Optimal. Leaf size=346 \[ \frac{x \left (a+c x^2\right )^{3/2} \left (3 a^2 f h^2-8 a c \left (h (d h+2 e g)+f g^2\right )+48 c^2 d g^2\right )}{192 c^2}+\frac{a x \sqrt{a+c x^2} \left (3 a^2 f h^2-8 a c \left (h (d h+2 e g)+f g^2\right )+48 c^2 d g^2\right )}{128 c^2}+\frac{a^2 \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right ) \left (3 a^2 f h^2-8 a c \left (h (d h+2 e g)+f g^2\right )+48 c^2 d g^2\right )}{128 c^{5/2}}-\frac{\left (a+c x^2\right )^{5/2} \left (12 \left (8 a h^2 (e h+2 f g)+c g \left (5 f g^2-8 h (7 d h+e g)\right )\right )-5 h x \left (7 h^2 (8 c d-3 a f)-2 c g (5 f g-8 e h)\right )\right )}{1680 c^2 h}-\frac{\left (a+c x^2\right )^{5/2} (g+h x)^2 (5 f g-8 e h)}{56 c h}+\frac{f \left (a+c x^2\right )^{5/2} (g+h x)^3}{8 c h} \]
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Rubi [A] time = 0.524336, antiderivative size = 345, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.207, Rules used = {1654, 833, 780, 195, 217, 206} \[ \frac{x \left (a+c x^2\right )^{3/2} \left (3 a^2 f h^2-8 a c \left (h (d h+2 e g)+f g^2\right )+48 c^2 d g^2\right )}{192 c^2}+\frac{a x \sqrt{a+c x^2} \left (3 a^2 f h^2-8 a c \left (h (d h+2 e g)+f g^2\right )+48 c^2 d g^2\right )}{128 c^2}+\frac{a^2 \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right ) \left (3 a^2 f h^2-8 a c \left (h (d h+2 e g)+f g^2\right )+48 c^2 d g^2\right )}{128 c^{5/2}}-\frac{\left (a+c x^2\right )^{5/2} \left (12 \left (8 a h^2 (e h+2 f g)-8 c g h (7 d h+e g)+5 c f g^3\right )-5 h x \left (7 h^2 (8 c d-3 a f)-2 c g (5 f g-8 e h)\right )\right )}{1680 c^2 h}-\frac{\left (a+c x^2\right )^{5/2} (g+h x)^2 (5 f g-8 e h)}{56 c h}+\frac{f \left (a+c x^2\right )^{5/2} (g+h x)^3}{8 c h} \]
Antiderivative was successfully verified.
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Rule 1654
Rule 833
Rule 780
Rule 195
Rule 217
Rule 206
Rubi steps
\begin{align*} \int (g+h x)^2 \left (a+c x^2\right )^{3/2} \left (d+e x+f x^2\right ) \, dx &=\frac{f (g+h x)^3 \left (a+c x^2\right )^{5/2}}{8 c h}+\frac{\int (g+h x)^2 \left ((8 c d-3 a f) h^2-c h (5 f g-8 e h) x\right ) \left (a+c x^2\right )^{3/2} \, dx}{8 c h^2}\\ &=-\frac{(5 f g-8 e h) (g+h x)^2 \left (a+c x^2\right )^{5/2}}{56 c h}+\frac{f (g+h x)^3 \left (a+c x^2\right )^{5/2}}{8 c h}+\frac{\int (g+h x) \left (c h^2 (56 c d g-11 a f g-16 a e h)+c h \left (7 (8 c d-3 a f) h^2-2 c g (5 f g-8 e h)\right ) x\right ) \left (a+c x^2\right )^{3/2} \, dx}{56 c^2 h^2}\\ &=-\frac{(5 f g-8 e h) (g+h x)^2 \left (a+c x^2\right )^{5/2}}{56 c h}+\frac{f (g+h x)^3 \left (a+c x^2\right )^{5/2}}{8 c h}-\frac{\left (12 \left (5 c f g^3-8 