Optimal. Leaf size=201 \[ \frac {\left (3 b^2-8 a c\right ) \sqrt {a x^2+b x^3+c x^4}}{c^2 x \left (b^2-4 a c\right )}+\frac {2 x^3 (2 a+b x)}{\left (b^2-4 a c\right ) \sqrt {a x^2+b x^3+c x^4}}-\frac {2 b \sqrt {a x^2+b x^3+c x^4}}{c \left (b^2-4 a c\right )}-\frac {3 b x \sqrt {a+b x+c x^2} \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{2 c^{5/2} \sqrt {a x^2+b x^3+c x^4}} \]
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Rubi [A] time = 0.30, antiderivative size = 201, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {1923, 1949, 12, 1914, 621, 206} \begin {gather*} \frac {\left (3 b^2-8 a c\right ) \sqrt {a x^2+b x^3+c x^4}}{c^2 x \left (b^2-4 a c\right )}+\frac {2 x^3 (2 a+b x)}{\left (b^2-4 a c\right ) \sqrt {a x^2+b x^3+c x^4}}-\frac {2 b \sqrt {a x^2+b x^3+c x^4}}{c \left (b^2-4 a c\right )}-\frac {3 b x \sqrt {a+b x+c x^2} \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{2 c^{5/2} \sqrt {a x^2+b x^3+c x^4}} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 206
Rule 621
Rule 1914
Rule 1923
Rule 1949
Rubi steps
\begin {align*} \int \frac {x^6}{\left (a x^2+b x^3+c x^4\right )^{3/2}} \, dx &=\frac {2 x^3 (2 a+b x)}{\left (b^2-4 a c\right ) \sqrt {a x^2+b x^3+c x^4}}-\frac {2 \int \frac {x^2 (4 a+2 b x)}{\sqrt {a x^2+b x^3+c x^4}} \, dx}{b^2-4 a c}\\ &=\frac {2 x^3 (2 a+b x)}{\left (b^2-4 a c\right ) \sqrt {a x^2+b x^3+c x^4}}-\frac {2 b \sqrt {a x^2+b x^3+c x^4}}{c \left (b^2-4 a c\right )}+\frac {\int \frac {x \left (2 a b+\left (3 b^2-8 a c\right ) x\right )}{\sqrt {a x^2+b x^3+c x^4}} \, dx}{c \left (b^2-4 a c\right )}\\ &=\frac {2 x^3 (2 a+b x)}{\left (b^2-4 a c\right ) \sqrt {a x^2+b x^3+c x^4}}-\frac {2 b \sqrt {a x^2+b x^3+c x^4}}{c \left (b^2-4 a c\right )}+\frac {\left (3 b^2-8 a c\right ) \sqrt {a x^2+b x^3+c x^4}}{c^2 \left (b^2-4 a c\right ) x}-\frac {\int \frac {3 b \left (b^2-4 a c\right ) x}{2 \sqrt {a x^2+b x^3+c x^4}} \, dx}{c^2 \left (b^2-4 a c\right )}\\ &=\frac {2 x^3 (2 a+b x)}{\left (b^2-4 a c\right ) \sqrt {a x^2+b x^3+c x^4}}-\frac {2 b \sqrt {a x^2+b x^3+c x^4}}{c \left (b^2-4 a c\right )}+\frac {\left (3 b^2-8 a c\right ) \sqrt {a x^2+b x^3+c x^4}}{c^2 \left (b^2-4 a c\right ) x}-\frac {(3 b) \int \frac {x}{\sqrt {a x^2+b x^3+c x^4}} \, dx}{2 c^2}\\ &=\frac {2 x^3 (2 a+b x)}{\left (b^2-4 a c\right ) \sqrt {a x^2+b x^3+c x^4}}-\frac {2 b \sqrt {a x^2+b x^3+c x^4}}{c \left (b^2-4 a c\right )}+\frac {\left (3 b^2-8 a c\right ) \sqrt {a x^2+b x^3+c x^4}}{c^2 \left (b^2-4 a c\right ) x}-\frac {\left (3 b x \sqrt {a+b x+c x^2}\right ) \int \frac {1}{\sqrt {a+b x+c x^2}} \, dx}{2 c^2 \sqrt {a x^2+b x^3+c x^4}}\\ &=\frac {2 x^3 (2 a+b x)}{\left (b^2-4 a c\right ) \sqrt {a x^2+b x^3+c x^4}}-\frac {2 b \sqrt {a x^2+b x^3+c x^4}}{c \left (b^2-4 a c\right )}+\frac {\left (3 b^2-8 a c\right ) \sqrt {a x^2+b x^3+c x^4}}{c^2 \left (b^2-4 a c\right ) x}-\frac {\left (3 b x \sqrt {a+b x+c x^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 )}{c^2 \sqrt {a x^2+b x^3+c x^4}}\\ &=\frac {2 x^3 (2 a+b x)}{\left (b^2-4 a c\right ) \sqrt {a x^2+b x^3+c x^4}}-\frac {2 b \sqrt {a x^2+b x^3+c x^4}}{c \left (b^2-4 a c\right )}+\frac {\left (3 b^2-8 a c\right ) \sqrt {a x^2+b x^3+c x^4}}{c^2 \left (b^2-4 a c\right ) x}-\frac {3 b x \sqrt {a+b x+c x^2} \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{2 c^{5/2} \sqrt {a x^2+b x^3+c x^4}}\\ \end {align*}
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Mathematica [A] time = 0.15, size = 141, normalized size = 0.70 \begin {gather*} \frac {x \left (2 \sqrt {c} \left (8 a^2 c+a \left (-3 b^2+10 b c x+4 c^2 x^2\right )-b^2 x (3 b+c x)\right )+3 b \left (b^2-4 a c\right ) \sqrt {a+x (b+c x)} \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+x (b+c x)}}\right )\right )}{2 c^{5/2} \left (4 a c-b^2\right ) \sqrt {x^2 (a+x (b+c x))}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 2.16, size = 145, normalized size = 0.72 \begin {gather*} \frac {\sqrt {a x^2+b x^3+c x^4} \left (8 a^2 c-3 a b^2+10 a b c x+4 a c^2 x^2-3 b^3 x-b^2 c x^2\right )}{c^2 x \left (4 a c-b^2\right ) \left (a+b x+c x^2\right )}+\frac {3 b \tanh ^{-1}\left (\frac {\sqrt {c} x^2}{\sqrt {a} x-\sqrt {a x^2+b x^3+c x^4}}\right )}{c^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.36, size = 486, normalized size = 2.42 \begin {gather*} \left [\frac {3 \, {\left ({\left (b^{3} c - 4 \, a b c^{2}\right )} x^{3} + {\left (b^{4} - 4 \, a b^{2} c\right )} x^{2} + {\left (a b^{3} - 4 \, a^{2} b c\right )} x\right )} \sqrt {c} \log \left (-\frac {8 \, c^{2} x^{3} + 8 \, b c x^{2} - 4 \, \sqrt {c x^{4} + b x^{3} + a x^{2}} {\left (2 \, c x + b\right )} \sqrt {c} + {\left (b^{2} + 4 \, a c\right )} x}{x}\right ) + 4 \, \sqrt {c x^{4} + b x^{3} + a x^{2}} {\left (3 \, a b^{2} c - 8 \, a^{2} c^{2} + {\left (b^{2} c^{2} - 4 \, a c^{3}\right )} x^{2} + {\left (3 \, b^{3} c - 10 \, a b c^{2}\right )} x\right )}}{4 \, {\left ({\left (b^{2} c^{4} - 4 \, a c^{5}\right )} x^{3} + {\left (b^{3} c^{3} - 4 \, a b c^{4}\right )} x^{2} + {\left (a b^{2} c^{3} - 4 \, a^{2} c^{4}\right )} x\right )}}, \frac {3 \, {\left ({\left (b^{3} c - 4 \, a b c^{2}\right )} x^{3} + {\left (b^{4} - 4 \, a b^{2} c\right )} x^{2} + {\left (a b^{3} - 4 \, a^{2} b c\right )} x\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {c x^{4} + b x^{3} + a x^{2}} {\left (2 \, c x + b\right )} \sqrt {-c}}{2 \, {\left (c^{2} x^{3} + b c x^{2} + a c x\right )}}\right ) + 2 \, \sqrt {c x^{4} + b x^{3} + a x^{2}} {\left (3 \, a b^{2} c - 8 \, a^{2} c^{2} + {\left (b^{2} c^{2} - 4 \, a c^{3}\right )} x^{2} + {\left (3 \, b^{3} c - 10 \, a b c^{2}\right )} x\right )}}{2 \, {\left ({\left (b^{2} c^{4} - 4 \, a c^{5}\right )} x^{3} + {\left (b^{3} c^{3} - 4 \, a b c^{4}\right )} x^{2} + {\left (a b^{2} c^{3} - 4 \, a^{2} c^{4}\right )} x\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.89, size = 195, normalized size = 0.97 \begin {gather*} \frac {2 \, {\left (\frac {b^{3} c^{2} - 3 \, a b c^{3}}{b^{2} c^{4} - 4 \, a c^{5}} + \frac {a b^{2} c^{2} - 2 \, a^{2} c^{3}}{{\left (b^{2} c^{4} - 4 \, a c^{5}\right )} x}\right )}}{\sqrt {c + \frac {b}{x} + \frac {a}{x^{2}}}} + \frac {3 \, b \arctan \left (\frac {\sqrt {c + \frac {b}{x} + \frac {a}{x^{2}}} - \frac {\sqrt {a}}{x}}{\sqrt {-c}}\right )}{\sqrt {-c} c^{2}} + \frac {b {\left (\sqrt {c + \frac {b}{x} + \frac {a}{x^{2}}} - \frac {\sqrt {a}}{x}\right )} - 2 \, \sqrt {a} c}{{\left ({\left (\sqrt {c + \frac {b}{x} + \frac {a}{x^{2}}} - \frac {\sqrt {a}}{x}\right )}^{2} - c\right )} c^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 199, normalized size = 0.99 \begin {gather*} \frac {\left (c \,x^{2}+b x +a \right ) \left (8 a \,c^{\frac {7}{2}} x^{2}-2 b^{2} c^{\frac {5}{2}} x^{2}+20 a b \,c^{\frac {5}{2}} x -6 b^{3} c^{\frac {3}{2}} x -12 \sqrt {c \,x^{2}+b x +a}\, a b \,c^{2} \ln \left (\frac {2 c x +b +2 \sqrt {c \,x^{2}+b x +a}\, \sqrt {c}}{2 \sqrt {c}}\right )+3 \sqrt {c \,x^{2}+b x +a}\, b^{3} c \ln \left (\frac {2 c x +b +2 \sqrt {c \,x^{2}+b x +a}\, \sqrt {c}}{2 \sqrt {c}}\right )+16 a^{2} c^{\frac {5}{2}}-6 a \,b^{2} c^{\frac {3}{2}}\right ) x^{3}}{2 \left (c \,x^{4}+b \,x^{3}+a \,x^{2}\right )^{\frac {3}{2}} \left (4 a c -b^{2}\right ) c^{\frac {7}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{6}}{{\left (c x^{4} + b x^{3} + a x^{2}\right )}^{\frac {3}{2}}}\,{d x} \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}{{\left (c\,x^4+b\,x^3+a\,x^2\right )}^{3/2}} \,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}}{\left (x^{2} \left (a + b x + c x^{2}\right )\right )^{\frac {3}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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