Optimal. Leaf size=61 \[ -\frac {\sqrt {a} d \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{c^{3/2}}-\frac {a e \log \left (a+c x^2\right )}{2 c^2}+\frac {d x}{c}+\frac {e x^2}{2 c} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.04, antiderivative size = 61, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {801, 635, 205, 260} \[ -\frac {\sqrt {a} d \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{c^{3/2}}-\frac {a e \log \left (a+c x^2\right )}{2 c^2}+\frac {d x}{c}+\frac {e x^2}{2 c} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 205
Rule 260
Rule 635
Rule 801
Rubi steps
\begin {align*} \int \frac {x^2 (d+e x)}{a+c x^2} \, dx &=\int \left (\frac {d}{c}+\frac {e x}{c}-\frac {a d+a e x}{c \left (a+c x^2\right )}\right ) \, dx\\ &=\frac {d x}{c}+\frac {e x^2}{2 c}-\frac {\int \frac {a d+a e x}{a+c x^2} \, dx}{c}\\ &=\frac {d x}{c}+\frac {e x^2}{2 c}-\frac {(a d) \int \frac {1}{a+c x^2} \, dx}{c}-\frac {(a e) \int \frac {x}{a+c x^2} \, dx}{c}\\ &=\frac {d x}{c}+\frac {e x^2}{2 c}-\frac {\sqrt {a} d \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{c^{3/2}}-\frac {a e \log \left (a+c x^2\right )}{2 c^2}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.02, size = 56, normalized size = 0.92 \[ \frac {-2 \sqrt {a} \sqrt {c} d \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right )-a e \log \left (a+c x^2\right )+c x (2 d+e x)}{2 c^2} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.92, size = 127, normalized size = 2.08 \[ \left [\frac {c e x^{2} + c d \sqrt {-\frac {a}{c}} \log \left (\frac {c x^{2} - 2 \, c x \sqrt {-\frac {a}{c}} - a}{c x^{2} + a}\right ) + 2 \, c d x - a e \log \left (c x^{2} + a\right )}{2 \, c^{2}}, \frac {c e x^{2} - 2 \, c d \sqrt {\frac {a}{c}} \arctan \left (\frac {c x \sqrt {\frac {a}{c}}}{a}\right ) + 2 \, c d x - a e \log \left (c x^{2} + a\right )}{2 \, c^{2}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.20, size = 56, normalized size = 0.92 \[ -\frac {a d \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{\sqrt {a c} c} - \frac {a e \log \left (c x^{2} + a\right )}{2 \, c^{2}} + \frac {c x^{2} e + 2 \, c d x}{2 \, c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.05, size = 53, normalized size = 0.87 \[ -\frac {a d \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{\sqrt {a c}\, c}+\frac {e \,x^{2}}{2 c}-\frac {a e \ln \left (c \,x^{2}+a \right )}{2 c^{2}}+\frac {d x}{c} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 1.16, size = 52, normalized size = 0.85 \[ -\frac {a d \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{\sqrt {a c} c} - \frac {a e \log \left (c x^{2} + a\right )}{2 \, c^{2}} + \frac {e x^{2} + 2 \, d x}{2 \, c} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 1.06, size = 49, normalized size = 0.80 \[ \frac {e\,x^2}{2\,c}+\frac {d\,x}{c}-\frac {\sqrt {a}\,d\,\mathrm {atan}\left (\frac {\sqrt {c}\,x}{\sqrt {a}}\right )}{c^{3/2}}-\frac {a\,e\,\ln \left (c\,x^2+a\right )}{2\,c^2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [B] time = 0.45, size = 151, normalized size = 2.48 \[ \left (- \frac {a e}{2 c^{2}} - \frac {d \sqrt {- a c^{5}}}{2 c^{4}}\right ) \log {\left (x + \frac {- a e - 2 c^{2} \left (- \frac {a e}{2 c^{2}} - \frac {d \sqrt {- a c^{5}}}{2 c^{4}}\right )}{c d} \right )} + \left (- \frac {a e}{2 c^{2}} + \frac {d \sqrt {- a c^{5}}}{2 c^{4}}\right ) \log {\left (x + \frac {- a e - 2 c^{2} \left (- \frac {a e}{2 c^{2}} + \frac {d \sqrt {- a c^{5}}}{2 c^{4}}\right )}{c d} \right )} + \frac {d x}{c} + \frac {e x^{2}}{2 c} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________