Optimal. Leaf size=397 \[ \sqrt {2} b^{3/4} d \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {\sqrt {a^2 x^2-b}+a x}}{\sqrt {a^2 x^2-b}+a x-\sqrt {b}}\right )+\sqrt {2} b^{3/4} d \tanh ^{-1}\left (\frac {\frac {\sqrt {a^2 x^2-b}}{\sqrt {2} \sqrt [4]{b}}+\frac {a x}{\sqrt {2} \sqrt [4]{b}}+\frac {\sqrt [4]{b}}{\sqrt {2}}}{\sqrt {\sqrt {a^2 x^2-b}+a x}}\right )+\frac {2 \left (5040 a^{10} c x^{10}+18480 a^{10} d x^6-10620 a^8 b c x^8-41580 a^8 b d x^4+4755 a^6 b^2 c x^6+24255 a^6 b^2 d x^2+3735 a^4 b^3 c x^4-2310 a^4 b^3 d-3078 a^2 b^4 c x^2+304 b^5 c\right )+2 \sqrt {a^2 x^2-b} \left (5040 a^9 c x^9+18480 a^9 d x^5-8100 a^7 b c x^7-32340 a^7 b d x^3+1335 a^5 b^2 c x^5+10395 a^5 b^2 d x+3705 a^3 b^3 c x^3-1368 a b^4 c x\right )}{3465 a^4 \left (\sqrt {a^2 x^2-b}+a x\right )^{9/2}} \]
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Rubi [A] time = 1.72, antiderivative size = 499, normalized size of antiderivative = 1.26, number of steps used = 18, number of rules used = 12, integrand size = 49, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.245, Rules used = {6742, 2120, 462, 459, 329, 297, 1162, 617, 204, 1165, 628, 448} \begin {gather*} -\frac {b^{3/4} d \log \left (\sqrt {a^2 x^2-b}-\sqrt {2} \sqrt [4]{b} \sqrt {\sqrt {a^2 x^2-b}+a x}+a x+\sqrt {b}\right )}{\sqrt {2}}+\frac {b^{3/4} d \log \left (\sqrt {a^2 x^2-b}+\sqrt {2} \sqrt [4]{b} \sqrt {\sqrt {a^2 x^2-b}+a x}+a x+\sqrt {b}\right )}{\sqrt {2}}+\sqrt {2} b^{3/4} d \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt {\sqrt {a^2 x^2-b}+a x}}{\sqrt [4]{b}}\right )-\sqrt {2} b^{3/4} d \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {\sqrt {a^2 x^2-b}+a x}}{\sqrt [4]{b}}+1\right )+\frac {1}{3} d \left (\sqrt {a^2 x^2-b}+a x\right )^{3/2}-\frac {b d}{\sqrt {\sqrt {a^2 x^2-b}+a x}}-\frac {b^5 c}{144 a^4 \left (\sqrt {a^2 x^2-b}+a x\right )^{9/2}}-\frac {b^4 c}{80 a^4 \left (\sqrt {a^2 x^2-b}+a x\right )^{5/2}}+\frac {b^3 c}{8 a^4 \sqrt {\sqrt {a^2 x^2-b}+a x}}-\frac {b^2 c \left (\sqrt {a^2 x^2-b}+a x\right )^{3/2}}{24 a^4}+\frac {c \left (\sqrt {a^2 x^2-b}+a x\right )^{11/2}}{176 a^4}+\frac {b c \left (\sqrt {a^2 x^2-b}+a x\right )^{7/2}}{112 a^4} \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+a^2 x^2} \left (d+c x^4\right ) \sqrt {a x+\sqrt {-b+a^2 x^2}}}{x} \, dx &=\int \left (\frac {d \sqrt {-b+a^2 x^2} \sqrt {a x+\sqrt {-b+a^2 x^2}}}{x}+c x^3 \sqrt {-b+a^2 x^2} \sqrt {a x+\sqrt {-b+a^2 x^2}}\right ) \, dx\\ &=c \int x^3 \sqrt {-b+a^2 x^2} \sqrt {a x+\sqrt {-b+a^2 x^2}} \, dx+d \int \frac {\sqrt {-b+a^2 x^2} \sqrt {a x+\sqrt {-b+a^2 x^2}}}{x} \, dx\\ &=\frac {c \operatorname {Subst}\left (\int \frac {\left (-b+x^2\right )^2 \left (b+x^2\right )^3}{x^{11/2}} \, dx,x,a x+\sqrt {-b+a^2 x^2}\right )}{32 a^4}+\frac {1}{2} d \operatorname {Subst}\left (\int \frac {\left (-b+x^2\right )^2}{x^{3/2} \left (b+x^2\right )} \, dx,x,a x+\sqrt {-b+a^2 x^2}\right )\\ &=-\frac {b d}{\sqrt {a x+\sqrt {-b+a^2 x^2}}}+\frac {c \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,a x+\sqrt {-b+a^2 x^2}\right )}{32 a^4}+\frac {d \operatorname {Subst}\left (\int \frac {\sqrt {x} \left (-\frac {3 b^2}{2}+\frac {b x^2}{2}\right )}{b+x^2} \, dx,x,a x+\sqrt {-b+a^2 x^2}\right )}{b}\\ &=-\frac {b^5 c}{144 a^4 \left (a x+\sqrt {-b+a^2 x^2}\right )^{9/2}}-\frac {b^4 c}{80 a^4 \left (a x+\sqrt {-b+a^2 x^2}\right )^{5/2}}+\frac {b^3 c}{8 a^4 \sqrt {a x+\sqrt {-b+a^2 x^2}}}-\frac {b d}{\sqrt {a x+\sqrt {-b+a^2 x^2}}}-\frac {b^2 c \left (a x+\sqrt {-b+a^2 x^2}\right )^{3/2}}{24 a^4}+\frac {1}{3} d \left (a x+\sqrt {-b+a^2 x^2}\right )^{3/2}+\frac {b c \left (a x+\sqrt {-b+a^2 x^2}\right )^{7/2}}{112 a^4}+\frac {c \left (a x+\sqrt {-b+a^2 x^2}\right )^{11/2}}{176 a^4}-(2 b d) \operatorname {Subst}\left (\int \frac {\sqrt {x}}{b+x^2} \, dx,x,a x+\sqrt {-b+a^2 x^2}\right )\\ &=-\frac {b^5 c}{144 a^4 \left (a x+\sqrt {-b+a^2 x^2}\right )^{9/2}}-\frac {b^4 c}{80 a^4 \left (a x+\sqrt {-b+a^2 x^2}\right )^{5/2}}+\frac {b^3 c}{8 a^4 \sqrt {a x+\sqrt {-b+a^2 x^2}}}-\frac {b d}{\sqrt {a x+\sqrt {-b+a^2 x^2}}}-\frac {b^2 c \left (a x+\sqrt {-b+a^2 x^2}\right )^{3/2}}{24 a^4}+\frac {1}{3} d \left (a x+\sqrt {-b+a^2 x^2}\right )^{3/2}+\frac {b c \left (a x+\sqrt {-b+a^2 x^2}\right )^{7/2}}{112 a^4}+\frac {c \left (a x+\sqrt {-b+a^2 x^2}\right )^{11/2}}{176 a^4}-(4 b d) \operatorname {Subst}\left (\int \frac {x^2}{b+x^4} \, dx,x,\sqrt {a x+\sqrt {-b+a^2 x^2}}\right )\\ &=-\frac {b^5 c}{144 a^4 \left (a x+\sqrt {-b+a^2 x^2}\right )^{9/2}}-\frac {b^4 c}{80 a^4 \left (a x+\sqrt {-b+a^2 x^2}\right )^{5/2}}+\frac {b^3 c}{8 a^4 \sqrt {a x+\sqrt {-b+a^2 x^2}}}-\frac {b d}{\sqrt {a x+\sqrt {-b+a^2 x^2}}}-\frac {b^2 c \left (a x+\sqrt {-b+a^2 x^2}\right )^{3/2}}{24 a^4}+\frac {1}{3} d \left (a x+\sqrt {-b+a^2 x^2}\right )^{3/2}+\frac {b c \left (a x+\sqrt {-b+a^2 x^2}\right )^{7/2}}{112 a^4}+\frac {c \left (a x+\sqrt {-b+a^2 x^2}\right )^{11/2}}{176 a^4}+(2 b d) \operatorname {Subst}\left (\int \frac {\sqrt {b}-x^2}{b+x^4} \, dx,x,\sqrt {a x+\sqrt {-b+a^2 x^2}}\right )-(2 b d) \operatorname {Subst}\left (\int \frac {\sqrt {b}+x^2}{b+x^4} \, dx,x,\sqrt {a x+\sqrt {-b+a^2 x^2}}\right )\\ &=-\frac {b^5 c}{144 a^4 \left (a x+\sqrt {-b+a^2 x^2}\right )^{9/2}}-\frac {b^4 c}{80 a^4 \left (a x+\sqrt {-b+a^2 x^2}\right )^{5/2}}+\frac {b^3 c}{8 a^4 \sqrt {a x+\sqrt {-b+a^2 x^2}}}-\frac {b d}{\sqrt {a x+\sqrt {-b+a^2 x^2}}}-\frac {b^2 c \left (a x+\sqrt {-b+a^2 x^2}\right )^{3/2}}{24 a^4}+\frac {1}{3} d \left (a x+\sqrt {-b+a^2 x^2}\right )^{3/2}+\frac {b c \left (a x+\sqrt {-b+a^2 x^2}\right )^{7/2}}{112 a^4}+\frac {c \left (a x+\sqrt {-b+a^2 x^2}\right )^{11/2}}{176 a^4}-\frac {\left (b^{3/4} d\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 {a x+\sqrt {-b+a^2 x^2}}\right )}{\sqrt {2}}-\frac {\left (b^{3/4} d\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 {a x+\sqrt {-b+a^2 x^2}}\right )}{\sqrt {2}}-(b d) \operatorname {Subst}\left (\int \frac {1}{\sqrt {b}-\sqrt {2} \sqrt [4]{b} x+x^2} \, dx,x,\sqrt {a x+\sqrt {-b+a^2 x^2}}\right )-(b d) \operatorname {Subst}\left (\int \frac {1}{\sqrt {b}+\sqrt {2} \sqrt [4]{b} x+x^2} \, dx,x,\sqrt {a x+\sqrt {-b+a^2 x^2}}\right )\\ &=-\frac {b^5 c}{144 a^4 \left (a x+\sqrt {-b+a^2 x^2}\right )^{9/2}}-\frac {b^4 c}{80 a^4 \left (a x+\sqrt {-b+a^2 x^2}\right )^{5/2}}+\frac {b^3 c}{8 a^4 \sqrt {a x+\sqrt {-b+a^2 x^2}}}-\frac {b d}{\sqrt {a x+\sqrt {-b+a^2 x^2}}}-\frac {b^2 c \left (a x+\sqrt {-b+a^2 x^2}\right )^{3/2}}{24 a^4}+\frac {1}{3} d \left (a x+\sqrt {-b+a^2 x^2}\right )^{3/2}+\frac {b c \left (a x+\sqrt {-b+a^2 x^2}\right )^{7/2}}{112 a^4}+\frac {c \left (a x+\sqrt {-b+a^2 x^2}\right )^{11/2}}{176 a^4}-\frac {b^{3/4} d \log \left (\sqrt {b}+a x+\sqrt {-b+a^2 x^2}-\sqrt {2} \sqrt [4]{b} \sqrt {a x+\sqrt {-b+a^2 x^2}}\right )}{\sqrt {2}}+\frac {b^{3/4} d \log \left (\sqrt {b}+a x+\sqrt {-b+a^2 x^2}+\sqrt {2} \sqrt [4]{b} \sqrt {a x+\sqrt {-b+a^2 x^2}}\right )}{\sqrt {2}}-\left (\sqrt {2} b^{3/4} d\right ) \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt {a x+\sqrt {-b+a^2 x^2}}}{\sqrt [4]{b}}\right )+\left (\sqrt {2} b^{3/4} d\right ) \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt {a x+\sqrt {-b+a^2 x^2}}}{\sqrt [4]{b}}\right )\\ &=-\frac {b^5 c}{144 a^4 \left (a x+\sqrt {-b+a^2 x^2}\right )^{9/2}}-\frac {b^4 c}{80 a^4 \left (a x+\sqrt {-b+a^2 x^2}\right )^{5/2}}+\frac {b^3 c}{8 a^4 \sqrt {a x+\sqrt {-b+a^2 x^2}}}-\frac {b d}{\sqrt {a x+\sqrt {-b+a^2 x^2}}}-\frac {b^2 c \left (a x+\sqrt {-b+a^2 x^2}\right )^{3/2}}{24 a^4}+\frac {1}{3} d \left (a x+\sqrt {-b+a^2 x^2}\right )^{3/2}+\frac {b c \left (a x+\sqrt {-b+a^2 x^2}\right )^{7/2}}{112 a^4}+\frac {c \left (a x+\sqrt {-b+a^2 x^2}\right )^{11/2}}{176 a^4}+\sqrt {2} b^{3/4} d \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt {a x+\sqrt {-b+a^2 x^2}}}{\sqrt [4]{b}}\right )-\sqrt {2} b^{3/4} d \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt {a x+\sqrt {-b+a^2 x^2}}}{\sqrt [4]{b}}\right )-\frac {b^{3/4} d \log \left (\sqrt {b}+a x+\sqrt {-b+a^2 x^2}-\sqrt {2} \sqrt [4]{b} \sqrt {a x+\sqrt {-b+a^2 x^2}}\right )}{\sqrt {2}}+\frac {b^{3/4} d \log \left (\sqrt {b}+a x+\sqrt {-b+a^2 x^2}+\sqrt {2} \sqrt [4]{b} \sqrt {a x+\sqrt {-b+a^2 x^2}}\right )}{\sqrt {2}}\\ \end {align*}
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Mathematica [B] time = 24.24, size = 9604, normalized size = 24.19 \begin {gather*} \text {Result too large to show} \end {gather*}
Warning: Unable to verify antiderivative.
