Optimal. Leaf size=660 \[ \frac{\sqrt{a+b x+c x^2} \left (2 c h x \left (8 c h (2 c g-b h) (b f g-2 c d h)-\left (-4 c h (2 b g-3 a h)-3 b^2 h^2+16 c^2 g^2\right ) (b f h-2 c e h+2 c f g)\right )+6 b^2 c h^3 (-2 a f h-b e h+b f g)-8 b c^2 h^2 \left (3 a h (f g-e h)-2 b \left (d h^2-e g h+f g^2\right )\right )-32 c^3 h (5 b g-4 a h) \left (f g^2-h (e g-d h)\right )+3 b^4 f h^4+128 c^4 g^2 \left (f g^2-h (e g-d h)\right )\right )}{128 c^3 h^5}-\frac{\tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right ) \left (4 c h (2 c g-b h) \left (8 c h (b g-2 a h) (b f g-2 c d h)-g \left (-4 a c h-3 b^2 h+8 b c g\right ) (b f h-2 c e h+2 c f g)\right )-2 \left (-2 c h (b g-a h)-\frac{b^2 h^2}{2}+4 c^2 g^2\right ) \left (8 c h (2 c g-b h) (b f g-2 c d h)-\left (-4 c h (2 b g-3 a h)-3 b^2 h^2+16 c^2 g^2\right ) (b f h-2 c e h+2 c f g)\right )\right )}{256 c^{7/2} h^6}-\frac{\left (a+b x+c x^2\right )^{3/2} (8 c h (b f g-2 c d h)+6 c h x (b f h-2 c e h+2 c f g)-(8 c g-3 b h) (b f h-2 c e h+2 c f g))}{48 c^2 h^3}+\frac{\left (a h^2-b g h+c g^2\right )^{3/2} \left (f g^2-h (e g-d h)\right ) \tanh ^{-1}\left (\frac{-2 a h+x (2 c g-b h)+b g}{2 \sqrt{a+b x+c x^2} \sqrt{a h^2-b g h+c g^2}}\right )}{h^6}+\frac{f \left (a+b x+c x^2\right )^{5/2}}{5 c h} \]
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Rubi [A] time = 1.82506, antiderivative size = 660, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.188, Rules used = {1653, 814, 843, 621, 206, 724} \[ \frac{\sqrt{a+b x+c x^2} \left (2 c h x \left (8 c h (2 c g-b h) (b f g-2 c d h)-\left (-4 c h (2 b g-3 a h)-3 b^2 h^2+16 c^2 g^2\right ) (b f h-2 c e h+2 c f g)\right )+6 b^2 c h^3 (-2 a f h-b e h+b f g)-8 b c^2 h^2 \left (3 a h (f g-e h)-2 b \left (d h^2-e g h+f g^2\right )\right )-32 c^3 h (5 b g-4 a h) \left (f g^2-h (e g-d h)\right )+3 b^4 f h^4+128 c^4 \left (f g^4-g^2 h (e g-d h)\right )\right )}{128 c^3 h^5}-\frac{\tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right ) \left (4 c h (2 c g-b h) \left (8 c h (b g-2 a h) (b f g-2 c d h)-g \left (-4 a c h-3 b^2 h+8 b c g\right ) (b f h-2 c e h+2 c f g)\right )-2 \left (-2 c h (b g-a h)-\frac{b^2 h^2}{2}+4 c^2 g^2\right ) \left (8 c h (2 c g-b h) (b f g-2 c d h)-\left (-4 c h (2 b g-3 a h)-3 b^2 h^2+16 c^2 g^2\right ) (b f h-2 c e h+2 c f g)\right )\right )}{256 c^{7/2} h^6}-\frac{\left (a+b x+c x^2\right )^{3/2} (8 c h (b f g-2 c d h)+6 c h x (b f h-2 c e h+2 c f g)-(8 c g-3 b h) (b f h-2 c e h+2 c f g))}{48 c^2 h^3}+\frac{\left (a h^2-b g h+c g^2\right )^{3/2} \left (f g^2-h (e g-d h)\right ) \tanh ^{-1}\left (\frac{-2 a h+x (2 c g-b h)+b g}{2 \sqrt{a+b x+c x^2} \sqrt{a h^2-b g h+c g^2}}\right )}{h^6}+\frac{f \left (a+b x+c x^2\right )^{5/2}}{5 c h} \]
Antiderivative was successfully verified.
