Optimal. Leaf size=246 \[ \frac {1}{8} a \text {RootSum}\left [-\text {$\#$1}^8+4 \text {$\#$1}^6-4 \text {$\#$1}^4+b\& ,\frac {5 \text {$\#$1}^4 \log \left (\sqrt {\sqrt {\sqrt {a x+b}+1}+1}-\text {$\#$1}\right )-9 \text {$\#$1}^2 \log \left (\sqrt {\sqrt {\sqrt {a x+b}+1}+1}-\text {$\#$1}\right )+2 \log \left (\sqrt {\sqrt {\sqrt {a x+b}+1}+1}-\text {$\#$1}\right )}{\text {$\#$1}^7-3 \text {$\#$1}^5+2 \text {$\#$1}^3}\& \right ]-\frac {\sqrt {a x+b} \sqrt {\sqrt {\sqrt {a x+b}+1}+1}}{x}+\frac {\sqrt {\sqrt {a x+b}+1} \sqrt {\sqrt {\sqrt {a x+b}+1}+1}}{x}-\frac {\sqrt {\sqrt {\sqrt {a x+b}+1}+1}}{x} \]
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Rubi [F] time = 3.12, 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 {\sqrt {b+a x} \sqrt {1+\sqrt {b+a x}}}{x^2 \sqrt {1+\sqrt {1+\sqrt {b+a x}}}} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {\sqrt {b+a x} \sqrt {1+\sqrt {b+a x}}}{x^2 \sqrt {1+\sqrt {1+\sqrt {b+a x}}}} \, dx &=(2 a) \operatorname {Subst}\left (\int \frac {x^2 \sqrt {1+x}}{\left (b-x^2\right )^2 \sqrt {1+\sqrt {1+x}}} \, dx,x,\sqrt {b+a x}\right )\\ &=(4 a) \operatorname {Subst}\left (\int \frac {x^2 \left (-1+x^2\right )^2}{\sqrt {1+x} \left (b-\left (-1+x^2\right )^2\right )^2} \, dx,x,\sqrt {1+\sqrt {b+a x}}\right )\\ &=(4 a) \operatorname {Subst}\left (\int \frac {(-1+x)^2 x^2 (1+x)^{3/2}}{\left (b-\left (-1+x^2\right )^2\right )^2} \, dx,x,\sqrt {1+\sqrt {b+a x}}\right )\\ &=(8 a) \operatorname {Subst}\left (\int \frac {x^4 \left (2-3 x^2+x^4\right )^2}{\left (b-x^4 \left (-2+x^2\right )^2\right )^2} \, dx,x,\sqrt {1+\sqrt {1+\sqrt {b+a x}}}\right )\\ &=-\frac {8 \left (1+\sqrt {1+\sqrt {b+a x}}\right )^{5/2}}{3 x}+\frac {1}{3} (8 a) \operatorname {Subst}\left (\int \frac {(12-5 b) x^4-36 x^6+43 x^8-14 x^{10}}{\left (b-x^4 \left (-2+x^2\right )^2\right )^2} \, dx,x,\sqrt {1+\sqrt {1+\sqrt {b+a x}}}\right )\\ &=\frac {112 \left (1+\sqrt {1+\sqrt {b+a x}}\right )^{3/2}}{15 x}-\frac {8 \left (1+\sqrt {1+\sqrt {b+a x}}\right )^{5/2}}{3 x}+\frac {1}{15} (8 a) \operatorname {Subst}\left (\int \frac {42 b x^2+5 (12-5 b) x^4-124 x^6+47 x^8}{\left (b-x^4 \left (-2+x^2\right )^2\right )^2} \, dx,x,\sqrt {1+\sqrt {1+\sqrt {b+a x}}}\right )\\ &=-\frac {376 \sqrt {1+\sqrt {1+\sqrt {b+a x}}}}{105 x}+\frac {112 \left (1+\sqrt {1+\sqrt {b+a x}}\right )^{3/2}}{15 x}-\frac {8 \left (1+\sqrt {1+\sqrt {b+a x}}\right )^{5/2}}{3 x}+\frac {1}{105} (8 a) \operatorname {Subst}\left (\int \frac {-47 b+294 b x^2-(144+175 b) x^4+72 x^6}{\left (b-x^4 \left (-2+x^2\right )^2\right )^2} \, dx,x,\sqrt {1+\sqrt {1+\sqrt {b+a x}}}\right )\\ &=-\frac {376 \sqrt {1+\sqrt {1+\sqrt {b+a x}}}}{105 x}+\frac {112 \left (1+\sqrt {1+\sqrt {b+a x}}\right )^{3/2}}{15 x}-\frac {8 \left (1+\sqrt {1+\sqrt {b+a x}}\right )^{5/2}}{3 x}+\frac {1}{105} (8 a) \operatorname {Subst}\left (\int \left (-\frac {47 b}{\left (b-4 x^4+4 x^6-x^8\right )^2}+\frac {294 b x^2}{\left (b-4 x^4+4 x^6-x^8\right )^2}-\frac {(144+175 b) x^4}{\left (b-4 x^4+4 x^6-x^8\right )^2}+\frac {72 x^6}{\left (-b+4 x^4-4 x^6+x^8\right )^2}\right ) \, dx,x,\sqrt {1+\sqrt {1+\sqrt {b+a x}}}\right )\\ &=-\frac {376 \sqrt {1+\sqrt {1+\sqrt {b+a x}}}}{105 x}+\frac {112 \left (1+\sqrt {1+\sqrt {b+a x}}\right )^{3/2}}{15 x}-\frac {8 \left (1+\sqrt {1+\sqrt {b+a x}}\right )^{5/2}}{3 x}+\frac {1}{35} (192 a) \operatorname {Subst}\left (\int \frac {x^6}{\left (-b+4 x^4-4 x^6+x^8\right )^2} \, dx,x,\sqrt {1+\sqrt {1+\sqrt {b+a x}}}\right )-\frac {1}{105} (376 a b) \operatorname {Subst}\left (\int \frac {1}{\left (b-4 x^4+4 x^6-x^8\right )^2} \, dx,x,\sqrt {1+\sqrt {1+\sqrt {b+a x}}}\right )+\frac {1}{5} (112 a b) \operatorname {Subst}\left (\int \frac {x^2}{\left (b-4 x^4+4 x^6-x^8\right )^2} \, dx,x,\sqrt {1+\sqrt {1+\sqrt {b+a x}}}\right )-\frac {1}{105} (8 a (144+175 b)) \operatorname {Subst}\left (\int \frac {x^4}{\left (b-4 x^4+4 x^6-x^8\right )^2} \, dx,x,\sqrt {1+\sqrt {1+\sqrt {b+a x}}}\right )\\ \end {align*}
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Mathematica [F] time = 1.64, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {b+a x} \sqrt {1+\sqrt {b+a x}}}{x^2 \sqrt {1+\sqrt {1+\sqrt {b+a x}}}} \, dx \end {gather*}
Verification is not applicable to the result.
