Optimal. Leaf size=63 \[ 2 \sqrt {3} \tanh ^{-1}\left (\frac {\sqrt {3} x}{\sqrt {a x^3+b x^2+c}}\right )-2 \sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {a x^3+b x^2+c}}\right ) \]
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Rubi [F] time = 3.01, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {\left (2 c-a x^3\right ) \sqrt {c+b x^2+a x^3}}{\left (c+(-3+b) x^2+a x^3\right ) \left (c+(-2+b) x^2+a x^3\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {\left (2 c-a x^3\right ) \sqrt {c+b x^2+a x^3}}{\left (c+(-3+b) x^2+a x^3\right ) \left (c+(-2+b) x^2+a x^3\right )} \, dx &=\int \frac {\left (2 c-a x^3\right ) \sqrt {c+b x^2+a x^3}}{\left (c-(2-b) x^2+a x^3\right ) \left (c-(3-b) x^2+a x^3\right )} \, dx\\ &=\int \left (\frac {(-4+2 b+3 a x) \sqrt {c+b x^2+a x^3}}{c-(2-b) x^2+a x^3}+\frac {(6-2 b-3 a x) \sqrt {c+b x^2+a x^3}}{c-(3-b) x^2+a x^3}\right ) \, dx\\ &=\int \frac {(-4+2 b+3 a x) \sqrt {c+b x^2+a x^3}}{c-(2-b) x^2+a x^3} \, dx+\int \frac {(6-2 b-3 a x) \sqrt {c+b x^2+a x^3}}{c-(3-b) x^2+a x^3} \, dx\\ &=\int \left (\frac {2 \left (1-\frac {3}{b}\right ) b \sqrt {c+b x^2+a x^3}}{-c+(3-b) x^2-a x^3}+\frac {3 a x \sqrt {c+b x^2+a x^3}}{-c+(3-b) x^2-a x^3}\right ) \, dx+\int \left (\frac {4 \left (1-\frac {b}{2}\right ) \sqrt {c+b x^2+a x^3}}{-c+(2-b) x^2-a x^3}+\frac {3 a x \sqrt {c+b x^2+a x^3}}{c-(2-b) x^2+a x^3}\right ) \, dx\\ &=(3 a) \int \frac {x \sqrt {c+b x^2+a x^3}}{-c+(3-b) x^2-a x^3} \, dx+(3 a) \int \frac {x \sqrt {c+b x^2+a x^3}}{c-(2-b) x^2+a x^3} \, dx+(2 (2-b)) \int \frac {\sqrt {c+b x^2+a x^3}}{-c+(2-b) x^2-a x^3} \, dx-(2 (3-b)) \int \frac {\sqrt {c+b x^2+a x^3}}{-c+(3-b) x^2-a x^3} \, dx\\ \end {align*}
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Mathematica [C] time = 6.92, size = 21715, normalized size = 344.68 \begin {gather*} \text {Result too large to show} \end {gather*}
Warning: Unable to verify antiderivative.
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IntegrateAlgebraic [A] time = 2.91, size = 63, normalized size = 1.00 \begin {gather*} 2 \sqrt {3} \tanh ^{-1}\left (\frac {\sqrt {3} x}{\sqrt {a x^3+b x^2+c}}\right )-2 \sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {a x^3+b x^2+c}}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.69, size = 293, normalized size = 4.65 \begin {gather*} \frac {1}{2} \, \sqrt {2} \log \left (\frac {a^{2} x^{6} + 2 \, {\left (a b + 6 \, a\right )} x^{5} + 2 \, a c x^{3} + {\left (b^{2} + 12 \, b + 4\right )} x^{4} + 2 \, {\left (b + 6\right )} c x^{2} - 4 \, \sqrt {2} {\left (a x^{4} + {\left (b + 2\right )} x^{3} + c x\right )} \sqrt {a x^{3} + b x^{2} + c} + c^{2}}{a^{2} x^{6} + 2 \, {\left (a b - 2 \, a\right )} x^{5} + 2 \, a c x^{3} + {\left (b^{2} - 4 \, b + 4\right )} x^{4} + 2 \, {\left (b - 2\right )} c x^{2} + c^{2}}\right ) + \frac {1}{2} \, \sqrt {3} \log \left (\frac {a^{2} x^{6} + 2 \, {\left (a b + 9 \, a\right )} x^{5} + 2 \, a c x^{3} + {\left (b^{2} + 18 \, b + 9\right )} x^{4} + 2 \, {\left (b + 9\right )} c x^{2} + 4 \, \sqrt {3} {\left (a x^{4} + {\left (b + 3\right )} x^{3} + c x\right )} \sqrt {a x^{3} + b x^{2} + c} + c^{2}}{a^{2} x^{6} + 2 \, {\left (a b - 3 \, a\right )} x^{5} + 2 \, a c x^{3} + {\left (b^{2} - 6 \, b + 9\right )} x^{4} + 2 \, {\left (b - 3\right )} c x^{2} + c^{2}}\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*} \int -\frac {\sqrt {a x^{3} + b x^{2} + c} {\left (a x^{3} - 2 \, c\right )}}{{\left (a x^{3} + {\left (b - 2\right )} x^{2} + c\right )} {\left (a x^{3} + {\left (b - 3\right )} x^{2} + c\right )}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 5.56, size = 16170, normalized size = 256.67 \begin {gather*} \text {output too large to display} \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 {\sqrt {a x^{3} + b x^{2} + c} {\left (a x^{3} - 2 \, c\right )}}{{\left (a x^{3} + {\left (b - 2\right )} x^{2} + c\right )} {\left (a x^{3} + {\left (b - 3\right )} x^{2} + c\right )}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 34.98, size = 125, normalized size = 1.98 \begin {gather*} \sqrt {2}\,\ln \left (\frac {c+a\,x^3+b\,x^2+2\,x^2-2\,\sqrt {2}\,x\,\sqrt {a\,x^3+b\,x^2+c}}{c+a\,x^3+b\,x^2-2\,x^2}\right )+\sqrt {3}\,\ln \left (\frac {c+a\,x^3+b\,x^2+3\,x^2+2\,\sqrt {3}\,x\,\sqrt {a\,x^3+b\,x^2+c}}{c+a\,x^3+b\,x^2-3\,x^2}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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