Optimal. Leaf size=195 \[ -\frac {2 \sqrt {-1+c_1} \tan ^{-1}\left (\frac {\sqrt {1-c_0} \sqrt {-1+c_1} \sqrt {\frac {c_3 x^2+c_0 x+c_4}{c_3 x^2+c_1 x+c_4}}}{-1+c_0}\right )}{\sqrt {1-c_0}}-\frac {4 \sqrt {1+c_1} \tan ^{-1}\left (\frac {\sqrt {-1-c_0} \sqrt {1+c_1} \sqrt {\frac {c_3 x^2+c_0 x+c_4}{c_3 x^2+c_1 x+c_4}}}{1+c_0}\right )}{\sqrt {-1-c_0}}+6 \tanh ^{-1}\left (\sqrt {\frac {c_3 x^2+c_0 x+c_4}{c_3 x^2+c_1 x+c_4}}\right ) \]
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Rubi [F] time = 15.75, 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 (x^2 c_3-c_4\right ) \left (x+3 x^2 c_3+3 c_4\right )}{x \sqrt {\frac {x c_0+x^2 c_3+c_4}{x c_1+x^2 c_3+c_4}} \left (-x^2+x^4 c_3{}^2+2 x^2 c_3 c_4+c_4{}^2\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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\begin {align*} \int \frac {\left (x^2 c_3-c_4\right ) \left (x+3 x^2 c_3+3 c_4\right )}{x \sqrt {\frac {x c_0+x^2 c_3+c_4}{x c_1+x^2 c_3+c_4}} \left (-x^2+x^4 c_3{}^2+2 x^2 c_3 c_4+c_4{}^2\right )} \, dx &=\int \frac {\left (x^2 c_3-c_4\right ) \left (x+3 x^2 c_3+3 c_4\right )}{x \sqrt {\frac {x c_0+x^2 c_3+c_4}{x c_1+x^2 c_3+c_4}} \left (x^4 c_3{}^2+c_4{}^2+x^2 (-1+2 c_3 c_4)\right )} \, dx\\ &=\frac {\sqrt {x c_0+x^2 c_3+c_4} \int \frac {\left (x^2 c_3-c_4\right ) \sqrt {x c_1+x^2 c_3+c_4} \left (x+3 x^2 c_3+3 c_4\right )}{x \sqrt {x c_0+x^2 c_3+c_4} \left (x^4 c_3{}^2+c_4{}^2+x^2 (-1+2 c_3 c_4)\right )} \, dx}{\sqrt {\frac {x c_0+x^2 c_3+c_4}{x c_1+x^2 c_3+c_4}} \sqrt {x c_1+x^2 c_3+c_4}}\\ &=\frac {\sqrt {x c_0+x^2 c_3+c_4} \int \left (-\frac {3 \sqrt {x c_1+x^2 c_3+c_4}}{x \sqrt {x c_0+x^2 c_3+c_4}}+\frac {2 (-1+2 x c_3) \sqrt {x c_1+x^2 c_3+c_4}}{\left (-x+x^2 c_3+c_4\right ) \sqrt {x c_0+x^2 c_3+c_4}}+\frac {(1+2 x c_3) \sqrt {x c_1+x^2 c_3+c_4}}{\left (x+x^2 c_3+c_4\right ) \sqrt {x c_0+x^2 c_3+c_4}}\right ) \, dx}{\sqrt {\frac {x c_0+x^2 c_3+c_4}{x c_1+x^2 c_3+c_4}} \sqrt {x c_1+x^2 c_3+c_4}}\\ &=\frac {\sqrt {x c_0+x^2 c_3+c_4} \int \frac {(1+2 x c_3) \sqrt {x c_1+x^2 c_3+c_4}}{\left (x+x^2 c_3+c_4\right ) \sqrt {x c_0+x^2 c_3+c_4}} \, dx}{\sqrt {\frac {x c_0+x^2 c_3+c_4}{x c_1+x^2 c_3+c_4}} \sqrt {x c_1+x^2 c_3+c_4}}+\frac {\left (2 \sqrt {x c_0+x^2 c_3+c_4}\right ) \int \frac {(-1+2 x c_3) \sqrt {x c_1+x^2 c_3+c_4}}{\left (-x+x^2 c_3+c_4\right ) \sqrt {x c_0+x^2 