Optimal. Leaf size=662 \[ -\frac{\tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right ) \left (16 b c^2 e^3 \left (3 a^2 e^2 g^2+3 a b e g (2 e f-d g)+b^2 (e f-d g)^2\right )+96 c^3 e^2 \left (-a^2 e^2 g (2 e f-d g)-2 a b e (e f-d g)^2+b^2 d (e f-d g)^2\right )-6 b^3 c e^4 g (4 a e g-b d g+2 b e f)-384 c^4 d e (b d-a e) (e f-d g)^2+3 b^5 e^5 g^2+256 c^5 d^3 (e f-d g)^2\right )}{256 c^{7/2} e^6}+\frac{\sqrt{a+b x+c x^2} \left (2 c e x \left (g \left (-4 c e (2 b d-3 a e)-3 b^2 e^2+16 c^2 d^2\right ) (-b e g-2 c d g+4 c e f)-8 c e (2 c d-b e) \left (2 c e f^2-b d g^2\right )\right )-6 b^2 c e^3 g (2 a e g-b d g+2 b e f)+8 b c^2 e^2 \left (3 a e g (2 e f-d g)+2 b (e f-d g)^2\right )-32 c^3 e (5 b d-4 a e) (e f-d g)^2+3 b^4 e^4 g^2+128 c^4 d^2 (e f-d g)^2\right )}{128 c^3 e^5}-\frac{\left (a+b x+c x^2\right )^{3/2} \left (3 b^2 e^2 g^2-6 c e g x (-b e g-2 c d g+4 c e f)-6 b c e g (2 e f-d g)-16 c^2 (e f-d g)^2\right )}{48 c^2 e^3}+\frac{(e f-d g)^2 \left (a e^2-b d e+c d^2\right )^{3/2} \tanh ^{-1}\left (\frac{-2 a e+x (2 c d-b e)+b d}{2 \sqrt{a+b x+c x^2} \sqrt{a e^2-b d e+c d^2}}\right )}{e^6}+\frac{g^2 \left (a+b x+c x^2\right )^{5/2}}{5 c e} \]
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Rubi [A] time = 1.55408, antiderivative size = 662, normalized size of antiderivative = 1., 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 = {1653, 814, 843, 621, 206, 724} \[ -\frac{\tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right ) \left (16 b c^2 e^3 \left (3 a^2 e^2 g^2+3 a b e g (2 e f-d g)+b^2 (e f-d g)^2\right )+96 c^3 e^2 \left (-a^2 e^2 g (2 e f-d g)-2 a b e (e f-d g)^2+b^2 d (e f-d g)^2\right )-6 b^3 c e^4 g (4 a e g-b d g+2 b e f)-384 c^4 d e (b d-a e) (e f-d g)^2+3 b^5 e^5 g^2+256 c^5 d^3 (e f-d g)^2\right )}{256 c^{7/2} e^6}+\frac{\sqrt{a+b x+c x^2} \left (2 c e x \left (g \left (-4 c e (2 b d-3 a e)-3 b^2 e^2+16 c^2 d^2\right ) (-b e g-2 c d g+4 c e f)-8 c e (2 c d-b e) \left (2 c e f^2-b d g^2\right )\right )-6 b^2 c e^3 g (2 a e g-b d g+2 b e f)+8 b c^2 e^2 \left (3 a e g (2 e f-d g)+2 b (e f-d g)^2\right )-32 c^3 e (5 b d-4 a e) (e f-d g)^2+3 b^4 e^4 g^2+128 c^4 d^2 (e f-d g)^2\right )}{128 c^3 e^5}-\frac{\left (a+b x+c x^2\right )^{3/2} \left (3 b^2 e^2 g^2-6 c e g x (-b e g-2 c d g+4 c e f)-6 b c e g (2 e f-d g)-16 c^2 (e f-d g)^2\right )}{48 c^2 e^3}+\frac{(e f-d g)^2 \left (a e^2-b d e+c d^2\right )^{3/2} \tanh ^{-1}\left (\frac{-2 a e+x (2 c d-b e)+b d}{2 \sqrt{a+b x+c x^2} \sqrt{a e^2-b d e+c d^2}}\right )}{e^6}+\frac{g^2 \left (a+b x+c x^2\right )^{5/2}}{5 c e} \]
