Optimal. Leaf size=78 \[ -\frac {3 \sqrt {a} e \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{2 c^{5/2}}+\frac {d \log \left (a+c x^2\right )}{2 c^2}-\frac {x^2 (d+e x)}{2 c \left (a+c x^2\right )}+\frac {3 e x}{2 c^2} \]
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Rubi [A] time = 0.04, antiderivative size = 78, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.278, Rules used = {819, 774, 635, 205, 260} \[ \frac {d \log \left (a+c x^2\right )}{2 c^2}-\frac {3 \sqrt {a} e \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{2 c^{5/2}}-\frac {x^2 (d+e x)}{2 c \left (a+c x^2\right )}+\frac {3 e x}{2 c^2} \]
Antiderivative was successfully verified.
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Rule 205
Rule 260
Rule 635
Rule 774
Rule 819
Rubi steps
\begin {align*} \int \frac {x^3 (d+e x)}{\left (a+c x^2\right )^2} \, dx &=-\frac {x^2 (d+e x)}{2 c \left (a+c x^2\right )}+\frac {\int \frac {x (2 a d+3 a e x)}{a+c x^2} \, dx}{2 a c}\\ &=\frac {3 e x}{2 c^2}-\frac {x^2 (d+e x)}{2 c \left (a+c x^2\right )}+\frac {\int \frac {-3 a^2 e+2 a c d x}{a+c x^2} \, dx}{2 a c^2}\\ &=\frac {3 e x}{2 c^2}-\frac {x^2 (d+e x)}{2 c \left (a+c x^2\right )}+\frac {d \int \frac {x}{a+c x^2} \, dx}{c}-\frac {(3 a e) \int \frac {1}{a+c x^2} \, dx}{2 c^2}\\ &=\frac {3 e x}{2 c^2}-\frac {x^2 (d+e x)}{2 c \left (a+c x^2\right )}-\frac {3 \sqrt {a} e \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{2 c^{5/2}}+\frac {d \log \left (a+c x^2\right )}{2 c^2}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 75, normalized size = 0.96 \[ -\frac {3 \sqrt {a} e \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{2 c^{5/2}}+\frac {a d+a e x}{2 c^2 \left (a+c x^2\right )}+\frac {d \log \left (a+c x^2\right )}{2 c^2}+\frac {e x}{c^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.77, size = 192, normalized size = 2.46 \[ \left [\frac {4 \, c e x^{3} + 6 \, a e x + 3 \, {\left (c e x^{2} + a e\right )} \sqrt {-\frac {a}{c}} \log \left (\frac {c x^{2} - 2 \, c x \sqrt {-\frac {a}{c}} - a}{c x^{2} + a}\right ) + 2 \, a d + 2 \, {\left (c d x^{2} + a d\right )} \log \left (c x^{2} + a\right )}{4 \, {\left (c^{3} x^{2} + a c^{2}\right )}}, \frac {2 \, c e x^{3} + 3 \, a e x - 3 \, {\left (c e x^{2} + a e\right )} \sqrt {\frac {a}{c}} \arctan \left (\frac {c x \sqrt {\frac {a}{c}}}{a}\right ) + a d + {\left (c d x^{2} + a d\right )} \log \left (c x^{2} + a\right )}{2 \, {\left (c^{3} x^{2} + a c^{2}\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 67, normalized size = 0.86 \[ -\frac {3 \, a \arctan \left (\frac {c x}{\sqrt {a c}}\right ) e}{2 \, \sqrt {a c} c^{2}} + \frac {x e}{c^{2}} + \frac {d \log \left (c x^{2} + a\right )}{2 \, c^{2}} + \frac {a x e + a d}{2 \, {\left (c x^{2} + a\right )} c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 76, normalized size = 0.97 \[ \frac {a e x}{2 \left (c \,x^{2}+a \right ) c^{2}}-\frac {3 a e \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{2 \sqrt {a c}\, c^{2}}+\frac {a d}{2 \left (c \,x^{2}+a \right ) c^{2}}+\frac {d \ln \left (c \,x^{2}+a \right )}{2 c^{2}}+\frac {e x}{c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.16, size = 67, normalized size = 0.86 \[ -\frac {3 \, a e \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{2 \, \sqrt {a c} c^{2}} + \frac {a e x + a d}{2 \, {\left (c^{3} x^{2} + a c^{2}\right )}} + \frac {e x}{c^{2}} + \frac {d \log \left (c x^{2} + a\right )}{2 \, c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.08, size = 65, normalized size = 0.83 \[ \frac {\frac {a\,d}{2}+\frac {a\,e\,x}{2}}{c^3\,x^2+a\,c^2}+\frac {d\,\ln \left (c\,x^2+a\right )}{2\,c^2}+\frac {e\,x}{c^2}-\frac {3\,\sqrt {a}\,e\,\mathrm {atan}\left (\frac {\sqrt {c}\,x}{\sqrt {a}}\right )}{2\,c^{5/2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.67, size = 162, normalized size = 2.08 \[ \left (\frac {d}{2 c^{2}} - \frac {3 e \sqrt {- a c^{5}}}{4 c^{5}}\right ) \log {\left (x + \frac {- 4 c^{2} \left (\frac {d}{2 c^{2}} - \frac {3 e \sqrt {- a c^{5}}}{4 c^{5}}\right ) + 2 d}{3 e} \right )} + \left (\frac {d}{2 c^{2}} + \frac {3 e \sqrt {- a c^{5}}}{4 c^{5}}\right ) \log {\left (x + \frac {- 4 c^{2} \left (\frac {d}{2 c^{2}} + \frac {3 e \sqrt {- a c^{5}}}{4 c^{5}}\right ) + 2 d}{3 e} \right )} + \frac {a d + a e x}{2 a c^{2} + 2 c^{3} x^{2}} + \frac {e x}{c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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