c_3+c_4}} \, dx}{\sqrt {\frac {x c_0+x^2 c_3+c_4}{x c_1+x^2 c_3+c_4}} \sqrt {x c_1+x^2 c_3+c_4}}-\frac {\left (3 \sqrt {x c_0+x^2 c_3+c_4}\right ) \int \frac {\sqrt {x c_1+x^2 c_3+c_4}}{x \sqrt {x c_0+x^2 c_3+c_4}} \, dx}{\sqrt {\frac {x c_0+x^2 c_3+c_4}{x c_1+x^2 c_3+c_4}} \sqrt {x c_1+x^2 c_3+c_4}}\\ &=\frac {\sqrt {x c_0+x^2 c_3+c_4} \int \left (\frac {2 c_3 \sqrt {x c_1+x^2 c_3+c_4}}{\sqrt {x c_0+x^2 c_3+c_4} \left (1+2 x c_3-\sqrt {1-4 c_3 c_4}\right )}+\frac {2 c_3 \sqrt {x c_1+x^2 c_3+c_4}}{\sqrt {x c_0+x^2 c_3+c_4} \left (1+2 x c_3+\sqrt {1-4 c_3 c_4}\right )}\right ) \, dx}{\sqrt {\frac {x c_0+x^2 c_3+c_4}{x c_1+x^2 c_3+c_4}} \sqrt {x c_1+x^2 c_3+c_4}}+\frac {\left (2 \sqrt {x c_0+x^2 c_3+c_4}\right ) \int \left (\frac {2 c_3 \sqrt {x c_1+x^2 c_3+c_4}}{\sqrt {x c_0+x^2 c_3+c_4} \left (-1+2 x c_3-\sqrt {1-4 c_3 c_4}\right )}+\frac {2 c_3 \sqrt {x c_1+x^2 c_3+c_4}}{\sqrt {x c_0+x^2 c_3+c_4} \left (-1+2 x c_3+\sqrt {1-4 c_3 c_4}\right )}\right ) \, dx}{\sqrt {\frac {x c_0+x^2 c_3+c_4}{x c_1+x^2 c_3+c_4}} \sqrt {x c_1+x^2 c_3+c_4}}-\frac {\left (3 \sqrt {x c_0+x^2 c_3+c_4}\right ) \int \frac {\sqrt {x c_1+x^2 c_3+c_4}}{x \sqrt {x c_0+x^2 c_3+c_4}} \, dx}{\sqrt {\frac {x c_0+x^2 c_3+c_4}{x c_1+x^2 c_3+c_4}} \sqrt {x c_1+x^2 c_3+c_4}}\\ &=-\frac {\left (3 \sqrt {x c_0+x^2 c_3+c_4}\right ) \int \frac {\sqrt {x c_1+x^2 c_3+c_4}}{x \sqrt {x c_0+x^2 c_3+c_4}} \, dx}{\sqrt {\frac {x c_0+x^2 c_3+c_4}{x c_1+x^2 c_3+c_4}} \sqrt {x c_1+x^2 c_3+c_4}}+\frac {\left (2 c_3 \sqrt {x c_0+x^2 c_3+c_4}\right ) \int \frac {\sqrt {x c_1+x^2 c_3+c_4}}{\sqrt {x c_0+x^2 c_3+c_4} \left (1+2 x c_3-\sqrt {1-4 c_3 c_4}\right )} \, dx}{\sqrt {\frac {x c_0+x^2 c_3+c_4}{x c_1+x^2 c_3+c_4}} \sqrt {x c_1+x^2 c_3+c_4}}+\frac {\left (2 c_3 \sqrt {x c_0+x^2 c_3+c_4}\right ) \int \frac {\sqrt {x c_1+x^2 c_3+c_4}}{\sqrt {x c_0+x^2 c_3+c_4} \left (1+2 x c_3+\sqrt {1-4 c_3 c_4}\right )} \, dx}{\sqrt {\frac {x c_0+x^2 c_3+c_4}{x c_1+x^2 c_3+c_4}} \sqrt {x c_1+x^2 c_3+c_4}}+\frac {\left (4 c_3 \sqrt {x c_0+x^2 c_3+c_4}\right ) \int \frac {\sqrt {x c_1+x^2 c_3+c_4}}{\sqrt {x c_0+x^2 c_3+c_4} \left (-1+2 x c_3-\sqrt {1-4 c_3 c_4}\right )} \, dx}{\sqrt {\frac {x c_0+x^2 c_3+c_4}{x c_1+x^2 c_3+c_4}} \sqrt {x c_1+x^2 c_3+c_4}}+\frac {\left (4 c_3 \sqrt {x c_0+x^2 c_3+c_4}\right ) \int \frac {\sqrt {x c_1+x^2 c_3+c_4}}{\sqrt {x c_0+x^2 c_3+c_4} \left (-1+2 x c_3+\sqrt {1-4 c_3 c_4}\right )} \, dx}{\sqrt {\frac {x c_0+x^2 c_3+c_4}{x c_1+x^2 c_3+c_4}} \sqrt {x c_1+x^2 c_3+c_4}}\\ \end {align*}
