Optimal. Leaf size=249 \[ -\frac{3 d^4 \left (b^2-4 a c\right )^4 (b+2 c x) \sqrt{a+b x+c x^2}}{8192 c^3}-\frac{d^4 \left (b^2-4 a c\right )^3 (b+2 c x)^3 \sqrt{a+b x+c x^2}}{4096 c^3}+\frac{d^4 \left (b^2-4 a c\right )^2 (b+2 c x)^5 \sqrt{a+b x+c x^2}}{1024 c^3}-\frac{d^4 \left (b^2-4 a c\right ) (b+2 c x)^5 \left (a+b x+c x^2\right )^{3/2}}{128 c^2}-\frac{3 d^4 \left (b^2-4 a c\right )^5 \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{16384 c^{7/2}}+\frac{d^4 (b+2 c x)^5 \left (a+b x+c x^2\right )^{5/2}}{20 c} \]
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Rubi [A] time = 0.170779, antiderivative size = 249, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 4, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {685, 692, 621, 206} \[ -\frac{3 d^4 \left (b^2-4 a c\right )^4 (b+2 c x) \sqrt{a+b x+c x^2}}{8192 c^3}-\frac{d^4 \left (b^2-4 a c\right )^3 (b+2 c x)^3 \sqrt{a+b x+c x^2}}{4096 c^3}+\frac{d^4 \left (b^2-4 a c\right )^2 (b+2 c x)^5 \sqrt{a+b x+c x^2}}{1024 c^3}-\frac{d^4 \left (b^2-4 a c\right ) (b+2 c x)^5 \left (a+b x+c x^2\right )^{3/2}}{128 c^2}-\frac{3 d^4 \left (b^2-4 a c\right )^5 \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{16384 c^{7/2}}+\frac{d^4 (b+2 c x)^5 \left (a+b x+c x^2\right )^{5/2}}{20 c} \]
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 )^{5/2} \, dx &=\frac{d^4 (b+2 c x)^5 \left (a+b x+c x^2\right )^{5/2}}{20 c}-\frac{\left (b^2-4 a c\right ) \int (b d+2 c d x)^4 \left (a+b x+c x^2\right )^{3/2} \, dx}{8 c}\\ &=-\frac{\left (b^2-4 a c\right ) d^4 (b+2 c x)^5 \left (a+b x+c x^2\right )^{3/2}}{128 c^2}+\frac{d^4 (b+2 c x)^5 \left (a+b x+c x^2\right )^{5/2}}{20 c}+\frac{\left (3 \left (b^2-4 a c\right )^2\right ) \int (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)^5 \sqrt{a+b x+c x^2}}{1024 c^3}-\frac{\left (b^2-4 a c\right ) d^4 (b+2 c x)^5 \left (a+b x+c x^2\right )^{3/2}}{128 c^2}+\frac{d^4 (b+2 c x)^5 \left (a+b x+c x^2\right )^{5/2}}{20 c}-\frac{\left (b^2-4 a c\right )^3 \int \frac{(b d+2 c d x)^4}{\sqrt{a+b x+c x^2}} \, dx}{2048 c^3}\\ &=-\frac{\left (b^2-4 a c\right )^3 d^4 (b+2 c x)^3 \sqrt{a+b x+c x^2}}{4096 c^3}+\frac{\left (b^2-4 a c\right )^2 d^4 (b+2 c x)^5 \sqrt{a+b x+c x^2}}{1024 c^3}-\frac{\left (b^2-4 a c\right ) d^4 (b+2 c x)^5 \left (a+b x+c x^2\right )^{3/2}}{128 c^2}+\frac{d^4 (b+2 c x)^5 \left (a+b x+c x^2\right )^{5/2}}{20 c}-\frac{\left (3 \left (b^2-4 a c\right )^4 d^2\right ) \int \frac{(b d+2 c d x)^2}{\sqrt{a+b x+c x^2}} \, dx}{8192 c^3}\\ &=-\frac{3 \left (b^2-4 a c\right )^4 d^4 (b+2 c x) \sqrt{a+b x+c x^2}}{8192 c^3}-\frac{\left (b^2-4 a c\right )^3 d^4 (b+2 c x)^3 \sqrt{a+b x+c x^2}}{4096 c^3}+\frac{\left (b^2-4 a c\right )^2 d^4 (b+2 c x)^5 \sqrt{a+b x+c x^2}}{1024 c^3}-\frac{\left (b^2-4 a c\right ) d^4 (b+2 c x)^5 \left (a+b x+c x^2\right )^{3/2}}{128 c^2}+\frac{d^4 (b+2 c x)^5 \left (a+b x+c x^2\right )^{5/2}}{20 c}-\frac{\left (3 \left (b^2-4 a c\right )^5 d^4\right ) \int \frac{1}{\sqrt{a+b x+c x^2}} \, dx}{16384 c^3}\\ &=-\frac{3 \left (b^2-4 a c\right )^4 d^4 (b+2 c x) \sqrt{a+b x+c x^2}}{8192 c^3}-\frac{\left (b^2-4 a c\right )^3 d^4 (b+2 c x)^3 \sqrt{a+b x+c x^2}}{4096 c^3}+\frac{\left (b^2-4 a c\right )^2 d^4 (b+2 c x)^5 \sqrt{a+b x+c x^2}}{1024 c^3}-\frac{\left (b^2-4 a c\right ) d^4 (b+2 c x)^5 \left (a+b x+c x^2\right )^{3/2}}{128 c^2}+\frac{d^4 (b+2 c x)^5 \left (a+b x+c x^2\right )^{5/2}}{20 c}-\frac{\left (3 \left (b^2-4 a c\right )^5 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 )}{8192 c^3}\\ &=-\frac{3 \left (b^2-4 a c\right )^4 d^4 (b+2 c x) \sqrt{a+b x+c x^2}}{8192 c^3}-\frac{\left (b^2-4 a c\right )^3 d^4 (b+2 c x)^3 \sqrt{a+b x+c x^2}}{4096 c^3}+\frac{\left (b^2-4 a c\right )^2 d^4 (b+2 c x)^5 \sqrt{a+b x+c x^2}}{1024 c^3}-\frac{\left (b^2-4 a c\right ) d^4 (b+2 c x)^5 \left (a+b x+c x^2\right )^{3/2}}{128 c^2}+\frac{d^4 (b+2 c x)^5 \left (a+b x+c x^2\right )^{5/2}}{20 c}-\frac{3 \left (b^2-4 a c\right )^5 d^4 \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{16384 c^{7/2}}\\ \end{align*}
Mathematica [A] time = 3.74033, size = 265, normalized size = 1.06 \[ \frac{1}{5} d^4 \left ((b+2 c x)^3 (a+x (b+c x))^{7/2}-\frac{3}{2} c \left (a-\frac{b^2}{4 c}\right ) (b+2 c x) \sqrt{a+x (b+c x)} \left (\frac{\left (b^2-4 a c\right ) \left (16 c^2 \left (33 a^2+26 a c x^2+8 c^2 x^4\right )+8 b^2 c \left (11 c x^2-20 a\right )+32 b c^2 x \left (13 a+8 c x^2\right )-40 b^3 c x+15 b^4\right )}{3072 c^3}-\frac{5 \sqrt{c} \sqrt{4 a-\frac{b^2}{c}} (a+x (b+c x))^3 \sinh ^{-1}\left (\frac{b+2 c x}{\sqrt{c} \sqrt{4 a-\frac{b^2}{c}}}\right )}{2048 (b+2 c x) \left (\frac{c (a+x (b+c x))}{4 a c-b^2}\right )^{7/2}}+(a+x (b+c x))^3\right )\right ) \]
Antiderivative was successfully verified.
