Optimal. Leaf size=153 \[ \frac {2 x^2 (2 a+b x)}{\left (b^2-4 a c\right ) \sqrt {a x^2+b x^3+c x^4}}-\frac {2 b \sqrt {a x^2+b x^3+c x^4}}{c x \left (b^2-4 a c\right )}+\frac {x \sqrt {a+b x+c x^2} \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{c^{3/2} \sqrt {a x^2+b x^3+c x^4}} \]
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Rubi [A] time = 0.18, antiderivative size = 153, normalized size of antiderivative = 1.00, number of steps used = 6, 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} \[ \frac {2 x^2 (2 a+b x)}{\left (b^2-4 a c\right ) \sqrt {a x^2+b x^3+c x^4}}-\frac {2 b \sqrt {a x^2+b x^3+c x^4}}{c x \left (b^2-4 a c\right )}+\frac {x \sqrt {a+b x+c x^2} \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{c^{3/2} \sqrt {a x^2+b x^3+c x^4}} \]
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^5}{\left (a x^2+b x^3+c x^4\right )^{3/2}} \, dx &=\frac {2 x^2 (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 a+b x)}{\sqrt {a x^2+b x^3+c x^4}} \, dx}{b^2-4 a c}\\ &=\frac {2 x^2 (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 ) x}+\frac {2 \int \frac {\left (b^2-4 a c\right ) x}{2 \sqrt {a x^2+b x^3+c x^4}} \, dx}{c \left (b^2-4 a c\right )}\\ &=\frac {2 x^2 (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 ) x}+\frac {\int \frac {x}{\sqrt {a x^2+b x^3+c x^4}} \, dx}{c}\\ &=\frac {2 x^2 (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 ) x}+\frac {\left (x \sqrt {a+b x+c x^2}\right ) \int \frac {1}{\sqrt {a+b x+c x^2}} \, dx}{c \sqrt {a x^2+b x^3+c x^4}}\\ &=\frac {2 x^2 (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 ) x}+\frac {\left (2 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 \sqrt {a x^2+b x^3+c x^4}}\\ &=\frac {2 x^2 (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 ) x}+\frac {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 )}{c^{3/2} \sqrt {a x^2+b x^3+c x^4}}\\ \end {align*}
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Mathematica [A] time = 0.12, size = 112, normalized size = 0.73 \[ -\frac {x \left (2 \sqrt {c} \left (-a b+2 a c x+b^2 (-x)\right )+\left (b^2-4 a c\right ) \sqrt {a+x (b+c x)} \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+x (b+c x)}}\right )\right )}{c^{3/2} \left (4 a c-b^2\right ) \sqrt {x^2 (a+x (b+c x))}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.77, size = 414, normalized size = 2.71 \[ \left [\frac {{\left ({\left (b^{2} c - 4 \, a c^{2}\right )} x^{3} + {\left (b^{3} - 4 \, a b c\right )} x^{2} + {\left (a b^{2} - 4 \, a^{2} c\right )} x\right )} \sqrt {c} \log \left (-\frac {8 \, c^{2} x^{3} + 8 \, b c x^{2} + 4 \, \sqrt {c x^{4} + b x^{3} + a x^{2}} {\left (2 \, c x + b\right )} \sqrt {c} + {\left (b^{2} + 4 \, a c\right )} x}{x}\right ) - 4 \, \sqrt {c x^{4} + b x^{3} + a x^{2}} {\left (a b c + {\left (b^{2} c - 2 \, a c^{2}\right )} x\right )}}{2 \, {\left ({\left (b^{2} c^{3} - 4 \, a c^{4}\right )} x^{3} + {\left (b^{3} c^{2} - 4 \, a b c^{3}\right )} x^{2} + {\left (a b^{2} c^{2} - 4 \, a^{2} c^{3}\right )} x\right )}}, -\frac {{\left ({\left (b^{2} c - 4 \, a c^{2}\right )} x^{3} + {\left (b^{3} - 4 \, a b c\right )} x^{2} + {\left (a b^{2} - 4 \, a^{2} c\right )} x\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {c x^{4} + b x^{3} + a x^{2}} {\left (2 \, c x + b\right )} \sqrt {-c}}{2 \, {\left (c^{2} x^{3} + b c x^{2} + a c x\right )}}\right ) + 2 \, \sqrt {c x^{4} + b x^{3} + a x^{2}} {\left (a b c + {\left (b^{2} c - 2 \, a c^{2}\right )} x\right )}}{{\left (b^{2} c^{3} - 4 \, a c^{4}\right )} x^{3} + {\left (b^{3} c^{2} - 4 \, a b c^{3}\right )} x^{2} + {\left (a b^{2} c^{2} - 4 \, a^{2} c^{3}\right )} x}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.98, size = 110, normalized size = 0.72 \[ -\frac {2 \, {\left (\frac {a b c}{{\left (b^{2} c^{2} - 4 \, a c^{3}\right )} x} + \frac {b^{2} c - 2 \, a c^{2}}{b^{2} c^{2} - 4 \, a c^{3}}\right )}}{\sqrt {c + \frac {b}{x} + \frac {a}{x^{2}}}} - \frac {2 \, \arctan \left (\frac {\sqrt {c + \frac {b}{x} + \frac {a}{x^{2}}} - \frac {\sqrt {a}}{x}}{\sqrt {-c}}\right )}{\sqrt {-c} c} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 166, normalized size = 1.08 \[ \frac {\left (c \,x^{2}+b x +a \right ) \left (-4 a \,c^{\frac {5}{2}} x +2 b^{2} c^{\frac {3}{2}} x +4 \sqrt {c \,x^{2}+b x +a}\, a \,c^{2} \ln \left (\frac {2 c x +b +2 \sqrt {c \,x^{2}+b x +a}\, \sqrt {c}}{2 \sqrt {c}}\right )-\sqrt {c \,x^{2}+b x +a}\, b^{2} c \ln \left (\frac {2 c x +b +2 \sqrt {c \,x^{2}+b x +a}\, \sqrt {c}}{2 \sqrt {c}}\right )+2 a b \,c^{\frac {3}{2}}\right ) x^{3}}{\left (c \,x^{4}+b \,x^{3}+a \,x^{2}\right )^{\frac {3}{2}} \left (4 a c -b^{2}\right ) c^{\frac {5}{2}}} \]
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 \[ \int \frac {x^{5}}{{\left (c x^{4} + b x^{3} + a x^{2}\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {x^5}{{\left (c\,x^4+b\,x^3+a\,x^2\right )}^{3/2}} \,d x \]
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 \[ \int \frac {x^{5}}{\left (x^{2} \left (a + b x + c x^{2}\right )\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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