Optimal. Leaf size=144 \[ \frac {3 b \tanh ^{-1}\left (\frac {x (2 a+b x)}{2 \sqrt {a} \sqrt {a x^2+b x^3+c x^4}}\right )}{2 a^{5/2}}-\frac {\left (3 b^2-8 a c\right ) \sqrt {a x^2+b x^3+c x^4}}{a^2 x^2 \left (b^2-4 a c\right )}+\frac {2 \left (-2 a c+b^2+b c x\right )}{a \left (b^2-4 a c\right ) \sqrt {a x^2+b x^3+c x^4}} \]
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Rubi [A] time = 0.16, antiderivative size = 144, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {1924, 1951, 12, 1904, 206} \[ -\frac {\left (3 b^2-8 a c\right ) \sqrt {a x^2+b x^3+c x^4}}{a^2 x^2 \left (b^2-4 a c\right )}+\frac {3 b \tanh ^{-1}\left (\frac {x (2 a+b x)}{2 \sqrt {a} \sqrt {a x^2+b x^3+c x^4}}\right )}{2 a^{5/2}}+\frac {2 \left (-2 a c+b^2+b c x\right )}{a \left (b^2-4 a c\right ) \sqrt {a x^2+b x^3+c x^4}} \]
Antiderivative was successfully verified.
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Rule 12
Rule 206
Rule 1904
Rule 1924
Rule 1951
Rubi steps
\begin {align*} \int \frac {x}{\left (a x^2+b x^3+c x^4\right )^{3/2}} \, dx &=\frac {2 \left (b^2-2 a c+b c x\right )}{a \left (b^2-4 a c\right ) \sqrt {a x^2+b x^3+c x^4}}-\frac {2 \int \frac {-\frac {3 b^2}{2}+4 a c-b c x}{x \sqrt {a x^2+b x^3+c x^4}} \, dx}{a \left (b^2-4 a c\right )}\\ &=\frac {2 \left (b^2-2 a c+b c x\right )}{a \left (b^2-4 a c\right ) \sqrt {a x^2+b x^3+c x^4}}-\frac {\left (3 b^2-8 a c\right ) \sqrt {a x^2+b x^3+c x^4}}{a^2 \left (b^2-4 a c\right ) x^2}+\frac {2 \int -\frac {3 b \left (b^2-4 a c\right )}{4 \sqrt {a x^2+b x^3+c x^4}} \, dx}{a^2 \left (b^2-4 a c\right )}\\ &=\frac {2 \left (b^2-2 a c+b c x\right )}{a \left (b^2-4 a c\right ) \sqrt {a x^2+b x^3+c x^4}}-\frac {\left (3 b^2-8 a c\right ) \sqrt {a x^2+b x^3+c x^4}}{a^2 \left (b^2-4 a c\right ) x^2}-\frac {(3 b) \int \frac {1}{\sqrt {a x^2+b x^3+c x^4}} \, dx}{2 a^2}\\ &=\frac {2 \left (b^2-2 a c+b c x\right )}{a \left (b^2-4 a c\right ) \sqrt {a x^2+b x^3+c x^4}}-\frac {\left (3 b^2-8 a c\right ) \sqrt {a x^2+b x^3+c x^4}}{a^2 \left (b^2-4 a c\right ) x^2}+\frac {(3 b) \operatorname {Subst}\left (\int \frac {1}{4 a-x^2} \, dx,x,\frac {x (2 a+b x)}{\sqrt {a x^2+b x^3+c x^4}}\right )}{a^2}\\ &=\frac {2 \left (b^2-2 a c+b c x\right )}{a \left (b^2-4 a c\right ) \sqrt {a x^2+b x^3+c x^4}}-\frac {\left (3 b^2-8 a c\right ) \sqrt {a x^2+b x^3+c x^4}}{a^2 \left (b^2-4 a c\right ) x^2}+\frac {3 b \tanh ^{-1}\left (\frac {x (2 a+b x)}{2 \sqrt {a} \sqrt {a x^2+b x^3+c x^4}}\right )}{2 a^{5/2}}\\ \end {align*}
