Optimal. Leaf size=586 \[ -\frac {\text {RootSum}\left [\text {$\#$1}^8+4 \text {$\#$1}^6 b-8 \text {$\#$1}^5 b+6 \text {$\#$1}^4 b^2+2 \text {$\#$1}^4 b+16 \text {$\#$1}^4-16 \text {$\#$1}^3 b^2-32 \text {$\#$1}^3+4 \text {$\#$1}^2 b^3+20 \text {$\#$1}^2 b^2+24 \text {$\#$1}^2-8 \text {$\#$1} b^3-8 \text {$\#$1} b^2-8 \text {$\#$1}+b^4+2 b^3+b^2+1\& ,\frac {2 \text {$\#$1}^5 \log \left (\text {$\#$1}+\sqrt {a x-b}-\sqrt {\sqrt {a x-b}+a x}\right )-5 \text {$\#$1}^4 \log \left (\text {$\#$1}+\sqrt {a x-b}-\sqrt {\sqrt {a x-b}+a x}\right )+4 \text {$\#$1}^3 b \log \left (\text {$\#$1}+\sqrt {a x-b}-\sqrt {\sqrt {a x-b}+a x}\right )+4 \text {$\#$1}^3 \log \left (\text {$\#$1}+\sqrt {a x-b}-\sqrt {\sqrt {a x-b}+a x}\right )-6 \text {$\#$1}^2 b \log \left (\text {$\#$1}+\sqrt {a x-b}-\sqrt {\sqrt {a x-b}+a x}\right )-\text {$\#$1}^2 \log \left (\text {$\#$1}+\sqrt {a x-b}-\sqrt {\sqrt {a x-b}+a x}\right )+2 \text {$\#$1} b^2 \log \left (\text {$\#$1}+\sqrt {a x-b}-\sqrt {\sqrt {a x-b}+a x}\right )-b^2 \log \left (\text {$\#$1}+\sqrt {a x-b}-\sqrt {\sqrt {a x-b}+a x}\right )+2 \text {$\#$1} b \log \left (\text {$\#$1}+\sqrt {a x-b}-\sqrt {\sqrt {a x-b}+a x}\right )}{\text {$\#$1}^7+3 \text {$\#$1}^5 b-5 \text {$\#$1}^4 b+3 \text {$\#$1}^3 b^2+\text {$\#$1}^3 b+8 \text {$\#$1}^3-6 \text {$\#$1}^2 b^2-12 \text {$\#$1}^2+\text {$\#$1} b^3+5 \text {$\#$1} b^2+6 \text {$\#$1}-b^3-b^2-1}\& \right ]}{2 a} \]
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Rubi [C] time = 1.91, antiderivative size = 590, normalized size of antiderivative = 1.01, number of steps used = 19, number of rules used = 10, integrand size = 42, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {6728, 990, 621, 206, 1033, 724, 1036, 1030, 205, 208} \begin {gather*} -\frac {\sqrt {-i b+\sqrt {b-(1-i)}+(1+i)} \tan ^{-1}\left (\frac {\left (b+i \sqrt {b-(1-i)}-(1-i)\right ) \sqrt {a x-b}+\sqrt {b-(1-i)} (b+i)}{\sqrt {2} \sqrt {1-i b} \sqrt {-i b+\sqrt {b-(1-i)}+(1+i)} \sqrt [4]{b-(1-i)} \sqrt {\sqrt {a x-b}+a x}}\right )}{\sqrt {2} a \sqrt {1-i b} \sqrt [4]{b-(1-i)}}+\frac {i \sqrt {i-\sqrt {-b+i}} \tanh ^{-1}\left (\frac {\left (1-2 \sqrt {-b+i}\right ) \left (-\sqrt {a x-b}\right )-2 b+\sqrt {-b+i}}{2 \sqrt {i-\sqrt {-b+i}} \sqrt {\sqrt {a x-b}+a x}}\right )}{2 a \sqrt {-b+i}}+\frac {i \sqrt {\sqrt {-b+i}+i} \tanh ^{-1}\left (\frac {\left (1+2 \sqrt {-b+i}\right ) \sqrt {a x-b}+2 b+\sqrt {-b+i}}{2 \sqrt {\sqrt {-b+i}+i} \sqrt {\sqrt {a x-b}+a x}}\right )}{2 a \sqrt {-b+i}}+\frac {i \sqrt {b-i \sqrt {b-(1-i)}-(1-i)} \tanh ^{-1}\left (\frac {\left (-b+i \sqrt {b-(1-i)}+(1-i)\right ) \sqrt {a x-b}+\sqrt {b-(1-i)} (b+i)}{\sqrt {2} \sqrt {-b-i} \sqrt [4]{b-(1-i)} \sqrt {b-i \sqrt {b-(1-i)}-(1-i)} \sqrt {\sqrt {a x-b}+a x}}\right )}{\sqrt {2} a \sqrt {-b-i} \sqrt [4]{b-(1-i)}} \end {gather*}
Antiderivative was successfully verified.
