Optimal. Leaf size=475 \[ \frac{c \tanh ^{-1}\left (\frac{a h-c g x}{\sqrt{a+c x^2} \sqrt{a h^2+c g^2}}\right ) \left (3 a^2 h^4 (4 f g-e h)+3 a c g h^2 \left (11 f g^2-h (4 e g-d h)\right )+2 c^2 g^3 \left (10 f g^2-h (4 e g-d h)\right )\right )}{2 h^6 \left (a h^2+c g^2\right )^{3/2}}-\frac{\left (a+c x^2\right )^{5/2} \left (d h^2-e g h+f g^2\right )}{3 h (g+h x)^3 \left (a h^2+c g^2\right )}-\frac{\left (a+c x^2\right )^{3/2} \left (-x \left (3 a f h^2+c \left (5 f g^2-2 h (e g-d h)\right )\right )-3 a h (3 f g-e h)+c g \left (-d h+4 e g-\frac{10 f g^2}{h}\right )\right )}{6 h^2 (g+h x)^2 \left (a h^2+c g^2\right )}-\frac{\sqrt{a+c x^2} \left (c h x \left (3 a h^2 (3 f g-e h)+c g \left (10 f g^2-h (4 e g-d h)\right )\right )+\left (a h^2+c g^2\right ) \left (3 a f h^2+2 c \left (10 f g^2-h (4 e g-d h)\right )\right )\right )}{2 h^5 (g+h x) \left (a h^2+c g^2\right )}+\frac{\sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right ) \left (3 a f h^2+2 c \left (10 f g^2-h (4 e g-d h)\right )\right )}{2 h^6} \]
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Rubi [A] time = 0.844642, antiderivative size = 469, 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, 813, 844, 217, 206, 725} \[ \frac{c \tanh ^{-1}\left (\frac{a h-c g x}{\sqrt{a+c x^2} \sqrt{a h^2+c g^2}}\right ) \left (3 a^2 h^4 (4 f g-e h)+3 a c g h^2 \left (11 f g^2-h (4 e g-d h)\right )+2 c^2 \left (10 f g^5-g^3 h (4 e g-d h)\right )\right )}{2 h^6 \left (a h^2+c g^2\right )^{3/2}}-\frac{\left (a+c x^2\right )^{5/2} \left (d h^2-e g h+f g^2\right )}{3 h (g+h x)^3 \left (a h^2+c g^2\right )}-\frac{\left (a+c x^2\right )^{3/2} \left (-x \left (3 a f h^2-2 c h (e g-d h)+5 c f g^2\right )-3 a h (3 f g-e h)+c g \left (-d h+4 e g-\frac{10 f g^2}{h}\right )\right )}{6 h^2 (g+h x)^2 \left (a h^2+c g^2\right )}-\frac{\sqrt{a+c x^2} \left (c h x \left (3 a h^2 (3 f g-e h)-c g h (4 e g-d h)+10 c f g^3\right )+\left (a h^2+c g^2\right ) \left (3 a f h^2-2 c h (4 e g-d h)+20 c f g^2\right )\right )}{2 h^5 (g+h x) \left (a h^2+c g^2\right )}+\frac{\sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right ) \left (3 a f h^2-2 c h (4 e g-d h)+20 c f g^2\right )}{2 h^6} \]
Antiderivative was successfully verified.
