Optimal. Leaf size=142 \[ \frac {a e+c d x}{2 a \left (c d^2+a e^2\right ) \left (a+c x^2\right )}+\frac {\sqrt {c} d \left (c d^2+3 a e^2\right ) \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{2 a^{3/2} \left (c d^2+a e^2\right )^2}+\frac {e^3 \log (d+e x)}{\left (c d^2+a e^2\right )^2}-\frac {e^3 \log \left (a+c x^2\right )}{2 \left (c d^2+a e^2\right )^2} \]
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Rubi [A]
time = 0.09, antiderivative size = 142, 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 = {755, 815, 649,
211, 266} \begin {gather*} \frac {\sqrt {c} d \text {ArcTan}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right ) \left (3 a e^2+c d^2\right )}{2 a^{3/2} \left (a e^2+c d^2\right )^2}+\frac {a e+c d x}{2 a \left (a+c x^2\right ) \left (a e^2+c d^2\right )}-\frac {e^3 \log \left (a+c x^2\right )}{2 \left (a e^2+c d^2\right )^2}+\frac {e^3 \log (d+e x)}{\left (a e^2+c d^2\right )^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 211
Rule 266
Rule 649
Rule 755
Rule 815
Rubi steps
\begin {align*} \int \frac {1}{(d+e x) \left (a+c x^2\right )^2} \, dx &=\frac {a e+c d x}{2 a \left (c d^2+a e^2\right ) \left (a+c x^2\right )}-\frac {\int \frac {-c d^2-2 a e^2-c d e x}{(d+e x) \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 ) \left (a+c x^2\right )}-\frac {\int \left (-\frac {2 a e^4}{\left (c d^2+a e^2\right ) (d+e x)}-\frac {c \left (c d^3+3 a d e^2-2 a e^3 x\right )}{\left (c d^2+a e^2\right ) \left (a+c x^2\right )}\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 ) \left (a+c x^2\right )}+\frac {e^3 \log (d+e x)}{\left (c d^2+a e^2\right )^2}+\frac {c \int \frac {c d^3+3 a d e^2-2 a e^3 x}{a+c x^2} \, dx}{2 a \left (c d^2+a e^2\right )^2}\\ &=\frac {a e+c d x}{2 a \left (c d^2+a e^2\right ) \left (a+c x^2\right )}+\frac {e^3 \log (d+e x)}{\left (c d^2+a e^2\right )^2}-\frac {\left (c e^3\right ) \int \frac {x}{a+c x^2} \, dx}{\left (c d^2+a e^2\right )^2}+\frac {\left (c d \left (c d^2+3 a e^2\right )\right ) \int \frac {1}{a+c x^2} \, dx}{2 a \left (c d^2+a e^2\right )^2}\\ &=\frac {a e+c d x}{2 a \left (c d^2+a e^2\right ) \left (a+c x^2\right )}+\frac {\sqrt {c} d \left (c d^2+3 a e^2\right ) \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{2 a^{3/2} \left (c d^2+a e^2\right )^2}+\frac {e^3 \log (d+e x)}{\left (c d^2+a e^2\right )^2}-\frac {e^3 \log \left (a+c x^2\right )}{2 \left (c d^2+a e^2\right )^2}\\ \end {align*}
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Mathematica [A]
time = 0.07, size = 138, normalized size = 0.97 \begin {gather*} \frac {\sqrt {c} d \left (c d^2+3 a e^2\right ) \left (a+c x^2\right ) \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right )+\sqrt {a} \left (\left (c d^2+a e^2\right ) (a e+c d x)+2 a e^3 \left (a+c x^2\right ) \log (d+e x)-a e^3 \left (a+c x^2\right ) \log \left (a+c x^2\right )\right )}{2 a^{3/2} \left (c d^2+a e^2\right )^2 \left (a+c x^2\right )} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.60, size = 138, normalized size = 0.97
