Optimal. Leaf size=269 \[ \frac {8}{15} \left (3 c+8 d^2\right ) \sqrt {d+\sqrt {c+\sqrt {b+a x}}}+\frac {8}{5} \sqrt {b+a x} \sqrt {d+\sqrt {c+\sqrt {b+a x}}}-\frac {32}{15} d \sqrt {c+\sqrt {b+a x}} \sqrt {d+\sqrt {c+\sqrt {b+a x}}}-b \text {RootSum}\left [b-c^2+2 c d^2-d^4-4 c d \text {$\#$1}^2+4 d^3 \text {$\#$1}^2+2 c \text {$\#$1}^4-6 d^2 \text {$\#$1}^4+4 d \text {$\#$1}^6-\text {$\#$1}^8\& ,\frac {d \log \left (\sqrt {d+\sqrt {c+\sqrt {b+a x}}}-\text {$\#$1}\right )-\log \left (\sqrt {d+\sqrt {c+\sqrt {b+a x}}}-\text {$\#$1}\right ) \text {$\#$1}^2}{-c \text {$\#$1}+d^2 \text {$\#$1}-2 d \text {$\#$1}^3+\text {$\#$1}^5}\& \right ] \]
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Rubi [B] Leaf count is larger than twice the leaf count of optimal. \(790\) vs. \(2(269)=538\).
time = 3.52, antiderivative size = 790, normalized size of antiderivative = 2.94, number of steps
used = 26, number of rules used = 13, integrand size = 49, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.265, Rules used = {1607, 6872,
200, 6820, 12, 2013, 1107, 210, 212, 1108, 648, 632, 642} \begin {gather*} -\frac {2 \sqrt {b} \sqrt {\sqrt {b}+c} \text {ArcTan}\left (\frac {\sqrt {\sqrt {\sqrt {a x+b}+c}+d}}{\sqrt {\sqrt {\sqrt {b}+c}-d}}\right )}{\sqrt {\sqrt {\sqrt {b}+c}-d}}+8 d^2 \sqrt {\sqrt {\sqrt {a x+b}+c}+d}-\frac {\sqrt {b} \left (\sqrt {b}-c\right ) \log \left (-\sqrt {2} \sqrt {\sqrt {\sqrt {b}-c+d^2}+d} \sqrt {\sqrt {\sqrt {a x+b}+c}+d}+\sqrt {\sqrt {a x+b}+c}+\sqrt {\sqrt {b}-c+d^2}+d\right )}{\sqrt {2} \sqrt {\sqrt {b}-c+d^2} \sqrt {\sqrt {\sqrt {b}-c+d^2}+d}}+\frac {\sqrt {b} \left (\sqrt {b}-c\right ) \log \left (\sqrt {2} \sqrt {\sqrt {\sqrt {b}-c+d^2}+d} \sqrt {\sqrt {\sqrt {a x+b}+c}+d}+\sqrt {\sqrt {a x+b}+c}+\sqrt {\sqrt {b}-c+d^2}+d\right )}{\sqrt {2} \sqrt {\sqrt {b}-c+d^2} \sqrt {\sqrt {\sqrt {b}-c+d^2}+d}}+\frac {\sqrt {2} \sqrt {b} \left (\sqrt {b}-c\right ) \tanh ^{-1}\left (\frac {\sqrt {\sqrt {\sqrt {b}-c+d^2}+d}-\sqrt {2} \sqrt {\sqrt {\sqrt {a x+b}+c}+d}}{\sqrt {d-\sqrt {\sqrt {b}-c+d^2}}}\right )}{\sqrt {\sqrt {b}-c+d^2} \sqrt {d-\sqrt {\sqrt {b}-c+d^2}}}-\frac {\sqrt {2} \sqrt {b} \left (\sqrt {b}-c\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {\sqrt {\sqrt {a x+b}+c}+d}+\sqrt {\sqrt {\sqrt {b}-c+d^2}+d}}{\sqrt {d-\sqrt {\sqrt {b}-c+d^2}}}\right )}{\sqrt {\sqrt {b}-c+d^2} \sqrt {d-\sqrt {\sqrt {b}-c+d^2}}}+\frac {8}{5} \left (\sqrt {\sqrt {a x+b}+c}+d\right )^{5/2}-\frac {16}{3} d \left (\sqrt {\sqrt {a x+b}+c}+d\right )^{3/2}-\frac {2 \sqrt {b} \sqrt {\sqrt {b}+c} \tanh ^{-1}\left (\frac {\sqrt {\sqrt {\sqrt {a x+b}+c}+d}}{\sqrt {\sqrt {\sqrt {b}+c}+d}}\right )}{\sqrt {\sqrt {\sqrt {b}+c}+d}} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 200
