Optimal. Leaf size=511 \[ \frac{\left (a+c x^2\right )^{3/2} \left (-3 h x \left (4 a^2 f h^4+a c h^2 \left (17 f g^2-h (5 e g-d h)\right )+2 c^2 g^2 \left (5 f g^2-h (d h+e g)\right )\right )+4 a^2 h^4 (f g-2 e h)-a c g h^2 \left (25 f g^2-h (5 e g-9 d h)\right )-4 c^2 g^4 (5 f g-e h)\right )}{24 h^3 (g+h x)^3 \left (a h^2+c g^2\right )^2}+\frac{c \sqrt{a+c x^2} \left (h x \left (12 a^2 f h^4+a c h^2 \left (35 f g^2-h (7 e g-3 d h)\right )+4 c^2 g^3 (5 f g-e h)\right )+8 \left (a h^2+c g^2\right )^2 (5 f g-e h)\right )}{8 h^5 (g+h x) \left (a h^2+c g^2\right )^2}-\frac{c \tanh ^{-1}\left (\frac{a h-c g x}{\sqrt{a+c x^2} \sqrt{a h^2+c g^2}}\right ) \left (3 a^2 c h^4 \left (25 f g^2-h (5 e g-d h)\right )+12 a^3 f h^6+20 a c^2 g^3 h^2 (5 f g-e h)+8 c^3 g^5 (5 f g-e h)\right )}{8 h^6 \left (a h^2+c g^2\right )^{5/2}}-\frac{c^{3/2} \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right ) (5 f g-e h)}{h^6}-\frac{\left (a+c x^2\right )^{5/2} \left (d h^2-e g h+f g^2\right )}{4 h (g+h x)^4 \left (a h^2+c g^2\right )} \]
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Rubi [A] time = 1.0919, antiderivative size = 511, normalized size of antiderivative = 1., 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, 811, 813, 844, 217, 206, 725} \[ \frac{\left (a+c x^2\right )^{3/2} \left (-3 x \left (4 a^2 f h^4+a c h^2 \left (17 f g^2-h (5 e g-d h)\right )+2 c^2 \left (5 f g^4-g^2 h (d h+e g)\right )\right )+4 a^2 h^3 (f g-2 e h)-a c g h \left (25 f g^2-h (5 e g-9 d h)\right )-\frac{4 c^2 g^4 (5 f g-e h)}{h}\right )}{24 h^2 (g+h x)^3 \left (a h^2+c g^2\right )^2}+\frac{c \sqrt{a+c x^2} \left (h x \left (12 a^2 f h^4+a c h^2 \left (35 f g^2-h (7 e g-3 d h)\right )+4 c^2 g^3 (5 f g-e h)\right )+8 \left (a h^2+c g^2\right )^2 (5 f g-e h)\right )}{8 h^5 (g+h x) \left (a h^2+c g^2\right )^2}-\frac{c \tanh ^{-1}\left (\frac{a h-c g x}{\sqrt{a+c x^2} \sqrt{a h^2+c g^2}}\right ) \left (3 a^2 c h^4 \left (25 f g^2-h (5 e g-d h)\right )+12 a^3 f h^6+20 a c^2 g^3 h^2 (5 f g-e h)+8 c^3 g^5 (5 f g-e h)\right )}{8 h^6 \left (a h^2+c g^2\right )^{5/2}}-\frac{c^{3/2} \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right ) (5 f g-e h)}{h^6}-\frac{\left (a+c x^2\right )^{5/2} \left (d h^2-e g h+f g^2\right )}{4 h (g+h x)^4 \left (a h^2+c g^2\right )} \]
Antiderivative was successfully verified.
