Optimal. Leaf size=87 \[ -\frac {3 \sqrt {c} d \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{2 a^{5/2}}-\frac {e \log \left (a+c x^2\right )}{2 a^2}-\frac {3 d}{2 a^2 x}+\frac {e \log (x)}{a^2}+\frac {d+e x}{2 a x \left (a+c x^2\right )} \]
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Rubi [A] time = 0.07, antiderivative size = 87, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.278, Rules used = {823, 801, 635, 205, 260} \begin {gather*} -\frac {3 \sqrt {c} d \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{2 a^{5/2}}-\frac {e \log \left (a+c x^2\right )}{2 a^2}-\frac {3 d}{2 a^2 x}+\frac {e \log (x)}{a^2}+\frac {d+e x}{2 a x \left (a+c x^2\right )} \end {gather*}
Antiderivative was successfully verified.
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Rule 205
Rule 260
Rule 635
Rule 801
Rule 823
Rubi steps
\begin {align*} \int \frac {d+e x}{x^2 \left (a+c x^2\right )^2} \, dx &=\frac {d+e x}{2 a x \left (a+c x^2\right )}-\frac {\int \frac {-3 a c d-2 a c e x}{x^2 \left (a+c x^2\right )} \, dx}{2 a^2 c}\\ &=\frac {d+e x}{2 a x \left (a+c x^2\right )}-\frac {\int \left (-\frac {3 c d}{x^2}-\frac {2 c e}{x}+\frac {c^2 (3 d+2 e x)}{a+c x^2}\right ) \, dx}{2 a^2 c}\\ &=-\frac {3 d}{2 a^2 x}+\frac {d+e x}{2 a x \left (a+c x^2\right )}+\frac {e \log (x)}{a^2}-\frac {c \int \frac {3 d+2 e x}{a+c x^2} \, dx}{2 a^2}\\ &=-\frac {3 d}{2 a^2 x}+\frac {d+e x}{2 a x \left (a+c x^2\right )}+\frac {e \log (x)}{a^2}-\frac {(3 c d) \int \frac {1}{a+c x^2} \, dx}{2 a^2}-\frac {(c e) \int \frac {x}{a+c x^2} \, dx}{a^2}\\ &=-\frac {3 d}{2 a^2 x}+\frac {d+e x}{2 a x \left (a+c x^2\right )}-\frac {3 \sqrt {c} d \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{2 a^{5/2}}+\frac {e \log (x)}{a^2}-\frac {e \log \left (a+c x^2\right )}{2 a^2}\\ \end {align*}
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Mathematica [A] time = 0.12, size = 81, normalized size = 0.93 \begin {gather*} -\frac {3 \sqrt {c} d \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{2 a^{5/2}}-\frac {\frac {2 a d-a e x+3 c d x^2}{a x+c x^3}+e \log \left (a+c x^2\right )-2 e \log (x)}{2 a^2} \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 {d+e x}{x^2 \left (a+c x^2\right )^2} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.43, size = 225, normalized size = 2.59 \begin {gather*} \left [-\frac {6 \, c d x^{2} - 2 \, a e x - 3 \, {\left (c d x^{3} + a d 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 \, a d + 2 \, {\left (c e x^{3} + a e x\right )} \log \left (c x^{2} + a\right ) - 4 \, {\left (c e x^{3} + a e x\right )} \log \relax (x)}{4 \, {\left (a^{2} c x^{3} + a^{3} x\right )}}, -\frac {3 \, c d x^{2} - a e x + 3 \, {\left (c d x^{3} + a d x\right )} \sqrt {\frac {c}{a}} \arctan \left (x \sqrt {\frac {c}{a}}\right ) + 2 \, a d + {\left (c e x^{3} + a e x\right )} \log \left (c x^{2} + a\right ) - 2 \, {\left (c e x^{3} + a e x\right )} \log \relax (x)}{2 \, {\left (a^{2} c