Optimal. Leaf size=213 \[ \frac{a^2 \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right ) (-a e h-a f g+6 c d g)}{16 c^{3/2}}-\frac{\left (a+c x^2\right )^{5/2} \left (6 \left (2 a f h^2+c \left (5 f g^2-7 h (d h+e g)\right )\right )+5 c h x (5 f g-7 e h)\right )}{210 c^2 h}+\frac{x \left (a+c x^2\right )^{3/2} (6 c d g-a (e h+f g))}{24 c}+\frac{a x \sqrt{a+c x^2} (-a e h-a f g+6 c d g)}{16 c}+\frac{f \left (a+c x^2\right )^{5/2} (g+h x)^2}{7 c h} \]
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Rubi [A] time = 0.270914, antiderivative size = 212, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.185, Rules used = {1654, 780, 195, 217, 206} \[ \frac{a^2 \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right ) (-a e h-a f g+6 c d g)}{16 c^{3/2}}-\frac{\left (a+c x^2\right )^{5/2} \left (6 \left (2 a f h^2-7 c h (d h+e g)+5 c f g^2\right )+5 c h x (5 f g-7 e h)\right )}{210 c^2 h}+\frac{x \left (a+c x^2\right )^{3/2} (6 c d g-a (e h+f g))}{24 c}+\frac{a x \sqrt{a+c x^2} (-a e h-a f g+6 c d g)}{16 c}+\frac{f \left (a+c x^2\right )^{5/2} (g+h x)^2}{7 c h} \]
Antiderivative was successfully verified.
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Rule 1654
Rule 780
Rule 195
Rule 217
Rule 206
Rubi steps
\begin{align*} \int (g+h x) \left (a+c x^2\right )^{3/2} \left (d+e x+f x^2\right ) \, dx &=\frac{f (g+h x)^2 \left (a+c x^2\right )^{5/2}}{7 c h}+\frac{\int (g+h x) \left ((7 c d-2 a f) h^2-c h (5 f g-7 e h) x\right ) \left (a+c x^2\right )^{3/2} \, dx}{7 c h^2}\\ &=\frac{f (g+h x)^2 \left (a+c x^2\right )^{5/2}}{7 c h}-\frac{\left (6 \left (5 c f g^2+2 a f h^2-7 c h (e g+d h)\right )+5 c h (5 f g-7 e h) x\right ) \left (a+c x^2\right )^{5/2}}{210 c^2 h}+\frac{(6 c d g-a f g-a e h) \int \left (a+c x^2\right )^{3/2} \, dx}{6 c}\\ &=\frac{(6 c d g-a (f g+e h)) x \left (a+c x^2\right )^{3/2}}{24 c}+\frac{f (g+h x)^2 \left (a+c x^2\right )^{5/2}}{7 c h}-\frac{\left (6 \left (5 c f g^2+2 a f h^2-7 c h (e g+d h)\right )+5 c h (5 f g-7 e h) x\right ) \left (a+c x^2\right )^{5/2}}{210 c^2 h}+\frac{(a (6 c d g-a f g-a e h)) \int \sqrt{a+c x^2} \, dx}{8 c}\\ &=\frac{a (6 c d g-a f g-a e h) x \sqrt{a+c x^2}}{16 c}+\frac{(6 c d g-a (f g+e h)) x \left (a+c x^2\right )^{3/2}}{24 c}+\frac{f (g+h x)^2 \left (a+c x^2\right )^{5/2}}{7 c h}-\frac{\left (6 \left (5 c f g^2+2 a f h^2-7 c h (e g+d h)\right )+5 c h (5 f g-7 e h) x\right ) \left (a+c x^2\right )^{5/2}}{210 c^2 h}+\frac{\left (a^2 (6 c d g-a f g-a e h)\right ) \int \frac{1}{\sqrt{a+c x^2}} \, dx}{16 c}\\ &=\frac{a (6 c d g-a f g-a e h) x \sqrt{a+c x^2}}{16 c}+\frac{(6 c d g-a (f g+e h)) x \left (a+c x^2\right )^{3/2}}{24 c}+\frac{f (g+h x)^2 \left (a+c x^2\right )^{5/2}}{7 c h}-\frac{\left (6 \left (5 c f g^2+2 a f h^2-7 c h (e g+d h)\right )+5 c h (5 f g-7 e h) x\right ) \left (a+c x^2\right )^{5/2}}{210 c^2 h}+\frac{\left (a^2 (6 c d g-a f g-a e h)\right ) \operatorname{Subst}\left (\int \frac{1}{1-c x^2} \, dx,x,\frac{x}{\sqrt{a+c x^2}}\right )}{16 c}\\ &=\frac{a (6 c d g-a f g-a e h) x \sqrt{a+c x^2}}{16 c}+\frac{(6 c d g-a (f g+e h)) x \left (a+c x^2\right )^{3/2}}{24 c}+\frac{f (g+h x)^2 \left (a+c x^2\right )^{5/2}}{7 c h}-\frac{\left (6 \left (5 c f g^2+2 a f h^2-7 c h (e g+d h)\right )+5 c h (5 f g-7 e h) x\right ) \left (a+c x^2\right )^{5/2}}{210 c^2 h}+\frac{a^2 (6 c d g-a f g-a e h) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )}{16 c^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.688368, size = 209, normalized size = 0.98 \[ \frac{\sqrt{a+c x^2} \left (-\frac{105 a^{5/2} \sqrt{\frac{c x^2}{a}+1} \sinh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right ) (a e h+a f g-6 c d g)}{c^{3/2} \left (a+c x^2\right )}-\frac{96 a^3 f h}{c^2}+\frac{3 a^2 (112 d h+7 e (16 g+5 h x)+f x (35 g+16 h x))}{c}+2 a x (21 d (25 g+16 h x)+x (7 e (48 g+35 h x)+f x (245 g+192 h x)))+4 c x^3 (21 d (5 g+4 h x)+2 x (7 e (6 g+5 h x)+5 f x (7 g+6 h x)))\right )}{1680} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.052, size = 287, normalized size = 1.4 \begin{align*}{\frac{fh{x}^{2}}{7\,c} \left ( c{x}^{2}+a \right ) ^{{\frac{5}{2}}}}-{\frac{2\,afh}{35\,{c}^{2}} \left ( c{x}^{2}+a \right ) ^{{\frac{5}{2}}}}+{\frac{ehx}{6\,c} \left ( c{x}^{2}+a \right ) ^{{\frac{5}{2}}}}+{\frac{fgx}{6\,c} \left ( c{x}^{2}+a \right ) ^{{\frac{5}{2}}}}-{\frac{aehx}{24\,c} \left ( c{x}^{2}+a \right ) ^{{\frac{3}{2}}}}-{\frac{afgx}{24\,c} \left ( c{x}^{2}+a \right ) ^{{\frac{3}{2}}}}-{\frac{{a}^{2}xeh}{16\,c}\sqrt{c{x}^{2}+a}}-{\frac{{a}^{2}xfg}{16\,c}\sqrt{c{x}^{2}+a}}-{\frac{{a}^{3}eh}{16}\ln \left ( x\sqrt{c}+\sqrt{c{x}^{2}+a} \right ){c}^{-{\frac{3}{2}}}}-{\frac{{a}^{3}fg}{16}\ln \left ( x\sqrt{c}+\sqrt{c{x}^{2}+a} \right ){c}^{-{\frac{3}{2}}}}+{\frac{dh}{5\,c} \left ( c{x}^{2}+a \right ) ^{{\frac{5}{2}}}}+{\frac{eg}{5\,c} \left ( c{x}^{2}+a \right ) ^{{\frac{5}{2}}}}+{\frac{dgx}{4} \left ( c{x}^{2}+a \right ) ^{{\frac{3}{2}}}}+{\frac{3\,adgx}{8}\sqrt{c{x}^{2}+a}}+{\frac{3\,dg{a}^{2}}{8}\ln \left ( x\sqrt{c}+\sqrt{c{x}^{2}+a} \right ){\frac{1}{\sqrt{c}}}} \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 = 1.47333, size = 1107, normalized size = 5.2 \begin{align*} \left [\frac{105 \,{\left (a^{3} e h -{\left (6 \, a^{2} c d - a^{3} f\right )} g\right )} \sqrt{c} \log \left (-2 \, c x^{2} + 2 \, \sqrt{c x^{2} + a} \sqrt{c} x - a\right ) + 2 \,{\left (240 \, c^{3} f h x^{6} + 280 \,{\left (c^{3} f g + c^{3} e h\right )} x^{5} + 336 \, a^{2} c e g + 48 \,{\left (7 \, c^{3} e g +{\left (7 \, c^{3} d + 8 \, a c^{2} f\right )} h\right )} x^{4} + 70 \,{\left (7 \, a c^{2} e h +{\left (6 \, c^{3} d + 7 \, a c^{2} f\right )} g\right )} x^{3} + 48 \,{\left (14 \, a c^{2} e g +{\left (14 \, a c^{2} d + a^{2} c f\right )} h\right )} x^{2} + 48 \,{\left (7 \, a^{2} c d - 2 \, a^{3} f\right )} h + 105 \,{\left (a^{2} c e h +{\left (10 \, a c^{2} d + a^{2} c f\right )} g\right )} x\right )} \sqrt{c x^{2} + a}}{3360 \, c^{2}}, \frac{105 \,{\left (a^{3} e h -{\left (6 \, a^{2} c d - a^{3} f\right )} g\right )} \sqrt{-c} \arctan \left (\frac{\sqrt{-c} x}{\sqrt{c x^{2} + a}}\right ) +{\left (240 \, c^{3} f h x^{6} + 280 \,{\left (c^{3} f g + c^{3} e h\right )} x^{5} + 336 \, a^{2} c e g + 48 \,{\left (7 \, c^{3} e g +{\left (7 \, c^{3} d + 8 \, a c^{2} f\right )} h\right )} x^{4} + 70 \,{\left (7 \, a c^{2} e h +{\left (6 \, c^{3} d + 7 \, a c^{2} f\right )} g\right )} x^{3} + 48 \,{\left (14 \, a c^{2} e g +{\left (14 \, a c^{2} d + a^{2} c f\right )} h\right )} x^{2} + 48 \,{\left (7 \, a^{2} c d - 2 \, a^{3} f\right )} h + 105 \,{\left (a^{2} c e h +{\left (10 \, a c^{2} d + a^{2} c f\right )} g\right )} x\right )} \sqrt{c x^{2} + a}}{1680 \, c^{2}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 25.5837, size = 768, normalized size = 3.61 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.17146, size = 356, normalized size = 1.67 \begin{align*} \frac{1}{1680} \, \sqrt{c x^{2} + a}{\left ({\left (2 \,{\left ({\left (4 \,{\left (5 \,{\left (6 \, c f h x + \frac{7 \,{\left (c^{6} f g + c^{6} h e\right )}}{c^{5}}\right )} x + \frac{6 \,{\left (7 \, c^{6} d h + 8 \, a c^{5} f h + 7 \, c^{6} g e\right )}}{c^{5}}\right )} x + \frac{35 \,{\left (6 \, c^{6} d g + 7 \, a c^{5} f g + 7 \, a c^{5} h e\right )}}{c^{5}}\right )} x + \frac{24 \,{\left (14 \, a c^{5} d h + a^{2} c^{4} f h + 14 \, a c^{5} g e\right )}}{c^{5}}\right )} x + \frac{105 \,{\left (10 \, a c^{5} d g + a^{2} c^{4} f g + a^{2} c^{4} h e\right )}}{c^{5}}\right )} x + \frac{48 \,{\left (7 \, a^{2} c^{4} d h - 2 \, a^{3} c^{3} f h + 7 \, a^{2} c^{4} g e\right )}}{c^{5}}\right )} - \frac{{\left (6 \, a^{2} c d g - a^{3} f g - a^{3} h e\right )} \log \left ({\left | -\sqrt{c} x + \sqrt{c x^{2} + a} \right |}\right )}{16 \, c^{\frac{3}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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