Optimal. Leaf size=488 \[ \frac{\sqrt{a+c x^2} \left (2 a^2 f h^4-c h x \left (a h^2 (7 f g-3 e h)+c g \left (10 f g^2-3 h (2 e g-d h)\right )\right )+a c h^2 \left (19 f g^2-3 h (3 e g-d h)\right )+2 c^2 g^2 \left (10 f g^2-3 h (2 e g-d h)\right )\right )}{2 h^5 \left (a h^2+c g^2\right )}-\frac{\tanh ^{-1}\left (\frac{a h-c g x}{\sqrt{a+c x^2} \sqrt{a h^2+c g^2}}\right ) \left (2 a^2 f h^4+a c h^2 \left (19 f g^2-3 h (3 e g-d h)\right )+2 c^2 g^2 \left (10 f g^2-3 h (2 e g-d h)\right )\right )}{2 h^6 \sqrt{a h^2+c g^2}}-\frac{\left (a+c x^2\right )^{5/2} \left (d h^2-e g h+f g^2\right )}{2 h (g+h x)^2 \left (a h^2+c g^2\right )}-\frac{\left (a+c x^2\right )^{3/2} \left (2 \left (c g \left (-3 d h+6 e g-\frac{10 f g^2}{h}\right )-a h (7 f g-3 e h)\right )-x \left (2 a f h^2+c \left (5 f g^2-3 h (e g-d h)\right )\right )\right )}{6 h^2 (g+h x) \left (a h^2+c g^2\right )}-\frac{\sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right ) \left (3 a h^2 (3 f g-e h)+2 c g \left (10 f g^2-3 h (2 e g-d h)\right )\right )}{2 h^6} \]
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Rubi [A] time = 0.921043, antiderivative size = 480, normalized size of antiderivative = 0.98, number of steps used = 8, number of rules used = 7, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.241, Rules used = {1651, 813, 815, 844, 217, 206, 725} \[ \frac{\sqrt{a+c x^2} \left (2 a^2 f h^3-c x \left (a h^2 (7 f g-3 e h)-3 c g h (2 e g-d h)+10 c f g^3\right )+a c h \left (19 f g^2-3 h (3 e g-d h)\right )-2 c^2 g^2 \left (-3 d h+6 e g-\frac{10 f g^2}{h}\right )\right )}{2 h^4 \left (a h^2+c g^2\right )}-\frac{\tanh ^{-1}\left (\frac{a h-c g x}{\sqrt{a+c x^2} \sqrt{a h^2+c g^2}}\right ) \left (2 a^2 f h^4+a c h^2 \left (19 f g^2-3 h (3 e g-d h)\right )+2 c^2 \left (10 f g^4-3 g^2 h (2 e g-d h)\right )\right )}{2 h^6 \sqrt{a h^2+c g^2}}-\frac{\left (a+c x^2\right )^{5/2} \left (d h^2-e g h+f g^2\right )}{2 h (g+h x)^2 \left (a h^2+c g^2\right )}-\frac{\left (a+c x^2\right )^{3/2} \left (2 \left (c g \left (-3 d h+6 e g-\frac{10 f g^2}{h}\right )-a h (7 f g-3 e h)\right )-x \left (2 a f h^2-3 c h (e g-d h)+5 c f g^2\right )\right )}{6 h^2 (g+h x) \left (a h^2+c g^2\right )}-\frac{\sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right ) \left (3 a h^2 (3 f g-e h)-6 c g h (2 e g-d h)+20 c f g^3\right )}{2 h^6} \]
Antiderivative was successfully verified.
