Optimal. Leaf size=561 \[ \frac {2 \tanh ^{-1}\left (\frac {\sqrt [4]{c_3} \sqrt {c_4+\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5}}{\sqrt {\sqrt {c_3} c_4-\sqrt {c_1} c_5}}\right ) \sqrt [4]{c_3}}{\sqrt {\sqrt {c_3} c_4-\sqrt {c_1} c_5} c_7}+\frac {2 \tanh ^{-1}\left (\frac {\sqrt [4]{c_3} \sqrt {c_4+\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5}}{\sqrt {\sqrt {c_3} c_4+\sqrt {c_1} c_5}}\right ) \sqrt [4]{c_3}}{\sqrt {\sqrt {c_3} c_4+\sqrt {c_1} c_5} c_7}+\frac {2 \text {ArcTan}\left (\frac {\sqrt {c_4+\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5} \sqrt {c_3 c_6-c_2 c_7}}{\sqrt {-c_3 c_4 c_6+c_2 c_4 c_7-c_5 \sqrt {c_1 c_6-c_0 c_7} \sqrt {c_3 c_6-c_2 c_7}}}\right ) \sqrt {c_3 c_6-c_2 c_7}}{c_7 \sqrt {-c_3 c_4 c_6+c_2 c_4 c_7-c_5 \sqrt {c_1 c_6-c_0 c_7} \sqrt {c_3 c_6-c_2 c_7}}}+\frac {2 \text {ArcTan}\left (\frac {\sqrt {c_4+\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5} \sqrt {c_3 c_6-c_2 c_7}}{\sqrt {-c_3 c_4 c_6+c_2 c_4 c_7+c_5 \sqrt {c_1 c_6-c_0 c_7} \sqrt {c_3 c_6-c_2 c_7}}}\right ) \sqrt {c_3 c_6-c_2 c_7}}{c_7 \sqrt {-c_3 c_4 c_6+c_2 c_4 c_7+c_5 \sqrt {c_1 c_6-c_0 c_7} \sqrt {c_3 c_6-c_2 c_7}}} \]
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Rubi [A]
time = 4.39, antiderivative size = 561, normalized size of antiderivative = 1.00, number of steps
used = 11, number of rules used = 4, integrand size = 41, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.098, Rules used = {6873, 6860,
1180, 214} \begin {gather*} \frac {2 \sqrt [4]{c_3} \tanh ^{-1}\left (\frac {\sqrt [4]{c_3} \sqrt {c_5 \sqrt {\frac {c_1 x+c_0}{c_3 x+c_2}}+c_4}}{\sqrt {\sqrt {c_3} c_4-\sqrt {c_1} c_5}}\right )}{\sqrt {\sqrt {c_3} c_4-\sqrt {c_1} c_5} c_7}-\frac {2 \sqrt {c_3 c_6-c_2 c_7} \tanh ^{-1}\left (\frac {\sqrt {c_3 c_6-c_2 c_7} \sqrt {c_5 \sqrt {\frac {c_1 x+c_0}{c_3 x+c_2}}+c_4}}{\sqrt {-\sqrt {c_1 c_6-c_0 c_7} \sqrt {c_3 c_6-c_2 c_7} c_5+c_3 c_4 c_6-c_2 c_4 c_7}}\right )}{c_7 \sqrt {-\sqrt {c_1 c_6-c_0 c_7} \sqrt {c_3 c_6-c_2 c_7} c_5+c_3 c_4 c_6-c_2 c_4 c_7}}-\frac {2 \sqrt {c_3 c_6-c_2 c_7} \tanh ^{-1}\left (\frac {\sqrt {c_3 c_6-c_2 c_7} \sqrt {c_5 \sqrt {\frac {c_1 x+c_0}{c_3 x+c_2}}+c_4}}{\sqrt {\sqrt {c_1 c_6-c_0 c_7} \sqrt {c_3 c_6-c_2 c_7} c_5+c_3 c_4 c_6-c_2 c_4 c_7}}\right )}{c_7 \sqrt {\sqrt {c_1 c_6-c_0 c_7} \sqrt {c_3 c_6-c_2 c_7} c_5+c_3 c_4 c_6-c_2 c_4 c_7}}+\frac {2 \sqrt [4]{c_3} \tanh ^{-1}\left (\frac {\sqrt [4]{c_3} \sqrt {c_5 \sqrt {\frac {c_1 x+c_0}{c_3 x+c_2}}+c_4}}{\sqrt {\sqrt {c_3} c_4+\sqrt {c_1} c_5}}\right )}{\sqrt {\sqrt {c_3} c_4+\sqrt {c_1} c_5} c_7} \end {gather*}
Antiderivative was successfully verified.
