Optimal. Leaf size=85 \[ -\frac {3 \sqrt {a} d \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{2 c^{5/2}}-\frac {a e \log \left (a+c x^2\right )}{c^3}-\frac {x^3 (d+e x)}{2 c \left (a+c x^2\right )}+\frac {3 d x}{2 c^2}+\frac {e x^2}{c^2} \]
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Rubi [A] time = 0.07, antiderivative size = 85, 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 = {819, 801, 635, 205, 260} \begin {gather*} -\frac {3 \sqrt {a} d \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{2 c^{5/2}}-\frac {a e \log \left (a+c x^2\right )}{c^3}-\frac {x^3 (d+e x)}{2 c \left (a+c x^2\right )}+\frac {3 d x}{2 c^2}+\frac {e x^2}{c^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 205
Rule 260
Rule 635
Rule 801
Rule 819
Rubi steps
\begin {align*} \int \frac {x^4 (d+e x)}{\left (a+c x^2\right )^2} \, dx &=-\frac {x^3 (d+e x)}{2 c \left (a+c x^2\right )}+\frac {\int \frac {x^2 (3 a d+4 a e x)}{a+c x^2} \, dx}{2 a c}\\ &=-\frac {x^3 (d+e x)}{2 c \left (a+c x^2\right )}+\frac {\int \left (\frac {3 a d}{c}+\frac {4 a e x}{c}-\frac {3 a^2 d+4 a^2 e x}{c \left (a+c x^2\right )}\right ) \, dx}{2 a c}\\ &=\frac {3 d x}{2 c^2}+\frac {e x^2}{c^2}-\frac {x^3 (d+e x)}{2 c \left (a+c x^2\right )}-\frac {\int \frac {3 a^2 d+4 a^2 e x}{a+c x^2} \, dx}{2 a c^2}\\ &=\frac {3 d x}{2 c^2}+\frac {e x^2}{c^2}-\frac {x^3 (d+e x)}{2 c \left (a+c x^2\right )}-\frac {(3 a d) \int \frac {1}{a+c x^2} \, dx}{2 c^2}-\frac {(2 a e) \int \frac {x}{a+c x^2} \, dx}{c^2}\\ &=\frac {3 d x}{2 c^2}+\frac {e x^2}{c^2}-\frac {x^3 (d+e x)}{2 c \left (a+c x^2\right )}-\frac {3 \sqrt {a} d \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{2 c^{5/2}}-\frac {a e \log \left (a+c x^2\right )}{c^3}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 77, normalized size = 0.91 \begin {gather*} \frac {\frac {a (c d x-a e)}{a+c x^2}-3 \sqrt {a} \sqrt {c} d \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right )-2 a e \log \left (a+c x^2\right )+2 c d x+c e x^2}{2 c^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 {x^4 (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 = 0.40, size = 248, normalized size = 2.92 \begin {gather*} \left [\frac {2 \, c^{2} e x^{4} + 4 \, c^{2} d x^{3} + 2 \, a c e x^{2} + 6 \, a c d x - 2 \, a^{2} e + 3 \, {\left (c^{2} d x^{2} + a c d\right )} \sqrt {-\frac {a}{c}} \log \left (\frac {c x^{2} - 2 \, c x \sqrt {-\frac {a}{c}} - a}{c x^{2} + a}\right ) - 4 \, {\left (a c e x^{2} + a^{2} e\right )} \log \left (c x^{2} + a\right )}{4 \, {\left (c^{4} x^{2} + a c^{3}\right )}}, \frac {c^{2} e x^{4} + 2 \, c^{2} d x^{3} + a c e x^{2} + 3 \, a c d x - a^{2} e - 3 \, {\left (c^{2} d x^{2} + a c d\right )} \sqrt {\frac {a}{c}} \arctan \left (\frac {c x \sqrt {\frac {a}{c}}}{a}\right ) - 2 \, {\left (a c e x^{2} + a^{2} e\right )} \log \left (c x^{2} + a\right )}{2 \, {\left (c^{4} x^{2} + a c^{3}\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.20, size = 87, normalized size = 1.02 \begin {gather*} -\frac {3 \, a d \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{2 \, \sqrt {a c} c^{2}} - \frac {a e \log \left (c x^{2} + a\right )}{c^{3}} + \frac {c^{2} x^{2} e + 2 \, c^{2} d x}{2 \, c^{4}} + \frac {a c d x - a^{2} e}{2 \, {\left (c x^{2} + a\right )} c^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 88, normalized size = 1.04 \begin {gather*} \frac {a d x}{2 \left (c \,x^{2}+a \right ) c^{2}}-\frac {3 a d \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{2 \sqrt {a c}\, c^{2}}+\frac {e \,x^{2}}{2 c^{2}}-\frac {a^{2} e}{2 \left (c \,x^{2}+a \right ) c^{3}}-\frac {a e \ln \left (c \,x^{2}+a \right )}{c^{3}}+\frac {d x}{c^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.22, size = 81, normalized size = 0.95 \begin {gather*} -\frac {3 \, a d \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{2 \, \sqrt {a c} c^{2}} + \frac {a c d x - a^{2} e}{2 \, {\left (c^{4} x^{2} + a c^{3}\right )}} - \frac {a e \log \left (c x^{2} + a\right )}{c^{3}} + \frac {e x^{2} + 2 \, d x}{2 \, c^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.06, size = 81, normalized size = 0.95 \begin {gather*} \frac {e\,x^2}{2\,c^2}-\frac {\frac {a^2\,e}{2\,c}-\frac {a\,d\,x}{2}}{c^3\,x^2+a\,c^2}+\frac {d\,x}{c^2}-\frac {3\,\sqrt {a}\,d\,\mathrm {atan}\left (\frac {\sqrt {c}\,x}{\sqrt {a}}\right )}{2\,c^{5/2}}-\frac {a\,e\,\ln \left (c\,x^2+a\right )}{c^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.72, size = 189, normalized size = 2.22 \begin {gather*} \left (- \frac {a e}{c^{3}} - \frac {3 d \sqrt {- a c^{7}}}{4 c^{6}}\right ) \log {\left (x + \frac {- 4 a e - 4 c^{3} \left (- \frac {a e}{c^{3}} - \frac {3 d \sqrt {- a c^{7}}}{4 c^{6}}\right )}{3 c d} \right )} + \left (- \frac {a e}{c^{3}} + \frac {3 d \sqrt {- a c^{7}}}{4 c^{6}}\right ) \log {\left (x + \frac {- 4 a e - 4 c^{3} \left (- \frac {a e}{c^{3}} + \frac {3 d \sqrt {- a c^{7}}}{4 c^{6}}\right )}{3 c d} \right )} + \frac {- a^{2} e + a c d x}{2 a c^{3} + 2 c^{4} x^{2}} + \frac {d x}{c^{2}} + \frac {e x^{2}}{2 c^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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