Optimal. Leaf size=207 \[ -\frac{5 d^2 \left (b^2-4 a c\right )^3 (b+2 c x) \sqrt{a+b x+c x^2}}{4096 c^3}+\frac{5 d^2 \left (b^2-4 a c\right )^2 (b+2 c x)^3 \sqrt{a+b x+c x^2}}{2048 c^3}-\frac{5 d^2 \left (b^2-4 a c\right ) (b+2 c x)^3 \left (a+b x+c x^2\right )^{3/2}}{384 c^2}-\frac{5 d^2 \left (b^2-4 a c\right )^4 \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{8192 c^{7/2}}+\frac{d^2 (b+2 c x)^3 \left (a+b x+c x^2\right )^{5/2}}{16 c} \]
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Rubi [A] time = 0.124275, 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} \[ -\frac{5 d^2 \left (b^2-4 a c\right )^3 (b+2 c x) \sqrt{a+b x+c x^2}}{4096 c^3}+\frac{5 d^2 \left (b^2-4 a c\right )^2 (b+2 c x)^3 \sqrt{a+b x+c x^2}}{2048 c^3}-\frac{5 d^2 \left (b^2-4 a c\right ) (b+2 c x)^3 \left (a+b x+c x^2\right )^{3/2}}{384 c^2}-\frac{5 d^2 \left (b^2-4 a c\right )^4 \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{8192 c^{7/2}}+\frac{d^2 (b+2 c x)^3 \left (a+b x+c x^2\right )^{5/2}}{16 c} \]
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)^2 \left (a+b x+c x^2\right )^{5/2} \, dx &=\frac{d^2 (b+2 c x)^3 \left (a+b x+c x^2\right )^{5/2}}{16 c}-\frac{\left (5 \left (b^2-4 a c\right )\right ) \int (b d+2 c d x)^2 \left (a+b x+c x^2\right )^{3/2} \, dx}{32 c}\\ &=-\frac{5 \left (b^2-4 a c\right ) d^2 (b+2 c x)^3 \left (a+b x+c x^2\right )^{3/2}}{384 c^2}+\frac{d^2 (b+2 c x)^3 \left (a+b x+c x^2\right )^{5/2}}{16 c}+\frac{\left (5 \left (b^2-4 a c\right )^2\right ) \int (b d+2 c d x)^2 \sqrt{a+b x+c x^2} \, dx}{256 c^2}\\ &=\frac{5 \left (b^2-4 a c\right )^2 d^2 (b+2 c x)^3 \sqrt{a+b x+c x^2}}{2048 c^3}-\frac{5 \left (b^2-4 a c\right ) d^2 (b+2 c x)^3 \left (a+b x+c x^2\right )^{3/2}}{384 c^2}+\frac{d^2 (b+2 c x)^3 \left (a+b x+c x^2\right )^{5/2}}{16 c}-\frac{\left (5 \left (b^2-4 a c\right )^3\right ) \int \frac{(b d+2 c d x)^2}{\sqrt{a+b x+c x^2}} \, dx}{4096 c^3}\\ &=-\frac{5 \left (b^2-4 a c\right )^3 d^2 (b+2 c x) \sqrt{a+b x+c x^2}}{4096 c^3}+\frac{5 \left (b^2-4 a c\right )^2 d^2 (b+2 c x)^3 \sqrt{a+b x+c x^2}}{2048 c^3}-\frac{5 \left (b^2-4 a c\right ) d^2 (b+2 c x)^3 \left (a+b x+c x^2\right )^{3/2}}{384 c^2}+\frac{d^2 (b+2 c x)^3 \left (a+b x+c x^2\right )^{5/2}}{16 c}-\frac{\left (5 \left (b^2-4 a c\right )^4 d^2\right ) \int \frac{1}{\sqrt{a+b x+c x^2}} \, dx}{8192 c^3}\\ &=-\frac{5 \left (b^2-4 a c\right )^3 d^2 (b+2 c x) \sqrt{a+b x+c x^2}}{4096 c^3}+\frac{5 \left (b^2-4 a c\right )^2 d^2 (b+2 c x)^3 \sqrt{a+b x+c x^2}}{2048 c^3}-\frac{5 \left (b^2-4 a c\right ) d^2 (b+2 c x)^3 \left (a+b x+c x^2\right )^{3/2}}{384 c^2}+\frac{d^2 (b+2 c x)^3 \left (a+b x+c x^2\right )^{5/2}}{16 c}-\frac{\left (5 \left (b^2-4 a c\right )^4 d^2\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 )}{4096 c^3}\\ &=-\frac{5 \left (b^2-4 a c\right )^3 d^2 (b+2 c x) \sqrt{a+b x+c x^2}}{4096 c^3}+\frac{5 \left (b^2-4 a c\right )^2 d^2 (b+2 c x)^3 \sqrt{a+b x+c x^2}}{2048 c^3}-\frac{5 \left (b^2-4 a c\right ) d^2 (b+2 c x)^3 \left (a+b x+c x^2\right )^{3/2}}{384 c^2}+\frac{d^2 (b+2 c x)^3 \left (a+b x+c x^2\right )^{5/2}}{16 c}-\frac{5 \left (b^2-4 a c\right )^4 d^2 \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{8192 c^{7/2}}\\ \end{align*}
Mathematica [A] time = 0.989312, size = 225, normalized size = 1.09 \[ \frac{1}{4} d^2 (b+2 c x) \sqrt{a+x (b+c x)} \left (\frac{\left (b^2-4 a c\right ) \left (16 c^2 \left (33 a^2+26 a c x^2+8 c^2 x^4\right )+8 b^2 c \left (11 c x^2-20 a\right )+32 b c^2 x \left (13 a+8 c x^2\right )-40 b^3 c x+15 b^4\right )}{3072 c^3}-\frac{5 \sqrt{c} \sqrt{4 a-\frac{b^2}{c}} (a+x (b+c x))^3 \sinh ^{-1}\left (\frac{b+2 c x}{\sqrt{c} \sqrt{4 a-\frac{b^2}{c}}}\right )}{2048 (b+2 c x) \left (\frac{c (a+x (b+c x))}{4 a c-b^2}\right )^{7/2}}+(a+x (b+c x))^3\right ) \]
Antiderivative was successfully verified.
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Maple [B] time = 0.049, size = 634, 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 = 3.12294, size = 1569, normalized size = 7.58 \begin{align*} \left [\frac{15 \,{\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^{2} \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 (6144 \, c^{8} d^{2} x^{7} + 21504 \, b c^{7} d^{2} x^{6} + 256 \,{\left (109 \, b^{2} c^{6} + 68 \, a c^{7}\right )} d^{2} x^{5} + 640 \,{\left (25 \, b^{3} c^{5} + 68 \, a b c^{6}\right )} d^{2} x^{4} + 16 \,{\left (219 \, b^{4} c^{4} + 2248 \, a b^{2} c^{5} + 944 \, a^{2} c^{6}\right )} d^{2} x^{3} + 8 \,{\left (b^{5} c^{3} + 1304 \, a b^{3} c^{4} + 2832 \, a^{2} b c^{5}\right )} d^{2} x^{2} - 2 \,{\left (5 \, b^{6} c^{2} - 68 \, a b^{4} c^{3} - 4944 \, a^{2} b^{2} c^{4} - 960 \, a^{3} c^{5}\right )} d^{2} x +{\left (15 \, b^{7} c - 220 \, a b^{5} c^{2} + 1168 \, a^{2} b^{3} c^{3} + 960 \, a^{3} b c^{4}\right )} d^{2}\right )} \sqrt{c x^{2} + b x + a}}{49152 \, c^{4}}, \frac{15 \,{\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^{2} \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 (6144 \, c^{8} d^{2} x^{7} + 21504 \, b c^{7} d^{2} x^{6} + 256 \,{\left (109 \, b^{2} c^{6} + 68 \, a c^{7}\right )} d^{2} x^{5} + 640 \,{\left (25 \, b^{3} c^{5} + 68 \, a b c^{6}\right )} d^{2} x^{4} + 16 \,{\left (219 \, b^{4} c^{4} + 2248 \, a b^{2} c^{5} + 944 \, a^{2} c^{6}\right )} d^{2} x^{3} + 8 \,{\left (b^{5} c^{3} + 1304 \, a b^{3} c^{4} + 2832 \, a^{2} b c^{5}\right )} d^{2} x^{2} - 2 \,{\left (5 \, b^{6} c^{2} - 68 \, a b^{4} c^{3} - 4944 \, a^{2} b^{2} c^{4} - 960 \, a^{3} c^{5}\right )} d^{2} x +{\left (15 \, b^{7} c - 220 \, a b^{5} c^{2} + 1168 \, a^{2} b^{3} c^{3} + 960 \, a^{3} b c^{4}\right )} d^{2}\right )} \sqrt{c x^{2} + b x + a}}{24576 \, c^{4}}\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^{2} \left (\int a^{2} b^{2} \sqrt{a + b x + c x^{2}}\, dx + \int b^{4} x^{2} \sqrt{a + b x + c x^{2}}\, dx + \int 4 c^{4} x^{6} \sqrt{a + b x + c x^{2}}\, dx + \int 2 a b^{3} x \sqrt{a + b x + c x^{2}}\, dx + \int 8 a c^{3} x^{4} \sqrt{a + b x + c x^{2}}\, dx + \int 4 a^{2} c^{2} x^{2} \sqrt{a + b x + c x^{2}}\, dx + \int 12 b c^{3} x^{5} \sqrt{a + b x + c x^{2}}\, dx + \int 13 b^{2} c^{2} x^{4} \sqrt{a + b x + c x^{2}}\, dx + \int 6 b^{3} c x^{3} \sqrt{a + b x + c x^{2}}\, dx + \int 16 a b c^{2} x^{3} \sqrt{a + b x + c x^{2}}\, dx + \int 10 a b^{2} c x^{2} \sqrt{a + b x + c x^{2}}\, dx + \int 4 a^{2} b 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 = 1.21522, size = 525, normalized size = 2.54 \begin{align*} \frac{1}{12288} \, \sqrt{c x^{2} + b x + a}{\left (2 \,{\left (4 \,{\left (2 \,{\left (8 \,{\left (2 \,{\left (12 \,{\left (2 \, c^{4} d^{2} x + 7 \, b c^{3} d^{2}\right )} x + \frac{109 \, b^{2} c^{9} d^{2} + 68 \, a c^{10} d^{2}}{c^{7}}\right )} x + \frac{5 \,{\left (25 \, b^{3} c^{8} d^{2} + 68 \, a b c^{9} d^{2}\right )}}{c^{7}}\right )} x + \frac{219 \, b^{4} c^{7} d^{2} + 2248 \, a b^{2} c^{8} d^{2} + 944 \, a^{2} c^{9} d^{2}}{c^{7}}\right )} x + \frac{b^{5} c^{6} d^{2} + 1304 \, a b^{3} c^{7} d^{2} + 2832 \, a^{2} b c^{8} d^{2}}{c^{7}}\right )} x - \frac{5 \, b^{6} c^{5} d^{2} - 68 \, a b^{4} c^{6} d^{2} - 4944 \, a^{2} b^{2} c^{7} d^{2} - 960 \, a^{3} c^{8} d^{2}}{c^{7}}\right )} x + \frac{15 \, b^{7} c^{4} d^{2} - 220 \, a b^{5} c^{5} d^{2} + 1168 \, a^{2} b^{3} c^{6} d^{2} + 960 \, a^{3} b c^{7} d^{2}}{c^{7}}\right )} + \frac{5 \,{\left (b^{8} d^{2} - 16 \, a b^{6} c d^{2} + 96 \, a^{2} b^{4} c^{2} d^{2} - 256 \, a^{3} b^{2} c^{3} d^{2} + 256 \, a^{4} c^{4} d^{2}\right )} \log \left ({\left | -2 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )} \sqrt{c} - b \right |}\right )}{8192 \, c^{\frac{7}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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