Optimal. Leaf size=72 \[ \frac{A c \tanh ^{-1}\left (\frac{\sqrt{a+c x^2}}{\sqrt{a}}\right )}{2 a^{3/2}}-\frac{A \sqrt{a+c x^2}}{2 a x^2}-\frac{B \sqrt{a+c x^2}}{a x} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0513389, antiderivative size = 72, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {835, 807, 266, 63, 208} \[ \frac{A c \tanh ^{-1}\left (\frac{\sqrt{a+c x^2}}{\sqrt{a}}\right )}{2 a^{3/2}}-\frac{A \sqrt{a+c x^2}}{2 a x^2}-\frac{B \sqrt{a+c x^2}}{a x} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 835
Rule 807
Rule 266
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{A+B x}{x^3 \sqrt{a+c x^2}} \, dx &=-\frac{A \sqrt{a+c x^2}}{2 a x^2}-\frac{\int \frac{-2 a B+A c x}{x^2 \sqrt{a+c x^2}} \, dx}{2 a}\\ &=-\frac{A \sqrt{a+c x^2}}{2 a x^2}-\frac{B \sqrt{a+c x^2}}{a x}-\frac{(A c) \int \frac{1}{x \sqrt{a+c x^2}} \, dx}{2 a}\\ &=-\frac{A \sqrt{a+c x^2}}{2 a x^2}-\frac{B \sqrt{a+c x^2}}{a x}-\frac{(A c) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+c x}} \, dx,x,x^2\right )}{4 a}\\ &=-\frac{A \sqrt{a+c x^2}}{2 a x^2}-\frac{B \sqrt{a+c x^2}}{a x}-\frac{A \operatorname{Subst}\left (\int \frac{1}{-\frac{a}{c}+\frac{x^2}{c}} \, dx,x,\sqrt{a+c x^2}\right )}{2 a}\\ &=-\frac{A \sqrt{a+c x^2}}{2 a x^2}-\frac{B \sqrt{a+c x^2}}{a x}+\frac{A c \tanh ^{-1}\left (\frac{\sqrt{a+c x^2}}{\sqrt{a}}\right )}{2 a^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.105219, size = 63, normalized size = 0.88 \[ \frac{\sqrt{a+c x^2} \left (\frac{A c \tanh ^{-1}\left (\sqrt{\frac{c x^2}{a}+1}\right )}{\sqrt{\frac{c x^2}{a}+1}}-\frac{a (A+2 B x)}{x^2}\right )}{2 a^2} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.009, size = 68, normalized size = 0.9 \begin{align*} -{\frac{A}{2\,a{x}^{2}}\sqrt{c{x}^{2}+a}}+{\frac{Ac}{2}\ln \left ({\frac{1}{x} \left ( 2\,a+2\,\sqrt{a}\sqrt{c{x}^{2}+a} \right ) } \right ){a}^{-{\frac{3}{2}}}}-{\frac{B}{ax}\sqrt{c{x}^{2}+a}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.74039, size = 306, normalized size = 4.25 \begin{align*} \left [\frac{A \sqrt{a} c x^{2} \log \left (-\frac{c x^{2} + 2 \, \sqrt{c x^{2} + a} \sqrt{a} + 2 \, a}{x^{2}}\right ) - 2 \,{\left (2 \, B a x + A a\right )} \sqrt{c x^{2} + a}}{4 \, a^{2} x^{2}}, -\frac{A \sqrt{-a} c x^{2} \arctan \left (\frac{\sqrt{-a}}{\sqrt{c x^{2} + a}}\right ) +{\left (2 \, B a x + A a\right )} \sqrt{c x^{2} + a}}{2 \, a^{2} x^{2}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 3.84549, size = 66, normalized size = 0.92 \begin{align*} - \frac{A \sqrt{c} \sqrt{\frac{a}{c x^{2}} + 1}}{2 a x} + \frac{A c \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{c} x} \right )}}{2 a^{\frac{3}{2}}} - \frac{B \sqrt{c} \sqrt{\frac{a}{c x^{2}} + 1}}{a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [B] time = 1.18456, size = 197, normalized size = 2.74 \begin{align*} -\frac{A c \arctan \left (-\frac{\sqrt{c} x - \sqrt{c x^{2} + a}}{\sqrt{-a}}\right )}{\sqrt{-a} a} + \frac{{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{3} A c + 2 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{2} B a \sqrt{c} +{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )} A a c - 2 \, B a^{2} \sqrt{c}}{{\left ({\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{2} - a\right )}^{2} a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]