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Maple [B] time = 0.056, size = 920, normalized size = 3.7 \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 = 3.82941, size = 2192, normalized size = 8.8 \begin{align*} \text{result too large to display} \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^{2} b^{4} \sqrt{a + b x + c x^{2}}\, dx + \int b^{6} x^{2} \sqrt{a + b x + c x^{2}}\, dx + \int 16 c^{6} x^{8} \sqrt{a + b x + c x^{2}}\, dx + \int 2 a b^{5} x \sqrt{a + b x + c x^{2}}\, dx + \int 32 a c^{5} x^{6} \sqrt{a + b x + c x^{2}}\, dx + \int 16 a^{2} c^{4} x^{4} \sqrt{a + b x + c x^{2}}\, dx + \int 64 b c^{5} x^{7} \sqrt{a + b x + c x^{2}}\, dx + \int 104 b^{2} c^{4} x^{6} \sqrt{a + b x + c x^{2}}\, dx + \int 88 b^{3} c^{3} x^{5} \sqrt{a + b x + c x^{2}}\, dx + \int 41 b^{4} c^{2} x^{4} \sqrt{a + b x + c x^{2}}\, dx + \int 10 b^{5} c x^{3} \sqrt{a + b x + c x^{2}}\, dx + \int 96 a b c^{4} x^{5} \sqrt{a + b x + c x^{2}}\, dx + \int 112 a b^{2} c^{3} x^{4} \sqrt{a + b x + c x^{2}}\, dx + \int 64 a b^{3} c^{2} x^{3} \sqrt{a + b x + c x^{2}}\, dx + \int 18 a b^{4} c x^{2} \sqrt{a + b x + c x^{2}}\, dx + \int 32 a^{2} b c^{3} x^{3} \sqrt{a + b x + c x^{2}}\, dx + \int 24 a^{2} b^{2} c^{2} x^{2} \sqrt{a + b x + c x^{2}}\, dx + \int 8 a^{2} 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 = 1.20954, size = 738, normalized size = 2.96 \begin{align*} \frac{1}{40960} \, \sqrt{c x^{2} + b x + a}{\left (2 \,{\left (4 \,{\left (2 \,{\left (8 \,{\left (2 \,{\left (4 \,{\left (2 \,{\left (16 \,{\left (2 \, c^{6} d^{4} x + 9 \, b c^{5} d^{4}\right )} x + \frac{3 \,{\left (89 \, b^{2} c^{13} d^{4} + 28 \, a c^{14} d^{4}\right )}}{c^{9}}\right )} x + \frac{21 \,{\left (25 \, b^{3} c^{12} d^{4} + 28 \, a b c^{13} d^{4}\right )}}{c^{9}}\right )} x + \frac{1165 \, b^{4} c^{11} d^{4} + 3280 \, a b^{2} c^{12} d^{4} + 496 \, a^{2} c^{13} d^{4}}{c^{9}}\right )} x + \frac{701 \, b^{5} c^{10} d^{4} + 4640 \, a b^{3} c^{11} d^{4} + 2480 \, a^{2} b c^{12} d^{4}}{c^{9}}\right )} x + \frac{731 \, b^{6} c^{9} d^{4} + 13660 \, a b^{4} c^{10} d^{4} + 19600 \, a^{2} b^{2} c^{11} d^{4} + 320 \, a^{3} c^{12} d^{4}}{c^{9}}\right )} x + \frac{b^{7} c^{8} d^{4} + 4372 \, a b^{5} c^{9} d^{4} + 19120 \, a^{2} b^{3} c^{10} d^{4} + 960 \, a^{3} b c^{11} d^{4}}{c^{9}}\right )} x - \frac{5 \, b^{8} c^{7} d^{4} - 88 \, a b^{6} c^{8} d^{4} - 16960 \, a^{2} b^{4} c^{9} d^{4} - 5760 \, a^{3} b^{2} c^{10} d^{4} + 3840 \, a^{4} c^{11} d^{4}}{c^{9}}\right )} x + \frac{15 \, b^{9} c^{6} d^{4} - 280 \, a b^{7} c^{7} d^{4} + 2048 \, a^{2} b^{5} c^{8} d^{4} + 4480 \, a^{3} b^{3} c^{9} d^{4} - 3840 \, a^{4} b c^{10} d^{4}}{c^{9}}\right )} + \frac{3 \,{\left (b^{10} d^{4} - 20 \, a b^{8} c d^{4} + 160 \, a^{2} b^{6} c^{2} d^{4} - 640 \, a^{3} b^{4} c^{3} d^{4} + 1280 \, a^{4} b^{2} c^{4} d^{4} - 1024 \, a^{5} c^{5} d^{4}\right )} \log \left ({\left | -2 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )} \sqrt{c} - b \right |}\right )}{16384 \, c^{\frac{7}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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