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Rule 214
Rule 1180
Rule 6860
Rule 6873
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {c_4+\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5} (c_6+x c_7)} \, dx &=(2 (c_1 c_2-c_0 c_3)) \text {Subst}\left (\int \frac {x}{\left (c_1-x^2 c_3\right ) \sqrt {c_4+x c_5} \left (c_1 c_6-c_0 c_7+x^2 (-c_3 c_6+c_2 c_7)\right )} \, dx,x,\sqrt {\frac {c_0+x c_1}{c_2+x c_3}}\right )\\ &=\frac {(4 (c_1 c_2-c_0 c_3)) \text {Subst}\left (\int \frac {x^2-c_4}{\left (c_1-\frac {c_3 \left (x^2-c_4\right ){}^2}{c_5{}^2}\right ) \left (c_1 c_6-c_0 c_7-\frac {\left (x^2-c_4\right ){}^2 (c_3 c_6-c_2 c_7)}{c_5{}^2}\right )} \, dx,x,\sqrt {c_4+\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5}\right )}{c_5{}^2}\\ &=\frac {(4 (c_1 c_2-c_0 c_3)) \text {Subst}\left (\int \frac {x^2-c_4}{\left (c_1-\frac {x^4 c_3}{c_5{}^2}+\frac {2 x^2 c_3 c_4}{c_5{}^2}-\frac {c_3 c_4{}^2}{c_5{}^2}\right ) \left (c_1 c_6-\frac {c_3 c_4{}^2 c_6}{c_5{}^2}-c_0 c_7+\frac {c_2 c_4{}^2 c_7}{c_5{}^2}-\frac {x^4 (c_3 c_6-c_2 c_7)}{c_5{}^2}+\frac {2 x^2 c_4 (c_3 c_6-c_2 c_7)}{c_5{}^2}\right )} \, dx,x,\sqrt {c_4+\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5}\right )}{c_5{}^2}\\ &=\frac {(4 (c_1 c_2-c_0 c_3)) \text {Subst}\left (\int \left (\frac {c_3 \left (x^2-c_4\right ) c_5{}^2}{(c_1 c_2-c_0 c_3) \left (-x^4 c_3+2 x^2 c_3 c_4-c_3 c_4{}^2+c_1 c_5{}^2\right ) c_7}+\frac {\left (x^2-c_4\right ) c_5{}^2 (c_3 c_6-c_2 c_7)}{(c_1 c_2-c_0 c_3) c_7 \left (c_3 c_4{}^2 c_6-c_1 c_5{}^2 c_6-\left (c_2 c_4{}^2-c_0 c_5{}^2\right ) c_7+x^4 (c_3 c_6-c_2 c_7)-2 x^2 c_4 (c_3 c_6-c_2 c_7)\right )}\right ) \, dx,x,\sqrt {c_4+\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5}\right )}{c_5{}^2}\\ &=\frac {(4 c_3) \text {Subst}\left (\int \frac {x^2-c_4}{-x^4 c_3+2 x^2 c_3 c_4-c_3 c_4{}^2+c_1 c_5{}^2} \, dx,x,\sqrt {c_4+\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5}\right )}{c_7}+\frac {(4 (c_3 c_6-c_2 c_7)) \text {Subst}\left (\int \frac {x^2-c_4}{c_3 c_4{}^2 c_6-c_1 c_5{}^2 c_6-\left (c_2 c_4{}^2-c_0 c_5{}^2\right ) c_7+x^4 (c_3 c_6-c_2 c_7)-2 x^2 c_4 (c_3 c_6-c_2 c_7)} \, dx,x,\sqrt {c_4+\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5}\right )}{c_7}\\ &=\frac {(2 c_3) \text {Subst}\left (\int \frac {1}{-x^2 c_3+c_3 c_4-\sqrt {c_1} \sqrt {c_3} c_5} \, dx,x,\sqrt {c_4+\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5}\right )}{c_7}+\frac {(2 c_3) \text {Subst}\left (\int \frac {1}{-x^2 c_3+c_3 