Optimal. Leaf size=325 \[ \frac{\sqrt{a+c x^2} \left (4 \left (16 a^2 f h^4-4 a c h^2 \left (5 h (d h+3 e g)+13 f g^2\right )-c^2 g^2 \left (3 f g^2-5 h (16 d h+3 e g)\right )\right )-c h x \left (a h^2 (45 e h+71 f g)+2 c g \left (3 f g^2-5 h (10 d h+3 e g)\right )\right )\right )}{120 c^3 h}+\frac{\tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right ) \left (3 a^2 h^2 (e h+3 f g)-4 a c g \left (3 h (d h+e g)+f g^2\right )+8 c^2 d g^3\right )}{8 c^{5/2}}+\frac{\sqrt{a+c x^2} (g+h x)^2 \left (4 h^2 (5 c d-4 a f)-3 c g (f g-5 e h)\right )}{60 c^2 h}-\frac{\sqrt{a+c x^2} (g+h x)^3 (f g-5 e h)}{20 c h}+\frac{f \sqrt{a+c x^2} (g+h x)^4}{5 c h} \]
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Rubi [A] time = 0.664388, antiderivative size = 323, normalized size of antiderivative = 0.99, number of steps used = 6, number of rules used = 5, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.172, Rules used = {1654, 833, 780, 217, 206} \[ \frac{\sqrt{a+c x^2} \left (4 \left (16 a^2 f h^4-4 a c h^2 \left (5 h (d h+3 e g)+13 f g^2\right )-c^2 g^2 \left (3 f g^2-5 h (16 d h+3 e g)\right )\right )-c h x \left (a h^2 (45 e h+71 f g)-10 c g h (10 d h+3 e g)+6 c f g^3\right )\right )}{120 c^3 h}+\frac{\tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right ) \left (3 a^2 h^2 (e h+3 f g)-4 a c g \left (3 h (d h+e g)+f g^2\right )+8 c^2 d g^3\right )}{8 c^{5/2}}+\frac{\sqrt{a+c x^2} (g+h x)^2 \left (4 h^2 (5 c d-4 a f)-3 c g (f g-5 e h)\right )}{60 c^2 h}-\frac{\sqrt{a+c x^2} (g+h x)^3 (f g-5 e h)}{20 c h}+\frac{f \sqrt{a+c x^2} (g+h x)^4}{5 c h} \]
Antiderivative was successfully verified.
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Rule 1654
Rule 833
Rule 780
Rule 217
Rule 206
Rubi steps
\begin{align*} \int \frac{(g+h x)^3 \left (d+e x+f x^2\right )}{\sqrt{a+c x^2}} \, dx &=\frac{f (g+h x)^4 \sqrt{a+c x^2}}{5 c h}+\frac{\int \frac{(g+h x)^3 \left ((5 c d-4 a f) h^2-c h (f g-5 e h) x\right )}{\sqrt{a+c x^2}} \, dx}{5 c h^2}\\ &=-\frac{(f g-5 e h) (g+h x)^3 \sqrt{a+c x^2}}{20 c h}+\frac{f (g+h x)^4 \sqrt{a+c x^2}}{5 c h}+\frac{\int \frac{(g+h x)^2 \left (c h^2 (20 c d g-13 a f g-15 a e h)+c h \left (4 (5 c d-4 a f) h^2-3 c g (f g-5 e h)\right ) x\right )}{\sqrt{a+c x^2}} \, dx}{20 c^2 h^2}\\ &=\frac{\left (4 (5 c d-4 a f) h^2-3 c g (f g-5 e h)\right ) (g+h x)^2 \sqrt{a+c x^2}}{60 c^2 h}-\frac{(f g-5 e h) (g+h x)^3 \sqrt{a+c x^2}}{20 