Optimal. Leaf size=145 \[ -\frac{5 \left (b^2-4 a c\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{128 c^{7/2} d^4}-\frac{5 \left (a+b x+c x^2\right )^{3/2}}{24 c^2 d^4 (b+2 c x)}+\frac{5 (b+2 c x) \sqrt{a+b x+c x^2}}{64 c^3 d^4}-\frac{\left (a+b x+c x^2\right )^{5/2}}{6 c d^4 (b+2 c x)^3} \]
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Rubi [A] time = 0.0695159, antiderivative size = 145, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {684, 612, 621, 206} \[ -\frac{5 \left (b^2-4 a c\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{128 c^{7/2} d^4}-\frac{5 \left (a+b x+c x^2\right )^{3/2}}{24 c^2 d^4 (b+2 c x)}+\frac{5 (b+2 c x) \sqrt{a+b x+c x^2}}{64 c^3 d^4}-\frac{\left (a+b x+c x^2\right )^{5/2}}{6 c d^4 (b+2 c x)^3} \]
Antiderivative was successfully verified.
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Rule 684
Rule 612
Rule 621
Rule 206
Rubi steps
\begin{align*} \int \frac{\left (a+b x+c x^2\right )^{5/2}}{(b d+2 c d x)^4} \, dx &=-\frac{\left (a+b x+c x^2\right )^{5/2}}{6 c d^4 (b+2 c x)^3}+\frac{5 \int \frac{\left (a+b x+c x^2\right )^{3/2}}{(b d+2 c d x)^2} \, dx}{12 c d^2}\\ &=-\frac{5 \left (a+b x+c x^2\right )^{3/2}}{24 c^2 d^4 (b+2 c x)}-\frac{\left (a+b x+c x^2\right )^{5/2}}{6 c d^4 (b+2 c x)^3}+\frac{5 \int \sqrt{a+b x+c x^2} \, dx}{16 c^2 d^4}\\ &=\frac{5 (b+2 c x) \sqrt{a+b x+c x^2}}{64 c^3 d^4}-\frac{5 \left (a+b x+c x^2\right )^{3/2}}{24 c^2 d^4 (b+2 c x)}-\frac{\left (a+b x+c x^2\right )^{5/2}}{6 c d^4 (b+2 c x)^3}-\frac{\left (5 \left (b^2-4 a c\right )\right ) \int \frac{1}{\sqrt{a+b x+c x^2}} \, dx}{128 c^3 d^4}\\ &=\frac{5 (b+2 c x) \sqrt{a+b x+c x^2}}{64 c^3 d^4}-\frac{5 \left (a+b x+c x^2\right )^{3/2}}{24 c^2 d^4 (b+2 c x)}-\frac{\left (a+b x+c x^2\right )^{5/2}}{6 c d^4 (b+2 c x)^3}-\frac{\left (5 \left (b^2-4 a c\right )\right ) \operatorname{Subst}\left (\int \frac{1}{4 c-x^2} \, dx,x,\frac{b+2 c x}{\sqrt{a+b x+c x^2}}\right )}{64 c^3 d^4}\\ &=\frac{5 (b+2 c x) \sqrt{a+b x+c x^2}}{64 c^3 d^4}-\frac{5 \left (a+b x+c x^2\right )^{3/2}}{24 c^2 d^4 (b+2 c x)}-\frac{\left (a+b x+c x^2\right )^{5/2}}{6 c d^4 (b+2 c x)^3}-\frac{5 \left (b^2-4 a c\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{128 c^{7/2} d^4}\\ \end{align*}
Mathematica [C] time = 0.0474012, size = 97, normalized size = 0.67 \[ -\frac{\left (b^2-4 a c\right )^2 \sqrt{a+x (b+c x)} \, _2F_1\left (-\frac{5}{2},-\frac{3}{2};-\frac{1}{2};\frac{(b+2 c x)^2}{b^2-4 a c}\right )}{192 c^3 d^4 (b+2 c x)^3 \sqrt{\frac{c (a+x (b+c x))}{4 a c-b^2}}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.196, size = 1022, normalized size = 7.1 \begin{align*} \text{result too large to display} \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 = 14.8094, size = 1170, normalized size = 8.07 \begin{align*} \left [-\frac{15 \,{\left (b^{5} - 4 \, a b^{3} c + 8 \,{\left (b^{2} c^{3} - 4 \, a c^{4}\right )} x^{3} + 12 \,{\left (b^{3} c^{2} - 4 \, a b c^{3}\right )} x^{2} + 6 \,{\left (b^{4} c - 4 \, a b^{2} c^{2}\right )} x\right )} \sqrt{c} \log \left (-8 \, c^{2} x^{2} - 8 \, b c x - b^{2} - 4 \, \sqrt{c x^{2} + b x + a}{\left (2 \, c x + b\right )} \sqrt{c} - 4 \, a c\right ) - 4 \,{\left (48 \, c^{5} x^{4} + 96 \, b c^{4} x^{3} + 15 \, b^{4} c - 40 \, a b^{2} c^{2} - 32 \, a^{2} c^{3} + 32 \,{\left (4 \, b^{2} c^{3} - 7 \, a c^{4}\right )} x^{2} + 16 \,{\left (5 \, b^{3} c^{2} - 14 \, a b c^{3}\right )} x\right )} \sqrt{c x^{2} + b x + a}}{768 \,{\left (8 \, c^{7} d^{4} x^{3} + 12 \, b c^{6} d^{4} x^{2} + 6 \, b^{2} c^{5} d^{4} x + b^{3} c^{4} d^{4}\right )}}, \frac{15 \,{\left (b^{5} - 4 \, a b^{3} c + 8 \,{\left (b^{2} c^{3} - 