method | result | size |
default | \(\frac {e^{3} \ln \left (e x +d \right )}{\left (e^{2} a +c \,d^{2}\right )^{2}}+\frac {c \left (\frac {\frac {d \left (e^{2} a +c \,d^{2}\right ) x}{2 a}+\frac {e \left (e^{2} a +c \,d^{2}\right )}{2 c}}{c \,x^{2}+a}+\frac {-\frac {a \,e^{3} \ln \left (c \,x^{2}+a \right )}{c}+\frac {\left (3 a d \,e^{2}+c \,d^{3}\right ) \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{\sqrt {a c}}}{2 a}\right )}{\left (e^{2} a +c \,d^{2}\right )^{2}}\) | \(138\) |
risch | \(\frac {\frac {c d x}{2 a \left (e^{2} a +c \,d^{2}\right )}+\frac {e}{2 e^{2} a +2 c \,d^{2}}}{c \,x^{2}+a}+\frac {e^{3} \ln \left (e x +d \right )}{a^{2} e^{4}+2 a c \,d^{2} e^{2}+c^{2} d^{4}}+\frac {\left (\munderset {\textit {\_R} =\RootOf \left (\left (a^{5} e^{4}+2 a^{4} c \,d^{2} e^{2}+a^{3} c^{2} d^{4}\right ) \textit {\_Z}^{2}+4 a^{3} e^{3} \textit {\_Z} +4 e^{2} a +c \,d^{2}\right )}{\sum }\textit {\_R} \ln \left (\left (\left (3 a^{5} e^{6}+5 a^{4} c \,d^{2} e^{4}+a^{3} c^{2} d^{4} e^{2}-a^{2} c^{3} d^{6}\right ) \textit {\_R}^{2}+\left (6 a^{3} e^{5}+8 d^{2} e^{3} a^{2} c +2 d^{4} e \,c^{2} a \right ) \textit {\_R} +2 d^{2} e^{2} c \right ) x +\left (4 a^{5} d \,e^{5}+8 a^{4} c \,d^{3} e^{3}+4 a^{3} c^{2} d^{5} e \right ) \textit {\_R}^{2}+\left (d \,e^{4} a^{3}+2 d^{3} e^{2} a^{2} c +d^{5} c^{2} a \right ) \textit {\_R} +4 a d \,e^{3}+2 c \,d^{3} e \right )\right )}{4}\) | \(329\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.56, size = 180, normalized size = 1.27 \begin {gather*} -\frac {e^{3} \log \left (c x^{2} + a\right )}{2 \, {\left (c^{2} d^{4} + 2 \, a c d^{2} e^{2} + a^{2} e^{4}\right )}} + \frac {e^{3} \log \left (x e + d\right )}{c^{2} d^{4} + 2 \, a c d^{2} e^{2} + a^{2} e^{4}} + \frac {{\left (c^{2} d^{3} + 3 \, a c d e^{2}\right )} \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{2 \, {\left (a c^{2} d^{4} + 2 \, a^{2} c d^{2} e^{2} + a^{3} e^{4}\right )} \sqrt {a c}} + \frac {c d x + a e}{2 \, {\left (a^{2} c d^{2} + a^{3} e^{2} + {\left (a c^{2} d^{2} + a^{2} c e^{2}\right )} x^{2}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.39, size = 412, normalized size = 2.90 \begin {gather*} \left [\frac {2 \, c^{2} d^{3} x + 2 \, a c d x e^{2} + 2 \, a c d^{2} e + 2 \, a^{2} e^{3} - 2 \, {\left (a c x^{2} + a^{2}\right )} e^{3} \log \left (c x^{2} + a\right ) + 4 \, {\left (a c x^{2} + a^{2}\right )} e^{3} \log \left (x e + d\right ) + {\left (c^{2} d^{3} x^{2} + a c d^{3} + 3 \, {\left (a c d x^{2} + a^{2} d\right )} e^{2}\right )} \sqrt {-\frac {c}{a}} \log \left (\frac {c x^{2} + 2 \, a x \sqrt {-\frac {c}{a}} - a}{c x^{2} + a}\right )}{4 \, {\left (a c^{3} d^{4} x^{2} + a^{2} c^{2} d^{4} + {\left (a^{3} c x^{2} + a^{4}\right )} e^{4} + 2 \, {\left (a^{2} c^{2} d^{2} x^{2} + a^{3} c d^{2}\right )} e^{2}\right )}}, \frac {c^{2} d^{3} x + a c d x e^{2} + a c d^{2} e + a^{2} e^{3} - {\left (a c x^{2} + a^{2}\right )} e^{3} \log \left (c x^{2} + a\right ) + 2 \, {\left (a c x^{2} + a^{2}\right )} e^{3} \log \left (x e + d\right ) + {\left (c^{2} d^{3} x^{2} + a c d^{3} + 3 \, {\left (a c d x^{2} + a^{2} d\right )} e^{2}\right )} \sqrt {\frac {c}{a}} \arctan \left (x \sqrt {\frac {c}{a}}\right )}{2 \, {\left (a c^{3} d^{4} x^{2} + a^{2} c^{2} d^{4} + {\left (a^{3} c x^{2} + a^{4}\right )} e^{4} + 2 \, {\left (a^{2} c^{2} d^{2} x^{2} + a^{3} c d^{2}\right )} e^{2}\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 = 2.07, size = 192, normalized size = 1.35 \begin {gather*} -\frac {e^{3} \log \left (c x^{2} + a\right )}{2 \, {\left (c^{2} d^{4} + 2 \, a c d^{2} e^{2} + a^{2} e^{4}\right )}} + \frac {e^{4} \log \left ({\left | x e + d \right |}\right )}{c^{2} d^{4} e + 2 \, a c d^{2} e^{3} + a^{2} e^{5}} + \frac {{\left (c^{2} d^{3} + 3 \, a c d e^{2}\right )} \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{2 \, {\left (a c^{2} d^{4} + 2 \, a^{2} c d^{2} e^{2} + a^{3} e^{4}\right )} \sqrt {a c}} + \frac {a c d^{2} e + a^{2} e^{3} + {\left (c^{2} d^{3} + a c d e^{2}\right )} x}{2 \, {\left (c d^{2} + a e^{2}\right )}^{2} {\left (c x^{2} + a\right )} a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.04, size = 609, normalized size = 4.29 \begin {gather*} \frac {\frac {e}{2\,\left (c\,d^2+a\,e^2\right )}+\frac {c\,d\,x}{2\,a\,\left (c\,d^2+a\,e^2\right )}}{c\,x^2+a}+\frac {e^3\,\ln \left (d+e\,x\right )}{{\left (c\,d^2+a\,e^2\right )}^2}+\frac {\ln \left (36\,a^7\,e^{10}\,\sqrt {-a^3\,c}+a^3\,c^6\,d^{10}\,x+a^2\,c^5\,d^{10}\,\sqrt {-a^3\,c}-81\,a^3\,d^2\,e^8\,{\left (-a^3\,c\right )}^{3/2}-8\,c^3\,d^8\,e^2\,{\left (-a^3\,c\right )}^{3/2}+36\,a^8\,c\,e^{10}\,x+8\,a^4\,c^5\,d^8\,e^2\,x+22\,a^5\,c^4\,d^6\,e^4\,x+60\,a^6\,c^3\,d^4\,e^6\,x+81\,a^7\,c^2\,d^2\,e^8\,x-22\,a\,c^2\,d^6\,e^4\,{\left (-a^3\,c\right )}^{3/2}-60\,a^2\,c\,d^4\,e^6\,{\left (-a^3\,c\right )}^{3/2}\right )\,\left (c\,d^3\,\sqrt {-a^3\,c}-2\,a^3\,e^3+3\,a\,d\,e^2\,\sqrt {-a^3\,c}\right )}{4\,\left (a^5\,e^4+2\,a^4\,c\,d^2\,e^2+a^3\,c^2\,d^4\right )}-\frac {\ln \left (a^3\,c^6\,d^{10}\,x-36\,a^7\,e^{10}\,\sqrt {-a^3\,c}-a^2\,c^5\,d^{10}\,\sqrt {-a^3\,c}+81\,a^3\,d^2\,e^8\,{\left (-a^3\,c\right )}^{3/2}+8\,c^3\,d^8\,e^2\,{\left (-a^3\,c\right )}^{3/2}+36\,a^8\,c\,e^{10}\,x+8\,a^4\,c^5\,d^8\,e^2\,x+22\,a^5\,c^4\,d^6\,e^4\,x+60\,a^6\,c^3\,d^4\,e^6\,x+81\,a^7\,c^2\,d^2\,e^8\,x+22\,a\,c^2\,d^6\,e^4\,{\left (-a^3\,c\right )}^{3/2}+60\,a^2\,c\,d^4\,e^6\,{\left (-a^3\,c\right )}^{3/2}\right )\,\left (2\,a^3\,e^3+c\,d^3\,\sqrt {-a^3\,c}+3\,a\,d\,e^2\,\sqrt {-a^3\,c}\right )}{4\,\left (a^5\,e^4+2\,a^4\,c\,d^2\,e^2+a^3\,c^2\,d^4\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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