Optimal. Leaf size=301 \[ \sqrt {2} b^{3/4} c \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {\sqrt {x^2-b}+x}}{\sqrt {x^2-b}-\sqrt {b}+x}\right )+\sqrt {2} b^{3/4} c \tanh ^{-1}\left (\frac {\frac {\sqrt {x^2-b}}{\sqrt {2} \sqrt [4]{b}}+\frac {x}{\sqrt {2} \sqrt [4]{b}}+\frac {\sqrt [4]{b}}{\sqrt {2}}}{\sqrt {\sqrt {x^2-b}+x}}\right )+\frac {2 \sqrt {x^2-b} \left (-1368 b^4 x+3705 b^3 x^3+10395 b^2 c x+1335 b^2 x^5-32340 b c x^3-8100 b x^7+18480 c x^5+5040 x^9\right )+2 \left (304 b^5-3078 b^4 x^2-2310 b^3 c+3735 b^3 x^4+24255 b^2 c x^2+4755 b^2 x^6-41580 b c x^4-10620 b x^8+18480 c x^6+5040 x^{10}\right )}{3465 \left (\sqrt {x^2-b}+x\right )^{9/2}} \]
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Rubi [A] time = 1.25, antiderivative size = 391, normalized size of antiderivative = 1.30, number of steps used = 18, number of rules used = 12, integrand size = 37, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.324, Rules used = {6742, 2120, 462, 459, 329, 297, 1162, 617, 204, 1165, 628, 448} \begin {gather*} -\frac {b^{3/4} c \log \left (\sqrt {x^2-b}-\sqrt {2} \sqrt [4]{b} \sqrt {\sqrt {x^2-b}+x}+\sqrt {b}+x\right )}{\sqrt {2}}+\frac {b^{3/4} c \log \left (\sqrt {x^2-b}+\sqrt {2} \sqrt [4]{b} \sqrt {\sqrt {x^2-b}+x}+\sqrt {b}+x\right )}{\sqrt {2}}+\sqrt {2} b^{3/4} c \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt {\sqrt {x^2-b}+x}}{\sqrt [4]{b}}\right )-\sqrt {2} b^{3/4} c \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {\sqrt {x^2-b}+x}}{\sqrt [4]{b}}+1\right )-\frac {b^5}{144 \left (\sqrt {x^2-b}+x\right )^{9/2}}-\frac {b^4}{80 \left (\sqrt {x^2-b}+x\right )^{5/2}}+\frac {b^3}{8 \sqrt {\sqrt {x^2-b}+x}}-\frac {1}{24} b^2 \left (\sqrt {x^2-b}+x\right )^{3/2}+\frac {1}{3} c \left (\sqrt {x^2-b}+x\right )^{3/2}-\frac {b c}{\sqrt {\sqrt {x^2-b}+x}}+\frac {1}{176} \left (\sqrt {x^2-b}+x\right )^{11/2}+\frac {1}{112} b \left (\sqrt {x^2-b}+x\right )^{7/2} \end {gather*}
Antiderivative was successfully verified.
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Rule 204
Rule 297
Rule 329
Rule 448
Rule 459
Rule 462
Rule 617
Rule 628
Rule 1162
Rule 1165
Rule 2120
Rule 6742
Rubi steps
\begin {align*} \int \frac {\sqrt {-b+x^2} \left (c+x^4\right ) \sqrt {x+\sqrt {-b+x^2}}}{x} \, dx &=\int \left (\frac {c \sqrt {-b+x^2} \sqrt {x+\sqrt {-b+x^2}}}{x}+x^3 \sqrt {-b+x^2} \sqrt {x+\sqrt {-b+x^2}}\right ) \, dx\\ &=c \int \frac {\sqrt {-b+x^2} \sqrt {x+\sqrt {-b+x^2}}}{x} \, dx+\int x^3 \sqrt {-b+x^2} \sqrt {x+\sqrt {-b+x^2}} \, dx\\ &=\frac {1}{32} \operatorname {Subst}\left (\int \frac {\left (-b+x^2\right )^2 \left (b+x^2\right )^3}{x^{11/2}} \, dx,x,x+\sqrt {-b+x^2}\right )+\frac {1}{2} c \operatorname {Subst}\left (\int \frac {\left (-b+x^2\right )^2}{x^{3/2} \left (b+x^2\right )} \, dx,x,x+\sqrt {-b+x^2}\right )\\ &=-\frac {b c}{\sqrt {x+\sqrt {-b+x^2}}}+\frac {1}{32} \operatorname {Subst}\left (\int \left (\frac {b^5}{x^{11/2}}+\frac {b^4}{x^{7/2}}-\frac {2 b^3}{x^{3/2}}-2 b^2 \sqrt {x}+b x^{5/2}+x^{9/2}\right ) \, dx,x,x+\sqrt {-b+x^2}\right )+\frac {c \operatorname {Subst}\left (\int \frac {\sqrt {x} \left (-\frac {3 b^2}{2}+\frac {b x^2}{2}\right )}{b+x^2} \, dx,x,x+\sqrt {-b+x^2}\right )}{b}\\ &=-\frac {b^5}{144 \left (x+\sqrt {-b+x^2}\right )^{9/2}}-\frac {b^4}{80 \left (x+\sqrt {-b+x^2}\right )^{5/2}}+\frac {b^3}{8 \sqrt {x+\sqrt {-b+x^2}}}-\frac {b c}{\sqrt {x+\sqrt {-b+x^2}}}-\frac {1}{24} b^2 \left (x+\sqrt {-b+x^2}\right )^{3/2}+\frac {1}{3} c \left (x+\sqrt {-b+x^2}\right )^{3/2}+\frac {1}{112} b \left (x+\sqrt {-b+x^2}\right )^{7/2}+\frac {1}{176} \left (x+\sqrt {-b+x^2}\right )^{11/2}-(2 b c) \operatorname {Subst}\left (\int \frac {\sqrt {x}}{b+x^2} \, dx,x,x+\sqrt {-b+x^2}\right )\\ &=-\frac {b^5}{144 \left (x+\sqrt {-b+x^2}\right )^{9/2}}-\frac {b^4}{80 \left (x+\sqrt {-b+x^2}\right )^{5/2}}+\frac {b^3}{8 \sqrt {x+\sqrt {-b+x^2}}}-\frac {b c}{\sqrt {x+\sqrt {-b+x^2}}}-\frac {1}{24} b^2 \left (x+\sqrt {-b+x^2}\right )^{3/2}+\frac {1}{3} c \left (x+\sqrt {-b+x^2}\right )^{3/2}+\frac {1}{112} b \left (x+\sqrt {-b+x^2}\right )^{7/2}+\frac {1}{176} \left (x+\sqrt {-b+x^2}\right )^{11/2}-(4 b c) \operatorname {Subst}\left (\int \frac {x^2}{b+x^4} \, dx,x,\sqrt {x+\sqrt {-b+x^2}}\right )\\ &=-\frac {b^5}{144 \left (x+\sqrt {-b+x^2}\right )^{9/2}}-\frac {b^4}{80 \left (x+\sqrt {-b+x^2}\right )^{5/2}}+\frac {b^3}{8 \sqrt {x+\sqrt {-b+x^2}}}-\frac {b c}{\sqrt {x+\sqrt {-b+x^2}}}-\frac {1}{24} b^2 \left (x+\sqrt {-b+x^2}\right )^{3/2}+\frac {1}{3} c \left (x+\sqrt {-b+x^2}\right )^{3/2}+\frac {1}{112} b \left (x+\sqrt {-b+x^2}\right )^{7/2}+\frac {1}{176} \left (x+\sqrt {-b+x^2}\right )^{11/2}+(2 b c) \operatorname {Subst}\left (\int \frac {\sqrt {b}-x^2}{b+x^4} \, dx,x,\sqrt {x+\sqrt {-b+x^2}}\right )-(2 b c) \operatorname {Subst}\left (\int \frac {\sqrt {b}+x^2}{b+x^4} \, dx,x,\sqrt {x+\sqrt {-b+x^2}}\right )\\ &=-\frac {b^5}{144 \left (x+\sqrt {-b+x^2}\right )^{9/2}}-\frac {b^4}{80 \left (x+\sqrt {-b+x^2}\right )^{5/2}}+\frac {b^3}{8 \sqrt {x+\sqrt {-b+x^2}}}-\frac {b c}{\sqrt {x+\sqrt {-b+x^2}}}-\frac {1}{24} b^2 \left (x+\sqrt {-b+x^2}\right )^{3/2}+\frac {1}{3} c \left (x+\sqrt {-b+x^2}\right )^{3/2}+\frac {1}{112} b \left (x+\sqrt {-b+x^2}\right )^{7/2}+\frac {1}{176} \left (x+\sqrt {-b+x^2}\right )^{11/2}-\frac {\left (b^{3/4} c\right ) \operatorname {Subst}\left (\int \frac {\sqrt {2} \sqrt [4]{b}+2 x}{-\sqrt {b}-\sqrt {2} \sqrt [4]{b} x-x^2} \, dx,x,\sqrt {x+\sqrt {-b+x^2}}\right )}{\sqrt {2}}-\frac {\left (b^{3/4} c\right ) \operatorname {Subst}\left (\int \frac {\sqrt {2} \sqrt [4]{b}-2 x}{-\sqrt {b}+\sqrt {2} \sqrt [4]{b} x-x^2} \, dx,x,\sqrt {x+\sqrt {-b+x^2}}\right )}{\sqrt {2}}-(b c) \operatorname {Subst}\left (\int \frac {1}{\sqrt {b}-\sqrt {2} \sqrt [4]{b} x+x^2} \, dx,x,\sqrt {x+\sqrt {-b+x^2}}\right )-(b c) \operatorname {Subst}\left (\int \frac {1}{\sqrt {b}+\sqrt {2} \sqrt [4]{b} x+x^2} \, dx,x,\sqrt {x+\sqrt {-b+x^2}}\right )\\ &=-\frac {b^5}{144 \left (x+\sqrt {-b+x^2}\right )^{9/2}}-\frac {b^4}{80 \left (x+\sqrt {-b+x^2}\right )^{5/2}}+\frac {b^3}{8 \sqrt {x+\sqrt {-b+x^2}}}-\frac {b c}{\sqrt {x+\sqrt {-b+x^2}}}-\frac {1}{24} b^2 \left (x+\sqrt {-b+x^2}\right )^{3/2}+\frac {1}{3} c \left (x+\sqrt {-b+x^2}\right )^{3/2}+\frac {1}{112} b \left (x+\sqrt {-b+x^2}\right )^{7/2}+\frac {1}{176} \left (x+\sqrt {-b+x^2}\right )^{11/2}-\frac {b^{3/4} c \log \left (\sqrt {b}+x+\sqrt {-b+x^2}-\sqrt {2} \sqrt [4]{b} \sqrt {x+\sqrt {-b+x^2}}\right )}{\sqrt {2}}+\frac {b^{3/4} c \log \left (\sqrt {b}+x+\sqrt {-b+x^2}+\sqrt {2} \sqrt [4]{b} \sqrt {x+\sqrt {-b+x^2}}\right )}{\sqrt {2}}-\left (\sqrt {2} b^{3/4} c\right ) \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt {x+\sqrt {-b+x^2}}}{\sqrt [4]{b}}\right )+\left (\sqrt {2} b^{3/4} c\right ) \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt {x+\sqrt {-b+x^2}}}{\sqrt [4]{b}}\right )\\ &=-\frac {b^5}{144 \left (x+\sqrt {-b+x^2}\right )^{9/2}}-\frac {b^4}{80 \left (x+\sqrt {-b+x^2}\right )^{5/2}}+\frac {b^3}{8 \sqrt {x+\sqrt {-b+x^2}}}-\frac {b c}{\sqrt {x+\sqrt {-b+x^2}}}-\frac {1}{24} b^2 \left (x+\sqrt {-b+x^2}\right )^{3/2}+\frac {1}{3} c \left (x+\sqrt {-b+x^2}\right )^{3/2}+\frac {1}{112} b \left (x+\sqrt {-b+x^2}\right )^{7/2}+\frac {1}{176} \left (x+\sqrt {-b+x^2}\right )^{11/2}+\sqrt {2} b^{3/4} c \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt {x+\sqrt {-b+x^2}}}{\sqrt [4]{b}}\right )-\sqrt {2} b^{3/4} c \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt {x+\sqrt {-b+x^2}}}{\sqrt [4]{b}}\right )-\frac {b^{3/4} c \log \left (\sqrt {b}+x+\sqrt {-b+x^2}-\sqrt {2} \sqrt [4]{b} \sqrt {x+\sqrt {-b+x^2}}\right )}{\sqrt {2}}+\frac {b^{3/4} c \log \left (\sqrt {b}+x+\sqrt {-b+x^2}+\sqrt {2} \sqrt [4]{b} \sqrt {x+\sqrt {-b+x^2}}\right )}{\sqrt {2}}\\ \end {align*}
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Mathematica [A] time = 8.63, size = 524, normalized size = 1.74 \begin {gather*} \frac {1}{6} c \left (-3 \sqrt {2} b^{3/4} \log \left (\sqrt {x^2-b}-\sqrt {2} \sqrt [4]{b} \sqrt {\sqrt {x^2-b}+x}+\sqrt {b}+x\right )+3 \sqrt {2} b^{3/4} \log \left (\sqrt {x^2-b}+\sqrt {2} \sqrt [4]{b} \sqrt {\sqrt {x^2-b}+x}+\sqrt {b}+x\right )+6 \sqrt {2} b^{3/4} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt {\sqrt {x^2-b}+x}}{\sqrt [4]{b}}\right )-6 \sqrt {2} b^{3/4} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {\sqrt {x^2-b}+x}}{\sqrt [4]{b}}+1\right )+\frac {4 x \left (\sqrt {x^2-b}+x\right )-8 b}{\sqrt {\sqrt {x^2-b}+x}}\right )+\frac {2 \sqrt {x^2-b} \left (304 b^5-342 b^4 x \left (4 \sqrt {x^2-b}+9 x\right )+15 b^3 x^3 \left (247 \sqrt {x^2-b}+249 x\right )+15 b^2 x^5 \left (89 \sqrt {x^2-b}+317 x\right )+5040 x^9 \left (\sqrt {x^2-b}+x\right )-180 b x^7 \left (45 \sqrt {x^2-b}+59 x\right )\right ) \left (\sqrt {x^2-b}+x\right )^{9/2}}{3465 \left (-b^5+b^4 x \left (9 \sqrt {x^2-b}+41 x\right )-40 b^3 x^3 \left (3 \sqrt {x^2-b}+7 x\right )+16 b^2 x^5 \left (27 \sqrt {x^2-b}+43 x\right )+256 x^9 \left (\sqrt {x^2-b}+x\right )-64 b x^7 \left (9 \sqrt {x^2-b}+11 x\right )\right )} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.64, size = 301, normalized size = 1.00 \begin {gather*} \frac {2 \sqrt {-b+x^2} \left (-1368 b^4 x+10395 b^2 c x+3705 b^3 x^3-32340 b c x^3+1335 b^2 x^5+18480 c x^5-8100 b x^7+5040 x^9\right )+2 \left (304 b^5-2310 b^3 c-3078 b^4 x^2+24255 b^2 c x^2+3735 b^3 x^4-41580 b c x^4+4755 b^2 x^6+18480 c x^6-10620 b x^8+5040 x^{10}\right )}{3465 \left (x+\sqrt {-b+x^2}\right )^{9/2}}+\sqrt {2} b^{3/4} c \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x+\sqrt {-b+x^2}}}{-\sqrt {b}+x+\sqrt {-b+x^2}}\right )+\sqrt {2} b^{3/4} c \tanh ^{-1}\left (\frac {\frac {\sqrt [4]{b}}{\sqrt {2}}+\frac {x}{\sqrt {2} \sqrt [4]{b}}+\frac {\sqrt {-b+x^2}}{\sqrt {2} \sqrt [4]{b}}}{\sqrt {x+\sqrt {-b+x^2}}}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.48, size = 269, normalized size = 0.89 \begin {gather*} -\frac {2}{3465} \, {\left (35 \, x^{5} - 19 \, b x^{3} - {\left (152 \, b^{2} - 1155 \, c\right )} x - 2 \, {\left (175 \, x^{4} - 57 \, b x^{2} - 152 \, b^{2} + 1155 \, c\right )} \sqrt {x^{2} - b}\right )} \sqrt {x + \sqrt {x^{2} - b}} + 4 \, \left (-b^{3} c^{4}\right )^{\frac {1}{4}} \arctan \left (-\frac {\left (-b^{3} c^{4}\right )^{\frac {1}{4}} b^{2} c^{3} \sqrt {x + \sqrt {x^{2} - b}} - \sqrt {b^{4} c^{6} x + \sqrt {x^{2} - b} b^{4} c^{6} - \sqrt {-b^{3} c^{4}} b^{3} c^{4}} \left (-b^{3} c^{4}\right )^{\frac {1}{4}}}{b^{3} c^{4}}\right ) - \left (-b^{3} c^{4}\right )^{\frac {1}{4}} \log \left (b^{2} c^{3} \sqrt {x + \sqrt {x^{2} - b}} + \left (-b^{3} c^{4}\right )^{\frac {3}{4}}\right ) + \left (-b^{3} c^{4}\right )^{\frac {1}{4}} \log \left (b^{2} c^{3} \sqrt {x + \sqrt {x^{2} - b}} - \left (-b^{3} c^{4}\right )^{\frac {3}{4}}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (x^{4} + c\right )} \sqrt {x^{2} - b} \sqrt {x + \sqrt {x^{2} - b}}}{x}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.05, size = 0, normalized size = 0.00 \[\int \frac {\sqrt {x^{2}-b}\, \left (x^{4}+c \right ) \sqrt {x +\sqrt {x^{2}-b}}}{x}\, dx\]
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*} \int \frac {{\left (x^{4} + c\right )} \sqrt {x^{2} - b} \sqrt {x + \sqrt {x^{2} - b}}}{x}\,{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 {x+\sqrt {x^2-b}}\,\left (x^4+c\right )\,\sqrt {x^2-b}}{x} \,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 {- b + x^{2}} \left (c + x^{4}\right ) \sqrt {x + \sqrt {- b + x^{2}}}}{x}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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