Optimal. Leaf size=261 \[ -\frac {\left (7 b \left (528 a^2 c^2-680 a b^2 c+165 b^4\right )-2 c x \left (400 a^2 c^2-1176 a b^2 c+385 b^4\right )\right ) \sqrt {a+b x+c x^2}}{2560 c^6}+\frac {\left (-320 a^3 c^3+1680 a^2 b^2 c^2-1260 a b^4 c+231 b^6\right ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{1024 c^{13/2}}-\frac {b x^2 \left (77 b^2-156 a c\right ) \sqrt {a+b x+c x^2}}{320 c^4}+\frac {x^3 \left (99 b^2-100 a c\right ) \sqrt {a+b x+c x^2}}{480 c^3}-\frac {11 b x^4 \sqrt {a+b x+c x^2}}{60 c^2}+\frac {x^5 \sqrt {a+b x+c x^2}}{6 c} \]
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Rubi [A] time = 0.38, antiderivative size = 261, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.278, Rules used = {742, 832, 779, 621, 206} \begin {gather*} -\frac {\left (7 b \left (528 a^2 c^2-680 a b^2 c+165 b^4\right )-2 c x \left (400 a^2 c^2-1176 a b^2 c+385 b^4\right )\right ) \sqrt {a+b x+c x^2}}{2560 c^6}+\frac {\left (1680 a^2 b^2 c^2-320 a^3 c^3-1260 a b^4 c+231 b^6\right ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{1024 c^{13/2}}+\frac {x^3 \left (99 b^2-100 a c\right ) \sqrt {a+b x+c x^2}}{480 c^3}-\frac {b x^2 \left (77 b^2-156 a c\right ) \sqrt {a+b x+c x^2}}{320 c^4}-\frac {11 b x^4 \sqrt {a+b x+c x^2}}{60 c^2}+\frac {x^5 \sqrt {a+b x+c x^2}}{6 c} \end {gather*}
Antiderivative was successfully verified.
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Rule 206
Rule 621
Rule 742
Rule 779
Rule 832
Rubi steps
\begin {align*} \int \frac {x^6}{\sqrt {a+b x+c x^2}} \, dx &=\frac {x^5 \sqrt {a+b x+c x^2}}{6 c}+\frac {\int \frac {x^4 \left (-5 a-\frac {11 b x}{2}\right )}{\sqrt {a+b x+c x^2}} \, dx}{6 c}\\ &=-\frac {11 b x^4 \sqrt {a+b x+c x^2}}{60 c^2}+\frac {x^5 \sqrt {a+b x+c x^2}}{6 c}+\frac {\int \frac {x^3 \left (22 a b+\frac {1}{4} \left (99 b^2-100 a c\right ) x\right )}{\sqrt {a+b x+c x^2}} \, dx}{30 c^2}\\ &=\frac {\left (99 b^2-100 a c\right ) x^3 \sqrt {a+b x+c x^2}}{480 c^3}-\frac {11 b x^4 \sqrt {a+b x+c x^2}}{60 c^2}+\frac {x^5 \sqrt {a+b x+c x^2}}{6 c}+\frac {\int \frac {x^2 \left (-\frac {3}{4} a \left (99 b^2-100 a c\right )-\frac {9}{8} b \left (77 b^2-156 a c\right ) x\right )}{\sqrt {a+b x+c x^2}} \, dx}{120 c^3}\\ &=-\frac {b \left (77 b^2-156 a c\right ) x^2 \sqrt {a+b x+c x^2}}{320 c^4}+\frac {\left (99 b^2-100 a c\right ) x^3 \sqrt {a+b x+c x^2}}{480 c^3}-\frac {11 b x^4 \sqrt {a+b x+c x^2}}{60 c^2}+\frac {x^5 \sqrt {a+b x+c x^2}}{6 c}+\frac {\int \frac {x \left (\frac {9}{4} a b \left (77 b^2-156 a c\right )+\frac {9}{16} \left (385 b^4-1176 a b^2 c+400 a^2 c^2\right ) x\right )}{\sqrt {a+b x+c x^2}} \, dx}{360 c^4}\\ &=-\frac {b \left (77 b^2-156 a c\right ) x^2 \sqrt {a+b x+c x^2}}{320 c^4}+\frac {\left (99 b^2-100 a c\right ) x^3 \sqrt {a+b x+c x^2}}{480 c^3}-\frac {11 b x^4 \sqrt {a+b x+c x^2}}{60 c^2}+\frac {x^5 \sqrt {a+b x+c x^2}}{6 c}-\frac {\left (7 b \left (165 b^4-680 a b^2 c+528 a^2 c^2\right )-2 c \left (385 b^4-1176 a b^2 c+400 a^2 