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Mathematica [B] time = 6.65, size = 43491, normalized size = 223.03 \begin {gather*} \text {Result too large to show} \end {gather*}
Warning: Unable to verify antiderivative.
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IntegrateAlgebraic [A] time = 1.02, size = 195, normalized size = 1.00 \begin {gather*} 6 \tanh ^{-1}\left (\sqrt {\frac {x c_0+x^2 c_3+c_4}{x c_1+x^2 c_3+c_4}}\right )-\frac {2 \tan ^{-1}\left (\frac {\sqrt {1-c_0} \sqrt {-1+c_1} \sqrt {\frac {x c_0+x^2 c_3+c_4}{x c_1+x^2 c_3+c_4}}}{-1+c_0}\right ) \sqrt {-1+c_1}}{\sqrt {1-c_0}}-\frac {4 \tan ^{-1}\left (\frac {\sqrt {-1-c_0} \sqrt {1+c_1} \sqrt {\frac {x c_0+x^2 c_3+c_4}{x c_1+x^2 c_3+c_4}}}{1+c_0}\right ) \sqrt {1+c_1}}{\sqrt {-1-c_0}} \end {gather*}
Antiderivative was successfully verified.
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fricas [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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giac [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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maple [B] time = 13.84, size = 35837869, normalized size = 183783.94
method | result | size |
default | \(\text {Expression too large to display}\) | \(35837869\) |
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 {{\left (3 \, \_{C_{3}} x^{2} + 3 \, \_{C_{4}} + x\right )} {\left (\_{C_{3}} x^{2} - \_{C_{4}}\right )}}{{\left (\_{C_{3}}^{2} x^{4} + 2 \, \_{C_{3}} \_{C_{4}} x^{2} + \_{C_{4}}^{2} - x^{2}\right )} x \sqrt {\frac {\_{C_{3}} x^{2} + \_{C_{0}} x + \_{C_{4}}}{\_{C_{3}} x^{2} + \_{C_{1}} x + \_{C_{4}}}}}\,{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 = -0.01 \begin {gather*} \int -\frac {\left (_{\mathrm {C4}}-_{\mathrm {C3}}\,x^2\right )\,\left (3\,_{\mathrm {C3}}\,x^2+x+3\,_{\mathrm {C4}}\right )}{x\,\sqrt {\frac {_{\mathrm {C3}}\,x^2+_{\mathrm {C0}}\,x+_{\mathrm {C4}}}{_{\mathrm {C3}}\,x^2+_{\mathrm {C1}}\,x+_{\mathrm {C4}}}}\,\left ({_{\mathrm {C3}}}^2\,x^4+2\,_{\mathrm {C3}}\,_{\mathrm {C4}}\,x^2+{_{\mathrm {C4}}}^2-x^2\right )} \,d x \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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