Optimal. Leaf size=151 \[ 2 \sqrt {d \sqrt {a x^2+b}+c}+\sqrt {\sqrt {b} d-c} \tan ^{-1}\left (\frac {\sqrt {\sqrt {b} d-c} \sqrt {d \sqrt {a x^2+b}+c}}{c-\sqrt {b} d}\right )+\sqrt {-\sqrt {b} d-c} \tan ^{-1}\left (\frac {\sqrt {-\sqrt {b} d-c} \sqrt {d \sqrt {a x^2+b}+c}}{\sqrt {b} d+c}\right ) \]
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Rubi [A] time = 0.39, antiderivative size = 122, normalized size of antiderivative = 0.81, number of steps used = 8, number of rules used = 6, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {371, 1398, 825, 827, 1166, 207} \begin {gather*} 2 \sqrt {d \sqrt {a x^2+b}+c}-\sqrt {c-\sqrt {b} d} \tanh ^{-1}\left (\frac {\sqrt {d \sqrt {a x^2+b}+c}}{\sqrt {c-\sqrt {b} d}}\right )-\sqrt {\sqrt {b} d+c} \tanh ^{-1}\left (\frac {\sqrt {d \sqrt {a x^2+b}+c}}{\sqrt {\sqrt {b} d+c}}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 207
Rule 371
Rule 825
Rule 827
Rule 1166
Rule 1398
Rubi steps
\begin {align*} \int \frac {\sqrt {c+d \sqrt {b+a x^2}}}{x} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {\sqrt {c+d \sqrt {b+a x}}}{x} \, dx,x,x^2\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {\sqrt {c+d \sqrt {x}}}{-b+x} \, dx,x,b+a x^2\right )\\ &=\operatorname {Subst}\left (\int \frac {x \sqrt {c+d x}}{-b+x^2} \, dx,x,\sqrt {b+a x^2}\right )\\ &=2 \sqrt {c+d \sqrt {b+a x^2}}+\operatorname {Subst}\left (\int \frac {b d+c x}{\sqrt {c+d x} \left (-b+x^2\right )} \, dx,x,\sqrt {b+a x^2}\right )\\ &=2 \sqrt {c+d \sqrt {b+a x^2}}+2 \operatorname {Subst}\left (\int \frac {-c^2+b d^2+c x^2}{c^2-b d^2-2 c x^2+x^4} \, dx,x,\sqrt {c+d \sqrt {b+a x^2}}\right )\\ &=2 \sqrt {c+d \sqrt {b+a x^2}}+\left (c-\sqrt {b} d\right ) \operatorname {Subst}\left (\int \frac {1}{-c+\sqrt {b} d+x^2} \, dx,x,\sqrt {c+d \sqrt {b+a x^2}}\right )+\left (c+\sqrt {b} d\right ) \operatorname {Subst}\left (\int \frac {1}{-c-\sqrt {b} d+x^2} \, dx,x,\sqrt {c+d \sqrt {b+a x^2}}\right )\\ &=2 \sqrt {c+d \sqrt {b+a x^2}}-\sqrt {c-\sqrt {b} d} \tanh ^{-1}\left (\frac {\sqrt {c+d \sqrt {b+a x^2}}}{\sqrt {c-\sqrt {b} d}}\right )-\sqrt {c+\sqrt {b} d} \tanh ^{-1}\left (\frac {\sqrt {c+d \sqrt {b+a x^2}}}{\sqrt {c+\sqrt {b} d}}\right )\\ \end {align*}
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Mathematica [A] time = 0.22, size = 122, normalized size = 0.81 \begin {gather*} 2 \sqrt {d \sqrt {a x^2+b}+c}-\sqrt {c-\sqrt {b} d} \tanh ^{-1}\left (\frac {\sqrt {d \sqrt {a x^2+b}+c}}{\sqrt {c-\sqrt {b} d}}\right )-\sqrt {\sqrt {b} d+c} \tanh ^{-1}\left (\frac {\sqrt {d \sqrt {a x^2+b}+c}}{\sqrt {\sqrt {b} d+c}}\right ) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.39, size = 151, normalized size = 1.00 \begin {gather*} 2 \sqrt {c+d \sqrt {b+a x^2}}+\sqrt {-c+\sqrt {b} d} \tan ^{-1}\left (\frac {\sqrt {-c+\sqrt {b} d} \sqrt {c+d \sqrt {b+a x^2}}}{c-\sqrt {b} d}\right )+\sqrt {-c-\sqrt {b} d} \tan ^{-1}\left (\frac {\sqrt {-c-\sqrt {b} d} \sqrt {c+d \sqrt {b+a x^2}}}{c+\sqrt {b} d}\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 [B] time = 0.20, size = 288, normalized size = 1.91 \begin {gather*} \frac {2 \, \sqrt {\sqrt {a x^{2} + b} d + c} d + \frac {{\left (\sqrt {b} c d^{3} \mathrm {sgn}\left ({\left (\sqrt {a x^{2} + b} d + c\right )} d - c d\right ) - b d^{3} {\left | d \right |} + c^{2} d {\left | d \right |} \mathrm {sgn}\left ({\left (\sqrt {a x^{2} + b} d + c\right )} d - c d\right ) - \sqrt {b} c d^{3}\right )} \arctan \left (\frac {\sqrt {\sqrt {a x^{2} + b} d + c}}{\sqrt {-c + \sqrt {b d^{2}}}}\right )}{{\left (\sqrt {b} d + c\right )} \sqrt {\sqrt {b} d - c} {\left | d \right |}} + \frac {{\left (\sqrt {b} c d^{3} \mathrm {sgn}\left ({\left (\sqrt {a x^{2} + b} d + c\right )} d - c d\right ) + b d^{3} {\left | d \right |} - c^{2} d {\left | d \right |} \mathrm {sgn}\left ({\left (\sqrt {a x^{2} + b} d + c\right )} d - c d\right ) - \sqrt {b} c d^{3}\right )} \arctan \left (\frac {\sqrt {\sqrt {a x^{2} + b} d + c}}{\sqrt {-c - \sqrt {b d^{2}}}}\right )}{{\left (\sqrt {b} d - c\right )} \sqrt {-\sqrt {b} d - c} {\left | d \right |}}}{d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.01, size = 0, normalized size = 0.00 \[\int \frac {\sqrt {c +d \sqrt {a \,x^{2}+b}}}{x}\, dx\]
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 {\sqrt {\sqrt {a x^{2} + b} d + c}}{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 = -0.01 \begin {gather*} \int \frac {\sqrt {c+d\,\sqrt {a\,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 {c + d \sqrt {a x^{2} + b}}}{x}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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