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Rule 205
Rule 206
Rule 208
Rule 621
Rule 724
Rule 990
Rule 1030
Rule 1033
Rule 1036
Rule 6728
Rubi steps
\begin {align*} \int \frac {\sqrt {a x+\sqrt {-b+a x}}}{\sqrt {-b+a x} \left (1+a^2 x^2\right )} \, dx &=\frac {2 \operatorname {Subst}\left (\int \frac {\sqrt {b+x+x^2}}{1+b^2+2 b x^2+x^4} \, dx,x,\sqrt {-b+a x}\right )}{a}\\ &=\frac {2 \operatorname {Subst}\left (\int \left (-\frac {i \sqrt {b+x+x^2}}{-2 i+2 b+2 x^2}+\frac {i \sqrt {b+x+x^2}}{2 i+2 b+2 x^2}\right ) \, dx,x,\sqrt {-b+a x}\right )}{a}\\ &=-\frac {(2 i) \operatorname {Subst}\left (\int \frac {\sqrt {b+x+x^2}}{-2 i+2 b+2 x^2} \, dx,x,\sqrt {-b+a x}\right )}{a}+\frac {(2 i) \operatorname {Subst}\left (\int \frac {\sqrt {b+x+x^2}}{2 i+2 b+2 x^2} \, dx,x,\sqrt {-b+a x}\right )}{a}\\ &=\frac {i \operatorname {Subst}\left (\int \frac {-2 i-2 x}{\sqrt {b+x+x^2} \left (-2 i+2 b+2 x^2\right )} \, dx,x,\sqrt {-b+a x}\right )}{a}-\frac {i \operatorname {Subst}\left (\int \frac {2 i-2 x}{\sqrt {b+x+x^2} \left (2 i+2 b+2 x^2\right )} \, dx,x,\sqrt {-b+a x}\right )}{a}\\ &=-\frac {\left (i-\frac {1}{\sqrt {i-b}}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (-2 \sqrt {i-b}+2 x\right ) \sqrt {b+x+x^2}} \, dx,x,\sqrt {-b+a x}\right )}{a}-\frac {\left (i+\frac {1}{\sqrt {i-b}}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (2 \sqrt {i-b}+2 x\right ) \sqrt {b+x+x^2}} \, dx,x,\sqrt {-b+a x}\right )}{a}-\frac {i \operatorname {Subst}\left (\int \frac {4 \left ((-1+i)+i \sqrt {(-1+i)+b}+b\right )-4 \sqrt {(-1+i)+b} x}{\sqrt {b+x+x^2} \left (2 i+2 b+2 x^2\right )} \, dx,x,\sqrt {-b+a x}\right )}{4 a \sqrt {(-1+i)+b}}+\frac {i \operatorname {Subst}\left (\int \frac {4 \left ((-1+i)-i \sqrt {(-1+i)+b}+b\right )+4 \sqrt {(-1+i)+b} x}{\sqrt {b+x+x^2} \left (2 i+2 b+2 x^2\right )} \, dx,x,\sqrt {-b+a x}\right )}{4 a \sqrt {(-1+i)+b}}\\ &=\frac {\left (2 \left (i-\frac {1}{\sqrt {i-b}}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{16 \sqrt {i-b}+16 (i-b)+16 b-x^2} \, dx,x,\frac {2 \sqrt {i-b}+4 b-\left (-2-4 \sqrt {i-b}\right ) \sqrt {-b+a x}}{\sqrt {a x+\sqrt {-b+a x}}}\right )}{a}+\frac {\left (2 \left (i+\frac {1}{\sqrt {i-b}}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{-16 \sqrt {i-b}+16 (i-b)+16 