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Rule 1651
Rule 813
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)^3} \, dx &=-\frac{\left (f g^2-e g h+d h^2\right ) \left (a+c x^2\right )^{5/2}}{2 h \left (c g^2+a h^2\right ) (g+h x)^2}-\frac{\int \frac{\left (-2 (c d g-a f g+a e h)-\left (2 a f h-c \left (3 e g-\frac{5 f g^2}{h}-3 d h\right )\right ) x\right ) \left (a+c x^2\right )^{3/2}}{(g+h x)^2} \, dx}{2 \left (c g^2+a h^2\right )}\\ &=-\frac{\left (2 \left (c g \left (6 e g-\frac{10 f g^2}{h}-3 d h\right )-a h (7 f g-3 e h)\right )-\left (5 c f g^2+2 a f h^2-3 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)}-\frac{\left (f g^2-e g h+d h^2\right ) \left (a+c x^2\right )^{5/2}}{2 h \left (c g^2+a h^2\right ) (g+h x)^2}+\frac{\int \frac{\left (2 a \left (5 c f g^2+2 a f h^2-3 c h (e g-d h)\right )-\frac{4 c \left (10 c f g^3-3 c g h (2 e g-d h)+a h^2 (7 f g-3 e h)\right ) x}{h}\right ) \sqrt{a+c x^2}}{g+h x} \, dx}{4 h^2 \left (c g^2+a h^2\right )}\\ &=\frac{\left (2 a^2 f h^3-2 c^2 g^2 \left (6 e g-\frac{10 f g^2}{h}-3 d h\right )+a c h \left (19 f g^2-3 h (3 e g-d h)\right )-c \left (10 c f g^3-3 c g h (2 e g-d h)+a h^2 (7 f g-3 e h)\right ) x\right ) \sqrt{a+c x^2}}{2 h^4 \left (c g^2+a h^2\right )}-\frac{\left (2 \left (c g \left (6 e g-\frac{10 f g^2}{h}-3 d h\right )-a h (7 f g-3 e h)\right )-\left (5 c f g^2+2 a f h^2-3 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)}-\frac{\left (f g^2-e g h+d h^2\right ) \left (a+c x^2\right )^{5/2}}{2 h \left (c g^2+a h^2\right ) (g+h x)^2}+\frac{\int \frac{4 a c \left (c g^2+a h^2\right ) \left (10 c f g^2+2 a f h^2-3 c h (2 e g-d h)\right )-\frac{4 c^2 \left (c g^2+a h^2\right ) \left (20 c f g^3-6 c g h (2 e g-d h)+3 a h^2 (3 f g-e h)\right ) x}{h}}{(g+h x) \sqrt{a+c x^2}} \, dx}{8 c h^4 \left (c g^2+a h^2\right )}\\ &=\frac{\left (2 a^2 f h^3-2 c^2 g^2 \left (6 e g-\frac{10 f g^2}{h}-3 d h\right )+a c h \left (19 f g^2-3 h (3 e g-d h)\right )-c \left (10 c f g^3-3 c g h (2 e g-d h)+a h^2 (7 f g-3 e h)\right ) x\right ) \sqrt{a+c x^2}}{2 h^4 \left (c g^2+a h^2\right )}-\frac{\left (2 \left (c g \left (6 e g-\frac{10 f g^2}{h}-3 d h\right )-a h (7 f g-3 e h)\right )-\left (5 c f g^2+2 a f h^2-3 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)}-\frac{\left (f g^2-e g h+d h^2\right ) \left (a+c x^2\right )^{5/2}}{2 h \left (c g^2+a h^2\right ) (g+h x)^2}-\frac{\left (c \left (20 c f g^3-6 c g h (2 e g-d h)+3 a h^2 (3 f g-e h)\right )\right ) \int \frac{1}{\sqrt{a+c x^2}} \, dx}{2 h^6}+\frac{\left (2 a^2 f h^4+2 c^2 \left (10 f g^4-3 g^2 h (2 e g-d h)\right )+a c h^2 \left (19 f g^2-3 h (3 e g-d h)\right )\right ) \int \frac{1}{(g+h x) \sqrt{a+c x^2}} \, dx}{2 h^6}\\ &=\frac{\left (2 a^2 f h^3-2 c^2 g^2 \left (6 e g-\frac{10 f g^2}{h}-3 d h\right )+a c h \left (19 f g^2-3 h (3 e g-d h)\right )-c \left (10 c f