2+a\,e^2\right )}^2}-\frac {c\,d\,x}{2\,a\,\left (c\,d^2+a\,e^2\right )}+\frac {c\,x^2\,\left (3\,a\,e^3-c\,d^2\,e\right )}{2\,a\,\left (a^2\,e^4+2\,a\,c\,d^2\,e^2+c^2\,d^4\right )}}{c\,e\,x^3+c\,d\,x^2+a\,e\,x+a\,d} \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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