4 \, a c^{4}\right )} x^{3} + 12 \,{\left (b^{3} c^{2} - 4 \, a b c^{3}\right )} x^{2} + 6 \,{\left (b^{4} c - 4 \, a b^{2} c^{2}\right )} x\right )} \sqrt{-c} \arctan \left (\frac{\sqrt{c x^{2} + b x + a}{\left (2 \, c x + b\right )} \sqrt{-c}}{2 \,{\left (c^{2} x^{2} + b c x + a c\right )}}\right ) + 2 \,{\left (48 \, c^{5} x^{4} + 96 \, b c^{4} x^{3} + 15 \, b^{4} c - 40 \, a b^{2} c^{2} - 32 \, a^{2} c^{3} + 32 \,{\left (4 \, b^{2} c^{3} - 7 \, a c^{4}\right )} x^{2} + 16 \,{\left (5 \, b^{3} c^{2} - 14 \, a b c^{3}\right )} x\right )} \sqrt{c x^{2} + b x + a}}{384 \,{\left (8 \, c^{7} d^{4} x^{3} + 12 \, b c^{6} d^{4} x^{2} + 6 \, b^{2} c^{5} d^{4} x + b^{3} c^{4} d^{4}\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{a^{2} \sqrt{a + b x + c x^{2}}}{b^{4} + 8 b^{3} c x + 24 b^{2} c^{2} x^{2} + 32 b c^{3} x^{3} + 16 c^{4} x^{4}}\, dx + \int \frac{b^{2} x^{2} \sqrt{a + b x + c x^{2}}}{b^{4} + 8 b^{3} c x + 24 b^{2} c^{2} x^{2} + 32 b c^{3} x^{3} + 16 c^{4} x^{4}}\, dx + \int \frac{c^{2} x^{4} \sqrt{a + b x + c x^{2}}}{b^{4} + 8 b^{3} c x + 24 b^{2} c^{2} x^{2} + 32 b c^{3} x^{3} + 16 c^{4} x^{4}}\, dx + \int \frac{2 a b x \sqrt{a + b x + c x^{2}}}{b^{4} + 8 b^{3} c x + 24 b^{2} c^{2} x^{2} + 32 b c^{3} x^{3} + 16 c^{4} x^{4}}\, dx + \int \frac{2 a c x^{2} \sqrt{a + b x + c x^{2}}}{b^{4} + 8 b^{3} c x + 24 b^{2} c^{2} x^{2} + 32 b c^{3} x^{3} + 16 c^{4} x^{4}}\, dx + \int \frac{2 b c x^{3} \sqrt{a + b x + c x^{2}}}{b^{4} + 8 b^{3} c x + 24 b^{2} c^{2} x^{2} + 32 b c^{3} x^{3} + 16 c^{4} x^{4}}\, dx}{d^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.92642, size = 841, normalized size = 5.8 \begin{align*} \frac{1}{64} \, \sqrt{c x^{2} + b x + a}{\left (\frac{2 \, x}{c^{2} d^{4}} + \frac{b}{c^{3} d^{4}}\right )} + \frac{5 \,{\left (b^{2} - 4 \, a c\right )} \log \left ({\left | 2 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )} \sqrt{c} + b \right |}\right )}{128 \, c^{\frac{7}{2}} d^{4}} + \frac{36 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )}^{4} b^{4} c^{\frac{5}{2}} - 288 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )}^{4} a b^{2} c^{\frac{7}{2}} + 576 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )}^{4} a^{2} c^{\frac{9}{2}} + 72 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )}^{3} b^{5} c^{2} - 576 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )}^{3} a b^{3} c^{3} + 1152 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )}^{3} a^{2} b c^{4} + 66 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )}^{2} b^{6} c^{\frac{3}{2}} - 576 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )}^{2} a b^{4} c^{\frac{5}{2}} + 1440 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )}^{2} a^{2} b^{2} c^{\frac{7}{2}} - 768 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )}^{2} a^{3} c^{\frac{9}{2}} + 30 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )} b^{7} c - 288 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )} a b^{5} c^{2} + 864 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )} a^{2} b^{3} c^{3} - 768 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )} a^{3} b c^{4} + 7 \, b^{8} \sqrt{c} - 82 \, a b^{6} c^{\frac{3}{2}} + 348 \, a^{2} b^{4} c^{\frac{5}{2}} - 640 \, a^{3} b^{2} c^{\frac{7}{2}} + 448 \, a^{4} c^{\frac{9}{2}}}{192 \,{\left (2 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )}^{2} c^{\frac{3}{2}} + 2 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )} b c + b^{2} \sqrt{c} - 2 \, a c^{\frac{3}{2}}\right )}^{3} c^{\frac{5}{2}} d^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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