Optimal. Leaf size=207 \[ -\frac{d^4 \left (b^2-4 a c\right ) (b+2 c x)^5 \sqrt{a+b x+c x^2}}{128 c^2}+\frac{d^4 \left (b^2-4 a c\right )^2 (b+2 c x)^3 \sqrt{a+b x+c x^2}}{512 c^2}+\frac{3 d^4 \left (b^2-4 a c\right )^3 (b+2 c x) \sqrt{a+b x+c x^2}}{1024 c^2}+\frac{3 d^4 \left (b^2-4 a c\right )^4 \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{2048 c^{5/2}}+\frac{d^4 (b+2 c x)^5 \left (a+b x+c x^2\right )^{3/2}}{16 c} \]
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Rubi [A] time = 0.128642, antiderivative size = 207, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {685, 692, 621, 206} \[ -\frac{d^4 \left (b^2-4 a c\right ) (b+2 c x)^5 \sqrt{a+b x+c x^2}}{128 c^2}+\frac{d^4 \left (b^2-4 a c\right )^2 (b+2 c x)^3 \sqrt{a+b x+c x^2}}{512 c^2}+\frac{3 d^4 \left (b^2-4 a c\right )^3 (b+2 c x) \sqrt{a+b x+c x^2}}{1024 c^2}+\frac{3 d^4 \left (b^2-4 a c\right )^4 \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{2048 c^{5/2}}+\frac{d^4 (b+2 c x)^5 \left (a+b x+c x^2\right )^{3/2}}{16 c} \]
Antiderivative was successfully verified.
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Rule 685
Rule 692
Rule 621
Rule 206
Rubi steps
\begin{align*} \int (b d+2 c d x)^4 \left (a+b x+c x^2\right )^{3/2} \, dx &=\frac{d^4 (b+2 c x)^5 \left (a+b x+c x^2\right )^{3/2}}{16 c}-\frac{\left (3 \left (b^2-4 a c\right )\right ) \int (b d+2 c d x)^4 \sqrt{a+b x+c x^2} \, dx}{32 c}\\ &=-\frac{\left (b^2-4 a c\right ) d^4 (b+2 c x)^5 \sqrt{a+b x+c x^2}}{128 c^2}+\frac{d^4 (b+2 c x)^5 \left (a+b x+c x^2\right )^{3/2}}{16 c}+\frac{\left (b^2-4 a c\right )^2 \int \frac{(b d+2 c d x)^4}{\sqrt{a+b x+c x^2}} \, dx}{256 c^2}\\ &=\frac{\left (b^2-4 a c\right )^2 d^4 (b+2 c x)^3 \sqrt{a+b x+c x^2}}{512 c^2}-\frac{\left (b^2-4 a c\right ) d^4 (b+2 c x)^5 \sqrt{a+b x+c x^2}}{128 c^2}+\frac{d^4 (b+2 c x)^5 \left (a+b x+c x^2\right )^{3/2}}{16 c}+\frac{\left (3 \left (b^2-4 a c\right )^3 d^2\right ) \int \frac{(b d+2 c d x)^2}{\sqrt{a+b x+c x^2}} \, dx}{1024 c^2}\\ &=\frac{3 \left (b^2-4 a c\right )^3 d^4 (b+2 c x) \sqrt{a+b x+c x^2}}{1024 c^2}+\frac{\left (b^2-4 a c\right )^2 d^4 (b+2 c x)^3 \sqrt{a+b x+c x^2}}{512 c^2}-\frac{\left (b^2-4 a c\right ) d^4 (b+2 c x)^5 \sqrt{a+b x+c x^2}}{128 c^2}+\frac{d^4 (b+2 c x)^5 \left (a+b x+c x^2\right )^{3/2}}{16 c}+\frac{\left (3 \left (b^2-4 a c\right )^4 d^4\right ) \int \frac{1}{\sqrt{a+b x+c x^2}} \, dx}{2048 c^2}\\ &=\frac{3 \left (b^2-4 a c\right )^3 d^4 (b+2 c x) \sqrt{a+b x+c x^2}}{1024 c^2}+\frac{\left (b^2-4 a c\right )^2 d^4 (b+2 c x)^3 \sqrt{a+b x+c x^2}}{512 c^2}-\frac{\left (b^2-4 a c\right ) d^4 (b+2 c x)^5 \sqrt{a+b x+c x^2}}{128 