c g h (e g+7 d h)+8 a h^2 (2 f g+e h)\right )-5 h \left (7 (8 c d-3 a f) h^2-2 c g (5 f g-8 e h)\right ) x\right ) \left (a+c x^2\right )^{5/2}}{1680 c^2 h}+\frac{\left (48 c^2 d g^2+3 a^2 f h^2-8 a c \left (f g^2+h (2 e g+d h)\right )\right ) \int \left (a+c x^2\right )^{3/2} \, dx}{48 c^2}\\ &=\frac{\left (48 c^2 d g^2+3 a^2 f h^2-8 a c \left (f g^2+h (2 e g+d h)\right )\right ) x \left (a+c x^2\right )^{3/2}}{192 c^2}-\frac{(5 f g-8 e h) (g+h x)^2 \left (a+c x^2\right )^{5/2}}{56 c h}+\frac{f (g+h x)^3 \left (a+c x^2\right )^{5/2}}{8 c h}-\frac{\left (12 \left (5 c f g^3-8 c g h (e g+7 d h)+8 a h^2 (2 f g+e h)\right )-5 h \left (7 (8 c d-3 a f) h^2-2 c g (5 f g-8 e h)\right ) x\right ) \left (a+c x^2\right )^{5/2}}{1680 c^2 h}+\frac{\left (a \left (48 c^2 d g^2+3 a^2 f h^2-8 a c \left (f g^2+h (2 e g+d h)\right )\right )\right ) \int \sqrt{a+c x^2} \, dx}{64 c^2}\\ &=\frac{a \left (48 c^2 d g^2+3 a^2 f h^2-8 a c \left (f g^2+h (2 e g+d h)\right )\right ) x \sqrt{a+c x^2}}{128 c^2}+\frac{\left (48 c^2 d g^2+3 a^2 f h^2-8 a c \left (f g^2+h (2 e g+d h)\right )\right ) x \left (a+c x^2\right )^{3/2}}{192 c^2}-\frac{(5 f g-8 e h) (g+h x)^2 \left (a+c x^2\right )^{5/2}}{56 c h}+\frac{f (g+h x)^3 \left (a+c x^2\right )^{5/2}}{8 c h}-\frac{\left (12 \left (5 c f g^3-8 c g h (e g+7 d h)+8 a h^2 (2 f g+e h)\right )-5 h \left (7 (8 c d-3 a f) h^2-2 c g (5 f g-8 e h)\right ) x\right ) \left (a+c x^2\right )^{5/2}}{1680 c^2 h}+\frac{\left (a^2 \left (48 c^2 d g^2+3 a^2 f h^2-8 a c \left (f g^2+h (2 e g+d h)\right )\right )\right ) \int \frac{1}{\sqrt{a+c x^2}} \, dx}{128 c^2}\\ &=\frac{a \left (48 c^2 d g^2+3 a^2 f h^2-8 a c \left (f g^2+h (2 e g+d h)\right )\right ) x \sqrt{a+c x^2}}{128 c^2}+\frac{\left (48 c^2 d g^2+3 a^2 f h^2-8 a c \left (f g^2+h (2 e g+d h)\right )\right ) x \left (a+c x^2\right )^{3/2}}{192 c^2}-\frac{(5 f g-8 e h) (g+h x)^2 \left (a+c x^2\right )^{5/2}}{56 c h}+\frac{f (g+h x)^3 \left (a+c x^2\right )^{5/2}}{8 c h}-\frac{\left (12 \left (5 c f g^3-8 c g h (e g+7 d h)+8 a h^2 (2 f g+e h)\right )-5 h \left (7 (8 c d-3 a f) h^2-2 c g (5 f g-8 e h)\right ) x\right ) \left (a+c x^2\right )^{5/2}}{1680 c^2 h}+\frac{\left (a^2 \left (48 c^2 d g^2+3 a^2 f h^2-8 a c \left (f g^2+h (2 e g+d h)\right )\right )\right ) \operatorname{Subst}\left (\int \frac{1}{1-c x^2} \, dx,x,\frac{x}{\sqrt{a+c x^2}}\right )}{128 c^2}\\ &=\frac{a \left (48 c^2 d g^2+3 a^2 f h^2-8 a c \left (f g^2+h (2 e g+d h)\right )\right ) x \sqrt{a+c x^2}}{128 c^2}+\frac{\left (48 c^2 d g^2+3 a^2 f h^2-8 a c \left (f g^2+h (2 e g+d h)\right )\right ) x \left (a+c x^2\right )^{3/2}}{192 