Optimal. Leaf size=326 \[ -\frac{\tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right ) \left (3 a^2 h^4 (f g-e h)+12 a c g h^2 \left (f g^2-h (e g-d h)\right )+8 c^2 g^3 \left (f g^2-h (e g-d h)\right )\right )}{8 \sqrt{c} h^6}+\frac{\left (a+c x^2\right )^{3/2} \left (4 \left (d h^2-e g h+f g^2\right )-3 h x (f g-e h)\right )}{12 h^3}+\frac{\sqrt{a+c x^2} \left (8 \left (a h^2+c g^2\right ) \left (d h^2-e g h+f g^2\right )-h x \left (\left (3 a h^2+4 c g^2\right ) (f g-e h)+4 c d g h^2\right )\right )}{8 h^5}-\frac{\left (a h^2+c g^2\right )^{3/2} \left (d h^2-e g h+f g^2\right ) \tanh ^{-1}\left (\frac{a h-c g x}{\sqrt{a+c x^2} \sqrt{a h^2+c g^2}}\right )}{h^6}+\frac{f \left (a+c x^2\right )^{5/2}}{5 c h} \]
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Rubi [A] time = 0.76619, antiderivative size = 326, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.207, Rules used = {1654, 815, 844, 217, 206, 725} \[ -\frac{\tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right ) \left (3 a^2 h^4 (f g-e h)+12 a c g h^2 \left (f g^2-h (e g-d h)\right )+8 c^2 \left (f g^5-g^3 h (e g-d h)\right )\right )}{8 \sqrt{c} h^6}+\frac{\left (a+c x^2\right )^{3/2} \left (4 \left (d h^2-e g h+f g^2\right )-3 h x (f g-e h)\right )}{12 h^3}+\frac{\sqrt{a+c x^2} \left (8 \left (a h^2+c g^2\right ) \left (d h^2-e g h+f g^2\right )-h x \left (\left (3 a h^2+4 c g^2\right ) (f g-e h)+4 c d g h^2\right )\right )}{8 h^5}-\frac{\left (a h^2+c g^2\right )^{3/2} \left (d h^2-e g h+f g^2\right ) \tanh ^{-1}\left (\frac{a h-c g x}{\sqrt{a+c x^2} \sqrt{a h^2+c g^2}}\right )}{h^6}+\frac{f \left (a+c x^2\right )^{5/2}}{5 c h} \]
Antiderivative was successfully verified.
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Rule 1654
Rule 815
Rule 844
Rule 217
Rule 206
Rule 725
Rubi steps
\begin{align*} \int \frac{\left (a+c x^2\right )^{3/2} \left (d+e x+f x^2\right )}{g+h x} \, dx &=\frac{f \left (a+c x^2\right )^{5/2}}{5 c h}+\frac{\int \frac{\left (5 c d h^2-5 c h (f g-e h) x\right ) \left (a+c x^2\right )^{3/2}}{g+h x} \, dx}{5 c h^2}\\ &=\frac{\left (4 \left (f g^2-e g h+d h^2\right )-3 h (f g-e h) x\right ) \left (a+c x^2\right )^{3/2}}{12 h^3}+\frac{f \left (a+c x^2\right )^{5/2}}{5 c h}+\frac{\int \frac{\left (5 a c^2 h^2 \left (f g^2-h (e g-4 d h)\right )-5 c^2 h \left (4 c d g h^2+(f g-e h) \left (4 c g^2+3 a h^2\right )\right ) x\right ) \sqrt{a+c x^2}}{g+h x} \, dx}{20 c^2 h^4}\\ &=\frac{\left (8 \left (c g^2+a h^2\right ) \left (f g^2-e g h+d h^2\right )-h \left (4 c d g h^2+(f g-e h) \left (4 c g^2+3 a h^2\right )\right ) x\right ) \sqrt{a+c x^2}}{8 h^5}+\frac{\left (4 \left (f g^2-e g h+d h^2\right )-3 h (f g-e h) x\right ) \left (a+c x^2\right )^{3/2}}{12 h^3}+\frac{f \left (a+c x^2\right )^{5/2}}{5 c h}+\frac{\int \frac{5 a c^3 h^2 \left (a h^2 \left (5 f g^2-h (5 e g-8 d h)\right )+4 c \left (f g^4-g^2 h (e g-d h)\right )\right )-5 c^3 h \left (3 a^2 h^4 (f g-e h)+12 a c g h^2 \left (f g^2-h (e g-d h)\right )+8 c^2 \left (f g^5-g^3 h (e g-d h)\right )\right ) x}{(g+h x) \sqrt{a+c x^2}} \, dx}{40 c^3 h^6}\\ &=\frac{\left (8 \left (c g^2+a h^2\right ) \left (f g^2-e g h+d h^2\right )-h \left (4 c d g h^2+(f g-e h) \left (4 c g^2+3 a h^2\right )\right ) x\right ) \sqrt{a+c x^2}}{8 h^5}+\frac{\left (4 \left (f g^2-e g h+d h^2\right )-3 h (f g-e h) x\right ) \left (a+c x^2\right )^{3/2}}{12 h^3}+\frac{f \left (a+c x^2\right )^{5/2}}{5 c h}+\frac{\left (\left (c g^2+a h^2\right )^2 \left (f g^2-e g h+d h^2\right )\right ) \int \frac{1}{(g+h x) \sqrt{a+c x^2}} \, dx}{h^6}-\frac{\left (3 a^2 h^4 (f g-e h)+12 a c g h^2 \left (f g^2-h (e g-d h)\right )+8 c^2 \left (f g^5-g^3 h (e g-d h)\right )\right ) \int \frac{1}{\sqrt{a+c x^2}} \, dx}{8 h^6}\\ &=\frac{\left (8 \left (c g^2+a h^2\right ) \left (f g^2-e g h+d h^2\right )-h \left (4 c d g h^2+(f g-e h) \left (4 c g^2+3 a h^2\right )\right ) x\right ) \sqrt{a+c x^2}}{8 h^5}+\frac{\left (4 \left (f g^2-e g h+d h^2\right )-3 h (f g-e h) x\right ) \left (a+c x^2\right )^{3/2}}{12 h^3}+\frac{f \left (a+c x^2\right )^{5/2}}{5 c h}-\frac{\left (\left (c g^2+a h^2\right )^2 \left (f g^2-e g h+d h^2\right )\right ) \operatorname{Subst}\left (\int \frac{1}{c g^2+a h^2-x^2} \, dx,x,\frac{a h-c g x}{\sqrt{a+c x^2}}\right )}{h^6}-\frac{\left (3 a^2 h^4 (f g-e h)+12 a c g h^2 \left (f g^2-h (e g-d h)\right )+8 c^2 \left (f g^5-g^3 h (e g-d h)\right )\right ) \operatorname{Subst}\left (\int \frac{1}{1-c x^2} \, dx,x,\frac{x}{\sqrt{a+c x^2}}\right )}{8 h^6}\\ &=\frac{\left (8 \left (c g^2+a h^2\right ) \left (f g^2-e g h+d h^2\right )-h \left (4 c d g h^2+(f g-e h) \left (4 c g^2+3 a h^2\right )\right ) x\right ) \sqrt{a+c x^2}}{8 h^5}+\frac{\left (4 \left (f g^2-e g h+d h^2\right )-3 h (f g-e h) x\right ) \left (a+c x^2\right )^{3/2}}{12 h^3}+\frac{f \left (a+c x^2\right )^{5/2}}{5 c h}-\frac{\left (3 a^2 h^4 (f g-e h)+12 a c g h^2 \left (f g^2-h (e g-d h)\right )+8 c^2 \left (f g^5-g^3 h (e g-d h)\right )\right ) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )}{8 \sqrt{c} h^6}-\frac{\left (c g^2+a h^2\right )^{3/2} \left (f g^2-e g h+d h^2\right ) \tanh ^{-1}\left (\frac{a h-c g x}{\sqrt{c g^2+a h^2} \sqrt{a+c x^2}}\right )}{h^6}\\ \end{align*}
