Optimal. Leaf size=229 \[ -\frac{h \sqrt{a+c x^2} \left (4 \left (4 a^2 f h^2-a c \left (3 h (d h+3 e g)+7 f g^2\right )+3 c^2 d g^2\right )+c h x (-9 a e h-11 a f g+6 c d g)\right )}{6 a c^3}-\frac{\tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right ) \left (3 a h^2 (e h+3 f g)-2 c g \left (3 h (d h+e g)+f g^2\right )\right )}{2 c^{5/2}}-\frac{h \sqrt{a+c x^2} (g+h x)^2 (3 c d-4 a f)}{3 a c^2}-\frac{(g+h x)^3 (a e-x (c d-a f))}{a c \sqrt{a+c x^2}} \]
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Rubi [A] time = 0.324773, antiderivative size = 228, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.172, Rules used = {1645, 833, 780, 217, 206} \[ -\frac{h \sqrt{a+c x^2} \left (4 \left (4 a^2 f h^2-a c \left (3 h (d h+3 e g)+7 f g^2\right )+3 c^2 d g^2\right )+c h x (-9 a e h-11 a f g+6 c d g)\right )}{6 a c^3}+\frac{\tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right ) \left (-3 a h^2 (e h+3 f g)+6 c g h (d h+e g)+2 c f g^3\right )}{2 c^{5/2}}-\frac{h \sqrt{a+c x^2} (g+h x)^2 (3 c d-4 a f)}{3 a c^2}-\frac{(g+h x)^3 (a e-x (c d-a f))}{a c \sqrt{a+c x^2}} \]
Antiderivative was successfully verified.
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Rule 1645
Rule 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 )}{\left (a+c x^2\right )^{3/2}} \, dx &=-\frac{(a e-(c d-a f) x) (g+h x)^3}{a c \sqrt{a+c x^2}}-\frac{\int \frac{(g+h x)^2 (-a (f g+3 e h)+(3 c d-4 a f) h x)}{\sqrt{a+c x^2}} \, dx}{a c}\\ &=-\frac{(a e-(c d-a f) x) (g+h x)^3}{a c \sqrt{a+c x^2}}-\frac{(3 c d-4 a f) h (g+h x)^2 \sqrt{a+c x^2}}{3 a c^2}-\frac{\int \frac{(g+h x) \left (-a \left (2 (3 c d-4 a f) h^2+3 c g (f g+3 e h)\right )+c h (6 c d g-11 a f g-9 a e h) x\right )}{\sqrt{a+c x^2}} \, dx}{3 a c^2}\\ &=-\frac{(a e-(c d-a f) x) (g+h x)^3}{a c \sqrt{a+c x^2}}-\frac{(3 c d-4 a f) h (g+h x)^2 \sqrt{a+c x^2}}{3 a c^2}-\frac{h \left (4 \left (3 c^2 d g^2+4 a^2 f h^2-a c \left (7 f g^2+3 h (3 e g+d h)\right )\right )+c h (6 c d g-11 a f g-9 a e h) x\right ) \sqrt{a+c x^2}}{6 a c^3}+\frac{\left (2 c f g^3+6 c g h (e g+d h)-3 a h^2 (3 f g+e h)\right ) \int \frac{1}{\sqrt{a+c x^2}} \, dx}{2 c^2}\\ &=-\frac{(a e-(c d-a f) x) (g+h x)^3}{a c \sqrt{a+c x^2}}-\frac{(3 c d-4 a f) h (g+h x)^2 \sqrt{a+c x^2}}{3 a c^2}-\frac{h \left (4 \left (3 c^2 d g^2+4 a^2 f h^2-a c \left (7 f g^2+3 h (3 e g+d h)\right )\right )+c h (6 c d g-11 a f g-9 a e h) x\right ) \sqrt{a+c x^2}}{6 a c^3}+\frac{\left (2 c f g^3+6 c g h (e g+d h)-3 a h^2 (3 f g+e h)\right ) \operatorname{Subst}\left (\int \frac{1}{1-c x^2} \, dx,x,\frac{x}{\sqrt{a+c x^2}}\right )}{2 c^2}\\ &=-\frac{(a e-(c d-a f) x) (g+h x)^3}{a c \sqrt{a+c x^2}}-\frac{(3 c d-4 a f) h (g+h x)^2 \sqrt{a+c x^2}}{3 a c^2}-\frac{h \left (4 \left (3 c^2 d g^2+4 a^2 f h^2-a c \left (7 f g^2+3 h (3 e g+d h)\right )\right )+c h (6 c d g-11 a f g-9 a e h) x\right ) \sqrt{a+c x^2}}{6 a c^3}+\frac{\left (2 c f g^3+6 c g h (e g+d h)-3 a h^2 (3 f g+e h)\right ) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )}{2 c^{5/2}}\\ \end{align*}