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IntegrateAlgebraic [A] time = 0.00, size = 328, normalized size = 1.33 \begin {gather*} \frac {\sqrt {1+\sqrt {b+a x}} \sqrt {1+\sqrt {1+\sqrt {b+a x}}}}{x}-\frac {\left (a+a \sqrt {b+a x}\right ) \sqrt {1+\sqrt {1+\sqrt {b+a x}}}}{a x}+a \text {RootSum}\left [b-4 \text {$\#$1}^4+4 \text {$\#$1}^6-\text {$\#$1}^8\&,\frac {-\log \left (\sqrt {1+\sqrt {1+\sqrt {b+a x}}}-\text {$\#$1}\right )+\log \left (\sqrt {1+\sqrt {1+\sqrt {b+a x}}}-\text {$\#$1}\right ) \text {$\#$1}^2}{-2 \text {$\#$1}^3+\text {$\#$1}^5}\&\right ]-\frac {1}{8} a \text {RootSum}\left [b-4 \text {$\#$1}^4+4 \text {$\#$1}^6-\text {$\#$1}^8\&,\frac {6 \log \left (\sqrt {1+\sqrt {1+\sqrt {b+a x}}}-\text {$\#$1}\right )-7 \log \left (\sqrt {1+\sqrt {1+\sqrt {b+a x}}}-\text {$\#$1}\right ) \text {$\#$1}^2+3 \log \left (\sqrt {1+\sqrt {1+\sqrt {b+a x}}}-\text {$\#$1}\right ) \text {$\#$1}^4}{2 \text {$\#$1}^3-3 \text {$\#$1}^5+\text {$\#$1}^7}\&\right ] \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 = 0.45, size = 188, normalized size = 0.76
method | result | size |
derivativedivides | \(2 a \left (\frac {\frac {\left (1+\sqrt {1+\sqrt {a x +b}}\right )^{\frac {5}{2}}}{2}-\frac {3 \left (1+\sqrt {1+\sqrt {a x +b}}\right )^{\frac {3}{2}}}{2}+\sqrt {1+\sqrt {1+\sqrt {a x +b}}}}{-\left (1+\sqrt {1+\sqrt {a x +b}}\right )^{4}+4 \left (1+\sqrt {1+\sqrt {a x +b}}\right )^{3}-4 \left (1+\sqrt {1+\sqrt {a x +b}}\right )^{2}+b}+\frac {\left (\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{8}-4 \textit {\_Z}^{6}+4 \textit {\_Z}^{4}-b \right )}{\sum }\frac {\left (-5 \textit {\_R}^{4}+9 \textit {\_R}^{2}-2\right ) \ln \left (\sqrt {1+\sqrt {1+\sqrt {a x +b}}}-\textit {\_R} \right )}{-\textit {\_R}^{7}+3 \textit {\_R}^{5}-2 \textit {\_R}^{3}}\right )}{16}\right )\) | \(188\) |
default | \(2 a \left (\frac {\frac {\left (1+\sqrt {1+\sqrt {a x +b}}\right )^{\frac {5}{2}}}{2}-\frac {3 \left (1+\sqrt {1+\sqrt {a x +b}}\right )^{\frac {3}{2}}}{2}+\sqrt {1+\sqrt {1+\sqrt {a x +b}}}}{-\left (1+\sqrt {1+\sqrt {a x +b}}\right )^{4}+4 \left (1+\sqrt {1+\sqrt {a x +b}}\right )^{3}-4 \left (1+\sqrt {1+\sqrt {a x +b}}\right )^{2}+b}+\frac {\left (\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{8}-4 \textit {\_Z}^{6}+4 \textit {\_Z}^{4}-b \right )}{\sum }\frac {\left (-5 \textit {\_R}^{4}+9 \textit {\_R}^{2}-2\right ) \ln \left (\sqrt {1+\sqrt {1+\sqrt {a x +b}}}-\textit {\_R} \right )}{-\textit {\_R}^{7}+3 \textit {\_R}^{5}-2 \textit {\_R}^{3}}\right )}{16}\right )\) | \(188\) |
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 + b} \sqrt {\sqrt {a x + b} + 1}}{x^{2} \sqrt {\sqrt {\sqrt {a x + b} + 1} + 1}}\,{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 {\sqrt {b+a\,x}+1}\,\sqrt {b+a\,x}}{x^2\,\sqrt {\sqrt {\sqrt {b+a\,x}+1}+1}} \,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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