Optimal. Leaf size=209 \[ -\frac {3 \left (5 b^2-4 a c\right ) \tanh ^{-1}\left (\frac {x (2 a+b x)}{2 \sqrt {a} \sqrt {a x^2+b x^3+c x^4}}\right )}{8 a^{7/2}}+\frac {b \left (15 b^2-52 a c\right ) \sqrt {a x^2+b x^3+c x^4}}{4 a^3 x^2 \left (b^2-4 a c\right )}-\frac {\left (5 b^2-12 a c\right ) \sqrt {a x^2+b x^3+c x^4}}{2 a^2 x^3 \left (b^2-4 a c\right )}+\frac {2 \left (-2 a c+b^2+b c x\right )}{a x \left (b^2-4 a c\right ) \sqrt {a x^2+b x^3+c x^4}} \]
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Rubi [A] time = 0.29, antiderivative size = 209, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {1907, 1951, 12, 1904, 206} \begin {gather*} \frac {b \left (15 b^2-52 a c\right ) \sqrt {a x^2+b x^3+c x^4}}{4 a^3 x^2 \left (b^2-4 a c\right )}-\frac {\left (5 b^2-12 a c\right ) \sqrt {a x^2+b x^3+c x^4}}{2 a^2 x^3 \left (b^2-4 a c\right )}-\frac {3 \left (5 b^2-4 a c\right ) \tanh ^{-1}\left (\frac {x (2 a+b x)}{2 \sqrt {a} \sqrt {a x^2+b x^3+c x^4}}\right )}{8 a^{7/2}}+\frac {2 \left (-2 a c+b^2+b c x\right )}{a x \left (b^2-4 a c\right ) \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 1904
Rule 1907
Rule 1951
Rubi steps
\begin {align*} \int \frac {1}{\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 ) x \sqrt {a x^2+b x^3+c x^4}}-\frac {2 \int \frac {-2 \left (b^2-2 a c\right )+\frac {1}{2} \left (-b^2+4 a c\right )-2 b c x}{x^2 \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 ) x \sqrt {a x^2+b x^3+c x^4}}-\frac {\left (5 b^2-12 a c\right ) \sqrt {a x^2+b x^3+c x^4}}{2 a^2 \left (b^2-4 a c\right ) x^3}+\frac {\int \frac {-\frac {1}{4} b \left (15 b^2-52 a c\right )-\frac {1}{2} c \left (5 b^2-12 a c\right ) x}{x \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 ) x \sqrt {a x^2+b x^3+c x^4}}-\frac {\left (5 b^2-12 a c\right ) \sqrt {a x^2+b x^3+c x^4}}{2 a^2 \left (b^2-4 a c\right ) x^3}+\frac {b \left (15 b^2-52 a c\right ) \sqrt {a x^2+b x^3+c x^4}}{4 a^3 \left (b^2-4 a c\right ) x^2}-\frac {\int -\frac {3 \left (b^2-4 a c\right ) \left (5 b^2-4 a c\right )}{8 \sqrt {a x^2+b x^3+c x^4}} \, dx}{a^3 \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 ) x \sqrt {a x^2+b x^3+c x^4}}-\frac {\left (5 b^2-12 a c\right ) \sqrt {a x^2+b x^3+c x^4}}{2 a^2 \left (b^2-4 a c\right ) x^3}+\frac {b \left (15 b^2-52 a c\right ) \sqrt {a x^2+b x^3+c x^4}}{4 a^3 \left (b^2-4 a c\right ) x^2}+\frac {\left (3 \left (5 b^2-4 a c\right )\right ) \int \frac {1}{\sqrt {a x^2+b x^3+c x^4}} \, dx}{8 a^3}\\ &=\frac {2 \left (b^2-2 a c+b c x\right )}{a \left (b^2-4 a c\right ) x \sqrt {a x^2+b x^3+c x^4}}-\frac {\left (5 b^2-12 a c\right ) \sqrt {a x^2+b x^3+c x^4}}{2 a^2 \left (b^2-4 a c\right ) x^3}+\frac {b \left (15 b^2-52 a c\right ) \sqrt {a x^2+b x^3+c x^4}}{4 a^3 \left (b^2-4 a c\right ) x^2}-\frac {\left (3 \left (5 b^2-4 a c\right )\right ) \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 )}{4 a^3}\\ &=\frac {2 \left (b^2-2 a c+b c x\right )}{a \left (b^2-4 a c\right ) x \sqrt {a x^2+b x^3+c x^4}}-\frac {\left (5 b^2-12 a c\right ) \sqrt {a x^2+b x^3+c x^4}}{2 a^2 \left (b^2-4 a c\right ) x^3}+\frac {b \left (15 b^2-52 a c\right ) \sqrt {a x^2+b x^3+c x^4}}{4 a^3 \left (b^2-4 a c\right ) x^2}-\frac {3 \left (5 b^2-4 a c\right ) \tanh ^{-1}\left (\frac {x (2 a+b x)}{2 \sqrt {a} \sqrt {a x^2+b x^3+c x^4}}\right )}{8 a^{7/2}}\\ \end {align*}
