Optimal. Leaf size=50 \[ \frac {e \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{2 \sqrt {a} c^{3/2}}-\frac {d+e x}{2 c \left (a+c x^2\right )} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.02, antiderivative size = 50, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {778, 205} \[ \frac {e \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{2 \sqrt {a} c^{3/2}}-\frac {d+e x}{2 c \left (a+c x^2\right )} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 205
Rule 778
Rubi steps
\begin {align*} \int \frac {x (d+e x)}{\left (a+c x^2\right )^2} \, dx &=-\frac {d+e x}{2 c \left (a+c x^2\right )}+\frac {e \int \frac {1}{a+c x^2} \, dx}{2 c}\\ &=-\frac {d+e x}{2 c \left (a+c x^2\right )}+\frac {e \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{2 \sqrt {a} c^{3/2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.03, size = 53, normalized size = 1.06 \[ \frac {e \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{2 \sqrt {a} c^{3/2}}+\frac {-d-e x}{2 c \left (a+c x^2\right )} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.87, size = 137, normalized size = 2.74 \[ \left [-\frac {2 \, a c e x + 2 \, a c d + {\left (c e x^{2} + a e\right )} \sqrt {-a c} \log \left (\frac {c x^{2} - 2 \, \sqrt {-a c} x - a}{c x^{2} + a}\right )}{4 \, {\left (a c^{3} x^{2} + a^{2} c^{2}\right )}}, -\frac {a c e x + a c d - {\left (c e x^{2} + a e\right )} \sqrt {a c} \arctan \left (\frac {\sqrt {a c} x}{a}\right )}{2 \, {\left (a c^{3} x^{2} + a^{2} c^{2}\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.17, size = 42, normalized size = 0.84 \[ \frac {\arctan \left (\frac {c x}{\sqrt {a c}}\right ) e}{2 \, \sqrt {a c} c} - \frac {x e + d}{2 \, {\left (c x^{2} + a\right )} c} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.05, size = 46, normalized size = 0.92 \[ \frac {e \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{2 \sqrt {a c}\, c}+\frac {-\frac {e x}{2 c}-\frac {d}{2 c}}{c \,x^{2}+a} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 1.21, size = 41, normalized size = 0.82 \[ \frac {e \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{2 \, \sqrt {a c} c} - \frac {e x + d}{2 \, {\left (c^{2} x^{2} + a c\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.05, size = 44, normalized size = 0.88 \[ \frac {e\,\mathrm {atan}\left (\frac {\sqrt {c}\,x}{\sqrt {a}}\right )}{2\,\sqrt {a}\,c^{3/2}}-\frac {\frac {d}{2\,c}+\frac {e\,x}{2\,c}}{c\,x^2+a} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 0.36, size = 85, normalized size = 1.70 \[ e \left (- \frac {\sqrt {- \frac {1}{a c^{3}}} \log {\left (- a c \sqrt {- \frac {1}{a c^{3}}} + x \right )}}{4} + \frac {\sqrt {- \frac {1}{a c^{3}}} \log {\left (a c \sqrt {- \frac {1}{a c^{3}}} + x \right )}}{4}\right ) + \frac {- d - e x}{2 a c + 2 c^{2} x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________