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{(f+g x)^2 \left (a+b x+c x^2\right )^{3/2}}{d+e x} \, dx &=\frac{g^2 \left (a+b x+c x^2\right )^{5/2}}{5 c e}+\frac{\int \frac{\left (\frac{5}{2} e \left (2 c e f^2-b d g^2\right )+\frac{5}{2} e g (4 c e f-2 c d g-b e g) x\right ) \left (a+b x+c x^2\right )^{3/2}}{d+e x} \, dx}{5 c e^2}\\ &=-\frac{\left (3 b^2 e^2 g^2-16 c^2 (e f-d g)^2-6 b c e g (2 e f-d g)-6 c e g (4 c e f-2 c d g-b e g) x\right ) \left (a+b x+c x^2\right )^{3/2}}{48 c^2 e^3}+\frac{g^2 \left (a+b x+c x^2\right )^{5/2}}{5 c e}-\frac{\int \frac{\left (-\frac{5}{4} e \left (d \left (8 b c d-3 b^2 e-4 a c e\right ) g (4 c e f-2 c d g-b e g)-8 c e (b d-2 a e) \left (2 c e f^2-b d g^2\right )\right )-\frac{5}{4} e \left (\left (16 c^2 d^2-3 b^2 e^2-4 c e (2 b d-3 a e)\right ) g (4 c e f-2 c d g-b e g)-8 c e (2 c d-b e) \left (2 c e f^2-b d g^2\right )\right ) x\right ) \sqrt{a+b x+c x^2}}{d+e x} \, dx}{40 c^2 e^4}\\ &=\frac{\left (3 b^4 e^4 g^2+128 c^4 d^2 (e f-d g)^2-32 c^3 e (5 b d-4 a e) (e f-d g)^2-6 b^2 c e^3 g (2 b e f-b d g+2 a e g)+8 b c^2 e^2 \left (2 b (e f-d g)^2+3 a e g (2 e f-d g)\right )+2 c e \left (\left (16 c^2 d^2-3 b^2 e^2-4 c e (2 b d-3 a e)\right ) g (4 c e f-2 c d g-b e g)-8 c e (2 c d-b e) \left (2 c e f^2-b d g^2\right )\right ) x\right ) \sqrt{a+b x+c x^2}}{128 c^3 e^5}-\frac{\left (3 b^2 e^2 g^2-16 c^2 (e f-d g)^2-6 b c e g (2 e f-d g)-6 c e g (4 c e f-2 c d g-b e g) x\right ) \left (a+b x+c x^2\right )^{3/2}}{48 c^2 e^3}+\frac{g^2 \left (a+b x+c x^2\right )^{5/2}}{5 c e}+\frac{\int \frac{-\frac{5}{8} e \left (4 c e (b d-2 a e) \left (d \left (8 b c d-3 b^2 e-4 a c e\right ) g (4 c e f-2 c d g-b e g)-8 c e (b d-2 a e) \left (2 c e f^2-b d g^2\right )\right )-d \left (4 b c d-b^2 e-4 a c e\right ) \left (\left (16 c^2 d^2-3 b^2 e^2-4 c e (2 b d-3 a e)\right ) g (4 c e f-2 c d g-b e g)-8 c e (2 c d-b e) \left (2 c e f^2-b d g^2\right )\right )\right )-\frac{5}{8} e \left (4 c e (2 c d-b e) \left (d \left (8 b c d-3 b^2 e-4 a c e\right ) g (4 c e f-2 c d g-b e g)-8 c e (b d-2 a e) \left (2 c e f^2-b d g^2\right )\right )-2 \left (4 c^2 d^2-\frac{b^2 e^2}{2}-2 c e (b d-a e)\right ) \left (\left (16 c^2 d^2-3 b^2 e^2-4 c e (2 b d-3 a e)\right ) g (4 c e f-2 c d g-b e g)-8 c e (2 c d-b e) \left (2 c e f^2-b d g^2\right )\right )\right ) x}{(d+e x) \sqrt{a+b x+c x^2}} \, dx}{160 c^3 e^6}\\ &=\frac{\left (3 b^4 e^4 g^2+128 c^4 d^2 (e f-d g)^2-32 c^3 e (5 b d-4 a e) (e f-d g)^2-6 b^2 c e^3 g (2 b e f-b d g+2 a e g)+8 b c^2 e^2 \left (2 b (e f-d g)^2+3 a e g (2 e f-d g)\right )+2 c e \left (\left (16 c^2 d^2-3 b^2 e^2-4 c e (2 b d-3 a e)\right ) g (4 c e f-2 c d g-b e g)-8 c e (2 c d-b e) \left (2 c e f^2-b d g^2\right )\right ) x\right ) \sqrt{a+b x+c x^2}}{128 c^3 e^5}-\frac{\left (3 b^2 e^2 g^2-16 c^2 (e f-d g)^2-6 b c e g (2 e f-d g)-6 c e g (4 c e f-2 c d g-b e g) x\right ) \left (a+b x+c x^2\right )^{3/2}}{48 c^2 e^3}+\frac{g^2 \left (a+b x+c x^2\right )^{5/2}}{5 c e}+\frac{\left (\left (c d^2-b d e+a e^2\right )^2 (e f-d g)^2\right ) \int \frac{1}{(d+e x) \sqrt{a+b x+c x^2}} \, dx}{e^6}-\frac{\left (3 b^5 e^5 g^2+256 c^5 d^3 (e f-d g)^2-384 c^4 d e (b d-a e) (e f-d g)^2-6 b^3 c e^4 g (2 b e f-b d g+4 a e g)+16 b c^2 e^3 \left (3 a^2 e^2 g^2+b^2 (e f-d g)^2+3 a b e g (2 e f-d g)\right )+96 c^3 e^2 \left (b^2 d (e f-d g)^2-2 a b e (e f-d g)^2-a^2 e^2 g (2 e f-d g)\right )\right ) \int \frac{1}{\sqrt{a+b x+c x^2}} \, dx}{256 c^3 e^6}\\ &=\frac{\left (3 b^4 e^4 g^2+128 c^4 d^2 (e f-d g)^2-32 c^3 e (5 b d-4 a e) (e f-d g)^2-6 b^2 c e^3 g (2 b e f-b d g+2 a e g)+8 b c^2 e^2 \left (2 b (e f-d g)^2+3 a e g (2 e f-d g)\right )+2 c e \left (\left (16 c^2 d^2-3 b^2 e^2-4 c e (2 b d-3 a e)\right ) g (4 c e f-2 c d g-b e g)-8 c e (2 c d-b e) \left (2 c e f^2-b d g^2\right )\right ) x\right ) \sqrt{a+b x+c x^2}}{128 c^3 e^5}-\frac{\left (3 b^2 e^2 g^2-16 c^2 (e f-d g)^2-6 b c e g (2 e f-d g)-6 c e g (4 c e f-2 c d g-b e g) x\right ) \left (a+b x+c x^2\right )^{3/2}}{48 c^2 e^3}+\frac{g^2 \left (a+b x+c x^2\right )^{5/2}}{5 c e}-\frac{\left (2 \left (c d^2-b d e+a e^2\right )^2 (e f-d g)^2\right ) \operatorname{Subst}\left (\int \frac{1}{4 c d^2-4 b d e+4 a e^2-x^2} \, dx,x,\frac{-b d+2 a e-(2 c d-b e) x}{\sqrt{a+b x+c x^2}}\right )}{e^6}-\frac{\left (3 b^5 e^5 g^2+256 c^5 d^3 (e f-d g)^2-384 c^4 d e (b d-a e) (e f-d g)^2-6 b^3 c e^4 g (2 b e f-b d g+4 a e g)+16 b c^2 e^3 \left (3 a^2 e^2 g^2+b^2 (e f-d g)^2+3 a b e g (2 e f-d g)\right )+96 c^3 e^2 \left (b^2 d (e f-d g)^2-2 a b e (e f-d g)^2-a^2 e^2 g (2 e f-d g)\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 e^6}\\ &=\frac{\left (3 b^4 e^4 g^2+128 c^4 d^2 (e f-d g)^2-32 c^3 e (5 b d-4 a e) (e f-d g)^2-6 b^2 c e^3 g (2 b e f-b d g+2 a e g)+8 b c^2 e^2 \left (2 b (e f-d g)^2+3 