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Mathematica [A] time = 0.16, size = 181, normalized size = 0.87 \begin {gather*} -\frac {3 x^2 \left (16 a^2 c^2-24 a b^2 c+5 b^4\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 \sqrt {a} \left (-8 a^3 c+2 a^2 \left (b^2+10 b c x-12 c^2 x^2\right )+a b x \left (-5 b^2+62 b c x+52 c^2 x^2\right )-15 b^3 x^2 (b+c x)\right )}{8 a^{7/2} x \left (b^2-4 a c\right ) \sqrt {x^2 (a+x (b+c x))}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 3.25, size = 216, normalized size = 1.03 \begin {gather*} \frac {3 \log (x) \left (4 a c-5 b^2\right )}{4 a^{7/2}}+\frac {\sqrt {a x^2+b x^3+c x^4} \left (-8 a^3 c+2 a^2 b^2+20 a^2 b c x-24 a^2 c^2 x^2-5 a b^3 x+62 a b^2 c x^2+52 a b c^2 x^3-15 b^4 x^2-15 b^3 c x^3\right )}{4 a^3 x^3 \left (4 a c-b^2\right ) \left (a+b x+c x^2\right )}-\frac {3 \left (4 a c-5 b^2\right ) \log \left (-2 a^{7/2} \sqrt {a x^2+b x^3+c x^4}+2 a^4 x+a^3 b x^2\right )}{8 a^{7/2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.56, size = 630, normalized size = 3.01 \begin {gather*} \left [-\frac {3 \, {\left ({\left (5 \, b^{4} c - 24 \, a b^{2} c^{2} + 16 \, a^{2} c^{3}\right )} x^{5} + {\left (5 \, b^{5} - 24 \, a b^{3} c + 16 \, a^{2} b c^{2}\right )} x^{4} + {\left (5 \, a b^{4} - 24 \, a^{2} b^{2} c + 16 \, a^{3} c^{2}\right )} x^{3}\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 \, {\left (2 \, a^{3} b^{2} - 8 \, a^{4} c - {\left (15 \, a b^{3} c - 52 \, a^{2} b c^{2}\right )} x^{3} - {\left (15 \, a b^{4} - 62 \, a^{2} b^{2} c + 24 \, a^{3} c^{2}\right )} x^{2} - 5 \, {\left (a^{2} b^{3} - 4 \, a^{3} b c\right )} x\right )} \sqrt {c x^{4} + b x^{3} + a x^{2}}}{16 \, {\left ({\left (a^{4} b^{2} c - 4 \, a^{5} c^{2}\right )} x^{5} + {\left (a^{4} b^{3} - 4 \, a^{5} b c\right )} x^{4} + {\left (a^{5} b^{2} - 4 \, a^{6} c\right )} x^{3}\right )}}, \frac {3 \, {\left ({\left (5 \, b^{4} c - 24 \, a b^{2} c^{2} + 16 \, a^{2} c^{3}\right )} x^{5} + {\left (5 \, b^{5} - 24 \, a b^{3} c + 16 \, a^{2} b c^{2}\right )} x^{4} + {\left (5 \, a b^{4} - 24 \, a^{2} b^{2} c + 16 \, a^{3} c^{2}\right )} x^{3}\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 \, {\left (2 \, a^{3} b^{2} - 8 \, a^{4} c - {\left (15 \, a b^{3} c - 52 \, a^{2} b c^{2}\right )} x^{3} - {\left (15 \, a b^{4} - 62 \, a^{2} b^{2} c + 24 \, a^{3} c^{2}\right )} x^{2} - 5 \, {\left (a^{2} b^{3} - 4 \, a^{3} b c\right )} x\right )} \sqrt {c x^{4} + b x^{3} + a x^{2}}}{8 \, {\left ({\left (a^{4} b^{2} c - 4 \, a^{5} c^{2}\right )} x^{5} + {\left (a^{4} b^{3} - 4 \, a^{5} b c\right )} x^{4} + {\left (a^{5} b^{2} - 4 \, a^{6} c\right )} x^{3}\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \mathit {sage}_{0} x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 292, normalized size = 1.40 \begin {gather*} -\frac {\left (c \,x^{2}+b x +a \right ) \left (-104 a^{\frac {5}{2}} b \,c^{2} x^{3}+30 a^{\frac {3}{2}} b^{3} c \,x^{3}-48 \sqrt {c \,x^{2}+b x +a}\, a^{3} c^{2} x^{2} \ln \left (\frac {b x +2 a +2 \sqrt {c \,x^{2}+b x +a}\, \sqrt {a}}{x}\right )+72 \sqrt {c \,x^{2}+b x +a}\, a^{2} b^{2} c \,x^{2} \ln \left (\frac {b x +2 a +2 \sqrt {c \,x^{2}+b x +a}\, \sqrt {a}}{x}\right )-15 \sqrt {c \,x^{2}+b x +a}\, a \,b^{4} x^{2} \ln \left (\frac {b x +2 a +2 \sqrt {c \,x^{2}+b x +a}\, \sqrt {a}}{x}\right )+48 a^{\frac {7}{2}} c^{2} x^{2}-124 a^{\frac {5}{2}} b^{2} c \,x^{2}+30 a^{\frac {3}{2}} b^{4} x^{2}-40 a^{\frac {7}{2}} b c x +10 a^{\frac {5}{2}} b^{3} x +16 a^{\frac {9}{2}} c -4 a^{\frac {7}{2}} b^{2}\right ) x}{8 \left (c \,x^{4}+b \,x^{3}+a \,x^{2}\right )^{\frac {3}{2}} \left (4 a c -b^{2}\right ) a^{\frac {9}{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 {1}{{\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 {1}{{\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 {1}{\left (a x^{2} + b x^{3} + c x^{4}\right )^{\frac {3}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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