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Mathematica [A] time = 0.10, size = 138, normalized size = 0.96 \[ \frac {2 \sqrt {a} \left (-4 a^2 c+a \left (b^2-10 b c x-8 c^2 x^2\right )+3 b^2 x (b+c x)\right )-3 b x \left (b^2-4 a c\right ) \sqrt {a+x (b+c x)} \tanh ^{-1}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+x (b+c x)}}\right )}{2 a^{5/2} \left (4 a c-b^2\right ) \sqrt {x^2 (a+x (b+c x))}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.62, size = 496, normalized size = 3.44 \[ \left [\frac {3 \, {\left ({\left (b^{3} c - 4 \, a b c^{2}\right )} x^{4} + {\left (b^{4} - 4 \, a b^{2} c\right )} x^{3} + {\left (a b^{3} - 4 \, a^{2} b c\right )} x^{2}\right )} \sqrt {a} \log \left (-\frac {8 \, a b x^{2} + {\left (b^{2} + 4 \, a c\right )} x^{3} + 8 \, a^{2} x + 4 \, \sqrt {c x^{4} + b x^{3} + a x^{2}} {\left (b x + 2 \, a\right )} \sqrt {a}}{x^{3}}\right ) - 4 \, \sqrt {c x^{4} + b x^{3} + a x^{2}} {\left (a^{2} b^{2} - 4 \, a^{3} c + {\left (3 \, a b^{2} c - 8 \, a^{2} c^{2}\right )} x^{2} + {\left (3 \, a b^{3} - 10 \, a^{2} b c\right )} x\right )}}{4 \, {\left ({\left (a^{3} b^{2} c - 4 \, a^{4} c^{2}\right )} x^{4} + {\left (a^{3} b^{3} - 4 \, a^{4} b c\right )} x^{3} + {\left (a^{4} b^{2} - 4 \, a^{5} c\right )} x^{2}\right )}}, -\frac {3 \, {\left ({\left (b^{3} c - 4 \, a b c^{2}\right )} x^{4} + {\left (b^{4} - 4 \, a b^{2} c\right )} x^{3} + {\left (a b^{3} - 4 \, a^{2} b c\right )} x^{2}\right )} \sqrt {-a} \arctan \left (\frac {\sqrt {c x^{4} + b x^{3} + a x^{2}} {\left (b x + 2 \, a\right )} \sqrt {-a}}{2 \, {\left (a c x^{3} + a b x^{2} + a^{2} x\right )}}\right ) + 2 \, \sqrt {c x^{4} + b x^{3} + a x^{2}} {\left (a^{2} b^{2} - 4 \, a^{3} c + {\left (3 \, a b^{2} c - 8 \, a^{2} c^{2}\right )} x^{2} + {\left (3 \, a b^{3} - 10 \, a^{2} b c\right )} x\right )}}{2 \, {\left ({\left (a^{3} b^{2} c - 4 \, a^{4} c^{2}\right )} x^{4} + {\left (a^{3} b^{3} - 4 \, a^{4} b c\right )} x^{3} + {\left (a^{4} b^{2} - 4 \, a^{5} c\right )} x^{2}\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: NotImplementedError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 201, normalized size = 1.40 \[ \frac {\left (c \,x^{2}+b x +a \right ) \left (-16 a^{\frac {5}{2}} c^{2} x^{2}+6 a^{\frac {3}{2}} b^{2} c \,x^{2}+12 \sqrt {c \,x^{2}+b x +a}\, a^{2} b c x \ln \left (\frac {b x +2 a +2 \sqrt {c \,x^{2}+b x +a}\, \sqrt {a}}{x}\right )-3 \sqrt {c \,x^{2}+b x +a}\, a \,b^{3} x \ln \left (\frac {b x +2 a +2 \sqrt {c \,x^{2}+b x +a}\, \sqrt {a}}{x}\right )-20 a^{\frac {5}{2}} b c x +6 a^{\frac {3}{2}} b^{3} x -8 a^{\frac {7}{2}} c +2 a^{\frac {5}{2}} b^{2}\right ) x^{2}}{2 \left (c \,x^{4}+b \,x^{3}+a \,x^{2}\right )^{\frac {3}{2}} \left (4 a c -b^{2}\right ) a^{\frac {7}{2}}} \]
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 \[ \int \frac {x}{{\left (c x^{4} + b x^{3} + a x^{2}\right )}^{\frac {3}{2}}}\,{d x} \]
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 \[ \int \frac {x}{{\left (c\,x^4+b\,x^3+a\,x^2\right )}^{3/2}} \,d x \]
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 \[ \int \frac {x}{\left (x^{2} \left (a + b x + c x^{2}\right )\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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