Optimal. Leaf size=83 \[ \frac {e^2 \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{c^{3/2}}-\frac {d e \sqrt {a+c x^2}}{a c}-\frac {(d+e x) (a e-c d x)}{a c \sqrt {a+c x^2}} \]
________________________________________________________________________________________
Rubi [A] time = 0.03, antiderivative size = 83, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.210, Rules used = {739, 641, 217, 206} \begin {gather*} \frac {e^2 \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{c^{3/2}}-\frac {d e \sqrt {a+c x^2}}{a c}-\frac {(d+e x) (a e-c d x)}{a c \sqrt {a+c x^2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 206
Rule 217
Rule 641
Rule 739
Rubi steps
\begin {align*} \int \frac {(d+e x)^2}{\left (a+c x^2\right )^{3/2}} \, dx &=-\frac {(a e-c d x) (d+e x)}{a c \sqrt {a+c x^2}}+\frac {\int \frac {a e^2-c d e x}{\sqrt {a+c x^2}} \, dx}{a c}\\ &=-\frac {(a e-c d x) (d+e x)}{a c \sqrt {a+c x^2}}-\frac {d e \sqrt {a+c x^2}}{a c}+\frac {e^2 \int \frac {1}{\sqrt {a+c x^2}} \, dx}{c}\\ &=-\frac {(a e-c d x) (d+e x)}{a c \sqrt {a+c x^2}}-\frac {d e \sqrt {a+c x^2}}{a c}+\frac {e^2 \operatorname {Subst}\left (\int \frac {1}{1-c x^2} \, dx,x,\frac {x}{\sqrt {a+c x^2}}\right )}{c}\\ &=-\frac {(a e-c d x) (d+e x)}{a c \sqrt {a+c x^2}}-\frac {d e \sqrt {a+c x^2}}{a c}+\frac {e^2 \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{c^{3/2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.07, size = 69, normalized size = 0.83 \begin {gather*} \frac {e^2 \log \left (\sqrt {c} \sqrt {a+c x^2}+c x\right )}{c^{3/2}}+\frac {-2 a d e-a e^2 x+c d^2 x}{a c \sqrt {a+c x^2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [A] time = 0.39, size = 69, normalized size = 0.83 \begin {gather*} \frac {-2 a d e-a e^2 x+c d^2 x}{a c \sqrt {a+c x^2}}-\frac {e^2 \log \left (\sqrt {a+c x^2}-\sqrt {c} x\right )}{c^{3/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.44, size = 201, normalized size = 2.42 \begin {gather*} \left [\frac {{\left (a c e^{2} x^{2} + a^{2} e^{2}\right )} \sqrt {c} \log \left (-2 \, c x^{2} - 2 \, \sqrt {c x^{2} + a} \sqrt {c} x - a\right ) - 2 \, {\left (2 \, a c d e - {\left (c^{2} d^{2} - a c e^{2}\right )} x\right )} \sqrt {c x^{2} + a}}{2 \, {\left (a c^{3} x^{2} + a^{2} c^{2}\right )}}, -\frac {{\left (a c e^{2} x^{2} + a^{2} e^{2}\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {-c} x}{\sqrt {c x^{2} + a}}\right ) + {\left (2 \, a c d e - {\left (c^{2} d^{2} - a c e^{2}\right )} x\right )} \sqrt {c x^{2} + a}}{a c^{3} x^{2} + a^{2} c^{2}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.22, size = 69, normalized size = 0.83 \begin {gather*} -\frac {\frac {2 \, d e}{c} - \frac {{\left (c^{2} d^{2} - a c e^{2}\right )} x}{a c^{2}}}{\sqrt {c x^{2} + a}} - \frac {e^{2} \log \left ({\left | -\sqrt {c} x + \sqrt {c x^{2} + a} \right |}\right )}{c^{\frac {3}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.05, size = 76, normalized size = 0.92 \begin {gather*} \frac {d^{2} x}{\sqrt {c \,x^{2}+a}\, a}-\frac {e^{2} x}{\sqrt {c \,x^{2}+a}\, c}+\frac {e^{2} \ln \left (\sqrt {c}\, x +\sqrt {c \,x^{2}+a}\right )}{c^{\frac {3}{2}}}-\frac {2 d e}{\sqrt {c \,x^{2}+a}\, c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 1.39, size = 68, normalized size = 0.82 \begin {gather*} \frac {d^{2} x}{\sqrt {c x^{2} + a} a} - \frac {e^{2} x}{\sqrt {c x^{2} + a} c} + \frac {e^{2} \operatorname {arsinh}\left (\frac {c x}{\sqrt {a c}}\right )}{c^{\frac {3}{2}}} - \frac {2 \, d e}{\sqrt {c x^{2} + a} c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.63, size = 75, normalized size = 0.90 \begin {gather*} \frac {e^2\,\ln \left (\sqrt {c}\,x+\sqrt {c\,x^2+a}\right )}{c^{3/2}}+\frac {d^2\,x}{a\,\sqrt {c\,x^2+a}}-\frac {e^2\,x}{c\,\sqrt {c\,x^2+a}}-\frac {2\,d\,e}{c\,\sqrt {c\,x^2+a}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (d + e x\right )^{2}}{\left (a + c x^{2}\right )^{\frac {3}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________