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Rule 1651
Rule 813
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)^4} \, dx &=-\frac{\left (f g^2-e g h+d h^2\right ) \left (a+c x^2\right )^{5/2}}{3 h \left (c g^2+a h^2\right ) (g+h x)^3}-\frac{\int \frac{\left (-3 (c d g-a f g+a e h)-\left (3 a f h-c \left (2 e g-\frac{5 f g^2}{h}-2 d h\right )\right ) x\right ) \left (a+c x^2\right )^{3/2}}{(g+h x)^3} \, dx}{3 \left (c g^2+a h^2\right )}\\ &=-\frac{\left (c g \left (4 e g-\frac{10 f g^2}{h}-d h\right )-3 a h (3 f g-e h)-\left (5 c f g^2+3 a f h^2-2 c h (e g-d h)\right ) x\right ) \left (a+c x^2\right )^{3/2}}{6 h^2 \left (c g^2+a h^2\right ) (g+h x)^2}-\frac{\left (f g^2-e g h+d h^2\right ) \left (a+c x^2\right )^{5/2}}{3 h \left (c g^2+a h^2\right ) (g+h x)^3}+\frac{\int \frac{\left (4 a \left (5 c f g^2+3 a f h^2-2 c h (e g-d h)\right )-\frac{4 c \left (10 c f g^3-c g h (4 e g-d h)+3 a h^2 (3 f g-e h)\right ) x}{h}\right ) \sqrt{a+c x^2}}{(g+h x)^2} \, dx}{8 h^2 \left (c g^2+a h^2\right )}\\ &=-\frac{\left (\left (c g^2+a h^2\right ) \left (20 c f g^2+3 a f h^2-2 c h (4 e g-d h)\right )+c h \left (10 c f g^3-c g h (4 e g-d h)+3 a h^2 (3 f g-e h)\right ) x\right ) \sqrt{a+c x^2}}{2 h^5 \left (c g^2+a h^2\right ) (g+h x)}-\frac{\left (c g \left (4 e g-\frac{10 f g^2}{h}-d h\right )-3 a h (3 f g-e h)-\left (5 c f g^2+3 a f h^2-2 c h (e g-d h)\right ) x\right ) \left (a+c x^2\right )^{3/2}}{6 h^2 \left (c g^2+a h^2\right ) (g+h x)^2}-\frac{\left (f g^2-e g h+d h^2\right ) \left (a+c x^2\right )^{5/2}}{3 h \left (c g^2+a h^2\right ) (g+h x)^3}-\frac{\int \frac{8 a c \left (10 c f g^3-c g h (4 e g-d h)+3 a h^2 (3 f g-e h)\right )-\frac{8 c \left (c g^2+a h^2\right ) \left (20 c f g^2+3 a f h^2-2 c h (4 e g-d h)\right ) x}{h}}{(g+h x) \sqrt{a+c x^2}} \, dx}{16 h^4 \left (c g^2+a h^2\right )}\\ &=-\frac{\left (\left (c g^2+a h^2\right ) \left (20 c f g^2+3 a f h^2-2 c h (4 e g-d h)\right )+c h \left (10 c f g^3-c g h (4 e g-d h)+3 a h^2 (3 f g-e h)\right ) x\right ) \sqrt{a+c x^2}}{2 h^5 \left (c g^2+a h^2\right ) (g+h x)}-\frac{\left (c g \left (4 e g-\frac{10 f g^2}{h}-d h\right )-3 a h (3 f g-e h)-\left (5 c f g^2+3 a f h^2-2 c h (e g-d h)\right ) x\right ) \left (a+c x^2\right )^{3/2}}{6 h^2 \left (c g^2+a h^2\right ) (g+h x)^2}-\frac{\left (f g^2-e g h+d h^2\right ) \left (a+c x^2\right )^{5/2}}{3 h \left (c g^2+a h^2\right ) (g+h x)^3}+\frac{\left (c \left (20 c f g^2+3 a f h^2-2 c h (4 e g-d h)\right )\right ) \int \frac{1}{\sqrt{a+c x^2}} \, dx}{2 h^6}-\frac{\left (c \left (3 a^2 h^4 (4 f g-e h)+3 a c g h^2 \left (11 f g^2-h (4 e g-d h)\right )+2 c^2 \left (10 f g^5-g^3 h (4 e g-d h)\right )\right )\right ) \int \frac{1}{(g+h x) \sqrt{a+c x^2}} \, dx}{2 h^6 \left (c g^2+a h^2\right )}\\ &=-\frac{\left (\left (c g^2+a h^2\right ) \left (20 c f g^2+3 a f h^2-2 c h (4 e g-d h)\right )+c h \left (10 c f g^3-c g h (4 e g-d h)+3 a h^2 (3 f g-e h)\right ) x\right ) \sqrt{a+c x^2}}{2 h^5 \left (c g^2+a h^2\right ) (g+h x)}-\frac{\left (c g \left (4 e g-\frac{10 f g^2}{h}-d h\right )-3 a