g^3-3 c g h (2 e g-d h)+a h^2 (7 f g-3 e h)\right ) x\right ) \sqrt{a+c x^2}}{2 h^4 \left (c g^2+a h^2\right )}-\frac{\left (2 \left (c g \left (6 e g-\frac{10 f g^2}{h}-3 d h\right )-a h (7 f g-3 e h)\right )-\left (5 c f g^2+2 a f h^2-3 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)}-\frac{\left (f g^2-e g h+d h^2\right ) \left (a+c x^2\right )^{5/2}}{2 h \left (c g^2+a h^2\right ) (g+h x)^2}-\frac{\left (c \left (20 c f g^3-6 c g h (2 e g-d h)+3 a h^2 (3 f g-e 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 (2 a^2 f h^4+2 c^2 \left (10 f g^4-3 g^2 h (2 e g-d h)\right )+a c h^2 \left (19 f g^2-3 h (3 e g-d 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 )}{2 h^6}\\ &=\frac{\left (2 a^2 f h^3-2 c^2 g^2 \left (6 e g-\frac{10 f g^2}{h}-3 d h\right )+a c h \left (19 f g^2-3 h (3 e g-d h)\right )-c \left (10 c f g^3-3 c g h (2 e g-d h)+a h^2 (7 f g-3 e h)\right ) x\right ) \sqrt{a+c x^2}}{2 h^4 \left (c g^2+a h^2\right )}-\frac{\left (2 \left (c g \left (6 e g-\frac{10 f g^2}{h}-3 d h\right )-a h (7 f g-3 e h)\right )-\left (5 c f g^2+2 a f h^2-3 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)}-\frac{\left (f g^2-e g h+d h^2\right ) \left (a+c x^2\right )^{5/2}}{2 h \left (c g^2+a h^2\right ) (g+h x)^2}-\frac{\sqrt{c} \left (20 c f g^3-6 c g h (2 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 h^6}-\frac{\left (2 a^2 f h^4+2 c^2 \left (10 f g^4-3 g^2 h (2 e g-d h)\right )+a c h^2 \left (19 f g^2-3 h (3 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 \sqrt{c g^2+a h^2}}\\ \end{align*}
Mathematica [A] time = 0.706477, size = 435, normalized size = 0.89 \[ \frac{-\frac{3 \log \left (\sqrt{a+c x^2} \sqrt{a h^2+c g^2}+a h-c g x\right ) \left (2 a^2 f h^4+a c h^2 \left (3 h (d h-3 e g)+19 f g^2\right )+2 c^2 \left (3 g^2 h (d h-2 e g)+10 f g^4\right )\right )}{\sqrt{a h^2+c g^2}}+\frac{3 \log (g+h x) \left (2 a^2 f h^4+a c h^2 \left (3 h (d h-3 e g)+19 f g^2\right )+2 c^2 \left (3 g^2 h (d h-2 e g)+10 f g^4\right )\right )}{\sqrt{a h^2+c g^2}}+\frac{h \sqrt{a+c x^2} \left (a h^2 \left (f \left (17 g^2+28 g h x+8 h^2 x^2\right )-3 h (d h+e g+2 e h x)\right )+c \left (3 h \left (d h \left (6 g^2+9 g h x+2 h^2 x^2\right )+e \left (-18 g^2 h x-12 g^3-4 g h^2 x^2+h^3 x^3\right )\right )+f \left (20 g^2 h^2 x^2+90 g^3 h x+60 g^4-5 g h^3 x^3+2 h^4 x^4\right )\right )\right )}{(g+h x)^2}-3 \sqrt{c} \log \left (\sqrt{c} \sqrt{a+c x^2}+c x\right ) \left (-3 a h^2 (e h-3 f g)+6 c g h (d h-2 e g)+20 c f g^3\right )}{6 h^6} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.239, size = 7817, normalized size = 16. \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 )^{3}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.4696, size = 1399, normalized size = 2.87 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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