Optimal. Leaf size=132 \[ -\frac{12 c \sqrt{a+b x+c x^2}}{d^3 \left (b^2-4 a c\right )^2 (b+2 c x)^2}-\frac{2}{d^3 \left (b^2-4 a c\right ) (b+2 c x)^2 \sqrt{a+b x+c x^2}}-\frac{6 \sqrt{c} \tan ^{-1}\left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}\right )}{d^3 \left (b^2-4 a c\right )^{5/2}} \]
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Rubi [A] time = 0.0845561, antiderivative size = 132, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {687, 693, 688, 205} \[ -\frac{12 c \sqrt{a+b x+c x^2}}{d^3 \left (b^2-4 a c\right )^2 (b+2 c x)^2}-\frac{2}{d^3 \left (b^2-4 a c\right ) (b+2 c x)^2 \sqrt{a+b x+c x^2}}-\frac{6 \sqrt{c} \tan ^{-1}\left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}\right )}{d^3 \left (b^2-4 a c\right )^{5/2}} \]
Antiderivative was successfully verified.
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Rule 687
Rule 693
Rule 688
Rule 205
Rubi steps
\begin{align*} \int \frac{1}{(b d+2 c d x)^3 \left (a+b x+c x^2\right )^{3/2}} \, dx &=-\frac{2}{\left (b^2-4 a c\right ) d^3 (b+2 c x)^2 \sqrt{a+b x+c x^2}}-\frac{(12 c) \int \frac{1}{(b d+2 c d x)^3 \sqrt{a+b x+c x^2}} \, dx}{b^2-4 a c}\\ &=-\frac{2}{\left (b^2-4 a c\right ) d^3 (b+2 c x)^2 \sqrt{a+b x+c x^2}}-\frac{12 c \sqrt{a+b x+c x^2}}{\left (b^2-4 a c\right )^2 d^3 (b+2 c x)^2}-\frac{(6 c) \int \frac{1}{(b d+2 c d x) \sqrt{a+b x+c x^2}} \, dx}{\left (b^2-4 a c\right )^2 d^2}\\ &=-\frac{2}{\left (b^2-4 a c\right ) d^3 (b+2 c x)^2 \sqrt{a+b x+c x^2}}-\frac{12 c \sqrt{a+b x+c x^2}}{\left (b^2-4 a c\right )^2 d^3 (b+2 c x)^2}-\frac{\left (24 c^2\right ) \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 )}{\left (b^2-4 a c\right )^2 d^2}\\ &=-\frac{2}{\left (b^2-4 a c\right ) d^3 (b+2 c x)^2 \sqrt{a+b x+c x^2}}-\frac{12 c \sqrt{a+b x+c x^2}}{\left (b^2-4 a c\right )^2 d^3 (b+2 c x)^2}-\frac{6 \sqrt{c} \tan ^{-1}\left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{5/2} d^3}\\ \end{align*}
Mathematica [C] time = 0.0285623, size = 60, normalized size = 0.45 \[ -\frac{2 \, _2F_1\left (-\frac{1}{2},2;\frac{1}{2};\frac{4 c (a+x (b+c x))}{4 a c-b^2}\right )}{d^3 \left (b^2-4 a c\right )^2 \sqrt{a+x (b+c x)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.196, size = 218, normalized size = 1.7 \begin{align*} -{\frac{1}{4\,{c}^{2}{d}^{3} \left ( 4\,ac-{b}^{2} \right ) } \left ( x+{\frac{b}{2\,c}} \right ) ^{-2}{\frac{1}{\sqrt{ \left ( x+{\frac{b}{2\,c}} \right ) ^{2}c+{\frac{4\,ac-{b}^{2}}{4\,c}}}}}}-3\,{\frac{1}{{d}^{3} \left ( 4\,ac-{b}^{2} \right ) ^{2}}{\frac{1}{\sqrt{ \left ( x+1/2\,{\frac{b}{c}} \right ) ^{2}c+1/4\,{\frac{4\,ac-{b}^{2}}{c}}}}}}+6\,{\frac{1}{{d}^{3} \left ( 4\,ac-{b}^{2} \right ) ^{2}}\ln \left ({ \left ( 