Optimal. Leaf size=432 \[ \frac{\tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right ) \left (3 a^2 f h^4+12 a c h^2 \left (3 f g^2-h (2 e g-d h)\right )+8 c^2 g^2 \left (5 f g^2-h (4 e g-3 d h)\right )\right )}{8 \sqrt{c} h^6}-\frac{\left (a+c x^2\right )^{5/2} \left (d h^2-e g h+f g^2\right )}{h (g+h x) \left (a h^2+c g^2\right )}-\frac{\left (a+c x^2\right )^{3/2} \left (4 \left (a h^2 (2 f g-e h)+c g \left (5 f g^2-h (4 e g-3 d h)\right )\right )-3 h x \left (a f h^2+c \left (5 f g^2-4 h (e g-d h)\right )\right )\right )}{12 h^3 \left (a h^2+c g^2\right )}-\frac{\sqrt{a+c x^2} \left (8 \left (a h^2 (2 f g-e h)+c g \left (5 f g^2-h (4 e g-3 d h)\right )\right )-h x \left (3 a f h^2+12 c d h^2-16 c e g h+20 c f g^2\right )\right )}{8 h^5}+\frac{\sqrt{a h^2+c g^2} \tanh ^{-1}\left (\frac{a h-c g x}{\sqrt{a+c x^2} \sqrt{a h^2+c g^2}}\right ) \left (a h^2 (2 f g-e h)+c g \left (5 f g^2-h (4 e g-3 d h)\right )\right )}{h^6} \]
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Rubi [A] time = 0.90139, antiderivative size = 428, normalized size of antiderivative = 0.99, 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 = {1651, 815, 844, 217, 206, 725} \[ \frac{\tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right ) \left (3 a^2 f h^4+12 a c h^2 \left (3 f g^2-h (2 e g-d h)\right )+8 c^2 \left (5 f g^4-g^2 h (4 e g-3 d h)\right )\right )}{8 \sqrt{c} h^6}-\frac{\left (a+c x^2\right )^{5/2} \left (d h^2-e g h+f g^2\right )}{h (g+h x) \left (a h^2+c g^2\right )}-\frac{\left (a+c x^2\right )^{3/2} \left (4 \left (a h^2 (2 f g-e h)-c g h (4 e g-3 d h)+5 c f g^3\right )-3 h x \left (a f h^2-4 c h (e g-d h)+5 c f g^2\right )\right )}{12 h^3 \left (a h^2+c g^2\right )}-\frac{\sqrt{a+c x^2} \left (8 \left (a h^2 (2 f g-e h)-c g h (4 e g-3 d h)+5 c f g^3\right )-h x \left (3 a f h^2+12 c d h^2-16 c e g h+20 c f g^2\right )\right )}{8 h^5}+\frac{\sqrt{a h^2+c g^2} \tanh ^{-1}\left (\frac{a h-c g x}{\sqrt{a+c x^2} \sqrt{a h^2+c g^2}}\right ) \left (a h^2 (2 f g-e h)-c g h (4 e g-3 d h)+5 c f g^3\right )}{h^6} \]
Antiderivative was successfully verified.
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Rule 1651
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)^2} \, dx &=-\frac{\left (f g^2-e g h+d h^2\right ) \left (a+c x^2\right )^{5/2}}{h \left (c g^2+a h^2\right ) (g+h x)}-\frac{\int \frac{\left (-c d g+a f g-a e h-\left (a f h-c \left (4 e g-\frac{5 f g^2}{h}-4 d h\right )\right ) x\right ) \left (a+c x^2\right )^{3/2}}{g+h x} \, dx}{c g^2+a h^2}\\ &=-\frac{\left (4 \left (5 c f g^3-c g h (4 e g-3 d h)+a h^2 (2 f g-e h)\right )-3 h \left (5 c f g^2+a f h^2-4 c h (e g-d h)\right ) x\right ) \left (a+c x^2\right )^{3/2}}{12 h^3 \left (c g^2+a h^2\right )}-\frac{\left (f g^2-e g h+d h^2\right ) \left (a+c x^2\right )^{5/2}}{h \left (c g^2+a h^2\right ) (g+h x)}-\frac{\int \frac{\left (a c (5 f g-4 e h) \left (c g^2+a h^2\right )-\frac{c \left (c g^2+a h^2\right ) \left (20 c f g^2-16 c e g h+12 c d h^2+3 a f h^2\right ) x}{h}\right ) \sqrt{a+c x^2}}{g+h x} \, dx}{4 c h^2 \left (c g^2+a h^2\right )}\\ &=-\frac{\left (8 \left (5 c f g^3-c g h (4 e g-3 d h)+a h^2 (2 f g-e h)\right )-h \left (20 c f g^2-16 c e g h+12 c d h^2+3 a f h^2\right ) x\right ) \sqrt{a+c x^2}}{8 h^5}-\frac{\left (4 \left (5 c f g^3-c g h (4 e g-3 d h)+a h^2 (2 f g-e h)\right )-3 h \left (5 c f g^2+a f h^2-4 c h (e g-d h)\right ) x\right ) \left (a+c x^2\right )^{3/2}}{12 h^3 \left (c g^2+a h^2\right )}-\frac{\left (f g^2-e g h+d h^2\right ) \left (a+c x^2\right )^{5/2}}{h \left (c g^2+a h^2\right ) (g+h x)}-\frac{\int \frac{a c^2 \left (c g^2+a h^2\right ) \left (a h^2 (13 f g-8 e h)+4 c \left (5 f g^3-g h (4 e g-3 d h)\right )\right )-\frac{c^2 \left (c g^2+a h^2\right ) \left (3 a^2 f h^4+8 c^2 \left (5 f g^4-g^2 h (4 e g-3 d h)\right )+12 a c h^2 \left (3 f g^2-h (2 e g-d h)\right )\right ) x}{h}}{(g+h x) \sqrt{a+c x^2}} \, dx}{8 c^2 h^4 \left (c g^2+a h^2\right )}\\ &=-\frac{\left (8 \left (5 c f g^3-c g h (4 e g-3 d h)+a h^2 (2 f g-e h)\right )-h \left (20 c f g^2-16 c e g h+12 c d h^2+3 a f h^2\right ) x\right ) \sqrt{a+c x^2}}{8 h^5}-\frac{\left (4 \left (5 c f g^3-c g h (4 e g-3 d h)+a h^2 (2 f g-e h)\right )-3 h \left (5 c f g^2+a f h^2-4 c h (e g-d h)\right ) x\right ) \left (a+c x^2\right )^{3/2}}{12 h^3 \left (c g^2+a h^2\right )}-\frac{\left (f g^2-e g h+d h^2\right ) \left (a+c x^2\right )^{5/2}}{h \left (c g^2+a h^2\right ) (g+h x)}-\frac{\left (\left (c g^2+a h^2\right ) \left (5 c f g^3-c g h (4 e g-3 d h)+a h^2 (2 f g-e h)\right )\right ) \int \frac{1}{(g+h x) \sqrt{a+c x^2}} \, dx}{h^6}+\frac{\left (3 a^2 f h^4+8 c^2 \left (5 f g^4-g^2 h (4 e g-3 d h)\right )+12 a c h^2 \left (3 f g^2-h (2 e g-d h)\right )\right ) \int \frac{1}{\sqrt{a+c x^2}} \, dx}{8 h^6}\\ &=-\frac{\left (8 \left (5 c f g^3-c g h (4 e g-3 d h)+a h^2 (2 f g-e h)\right )-h \left (20 c f g^2-16 c e g h+12 c d h^2+3 a f h^2\right ) x\right ) \sqrt{a+c x^2}}{8 h^5}-\frac{\left (4 \left (5 c f g^3-c g h (4 e g-3 d h)+a h^2 (2 f g-e h)\right )-3 h \left (5 c f g^2+a f h^2-4 c h (e g-d h)\right ) x\right ) \left (a+c x^2\right )^{3/2}}{12 h^3 \left (c g^2+a h^2\right )}-\frac{\left (f g^2-e g h+d h^2\right ) \left (a+c x^2\right )^{5/2}}{h \left (c g^2+a h^2\right ) (g+h x)}+\frac{\left (\left (c g^2+a h^2\right ) \left (5 c f g^3-c g h (4 e g-3 d h)+a h^2 (2 f g-e h)\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 f h^4+8 c^2 \left (5 f g^4-g^2 h (4 e g-3 d h)\right )+12 a c h^2 \left (3 f g^2-h (2 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 (5 c f g^3-c g h (4 e g-3 d h)+a h^2 (2 f g-e h)\right )-h \left (20 c f g^2-16 c e g h+12 c d h^2+3 a f h^2\right ) x\right ) \sqrt{a+c x^2}}{8 h^5}-\frac{\left (4 \left (5 c f g^3-c g h (4 e g-3 d h)+a h^2 (2 f g-e h)\right )-3 h \left (5 c f g^2+a f h^2-4 c h (e g-d h)\right ) x\right ) \left (a+c x^2\right )^{3/2}}{12 h^3 \left (c g^2+a h^2\right )}-\frac{\left (f g^2-e g h+d h^2\right ) \left (a+c x^2\right )^{5/2}}{h \left (c g^2+a h^2\right ) (g+h x)}+\frac{\left (3 a^2 f h^4+8 c^2 \left (5 f g^4-g^2 h (4 e g-3 d h)\right )+12 a c h^2 \left (3 f g^2-h (2 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{\sqrt{c g^2+a h^2} \left (5 c f g^3-c g h (4 e g-3 d h)+a h^2 (2 f g-e h)\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 = 0.582258, size = 392, normalized size = 0.91 \[ \frac{\frac{3 \log \left (\sqrt{c} \sqrt{a+c x^2}+c x\right ) \left (3 a^2 f h^4+12 a c h^2 \left (h (d h-2 e g)+3 f g^2\right )+8 c^2 \left (g^2 h (3 d h-4 e g)+5 f g^4\right )\right )}{\sqrt{c}}+h \sqrt{a+c x^2} \left (3 h x \left (5 a f h^2+4 c \left (h (d h-2 e g)+3 f g^2\right )\right )-\frac{24 \left (a h^2+c g^2\right ) \left (h (d h-e g)+f g^2\right )}{g+h x}+8 \left (4 a h^2 (e h-2 f g)-3 c \left (g h (2 d h-3 e g)+4 f g^3\right )\right )+8 c h^2 x^2 (e h-2 f g)+6 c f h^3 x^3\right )+24 \sqrt{a h^2+c g^2} \log \left (\sqrt{a+c x^2} \sqrt{a h^2+c g^2}+a h-c g x\right ) \left (a h^2 (2 f g-e h)+c g h (3 d h-4 e g)+5 c f g^3\right )-24 \sqrt{a h^2+c g^2} \log (g+h x) \left (a h^2 (2 f g-e h)+c g h (3 d h-4 e g)+5 c f g^3\right )}{24 h^6} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.226, size = 5121, normalized size = 11.9 \begin{align*} \text{output 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 )}{\left (g + h x\right )^{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [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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