c h}+\frac{f (g+h x)^4 \sqrt{a+c x^2}}{5 c h}+\frac{\int \frac{(g+h x) \left (c h^2 \left (60 c^2 d g^2+32 a^2 f h^2-a c \left (33 f g^2+5 h (15 e g+8 d h)\right )\right )-c^2 h \left (6 c f g^3-10 c g h (3 e g+10 d h)+a h^2 (71 f g+45 e h)\right ) x\right )}{\sqrt{a+c x^2}} \, dx}{60 c^3 h^2}\\ &=\frac{\left (4 (5 c d-4 a f) h^2-3 c g (f g-5 e h)\right ) (g+h x)^2 \sqrt{a+c x^2}}{60 c^2 h}-\frac{(f g-5 e h) (g+h x)^3 \sqrt{a+c x^2}}{20 c h}+\frac{f (g+h x)^4 \sqrt{a+c x^2}}{5 c h}+\frac{\left (4 \left (16 a^2 f h^4-4 a c h^2 \left (13 f g^2+5 h (3 e g+d h)\right )-c^2 g^2 \left (3 f g^2-5 h (3 e g+16 d h)\right )\right )-c h \left (6 c f g^3-10 c g h (3 e g+10 d h)+a h^2 (71 f g+45 e h)\right ) x\right ) \sqrt{a+c x^2}}{120 c^3 h}+\frac{\left (8 c^2 d g^3+3 a^2 h^2 (3 f g+e h)-4 a c g \left (f g^2+3 h (e g+d h)\right )\right ) \int \frac{1}{\sqrt{a+c x^2}} \, dx}{8 c^2}\\ &=\frac{\left (4 (5 c d-4 a f) h^2-3 c g (f g-5 e h)\right ) (g+h x)^2 \sqrt{a+c x^2}}{60 c^2 h}-\frac{(f g-5 e h) (g+h x)^3 \sqrt{a+c x^2}}{20 c h}+\frac{f (g+h x)^4 \sqrt{a+c x^2}}{5 c h}+\frac{\left (4 \left (16 a^2 f h^4-4 a c h^2 \left (13 f g^2+5 h (3 e g+d h)\right )-c^2 g^2 \left (3 f g^2-5 h (3 e g+16 d h)\right )\right )-c h \left (6 c f g^3-10 c g h (3 e g+10 d h)+a h^2 (71 f g+45 e h)\right ) x\right ) \sqrt{a+c x^2}}{120 c^3 h}+\frac{\left (8 c^2 d g^3+3 a^2 h^2 (3 f g+e h)-4 a c g \left (f g^2+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 c^2}\\ &=\frac{\left (4 (5 c d-4 a f) h^2-3 c g (f g-5 e h)\right ) (g+h x)^2 \sqrt{a+c x^2}}{60 c^2 h}-\frac{(f g-5 e h) (g+h x)^3 \sqrt{a+c x^2}}{20 c h}+\frac{f (g+h x)^4 \sqrt{a+c x^2}}{5 c h}+\frac{\left (4 \left (16 a^2 f h^4-4 a c h^2 \left (13 f g^2+5 h (3 e g+d h)\right )-c^2 g^2 \left (3 f g^2-5 h (3 e g+16 d h)\right )\right )-c h \left (6 c f g^3-10 c g h (3 e g+10 d h)+a h^2 (71 f g+45 e h)\right ) x\right ) \sqrt{a+c x^2}}{120 c^3 h}+\frac{\left (8 c^2 d g^3+3 a^2 h^2 (3 f g+e h)-4 a c g \left (f g^2+3 h (e g+d h)\right )\right ) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )}{8 c^{5/2}}\\ \end{align*}
