Optimal. Leaf size=270 \[ \frac{g \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right ) \left (-4 c e g (a e g-b d g+3 b e f)+3 b^2 e^2 g^2+8 c^2 \left (d^2 g^2-3 d e f g+3 e^2 f^2\right )\right )}{8 c^{5/2} e^3}+\frac{3 g^2 \sqrt{a+b x+c x^2} (-b e g-2 c d g+4 c e f)}{4 c^2 e^2}+\frac{(e f-d g)^3 \tanh ^{-1}\left (\frac{-2 a e+x (2 c d-b e)+b d}{2 \sqrt{a+b x+c x^2} \sqrt{a e^2-b d e+c d^2}}\right )}{e^3 \sqrt{a e^2-b d e+c d^2}}+\frac{g^3 (d+e x) \sqrt{a+b x+c x^2}}{2 c e^2} \]
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Rubi [A] time = 0.708168, antiderivative size = 270, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.172, Rules used = {1653, 843, 621, 206, 724} \[ \frac{g \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right ) \left (-4 c e g (a e g-b d g+3 b e f)+3 b^2 e^2 g^2+8 c^2 \left (d^2 g^2-3 d e f g+3 e^2 f^2\right )\right )}{8 c^{5/2} e^3}+\frac{3 g^2 \sqrt{a+b x+c x^2} (-b e g-2 c d g+4 c e f)}{4 c^2 e^2}+\frac{(e f-d g)^3 \tanh ^{-1}\left (\frac{-2 a e+x (2 c d-b e)+b d}{2 \sqrt{a+b x+c x^2} \sqrt{a e^2-b d e+c d^2}}\right )}{e^3 \sqrt{a e^2-b d e+c d^2}}+\frac{g^3 (d+e x) \sqrt{a+b x+c x^2}}{2 c e^2} \]
Antiderivative was successfully verified.
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Rule 1653
Rule 843
Rule 621
Rule 206
Rule 724
Rubi steps
\begin{align*} \int \frac{(f+g x)^3}{(d+e x) \sqrt{a+b x+c x^2}} \, dx &=\frac{g^3 (d+e x) \sqrt{a+b x+c x^2}}{2 c e^2}+\frac{\int \frac{\frac{1}{2} e \left (4 c e^2 f^3-d (b d+2 a e) g^3\right )-e g \left (e (2 b d+a e) g^2-c \left (6 e^2 f^2-d^2 g^2\right )\right ) x+\frac{3}{2} e^2 g^2 (4 c e f-2 c d g-b e g) x^2}{(d+e x) \sqrt{a+b x+c x^2}} \, dx}{2 c e^3}\\ &=\frac{3 g^2 (4 c e f-2 c d g-b e g) \sqrt{a+b x+c x^2}}{4 c^2 e^2}+\frac{g^3 (d+e x) \sqrt{a+b x+c x^2}}{2 c e^2}+\frac{\int \frac{\frac{1}{4} e^3 \left (8 c^2 e^2 f^3+3 b^2 d e g^3-4 c d g^2 (3 b e f-b d g+a e g)\right )+\frac{1}{4} e^3 g \left (3 b^2 e^2 g^2-4 c e g (3 b e f-b d g+a e g)+8 c^2 \left (3 e^2 f^2-3 d e f g+d^2 g^2\right )\right ) x}{(d+e x) \sqrt{a+b x+c x^2}} \, dx}{2 c^2 e^5}\\ &=\frac{3 g^2 (4 c e f-2 c d g-b e g) \sqrt{a+b x+c x^2}}{4 c^2 e^2}+\frac{g^3 (d+e x) \sqrt{a+b x+c x^2}}{2 c e^2}+\frac{(e f-d g)^3 \int \frac{1}{(d+e x) \sqrt{a+b x+c x^2}} \, dx}{e^3}+\frac{\left (g \left (3 b^2 e^2 g^2-4 c e g (3 b e f-b d g+a e g)+8 c^2 \left (3 e^2 f^2-3 d e f g+d^2 g^2\right )\right )\right ) \int \frac{1}{\sqrt{a+b x+c x^2}} \, dx}{8 c^2 e^3}\\ &=\frac{3 g^2 (4 c e f-2 c d g-b e g) \sqrt{a+b x+c x^2}}{4 c^2 e^2}+\frac{g^3 (d+e x) \sqrt{a+b x+c x^2}}{2 c e^2}-\frac{\left (2 (e f-d g)^3\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^3}+\frac{\left (g \left (3 b^2 e^2 g^2-4 c e g (3 b e f-b d g+a e g)+8 c^2 \left (3 e^2 f^2-3 d e f g+d^2 g^2\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 )}{4 c^2 e^3}\\ &=\frac{3 g^2 (4 c e f-2 c d g-b e g) \sqrt{a+b x+c x^2}}{4 c^2 e^2}+\frac{g^3 (d+e x) \sqrt{a+b x+c x^2}}{2 c e^2}+\frac{g \left (3 b^2 e^2 g^2-4 c e g (3 b e f-b d g+a e g)+8 c^2 \left (3 e^2 f^2-3 d e f g+d^2 g^2\right )\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{8 c^{5/2} e^3}+\frac{(e f-d g)^3 \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^3 \sqrt{c d^2-b d e+a e^2}}\\ \end{align*}
Mathematica [A] time = 0.400689, size = 358, normalized size = 1.33 \[ \frac{\frac{e^2 g \left (-4 c g (a g+2 b f)+3 b^2 g^2+8 c^2 f^2\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+x (b+c x)}}\right )}{c^{5/2}}+\frac{4 e g (2 c f-b g) (e f-d g) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+x (b+c x)}}\right )}{c^{3/2}}+\frac{6 e^2 g^2 \sqrt{a+x (b+c x)} (2 c f-b g)}{c^2}+\frac{8 (e f-d g)^3 \tanh ^{-1}\left (\frac{-2 a e+b (d-e x)+2 c d x}{2 \sqrt{a+x (b+c x)} \sqrt{e (a e-b d)+c d^2}}\right )}{\sqrt{e (a e-b d)+c d^2}}+\frac{8 e g^2 \sqrt{a+x (b+c x)} (e f-d g)}{c}+\frac{8 g (e f-d g)^2 \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+x (b+c x)}}\right )}{\sqrt{c}}+\frac{4 e^2 g^2 (f+g x) \sqrt{a+x (b+c x)}}{c}}{8 e^3} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.301, size = 1007, normalized size = 3.7 \begin{align*} \text{result 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 )^{3}}{\left (d + e x\right ) \sqrt{a + b x + c x^{2}}}\, 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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