Optimal. Leaf size=116 \[ \frac{\tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right ) \left (4 c (2 A c d-a B e)-4 b c (A e+B d)+3 b^2 B e\right )}{8 c^{5/2}}-\frac{\sqrt{a+b x+c x^2} (-4 c (A e+B d)+3 b B e-2 B c e x)}{4 c^2} \]
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Rubi [A] time = 0.0876261, antiderivative size = 115, normalized size of antiderivative = 0.99, number of steps used = 3, number of rules used = 3, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.12, Rules used = {779, 621, 206} \[ \frac{\tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right ) \left (-4 a B c e-4 b c (A e+B d)+8 A c^2 d+3 b^2 B e\right )}{8 c^{5/2}}-\frac{\sqrt{a+b x+c x^2} (-4 c (A e+B d)+3 b B e-2 B c e x)}{4 c^2} \]
Antiderivative was successfully verified.
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Rule 779
Rule 621
Rule 206
Rubi steps
\begin{align*} \int \frac{(A+B x) (d+e x)}{\sqrt{a+b x+c x^2}} \, dx &=-\frac{(3 b B e-4 c (B d+A e)-2 B c e x) \sqrt{a+b x+c x^2}}{4 c^2}+\frac{\left (8 A c^2 d+3 b^2 B e-4 a B c e-4 b c (B d+A e)\right ) \int \frac{1}{\sqrt{a+b x+c x^2}} \, dx}{8 c^2}\\ &=-\frac{(3 b B e-4 c (B d+A e)-2 B c e x) \sqrt{a+b x+c x^2}}{4 c^2}+\frac{\left (8 A c^2 d+3 b^2 B e-4 a B c e-4 b c (B d+A e)\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}\\ &=-\frac{(3 b B e-4 c (B d+A e)-2 B c e x) \sqrt{a+b x+c x^2}}{4 c^2}+\frac{\left (8 A c^2 d+3 b^2 B e-4 a B c e-4 b c (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 = 0.0827641, size = 115, normalized size = 0.99 \[ \frac{\tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+x (b+c x)}}\right ) \left (-2 a B c e-2 b c (A e+B d)+4 A c^2 d+\frac{3}{2} b^2 B e\right )+\sqrt{c} \sqrt{a+x (b+c x)} (4 A c e+B (-3 b e+4 c d+2 c e x))}{4 c^{5/2}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.006, size = 243, normalized size = 2.1 \begin{align*}{\frac{Bex}{2\,c}\sqrt{c{x}^{2}+bx+a}}-{\frac{3\,bBe}{4\,{c}^{2}}\sqrt{c{x}^{2}+bx+a}}+{\frac{3\,{b}^{2}Be}{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{aBe}{2}\ln \left ({ \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx+a} \right ){c}^{-{\frac{3}{2}}}}+{\frac{Ae}{c}\sqrt{c{x}^{2}+bx+a}}+{\frac{Bd}{c}\sqrt{c{x}^{2}+bx+a}}-{\frac{Abe}{2}\ln \left ({ \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx+a} \right ){c}^{-{\frac{3}{2}}}}-{\frac{Bbd}{2}\ln \left ({ \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx+a} \right ){c}^{-{\frac{3}{2}}}}+{Ad\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 = 1.60711, size = 639, normalized size = 5.51 \begin{align*} \left [\frac{{\left (4 \,{\left (B b c - 2 \, A c^{2}\right )} d -{\left (3 \, B b^{2} - 4 \,{\left (B a + A b\right )} c\right )} e\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 \, B c^{2} e x + 4 \, B c^{2} d -{\left (3 \, B b c - 4 \, A c^{2}\right )} e\right )} \sqrt{c x^{2} + b x + a}}{16 \, c^{3}}, \frac{{\left (4 \,{\left (B b c - 2 \, A c^{2}\right )} d -{\left (3 \, B b^{2} - 4 \,{\left (B a + A b\right )} c\right )} e\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 \, B c^{2} e x + 4 \, B c^{2} d -{\left (3 \, B b c - 4 \, A c^{2}\right )} e\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 (A + B x\right ) \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 [A] time = 1.19437, size = 161, normalized size = 1.39 \begin{align*} \frac{1}{4} \, \sqrt{c x^{2} + b x + a}{\left (\frac{2 \, B x e}{c} + \frac{4 \, B c d - 3 \, B b e + 4 \, A c e}{c^{2}}\right )} + \frac{{\left (4 \, B b c d - 8 \, A c^{2} d - 3 \, B b^{2} e + 4 \, B a c e + 4 \, A b c e\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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