c^2}-\frac{(5 f g-8 e h) (g+h x)^2 \left (a+c x^2\right )^{5/2}}{56 c h}+\frac{f (g+h x)^3 \left (a+c x^2\right )^{5/2}}{8 c h}-\frac{\left (12 \left (5 c f g^3-8 c g h (e g+7 d h)+8 a h^2 (2 f g+e h)\right )-5 h \left (7 (8 c d-3 a f) h^2-2 c g (5 f g-8 e h)\right ) x\right ) \left (a+c x^2\right )^{5/2}}{1680 c^2 h}+\frac{a^2 \left (48 c^2 d g^2+3 a^2 f h^2-8 a c \left (f g^2+h (2 e g+d h)\right )\right ) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )}{128 c^{5/2}}\\ \end{align*}
Mathematica [A] time = 1.14949, size = 346, normalized size = 1. \[ \frac{\sqrt{a+c x^2} \left (-\frac{280 a \left (3 a^{3/2} \sinh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )+\sqrt{c} x \left (5 a+2 c x^2\right ) \sqrt{\frac{c x^2}{a}+1}\right ) \left (h (d h+2 e g)+f g^2\right )}{c^{3/2} \sqrt{\frac{c x^2}{a}+1}}+\frac{105 a f h^2 \left (\frac{3 a^{5/2} \sinh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{\sqrt{\frac{c x^2}{a}+1}}-\sqrt{c} x \left (3 a^2+14 a c x^2+8 c^2 x^4\right )\right )}{c^{5/2}}+1680 d g^2 \left (\frac{3 a^{3/2} \sinh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{\sqrt{c} \sqrt{\frac{c x^2}{a}+1}}+5 a x+2 c x^3\right )+\frac{384 h \left (a+c x^2\right )^2 \left (5 c x^2-2 a\right ) (e h+2 f g)}{c^2}+\frac{2240 x \left (a+c x^2\right )^2 \left (h (d h+2 e g)+f g^2\right )}{c}+\frac{2688 g \left (a+c x^2\right )^2 (2 d h+e g)}{c}+\frac{1680 f h^2 x^3 \left (a+c x^2\right )^2}{c}\right )}{13440} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.057, size = 552, normalized size = 1.6 \begin{align*} -{\frac{{a}^{3}egh}{8}\ln \left ( x\sqrt{c}+\sqrt{c{x}^{2}+a} \right ){c}^{-{\frac{3}{2}}}}+{\frac{2\,fg{x}^{2}h}{7\,c} \left ( c{x}^{2}+a \right ) ^{{\frac{5}{2}}}}-{\frac{4\,afgh}{35\,{c}^{2}} \left ( c{x}^{2}+a \right ) ^{{\frac{5}{2}}}}-{\frac{af{h}^{2}x}{16\,{c}^{2}} \left ( c{x}^{2}+a \right ) ^{{\frac{5}{2}}}}-{\frac{aexgh}{12\,c} \left ( c{x}^{2}+a \right ) ^{{\frac{3}{2}}}}-{\frac{{a}^{2}xegh}{8\,c}\sqrt{c{x}^{2}+a}}+{\frac{3\,d{g}^{2}{a}^{2}}{8}\ln \left ( x\sqrt{c}+\sqrt{c{x}^{2}+a} \right ){\frac{1}{\sqrt{c}}}}+{\frac{dx{h}^{2}}{6\,c} \left ( c{x}^{2}+a \right ) ^{{\frac{5}{2}}}}+{\frac{fx{g}^{2}}{6\,c} \left ( c{x}^{2}+a \right ) ^{{\frac{5}{2}}}}-{\frac{{a}^{3}d{h}^{2}}{16}\ln \left ( x\sqrt{c}+\sqrt{c{x}^{2}+a} \right ){c}^{-{\frac{3}{2}}}}-{\frac{{a}^{3}f{g}^{2}}{16}\ln \left ( x\sqrt{c}+\sqrt{c{x}^{2}+a} \right ){c}^{-{\frac{3}{2}}}}+{\frac{2\,dgh}{5\,c} \left ( c{x}^{2}+a \right ) ^{{\frac{5}{2}}}}+{\frac{3\,d{g}^{2}ax}{8}\sqrt{c{x}^{2}+a}}+{\frac{f{h}^{2}{x}^{3}}{8\,c} \left ( c{x}^{2}+a \right ) ^{{\frac{5}{2}}}}+{\frac{3\,f{h}^{2}{a}^{4}}{128}\ln \left ( x\sqrt{c}+\sqrt{c{x}^{2}+a} \right ){c}^{-{\frac{5}{2}}}}+{\frac{e{x}^{2}{h}^{2}}{7\,c} \left ( c{x}^{2}+a \right ) ^{{\frac{5}{2}}}}-{\frac{2\,ae{h}^{2}}{35\,{c}^{2}} \left ( c{x}^{2}+a \right ) ^{{\frac{5}{2}}}}+{\frac{e{g}^{2}}{5\,c} \left ( c{x}^{2}+a \right ) ^{{\frac{5}{2}}}}+{\frac{d{g}^{2}x}{4} \left ( c{x}^{2}+a \right ) ^{{\frac{3}{2}}}}-{\frac{{a}^{2}xf{g}^{2}}{16\,c}\sqrt{c{x}^{2}+a}}+{\frac{{a}^{2}f{h}^{2}x}{64\,{c}^{2}} \left ( c{x}^{2}+a \right ) ^{{\frac{3}{2}}}}+{\frac{3\,f{h}^{2}{a}^{3}x}{128\,{c}^{2}}\sqrt{c{x}^{2}+a}}+{\frac{egxh}{3\,c} \left ( c{x}^{2}+a \right ) ^{{\frac{5}{2}}}}-{\frac{axd{h}^{2}}{24\,c} \left ( c{x}^{2}+a \right ) ^{{\frac{3}{2}}}}-{\frac{{a}^{2}xd{h}^{2}}{16\,c}\sqrt{c{x}^{2}+a}}-{\frac{axf{g}^{2}}{24\,c} \left ( c{x}^{2}+a \right ) ^{{\frac{3}{2}}}} \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 = 2.11568, size = 1878, normalized size = 5.43 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 50.0334, size = 1304, normalized size = 3.77 \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 = 1.18857, size = 610, normalized size = 1.76 \begin{align*} \frac{1}{13440} \, \sqrt{c x^{2} + a}{\left ({\left (2 \,{\left ({\left (4 \,{\left (5 \,{\left (6 \,{\left (7 \, c f h^{2} x + \frac{8 \,{\left (2 \, c^{7} f g h + c^{7} h^{2} e\right )}}{c^{6}}\right )} x + \frac{7 \,{\left (8 \, c^{7} f g^{2} + 8 \, c^{7} d h^{2} + 9 \, a c^{6} f h^{2} + 16 \, c^{7} g h e\right )}}{c^{6}}\right )} x + \frac{48 \,{\left (14 \, c^{7} d g h + 16 \, a c^{6} f g h + 7 \, c^{7} g^{2} e + 8 \, a c^{6} h^{2} e\right )}}{c^{6}}\right )} x + \frac{35 \,{\left (48 \, c^{7} d g^{2} + 56 \, a c^{6} f g^{2} + 56 \, a c^{6} d h^{2} + 3 \, a^{2} c^{5} f h^{2} + 112 \, a c^{6} g h e\right )}}{c^{6}}\right )} x + \frac{192 \,{\left (28 \, a c^{6} d g h + 2 \, a^{2} c^{5} f g h + 14 \, a c^{6} g^{2} e + a^{2} c^{5} h^{2} e\right )}}{c^{6}}\right )} x + \frac{105 \,{\left (80 \, a c^{6} d g^{2} + 8 \, a^{2} c^{5} f g^{2} + 8 \, a^{2} c^{5} d h^{2} - 3 \, a^{3} c^{4} f h^{2} + 16 \, a^{2} c^{5} g h e\right )}}{c^{6}}\right )} x + \frac{384 \,{\left (14 \, a^{2} c^{5} d g h - 4 \, a^{3} c^{4} f g h + 7 \, a^{2} c^{5} g^{2} e - 2 \, a^{3} c^{4} h^{2} e\right )}}{c^{6}}\right )} - \frac{{\left (48 \, a^{2} c^{2} d g^{2} - 8 \, a^{3} c f g^{2} - 8 \, a^{3} c d h^{2} + 3 \, a^{4} f h^{2} - 16 \, a^{3} c g h e\right )} \log \left ({\left | -\sqrt{c} x + \sqrt{c x^{2} + a} \right |}\right )}{128 \, c^{\frac{5}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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