Optimal. Leaf size=122 \[ \frac{256 c^2 \sqrt{a+b x+c x^2}}{3 d^2 \left (b^2-4 a c\right )^3 (b+2 c x)}+\frac{32 c}{3 d^2 \left (b^2-4 a c\right )^2 (b+2 c x) \sqrt{a+b x+c x^2}}-\frac{2}{3 d^2 \left (b^2-4 a c\right ) (b+2 c x) \left (a+b x+c x^2\right )^{3/2}} \]
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Rubi [A] time = 0.0557404, antiderivative size = 122, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {687, 682} \[ \frac{256 c^2 \sqrt{a+b x+c x^2}}{3 d^2 \left (b^2-4 a c\right )^3 (b+2 c x)}+\frac{32 c}{3 d^2 \left (b^2-4 a c\right )^2 (b+2 c x) \sqrt{a+b x+c x^2}}-\frac{2}{3 d^2 \left (b^2-4 a c\right ) (b+2 c x) \left (a+b x+c x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 687
Rule 682
Rubi steps
\begin{align*} \int \frac{1}{(b d+2 c d x)^2 \left (a+b x+c x^2\right )^{5/2}} \, dx &=-\frac{2}{3 \left (b^2-4 a c\right ) d^2 (b+2 c x) \left (a+b x+c x^2\right )^{3/2}}-\frac{(16 c) \int \frac{1}{(b d+2 c d x)^2 \left (a+b x+c x^2\right )^{3/2}} \, dx}{3 \left (b^2-4 a c\right )}\\ &=-\frac{2}{3 \left (b^2-4 a c\right ) d^2 (b+2 c x) \left (a+b x+c x^2\right )^{3/2}}+\frac{32 c}{3 \left (b^2-4 a c\right )^2 d^2 (b+2 c x) \sqrt{a+b x+c x^2}}+\frac{\left (128 c^2\right ) \int \frac{1}{(b d+2 c d x)^2 \sqrt{a+b x+c x^2}} \, dx}{3 \left (b^2-4 a c\right )^2}\\ &=-\frac{2}{3 \left (b^2-4 a c\right ) d^2 (b+2 c x) \left (a+b x+c x^2\right )^{3/2}}+\frac{32 c}{3 \left (b^2-4 a c\right )^2 d^2 (b+2 c x) \sqrt{a+b x+c x^2}}+\frac{256 c^2 \sqrt{a+b x+c x^2}}{3 \left (b^2-4 a c\right )^3 d^2 (b+2 c x)}\\ \end{align*}
Mathematica [A] time = 0.0516286, size = 108, normalized size = 0.89 \[ \frac{32 c^2 \left (3 a^2+12 a c x^2+8 c^2 x^4\right )+48 b^2 c \left (a+6 c x^2\right )+128 b c^2 x \left (3 a+4 c x^2\right )+32 b^3 c x-2 b^4}{3 d^2 \left (b^2-4 a c\right )^3 (b+2 c x) (a+x (b+c x))^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.048, size = 133, normalized size = 1.1 \begin{align*} -{\frac{256\,{c}^{4}{x}^{4}+512\,b{c}^{3}{x}^{3}+384\,a{c}^{3}{x}^{2}+288\,{b}^{2}{c}^{2}{x}^{2}+384\,ab{c}^{2}x+32\,{b}^{3}cx+96\,{a}^{2}{c}^{2}+48\,ac{b}^{2}-2\,{b}^{4}}{ \left ( 6\,cx+3\,b \right ){d}^{2} \left ( 64\,{a}^{3}{c}^{3}-48\,{a}^{2}{b}^{2}{c}^{2}+12\,a{b}^{4}c-{b}^{6} \right ) } \left ( c{x}^{2}+bx+a \right ) ^{-{\frac{3}{2}}}} \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 = 22.3653, size = 794, normalized size = 6.51 \begin{align*} \frac{2 \,{\left (128 \, c^{4} x^{4} + 256 \, b c^{3} x^{3} - b^{4} + 24 \, a b^{2} c + 48 \, a^{2} c^{2} + 48 \,{\left (3 \, b^{2} c^{2} + 4 \, a c^{3}\right )} x^{2} + 16 \,{\left (b^{3} c + 12 \, a b c^{2}\right )} x\right )} \sqrt{c x^{2} + b x + a}}{3 \,{\left (2 \,{\left (b^{6} c^{3} - 12 \, a b^{4} c^{4} + 48 \, a^{2} b^{2} c^{5} - 64 \, a^{3} c^{6}\right )} d^{2} x^{5} + 5 \,{\left (b^{7} c^{2} - 12 \, a b^{5} c^{3} + 48 \, a^{2} b^{3} c^{4} - 64 \, a^{3} b c^{5}\right )} d^{2} x^{4} + 4 \,{\left (b^{8} c - 11 \, a b^{6} c^{2} + 36 \, a^{2} b^{4} c^{3} - 16 \, a^{3} b^{2} c^{4} - 64 \, a^{4} c^{5}\right )} d^{2} x^{3} +{\left (b^{9} - 6 \, a b^{7} c - 24 \, a^{2} b^{5} c^{2} + 224 \, a^{3} b^{3} c^{3} - 384 \, a^{4} b c^{4}\right )} d^{2} x^{2} + 2 \,{\left (a b^{8} - 11 \, a^{2} b^{6} c + 36 \, a^{3} b^{4} c^{2} - 16 \, a^{4} b^{2} c^{3} - 64 \, a^{5} c^{4}\right )} d^{2} x +{\left (a^{2} b^{7} - 12 \, a^{3} b^{5} c + 48 \, a^{4} b^{3} c^{2} - 64 \, a^{5} b c^{3}\right )} d^{2}\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^{2} \sqrt{a + b x + c x^{2}} + 4 a^{2} b c x \sqrt{a + b x + c x^{2}} + 4 a^{2} c^{2} x^{2} \sqrt{a + b x + c x^{2}} + 2 a b^{3} x \sqrt{a + b x + c x^{2}} + 10 a b^{2} c x^{2} \sqrt{a + b x + c x^{2}} + 16 a b c^{2} x^{3} \sqrt{a + b x + c x^{2}} + 8 a c^{3} x^{4} \sqrt{a + b x + c x^{2}} + b^{4} x^{2} \sqrt{a + b x + c x^{2}} + 6 b^{3} c x^{3} \sqrt{a + b x + c x^{2}} + 13 b^{2} c^{2} x^{4} \sqrt{a + b x + c x^{2}} + 12 b c^{3} x^{5} \sqrt{a + b x + c x^{2}} + 4 c^{4} x^{6} \sqrt{a + b x + c x^{2}}}\, dx}{d^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (2 \, c d x + b d\right )}^{2}{\left (c x^{2} + b x + a\right )}^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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