Optimal. Leaf size=67 \[ \frac {d \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{2 \sqrt {a} c^{3/2}}+\frac {e \log \left (a+c x^2\right )}{2 c^2}-\frac {x (d+e x)}{2 c \left (a+c x^2\right )} \]
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Rubi [A] time = 0.03, antiderivative size = 67, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {819, 635, 205, 260} \begin {gather*} \frac {d \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{2 \sqrt {a} c^{3/2}}+\frac {e \log \left (a+c x^2\right )}{2 c^2}-\frac {x (d+e x)}{2 c \left (a+c x^2\right )} \end {gather*}
Antiderivative was successfully verified.
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Rule 205
Rule 260
Rule 635
Rule 819
Rubi steps
\begin {align*} \int \frac {x^2 (d+e x)}{\left (a+c x^2\right )^2} \, dx &=-\frac {x (d+e x)}{2 c \left (a+c x^2\right )}+\frac {\int \frac {a d+2 a e x}{a+c x^2} \, dx}{2 a c}\\ &=-\frac {x (d+e x)}{2 c \left (a+c x^2\right )}+\frac {d \int \frac {1}{a+c x^2} \, dx}{2 c}+\frac {e \int \frac {x}{a+c x^2} \, dx}{c}\\ &=-\frac {x (d+e x)}{2 c \left (a+c x^2\right )}+\frac {d \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{2 \sqrt {a} c^{3/2}}+\frac {e \log \left (a+c x^2\right )}{2 c^2}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 62, normalized size = 0.93 \begin {gather*} \frac {\frac {a e-c d x}{a+c x^2}+\frac {\sqrt {c} d \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{\sqrt {a}}+e \log \left (a+c x^2\right )}{2 c^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 {x^2 (d+e x)}{\left (a+c x^2\right )^2} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.39, size = 186, normalized size = 2.78 \begin {gather*} \left [-\frac {2 \, a c d x - 2 \, a^{2} e + {\left (c d x^{2} + a d\right )} \sqrt {-a c} \log \left (\frac {c x^{2} - 2 \, \sqrt {-a c} x - a}{c x^{2} + a}\right ) - 2 \, {\left (a c e x^{2} + a^{2} e\right )} \log \left (c x^{2} + a\right )}{4 \, {\left (a c^{3} x^{2} + a^{2} c^{2}\right )}}, -\frac {a c d x - a^{2} e - {\left (c d x^{2} + a d\right )} \sqrt {a c} \arctan \left (\frac {\sqrt {a c} x}{a}\right ) - {\left (a c e x^{2} + a^{2} e\right )} \log \left (c x^{2} + a\right )}{2 \, {\left (a c^{3} x^{2} + a^{2} c^{2}\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.19, size = 62, normalized size = 0.93 \begin {gather*} \frac {d \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{2 \, \sqrt {a c} c} + \frac {e \log \left (c x^{2} + a\right )}{2 \, c^{2}} - \frac {d x - \frac {a e}{c}}{2 \, {\left (c x^{2} + a\right )} c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 61, normalized size = 0.91 \begin {gather*} \frac {d \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{2 \sqrt {a c}\, c}+\frac {e \ln \left (c \,x^{2}+a \right )}{2 c^{2}}+\frac {-\frac {d x}{2 c}+\frac {a e}{2 c^{2}}}{c \,x^{2}+a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.16, size = 61, normalized size = 0.91 \begin {gather*} \frac {d \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{2 \, \sqrt {a c} c} - \frac {c d x - a e}{2 \, {\left (c^{3} x^{2} + a c^{2}\right )}} + \frac {e \log \left (c x^{2} + a\right )}{2 \, c^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.04, size = 72, normalized size = 1.07 \begin {gather*} \frac {e\,\ln \left (c\,x^2+a\right )}{2\,c^2}-\frac {d\,x}{2\,\left (c^2\,x^2+a\,c\right )}+\frac {a\,e}{2\,\left (c^3\,x^2+a\,c^2\right )}+\frac {d\,\mathrm {atan}\left (\frac {\sqrt {c}\,x}{\sqrt {a}}\right )}{2\,\sqrt {a}\,c^{3/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.58, size = 162, normalized size = 2.42 \begin {gather*} \left (\frac {e}{2 c^{2}} - \frac {d \sqrt {- a c^{5}}}{4 a c^{4}}\right ) \log {\left (x + \frac {4 a c^{2} \left (\frac {e}{2 c^{2}} - \frac {d \sqrt {- a c^{5}}}{4 a c^{4}}\right ) - 2 a e}{c d} \right )} + \left (\frac {e}{2 c^{2}} + \frac {d \sqrt {- a c^{5}}}{4 a c^{4}}\right ) \log {\left (x + \frac {4 a c^{2} \left (\frac {e}{2 c^{2}} + \frac {d \sqrt {- a c^{5}}}{4 a c^{4}}\right ) - 2 a e}{c d} \right )} + \frac {a e - c d x}{2 a c^{2} + 2 c^{3} x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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