1/2\,{\frac{4\,ac-{b}^{2}}{c}}+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+1/2\,{\frac{b}{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 [B] time = 13.6228, size = 1496, normalized size = 11.33 \begin{align*} \left [\frac{3 \,{\left (4 \, c^{3} x^{4} + 8 \, b c^{2} x^{3} + a b^{2} +{\left (5 \, b^{2} c + 4 \, a c^{2}\right )} x^{2} +{\left (b^{3} + 4 \, a b c\right )} x\right )} \sqrt{-\frac{c}{b^{2} - 4 \, a 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}{\left (b^{2} - 4 \, a c\right )} \sqrt{-\frac{c}{b^{2} - 4 \, a c}}}{4 \, c^{2} x^{2} + 4 \, b c x + b^{2}}\right ) - 2 \,{\left (6 \, c^{2} x^{2} + 6 \, b c x + b^{2} + 2 \, a c\right )} \sqrt{c x^{2} + b x + a}}{4 \,{\left (b^{4} c^{3} - 8 \, a b^{2} c^{4} + 16 \, a^{2} c^{5}\right )} d^{3} x^{4} + 8 \,{\left (b^{5} c^{2} - 8 \, a b^{3} c^{3} + 16 \, a^{2} b c^{4}\right )} d^{3} x^{3} +{\left (5 \, b^{6} c - 36 \, a b^{4} c^{2} + 48 \, a^{2} b^{2} c^{3} + 64 \, a^{3} c^{4}\right )} d^{3} x^{2} +{\left (b^{7} - 4 \, a b^{5} c - 16 \, a^{2} b^{3} c^{2} + 64 \, a^{3} b c^{3}\right )} d^{3} x +{\left (a b^{6} - 8 \, a^{2} b^{4} c + 16 \, a^{3} b^{2} c^{2}\right )} d^{3}}, -\frac{2 \,{\left (3 \,{\left (4 \, c^{3} x^{4} + 8 \, b c^{2} x^{3} + a b^{2} +{\left (5 \, b^{2} c + 4 \, a c^{2}\right )} x^{2} +{\left (b^{3} + 4 \, a b c\right )} x\right )} \sqrt{\frac{c}{b^{2} - 4 \, a c}} \arctan \left (-\frac{\sqrt{c x^{2} + b x + a}{\left (b^{2} - 4 \, a c\right )} \sqrt{\frac{c}{b^{2} - 4 \, a c}}}{2 \,{\left (c^{2} x^{2} + b c x + a c\right )}}\right ) +{\left (6 \, c^{2} x^{2} + 6 \, b c x + b^{2} + 2 \, a c\right )} \sqrt{c x^{2} + b x + a}\right )}}{4 \,{\left (b^{4} c^{3} - 8 \, a b^{2} c^{4} + 16 \, a^{2} c^{5}\right )} d^{3} x^{4} + 8 \,{\left (b^{5} c^{2} - 8 \, a b^{3} c^{3} + 16 \, a^{2} b c^{4}\right )} d^{3} x^{3} +{\left (5 \, b^{6} c - 36 \, a b^{4} c^{2} + 48 \, a^{2} b^{2} c^{3} + 64 \, a^{3} c^{4}\right )} d^{3} x^{2} +{\left (b^{7} - 4 \, a b^{5} c - 16 \, a^{2} b^{3} c^{2} + 64 \, a^{3} b c^{3}\right )} d^{3} x +{\left (a b^{6} - 8 \, a^{2} b^{4} c + 16 \, a^{3} b^{2} c^{2}\right )} d^{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*} \frac{\int \frac{1}{a b^{3} \sqrt{a + b x + c x^{2}} + 6 a b^{2} c x \sqrt{a + b x + c x^{2}} + 12 a b c^{2} x^{2} \sqrt{a + b x + c x^{2}} + 8 a c^{3} x^{3} \sqrt{a + b x + c x^{2}} + b^{4} x \sqrt{a + b x + c x^{2}} + 7 b^{3} c x^{2} \sqrt{a + b x + c x^{2}} + 18 b^{2} c^{2} x^{3} \sqrt{a + b x + c x^{2}} + 20 b c^{3} x^{4} \sqrt{a + b x + c x^{2}} + 8 c^{4} x^{5} \sqrt{a + b x + c x^{2}}}\, dx}{d^{3}} \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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