Optimal. Leaf size=202 \[ -\frac {b \left (240 a^2 c^2-280 a b^2 c+63 b^4\right ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{256 c^{11/2}}+\frac {\left (1024 a^2 c^2-14 b c x \left (45 b^2-92 a c\right )-2940 a b^2 c+945 b^4\right ) \sqrt {a+b x+c x^2}}{1920 c^5}+\frac {x^2 \left (63 b^2-64 a c\right ) \sqrt {a+b x+c x^2}}{240 c^3}-\frac {9 b x^3 \sqrt {a+b x+c x^2}}{40 c^2}+\frac {x^4 \sqrt {a+b x+c x^2}}{5 c} \]
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Rubi [A] time = 0.22, antiderivative size = 202, normalized size of antiderivative = 1.00, number of steps used = 6, 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} \[ \frac {\left (1024 a^2 c^2-14 b c x \left (45 b^2-92 a c\right )-2940 a b^2 c+945 b^4\right ) \sqrt {a+b x+c x^2}}{1920 c^5}-\frac {b \left (240 a^2 c^2-280 a b^2 c+63 b^4\right ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{256 c^{11/2}}+\frac {x^2 \left (63 b^2-64 a c\right ) \sqrt {a+b x+c x^2}}{240 c^3}-\frac {9 b x^3 \sqrt {a+b x+c x^2}}{40 c^2}+\frac {x^4 \sqrt {a+b x+c x^2}}{5 c} \]
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^5}{\sqrt {a+b x+c x^2}} \, dx &=\frac {x^4 \sqrt {a+b x+c x^2}}{5 c}+\frac {\int \frac {x^3 \left (-4 a-\frac {9 b x}{2}\right )}{\sqrt {a+b x+c x^2}} \, dx}{5 c}\\ &=-\frac {9 b x^3 \sqrt {a+b x+c x^2}}{40 c^2}+\frac {x^4 \sqrt {a+b x+c x^2}}{5 c}+\frac {\int \frac {x^2 \left (\frac {27 a b}{2}+\frac {1}{4} \left (63 b^2-64 a c\right ) x\right )}{\sqrt {a+b x+c x^2}} \, dx}{20 c^2}\\ &=\frac {\left (63 b^2-64 a c\right ) x^2 \sqrt {a+b x+c x^2}}{240 c^3}-\frac {9 b x^3 \sqrt {a+b x+c x^2}}{40 c^2}+\frac {x^4 \sqrt {a+b x+c x^2}}{5 c}+\frac {\int \frac {x \left (-\frac {1}{2} a \left (63 b^2-64 a c\right )-\frac {7}{8} b \left (45 b^2-92 a c\right ) x\right )}{\sqrt {a+b x+c x^2}} \, dx}{60 c^3}\\ &=\frac {\left (63 b^2-64 a c\right ) x^2 \sqrt {a+b x+c x^2}}{240 c^3}-\frac {9 b x^3 \sqrt {a+b x+c x^2}}{40 c^2}+\frac {x^4 \sqrt {a+b x+c x^2}}{5 c}+\frac {\left (945 b^4-2940 a b^2 c+1024 a^2 c^2-14 b c \left (45 b^2-92 a c\right ) x\right ) \sqrt {a+b x+c x^2}}{1920 c^5}-\frac {\left (b \left (63 b^4-280 a b^2 c+240 a^2 c^2\right )\right ) \int \frac {1}{\sqrt {a+b x+c x^2}} \, dx}{256 c^5}\\ &=\frac {\left (63 b^2-64 a c\right ) x^2 \sqrt {a+b x+c x^2}}{240 c^3}-\frac {9 b x^3 \sqrt {a+b x+c x^2}}{40 c^2}+\frac {x^4 \sqrt {a+b x+c x^2}}{5 c}+\frac {\left (945 b^4-2940 a b^2 c+1024 a^2 c^2-14 b c \left (45 b^2-92 a c\right ) x\right ) \sqrt {a+b x+c x^2}}{1920 c^5}-\frac {\left (b \left (63 b^4-280 a b^2 c+240 a^2 c^2\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 )}{128 c^5}\\ &=\frac {\left (63 b^2-64 a c\right ) x^2 \sqrt {a+b x+c x^2}}{240 c^3}-\frac {9 b x^3 \sqrt {a+b x+c x^2}}{40 c^2}+\frac {x^4 \sqrt {a+b x+c x^2}}{5 c}+\frac {\left (945 b^4-2940 a b^2 c+1024 a^2 c^2-14 b c \left (45 b^2-92 a c\right ) x\right ) \sqrt {a+b x+c x^2}}{1920 c^5}-\frac {b \left (63 b^4-280 a b^2 c+240 a^2 c^2\right ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{256 c^{11/2}}\\ \end {align*}
