Optimal. Leaf size=276 \[ \frac {b^2 \log \left (\sqrt {a^2 x^2+b^2}+a x\right )}{2 a c^3}-\frac {b^2 \log \left (\sqrt {\sqrt {a^2 x^2+b^2}+a x}+c\right )}{a c^3}+\frac {\sqrt {a^2 x^2+b^2} \left (\left (3 b^2+2 c^4\right ) \sqrt {\sqrt {a^2 x^2+b^2}+a x}+4 a c^3 x+2 b^2 c\right )+\sqrt {\sqrt {a^2 x^2+b^2}+a x} \left (3 a b^2 x+2 a c^4 x-b^2 c^2\right )+4 a^2 c^3 x^2+2 a b^2 c x+2 b^2 c^3}{2 a c^3 \left (\sqrt {a^2 x^2+b^2}+a x\right )^{3/2}}-\frac {c \log \left (\sqrt {\sqrt {a^2 x^2+b^2}+a x}+c\right )}{a} \]
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Rubi [A] time = 0.13, antiderivative size = 169, normalized size of antiderivative = 0.61, number of steps used = 4, number of rules used = 3, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {2117, 1821, 1620} \begin {gather*} \frac {b^2 \log \left (\sqrt {a^2 x^2+b^2}+a x\right )}{2 a c^3}+\frac {b^2}{a c^2 \sqrt {\sqrt {a^2 x^2+b^2}+a x}}-\frac {\left (b^2+c^4\right ) \log \left (\sqrt {\sqrt {a^2 x^2+b^2}+a x}+c\right )}{a c^3}-\frac {b^2}{2 a c \left (\sqrt {a^2 x^2+b^2}+a x\right )}+\frac {\sqrt {\sqrt {a^2 x^2+b^2}+a x}}{a} \end {gather*}
Antiderivative was successfully verified.
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Rule 1620
Rule 1821
Rule 2117
Rubi steps
\begin {align*} \int \frac {1}{c+\sqrt {a x+\sqrt {b^2+a^2 x^2}}} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {b^2+x^2}{\left (c+\sqrt {x}\right ) x^2} \, dx,x,a x+\sqrt {b^2+a^2 x^2}\right )}{2 a}\\ &=\frac {\operatorname {Subst}\left (\int \frac {b^2+x^4}{x^3 (c+x)} \, dx,x,\sqrt {a x+\sqrt {b^2+a^2 x^2}}\right )}{a}\\ &=\frac {\operatorname {Subst}\left (\int \left (1+\frac {b^2}{c x^3}-\frac {b^2}{c^2 x^2}+\frac {b^2}{c^3 x}+\frac {-b^2-c^4}{c^3 (c+x)}\right ) \, dx,x,\sqrt {a x+\sqrt {b^2+a^2 x^2}}\right )}{a}\\ &=-\frac {b^2}{2 a c \left (a x+\sqrt {b^2+a^2 x^2}\right )}+\frac {b^2}{a c^2 \sqrt {a x+\sqrt {b^2+a^2 x^2}}}+\frac {\sqrt {a x+\sqrt {b^2+a^2 x^2}}}{a}+\frac {b^2 \log \left (a x+\sqrt {b^2+a^2 x^2}\right )}{2 a c^3}-\frac {\left (b^2+c^4\right ) \log \left (c+\sqrt {a x+\sqrt {b^2+a^2 x^2}}\right )}{a c^3}\\ \end {align*}
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Mathematica [A] time = 0.19, size = 157, normalized size = 0.57 \begin {gather*} \frac {\frac {b^2 \log \left (\sqrt {a^2 x^2+b^2}+a x\right )}{2 c^3}+\frac {b^2}{c^2 \sqrt {\sqrt {a^2 x^2+b^2}+a x}}-\frac {\left (b^2+c^4\right ) \log \left (\sqrt {\sqrt {a^2 x^2+b^2}+a x}+c\right )}{c^3}-\frac {b^2}{2 c \left (\sqrt {a^2 x^2+b^2}+a x\right )}+\sqrt {\sqrt {a^2 x^2+b^2}+a x}}{a} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.45, size = 276, normalized size = 1.00 \begin {gather*} \frac {2 b^2 c^3+2 a b^2 c x+4 a^2 c^3 x^2+\left (-b^2 c^2+3 a b^2 x+2 a c^4 x\right ) \sqrt {a x+\sqrt {b^2+a^2 x^2}}+\sqrt {b^2+a^2 x^2} \left (2 b^2 c+4 a c^3 x+\left (3 b^2+2 c^4\right ) \sqrt {a x+\sqrt {b^2+a^2 x^2}}\right )}{2 a c^3 \left (a x+\sqrt {b^2+a^2 x^2}\right )^{3/2}}+\frac {b^2 \log \left (a x+\sqrt {b^2+a^2 x^2}\right )}{2 a c^3}-\frac {b^2 \log \left (c+\sqrt {a x+\sqrt {b^2+a^2 x^2}}\right )}{a c^3}-\frac {c \log \left (c+\sqrt {a x+\sqrt {b^2+a^2 x^2}}\right )}{a} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.64, size = 134, normalized size = 0.49 \begin {gather*} \frac {a c^{2} x + 2 \, b^{2} \log \left (\sqrt {a x + \sqrt {a^{2} x^{2} + b^{2}}}\right ) - \sqrt {a^{2} x^{2} + b^{2}} c^{2} - 2 \, {\left (c^{4} + b^{2}\right )} \log \left (c + \sqrt {a x + \sqrt {a^{2} x^{2} + b^{2}}}\right ) + 2 \, {\left (c^{3} - a c x + \sqrt {a^{2} x^{2} + b^{2}} c\right )} \sqrt {a x + \sqrt {a^{2} x^{2} + b^{2}}}}{2 \, a c^{3}} \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 {1}{c + \sqrt {a x + \sqrt {a^{2} x^{2} + b^{2}}}}\,{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 \[\int \frac {1}{c +\sqrt {a x +\sqrt {a^{2} x^{2}+b^{2}}}}\, dx\]
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 {1}{c + \sqrt {a x + \sqrt {a^{2} x^{2} + b^{2}}}}\,{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 {1}{c+\sqrt {a\,x+\sqrt {a^2\,x^2+b^2}}} \,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 {1}{c + \sqrt {a x + \sqrt {a^{2} x^{2} + b^{2}}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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