c^2}+\frac{d^4 (b+2 c x)^5 \left (a+b x+c x^2\right )^{3/2}}{16 c}+\frac{\left (3 \left (b^2-4 a c\right )^4 d^4\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 )}{1024 c^2}\\ &=\frac{3 \left (b^2-4 a c\right )^3 d^4 (b+2 c x) \sqrt{a+b x+c x^2}}{1024 c^2}+\frac{\left (b^2-4 a c\right )^2 d^4 (b+2 c x)^3 \sqrt{a+b x+c x^2}}{512 c^2}-\frac{\left (b^2-4 a c\right ) d^4 (b+2 c x)^5 \sqrt{a+b x+c x^2}}{128 c^2}+\frac{d^4 (b+2 c x)^5 \left (a+b x+c x^2\right )^{3/2}}{16 c}+\frac{3 \left (b^2-4 a c\right )^4 d^4 \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{2048 c^{5/2}}\\ \end{align*}
Mathematica [A] time = 2.10874, size = 249, normalized size = 1.2 \[ \frac{1}{4} d^4 \left ((b+2 c x)^3 (a+x (b+c x))^{5/2}-2 c \left (a-\frac{b^2}{4 c}\right ) (b+2 c x) \sqrt{a+x (b+c x)} \left ((a+x (b+c x))^2-\frac{(a+x (b+c x)) \left (2 (b+2 c x) \sqrt{\frac{c (a+x (b+c x))}{4 a c-b^2}} \left (4 c \left (5 a+2 c x^2\right )-3 b^2+8 b c x\right )+3 \sqrt{c} \sqrt{4 a-\frac{b^2}{c}} \left (4 a c-b^2\right ) \sinh ^{-1}\left (\frac{b+2 c x}{\sqrt{c} \sqrt{4 a-\frac{b^2}{c}}}\right )\right )}{256 c (b+2 c x) \left (\frac{c (a+x (b+c x))}{4 a c-b^2}\right )^{3/2}}\right )\right ) \]
Antiderivative was successfully verified.
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Maple [B] time = 0.056, size = 641, normalized size = 3.1 \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 [A] time = 2.80686, size = 1540, normalized size = 7.44 \begin{align*} \left [\frac{3 \,{\left (b^{8} - 16 \, a b^{6} c + 96 \, a^{2} b^{4} c^{2} - 256 \, a^{3} b^{2} c^{3} + 256 \, a^{4} c^{4}\right )} \sqrt{c} d^{4} \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 (2048 \, c^{8} d^{4} x^{7} + 7168 \, b c^{7} d^{4} x^{6} + 768 \,{\left (13 \, b^{2} c^{6} + 4 \, a c^{7}\right )} d^{4} x^{5} + 640 \,{\left (11 \, b^{3} c^{5} + 12 \, a b c^{6}\right )} d^{4} x^{4} + 16 \,{\left (161 \, b^{4} c^{4} + 472 \, a b^{2} c^{5} + 16 \, a^{2} c^{6}\right )} d^{4} x^{3} + 24 \,{\left (17 \, b^{5} c^{3} + 152 \, a b^{3} c^{4} + 16 \, a^{2} b c^{5}\right )} d^{4} x^{2} + 2 \,{\left (b^{6} c^{2} + 396 \, a b^{4} c^{3} + 240 \, a^{2} b^{2} c^{4} - 192 \, a^{3} c^{5}\right )} d^{4} x -{\left (3 \, b^{7} c - 44 \, a b^{5} c^{2} - 176 \, a^{2} b^{3} c^{3} + 192 \, a^{3} b c^{4}\right )} d^{4}\right )} \sqrt{c x^{2} + b x + a}}{4096 \, c^{3}}, -\frac{3 \,{\left (b^{8} - 16 \, a b^{6} c + 96 \, a^{2} b^{4} c^{2} - 256 \, a^{3} b^{2} c^{3} + 256 \, a^{4} c^{4}\right )} \sqrt{-c} d^{4} \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 (2048 \, c^{8} d^{4} x^{7} + 7168 \, b c^{7} d^{4} x^{6} + 768 \,{\left (13 \, b^{2} c^{6} + 4 \, a c^{7}\right )} d^{4} x^{5} + 640 \,{\left (11 \, b^{3} c^{5} + 12 \, a b c^{6}\right )} d^{4} x^{4} + 16 \,{\left (161 \, b^{4} c^{4} + 472 \, a b^{2} c^{5} + 16 \, a^{2} c^{6}\right )} d^{4} x^{3} + 24 \,{\left (17 \, b^{5} c^{3} + 152 \, a b^{3} c^{4} + 16 \, a^{2} b c^{5}\right )} d^{4} x^{2} + 2 \,{\left (b^{6} c^{2} + 396 \, a b^{4} c^{3} + 240 \, a^{2} b^{2} c^{4} - 192 \, a^{3} c^{5}\right )} d^{4} x -{\left (3 \, b^{7} c - 44 \, a b^{5} c^{2} - 176 \, a^{2} b^{3} c^{3} + 192 \, a^{3} b c^{4}\right )} d^{4}\right )} \sqrt{c x^{2} + b x + a}}{2048 \, c^{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*} d^{4} \left (\int a b^{4} \sqrt{a + b x + c x^{2}}\, dx + \int b^{5} x \sqrt{a + b x + c x^{2}}\, dx + \int 16 c^{5} x^{6} \sqrt{a + b x + c x^{2}}\, dx + \int 16 a c^{4} x^{4} \sqrt{a + b x + c x^{2}}\, dx + \int 48 b c^{4} x^{5} \sqrt{a + b x + c x^{2}}\, dx + \int 56 b^{2} c^{3} x^{4} \sqrt{a + b x + c x^{2}}\, dx + \int 32 b^{3} c^{2} x^{3} \sqrt{a + b x + c x^{2}}\, dx + \int 9 b^{4} c x^{2} \sqrt{a + b x + c x^{2}}\, dx + \int 32 a b c^{3} x^{3} \sqrt{a + b x + c x^{2}}\, dx + \int 24 a b^{2} c^{2} x^{2} \sqrt{a + b x + c x^{2}}\, dx + \int 8 a b^{3} c x \sqrt{a + b x + c x^{2}}\, dx\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.19455, size = 528, normalized size = 2.55 \begin{align*} \frac{1}{1024} \, \sqrt{c x^{2} + b x + a}{\left (2 \,{\left (4 \,{\left (2 \,{\left (8 \,{\left (2 \,{\left (4 \,{\left (2 \, c^{5} d^{4} x + 7 \, b c^{4} d^{4}\right )} x + \frac{3 \,{\left (13 \, b^{2} c^{10} d^{4} + 4 \, a c^{11} d^{4}\right )}}{c^{7}}\right )} x + \frac{5 \,{\left (11 \, b^{3} c^{9} d^{4} + 12 \, a b c^{10} d^{4}\right )}}{c^{7}}\right )} x + \frac{161 \, b^{4} c^{8} d^{4} + 472 \, a b^{2} c^{9} d^{4} + 16 \, a^{2} c^{10} d^{4}}{c^{7}}\right )} x + \frac{3 \,{\left (17 \, b^{5} c^{7} d^{4} + 152 \, a b^{3} c^{8} d^{4} + 16 \, a^{2} b c^{9} d^{4}\right )}}{c^{7}}\right )} x + \frac{b^{6} c^{6} d^{4} + 396 \, a b^{4} c^{7} d^{4} + 240 \, a^{2} b^{2} c^{8} d^{4} - 192 \, a^{3} c^{9} d^{4}}{c^{7}}\right )} x - \frac{3 \, b^{7} c^{5} d^{4} - 44 \, a b^{5} c^{6} d^{4} - 176 \, a^{2} b^{3} c^{7} d^{4} + 192 \, a^{3} b c^{8} d^{4}}{c^{7}}\right )} - \frac{3 \,{\left (b^{8} d^{4} - 16 \, a b^{6} c d^{4} + 96 \, a^{2} b^{4} c^{2} d^{4} - 256 \, a^{3} b^{2} c^{3} d^{4} + 256 \, a^{4} c^{4} d^{4}\right )} \log \left ({\left | -2 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )} \sqrt{c} - b \right |}\right )}{2048 \, c^{\frac{5}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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