b-x^2} \, dx,x,\frac {-2 \sqrt {i-b}+4 b-\left (-2+4 \sqrt {i-b}\right ) \sqrt {-b+a x}}{\sqrt {a x+\sqrt {-b+a x}}}\right )}{a}-\frac {\left (16 \left ((-1-i)+\sqrt {(-1+i)+b}+i b\right ) (i+b)\right ) \operatorname {Subst}\left (\int \frac {1}{256 \sqrt {(-1+i)+b} (i+b)^2 \left ((-1+i)-i \sqrt {(-1+i)+b}+b\right )+2 (i+b) x^2} \, dx,x,\frac {8 \sqrt {(-1+i)+b} (i+b)-8 \left ((-1+i)-i \sqrt {(-1+i)+b}+b\right ) \sqrt {-b+a x}}{\sqrt {a x+\sqrt {-b+a x}}}\right )}{a}-\frac {\left (16 i (i+b) \left ((-1+i)+i \sqrt {(-1+i)+b}+b\right )\right ) \operatorname {Subst}\left (\int \frac {1}{-256 \sqrt {(-1+i)+b} (i+b)^2 \left (i \left ((1+i)+\sqrt {(-1+i)+b}\right )+b\right )+2 (i+b) x^2} \, dx,x,\frac {-8 \sqrt {(-1+i)+b} (i+b)-8 \left ((-1+i)+i \sqrt {(-1+i)+b}+b\right ) \sqrt {-b+a x}}{\sqrt {a x+\sqrt {-b+a x}}}\right )}{a}\\ &=-\frac {\sqrt {(1+i)+\sqrt {(-1+i)+b}-i b} \tan ^{-1}\left (\frac {\sqrt {(-1+i)+b} (i+b)+\left ((-1+i)+i \sqrt {(-1+i)+b}+b\right ) \sqrt {-b+a x}}{\sqrt {2} \sqrt {1-i b} \sqrt {(1+i)+\sqrt {(-1+i)+b}-i b} \sqrt [4]{(-1+i)+b} \sqrt {a x+\sqrt {-b+a x}}}\right )}{\sqrt {2} a \sqrt {1-i b} \sqrt [4]{(-1+i)+b}}-\frac {\left (i+\frac {1}{\sqrt {i-b}}\right ) \tanh ^{-1}\left (\frac {\sqrt {i-b}-2 b-\left (1-2 \sqrt {i-b}\right ) \sqrt {-b+a x}}{2 \sqrt {i-\sqrt {i-b}} \sqrt {a x+\sqrt {-b+a x}}}\right )}{2 a \sqrt {i-\sqrt {i-b}}}+\frac {i \sqrt {i+\sqrt {i-b}} \tanh ^{-1}\left (\frac {\sqrt {i-b}+2 b+\left (1+2 \sqrt {i-b}\right ) \sqrt {-b+a x}}{2 \sqrt {i+\sqrt {i-b}} \sqrt {a x+\sqrt {-b+a x}}}\right )}{2 a \sqrt {i-b}}+\frac {i \sqrt {(-1+i)-i \sqrt {(-1+i)+b}+b} \tanh ^{-1}\left (\frac {\sqrt {(-1+i)+b} (i+b)-\left ((-1+i)-i \sqrt {(-1+i)+b}+b\right ) \sqrt {-b+a x}}{\sqrt {2} \sqrt {-i-b} \sqrt [4]{(-1+i)+b} \sqrt {(-1+i)-i \sqrt {(-1+i)+b}+b} \sqrt {a x+\sqrt {-b+a x}}}\right )}{\sqrt {2} a \sqrt {-i-b} \sqrt [4]{(-1+i)+b}}\\ \end {align*}
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Mathematica [C] time = 0.85, size = 475, normalized size = 0.81 \begin {gather*} -\frac {i \left (\frac {\sqrt {\sqrt {-b-i}+i} \tan ^{-1}\left (\frac {\left (-1+2 \sqrt {-b-i}\right ) \sqrt {a x-b}-2 b+\sqrt {-b-i}}{2 \sqrt {\sqrt {-b-i}+i} \sqrt {\sqrt {a x-b}+a x}}\right )+\sqrt {i-\sqrt {-b-i}} \tan ^{-1}\left (\frac {\left (1+2 \sqrt {-b-i}\right ) \sqrt {a x-b}+2 b+\sqrt {-b-i}}{2 \sqrt {i-\sqrt {-b-i}} \sqrt {\sqrt {a x-b}+a x}}\right )}{\sqrt {-b-i}}-\frac {\sqrt {i-\sqrt {-b+i}} \tanh ^{-1}\left (\frac {\left (-1+2 \sqrt {-b+i}\right ) \sqrt {a x-b}-2 b+\sqrt {-b+i}}{2 \sqrt {i-\sqrt {-b+i}} \sqrt {\sqrt {a x-b}+a x}}\right )}{\sqrt {-b+i}}-\frac {\sqrt {\sqrt {-b+i}+i} \tanh ^{-1}\left (\frac {\left (1+2 \sqrt {-b+i}\right ) \sqrt {a x-b}+2 b+\sqrt {-b+i}}{2 \sqrt {\sqrt {-b+i}+i} \sqrt {\sqrt {a x-b}+a x}}\right )}{\sqrt {-b+i}}\right )}{2 a} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.76, size = 604, normalized size = 1.03 \begin {gather*} -\frac {\text {RootSum}\left [1+b^2+2 b^3+b^4-8 \text {$\#$1}-8 b^2 \text {$\#$1}-8 b^3 \text {$\#$1}+24 \text {$\#$1}^2+20 b^2 \text {$\#$1}^2+4 b^3 \text {$\#$1}^2-32 \text {$\#$1}^3-16 b^2 \text {$\#$1}^3+16 \text {$\#$1}^4+2 b \text {$\#$1}^4+6 b^2 \text {$\#$1}^4-8 b \text {$\#$1}^5+4 b \text {$\#$1}^6+\text {$\#$1}^8\&,\frac {-b^2 \log \left (-\sqrt {-b+a x}+\sqrt {a x+\sqrt {-b+a x}}-\text {$\#$1}\right )+2 b \log \left (-\sqrt {-b+a x}+\sqrt {a x+\sqrt {-b+a x}}-\text {$\#$1}\right ) \text {$\#$1}+2 b^2 \log \left (-\sqrt {-b+a x}+\sqrt {a x+\sqrt {-b+a x}}-\text {$\#$1}\right ) \text {$\#$1}-\log \left (-\sqrt {-b+a x}+\sqrt {a x+\sqrt {-b+a x}}-\text {$\#$1}\right ) \text {$\#$1}^2-6 b \log \left (-\sqrt {-b+a x}+\sqrt {a x+\sqrt {-b+a x}}-\text {$\#$1}\right ) \text {$\#$1}^2+4 \log \left (-\sqrt {-b+a x}+\sqrt {a x+\sqrt {-b+a x}}-\text {$\#$1}\right ) \text {$\#$1}^3+4 b \log \left (-\sqrt {-b+a x}+\sqrt {a x+\sqrt {-b+a x}}-\text {$\#$1}\right ) \text {$\#$1}^3-5 \log \left (-\sqrt {-b+a x}+\sqrt {a x+\sqrt {-b+a x}}-\text {$\#$1}\right ) \text {$\#$1}^4+2 \log \left (-\sqrt {-b+a x}+\sqrt {a x+\sqrt {-b+a x}}-\text {$\#$1}\right ) \text {$\#$1}^5}{-1-b^2-b^3+6 \text {$\#$1}+5 b^2 \text {$\#$1}+b^3 \text {$\#$1}-12 \text {$\#$1}^2-6 b^2 \text {$\#$1}^2+8 \text {$\#$1}^3+b \text {$\#$1}^3+3 b^2 \text {$\#$1}^3-5 b \text {$\#$1}^4+3 b \text {$\#$1}^5+\text {$\#$1}^7}\&\right ]}{2 a} \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 = 0.13, size = 239, normalized size = 0.41
method | result | size |