a e g (2 e f-d g)\right )+2 c e \left (\left (16 c^2 d^2-3 b^2 e^2-4 c e (2 b d-3 a e)\right ) g (4 c e f-2 c d g-b e g)-8 c e (2 c d-b e) \left (2 c e f^2-b d g^2\right )\right ) x\right ) \sqrt{a+b x+c x^2}}{128 c^3 e^5}-\frac{\left (3 b^2 e^2 g^2-16 c^2 (e f-d g)^2-6 b c e g (2 e f-d g)-6 c e g (4 c e f-2 c d g-b e g) x\right ) \left (a+b x+c x^2\right )^{3/2}}{48 c^2 e^3}+\frac{g^2 \left (a+b x+c x^2\right )^{5/2}}{5 c e}-\frac{\left (3 b^5 e^5 g^2+256 c^5 d^3 (e f-d g)^2-384 c^4 d e (b d-a e) (e f-d g)^2-6 b^3 c e^4 g (2 b e f-b d g+4 a e g)+16 b c^2 e^3 \left (3 a^2 e^2 g^2+b^2 (e f-d g)^2+3 a b e g (2 e f-d g)\right )+96 c^3 e^2 \left (b^2 d (e f-d g)^2-2 a b e (e f-d g)^2-a^2 e^2 g (2 e f-d g)\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} e^6}+\frac{\left (c d^2-b d e+a e^2\right )^{3/2} (e f-d g)^2 \tanh ^{-1}\left (\frac{b d-2 a e+(2 c d-b e) x}{2 \sqrt{c d^2-b d e+a e^2} \sqrt{a+b x+c x^2}}\right )}{e^6}\\ \end{align*}
Mathematica [A] time = 1.32335, size = 536, normalized size = 0.81 \[ \frac{\frac{240 (e f-d g)^2 \left (-(2 c d-b e) \left (4 c e (3 a e-2 b d)-b^2 e^2+8 c^2 d^2\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+x (b+c x)}}\right )-2 \sqrt{c} \left (e \sqrt{a+x (b+c x)} \left (-2 c e (4 a e-5 b d+b e x)-b^2 e^2+4 c^2 d (e x-2 d)\right )+8 c \left (e (a e-b d)+c d^2\right )^{3/2} \tanh ^{-1}\left (\frac{2 a e-b d+b e x-2 c d x}{2 \sqrt{a+x (b+c x)} \sqrt{e (a e-b d)+c d^2}}\right )\right )\right )}{c^{3/2} e^3}+\frac{90 e g \left (b^2-4 a c\right ) (e f-d g) \left (\left (b^2-4 a c\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+x (b+c x)}}\right )-2 \sqrt{c} (b+2 c x) \sqrt{a+x (b+c x)}\right )}{c^{5/2}}+\frac{15 e^2 g (2 c f-b g) \left (\frac{3 \left (b^2-4 a c\right ) \left (\left (b^2-4 a c\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+x (b+c x)}}\right )-2 \sqrt{c} (b+2 c x) \sqrt{a+x (b+c x)}\right )}{c^{5/2}}+\frac{16 (b+2 c x) (a+x (b+c x))^{3/2}}{c}\right )}{c}+1280 (a+x (b+c x))^{3/2} (e f-d g)^2+\frac{480 e g (b+2 c x) (a+x (b+c x))^{3/2} (e f-d g)}{c}+\frac{768 e^2 g^2 (a+x (b+c x))^{5/2}}{c}}{3840 e^3} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.287, size = 6860, normalized size = 10.4 \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 (f + g x\right )^{2} \left (a + b x + c x^{2}\right )^{\frac{3}{2}}}{d + e 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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