Optimal. Leaf size=115 \[ -\frac{\left (b^2-4 a c\right ) \sqrt{a+b x+c x^2}}{8 c^2 d}+\frac{\left (b^2-4 a c\right )^{3/2} \tan ^{-1}\left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}\right )}{16 c^{5/2} d}+\frac{\left (a+b x+c x^2\right )^{3/2}}{6 c d} \]
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Rubi [A] time = 0.0827866, antiderivative size = 115, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.115, Rules used = {685, 688, 205} \[ -\frac{\left (b^2-4 a c\right ) \sqrt{a+b x+c x^2}}{8 c^2 d}+\frac{\left (b^2-4 a c\right )^{3/2} \tan ^{-1}\left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}\right )}{16 c^{5/2} d}+\frac{\left (a+b x+c x^2\right )^{3/2}}{6 c d} \]
Antiderivative was successfully verified.
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Rule 685
Rule 688
Rule 205
Rubi steps
\begin{align*} \int \frac{\left (a+b x+c x^2\right )^{3/2}}{b d+2 c d x} \, dx &=\frac{\left (a+b x+c x^2\right )^{3/2}}{6 c d}-\frac{\left (b^2-4 a c\right ) \int \frac{\sqrt{a+b x+c x^2}}{b d+2 c d x} \, dx}{4 c}\\ &=-\frac{\left (b^2-4 a c\right ) \sqrt{a+b x+c x^2}}{8 c^2 d}+\frac{\left (a+b x+c x^2\right )^{3/2}}{6 c d}+\frac{\left (b^2-4 a c\right )^2 \int \frac{1}{(b d+2 c d x) \sqrt{a+b x+c x^2}} \, dx}{16 c^2}\\ &=-\frac{\left (b^2-4 a c\right ) \sqrt{a+b x+c x^2}}{8 c^2 d}+\frac{\left (a+b x+c x^2\right )^{3/2}}{6 c d}+\frac{\left (b^2-4 a c\right )^2 \operatorname{Subst}\left (\int \frac{1}{2 b^2 c d-8 a c^2 d+8 c^2 d x^2} \, dx,x,\sqrt{a+b x+c x^2}\right )}{4 c}\\ &=-\frac{\left (b^2-4 a c\right ) \sqrt{a+b x+c x^2}}{8 c^2 d}+\frac{\left (a+b x+c x^2\right )^{3/2}}{6 c d}+\frac{\left (b^2-4 a c\right )^{3/2} \tan ^{-1}\left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}\right )}{16 c^{5/2} d}\\ \end{align*}
Mathematica [A] time = 0.105074, size = 103, normalized size = 0.9 \[ \frac{2 \sqrt{c} \sqrt{a+x (b+c x)} \left (4 c \left (4 a+c x^2\right )-3 b^2+4 b c x\right )+3 \left (b^2-4 a c\right )^{3/2} \tan ^{-1}\left (\frac{2 \sqrt{c} \sqrt{a+x (b+c x)}}{\sqrt{b^2-4 a c}}\right )}{48 c^{5/2} d} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.19, size = 430, normalized size = 3.7 \begin{align*}{\frac{1}{6\,cd} \left ( \left ( x+{\frac{b}{2\,c}} \right ) ^{2}c+{\frac{4\,ac-{b}^{2}}{4\,c}} \right ) ^{{\frac{3}{2}}}}+{\frac{a}{4\,cd}\sqrt{4\, \left ( x+1/2\,{\frac{b}{c}} \right ) ^{2}c+{\frac{4\,ac-{b}^{2}}{c}}}}-{\frac{{b}^{2}}{16\,{c}^{2}d}\sqrt{4\, \left ( x+1/2\,{\frac{b}{c}} \right ) ^{2}c+{\frac{4\,ac-{b}^{2}}{c}}}}-{\frac{{a}^{2}}{cd}\ln \left ({ \left ({\frac{4\,ac-{b}^{2}}{2\,c}}+{\frac{1}{2}\sqrt{{\frac{4\,ac-{b}^{2}}{c}}}\sqrt{4\, \left ( x+1/2\,{\frac{b}{c}} \right ) ^{2}c+{\frac{4\,ac-{b}^{2}}{c}}}} \right ) \left ( x+{\frac{b}{2\,c}} \right ) ^{-1}} \right ){\frac{1}{\sqrt{{\frac{4\,ac-{b}^{2}}{c}}}}}}+{\frac{{b}^{2}a}{2\,{c}^{2}d}\ln \left ({ \left ({\frac{4\,ac-{b}^{2}}{2\,c}}+{\frac{1}{2}\sqrt{{\frac{4\,ac-{b}^{2}}{c}}}\sqrt{4\, \left ( x+1/2\,{\frac{b}{c}} \right ) ^{2}c+{\frac{4\,ac-{b}^{2}}{c}}}} \right ) \left ( x+{\frac{b}{2\,c}} \right ) ^{-1}} \right ){\frac{1}{\sqrt{{\frac{4\,ac-{b}^{2}}{c}}}}}}-{\frac{{b}^{4}}{16\,d{c}^{3}}\ln \left ({ \left ({\frac{4\,ac-{b}^{2}}{2\,c}}+{\frac{1}{2}\sqrt{{\frac{4\,ac-{b}^{2}}{c}}}\sqrt{4\, \left ( x+1/2\,{\frac{b}{c}} \right ) ^{2}c+{\frac{4\,ac-{b}^{2}}{c}}}} \right ) \left ( x+{\frac{b}{2\,c}} \right ) ^{-1}} \right ){\frac{1}{\sqrt{{\frac{4\,ac-{b}^{2}}{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 = 3.69747, size = 570, normalized size = 4.96 \begin{align*} \left [-\frac{3 \,{\left (b^{2} - 4 \, a c\right )} \sqrt{-\frac{b^{2} - 4 \, a c}{c}} \log \left (-\frac{4 \, c^{2} x^{2} + 4 \, b c x - b^{2} + 8 \, a c - 4 \, \sqrt{c x^{2} + b x + a} c \sqrt{-\frac{b^{2} - 4 \, a c}{c}}}{4 \, c^{2} x^{2} + 4 \, b c x + b^{2}}\right ) - 4 \,{\left (4 \, c^{2} x^{2} + 4 \, b c x - 3 \, b^{2} + 16 \, a c\right )} \sqrt{c x^{2} + b x + a}}{96 \, c^{2} d}, -\frac{3 \,{\left (b^{2} - 4 \, a c\right )} \sqrt{\frac{b^{2} - 4 \, a c}{c}} \arctan \left (\frac{\sqrt{\frac{b^{2} - 4 \, a c}{c}}}{2 \, \sqrt{c x^{2} + b x + a}}\right ) - 2 \,{\left (4 \, c^{2} x^{2} + 4 \, b c x - 3 \, b^{2} + 16 \, a c\right )} \sqrt{c x^{2} + b x + a}}{48 \, c^{2} d}\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*} \frac{\int \frac{a \sqrt{a + b x + c x^{2}}}{b + 2 c x}\, dx + \int \frac{b x \sqrt{a + b x + c x^{2}}}{b + 2 c x}\, dx + \int \frac{c x^{2} \sqrt{a + b x + c x^{2}}}{b + 2 c x}\, dx}{d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.15905, size = 201, normalized size = 1.75 \begin{align*} \frac{1}{24} \, \sqrt{c x^{2} + b x + a}{\left (4 \, x{\left (\frac{x}{d} + \frac{b}{c d}\right )} - \frac{3 \, b^{2} c^{3} d^{3} - 16 \, a c^{4} d^{3}}{c^{5} d^{4}}\right )} + \frac{{\left (b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}\right )} \arctan \left (-\frac{2 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )} c + b \sqrt{c}}{\sqrt{b^{2} c - 4 \, a c^{2}}}\right )}{8 \, \sqrt{b^{2} c - 4 \, a c^{2}} c^{2} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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