Optimal. Leaf size=176 \[ \frac {c^{3/2} d \left (c d^2-3 a e^2\right ) \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{\sqrt {a} \left (a e^2+c d^2\right )^3}-\frac {c e \left (3 c d^2-a e^2\right ) \log \left (a+c x^2\right )}{2 \left (a e^2+c d^2\right )^3}-\frac {2 c d e}{(d+e x) \left (a e^2+c d^2\right )^2}-\frac {e}{2 (d+e x)^2 \left (a e^2+c d^2\right )}+\frac {c e \left (3 c d^2-a e^2\right ) \log (d+e x)}{\left (a e^2+c d^2\right )^3} \]
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Rubi [A] time = 0.17, antiderivative size = 176, 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 = {710, 801, 635, 205, 260} \begin {gather*} \frac {c^{3/2} d \left (c d^2-3 a e^2\right ) \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{\sqrt {a} \left (a e^2+c d^2\right )^3}-\frac {c e \left (3 c d^2-a e^2\right ) \log \left (a+c x^2\right )}{2 \left (a e^2+c d^2\right )^3}-\frac {2 c d e}{(d+e x) \left (a e^2+c d^2\right )^2}-\frac {e}{2 (d+e x)^2 \left (a e^2+c d^2\right )}+\frac {c e \left (3 c d^2-a e^2\right ) \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 710
Rule 801
Rubi steps
\begin {align*} \int \frac {1}{(d+e x)^3 \left (a+c x^2\right )} \, dx &=-\frac {e}{2 \left (c d^2+a e^2\right ) (d+e x)^2}+\frac {c \int \frac {d-e x}{(d+e x)^2 \left (a+c x^2\right )} \, dx}{c d^2+a e^2}\\ &=-\frac {e}{2 \left (c d^2+a e^2\right ) (d+e x)^2}+\frac {c \int \left (\frac {2 d e^2}{\left (c d^2+a e^2\right ) (d+e x)^2}+\frac {3 c d^2 e^2-a e^4}{\left (c d^2+a e^2\right )^2 (d+e x)}+\frac {c \left (d \left (c d^2-3 a e^2\right )-e \left (3 c d^2-a e^2\right ) x\right )}{\left (c d^2+a e^2\right )^2 \left (a+c x^2\right )}\right ) \, dx}{c d^2+a e^2}\\ &=-\frac {e}{2 \left (c d^2+a e^2\right ) (d+e x)^2}-\frac {2 c d e}{\left (c d^2+a e^2\right )^2 (d+e x)}+\frac {c e \left (3 c d^2-a e^2\right ) \log (d+e x)}{\left (c d^2+a e^2\right )^3}+\frac {c^2 \int \frac {d \left (c d^2-3 a e^2\right )-e \left (3 c d^2-a e^2\right ) x}{a+c x^2} \, dx}{\left (c d^2+a e^2\right )^3}\\ &=-\frac {e}{2 \left (c d^2+a e^2\right ) (d+e x)^2}-\frac {2 c d e}{\left (c d^2+a e^2\right )^2 (d+e x)}+\frac {c e \left (3 c d^2-a e^2\right ) \log (d+e x)}{\left (c d^2+a e^2\right )^3}+\frac {\left (c^2 d \left (c d^2-3 a e^2\right )\right ) \int \frac {1}{a+c x^2} \, dx}{\left (c d^2+a e^2\right )^3}-\frac {\left (c^2 e \left (3 c d^2-a e^2\right )\right ) \int \frac {x}{a+c x^2} \, dx}{\left (c d^2+a e^2\right )^3}\\ &=-\frac {e}{2 \left (c d^2+a e^2\right ) (d+e x)^2}-\frac {2 c d e}{\left (c d^2+a e^2\right )^2 (d+e x)}+\frac {c^{3/2} d \left (c d^2-3 a e^2\right ) \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{\sqrt {a} \left (c d^2+a e^2\right )^3}+\frac {c e \left (3 c d^2-a e^2\right ) \log (d+e x)}{\left (c d^2+a e^2\right )^3}-\frac {c e \left (3 c d^2-a e^2\right ) \log \left (a+c x^2\right )}{2 \left (c d^2+a e^2\right )^3}\\ \end {align*}