Rule 210
Rule 212
Rule 632
Rule 642
Rule 648
Rule 1107
Rule 1108
Rule 1607
Rule 2013
Rule 6820
Rule 6872
Rubi steps
\begin {align*} \int \frac {\sqrt {b+a x} \sqrt {c+\sqrt {b+a x}}}{x \sqrt {d+\sqrt {c+\sqrt {b+a x}}}} \, dx &=2 \text {Subst}\left (\int \frac {x^2 \sqrt {c+x}}{\left (-b+x^2\right ) \sqrt {d+\sqrt {c+x}}} \, dx,x,\sqrt {b+a x}\right )\\ &=4 \text {Subst}\left (\int \frac {\left (-c x+x^3\right )^2}{\sqrt {d+x} \left (-b+\left (c-x^2\right )^2\right )} \, dx,x,\sqrt {c+\sqrt {b+a x}}\right )\\ &=4 \text {Subst}\left (\int \frac {x^2 \left (-c+x^2\right )^2}{\sqrt {d+x} \left (-b+\left (c-x^2\right )^2\right )} \, dx,x,\sqrt {c+\sqrt {b+a x}}\right )\\ &=8 \text {Subst}\left (\int \frac {\left (d-x^2\right )^2 \left (c-\left (d-x^2\right )^2\right )^2}{-b+\left (c-\left (d-x^2\right )^2\right )^2} \, dx,x,\sqrt {d+\sqrt {c+\sqrt {b+a x}}}\right )\\ &=8 \text {Subst}\left (\int \left (\left (d-x^2\right )^2+\frac {b c-b \left (c-\left (d-x^2\right )^2\right )}{-b+\left (c-\left (d-x^2\right )^2\right )^2}\right ) \, dx,x,\sqrt {d+\sqrt {c+\sqrt {b+a x}}}\right )\\ &=8 \text {Subst}\left (\int \left (d-x^2\right )^2 \, dx,x,\sqrt {d+\sqrt {c+\sqrt {b+a x}}}\right )+8 \text {Subst}\left (\int \frac {b c-b \left (c-\left (d-x^2\right )^2\right )}{-b+\left (c-\left (d-x^2\right )^2\right )^2} \, dx,x,\sqrt {d+\sqrt {c+\sqrt {b+a x}}}\right )\\ &=8 \text {Subst}\left (\int \left (d^2-2 d x^2+x^4\right ) \, dx,x,\sqrt {d+\sqrt {c+\sqrt {b+a x}}}\right )+8 \text {Subst}\left (\int \frac {b \left (d-x^2\right )^2}{-b+\left (c-\left (d-x^2\right )^2\right )^2} \, dx,x,\sqrt {d+\sqrt {c+\sqrt {b+a x}}}\right )\\ &=8 d^2 \sqrt {d+\sqrt {c+\sqrt {b+a x}}}-\frac {16}{3} d \left (d+\sqrt {c+\sqrt {b+a x}}\right )^{3/2}+\frac {8}{5} \left (d+\sqrt {c+\sqrt {b+a x}}\right )^{5/2}+(8 b) \text {Subst}\left (\int \frac {\left (d-x^2\right )^2}{-b+\left (c-\left (d-x^2\right )^2\right )^2} \, dx,x,\sqrt {d+\sqrt {c+\sqrt {b+a x}}}\right )\\ &=8 d^2 \sqrt {d+\sqrt {c+\sqrt {b+a x}}}-\frac {16}{3} d \left (d+\sqrt {c+\sqrt {b+a x}}\right )^{3/2}+\frac {8}{5} \left (d+\sqrt {c+\sqrt {b+a x}}\right )^{5/2}+(8 b) \text {Subst}\left (\int \left (-\frac {b+\sqrt {b} c}{2 b \left (\sqrt {b}+c-\left (d-x^2\right )^2\right )}-\frac {-b+\sqrt {b} c}{2 b \left (\sqrt {b}-c+\left (d-x^2\right )^2\right )}\right ) \, dx,x,\sqrt {d+\sqrt {c+\sqrt {b+a x}}}\right )\\ &=8 d^2 \sqrt {d+\sqrt {c+\sqrt {b+a x}}}-\frac {16}{3} d \left (d+\sqrt {c+\sqrt {b+a x}}\right )^{3/2}+\frac {8}{5} \left (d+\sqrt {c+\sqrt {b+a x}}\right )^{5/2}+\left (4 b \left (1-\frac {c}{\sqrt {b}}\right )\right ) \text {Subst}\left (\int \frac {1}{\sqrt {b}-c+\left (d-x^2\right )^2} \, dx,x,\sqrt {d+\sqrt {c+\sqrt {b+a x}}}\right )-\left (4 b \left (1+\frac {c}{\sqrt {b}}\right )\right ) \text {Subst}\left (\int \frac {1}{\sqrt {b}+c-\left (d-x^2\right )^2} \, dx,x,\sqrt {d+\sqrt {c+\sqrt {b+a x}}}\right )\\ &=8 d^2 \sqrt {d+\sqrt {c+\sqrt {b+a x}}}-\frac {16}{3} d \left (d+\sqrt {c+\sqrt {b+a x}}\right )^{3/2}+\frac {8}{5} \left (d+\sqrt {c+\sqrt {b+a x}}\right )^{5/2}+\left (4 b \left (1-\frac {c}{\sqrt {b}}\right )\right ) \text {Subst}\left (\int \frac {1}{\sqrt {b}-c+d^2-2 d x^2+x^4} \, dx,x,\sqrt {d+\sqrt {c+\sqrt {b+a x}}}\right )-\left (4 b \left (1+\frac {c}{\sqrt {b}}\right )\right ) \text {Subst}\left (\int \frac {1}{\sqrt {b}+c-d^2+2 d x^2-x^4} \, dx,x,\sqrt {d+\sqrt {c+\sqrt {b+a x}}}\right )\\ &=8 d^2 \sqrt {d+\sqrt {c+\sqrt {b+a x}}}-\frac {16}{3} d \left (d+\sqrt {c+\sqrt {b+a x}}\right )^{3/2}+\frac {8}{5} \left (d+\sqrt {c+\sqrt {b+a x}}\right )^{5/2}+\left (2 \sqrt {b} \sqrt {\sqrt {b}+c}\right ) \text {Subst}\left (\int \frac {1}{-\sqrt {\sqrt {b}+c}+d-x^2} \, dx,x,\sqrt {d+\sqrt {c+\sqrt {b+a x}}}\right )-\left (2 \sqrt {b} \sqrt {\sqrt {b}+c}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {\sqrt {b}+c}+d-x^2} \, dx,x,\sqrt {d+\sqrt {c+\sqrt {b+a x}}}\right )+\frac {\left (\sqrt {2} b \left (1-\frac {c}{\sqrt {b}}\right )\right ) \text {Subst}\left (\int \frac {\sqrt {2} \sqrt {d+\sqrt {\sqrt {b}-c+d^2}}-x}{\sqrt {\sqrt {b}-c+d^2}-\sqrt {2} \sqrt {d+\sqrt {\sqrt {b}-c+d^2}} x+x^2} \, dx,x,\sqrt {d+\sqrt {c+\sqrt {b+a x}}}\right )}{\sqrt {\sqrt {b}-c+d^2} \sqrt {d+\sqrt {\sqrt {b}-c+d^2}}}+\frac {\left (\sqrt {2} b \left (1-\frac {c}{\sqrt {b}}\right )\right ) \text {Subst}\left (\int \frac {\sqrt {2} \sqrt {d+\sqrt {\sqrt {b}-c+d^2}}+x}{\sqrt {\sqrt {b}-c+d^2}+\sqrt {2} \sqrt {d+\sqrt {\sqrt {b}-c+d^2}} x+x^2} \, dx,x,\sqrt {d+\sqrt {c+\sqrt {b+a x}}}\right )}{\sqrt {\sqrt {b}-c+d^2} \sqrt {d+\sqrt {\sqrt {b}-c+d^2}}}\\ &=8 d^2 \sqrt {d+\sqrt {c+\sqrt {b+a x}}}-\frac {16}{3} d \left (d+\sqrt {c+\sqrt {b+a x}}\right )^{3/2}+\frac {8}{5} \left (d+\sqrt {c+\sqrt {b+a x}}\right )^{5/2}-\frac {2 \sqrt {b} \sqrt {\sqrt {b}+c} \tan ^{-1}\left (\frac {\sqrt {d+\sqrt {c+\sqrt {b+a x}}}}{\sqrt {\sqrt {\sqrt {b}+c}-d}}\right )}{\sqrt {\sqrt {\sqrt {b}+c}-d}}-\frac {2 \sqrt {b} \sqrt {\sqrt {b}+c} \tanh ^{-1}\left (\frac {\sqrt {d+\sqrt {c+\sqrt {b+a x}}}}{\sqrt {\sqrt {\sqrt {b}+c}+d}}\right )}{\sqrt {\sqrt {\sqrt {b}+c}+d}}+\frac {\left (b \left (1-\frac {c}{\sqrt {b}}\right )\right ) \text {Subst}\left (\int \frac {1}{\sqrt {\sqrt {b}-c+d^2}-\sqrt {2} \sqrt {d+\sqrt {\sqrt {b}-c+d^2}} x+x^2} \, dx,x,\sqrt {d+\sqrt {c+\sqrt {b+a x}}}\right )}{\sqrt {\sqrt {b}-c+d^2}}+\frac {\left (b \left (1-\frac {c}{\sqrt {b}}\right )\right ) \text {Subst}\left (\int \frac {1}{\sqrt {\sqrt {b}-c+d^2}+\sqrt {2} \sqrt {d+\sqrt {\sqrt {b}-c+d^2}} x+x^2} \, dx,x,\sqrt {d+\sqrt {c+\sqrt {b+a x}}}\right )}{\sqrt {\sqrt {b}-c+d^2}}-\frac {\left (b \left (1-\frac {c}{\sqrt {b}}\right )\right ) \text {Subst}\left (\int \frac {-\sqrt {2} \sqrt {d+\sqrt {\sqrt {b}-c+d^2}}+2 x}{\sqrt {\sqrt {b}-c+d^2}-\sqrt {2} \sqrt {d+\sqrt {\sqrt {b}-c+d^2}} x+x^2} \, dx,x,\sqrt {d+\sqrt {c+\sqrt {b+a x}}}\right )}{\sqrt {2} \sqrt {\sqrt {b}-c+d^2} \sqrt {d+\sqrt {\sqrt {b}-c+d^2}}}+\frac {\left (b \left (1-\frac {c}{\sqrt {b}}\right )\right ) \text {Subst}\left (\int \frac {\sqrt {2} \sqrt {d+\sqrt {\sqrt {b}-c+d^2}}+2 x}{\sqrt {\sqrt {b}-c+d^2}+\sqrt {2} \sqrt {d+\sqrt {\sqrt {b}-c+d^2}} x+x^2} \, dx,x,\sqrt {d+\sqrt {c+\sqrt {b+a x}}}\right )}{\sqrt {2} \sqrt {\sqrt {b}-c+d^2} \sqrt {d+\sqrt {\sqrt {b}-c+d^2}}}\\ &=8 d^2 \sqrt {d+\sqrt {c+\sqrt {b+a x}}}-\frac {16}{3} d \left (d+\sqrt {c+\sqrt {b+a x}}\right )^{3/2}+\frac {8}{5} \left (d+\sqrt {c+\sqrt {b+a x}}\right )^{5/2}-\frac {2 \sqrt {b} \sqrt {\sqrt {b}+c} \tan ^{-1}\left (\frac {\sqrt {d+\sqrt {c+\sqrt {b+a x}}}}{\sqrt {\sqrt {\sqrt {b}+c}-d}}\right )}{\sqrt {\sqrt {\sqrt {b}+c}-d}}-\frac {2 \sqrt {b} \sqrt {\sqrt {b}+c} \tanh ^{-1}\left (\frac {\sqrt {d+\sqrt {c+\sqrt {b+a x}}}}{\sqrt {\sqrt {\sqrt {b}+c}+d}}\right )}{\sqrt {\sqrt {\sqrt {b}+c}+d}}-\frac {b \left (1-\frac {c}{\sqrt {b}}\right ) \log \left (d+\sqrt {\sqrt {b}-c+d^2}+\sqrt {c+\sqrt {b+a x}}-\sqrt {2} \sqrt {d+\sqrt {\sqrt {b}-c+d^2}} \sqrt {d+\sqrt {c+\sqrt {b+a x}}}\right )}{\sqrt {2} \sqrt {\sqrt {b}-c+d^2} \sqrt {d+\sqrt {\sqrt {b}-c+d^2}}}+\frac {b \left (1-\frac {c}{\sqrt {b}}\right ) \log \left (d+\sqrt {\sqrt {b}-c+d^2}+\sqrt {c+\sqrt {b+a x}}+\sqrt {2} \sqrt {d+\sqrt {\sqrt {b}-c+d^2}} \sqrt {d+\sqrt {c+\sqrt {b+a x}}}\right )}{\sqrt {2} \sqrt {\sqrt {b}-c+d^2} \sqrt {d+\sqrt {\sqrt {b}-c+d^2}}}-\frac {\left (2 b \left (1-\frac {c}{\sqrt {b}}\right )\right ) \text {Subst}\left (\int \frac {1}{2 \left (d-\sqrt {\sqrt {b}-c+d^2}\right )-x^2} \, dx,x,-\sqrt {2} \sqrt {d+\sqrt {\sqrt {b}-c+d^2}}+2 \sqrt {d+\sqrt {c+\sqrt {b+a x}}}\right )}{\sqrt {\sqrt {b}-c+d^2}}-\frac {\left (2 b \left (1-\frac {c}{\sqrt {b}}\right )\right ) \text {Subst}\left (\int \frac {1}{2 \left (d-\sqrt {\sqrt {b}-c+d^2}\right )-x^2} \, dx,x,\sqrt {2} \sqrt {d+\sqrt {\sqrt {b}-c+d^2}}+2 \sqrt {d+\sqrt {c+\sqrt {b+a x}}}\right )}{\sqrt {\sqrt {b}-c+d^2}}\\ &=8 d^2 \sqrt {d+\sqrt {c+\sqrt {b+a x}}}-\frac {16}{3} d \left (d+\sqrt {c+\sqrt {b+a x}}\right )^{3/2}+\frac {8}{5} \left (d+\sqrt {c+\sqrt {b+a x}}\right )^{5/2}-\frac {2 \sqrt {b} \sqrt {\sqrt {b}+c} \tan ^{-1}\left (\frac {\sqrt {d+\sqrt {c+\sqrt {b+a x}}}}{\sqrt {\sqrt {\sqrt {b}+c}-d}}\right )}{\sqrt {\sqrt {\sqrt {b}+c}-d}}-\frac {2 \sqrt {b} \sqrt {\sqrt {b}+c} \tanh ^{-1}\left (\frac {\sqrt {d+\sqrt {c+\sqrt {b+a x}}}}{\sqrt {\sqrt {\sqrt {b}+c}+d}}\right )}{\sqrt {\sqrt {\sqrt {b}+c}+d}}+\frac {\sqrt {2} b \left (1-\frac {c}{\sqrt {b}}\right ) \tanh ^{-1}\left (\frac {\sqrt {d+\sqrt {\sqrt {b}-c+d^2}}-\sqrt {2} \sqrt {d+\sqrt {c+\sqrt {b+a x}}}}{\sqrt {d-\sqrt {\sqrt {b}-c+d^2}}}\right )}{\sqrt {\sqrt {b}-c+d^2} \sqrt {d-\sqrt {\sqrt {b}-c+d^2}}}-\frac {\sqrt {2} b \left (1-\frac {c}{\sqrt {b}}\right ) \tanh ^{-1}\left (\frac {\sqrt {d+\sqrt {\sqrt {b}-c+d^2}}+\sqrt {2} \sqrt {d+\sqrt {c+\sqrt {b+a x}}}}{\sqrt {d-\sqrt {\sqrt {b}-c+d^2}}}\right )}{\sqrt {\sqrt {b}-c+d^2} \sqrt {d-\sqrt {\sqrt {b}-c+d^2}}}-\frac {b \left (1-\frac {c}{\sqrt {b}}\right ) \log \left (d+\sqrt {\sqrt {b}-c+d^2}+\sqrt {c+\sqrt {b+a x}}-\sqrt {2} \sqrt {d+\sqrt {\sqrt {b}-c+d^2}} \sqrt {d+\sqrt {c+\sqrt {b+a x}}}\right )}{\sqrt {2} \sqrt {\sqrt {b}-c+d^2} \sqrt {d+\sqrt {\sqrt {b}-c+d^2}}}+\frac {b \left (1-\frac {c}{\sqrt {b}}\right ) \log \left (d+\sqrt {\sqrt {b}-c+d^2}+\sqrt {c+\sqrt {b+a x}}+\sqrt {2} \sqrt {d+\sqrt {\sqrt {b}-c+d^2}} \sqrt {d+\sqrt {c+\sqrt {b+a x}}}\right )}{\sqrt {2} \sqrt {\sqrt {b}-c+d^2} \sqrt {d+\sqrt {\sqrt {b}-c+d^2}}}\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 223, normalized size = 0.83 \begin {gather*} \frac {8}{15} \sqrt {d+\sqrt {c+\sqrt {b+a x}}} \left (3 c+8 d^2+3 \sqrt {b+a x}-4 d \sqrt {c+\sqrt {b+a x}}\right )-b \text {RootSum}\left [b-c^2+2 c d^2-d^4-4 c d \text {$\#$1}^2+4 d^3 \text {$\#$1}^2+2 c \text {$\#$1}^4-6 d^2 \text {$\#$1}^4+4 d \text {$\#$1}^6-\text {$\#$1}^8\&,\frac {d \log \left (\sqrt {d+\sqrt {c+\sqrt {b+a x}}}-\text {$\#$1}\right )-\log \left (\sqrt {d+\sqrt {c+\sqrt {b+a x}}}-\text {$\#$1}\right ) \text {$\#$1}^2}{-c \text {$\#$1}+d^2 \text {$\#$1}-2 d \text {$\#$1}^3+\text {$\#$1}^5}\&\right ] \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
1.
time = 0.31, size = 190, normalized size = 0.71
method | result | size |
derivativedivides | \(\frac {8 \left (d +\sqrt {c +\sqrt {a x +b}}\right )^{\frac {5}{2}}}{5}-\frac {16 d \left (d +\sqrt {c +\sqrt {a x +b}}\right )^{\frac {3}{2}}}{3}+8 d^{2} \sqrt {d +\sqrt {c +\sqrt {a x +b}}}+b \left (\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{8}-4 d \,\textit {\_Z}^{6}+\left (6 d^{2}-2 c \right ) \textit {\_Z}^{4}+\left (-4 d^{3}+4 c d \right ) \textit {\_Z}^{2}+d^{4}-2 c \,d^{2}+c^{2}-b \right )}{\sum }\frac {\left (-\textit {\_R}^{4}+2 \textit {\_R}^{2} d -d^{2}\right ) \ln \left (\sqrt {d +\sqrt {c +\sqrt {a x +b}}}-\textit {\_R} \right )}{-\textit {\_R}^{7}+3 \textit {\_R}^{5} d -3 \textit {\_R}^{3} d^{2}+\textit {\_R}^{3} c +\textit {\_R} \,d^{3}-\textit {\_R} c d}\right )\) | \(190\) |
default | \(\frac {8 \left (d +\sqrt {c +\sqrt {a x +b}}\right )^{\frac {5}{2}}}{5}-\frac {16 d \left (d +\sqrt {c +\sqrt {a x +b}}\right )^{\frac {3}{2}}}{3}+8 d^{2} \sqrt {d +\sqrt {c +\sqrt {a x +b}}}+b \left (\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{8}-4 d \,\textit {\_Z}^{6}+\left (6 d^{2}-2 c \right ) \textit {\_Z}^{4}+\left (-4 d^{3}+4 c d \right ) \textit {\_Z}^{2}+d^{4}-2 c \,d^{2}+c^{2}-b \right )}{\sum }\frac {\left (-\textit {\_R}^{4}+2 \textit {\_R}^{2} d -d^{2}\right ) \ln \left (\sqrt {d +\sqrt {c +\sqrt {a x +b}}}-\textit {\_R} \right )}{-\textit {\_R}^{7}+3 \textit {\_R}^{5} d -3 \textit {\_R}^{3} d^{2}+\textit {\_R}^{3} c +\textit {\_R} \,d^{3}-\textit {\_R} c d}\right )\) | \(190\) |
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 [F(-1)] Timed out
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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Sympy [F(-1)] Timed out
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 [C] Result contains higher order function than in optimal. Order 3 vs. order
1.