h (3 f g-e h)-\left (5 c f g^2+3 a f h^2-2 c h (e g-d h)\right ) x\right ) \left (a+c x^2\right )^{3/2}}{6 h^2 \left (c g^2+a h^2\right ) (g+h x)^2}-\frac{\left (f g^2-e g h+d h^2\right ) \left (a+c x^2\right )^{5/2}}{3 h \left (c g^2+a h^2\right ) (g+h x)^3}+\frac{\left (c \left (20 c f g^2+3 a f h^2-2 c h (4 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 )}{2 h^6}+\frac{\left (c \left (3 a^2 h^4 (4 f g-e h)+3 a c g h^2 \left (11 f g^2-h (4 e g-d h)\right )+2 c^2 \left (10 f g^5-g^3 h (4 e g-d h)\right )\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 )}{2 h^6 \left (c g^2+a h^2\right )}\\ &=-\frac{\left (\left (c g^2+a h^2\right ) \left (20 c f g^2+3 a f h^2-2 c h (4 e g-d h)\right )+c h \left (10 c f g^3-c g h (4 e g-d h)+3 a h^2 (3 f g-e h)\right ) x\right ) \sqrt{a+c x^2}}{2 h^5 \left (c g^2+a h^2\right ) (g+h x)}-\frac{\left (c g \left (4 e g-\frac{10 f g^2}{h}-d h\right )-3 a h (3 f g-e h)-\left (5 c f g^2+3 a f h^2-2 c h (e g-d h)\right ) x\right ) \left (a+c x^2\right )^{3/2}}{6 h^2 \left (c g^2+a h^2\right ) (g+h x)^2}-\frac{\left (f g^2-e g h+d h^2\right ) \left (a+c x^2\right )^{5/2}}{3 h \left (c g^2+a h^2\right ) (g+h x)^3}+\frac{\sqrt{c} \left (20 c f g^2+3 a f h^2-2 c h (4 e g-d h)\right ) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )}{2 h^6}+\frac{c \left (3 a^2 h^4 (4 f g-e h)+3 a c g h^2 \left (11 f g^2-h (4 e g-d h)\right )+2 c^2 \left (10 f g^5-g^3 h (4 e g-d h)\right )\right ) \tanh ^{-1}\left (\frac{a h-c g x}{\sqrt{c g^2+a h^2} \sqrt{a+c x^2}}\right )}{2 h^6 \left (c g^2+a h^2\right )^{3/2}}\\ \end{align*}
Mathematica [A] time = 1.40119, size = 517, normalized size = 1.09 \[ \frac{-\frac{h \sqrt{a+c x^2} \left ((g+h x)^2 \left (6 a^2 f h^4+a c h^2 \left (h (8 d h-23 e g)+50 f g^2\right )+c^2 \left (g^2 h (11 d h-26 e g)+47 f g^4\right )\right )-(g+h x) \left (a h^2+c g^2\right ) \left (-3 a h^2 (e h-2 f g)+c g h (7 d h-10 e g)+13 c f g^3\right )+2 \left (a h^2+c g^2\right )^2 \left (h (d h-e g)+f g^2\right )+6 c (g+h x)^3 \left (a h^2+c g^2\right ) (4 f g-e h)-3 c f h x (g+h x)^3 \left (a h^2+c g^2\right )\right )}{(g+h x)^3 \left (a h^2+c g^2\right )}+\frac{3 c \log \left (\sqrt{a+c x^2} \sqrt{a h^2+c g^2}+a h-c g x\right ) \left (-3 a^2 h^4 (e h-4 f g)+3 a c g h^2 \left (h (d h-4 e g)+11 f g^2\right )+2 c^2 \left (g^3 h (d h-4 e g)+10 f g^5\right )\right )}{\left (a h^2+c g^2\right )^{3/2}}-\frac{3 c \log (g+h x) \left (-3 a^2 h^4 (e h-4 f g)+3 a c g h^2 \left (h (d h-4 e g)+11 f g^2\right )+2 c^2 \left (g^3 h (d h-4 e g)+10 f g^5\right )\right )}{\left (a h^2+c g^2\right )^{3/2}}+3 \sqrt{c} \log \left (\sqrt{c} \sqrt{a+c x^2}+c x\right ) \left (3 a f h^2+2 c h (d h-4 e g)+20 c f g^2\right )}{6 h^6} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.252, size = 9835, normalized size = 20.7 \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 )^{4}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.68426, size = 2565, normalized size = 5.4 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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