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Rule 1651
Rule 811
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)^5} \, dx &=-\frac{\left (f g^2-e g h+d h^2\right ) \left (a+c x^2\right )^{5/2}}{4 h \left (c g^2+a h^2\right ) (g+h x)^4}-\frac{\int \frac{\left (-4 (c d g-a f g+a e h)-\left (4 a f h-c \left (e g-\frac{5 f g^2}{h}-d h\right )\right ) x\right ) \left (a+c x^2\right )^{3/2}}{(g+h x)^4} \, dx}{4 \left (c g^2+a h^2\right )}\\ &=\frac{\left (4 a^2 h^3 (f g-2 e h)-\frac{4 c^2 g^4 (5 f g-e h)}{h}-a c g h \left (25 f g^2-h (5 e g-9 d h)\right )-3 \left (4 a^2 f h^4+a c h^2 \left (17 f g^2-h (5 e g-d h)\right )+2 c^2 \left (5 f g^4-g^2 h (e g+d h)\right )\right ) x\right ) \left (a+c x^2\right )^{3/2}}{24 h^2 \left (c g^2+a h^2\right )^2 (g+h x)^3}-\frac{\left (f g^2-e g h+d h^2\right ) \left (a+c x^2\right )^{5/2}}{4 h \left (c g^2+a h^2\right ) (g+h x)^4}+\frac{\int \frac{\left (-4 a c \left (5 c f g^3-c g h (e g+3 d h)+4 a h^2 (2 f g-e h)\right )+\frac{2 c \left (12 a^2 f h^4+4 c^2 g^3 (5 f g-e h)+a c h^2 \left (35 f g^2-h (7 e g-3 d h)\right )\right ) x}{h}\right ) \sqrt{a+c x^2}}{(g+h x)^2} \, dx}{16 h^2 \left (c g^2+a h^2\right )^2}\\ &=\frac{c \left (8 (5 f g-e h) \left (c g^2+a h^2\right )^2+h \left (12 a^2 f h^4+4 c^2 g^3 (5 f g-e h)+a c h^2 \left (35 f g^2-h (7 e g-3 d h)\right )\right ) x\right ) \sqrt{a+c x^2}}{8 h^5 \left (c g^2+a h^2\right )^2 (g+h x)}+\frac{\left (4 a^2 h^3 (f g-2 e h)-\frac{4 c^2 g^4 (5 f g-e h)}{h}-a c g h \left (25 f g^2-h (5 e g-9 d h)\right )-3 \left (4 a^2 f h^4+a c h^2 \left (17 f g^2-h (5 e g-d h)\right )+2 c^2 \left (5 f g^4-g^2 h (e g+d h)\right )\right ) x\right ) \left (a+c x^2\right )^{3/2}}{24 h^2 \left (c g^2+a h^2\right )^2 (g+h x)^3}-\frac{\left (f g^2-e g h+d h^2\right ) \left (a+c x^2\right )^{5/2}}{4 h \left (c g^2+a h^2\right ) (g+h x)^4}-\frac{\int \frac{-4 a c \left (12 a^2 f h^4+4 c^2 g^3 (5 f g-e h)+a c h^2 \left (35 f g^2-h (7 e g-3 d h)\right )\right )+\frac{32 c^2 (5 f g-e h) \left (c g^2+a h^2\right )^2 x}{h}}{(g+h x) \sqrt{a+c x^2}} \, dx}{32 h^4 \left (c g^2+a h^2\right )^2}\\ &=\frac{c \left (8 (5 f g-e h) \left (c g^2+a h^2\right )^2+h \left (12 a^2 f h^4+4 c^2 g^3 (5 f g-e h)+a c h^2 \left (35 f g^2-h (7 e g-3 d h)\right )\right ) x\right ) \sqrt{a+c x^2}}{8 h^5 \left (c g^2+a h^2\right )^2 (g+h x)}+\frac{\left (4 a^2 h^3 (f g-2 e h)-\frac{4 c^2 g^4 (5 f g-e h)}{h}-a c g h \left (25 f g^2-h (5 e g-9 d h)\right )-3 \left (4 a^2 f h^4+a c h^2 \left (17 f g^2-h (5 e g-d h)\right )+2 c^2 \left (5 f g^4-g^2 h (e g+d h)\right )\right ) x\right ) \left (a+c x^2\right )^{3/2}}{24 h^2 \left (c g^2+a h^2\right )^2 (g+h x)^3}-\frac{\left (f g^2-e g h+d h^2\right ) \left (a+c x^2\right )^{5/2}}{4 h \left (c g^2+a h^2\right ) (g+h x)^4}-\frac{\left (c^2 (5 f g-e h)\right ) \int \frac{1}{\sqrt{a+c x^2}} \, dx}{h^6}+\frac{\left (c \left (12 a^3 f h^6+8 c^3 g^5 (5 f g-e h)+20 a c^2 g^3 h^2 (5 f g-e h)+3 a^2 c h^4 \left (25 f g^2-h (5 e g-d h)\right )\right )\right ) \int \frac{1}{(g+h x) \sqrt{a+c x^2}} \, dx}{8 h^6 \left (c g^2+a h^2\right )^2}\\ &=\frac{c \left (8 (5 f g-e h) \left (c g^2+a h^2\right )^2+h \left (12 a^2 f h^4+4 c^2 g^3 (5 f g-e h)+a c h^2 \left (35 f g^2-h (7 e g-3 d h)\right )\right ) x\right ) \sqrt{a+c x^2}}{8 h^5 \left (c g^2+a h^2\right )^2 (g+h x)}+\frac{\left (4 a^2 h^3 (f g-2 e h)-\frac{4 c^2 g^4 (5 f g-e h)}{h}-a c g h \left (25 f g^2-h (5 e g-9 d h)\right )-3 \left (4 a^2 f h^4+a c h^2 \left (17 f g^2-h (5 e g-d h)\right )+2 c^2 \left (5 f g^4-g^2 h (e g+d h)\right )\right ) x\right ) \left (a+c x^2\right )^{3/2}}{24 h^2 \left (c g^2+a h^2\right )^2 (g+h x)^3}-\frac{\left (f g^2-e g h+d h^2\right ) \left (a+c x^2\right )^{5/2}}{4 h \left (c g^2+a h^2\right ) (g+h x)^4}-\frac{\left (c^2 (5 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 )}{h^6}-\frac{\left (c \left (12 a^3 f h^6+8 c^3 g^5 (5 f g-e h)+20 a c^2 g^3 h^2 (5 f g-e h)+3 a^2 c h^4 \left (25 f g^2-h (5 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 )}{8 h^6 \left (c g^2+a h^2\right )^2}\\ &=\frac{c \left (8 (5 f g-e h) \left (c g^2+a h^2\right )^2+h \left (12 a^2 f h^4+4 c^2 g^3 (5 f g-e h)+a c h^2 \left (35 f g^2-h (7 e g-3 d h)\right )\right ) x\right ) \sqrt{a+c x^2}}{8 h^5 \left (c g^2+a h^2\right )^2 (g+h x)}+\frac{\left (4 a^2 h^3 (f g-2 e h)-\frac{4 c^2 g^4 (5 f g-e h)}{h}-a c g h \left (25 f g^2-h (5 e g-9 d h)\right )-3 \left (4 a^2 f h^4+a c h^2 \left (17 f g^2-h (5 e g-d h)\right )+2 c^2 \left (5 f g^4-g^2 h (e g+d h)\right )\right ) x\right ) \left (a+c x^2\right )^{3/2}}{24 h^2 \left (c g^2+a h^2\right )^2 (g+h x)^3}-\frac{\left (f g^2-e g h+d h^2\right ) \left (a+c x^2\right )^{5/2}}{4 h \left (c g^2+a h^2\right ) (g+h x)^4}-\frac{c^{3/2} (5 f g-e h) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )}{h^6}-\frac{c \left (12 a^3 f h^6+8 c^3 g^5 (5 f g-e h)+20 a c^2 g^3 h^2 (5 f g-e h)+3 a^2 c h^4 \left (25 f g^2-h (5 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 )}{8 h^6 \left (c g^2+a h^2\right )^{5/2}}\\ \end{align*}
Mathematica [A] time = 2.52245, size = 575, normalized size = 1.13 \[ -\frac{\frac{h \sqrt{a+c x^2} \left (-c (g+h x)^3 \left (4 a^2 h^4 (31 f g-8 e h)+a c g h^2 \left (h (15 d h-91 e g)+287 f g^2\right )+2 c^2 \left (g^3 h (3 d h-25 e g)+77 f g^5\right )\right )+(g+h x)^2 \left (a h^2+c g^2\right ) \left (12 a^2 f h^4+a c h^2 \left (h (15 d h-43 e g)+95 f g^2\right )+2 c^2 \left (g^2 h (9 d h-23 e g)+43 f g^4\right )\right )-2 (g+h x) \left (a h^2+c g^2\right )^2 \left (-4 a h^2 (e h-2 f g)+c g h (9 d h-13 e g)+17 c f g^3\right )+6 \left (a h^2+c g^2\right )^3 \left (h (d h-e g)+f g^2\right )-24 c f (g+h x)^4 \left (a h^2+c g^2\right )^2\right )}{(g+h x)^4 \left (a h^2+c g^2\right )^2}+\frac{3 c \log \left (\sqrt{a+c x^2} \sqrt{a h^2+c g^2}+a h-c g x\right ) \left (3 a^2 c h^4 \left (h (d h-5 e g)+25 f g^2\right )+12 a^3 f h^6+20 a c^2 g^3 h^2 (5 f g-e h)+8 c^3 g^5 (5 f g-e h)\right )}{\left (a h^2+c g^2\right )^{5/2}}-\frac{3 c \log (g+h x) \left (3 a^2 c h^4 \left (h (d h-5 e g)+25 f g^2\right )+12 a^3 f h^6+20 a c^2 g^3 h^2 (5 f g-e h)+8 c^3 g^5 (5 f g-e h)\right )}{\left (a h^2+c g^2\right )^{5/2}}+24 c^{3/2} \log \left (\sqrt{c} \sqrt{a+c x^2}+c x\right ) (5 f g-e h)}{24 h^6} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.254, size = 12481, normalized size = 24.4 \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 )^{5}}\, 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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