c^2\right ) x\right ) \sqrt {a+b x+c x^2}}{2560 c^6}+\frac {\left (231 b^6-1260 a b^4 c+1680 a^2 b^2 c^2-320 a^3 c^3\right ) \int \frac {1}{\sqrt {a+b x+c x^2}} \, dx}{1024 c^6}\\ &=-\frac {b \left (77 b^2-156 a c\right ) x^2 \sqrt {a+b x+c x^2}}{320 c^4}+\frac {\left (99 b^2-100 a c\right ) x^3 \sqrt {a+b x+c x^2}}{480 c^3}-\frac {11 b x^4 \sqrt {a+b x+c x^2}}{60 c^2}+\frac {x^5 \sqrt {a+b x+c x^2}}{6 c}-\frac {\left (7 b \left (165 b^4-680 a b^2 c+528 a^2 c^2\right )-2 c \left (385 b^4-1176 a b^2 c+400 a^2 c^2\right ) x\right ) \sqrt {a+b x+c x^2}}{2560 c^6}+\frac {\left (231 b^6-1260 a b^4 c+1680 a^2 b^2 c^2-320 a^3 c^3\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 )}{512 c^6}\\ &=-\frac {b \left (77 b^2-156 a c\right ) x^2 \sqrt {a+b x+c x^2}}{320 c^4}+\frac {\left (99 b^2-100 a c\right ) x^3 \sqrt {a+b x+c x^2}}{480 c^3}-\frac {11 b x^4 \sqrt {a+b x+c x^2}}{60 c^2}+\frac {x^5 \sqrt {a+b x+c x^2}}{6 c}-\frac {\left (7 b \left (165 b^4-680 a b^2 c+528 a^2 c^2\right )-2 c \left (385 b^4-1176 a b^2 c+400 a^2 c^2\right ) x\right ) \sqrt {a+b x+c x^2}}{2560 c^6}+\frac {\left (231 b^6-1260 a b^4 c+1680 a^2 b^2 c^2-320 a^3 c^3\right ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{1024 c^{13/2}}\\ \end {align*}
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Mathematica [A] time = 0.32, size = 263, normalized size = 1.01 \begin {gather*} \frac {\left (-320 a^3 c^3+1680 a^2 b^2 c^2-1260 a b^4 c+231 b^6\right ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+x (b+c x)}}\right )}{1024 c^{13/2}}+\frac {48 a^3 c^2 (50 c x-231 b)+8 a^2 c \left (1785 b^3-2268 b^2 c x-618 b c^2 x^2+100 c^3 x^3\right )+a \left (-3465 b^5+16590 b^4 c x+5376 b^3 c^2 x^2-1728 b^2 c^3 x^3+736 b c^4 x^4-320 c^5 x^5\right )+x \left (-3465 b^6-1155 b^5 c x+462 b^4 c^2 x^2-264 b^3 c^3 x^3+176 b^2 c^4 x^4-128 b c^5 x^5+1280 c^6 x^6\right )}{7680 c^6 \sqrt {a+x (b+c x)}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.62, size = 197, normalized size = 0.75 \begin {gather*} \frac {\sqrt {a+b x+c x^2} \left (-11088 a^2 b c^2+2400 a^2 c^3 x+14280 a b^3 c-7056 a b^2 c^2 x+3744 a b c^3 x^2-1600 a c^4 x^3-3465 b^5+2310 b^4 c x-1848 b^3 c^2 x^2+1584 b^2 c^3 x^3-1408 b c^4 x^4+1280 c^5 x^5\right )}{7680 c^6}+\frac {\left (320 a^3 c^3-1680 a^2 b^2 c^2+1260 a b^4 c-231 b^6\right ) \log \left (-2 \sqrt {c} \sqrt {a+b x+c x^2}+b+2 c x\right )}{1024 c^{13/2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 431, normalized size = 1.65 \begin {gather*} \left [-\frac {15 \, {\left (231 \, b^{6} - 1260 \, a b^{4} c + 1680 \, a^{2} b^{2} c^{2} - 320 \, a^{3} c^{3}\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 (1280 \, c^{6} x^{5} - 1408 \, b c^{5} x^{4} - 3465 \, b^{5} c + 14280 \, a b^{3} c^{2} - 11088 \, a^{2} b c^{3} + 16 \, {\left (99 \, b^{2} c^{4} - 100 \, a c^{5}\right )} x^{3} - 24 \, {\left (77 \, b^{3} c^{3} - 156 \, a b c^{4}\right )} x^{2} + 6 \, {\left (385 \, b^{4} c^{2} - 1176 \, a b^{2} c^{3} + 