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Rule 1653
Rule 814
Rule 843
Rule 621
Rule 206
Rule 724
Rubi steps
\begin{align*} \int \frac{\left (a+b x+c x^2\right )^{3/2} \left (d+e x+f x^2\right )}{g+h x} \, dx &=\frac{f \left (a+b x+c x^2\right )^{5/2}}{5 c h}+\frac{\int \frac{\left (-\frac{5}{2} h (b f g-2 c d h)-\frac{5}{2} h (2 c f g-2 c e h+b f h) x\right ) \left (a+b x+c x^2\right )^{3/2}}{g+h x} \, dx}{5 c h^2}\\ &=-\frac{(8 c h (b f g-2 c d h)-(8 c g-3 b h) (2 c f g-2 c e h+b f h)+6 c h (2 c f g-2 c e h+b f h) x) \left (a+b x+c x^2\right )^{3/2}}{48 c^2 h^3}+\frac{f \left (a+b x+c x^2\right )^{5/2}}{5 c h}-\frac{\int \frac{\left (-\frac{5}{4} h \left (8 c h (b g-2 a h) (b f g-2 c d h)-g \left (8 b c g-3 b^2 h-4 a c h\right ) (2 c f g-2 c e h+b f h)\right )-\frac{5}{4} h \left (8 c h (2 c g-b h) (b f g-2 c d h)-2 (2 c f g-2 c e h+b f h) \left (8 c^2 g^2-\frac{3 b^2 h^2}{2}-2 c h (2 b g-3 a h)\right )\right ) x\right ) \sqrt{a+b x+c x^2}}{g+h x} \, dx}{40 c^2 h^4}\\ &=\frac{\left (3 b^4 f h^4+6 b^2 c h^3 (b f g-b e h-2 a f h)-32 c^3 h (5 b g-4 a h) \left (f g^2-h (e g-d h)\right )+128 c^4 \left (f g^4-g^2 h (e g-d h)\right )-8 b c^2 h^2 \left (3 a h (f g-e h)-2 b \left (f g^2-e g h+d h^2\right )\right )+2 c h \left (8 c h (2 c g-b h) (b f g-2 c d h)-(2 c f g-2 c e h+b f h) \left (16 c^2 g^2-3 b^2 h^2-4 c h (2 b g-3 a h)\right )\right ) x\right ) \sqrt{a+b x+c x^2}}{128 c^3 h^5}-\frac{(8 c h (b f g-2 c d h)-(8 c g-3 b h) (2 c f g-2 c e h+b f h)+6 c h (2 c f g-2 c e h+b f h) x) \left (a+b x+c x^2\right )^{3/2}}{48 c^2 h^3}+\frac{f \left (a+b x+c x^2\right )^{5/2}}{5 c h}+\frac{\int \frac{-\frac{5}{8} h \left (4 c h (b g-2 a h) \left (8 c h (b g-2 a h) (b f g-2 c d h)-g \left (8 b c g-3 b^2 h-4 a c h\right ) (2 c f g-2 c e h+b f h)\right )-g \left (4 b c g-b^2 h-4 a c h\right ) \left (8 c h (2 c g-b h) (b f g-2 c d h)-(2 c f g-2 c e h+b f h) \left (16 c^2 g^2-3 b^2 h^2-4 c h (2 b g-3 a h)\right )\right )\right )-\frac{5}{8} h \left (4 c h (2 c g-b h) \left (8 c h (b g-2 a h) (b f g-2 c d h)-g \left (8 b c g-3 b^2 h-4 a c h\right ) (2 c f g-2 c e h+b f h)\right )-2 \left (4 c^2 g^2-\frac{b^2 h^2}{2}-2 c h (b g-a h)\right ) \left (8 c h (2 c g-b h) (b f g-2 c d h)-(2 c f g-2 c e h+b f h) \left (16 c^2 g^2-3 b^2 h^2-4 c h (2 b g-3 a h)\right )\right )\right ) x}{(g+h x) \sqrt{a+b x+c x^2}} \, dx}{160 c^3 h^6}\\ &=\frac{\left (3 b^4 f h^4+6 b^2 c h^3 (b f g-b e h-2 a f h)-32 c^3 h (5 b g-4 a h) \left (f g^2-h (e g-d h)\right )+128 c^4 \left (f g^4-g^2 h (e g-d h)\right )-8 b c^2 h^2 \left (3 a h (f g-e h)-2 b \left (f g^2-e g h+d h^2\right )\right )+2 c h \left (8 c h (2 c g-b h) (b f g-2 c d h)-(2 c f g-2 c e h+b f h) \left (16 c^2 g^2-3 b^2 h^2-4 c h (2 b g-3 a h)\right )\right ) x\right ) \sqrt{a+b x+c x^2}}{128 c^3 h^5}-\frac{(8 c h (b f g-2 c d h)-(8 c g-3 b h) (2 c f g-2 c e h+b f h)+6 c h (2 c f g-2 c e h+b f h) x) \left (a+b x+c x^2\right )^{3/2}}{48 c^2 h^3}+\frac{f \left (a+b x+c x^2\right )^{5/2}}{5 c h}+\frac{\left (\left (c g^2-b g h+a h^2\right )^2 \left (f g^2-h (e g-d h)\right )\right ) \int \frac{1}{(g+h x) \sqrt{a+b x+c x^2}} \, dx}{h^6}-\frac{\left (4 c h (2 c g-b h) \left (8 c h (b g-2 a h) (b f g-2 c d h)-g \left (8 b c g-3 b^2 h-4 a c h\right ) (2 c f g-2 c e h+b f h)\right )-2 \left (4 c^2 g^2-\frac{b^2 h^2}{2}-2 c h (b g-a h)\right ) \left (8 c h (2 c g-b h) (b f g-2 c d h)-(2 c f g-2 c e h+b f h) \left (16 c^2 g^2-3 b^2 h^2-4 c h (2 b g-3 a h)\right )\right )\right ) \int \frac{1}{\sqrt{a+b x+c x^2}} \, dx}{256 c^3 h^6}\\ &=\frac{\left (3 b^4 f h^4+6 b^2 c h^3 (b f g-b e h-2 a f h)-32 c^3 h (5 b g-4 a h) \left (f g^2-h (e g-d h)\right )+128 c^4 \left (f g^4-g^2 h (e g-d h)\right )-8 b c^2 h^2 \left (3 a h (f g-e h)-2 b \left (f g^2-e g h+d h^2\right )\right )+2 c h \left (8 c h (2 c g-b h) (b f g-2 c d h)-(2 c f g-2 c e h+b f h) \left (16 c^2 g^2-3 b^2 h^2-4 c h (2 b g-3 a h)\right )\right ) x\right ) \sqrt{a+b x+c x^2}}{128 c^3 h^5}-\frac{(8 c h (b f g-2 c d h)-(8 c g-3 b h) (2 c f g-2 c e h+b f h)+6 c h (2 c f g-2 c e h+b f h) x) \left (a+b x+c x^2\right )^{3/2}}{48 c^2 h^3}+\frac{f \left (a+b x+c x^2\right )^{5/2}}{5 c h}-\frac{\left (2 \left (c g^2-b g h+a h^2\right )^2 \left (f g^2-h (e g-d h)\right )\right ) \operatorname{Subst}\left (\int \frac{1}{4 c g^2-4 b g h+4 a h^2-x^2} \, dx,x,\frac{-b g+2 a h-(2 c g-b h) x}{\sqrt{a+b x+c x^2}}\right )}{h^6}-\frac{\left (4 c h (2 c g-b h) \left (8 c h (b g-2 a h) (b f g-2 c d h)-g \left (8 b c g-3 b^2 h-4 a c h\right ) (2 c f g-2 c e h+b f h)\right )-2 \left (4 c^2 g^2-\frac{b^2 h^2}{2}-2 c h (b g-a h)\right ) \left (8 c h (2 c g-b h) (b f g-2 c d h)-(2 c f g-2 c e h+b f h) \left (16 c^2 g^2-3 b^2 h^2-4 c h (2 b g-3 a h)\right )\right )\right ) \operatorname{Subst}\left (\int \frac{1}{4 c-x^2} \, dx,x,\frac{b+2 c x}{\sqrt{a+b x+c x^2}}\right )}{128 c^3 h^6}\\ &=\frac{\left (3 b^4 f h^4+6 b^2 c h^3 (b f g-b e h-2 a f h)-32 c^3 h (5 b g-4 a h) \left (f g^2-h (e g-d h)\right )+128 c^4 \left (f g^4-g^2 h (e g-d h)\right )-8 b c^2 h^2 \left (3 a h (f g-e h)-2 b \left (f g^2-e g h+d h^2\right )\right )+2 c h \left (8 c h (2 c g-b h) (b f g-2 c d h)-(2 c f g-2 c e h+b f h) \left (16 c^2 g^2-3 b^2 h^2-4 c h (2 b g-3 a h)\right )\right ) x\right ) \sqrt{a+b x+c x^2}}{128 c^3 h^5}-\frac{(8 c h (b f g-2 c d h)-(8 c g-3 b h) (2 c f g-2 c e h+b f h)+6 c h (2 c f g-2 c e h+b f h) x) \left (a+b x+c x^2\right )^{3/2}}{48 c^2 h^3}+\frac{f \left (a+b x+c x^2\right )^{5/2}}{5 c h}-\frac{\left (4 c h (2 c g-b h) \left (8 c h (b g-2 a h) (b f g-2 c d h)-g \left (8 b c g-3 b^2 h-4 a c h\right ) (2 c f g-2 c e h+b f h)\right )-2 \left (4 c^2 g^2-\frac{b^2 h^2}{2}-2 c h (b g-a h)\right ) \left (8 c h (2 c g-b h) (b f g-2 c d h)-(2 c f g-2 c e h+b f h) \left (16 c^2 g^2-3 b^2 h^2-4 c h (2 b g-3 a h)\right )\right )\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{256 c^{7/2} h^6}+\frac{\left (c g^2-b g h+a h^2\right )^{3/2} \left (f g^2-h (e g-d h)\right ) \tanh ^{-1}\left (\frac{b g-2 a h+(2 c g-b h) x}{2 \sqrt{c g^2-b g h+a h^2} \sqrt{a+b x+c x^2}}\right )}{h^6}\\ \end{align*}
Mathematica [A] time = 2.40673, size = 635, normalized size = 0.96 \[ \frac{\sqrt{c} h \sqrt{a+x (b+c x)} \left (6 b^2 c h^3 (-2 a f h-b e h+b f (g+h x))-16 c^3 h \left (a h (h (-8 d h+8 e g-3 e h x)+f g (3 h x-8 g))+2 b (5 g-h x) \left (h (d h-e g)+f g^2\right )\right )+4 b c^2 h^2 \left (6 a e h^2-6 a f h (g+h x)+b h (4 d h-4 e g-3 e h x)+b f g (4 g+3 h x)\right )+3 b^4 f h^4+64 c^4 g (2 g-h x) \left (h (d h-e g)+f g^2\right )\right )-\tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+x (b+c x)}}\right ) \left (\left (2 c h (b g-a h)+\frac{b^2 h^2}{2}-4 c^2 g^2\right ) \left (8 c h (2 c g-b h) (b f g-2 c d h)-\left (4 c h (3 a h-2 b g)-3 b^2 h^2+16 c^2 g^2\right ) (b f h-2 c e h+2 c f g)\right )+2 c h (2 c g-b h) \left (8 c h (b g-2 a h) (b f g-2 c d h)-g \left (-4 a c h-3 b^2 h+8 b c g\right ) (b f h-2 c e h+2 c f g)\right )\right )-128 c^{7/2} \left (h (a h-b g)+c g^2\right )^{3/2} \left (h (d h-e g)+f g^2\right ) \tanh ^{-1}\left (\frac{2 a h-b g+b h x-2 c g x}{2 \sqrt{a+x (b+c x)} \sqrt{h (a h-b g)+c g^2}}\right )}{128 c^{7/2} h^6}-\frac{(a+x (b+c x))^{3/2} \left (3 b^2 f h^2+6 b c h (f (g+h x)-e h)-4 c^2 (h (4 d h-4 e g+3 e h x)+f g (4 g-3 h x))\right )}{48 c^2 h^3}+\frac{f (a+x (b+c x))^{5/2}}{5 c h} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.263, size = 6715, normalized size = 10.2 \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 + b x + c x^{2}\right )^{\frac{3}{2}} \left (d + e x + f x^{2}\right )}{g + h x}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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