Optimal. Leaf size=127 \[ \frac{\left (-4 c e (a e+2 b d)+3 b^2 e^2+8 c^2 d^2\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{8 c^{5/2}}+\frac{3 e \sqrt{a+b x+c x^2} (2 c d-b e)}{4 c^2}+\frac{e (d+e x) \sqrt{a+b x+c x^2}}{2 c} \]
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Rubi [A] time = 0.116312, antiderivative size = 127, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {742, 640, 621, 206} \[ \frac{\left (-4 c e (a e+2 b d)+3 b^2 e^2+8 c^2 d^2\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{8 c^{5/2}}+\frac{3 e \sqrt{a+b x+c x^2} (2 c d-b e)}{4 c^2}+\frac{e (d+e x) \sqrt{a+b x+c x^2}}{2 c} \]
Antiderivative was successfully verified.
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Rule 742
Rule 640
Rule 621
Rule 206
Rubi steps
\begin{align*} \int \frac{(d+e x)^2}{\sqrt{a+b x+c x^2}} \, dx &=\frac{e (d+e x) \sqrt{a+b x+c x^2}}{2 c}+\frac{\int \frac{\frac{1}{2} \left (4 c d^2-e (b d+2 a e)\right )+\frac{3}{2} e (2 c d-b e) x}{\sqrt{a+b x+c x^2}} \, dx}{2 c}\\ &=\frac{3 e (2 c d-b e) \sqrt{a+b x+c x^2}}{4 c^2}+\frac{e (d+e x) \sqrt{a+b x+c x^2}}{2 c}+\frac{\left (-\frac{3}{2} b e (2 c d-b e)+c \left (4 c d^2-e (b d+2 a e)\right )\right ) \int \frac{1}{\sqrt{a+b x+c x^2}} \, dx}{4 c^2}\\ &=\frac{3 e (2 c d-b e) \sqrt{a+b x+c x^2}}{4 c^2}+\frac{e (d+e x) \sqrt{a+b x+c x^2}}{2 c}+\frac{\left (-\frac{3}{2} b e (2 c d-b e)+c \left (4 c d^2-e (b d+2 a e)\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 )}{2 c^2}\\ &=\frac{3 e (2 c d-b e) \sqrt{a+b x+c x^2}}{4 c^2}+\frac{e (d+e x) \sqrt{a+b x+c x^2}}{2 c}+\frac{\left (8 c^2 d^2+3 b^2 e^2-4 c e (2 b d+a e)\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{8 c^{5/2}}\\ \end{align*}
Mathematica [A] time = 0.120862, size = 103, normalized size = 0.81 \[ \frac{\left (-4 c e (a e+2 b d)+3 b^2 e^2+8 c^2 d^2\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+x (b+c x)}}\right )}{8 c^{5/2}}+\frac{e \sqrt{a+x (b+c x)} (-3 b e+8 c d+2 c e x)}{4 c^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.049, size = 198, normalized size = 1.6 \begin{align*}{\frac{{e}^{2}x}{2\,c}\sqrt{c{x}^{2}+bx+a}}-{\frac{3\,b{e}^{2}}{4\,{c}^{2}}\sqrt{c{x}^{2}+bx+a}}+{\frac{3\,{b}^{2}{e}^{2}}{8}\ln \left ({ \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx+a} \right ){c}^{-{\frac{5}{2}}}}-{\frac{a{e}^{2}}{2}\ln \left ({ \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx+a} \right ){c}^{-{\frac{3}{2}}}}+2\,{\frac{de\sqrt{c{x}^{2}+bx+a}}{c}}-{bde\ln \left ({ \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx+a} \right ){c}^{-{\frac{3}{2}}}}+{{d}^{2}\ln \left ({ \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx+a} \right ){\frac{1}{\sqrt{c}}}} \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 [A] time = 2.52757, size = 582, normalized size = 4.58 \begin{align*} \left [-\frac{{\left (8 \, c^{2} d^{2} - 8 \, b c d e +{\left (3 \, b^{2} - 4 \, a c\right )} e^{2}\right )} \sqrt{c} \log \left (-8 \, c^{2} x^{2} - 8 \, b c x - b^{2} + 4 \, \sqrt{c x^{2} + b x + a}{\left (2 \, c x + b\right )} \sqrt{c} - 4 \, a c\right ) - 4 \,{\left (2 \, c^{2} e^{2} x + 8 \, c^{2} d e - 3 \, b c e^{2}\right )} \sqrt{c x^{2} + b x + a}}{16 \, c^{3}}, -\frac{{\left (8 \, c^{2} d^{2} - 8 \, b c d e +{\left (3 \, b^{2} - 4 \, a c\right )} e^{2}\right )} \sqrt{-c} \arctan \left (\frac{\sqrt{c x^{2} + b x + a}{\left (2 \, c x + b\right )} \sqrt{-c}}{2 \,{\left (c^{2} x^{2} + b c x + a c\right )}}\right ) - 2 \,{\left (2 \, c^{2} e^{2} x + 8 \, c^{2} d e - 3 \, b c e^{2}\right )} \sqrt{c x^{2} + b x + a}}{8 \, c^{3}}\right ] \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 (d + e x\right )^{2}}{\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 [A] time = 1.12988, size = 142, normalized size = 1.12 \begin{align*} \frac{1}{4} \, \sqrt{c x^{2} + b x + a}{\left (\frac{2 \, x e^{2}}{c} + \frac{8 \, c d e - 3 \, b e^{2}}{c^{2}}\right )} - \frac{{\left (8 \, c^{2} d^{2} - 8 \, b c d e + 3 \, b^{2} e^{2} - 4 \, a c e^{2}\right )} \log \left ({\left | -2 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )} \sqrt{c} - b \right |}\right )}{8 \, c^{\frac{5}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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