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IntegrateAlgebraic [A] time = 0.93, size = 397, normalized size = 1.00 \begin {gather*} \frac {2 \sqrt {-b+a^2 x^2} \left (-1368 a b^4 c x+10395 a^5 b^2 d x+3705 a^3 b^3 c x^3-32340 a^7 b d x^3+1335 a^5 b^2 c x^5+18480 a^9 d x^5-8100 a^7 b c x^7+5040 a^9 c x^9\right )+2 \left (304 b^5 c-2310 a^4 b^3 d-3078 a^2 b^4 c x^2+24255 a^6 b^2 d x^2+3735 a^4 b^3 c x^4-41580 a^8 b d x^4+4755 a^6 b^2 c x^6+18480 a^{10} d x^6-10620 a^8 b c x^8+5040 a^{10} c x^{10}\right )}{3465 a^4 \left (a x+\sqrt {-b+a^2 x^2}\right )^{9/2}}+\sqrt {2} b^{3/4} d \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {a x+\sqrt {-b+a^2 x^2}}}{-\sqrt {b}+a x+\sqrt {-b+a^2 x^2}}\right )+\sqrt {2} b^{3/4} d \tanh ^{-1}\left (\frac {\frac {\sqrt [4]{b}}{\sqrt {2}}+\frac {a x}{\sqrt {2} \sqrt [4]{b}}+\frac {\sqrt {-b+a^2 x^2}}{\sqrt {2} \sqrt [4]{b}}}{\sqrt {a x+\sqrt {-b+a^2 x^2}}}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.51, size = 341, normalized size = 0.86 \begin {gather*} \frac {13860 \, \left (-b^{3} d^{4}\right )^{\frac {1}{4}} a^{4} \arctan \left (-\frac {\left (-b^{3} d^{4}\right )^{\frac {1}{4}} \sqrt {a x + \sqrt {a^{2} x^{2} - b}} b^{2} d^{3} - \sqrt {a b^{4} d^{6} x + \sqrt {a^{2} x^{2} - b} b^{4} d^{6} - \sqrt {-b^{3} d^{4}} b^{3} d^{4}} \left (-b^{3} d^{4}\right )^{\frac {1}{4}}}{b^{3} d^{4}}\right ) - 3465 \, \left (-b^{3} d^{4}\right )^{\frac {1}{4}} a^{4} \log \left (\sqrt {a x + \sqrt {a^{2} x^{2} - b}} b^{2} d^{3} + \left (-b^{3} d^{4}\right )^{\frac {3}{4}}\right ) + 3465 \, \left (-b^{3} d^{4}\right )^{\frac {1}{4}} a^{4} \log \left (\sqrt {a x + \sqrt {a^{2} x^{2} - b}} b^{2} d^{3} - \left (-b^{3} d^{4}\right )^{\frac {3}{4}}\right ) - 2 \, {\left (35 \, a^{5} c x^{5} - 19 \, a^{3} b c x^{3} + {\left (1155 \, a^{5} d - 152 \, a b^{2} c\right )} x - 2 \, {\left (175 \, a^{4} c x^{4} - 57 \, a^{2} b c x^{2} + 1155 \, a^{4} d - 152 \, b^{2} c\right )} \sqrt {a^{2} x^{2} - b}\right )} \sqrt {a x + \sqrt {a^{2} x^{2} - b}}}{3465 \, a^{4}} \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 (c x^{4} + d\right )} \sqrt {a^{2} x^{2} - b} \sqrt {a x + \sqrt {a^{2} 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 = 180.00, size = 0, normalized size = 0.00 \[\int \frac {\sqrt {a^{2} x^{2}-b}\, \left (c \,x^{4}+d \right ) \sqrt {a x +\sqrt {a^{2} x^{2}-b}}}{x}\, dx\]
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 (c x^{4} + d\right )} \sqrt {a^{2} x^{2} - b} \sqrt {a x + \sqrt {a^{2} 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 {a\,x+\sqrt {a^2\,x^2-b}}\,\left (c\,x^4+d\right )\,\sqrt {a^2\,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 {a x + \sqrt {a^{2} x^{2} - b}} \sqrt {a^{2} x^{2} - b} \left (c x^{4} + d\right )}{x}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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