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Mathematica [A] time = 0.20, size = 213, normalized size = 1.05 \[ \frac {1024 a^3 c^2+4 a^2 c \left (-735 b^2+578 b c x+128 c^2 x^2\right )+a \left (945 b^4-3570 b^3 c x-1148 b^2 c^2 x^2+344 b c^3 x^3-128 c^4 x^4\right )+3 x \left (315 b^5+105 b^4 c x-42 b^3 c^2 x^2+24 b^2 c^3 x^3-16 b c^4 x^4+128 c^5 x^5\right )}{1920 c^5 \sqrt {a+x (b+c x)}}-\frac {b \left (240 a^2 c^2-280 a b^2 c+63 b^4\right ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+x (b+c x)}}\right )}{256 c^{11/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.98, size = 347, normalized size = 1.72 \[ \left [\frac {15 \, {\left (63 \, b^{5} - 280 \, a b^{3} c + 240 \, a^{2} b c^{2}\right )} \sqrt {c} \log \left (-8 \, c^{2} x^{2} - 8 \, b c x - b^{2} + 4 \, \sqrt {c x^{2} + b x + a} {\left (2 \, c x + b\right )} \sqrt {c} - 4 \, a c\right ) + 4 \, {\left (384 \, c^{5} x^{4} - 432 \, b c^{4} x^{3} + 945 \, b^{4} c - 2940 \, a b^{2} c^{2} + 1024 \, a^{2} c^{3} + 8 \, {\left (63 \, b^{2} c^{3} - 64 \, a c^{4}\right )} x^{2} - 14 \, {\left (45 \, b^{3} c^{2} - 92 \, a b c^{3}\right )} x\right )} \sqrt {c x^{2} + b x + a}}{7680 \, c^{6}}, \frac {15 \, {\left (63 \, b^{5} - 280 \, a b^{3} c + 240 \, a^{2} b c^{2}\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {c x^{2} + b x + a} {\left (2 \, c x + b\right )} \sqrt {-c}}{2 \, {\left (c^{2} x^{2} + b c x + a c\right )}}\right ) + 2 \, {\left (384 \, c^{5} x^{4} - 432 \, b c^{4} x^{3} + 945 \, b^{4} c - 2940 \, a b^{2} c^{2} + 1024 \, a^{2} c^{3} + 8 \, {\left (63 \, b^{2} c^{3} - 64 \, a c^{4}\right )} x^{2} - 14 \, {\left (45 \, b^{3} c^{2} - 92 \, a b c^{3}\right )} x\right )} \sqrt {c x^{2} + b x + a}}{3840 \, c^{6}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.23, size = 161, normalized size = 0.80 \[ \frac {1}{1920} \, \sqrt {c x^{2} + b x + a} {\left (2 \, {\left (4 \, {\left (6 \, x {\left (\frac {8 \, x}{c} - \frac {9 \, b}{c^{2}}\right )} + \frac {63 \, b^{2} c^{2} - 64 \, a c^{3}}{c^{5}}\right )} x - \frac {7 \, {\left (45 \, b^{3} c - 92 \, a b c^{2}\right )}}{c^{5}}\right )} x + \frac {945 \, b^{4} - 2940 \, a b^{2} c + 1024 \, a^{2} c^{2}}{c^{5}}\right )} + \frac {{\left (63 \, b^{5} - 280 \, a b^{3} c + 240 \, a^{2} b c^{2}\right )} \log \left ({\left | -2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} \sqrt {c} - b \right |}\right )}{256 \, c^{\frac {11}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 290, normalized size = 1.44 \[ \frac {\sqrt {c \,x^{2}+b x +a}\, x^{4}}{5 c}-\frac {9 \sqrt {c \,x^{2}+b x +a}\, b \,x^{3}}{40 c^{2}}-\frac {4 \sqrt {c \,x^{2}+b x +a}\, a \,x^{2}}{15 c^{2}}+\frac {21 \sqrt {c \,x^{2}+b x +a}\, b^{2} x^{2}}{80 c^{3}}-\frac {15 a^{2} b \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{16 c^{\frac {7}{2}}}+\frac {35 a \,b^{3} \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{32 c^{\frac {9}{2}}}-\frac {63 b^{5} \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{256 c^{\frac {11}{2}}}+\frac {161 \sqrt {c \,x^{2}+b x +a}\, a b x}{240 c^{3}}-\frac {21 \sqrt {c \,x^{2}+b x +a}\, b^{3} x}{64 c^{4}}+\frac {8 \sqrt {c \,x^{2}+b x +a}\, a^{2}}{15 c^{3}}-\frac {49 \sqrt {c \,x^{2}+b x +a}\, a \,b^{2}}{32 c^{4}}+\frac {63 \sqrt {c \,x^{2}+b x +a}\, b^{4}}{128 c^{5}} \]
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 \[ \text {Exception raised: ValueError} \]
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 \[ \int \frac {x^5}{\sqrt {c\,x^2+b\,x+a}} \,d x \]
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 \[ \int \frac {x^{5}}{\sqrt {a + b x + c x^{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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