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Mathematica [A] time = 0.29, size = 140, normalized size = 0.80 \begin {gather*} \frac {\frac {2 c^{3/2} d \left (c d^2-3 a e^2\right ) \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{\sqrt {a}}+e \left (c \left (a e^2-3 c d^2\right ) \log \left (a+c x^2\right )-\frac {\left (a e^2+c d^2\right ) \left (a e^2+c d (5 d+4 e x)\right )}{(d+e x)^2}+2 c \left (3 c d^2-a e^2\right ) \log (d+e x)\right )}{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)^3 \left (a+c x^2\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [B] time = 1.36, size = 853, normalized size = 4.85 \begin {gather*} \left [-\frac {5 \, c^{2} d^{4} e + 6 \, a c d^{2} e^{3} + a^{2} e^{5} + {\left (c^{2} d^{5} - 3 \, a c d^{3} e^{2} + {\left (c^{2} d^{3} e^{2} - 3 \, a c d e^{4}\right )} x^{2} + 2 \, {\left (c^{2} d^{4} e - 3 \, a c d^{2} e^{3}\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 ) + 4 \, {\left (c^{2} d^{3} e^{2} + a c d e^{4}\right )} x + {\left (3 \, c^{2} d^{4} e - a c d^{2} e^{3} + {\left (3 \, c^{2} d^{2} e^{3} - a c e^{5}\right )} x^{2} + 2 \, {\left (3 \, c^{2} d^{3} e^{2} - a c d e^{4}\right )} x\right )} \log \left (c x^{2} + a\right ) - 2 \, {\left (3 \, c^{2} d^{4} e - a c d^{2} e^{3} + {\left (3 \, c^{2} d^{2} e^{3} - a c e^{5}\right )} x^{2} + 2 \, {\left (3 \, c^{2} d^{3} e^{2} - a c d e^{4}\right )} x\right )} \log \left (e x + d\right )}{2 \, {\left (c^{3} d^{8} + 3 \, a c^{2} d^{6} e^{2} + 3 \, a^{2} c d^{4} e^{4} + a^{3} d^{2} e^{6} + {\left (c^{3} d^{6} e^{2} + 3 \, a c^{2} d^{4} e^{4} + 3 \, a^{2} c d^{2} e^{6} + a^{3} e^{8}\right )} x^{2} + 2 \, {\left (c^{3} d^{7} e + 3 \, a c^{2} d^{5} e^{3} + 3 \, a^{2} c d^{3} e^{5} + a^{3} d e^{7}\right )} x\right )}}, -\frac {5 \, c^{2} d^{4} e + 6 \, a c d^{2} e^{3} + a^{2} e^{5} - 2 \, {\left (c^{2} d^{5} - 3 \, a c d^{3} e^{2} + {\left (c^{2} d^{3} e^{2} - 3 \, a c d e^{4}\right )} x^{2} + 2 \, {\left (c^{2} d^{4} e - 3 \, a c d^{2} e^{3}\right )} x\right )} \sqrt {\frac {c}{a}} \arctan \left (x \sqrt {\frac {c}{a}}\right ) + 4 \, {\left (c^{2} d^{3} e^{2} + a c d e^{4}\right )} x + {\left (3 \, c^{2} d^{4} e - a c d^{2} e^{3} + {\left (3 \, c^{2} d^{2} e^{3} - a c e^{5}\right )} x^{2} + 2 \, {\left (3 \, c^{2} d^{3} e^{2} - a c d e^{4}\right )} x\right )} \log \left (c x^{2} + a\right ) - 2 \, {\left (3 \, c^{2} d^{4} e - a c d^{2} e^{3} + {\left (3 \, c^{2} d^{2} e^{3} - a c e^{5}\right )} x^{2} + 2 \, {\left (3 \, c^{2} d^{3} e^{2} - a c d e^{4}\right )} x\right )} \log \left (e x + d\right )}{2 \, {\left (c^{3} d^{8} + 3 \, a c^{2} d^{6} e^{2} + 3 \, a^{2} c d^{4} e^{4} + a^{3} d^{2} e^{6} + {\left (c^{3} d^{6} e^{2} + 3 \, a c^{2} d^{4} e^{4} + 3 \, a^{2} c d^{2} e^{6} + a^{3} e^{8}\right )} x^{2} + 2 \, {\left (c^{3} d^{7} e + 3 \, a c^{2} d^{5} e^{3} + 3 \, a^{2} c d^{3} e^{5} + a^{3} d e^{7}\right )} x\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.20, size = 269, normalized size = 1.53 \begin {gather*} -\frac {{\left (3 \, c^{2} d^{2} e - a c e^{3}\right )} \log \left (c x^{2} + a\right )}{2 \, {\left (c^{3} d^{6} + 3 \, a c^{2} d^{4} e^{2} + 3 \, a^{2} c d^{2} e^{4} + a^{3} e^{6}\right )}} + \frac {{\left (3 \, c^{2} d^{2} e^{2} - a c e^{4}\right )} \log \left ({\left | x e + d \right |}\right )}{c^{3} d^{6} e + 3 \, a c^{2} d^{4} e^{3} + 3 \, a^{2} c d^{2} e^{5} + a^{3} e^{7}} + \frac {{\left (c^{3} d^{3} - 3 \, a c^{2} d e^{2}\right )} \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{{\left (c^{3} d^{6} + 3 \, a c^{2} d^{4} e^{2} + 3 \, a^{2} c d^{2} e^{4} + a^{3} e^{6}\right )} \sqrt {a c}} - \frac {5 \, c^{2} d^{4} e + 6 \, a c d^{2} e^{3} + a^{2} e^{5} + 4 \, {\left (c^{2} d^{3} e^{2} + a c d e^{4}\right )} x}{2 \, {\left (c d^{2} + a e^{2}\right )}^{3} {\left (x e + d\right )}^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 233, normalized size = 1.32 \begin {gather*} -\frac {3 a \,c^{2} d \,e^{2} \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{\left (a \,e^{2}+c \,d^{2}\right )^{3} \sqrt {a c}}+\frac {c^{3} d^{3} \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{\left (a \,e^{2}+c \,d^{2}\right )^{3} \sqrt {a c}}+\frac {a c \,e^{3} \ln \left (c \,x^{2}+a \right )}{2 \left (a \,e^{2}+c \,d^{2}\right )^{3}}-\frac {a c \,e^{3} \ln \left (e x +d \right )}{\left (a \,e^{2}+c \,d^{2}\right )^{3}}-\frac {3 c^{2} d^{2} e \ln \left (c \,x^{2}+a \right )}{2 \left (a \,e^{2}+c \,d^{2}\right )^{3}}+\frac {3 c^{2} d^{2} e \ln \left (e x +d \right )}{\left (a \,e^{2}+c \,d^{2}\right )^{3}}-\frac {2 c d e}{\left (a \,e^{2}+c \,d^{2}\right )^{2} \left (e x +d \right )}-\frac {e}{2 \left (a \,e^{2}+c \,d^{2}\right ) \left (e x +d \right )^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 3.03, size = 323, normalized size = 1.84 \begin {gather*} -\frac {{\left (3 \, c^{2} d^{2} e - a c e^{3}\right )} \log \left (c x^{2} + a\right )}{2 \, {\left (c^{3} d^{6} + 3 \, a c^{2} d^{4} e^{2} + 3 \, a^{2} c d^{2} e^{4} + a^{3} e^{6}\right )}} + \frac {{\left (3 \, c^{2} d^{2} e - a c e^{3}\right )} \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^{3} - 3 \, a c^{2} d e^{2}\right )} \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{{\left (c^{3} d^{6} + 3 \, a c^{2} d^{4} e^{2} + 3 \, a^{2} c d^{2} e^{4} + a^{3} e^{6}\right )} \sqrt {a c}} - \frac {4 \, c d e^{2} x + 5 \, c d^{2} e + a e^{3}}{2 \, {\left (c^{2} d^{6} + 2 \, a c d^{4} e^{2} + a^{2} d^{2} e^{4} + {\left (c^{2} d^{4} e^{2} + 2 \, a c d^{2} e^{4} + a^{2} e^{6}\right )} x^{2} + 2 \, {\left (c^{2} d^{5} e + 2 \, a c d^{3} e^{3} + a^{2} d e^{5}\right )} x\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.14, size = 745, normalized size = 4.23 \begin {gather*} \frac {\ln \left (d+e\,x\right )\,\left (3\,c^2\,d^2\,e-a\,c\,e^3\right )}{a^3\,e^6+3\,a^2\,c\,d^2\,e^4+3\,a\,c^2\,d^4\,e^2+c^3\,d^6}-\frac {\ln \left (c^2\,d^{10}\,{\left (-a\,c^3\right )}^{3/2}-9\,a^6\,e^{10}\,\sqrt {-a\,c^3}+9\,a^6\,c^2\,e^{10}\,x+106\,a^2\,d^6\,e^4\,{\left (-a\,c^3\right )}^{3/2}+a\,c^7\,d^{10}\,x+6\,a^4\,c^2\,d^4\,e^6\,\sqrt {-a\,c^3}+77\,a\,c\,d^8\,e^2\,{\left (-a\,c^3\right )}^{3/2}+77\,a^2\,c^6\,d^8\,e^2\,x+106\,a^3\,c^5\,d^6\,e^4\,x-6\,a^4\,c^4\,d^4\,e^6\,x-27\,a^5\,c^3\,d^2\,e^8\,x+27\,a^5\,c\,d^2\,e^8\,\sqrt {-a\,c^3}\right )\,\left (c\,\left (\frac {d^3\,\sqrt {-a\,c^3}}{2}-\frac {a^2\,e^3}{2}\right )+\frac {3\,a\,c^2\,d^2\,e}{2}-\frac {3\,a\,d\,e^2\,\sqrt {-a\,c^3}}{2}\right )}{a^4\,e^6+3\,a^3\,c\,d^2\,e^4+3\,a^2\,c^2\,d^4\,e^2+a\,c^3\,d^6}-\frac {\ln \left (c^2\,d^{10}\,{\left (-a\,c^3\right )}^{3/2}-9\,a^6\,e^{10}\,\sqrt {-a\,c^3}-9\,a^6\,c^2\,e^{10}\,x+106\,a^2\,d^6\,e^4\,{\left (-a\,c^3\right )}^{3/2}-a\,c^7\,d^{10}\,x+6\,a^4\,c^2\,d^4\,e^6\,\sqrt {-a\,c^3}+77\,a\,c\,d^8\,e^2\,{\left (-a\,c^3\right )}^{3/2}-77\,a^2\,c^6\,d^8\,e^2\,x-106\,a^3\,c^5\,d^6\,e^4\,x+6\,a^4\,c^4\,d^4\,e^6\,x+27\,a^5\,c^3\,d^2\,e^8\,x+27\,a^5\,c\,d^2\,e^8\,\sqrt {-a\,c^3}\right )\,\left (\frac {3\,a\,c^2\,d^2\,e}{2}-c\,\left (\frac {d^3\,\sqrt {-a\,c^3}}{2}+\frac {a^2\,e^3}{2}\right )+\frac {3\,a\,d\,e^2\,\sqrt {-a\,c^3}}{2}\right )}{a^4\,e^6+3\,a^3\,c\,d^2\,e^4+3\,a^2\,c^2\,d^4\,e^2+a\,c^3\,d^6}-\frac {\frac {5\,c\,d^2\,e+a\,e^3}{2\,\left (a^2\,e^4+2\,a\,c\,d^2\,e^2+c^2\,d^4\right )}+\frac {2\,c\,d\,e^2\,x}{a^2\,e^4+2\,a\,c\,d^2\,e^2+c^2\,d^4}}{d^2+2\,d\,e\,x+e^2\,x^2} \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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