Mathematica [A] time = 0.393918, size = 252, normalized size = 0.78 \[ \frac{\sqrt{a+c x^2} \left (8 \left (8 a^2 f h^3-10 a c h \left (h (d h+3 e g)+3 f g^2\right )+15 c^2 g^2 (3 d h+e g)\right )+8 c h x^2 \left (5 c \left (h (d h+3 e g)+3 f g^2\right )-4 a f h^2\right )+15 c x \left (4 c \left (3 g h (d h+e g)+f g^3\right )-3 a h^2 (e h+3 f g)\right )+30 c^2 h^2 x^3 (e h+3 f g)+24 c^2 f h^3 x^4\right )+15 \sqrt{c} \log \left (\sqrt{c} \sqrt{a+c x^2}+c x\right ) \left (3 a^2 h^2 (e h+3 f g)-4 a c g \left (3 h (d h+e g)+f g^2\right )+8 c^2 d g^3\right )}{120 c^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.062, size = 528, normalized size = 1.6 \begin{align*}{\frac{{h}^{3}f{x}^{4}}{5\,c}\sqrt{c{x}^{2}+a}}-{\frac{4\,a{h}^{3}f{x}^{2}}{15\,{c}^{2}}\sqrt{c{x}^{2}+a}}+{\frac{8\,{a}^{2}f{h}^{3}}{15\,{c}^{3}}\sqrt{c{x}^{2}+a}}+{\frac{{x}^{3}{h}^{3}e}{4\,c}\sqrt{c{x}^{2}+a}}+{\frac{3\,{x}^{3}g{h}^{2}f}{4\,c}\sqrt{c{x}^{2}+a}}-{\frac{3\,ax{h}^{3}e}{8\,{c}^{2}}\sqrt{c{x}^{2}+a}}-{\frac{9\,axg{h}^{2}f}{8\,{c}^{2}}\sqrt{c{x}^{2}+a}}+{\frac{3\,{a}^{2}e{h}^{3}}{8}\ln \left ( x\sqrt{c}+\sqrt{c{x}^{2}+a} \right ){c}^{-{\frac{5}{2}}}}+{\frac{9\,{a}^{2}g{h}^{2}f}{8}\ln \left ( x\sqrt{c}+\sqrt{c{x}^{2}+a} \right ){c}^{-{\frac{5}{2}}}}+{\frac{{x}^{2}{h}^{3}d}{3\,c}\sqrt{c{x}^{2}+a}}+{\frac{{x}^{2}g{h}^{2}e}{c}\sqrt{c{x}^{2}+a}}+{\frac{{g}^{2}{x}^{2}hf}{c}\sqrt{c{x}^{2}+a}}-{\frac{2\,a{h}^{3}d}{3\,{c}^{2}}\sqrt{c{x}^{2}+a}}-2\,{\frac{a\sqrt{c{x}^{2}+a}g{h}^{2}e}{{c}^{2}}}-2\,{\frac{a\sqrt{c{x}^{2}+a}{g}^{2}hf}{{c}^{2}}}+{\frac{3\,gx{h}^{2}d}{2\,c}\sqrt{c{x}^{2}+a}}+{\frac{3\,x{g}^{2}he}{2\,c}\sqrt{c{x}^{2}+a}}+{\frac{x{g}^{3}f}{2\,c}\sqrt{c{x}^{2}+a}}-{\frac{3\,ag{h}^{2}d}{2}\ln \left ( x\sqrt{c}+\sqrt{c{x}^{2}+a} \right ){c}^{-{\frac{3}{2}}}}-{\frac{3\,a{g}^{2}he}{2}\ln \left ( x\sqrt{c}+\sqrt{c{x}^{2}+a} \right ){c}^{-{\frac{3}{2}}}}-{\frac{a{g}^{3}f}{2}\ln \left ( x\sqrt{c}+\sqrt{c{x}^{2}+a} \right ){c}^{-{\frac{3}{2}}}}+3\,{\frac{\sqrt{c{x}^{2}+a}{g}^{2}hd}{c}}+{\frac{{g}^{3}e}{c}\sqrt{c{x}^{2}+a}}+{{g}^{3}d\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 = 2.08434, size = 1284, normalized size = 3.95 \begin{align*} \left [-\frac{15 \,{\left (12 \, a c e g^{2} h - 3 \, a^{2} e h^{3} - 4 \,{\left (2 \, c^{2} d - a c f\right )} g^{3} + 3 \,{\left (4 \, a c d - 3 \, a^{2} f\right )} g h^{2}\right )} \sqrt{c} \log \left (-2 \, c x^{2} - 2 \, \sqrt{c x^{2} + a} \sqrt{c} x - a\right ) - 2 \,{\left (24 \, c^{2} f h^{3} x^{4} + 120 \, c^{2} e g^{3} - 240 \, a c e g h^{2} + 120 \,{\left (3 \, c^{2} d - 2 \, a c f\right )} g^{2} h - 16 \,{\left (5 \, a c d - 4 \, a^{2} f\right )} h^{3} + 30 \,{\left (3 \, c^{2} f g h^{2} + c^{2} e h^{3}\right )} x^{3} + 8 \,{\left (15 \, c^{2} f g^{2} h + 15 \, c^{2} e g h^{2} +{\left (5 \, c^{2} d - 4 \, a c f\right )} h^{3}\right )} x^{2} + 15 \,{\left (4 \, c^{2} f g^{3} + 12 \, c^{2} e g^{2} h - 3 \, a c e h^{3} + 3 \,{\left (4 \, c^{2} d - 3 \, a c f\right )} g h^{2}\right )} x\right )} \sqrt{c x^{2} + a}}{240 \, c^{3}}, \frac{15 \,{\left (12 \, a c e g^{2} h - 3 \, a^{2} e h^{3} - 4 \,{\left (2 \, c^{2} d - a c f\right )} g^{3} + 3 \,{\left (4 \, a c d - 3 \, a^{2} f\right )} g h^{2}\right )} \sqrt{-c} \arctan \left (\frac{\sqrt{-c} x}{\sqrt{c x^{2} + a}}\right ) +{\left (24 \, c^{2} f h^{3} x^{4} + 120 \, c^{2} e g^{3} - 240 \, a c e g h^{2} + 120 \,{\left (3 \, c^{2} d - 2 \, a c f\right )} g^{2} h - 16 \,{\left (5 \, a c d - 4 \, a^{2} f\right )} h^{3} + 30 \,{\left (3 \, c^{2} f g h^{2} + c^{2} e h^{3}\right )} x^{3} + 8 \,{\left (15 \, c^{2} f g^{2} h + 15 \, c^{2} e g h^{2} +{\left (5 \, c^{2} d - 4 \, a c f\right )} h^{3}\right )} x^{2} + 15 \,{\left (4 \, c^{2} f g^{3} + 12 \, c^{2} e g^{2} h - 3 \, a c e h^{3} + 3 \,{\left (4 \, c^{2} d - 3 \, a c f\right )} g h^{2}\right )} x\right )} \sqrt{c x^{2} + a}}{120 \, c^{3}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 19.6929, size = 796, normalized size = 2.45 \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 = 1.17259, size = 424, normalized size = 1.3 \begin{align*} \frac{1}{120} \, \sqrt{c x^{2} + a}{\left ({\left (2 \,{\left (3 \,{\left (\frac{4 \, f h^{3} x}{c} + \frac{5 \,{\left (3 \, c^{4} f g h^{2} + c^{4} h^{3} e\right )}}{c^{5}}\right )} x + \frac{4 \,{\left (15 \, c^{4} f g^{2} h + 5 \, c^{4} d h^{3} - 4 \, a c^{3} f h^{3} + 15 \, c^{4} g h^{2} e\right )}}{c^{5}}\right )} x + \frac{15 \,{\left (4 \, c^{4} f g^{3} + 12 \, c^{4} d g h^{2} - 9 \, a c^{3} f g h^{2} + 12 \, c^{4} g^{2} h e - 3 \, a c^{3} h^{3} e\right )}}{c^{5}}\right )} x + \frac{8 \,{\left (45 \, c^{4} d g^{2} h - 30 \, a c^{3} f g^{2} h - 10 \, a c^{3} d h^{3} + 8 \, a^{2} c^{2} f h^{3} + 15 \, c^{4} g^{3} e - 30 \, a c^{3} g h^{2} e\right )}}{c^{5}}\right )} - \frac{{\left (8 \, c^{2} d g^{3} - 4 \, a c f g^{3} - 12 \, a c d g h^{2} + 9 \, a^{2} f g h^{2} - 12 \, a c g^{2} h e + 3 \, a^{2} h^{3} e\right )} \log \left ({\left | -\sqrt{c} x + \sqrt{c x^{2} + a} \right |}\right )}{8 \, c^{\frac{5}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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