Optimal. Leaf size=149 \[ \frac{\left (b^2-4 a c\right )^2 \sqrt{a+b x+c x^2}}{32 c^3 d}-\frac{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}}{24 c^2 d}-\frac{\left (b^2-4 a c\right )^{5/2} \tan ^{-1}\left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}\right )}{64 c^{7/2} d}+\frac{\left (a+b x+c x^2\right )^{5/2}}{10 c d} \]
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Rubi [A] time = 0.12067, antiderivative size = 149, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.115, Rules used = {685, 688, 205} \[ \frac{\left (b^2-4 a c\right )^2 \sqrt{a+b x+c x^2}}{32 c^3 d}-\frac{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}}{24 c^2 d}-\frac{\left (b^2-4 a c\right )^{5/2} \tan ^{-1}\left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}\right )}{64 c^{7/2} d}+\frac{\left (a+b x+c x^2\right )^{5/2}}{10 c d} \]
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 )^{5/2}}{b d+2 c d x} \, dx &=\frac{\left (a+b x+c x^2\right )^{5/2}}{10 c d}-\frac{\left (b^2-4 a c\right ) \int \frac{\left (a+b x+c x^2\right )^{3/2}}{b d+2 c d x} \, dx}{4 c}\\ &=-\frac{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}}{24 c^2 d}+\frac{\left (a+b x+c x^2\right )^{5/2}}{10 c d}+\frac{\left (b^2-4 a c\right )^2 \int \frac{\sqrt{a+b x+c x^2}}{b d+2 c d x} \, dx}{16 c^2}\\ &=\frac{\left (b^2-4 a c\right )^2 \sqrt{a+b x+c x^2}}{32 c^3 d}-\frac{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}}{24 c^2 d}+\frac{\left (a+b x+c x^2\right )^{5/2}}{10 c d}-\frac{\left (b^2-4 a c\right )^3 \int \frac{1}{(b d+2 c d x) \sqrt{a+b x+c x^2}} \, dx}{64 c^3}\\ &=\frac{\left (b^2-4 a c\right )^2 \sqrt{a+b x+c x^2}}{32 c^3 d}-\frac{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}}{24 c^2 d}+\frac{\left (a+b x+c x^2\right )^{5/2}}{10 c d}-\frac{\left (b^2-4 a c\right )^3 \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 )}{16 c^2}\\ &=\frac{\left (b^2-4 a c\right )^2 \sqrt{a+b x+c x^2}}{32 c^3 d}-\frac{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}}{24 c^2 d}+\frac{\left (a+b x+c x^2\right )^{5/2}}{10 c d}-\frac{\left (b^2-4 a c\right )^{5/2} \tan ^{-1}\left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}\right )}{64 c^{7/2} d}\\ \end{align*}
Mathematica [A] time = 0.174842, size = 150, normalized size = 1.01 \[ \frac{\frac{\sqrt{a+x (b+c x)} \left (16 c^2 \left (23 a^2+11 a c x^2+3 c^2 x^4\right )+28 b^2 c \left (c x^2-5 a\right )+16 b c^2 x \left (11 a+6 c x^2\right )-20 b^3 c x+15 b^4\right )}{480 c^3}-\frac{\left (b^2-4 a c\right )^{5/2} \tan ^{-1}\left (\frac{2 \sqrt{c} \sqrt{a+x (b+c x)}}{\sqrt{b^2-4 a c}}\right )}{64 c^{7/2}}}{d} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.19, size = 660, normalized size = 4.4 \begin{align*}{\frac{1}{10\,cd} \left ( \left ( x+{\frac{b}{2\,c}} \right ) ^{2}c+{\frac{4\,ac-{b}^{2}}{4\,c}} \right ) ^{{\frac{5}{2}}}}+{\frac{a}{6\,cd} \left ( \left ( x+{\frac{b}{2\,c}} \right ) ^{2}c+{\frac{4\,ac-{b}^{2}}{4\,c}} \right ) ^{{\frac{3}{2}}}}-{\frac{{b}^{2}}{24\,{c}^{2}d} \left ( \left ( x+{\frac{b}{2\,c}} \right ) ^{2}c+{\frac{4\,ac-{b}^{2}}{4\,c}} \right ) ^{{\frac{3}{2}}}}+{\frac{{a}^{2}}{4\,cd}\sqrt{4\, \left ( x+1/2\,{\frac{b}{c}} \right ) ^{2}c+{\frac{4\,ac-{b}^{2}}{c}}}}-{\frac{{b}^{2}a}{8\,{c}^{2}d}\sqrt{4\, \left ( x+1/2\,{\frac{b}{c}} \right ) ^{2}c+{\frac{4\,ac-{b}^{2}}{c}}}}+{\frac{{b}^{4}}{64\,d{c}^{3}}\sqrt{4\, \left ( x+1/2\,{\frac{b}{c}} \right ) ^{2}c+{\frac{4\,ac-{b}^{2}}{c}}}}-{\frac{{a}^{3}}{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{3\,{b}^{2}{a}^{2}}{4\,{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{3\,a{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}}}}}}+{\frac{{b}^{6}}{64\,d{c}^{4}}\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 = 4.46849, size = 852, normalized size = 5.72 \begin{align*} \left [\frac{15 \,{\left (b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}\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 (48 \, c^{4} x^{4} + 96 \, b c^{3} x^{3} + 15 \, b^{4} - 140 \, a b^{2} c + 368 \, a^{2} c^{2} + 4 \,{\left (7 \, b^{2} c^{2} + 44 \, a c^{3}\right )} x^{2} - 4 \,{\left (5 \, b^{3} c - 44 \, a b c^{2}\right )} x\right )} \sqrt{c x^{2} + b x + a}}{1920 \, c^{3} d}, \frac{15 \,{\left (b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}\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 (48 \, c^{4} x^{4} + 96 \, b c^{3} x^{3} + 15 \, b^{4} - 140 \, a b^{2} c + 368 \, a^{2} c^{2} + 4 \,{\left (7 \, b^{2} c^{2} + 44 \, a c^{3}\right )} x^{2} - 4 \,{\left (5 \, b^{3} c - 44 \, a b c^{2}\right )} x\right )} \sqrt{c x^{2} + b x + a}}{960 \, c^{3} 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^{2} \sqrt{a + b x + c x^{2}}}{b + 2 c x}\, dx + \int \frac{b^{2} x^{2} \sqrt{a + b x + c x^{2}}}{b + 2 c x}\, dx + \int \frac{c^{2} x^{4} \sqrt{a + b x + c x^{2}}}{b + 2 c x}\, dx + \int \frac{2 a b x \sqrt{a + b x + c x^{2}}}{b + 2 c x}\, dx + \int \frac{2 a c x^{2} \sqrt{a + b x + c x^{2}}}{b + 2 c x}\, dx + \int \frac{2 b c x^{3} \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 = 1.18425, size = 320, normalized size = 2.15 \begin{align*} \frac{1}{480} \, \sqrt{c x^{2} + b x + a}{\left (4 \,{\left ({\left (12 \,{\left (\frac{c x}{d} + \frac{2 \, b}{d}\right )} x + \frac{7 \, b^{2} c^{9} d^{5} + 44 \, a c^{10} d^{5}}{c^{10} d^{6}}\right )} x - \frac{5 \, b^{3} c^{8} d^{5} - 44 \, a b c^{9} d^{5}}{c^{10} d^{6}}\right )} x + \frac{15 \, b^{4} c^{7} d^{5} - 140 \, a b^{2} c^{8} d^{5} + 368 \, a^{2} c^{9} d^{5}}{c^{10} d^{6}}\right )} - \frac{{\left (b^{6} - 12 \, a b^{4} c + 48 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}\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 )}{32 \, \sqrt{b^{2} c - 4 \, a c^{2}} c^{3} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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