time = 34.17, size = 1404, normalized size = 5.22 \begin {gather*} \frac {8}{5} \, {\left (d + \sqrt {c + \sqrt {a x + b}}\right )}^{\frac {5}{2}} - \frac {16}{3} \, {\left (d + \sqrt {c + \sqrt {a x + b}}\right )}^{\frac {3}{2}} d + 8 \, \sqrt {d + \sqrt {c + \sqrt {a x + b}}} d^{2} - \frac {{\left (b {\left (d + \sqrt {c + \sqrt {b}}\right )}^{2} - 2 \, b {\left (d + \sqrt {c + \sqrt {b}}\right )} d + b d^{2}\right )} \log \left (\sqrt {d + \sqrt {c + \sqrt {a x + b}}} + \sqrt {d + \sqrt {c + \sqrt {b}}}\right )}{{\left (d + \sqrt {c + \sqrt {b}}\right )}^{\frac {7}{2}} - 3 \, {\left (d + \sqrt {c + \sqrt {b}}\right )}^{\frac {5}{2}} d + 3 \, {\left (d + \sqrt {c + \sqrt {b}}\right )}^{\frac {3}{2}} d^{2} - \sqrt {d + \sqrt {c + \sqrt {b}}} d^{3} - c {\left (d + \sqrt {c + \sqrt {b}}\right )}^{\frac {3}{2}} + c \sqrt {d + \sqrt {c + \sqrt {b}}} d} + \frac {{\left (b {\left (d + \sqrt {c + \sqrt {b}}\right )}^{2} - 2 \, b {\left (d + \sqrt {c + \sqrt {b}}\right )} d + b d^{2}\right )} \log \left (\sqrt {d + \sqrt {c + \sqrt {a x + b}}} - \sqrt {d + \sqrt {c + \sqrt {b}}}\right )}{{\left (d + \sqrt {c + \sqrt {b}}\right )}^{\frac {7}{2}} - 3 \, {\left (d + \sqrt {c + \sqrt {b}}\right )}^{\frac {5}{2}} d + 3 \, {\left (d + \sqrt {c + \sqrt {b}}\right )}^{\frac {3}{2}} d^{2} - \sqrt {d + \sqrt {c + \sqrt {b}}} d^{3} - c {\left (d + \sqrt {c + \sqrt {b}}\right )}^{\frac {3}{2}} + c \sqrt {d + \sqrt {c + \sqrt {b}}} d} - \frac {{\left (b {\left (d - \sqrt {c + \sqrt {b}}\right )}^{2} - 2 \, b {\left (d - \sqrt {c + \sqrt {b}}\right )} d + b d^{2}\right )} \log \left (\sqrt {d + \sqrt {c + \sqrt {a x + b}}} + \sqrt {d - \sqrt {c + \sqrt {b}}}\right )}{{\left (d - \sqrt {c + \sqrt {b}}\right )}^{\frac {7}{2}} - 3 \, {\left (d - \sqrt {c + \sqrt {b}}\right )}^{\frac {5}{2}} d + 3 \, {\left (d - \sqrt {c + \sqrt {b}}\right )}^{\frac {3}{2}} d^{2} - \sqrt {d - \sqrt {c + \sqrt {b}}} d^{3} - c {\left (d - \sqrt {c + \sqrt {b}}\right )}^{\frac {3}{2}} + c \sqrt {d - \sqrt {c + \sqrt {b}}} d} + \frac {{\left (b {\left (d - \sqrt {c + \sqrt {b}}\right )}^{2} - 2 \, b {\left (d - \sqrt {c + \sqrt {b}}\right )} d + b d^{2}\right )} \log \left (\sqrt {d + \sqrt {c + \sqrt {a x + b}}} - \sqrt {d - \sqrt {c + \sqrt {b}}}\right )}{{\left (d - \sqrt {c + \sqrt {b}}\right )}^{\frac {7}{2}} - 3 \, {\left (d - \sqrt {c + \sqrt {b}}\right )}^{\frac {5}{2}} d + 3 \, {\left (d - \sqrt {c + \sqrt {b}}\right )}^{\frac {3}{2}} d^{2} - \sqrt {d - \sqrt {c + \sqrt {b}}} d^{3} - c {\left (d - \sqrt {c + \sqrt {b}}\right )}^{\frac {3}{2}} + c \sqrt {d - \sqrt {c + \sqrt {b}}} d} - \frac {{\left (b {\left (d + \sqrt {c - \sqrt {b}}\right )}^{2} - 2 \, b {\left (d + \sqrt {c - \sqrt {b}}\right )} d + b d^{2}\right )} \log \left (\sqrt {d + \sqrt {c + \sqrt {a x + b}}} + \sqrt {d + \sqrt {c - \sqrt {b}}}\right )}{{\left (d + \sqrt {c - \sqrt {b}}\right )}^{\frac {7}{2}} - 3 \, {\left (d + \sqrt {c - \sqrt {b}}\right )}^{\frac {5}{2}} d + 3 \, {\left (d + \sqrt {c - \sqrt {b}}\right )}^{\frac {3}{2}} d^{2} - \sqrt {d + \sqrt {c - \sqrt {b}}} d^{3} - c {\left (d + \sqrt {c - \sqrt {b}}\right )}^{\frac {3}{2}} + c \sqrt {d + \sqrt {c - \sqrt {b}}} d} + \frac {{\left (b {\left (d + \sqrt {c - \sqrt {b}}\right )}^{2} - 2 \, b {\left (d + \sqrt {c - \sqrt {b}}\right )} d + b d^{2}\right )} \log \left (\sqrt {d + \sqrt {c + \sqrt {a x + b}}} - \sqrt {d + \sqrt {c - \sqrt {b}}}\right )}{{\left (d + \sqrt {c - \sqrt {b}}\right )}^{\frac {7}{2}} - 3 \, {\left (d + \sqrt {c - \sqrt {b}}\right )}^{\frac {5}{2}} d + 3 \, {\left (d + \sqrt {c - \sqrt {b}}\right )}^{\frac {3}{2}} d^{2} - \sqrt {d + \sqrt {c - \sqrt {b}}} d^{3} - c {\left (d + \sqrt {c - \sqrt {b}}\right )}^{\frac {3}{2}} + c \sqrt {d + \sqrt {c - \sqrt {b}}} d} - \frac {{\left (b {\left (d - \sqrt {c - \sqrt {b}}\right )}^{2} - 2 \, b {\left (d - \sqrt {c - \sqrt {b}}\right )} d + b d^{2}\right )} \log \left (\sqrt {d + \sqrt {c + \sqrt {a x + b}}} + \sqrt {d - \sqrt {c - \sqrt {b}}}\right )}{{\left (d - \sqrt {c - \sqrt {b}}\right )}^{\frac {7}{2}} - 3 \, {\left (d - \sqrt {c - \sqrt {b}}\right )}^{\frac {5}{2}} d + 3 \, {\left (d - \sqrt {c - \sqrt {b}}\right )}^{\frac {3}{2}} d^{2} - \sqrt {d - \sqrt {c - \sqrt {b}}} d^{3} - c {\left (d - \sqrt {c - \sqrt {b}}\right )}^{\frac {3}{2}} + c \sqrt {d - \sqrt {c - \sqrt {b}}} d} + \frac {{\left (b {\left (d - \sqrt {c - \sqrt {b}}\right )}^{2} - 2 \, b {\left (d - \sqrt {c - \sqrt {b}}\right )} d + b d^{2}\right )} \log \left (\sqrt {d + \sqrt {c + \sqrt {a x + b}}} - \sqrt {d - \sqrt {c - \sqrt {b}}}\right )}{{\left (d - \sqrt {c - \sqrt {b}}\right )}^{\frac {7}{2}} - 3 \, {\left (d - \sqrt {c - \sqrt {b}}\right )}^{\frac {5}{2}} d + 3 \, {\left (d - \sqrt {c - \sqrt {b}}\right )}^{\frac {3}{2}} d^{2} - \sqrt {d - \sqrt {c - \sqrt {b}}} d^{3} - c {\left (d - \sqrt {c - \sqrt {b}}\right )}^{\frac {3}{2}} + c \sqrt {d - \sqrt {c - \sqrt {b}}} d} \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 {c+\sqrt {b+a\,x}}\,\sqrt {b+a\,x}}{x\,\sqrt {d+\sqrt {c+\sqrt {b+a\,x}}}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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