Optimal. Leaf size=100 \[ -\frac{h \sqrt{a+c x^2} (c d-2 a f)}{a c^2}+\frac{\tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right ) (e h+f g)}{c^{3/2}}-\frac{(g+h x) (a e-x (c d-a f))}{a c \sqrt{a+c x^2}} \]
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Rubi [A] time = 0.0869532, antiderivative size = 100, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.148, Rules used = {1645, 641, 217, 206} \[ -\frac{h \sqrt{a+c x^2} (c d-2 a f)}{a c^2}+\frac{\tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right ) (e h+f g)}{c^{3/2}}-\frac{(g+h x) (a e-x (c d-a f))}{a c \sqrt{a+c x^2}} \]
Antiderivative was successfully verified.
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Rule 1645
Rule 641
Rule 217
Rule 206
Rubi steps
\begin{align*} \int \frac{(g+h x) \left (d+e x+f x^2\right )}{\left (a+c x^2\right )^{3/2}} \, dx &=-\frac{(a e-(c d-a f) x) (g+h x)}{a c \sqrt{a+c x^2}}-\frac{\int \frac{-a (f g+e h)+(c d-2 a f) h x}{\sqrt{a+c x^2}} \, dx}{a c}\\ &=-\frac{(a e-(c d-a f) x) (g+h x)}{a c \sqrt{a+c x^2}}-\frac{(c d-2 a f) h \sqrt{a+c x^2}}{a c^2}+\frac{(f g+e h) \int \frac{1}{\sqrt{a+c x^2}} \, dx}{c}\\ &=-\frac{(a e-(c d-a f) x) (g+h x)}{a c \sqrt{a+c x^2}}-\frac{(c d-2 a f) h \sqrt{a+c x^2}}{a c^2}+\frac{(f g+e h) \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-a f) x) (g+h x)}{a c \sqrt{a+c x^2}}-\frac{(c d-2 a f) h \sqrt{a+c x^2}}{a c^2}+\frac{(f g+e h) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )}{c^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.151096, size = 102, normalized size = 1.02 \[ \frac{a^{3/2} \sqrt{c} \sqrt{\frac{c x^2}{a}+1} \sinh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right ) (e h+f g)+2 a^2 f h-a c (d h+e (g+h x)+f x (g-h x))+c^2 d g x}{a c^2 \sqrt{a+c x^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.056, size = 163, normalized size = 1.6 \begin{align*}{\frac{fh{x}^{2}}{c}{\frac{1}{\sqrt{c{x}^{2}+a}}}}+2\,{\frac{afh}{{c}^{2}\sqrt{c{x}^{2}+a}}}-{\frac{ehx}{c}{\frac{1}{\sqrt{c{x}^{2}+a}}}}-{\frac{fgx}{c}{\frac{1}{\sqrt{c{x}^{2}+a}}}}+{eh\ln \left ( x\sqrt{c}+\sqrt{c{x}^{2}+a} \right ){c}^{-{\frac{3}{2}}}}+{fg\ln \left ( x\sqrt{c}+\sqrt{c{x}^{2}+a} \right ){c}^{-{\frac{3}{2}}}}-{\frac{dh}{c}{\frac{1}{\sqrt{c{x}^{2}+a}}}}-{\frac{eg}{c}{\frac{1}{\sqrt{c{x}^{2}+a}}}}+{\frac{dgx}{a}{\frac{1}{\sqrt{c{x}^{2}+a}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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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.
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Fricas [A] time = 1.9, size = 606, normalized size = 6.06 \begin{align*} \left [\frac{{\left (a^{2} f g + a^{2} e h +{\left (a c f g + a c e h\right )} x^{2}\right )} \sqrt{c} \log \left (-2 \, c x^{2} - 2 \, \sqrt{c x^{2} + a} \sqrt{c} x - a\right ) + 2 \,{\left (a c f h x^{2} - a c e g -{\left (a c d - 2 \, a^{2} f\right )} h -{\left (a c e h -{\left (c^{2} d - a c f\right )} g\right )} x\right )} \sqrt{c x^{2} + a}}{2 \,{\left (a c^{3} x^{2} + a^{2} c^{2}\right )}}, -\frac{{\left (a^{2} f g + a^{2} e h +{\left (a c f g + a c e h\right )} x^{2}\right )} \sqrt{-c} \arctan \left (\frac{\sqrt{-c} x}{\sqrt{c x^{2} + a}}\right ) -{\left (a c f h x^{2} - a c e g -{\left (a c d - 2 \, a^{2} f\right )} h -{\left (a c e h -{\left (c^{2} d - a c f\right )} g\right )} x\right )} \sqrt{c x^{2} + a}}{a c^{3} x^{2} + a^{2} c^{2}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 12.7682, size = 209, normalized size = 2.09 \begin{align*} d h \left (\begin{cases} - \frac{1}{c \sqrt{a + c x^{2}}} & \text{for}\: c \neq 0 \\\frac{x^{2}}{2 a^{\frac{3}{2}}} & \text{otherwise} \end{cases}\right ) + e g \left (\begin{cases} - \frac{1}{c \sqrt{a + c x^{2}}} & \text{for}\: c \neq 0 \\\frac{x^{2}}{2 a^{\frac{3}{2}}} & \text{otherwise} \end{cases}\right ) + e h \left (\frac{\operatorname{asinh}{\left (\frac{\sqrt{c} x}{\sqrt{a}} \right )}}{c^{\frac{3}{2}}} - \frac{x}{\sqrt{a} c \sqrt{1 + \frac{c x^{2}}{a}}}\right ) + f g \left (\frac{\operatorname{asinh}{\left (\frac{\sqrt{c} x}{\sqrt{a}} \right )}}{c^{\frac{3}{2}}} - \frac{x}{\sqrt{a} c \sqrt{1 + \frac{c x^{2}}{a}}}\right ) + f h \left (\begin{cases} \frac{2 a}{c^{2} \sqrt{a + c x^{2}}} + \frac{x^{2}}{c \sqrt{a + c x^{2}}} & \text{for}\: c \neq 0 \\\frac{x^{4}}{4 a^{\frac{3}{2}}} & \text{otherwise} \end{cases}\right ) + \frac{d g x}{a^{\frac{3}{2}} \sqrt{1 + \frac{c x^{2}}{a}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.16895, size = 157, normalized size = 1.57 \begin{align*} \frac{{\left (\frac{f h x}{c} + \frac{c^{3} d g - a c^{2} f g - a c^{2} h e}{a c^{3}}\right )} x - \frac{a c^{2} d h - 2 \, a^{2} c f h + a c^{2} g e}{a c^{3}}}{\sqrt{c x^{2} + a}} - \frac{{\left (f g + h e\right )} \log \left ({\left | -\sqrt{c} x + \sqrt{c x^{2} + a} \right |}\right )}{c^{\frac{3}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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