Optimal. Leaf size=118 \[ \frac{16 c^{3/2} \tan ^{-1}\left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}\right )}{d \left (b^2-4 a c\right )^{5/2}}+\frac{8 c}{d \left (b^2-4 a c\right )^2 \sqrt{a+b x+c x^2}}-\frac{2}{3 d \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}} \]
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Rubi [A] time = 0.0767907, antiderivative size = 118, 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 = {687, 688, 205} \[ \frac{16 c^{3/2} \tan ^{-1}\left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}\right )}{d \left (b^2-4 a c\right )^{5/2}}+\frac{8 c}{d \left (b^2-4 a c\right )^2 \sqrt{a+b x+c x^2}}-\frac{2}{3 d \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 687
Rule 688
Rule 205
Rubi steps
\begin{align*} \int \frac{1}{(b d+2 c d x) \left (a+b x+c x^2\right )^{5/2}} \, dx &=-\frac{2}{3 \left (b^2-4 a c\right ) d \left (a+b x+c x^2\right )^{3/2}}-\frac{(4 c) \int \frac{1}{(b d+2 c d x) \left (a+b x+c x^2\right )^{3/2}} \, dx}{b^2-4 a c}\\ &=-\frac{2}{3 \left (b^2-4 a c\right ) d \left (a+b x+c x^2\right )^{3/2}}+\frac{8 c}{\left (b^2-4 a c\right )^2 d \sqrt{a+b x+c x^2}}+\frac{\left (16 c^2\right ) \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}\\ &=-\frac{2}{3 \left (b^2-4 a c\right ) d \left (a+b x+c x^2\right )^{3/2}}+\frac{8 c}{\left (b^2-4 a c\right )^2 d \sqrt{a+b x+c x^2}}+\frac{\left (64 c^3\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}\\ &=-\frac{2}{3 \left (b^2-4 a c\right ) d \left (a+b x+c x^2\right )^{3/2}}+\frac{8 c}{\left (b^2-4 a c\right )^2 d \sqrt{a+b x+c x^2}}+\frac{16 c^{3/2} \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}\\ \end{align*}
Mathematica [C] time = 0.0360012, size = 62, normalized size = 0.53 \[ -\frac{2 \, _2F_1\left (-\frac{3}{2},1;-\frac{1}{2};\frac{4 c (a+x (b+c x))}{4 a c-b^2}\right )}{3 d \left (b^2-4 a c\right ) (a+x (b+c x))^{3/2}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.192, size = 207, normalized size = 1.8 \begin{align*}{\frac{2}{3\,d \left ( 4\,ac-{b}^{2} \right ) } \left ( \left ( x+{\frac{b}{2\,c}} \right ) ^{2}c+{\frac{4\,ac-{b}^{2}}{4\,c}} \right ) ^{-{\frac{3}{2}}}}+8\,{\frac{c}{d \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}}}}}}-16\,{\frac{c}{d \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 = 7.54834, size = 1341, normalized size = 11.36 \begin{align*} \left [\frac{2 \,{\left (12 \,{\left (c^{3} x^{4} + 2 \, b c^{2} x^{3} + 2 \, a b c x + a^{2} c +{\left (b^{2} c + 2 \, a c^{2}\right )} x^{2}\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 ) +{\left (12 \, c^{2} x^{2} + 12 \, b c x - b^{2} + 16 \, a c\right )} \sqrt{c x^{2} + b x + a}\right )}}{3 \,{\left ({\left (b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}\right )} d x^{4} + 2 \,{\left (b^{5} c - 8 \, a b^{3} c^{2} + 16 \, a^{2} b c^{3}\right )} d x^{3} +{\left (b^{6} - 6 \, a b^{4} c + 32 \, a^{3} c^{3}\right )} d x^{2} + 2 \,{\left (a b^{5} - 8 \, a^{2} b^{3} c + 16 \, a^{3} b c^{2}\right )} d x +{\left (a^{2} b^{4} - 8 \, a^{3} b^{2} c + 16 \, a^{4} c^{2}\right )} d\right )}}, \frac{2 \,{\left (24 \,{\left (c^{3} x^{4} + 2 \, b c^{2} x^{3} + 2 \, a b c x + a^{2} c +{\left (b^{2} c + 2 \, a c^{2}\right )} x^{2}\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 (12 \, c^{2} x^{2} + 12 \, b c x - b^{2} + 16 \, a c\right )} \sqrt{c x^{2} + b x + a}\right )}}{3 \,{\left ({\left (b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}\right )} d x^{4} + 2 \,{\left (b^{5} c - 8 \, a b^{3} c^{2} + 16 \, a^{2} b c^{3}\right )} d x^{3} +{\left (b^{6} - 6 \, a b^{4} c + 32 \, a^{3} c^{3}\right )} d x^{2} + 2 \,{\left (a b^{5} - 8 \, a^{2} b^{3} c + 16 \, a^{3} b c^{2}\right )} d x +{\left (a^{2} b^{4} - 8 \, a^{3} b^{2} c + 16 \, a^{4} c^{2}\right )} d\right )}}\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^{2} b \sqrt{a + b x + c x^{2}} + 2 a^{2} c x \sqrt{a + b x + c x^{2}} + 2 a b^{2} x \sqrt{a + b x + c x^{2}} + 6 a b c x^{2} \sqrt{a + b x + c x^{2}} + 4 a c^{2} x^{3} \sqrt{a + b x + c x^{2}} + b^{3} x^{2} \sqrt{a + b x + c x^{2}} + 4 b^{2} c x^{3} \sqrt{a + b x + c x^{2}} + 5 b c^{2} x^{4} \sqrt{a + b x + c x^{2}} + 2 c^{3} x^{5} \sqrt{a + b x + c x^{2}}}\, dx}{d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.21558, size = 753, normalized size = 6.38 \begin{align*} \frac{32 \, c^{2} \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 )}{{\left (b^{4} d - 8 \, a b^{2} c d + 16 \, a^{2} c^{2} d\right )} \sqrt{b^{2} c - 4 \, a c^{2}}} + \frac{12 \,{\left (\frac{{\left (b^{16} c^{2} d^{3} - 32 \, a b^{14} c^{3} d^{3} + 448 \, a^{2} b^{12} c^{4} d^{3} - 3584 \, a^{3} b^{10} c^{5} d^{3} + 17920 \, a^{4} b^{8} c^{6} d^{3} - 57344 \, a^{5} b^{6} c^{7} d^{3} + 114688 \, a^{6} b^{4} c^{8} d^{3} - 131072 \, a^{7} b^{2} c^{9} d^{3} + 65536 \, a^{8} c^{10} d^{3}\right )} x}{b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}} + \frac{b^{17} c d^{3} - 32 \, a b^{15} c^{2} d^{3} + 448 \, a^{2} b^{13} c^{3} d^{3} - 3584 \, a^{3} b^{11} c^{4} d^{3} + 17920 \, a^{4} b^{9} c^{5} d^{3} - 57344 \, a^{5} b^{7} c^{6} d^{3} + 114688 \, a^{6} b^{5} c^{7} d^{3} - 131072 \, a^{7} b^{3} c^{8} d^{3} + 65536 \, a^{8} b c^{9} d^{3}}{b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}}\right )} x - \frac{b^{18} d^{3} - 48 \, a b^{16} c d^{3} + 960 \, a^{2} b^{14} c^{2} d^{3} - 10752 \, a^{3} b^{12} c^{3} d^{3} + 75264 \, a^{4} b^{10} c^{4} d^{3} - 344064 \, a^{5} b^{8} c^{5} d^{3} + 1032192 \, a^{6} b^{6} c^{6} d^{3} - 1966080 \, a^{7} b^{4} c^{7} d^{3} + 2162688 \, a^{8} b^{2} c^{8} d^{3} - 1048576 \, a^{9} c^{9} d^{3}}{b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}}}{3 \,{\left (c x^{2} + b x + a\right )}^{\frac{3}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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