Mathematica [A] time = 0.503031, size = 246, normalized size = 1.07 \[ \frac{\frac{a^2 c h \left (3 h (4 d h+3 e (4 g+h x))+f \left (36 g^2+27 g h x-8 h^2 x^2\right )\right )-16 a^3 f h^3+a c^2 \left (6 d h \left (-3 g^2-3 g h x+h^2 x^2\right )-3 e \left (6 g^2 h x+2 g^3-6 g h^2 x^2-h^3 x^3\right )+f x \left (18 g^2 h x-6 g^3+9 g h^2 x^2+2 h^3 x^3\right )\right )+6 c^3 d g^3 x}{a \sqrt{a+c x^2}}+3 \sqrt{c} \log \left (\sqrt{c} \sqrt{a+c x^2}+c x\right ) \left (-3 a h^2 (e h+3 f g)+6 c g h (d h+e g)+2 c f g^3\right )}{6 c^3} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.06, size = 516, normalized size = 2.3 \begin{align*}{\frac{{h}^{3}f{x}^{4}}{3\,c}{\frac{1}{\sqrt{c{x}^{2}+a}}}}-{\frac{4\,a{h}^{3}f{x}^{2}}{3\,{c}^{2}}{\frac{1}{\sqrt{c{x}^{2}+a}}}}-{\frac{8\,{a}^{2}f{h}^{3}}{3\,{c}^{3}}{\frac{1}{\sqrt{c{x}^{2}+a}}}}+{\frac{{x}^{3}{h}^{3}e}{2\,c}{\frac{1}{\sqrt{c{x}^{2}+a}}}}+{\frac{3\,{x}^{3}g{h}^{2}f}{2\,c}{\frac{1}{\sqrt{c{x}^{2}+a}}}}+{\frac{3\,ax{h}^{3}e}{2\,{c}^{2}}{\frac{1}{\sqrt{c{x}^{2}+a}}}}+{\frac{9\,axg{h}^{2}f}{2\,{c}^{2}}{\frac{1}{\sqrt{c{x}^{2}+a}}}}-{\frac{3\,ae{h}^{3}}{2}\ln \left ( x\sqrt{c}+\sqrt{c{x}^{2}+a} \right ){c}^{-{\frac{5}{2}}}}-{\frac{9\,ag{h}^{2}f}{2}\ln \left ( x\sqrt{c}+\sqrt{c{x}^{2}+a} \right ){c}^{-{\frac{5}{2}}}}+{\frac{{x}^{2}{h}^{3}d}{c}{\frac{1}{\sqrt{c{x}^{2}+a}}}}+3\,{\frac{{x}^{2}g{h}^{2}e}{c\sqrt{c{x}^{2}+a}}}+3\,{\frac{{g}^{2}{x}^{2}hf}{c\sqrt{c{x}^{2}+a}}}+2\,{\frac{a{h}^{3}d}{{c}^{2}\sqrt{c{x}^{2}+a}}}+6\,{\frac{ag{h}^{2}e}{{c}^{2}\sqrt{c{x}^{2}+a}}}+6\,{\frac{a{g}^{2}hf}{{c}^{2}\sqrt{c{x}^{2}+a}}}-3\,{\frac{gx{h}^{2}d}{c\sqrt{c{x}^{2}+a}}}-3\,{\frac{x{g}^{2}he}{c\sqrt{c{x}^{2}+a}}}-{\frac{x{g}^{3}f}{c}{\frac{1}{\sqrt{c{x}^{2}+a}}}}+3\,{\frac{\ln \left ( x\sqrt{c}+\sqrt{c{x}^{2}+a} \right ) g{h}^{2}d}{{c}^{3/2}}}+3\,{\frac{\ln \left ( x\sqrt{c}+\sqrt{c{x}^{2}+a} \right ){g}^{2}he}{{c}^{3/2}}}+{{g}^{3}f\ln \left ( x\sqrt{c}+\sqrt{c{x}^{2}+a} \right ){c}^{-{\frac{3}{2}}}}-3\,{\frac{{g}^{2}hd}{c\sqrt{c{x}^{2}+a}}}-{\frac{{g}^{3}e}{c}{\frac{1}{\sqrt{c{x}^{2}+a}}}}+{\frac{{g}^{3}dx}{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 = 2.20619, size = 1635, normalized size = 7.14 \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 [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (g + h x\right )^{3} \left (d + e x + f x^{2}\right )}{\left (a + c x^{2}\right )^{\frac{3}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.20149, size = 458, normalized size = 2. \begin{align*} \frac{{\left ({\left ({\left (\frac{2 \, f h^{3} x}{c} + \frac{3 \,{\left (3 \, a c^{4} f g h^{2} + a c^{4} h^{3} e\right )}}{a c^{5}}\right )} x + \frac{2 \,{\left (9 \, a c^{4} f g^{2} h + 3 \, a c^{4} d h^{3} - 4 \, a^{2} c^{3} f h^{3} + 9 \, a c^{4} g h^{2} e\right )}}{a c^{5}}\right )} x + \frac{3 \,{\left (2 \, c^{5} d g^{3} - 2 \, a c^{4} f g^{3} - 6 \, a c^{4} d g h^{2} + 9 \, a^{2} c^{3} f g h^{2} - 6 \, a c^{4} g^{2} h e + 3 \, a^{2} c^{3} h^{3} e\right )}}{a c^{5}}\right )} x - \frac{2 \,{\left (9 \, a c^{4} d g^{2} h - 18 \, a^{2} c^{3} f g^{2} h - 6 \, a^{2} c^{3} d h^{3} + 8 \, a^{3} c^{2} f h^{3} + 3 \, a c^{4} g^{3} e - 18 \, a^{2} c^{3} g h^{2} e\right )}}{a c^{5}}}{6 \, \sqrt{c x^{2} + a}} - \frac{{\left (2 \, c f g^{3} + 6 \, c d g h^{2} - 9 \, a f g h^{2} + 6 \, c g^{2} h e - 3 \, a h^{3} e\right )} \log \left ({\left | -\sqrt{c} x + \sqrt{c x^{2} + a} \right |}\right )}{2 \, c^{\frac{5}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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