x^{3} + a^{3} x\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 80, normalized size = 0.92 \begin {gather*} -\frac {3 \, c d \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{2 \, \sqrt {a c} a^{2}} - \frac {e \log \left (c x^{2} + a\right )}{2 \, a^{2}} + \frac {e \log \left ({\left | x \right |}\right )}{a^{2}} - \frac {3 \, c d x^{2} - a x e + 2 \, a d}{2 \, {\left (c x^{3} + a x\right )} a^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.08, size = 85, normalized size = 0.98 \begin {gather*} -\frac {c d x}{2 \left (c \,x^{2}+a \right ) a^{2}}-\frac {3 c d \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{2 \sqrt {a c}\, a^{2}}+\frac {e}{2 \left (c \,x^{2}+a \right ) a}+\frac {e \ln \relax (x )}{a^{2}}-\frac {e \ln \left (c \,x^{2}+a \right )}{2 a^{2}}-\frac {d}{a^{2} x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.12, size = 78, normalized size = 0.90 \begin {gather*} -\frac {3 \, c d \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{2 \, \sqrt {a c} a^{2}} - \frac {3 \, c d x^{2} - a e x + 2 \, a d}{2 \, {\left (a^{2} c x^{3} + a^{3} x\right )}} - \frac {e \log \left (c x^{2} + a\right )}{2 \, a^{2}} + \frac {e \log \relax (x)}{a^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.22, size = 165, normalized size = 1.90 \begin {gather*} \frac {e\,\ln \relax (x)}{a^2}-\frac {\ln \left (2\,a^3\,e+d\,\sqrt {-a^5\,c}-2\,e\,x\,\sqrt {-a^5\,c}+a^2\,c\,d\,x\right )\,\left (2\,a^3\,e+3\,d\,\sqrt {-a^5\,c}\right )}{4\,a^5}-\frac {\ln \left (2\,a^3\,e-d\,\sqrt {-a^5\,c}+2\,e\,x\,\sqrt {-a^5\,c}+a^2\,c\,d\,x\right )\,\left (2\,a^3\,e-3\,d\,\sqrt {-a^5\,c}\right )}{4\,a^5}-\frac {\frac {d}{a}-\frac {e\,x}{2\,a}+\frac {3\,c\,d\,x^2}{2\,a^2}}{c\,x^3+a\,x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 1.81, size = 389, normalized size = 4.47 \begin {gather*} \left (- \frac {e}{2 a^{2}} - \frac {3 d \sqrt {- a^{5} c}}{4 a^{5}}\right ) \log {\left (x + \frac {32 a^{6} e \left (- \frac {e}{2 a^{2}} - \frac {3 d \sqrt {- a^{5} c}}{4 a^{5}}\right )^{2} - 16 a^{4} e^{2} \left (- \frac {e}{2 a^{2}} - \frac {3 d \sqrt {- a^{5} c}}{4 a^{5}}\right ) - 12 a^{3} c d^{2} \left (- \frac {e}{2 a^{2}} - \frac {3 d \sqrt {- a^{5} c}}{4 a^{5}}\right ) - 16 a^{2} e^{3} + 12 a c d^{2} e}{36 a c d e^{2} + 9 c^{2} d^{3}} \right )} + \left (- \frac {e}{2 a^{2}} + \frac {3 d \sqrt {- a^{5} c}}{4 a^{5}}\right ) \log {\left (x + \frac {32 a^{6} e \left (- \frac {e}{2 a^{2}} + \frac {3 d \sqrt {- a^{5} c}}{4 a^{5}}\right )^{2} - 16 a^{4} e^{2} \left (- \frac {e}{2 a^{2}} + \frac {3 d \sqrt {- a^{5} c}}{4 a^{5}}\right ) - 12 a^{3} c d^{2} \left (- \frac {e}{2 a^{2}} + \frac {3 d \sqrt {- a^{5} c}}{4 a^{5}}\right ) - 16 a^{2} e^{3} + 12 a c d^{2} e}{36 a c d e^{2} + 9 c^{2} d^{3}} \right )} + \frac {- 2 a d + a e x - 3 c d x^{2}}{2 a^{3} x + 2 a^{2} c x^{3}} + \frac {e \log {\relax (x )}}{a^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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