Mathematica [A] time = 1.42591, size = 348, normalized size = 1.07 \[ \frac{\sqrt{a+c x^2} \left (3 a^{3/2} \sinh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )+\sqrt{c} x \left (5 a+2 c x^2\right ) \sqrt{\frac{c x^2}{a}+1}\right ) (e h-f g)}{8 \sqrt{c} h^2 \sqrt{\frac{c x^2}{a}+1}}-\frac{\left (h (d h-e g)+f g^2\right ) \left (\sqrt{\frac{c x^2}{a}+1} \left (-h \sqrt{a+c x^2} \left (8 a h^2+6 c g^2-3 c g h x+2 c h^2 x^2\right )+6 \left (a h^2+c g^2\right )^{3/2} \tanh ^{-1}\left (\frac{a h-c g x}{\sqrt{a+c x^2} \sqrt{a h^2+c g^2}}\right )+6 \sqrt{c} g \left (a h^2+c g^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )\right )+3 \sqrt{a} \sqrt{c} g h^2 \sqrt{a+c x^2} \sinh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )\right )}{6 h^6 \sqrt{\frac{c x^2}{a}+1}}+\frac{f \left (a+c x^2\right )^{5/2}}{5 c h} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.228, size = 2420, normalized size = 7.4 \begin{align*} \text{result too large to display} \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 [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (a + c x^{2}\right )^{\frac{3}{2}} \left (d + e x + f x^{2}\right )}{g + h x}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.22853, size = 744, normalized size = 2.28 \begin{align*} \frac{1}{120} \, \sqrt{c x^{2} + a}{\left ({\left (2 \,{\left (3 \,{\left (\frac{4 \, c f x}{h} - \frac{5 \,{\left (c^{4} f g h^{19} - c^{4} h^{20} e\right )}}{c^{3} h^{21}}\right )} x + \frac{4 \,{\left (5 \, c^{4} f g^{2} h^{18} + 5 \, c^{4} d h^{20} + 6 \, a c^{3} f h^{20} - 5 \, c^{4} g h^{19} e\right )}}{c^{3} h^{21}}\right )} x - \frac{15 \,{\left (4 \, c^{4} f g^{3} h^{17} + 4 \, c^{4} d g h^{19} + 5 \, a c^{3} f g h^{19} - 4 \, c^{4} g^{2} h^{18} e - 5 \, a c^{3} h^{20} e\right )}}{c^{3} h^{21}}\right )} x + \frac{8 \,{\left (15 \, c^{4} f g^{4} h^{16} + 15 \, c^{4} d g^{2} h^{18} + 20 \, a c^{3} f g^{2} h^{18} + 20 \, a c^{3} d h^{20} + 3 \, a^{2} c^{2} f h^{20} - 15 \, c^{4} g^{3} h^{17} e - 20 \, a c^{3} g h^{19} e\right )}}{c^{3} h^{21}}\right )} + \frac{2 \,{\left (c^{2} f g^{6} + c^{2} d g^{4} h^{2} + 2 \, a c f g^{4} h^{2} + 2 \, a c d g^{2} h^{4} + a^{2} f g^{2} h^{4} + a^{2} d h^{6} - c^{2} g^{5} h e - 2 \, a c g^{3} h^{3} e - a^{2} g h^{5} e\right )} \arctan \left (-\frac{{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )} h + \sqrt{c} g}{\sqrt{-c g^{2} - a h^{2}}}\right )}{\sqrt{-c g^{2} - a h^{2}} h^{6}} + \frac{{\left (8 \, c^{\frac{5}{2}} f g^{5} + 8 \, c^{\frac{5}{2}} d g^{3} h^{2} + 12 \, a c^{\frac{3}{2}} f g^{3} h^{2} + 12 \, a c^{\frac{3}{2}} d g h^{4} + 3 \, a^{2} \sqrt{c} f g h^{4} - 8 \, c^{\frac{5}{2}} g^{4} h e - 12 \, a c^{\frac{3}{2}} g^{2} h^{3} e - 3 \, a^{2} \sqrt{c} h^{5} e\right )} \log \left ({\left | -\sqrt{c} x + \sqrt{c x^{2} + a} \right |}\right )}{8 \, c h^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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