Optimal. Leaf size=142 \[ \frac {\sqrt {c} d \left (3 a e^2+c d^2\right ) \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{2 a^{3/2} \left (a e^2+c d^2\right )^2}+\frac {a e+c d x}{2 a \left (a+c x^2\right ) \left (a e^2+c d^2\right )}-\frac {e^3 \log \left (a+c x^2\right )}{2 \left (a e^2+c d^2\right )^2}+\frac {e^3 \log (d+e x)}{\left (a e^2+c d^2\right )^2} \]
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Rubi [A] time = 0.13, 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 = {741, 801, 635, 205, 260} \[ \frac {\sqrt {c} d \left (3 a e^2+c d^2\right ) \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{2 a^{3/2} \left (a e^2+c d^2\right )^2}+\frac {a e+c d x}{2 a \left (a+c x^2\right ) \left (a e^2+c d^2\right )}-\frac {e^3 \log \left (a+c x^2\right )}{2 \left (a e^2+c d^2\right )^2}+\frac {e^3 \log (d+e x)}{\left (a e^2+c d^2\right )^2} \]
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) \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 = 0.10, size = 138, normalized size = 0.97 \[ \frac {\sqrt {c} d \left (a+c x^2\right ) \left (3 a e^2+c d^2\right ) \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right )+\sqrt {a} \left (\left (a e^2+c d^2\right ) (a e+c d x)+2 a e^3 \left (a+c x^2\right ) \log (d+e x)-a e^3 \left (a+c x^2\right ) \log \left (a+c x^2\right )\right )}{2 a^{3/2} \left (a+c x^2\right ) \left (a e^2+c d^2\right )^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 2.64, size = 441, normalized size = 3.11 \[ \left [\frac {2 \, a c d^{2} e + 2 \, a^{2} e^{3} + {\left (a c d^{3} + 3 \, a^{2} d e^{2} + {\left (c^{2} d^{3} + 3 \, a c d e^{2}\right )} x^{2}\right )} \sqrt {-\frac {c}{a}} \log \left (\frac {c x^{2} + 2 \, a x \sqrt {-\frac {c}{a}} - a}{c x^{2} + a}\right ) + 2 \, {\left (c^{2} d^{3} + a c d e^{2}\right )} x - 2 \, {\left (a c e^{3} x^{2} + a^{2} e^{3}\right )} \log \left (c x^{2} + a\right ) + 4 \, {\left (a c e^{3} x^{2} + a^{2} e^{3}\right )} \log \left (e x + d\right )}{4 \, {\left (a^{2} c^{2} d^{4} + 2 \, a^{3} c d^{2} e^{2} + a^{4} e^{4} + {\left (a c^{3} d^{4} + 2 \, a^{2} c^{2} d^{2} e^{2} + a^{3} c e^{4}\right )} x^{2}\right )}}, \frac {a c d^{2} e + a^{2} e^{3} + {\left (a c d^{3} + 3 \, a^{2} d e^{2} + {\left (c^{2} d^{3} + 3 \, a c d e^{2}\right )} x^{2}\right )} \sqrt {\frac {c}{a}} \arctan \left (x \sqrt {\frac {c}{a}}\right ) + {\left (c^{2} d^{3} + a c d e^{2}\right )} x - {\left (a c e^{3} x^{2} + a^{2} e^{3}\right )} \log \left (c x^{2} + a\right ) + 2 \, {\left (a c e^{3} x^{2} + a^{2} e^{3}\right )} \log \left (e x + d\right )}{2 \, {\left (a^{2} c^{2} d^{4} + 2 \, a^{3} c d^{2} e^{2} + a^{4} e^{4} + {\left (a c^{3} d^{4} + 2 \, a^{2} c^{2} d^{2} e^{2} + a^{3} c e^{4}\right )} x^{2}\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 192, normalized size = 1.35 \[ -\frac {e^{3} \log \left (c x^{2} + a\right )}{2 \, {\left (c^{2} d^{4} + 2 \, a c d^{2} e^{2} + a^{2} e^{4}\right )}} + \frac {e^{4} \log \left ({\left | x e + d \right |}\right )}{c^{2} d^{4} e + 2 \, a c d^{2} e^{3} + a^{2} e^{5}} + \frac {{\left (c^{2} d^{3} + 3 \, a c d e^{2}\right )} \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{2 \, {\left (a c^{2} d^{4} + 2 \, a^{2} c d^{2} e^{2} + a^{3} e^{4}\right )} \sqrt {a c}} + \frac {a c d^{2} e + a^{2} e^{3} + {\left (c^{2} d^{3} + a c d e^{2}\right )} x}{2 \, {\left (c d^{2} + a e^{2}\right )}^{2} {\left (c x^{2} + a\right )} a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 244, normalized size = 1.72 \[ \frac {c^{2} d^{3} x}{2 \left (a \,e^{2}+c \,d^{2}\right )^{2} \left (c \,x^{2}+a \right ) a}+\frac {c^{2} d^{3} \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{2 \left (a \,e^{2}+c \,d^{2}\right )^{2} \sqrt {a c}\, a}+\frac {c d \,e^{2} x}{2 \left (a \,e^{2}+c \,d^{2}\right )^{2} \left (c \,x^{2}+a \right )}+\frac {3 c d \,e^{2} \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{2 \left (a \,e^{2}+c \,d^{2}\right )^{2} \sqrt {a c}}+\frac {a \,e^{3}}{2 \left (a \,e^{2}+c \,d^{2}\right )^{2} \left (c \,x^{2}+a \right )}+\frac {c \,d^{2} e}{2 \left (a \,e^{2}+c \,d^{2}\right )^{2} \left (c \,x^{2}+a \right )}-\frac {e^{3} \ln \left (c \,x^{2}+a \right )}{2 \left (a \,e^{2}+c \,d^{2}\right )^{2}}+\frac {e^{3} \ln \left (e x +d \right )}{\left (a \,e^{2}+c \,d^{2}\right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.99, size = 189, normalized size = 1.33 \[ -\frac {e^{3} \log \left (c x^{2} + a\right )}{2 \, {\left (c^{2} d^{4} + 2 \, a c d^{2} e^{2} + a^{2} e^{4}\right )}} + \frac {e^{3} \log \left (e x + d\right )}{c^{2} d^{4} + 2 \, a c d^{2} e^{2} + a^{2} e^{4}} + \frac {{\left (c^{2} d^{3} + 3 \, a c d e^{2}\right )} \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{2 \, {\left (a c^{2} d^{4} + 2 \, a^{2} c d^{2} e^{2} + a^{3} e^{4}\right )} \sqrt {a c}} + \frac {c d x + a e}{2 \, {\left (a^{2} c d^{2} + a^{3} e^{2} + {\left (a c^{2} d^{2} + a^{2} c e^{2}\right )} x^{2}\right )}} \]
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 \[ \frac {\frac {e}{2\,\left (c\,d^2+a\,e^2\right )}+\frac {c\,d\,x}{2\,a\,\left (c\,d^2+a\,e^2\right )}}{c\,x^2+a}+\frac {e^3\,\ln \left (d+e\,x\right )}{{\left (c\,d^2+a\,e^2\right )}^2}+\frac {\ln \left (36\,a^7\,e^{10}\,\sqrt {-a^3\,c}+a^3\,c^6\,d^{10}\,x+a^2\,c^5\,d^{10}\,\sqrt {-a^3\,c}-81\,a^3\,d^2\,e^8\,{\left (-a^3\,c\right )}^{3/2}-8\,c^3\,d^8\,e^2\,{\left (-a^3\,c\right )}^{3/2}+36\,a^8\,c\,e^{10}\,x+8\,a^4\,c^5\,d^8\,e^2\,x+22\,a^5\,c^4\,d^6\,e^4\,x+60\,a^6\,c^3\,d^4\,e^6\,x+81\,a^7\,c^2\,d^2\,e^8\,x-22\,a\,c^2\,d^6\,e^4\,{\left (-a^3\,c\right )}^{3/2}-60\,a^2\,c\,d^4\,e^6\,{\left (-a^3\,c\right )}^{3/2}\right )\,\left (c\,d^3\,\sqrt {-a^3\,c}-2\,a^3\,e^3+3\,a\,d\,e^2\,\sqrt {-a^3\,c}\right )}{4\,\left (a^5\,e^4+2\,a^4\,c\,d^2\,e^2+a^3\,c^2\,d^4\right )}-\frac {\ln \left (a^3\,c^6\,d^{10}\,x-36\,a^7\,e^{10}\,\sqrt {-a^3\,c}-a^2\,c^5\,d^{10}\,\sqrt {-a^3\,c}+81\,a^3\,d^2\,e^8\,{\left (-a^3\,c\right )}^{3/2}+8\,c^3\,d^8\,e^2\,{\left (-a^3\,c\right )}^{3/2}+36\,a^8\,c\,e^{10}\,x+8\,a^4\,c^5\,d^8\,e^2\,x+22\,a^5\,c^4\,d^6\,e^4\,x+60\,a^6\,c^3\,d^4\,e^6\,x+81\,a^7\,c^2\,d^2\,e^8\,x+22\,a\,c^2\,d^6\,e^4\,{\left (-a^3\,c\right )}^{3/2}+60\,a^2\,c\,d^4\,e^6\,{\left (-a^3\,c\right )}^{3/2}\right )\,\left (2\,a^3\,e^3+c\,d^3\,\sqrt {-a^3\,c}+3\,a\,d\,e^2\,\sqrt {-a^3\,c}\right )}{4\,\left (a^5\,e^4+2\,a^4\,c\,d^2\,e^2+a^3\,c^2\,d^4\right )} \]
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 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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