400 \, a^{2} c^{4}\right )} x\right )} \sqrt {c x^{2} + b x + a}}{30720 \, c^{7}}, -\frac {15 \, {\left (231 \, b^{6} - 1260 \, a b^{4} c + 1680 \, a^{2} b^{2} c^{2} - 320 \, a^{3} c^{3}\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 (1280 \, c^{6} x^{5} - 1408 \, b c^{5} x^{4} - 3465 \, b^{5} c + 14280 \, a b^{3} c^{2} - 11088 \, a^{2} b c^{3} + 16 \, {\left (99 \, b^{2} c^{4} - 100 \, a c^{5}\right )} x^{3} - 24 \, {\left (77 \, b^{3} c^{3} - 156 \, a b c^{4}\right )} x^{2} + 6 \, {\left (385 \, b^{4} c^{2} - 1176 \, a b^{2} c^{3} + 400 \, a^{2} c^{4}\right )} x\right )} \sqrt {c x^{2} + b x + a}}{15360 \, c^{7}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.28, size = 208, normalized size = 0.80 \begin {gather*} \frac {1}{7680} \, \sqrt {c x^{2} + b x + a} {\left (2 \, {\left (4 \, {\left (2 \, {\left (8 \, x {\left (\frac {10 \, x}{c} - \frac {11 \, b}{c^{2}}\right )} + \frac {99 \, b^{2} c^{3} - 100 \, a c^{4}}{c^{6}}\right )} x - \frac {3 \, {\left (77 \, b^{3} c^{2} - 156 \, a b c^{3}\right )}}{c^{6}}\right )} x + \frac {3 \, {\left (385 \, b^{4} c - 1176 \, a b^{2} c^{2} + 400 \, a^{2} c^{3}\right )}}{c^{6}}\right )} x - \frac {21 \, {\left (165 \, b^{5} - 680 \, a b^{3} c + 528 \, a^{2} b c^{2}\right )}}{c^{6}}\right )} - \frac {{\left (231 \, b^{6} - 1260 \, a b^{4} c + 1680 \, a^{2} b^{2} c^{2} - 320 \, a^{3} c^{3}\right )} \log \left ({\left | -2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} \sqrt {c} - b \right |}\right )}{1024 \, c^{\frac {13}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.09, size = 394, normalized size = 1.51 \begin {gather*} \frac {\sqrt {c \,x^{2}+b x +a}\, x^{5}}{6 c}-\frac {11 \sqrt {c \,x^{2}+b x +a}\, b \,x^{4}}{60 c^{2}}-\frac {5 \sqrt {c \,x^{2}+b x +a}\, a \,x^{3}}{24 c^{2}}+\frac {33 \sqrt {c \,x^{2}+b x +a}\, b^{2} x^{3}}{160 c^{3}}+\frac {39 \sqrt {c \,x^{2}+b x +a}\, a b \,x^{2}}{80 c^{3}}-\frac {77 \sqrt {c \,x^{2}+b x +a}\, b^{3} x^{2}}{320 c^{4}}-\frac {5 a^{3} \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{16 c^{\frac {7}{2}}}+\frac {105 a^{2} b^{2} \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{64 c^{\frac {9}{2}}}-\frac {315 a \,b^{4} \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{256 c^{\frac {11}{2}}}+\frac {231 b^{6} \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{1024 c^{\frac {13}{2}}}+\frac {5 \sqrt {c \,x^{2}+b x +a}\, a^{2} x}{16 c^{3}}-\frac {147 \sqrt {c \,x^{2}+b x +a}\, a \,b^{2} x}{160 c^{4}}+\frac {77 \sqrt {c \,x^{2}+b x +a}\, b^{4} x}{256 c^{5}}-\frac {231 \sqrt {c \,x^{2}+b x +a}\, a^{2} b}{160 c^{4}}+\frac {119 \sqrt {c \,x^{2}+b x +a}\, a \,b^{3}}{64 c^{5}}-\frac {231 \sqrt {c \,x^{2}+b x +a}\, b^{5}}{512 c^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {x^6}{\sqrt {c\,x^2+b\,x+a}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{6}}{\sqrt {a + b x + c x^{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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