derivativedivides | \(-\frac {\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{8}+4 b \,\textit {\_Z}^{6}-8 b \,\textit {\_Z}^{5}+\left (6 b^{2}+2 b +16\right ) \textit {\_Z}^{4}+\left (-16 b^{2}-32\right ) \textit {\_Z}^{3}+\left (4 b^{3}+20 b^{2}+24\right ) \textit {\_Z}^{2}+\left (-8 b^{3}-8 b^{2}-8\right ) \textit {\_Z} +b^{4}+2 b^{3}+b^{2}+1\right )}{\sum }\frac {\left (2 \textit {\_R}^{5}-5 \textit {\_R}^{4}+4 \left (1+b \right ) \textit {\_R}^{3}+\left (-6 b -1\right ) \textit {\_R}^{2}+2 b \left (1+b \right ) \textit {\_R} -b^{2}\right ) \ln \left (\sqrt {a x +\sqrt {a x -b}}-\sqrt {a x -b}-\textit {\_R} \right )}{\textit {\_R}^{7}+3 \textit {\_R}^{5} b -5 \textit {\_R}^{4} b +3 \textit {\_R}^{3} b^{2}+\textit {\_R}^{3} b -6 \textit {\_R}^{2} b^{2}+\textit {\_R} \,b^{3}+8 \textit {\_R}^{3}+5 \textit {\_R} \,b^{2}-b^{3}-12 \textit {\_R}^{2}-b^{2}+6 \textit {\_R} -1}}{2 a}\) | \(239\) |
default | \(-\frac {\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{8}+4 b \,\textit {\_Z}^{6}-8 b \,\textit {\_Z}^{5}+\left (6 b^{2}+2 b +16\right ) \textit {\_Z}^{4}+\left (-16 b^{2}-32\right ) \textit {\_Z}^{3}+\left (4 b^{3}+20 b^{2}+24\right ) \textit {\_Z}^{2}+\left (-8 b^{3}-8 b^{2}-8\right ) \textit {\_Z} +b^{4}+2 b^{3}+b^{2}+1\right )}{\sum }\frac {\left (2 \textit {\_R}^{5}-5 \textit {\_R}^{4}+4 \left (1+b \right ) \textit {\_R}^{3}+\left (-6 b -1\right ) \textit {\_R}^{2}+2 b \left (1+b \right ) \textit {\_R} -b^{2}\right ) \ln \left (\sqrt {a x +\sqrt {a x -b}}-\sqrt {a x -b}-\textit {\_R} \right )}{\textit {\_R}^{7}+3 \textit {\_R}^{5} b -5 \textit {\_R}^{4} b +3 \textit {\_R}^{3} b^{2}+\textit {\_R}^{3} b -6 \textit {\_R}^{2} b^{2}+\textit {\_R} \,b^{3}+8 \textit {\_R}^{3}+5 \textit {\_R} \,b^{2}-b^{3}-12 \textit {\_R}^{2}-b^{2}+6 \textit {\_R} -1}}{2 a}\) | \(239\) |
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 + \sqrt {a x - b}}}{{\left (a^{2} x^{2} + 1\right )} \sqrt {a x - b}}\,{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 {\sqrt {a\,x+\sqrt {a\,x-b}}}{\left (a^2\,x^2+1\right )\,\sqrt {a\,x-b}} \,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 {\sqrt {a x + \sqrt {a x - b}}}{\sqrt {a x - b} \left (a^{2} x^{2} + 1\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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