c_4+\sqrt {c_1} \sqrt {c_3} c_5} \, dx,x,\sqrt {c_4+\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5}\right )}{c_7}+\frac {(2 (c_3 c_6-c_2 c_7)) \text {Subst}\left (\int \frac {1}{-c_5 \sqrt {c_1 c_6-c_0 c_7} \sqrt {c_3 c_6-c_2 c_7}+x^2 (c_3 c_6-c_2 c_7)-c_4 (c_3 c_6-c_2 c_7)} \, dx,x,\sqrt {c_4+\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5}\right )}{c_7}+\frac {(2 (c_3 c_6-c_2 c_7)) \text {Subst}\left (\int \frac {1}{c_5 \sqrt {c_1 c_6-c_0 c_7} \sqrt {c_3 c_6-c_2 c_7}+x^2 (c_3 c_6-c_2 c_7)-c_4 (c_3 c_6-c_2 c_7)} \, dx,x,\sqrt {c_4+\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5}\right )}{c_7}\\ &=\frac {2 \tanh ^{-1}\left (\frac {\sqrt [4]{c_3} \sqrt {c_4+\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5}}{\sqrt {\sqrt {c_3} c_4-\sqrt {c_1} c_5}}\right ) \sqrt [4]{c_3}}{\sqrt {\sqrt {c_3} c_4-\sqrt {c_1} c_5} c_7}+\frac {2 \tanh ^{-1}\left (\frac {\sqrt [4]{c_3} \sqrt {c_4+\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5}}{\sqrt {\sqrt {c_3} c_4+\sqrt {c_1} c_5}}\right ) \sqrt [4]{c_3}}{\sqrt {\sqrt {c_3} c_4+\sqrt {c_1} c_5} c_7}-\frac {2 \tanh ^{-1}\left (\frac {\sqrt {c_4+\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5} \sqrt {c_3 c_6-c_2 c_7}}{\sqrt {c_3 c_4 c_6-c_2 c_4 c_7-c_5 \sqrt {c_1 c_6-c_0 c_7} \sqrt {c_3 c_6-c_2 c_7}}}\right ) \sqrt {c_3 c_6-c_2 c_7}}{c_7 \sqrt {c_3 c_4 c_6-c_2 c_4 c_7-c_5 \sqrt {c_1 c_6-c_0 c_7} \sqrt {c_3 c_6-c_2 c_7}}}-\frac {2 \tanh ^{-1}\left (\frac {\sqrt {c_4+\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5} \sqrt {c_3 c_6-c_2 c_7}}{\sqrt {c_3 c_4 c_6-c_2 c_4 c_7+c_5 \sqrt {c_1 c_6-c_0 c_7} \sqrt {c_3 c_6-c_2 c_7}}}\right ) \sqrt {c_3 c_6-c_2 c_7}}{c_7 \sqrt {c_3 c_4 c_6-c_2 c_4 c_7+c_5 \sqrt {c_1 c_6-c_0 c_7} \sqrt {c_3 c_6-c_2 c_7}}}\\ \end {align*}
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Mathematica [A]
time = 56.01, size = 968, normalized size = 1.73 \begin {gather*} \frac {2 (c_1 c_2-c_0 c_3) \left (\frac {\text {ArcTan}\left (\frac {\sqrt {c_3 c_4{}^2-c_1 c_5{}^2}}{\sqrt {-c_3 c_4-\sqrt {c_1} \sqrt {c_3} c_5} \sqrt {c_4+\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5}}\right ) \sqrt {c_3} \left (\sqrt {c_3} c_4+\sqrt {c_1} c_5\right )}{\sqrt {-c_3 c_4-\sqrt {c_1} \sqrt {c_3} c_5} \sqrt {c_3 c_4{}^2-c_1 c_5{}^2}}+\frac {\text {ArcTan}\left (\frac {\sqrt {c_3 c_4{}^2-c_1 c_5{}^2}}{\sqrt {-c_3 c_4+\sqrt {c_1} \sqrt {c_3} c_5} \sqrt {c_4+\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5}}\right ) \sqrt {c_3} \left (\sqrt {c_3} c_4-\sqrt {c_1} c_5\right )}{\sqrt {-c_3 c_4+\sqrt {c_1} \sqrt {c_3} c_5} \sqrt {c_3 c_4{}^2-c_1 c_5{}^2}}-\frac {\text {ArcTan}\left (\frac {\sqrt {c_3 c_4{}^2 c_6-c_1 c_5{}^2 c_6-c_2 c_4{}^2 c_7+c_0 c_5{}^2 c_7}}{\sqrt {c_4+\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5} \sqrt {-c_3 c_4 c_6+c_2 c_4 c_7-c_5 \sqrt {c_1 c_6-c_0 c_7} \sqrt {c_3 c_6-c_2 c_7}}}\right ) \sqrt {c_3 c_6-c_2 c_7} \left (c_1 c_5 c_6-c_0 c_5 c_7+c_4 \sqrt {c_1 c_6-c_0 c_7} \sqrt {c_3 c_6-c_2 c_7}\right )}{\sqrt {c_1 c_6-c_0 c_7} \sqrt {c_3 c_4{}^2 c_6-c_1 c_5{}^2 c_6+\left (-c_2 c_4{}^2+c_0 c_5{}^2\right ) c_7} \sqrt {-c_3 c_4 c_6+c_2 c_4 c_7-c_5 \sqrt {c_1 c_6-c_0 c_7} \sqrt {c_3 c_6-c_2 c_7}}}-\frac {\text {ArcTan}\left (\frac {\sqrt {c_3 c_4{}^2 c_6-c_1 c_5{}^2 c_6-c_2 c_4{}^2 c_7+c_0 c_5{}^2 c_7}}{\sqrt {c_4+\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5} \sqrt {-c_3 c_4 c_6+c_2 c_4 c_7+c_5 \sqrt {c_1 c_6-c_0 c_7} \sqrt {c_3 c_6-c_2 c_7}}}\right ) \sqrt {c_3 c_6-c_2 c_7} \left (-c_1 c_5 c_6+c_0 c_5 c_7+c_4 \sqrt {c_1 c_6-c_0 c_7} \sqrt {c_3 c_6-c_2 c_7}\right )}{\sqrt {c_1 c_6-c_0 c_7} \sqrt {c_3 c_4{}^2 c_6-c_1 c_5{}^2 c_6+\left (-c_2 c_4{}^2+c_0 c_5{}^2\right ) c_7} \sqrt {-c_3 c_4 c_6+c_2 c_4 c_7+c_5 \sqrt {c_1 c_6-c_0 c_7} \sqrt {c_3 c_6-c_2 c_7}}}\right )}{(-c_1 c_2+c_0 c_3) c_7} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.05, size = 0, normalized size = 0.00 \[\int \frac {1}{\sqrt {\textit {\_C4} +\sqrt {\frac {\textit {\_C1} x +\textit {\_C0}}{\textit {\_C3} x +\textit {\_C2}}}\, \textit {\_C5}}\, \left (\textit {\_C7} x +\textit {\_C6} \right )}\, dx\]
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*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 3633 vs.
\(2 (357) = 714\).
time = 173.34, size = 3633, normalized size = 6.48 \begin {gather*} \text {Too large to display} \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}{\sqrt {_C4 + _C5 \sqrt {\frac {_C0}{_C2 + _C3 x} + \frac {_C1 x}{_C2 + _C3 x}}} \left (_C6 + _C7 x\right )}\, dx \end {gather*}
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 \begin {gather*} \text {Exception raised: TypeError} \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}{\sqrt {_{\mathrm {C4}}+_{\mathrm {C5}}\,\sqrt {\frac {_{\mathrm {C0}}+_{\mathrm {C1}}\,x}{_{\mathrm {C2}}+_{\mathrm {C3}}\,x}}}\,\left (_{\mathrm {C6}}+_{\mathrm {C7}}\,x\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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