Integrand size = 29, antiderivative size = 128 \[ \int \frac {d+c x}{\sqrt {a x+\sqrt {b^2+a^2 x^2}}} \, dx=\frac {b^4 c}{10 a^2 \left (a x+\sqrt {b^2+a^2 x^2}\right )^{5/2}}-\frac {b^2 d}{3 a \left (a x+\sqrt {b^2+a^2 x^2}\right )^{3/2}}+\frac {d \sqrt {a x+\sqrt {b^2+a^2 x^2}}}{a}+\frac {c \left (a x+\sqrt {b^2+a^2 x^2}\right )^{3/2}}{6 a^2} \]
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Time = 0.07 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.069, Rules used = {2144, 1642} \[ \int \frac {d+c x}{\sqrt {a x+\sqrt {b^2+a^2 x^2}}} \, dx=\frac {c \left (\sqrt {a^2 x^2+b^2}+a x\right )^{3/2}}{6 a^2}-\frac {b^2 d}{3 a \left (\sqrt {a^2 x^2+b^2}+a x\right )^{3/2}}+\frac {d \sqrt {\sqrt {a^2 x^2+b^2}+a x}}{a}+\frac {b^4 c}{10 a^2 \left (\sqrt {a^2 x^2+b^2}+a x\right )^{5/2}} \]
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Rule 1642
Rule 2144
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {\left (b^2+x^2\right ) \left (-b^2 c+2 a d x+c x^2\right )}{x^{7/2}} \, dx,x,a x+\sqrt {b^2+a^2 x^2}\right )}{4 a^2} \\ & = \frac {\text {Subst}\left (\int \left (-\frac {b^4 c}{x^{7/2}}+\frac {2 a b^2 d}{x^{5/2}}+\frac {2 a d}{\sqrt {x}}+c \sqrt {x}\right ) \, dx,x,a x+\sqrt {b^2+a^2 x^2}\right )}{4 a^2} \\ & = \frac {b^4 c}{10 a^2 \left (a x+\sqrt {b^2+a^2 x^2}\right )^{5/2}}-\frac {b^2 d}{3 a \left (a x+\sqrt {b^2+a^2 x^2}\right )^{3/2}}+\frac {d \sqrt {a x+\sqrt {b^2+a^2 x^2}}}{a}+\frac {c \left (a x+\sqrt {b^2+a^2 x^2}\right )^{3/2}}{6 a^2} \\ \end{align*}
Time = 0.19 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.86 \[ \int \frac {d+c x}{\sqrt {a x+\sqrt {b^2+a^2 x^2}}} \, dx=\frac {4 b^4 c+20 a^3 x^2 (3 d+c x) \left (a x+\sqrt {b^2+a^2 x^2}\right )+10 a b^2 \left (2 a x (2 d+c x)+(d+c x) \sqrt {b^2+a^2 x^2}\right )}{15 a^2 \left (a x+\sqrt {b^2+a^2 x^2}\right )^{5/2}} \]
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\[\int \frac {c x +d}{\sqrt {a x +\sqrt {a^{2} x^{2}+b^{2}}}}d x\]
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none
Time = 0.23 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.77 \[ \int \frac {d+c x}{\sqrt {a x+\sqrt {b^2+a^2 x^2}}} \, dx=-\frac {2 \, {\left (3 \, a^{3} c x^{3} + 5 \, a^{3} d x^{2} + a b^{2} c x - 5 \, a b^{2} d - {\left (3 \, a^{2} c x^{2} + 5 \, a^{2} d x + 2 \, b^{2} c\right )} \sqrt {a^{2} x^{2} + b^{2}}\right )} \sqrt {a x + \sqrt {a^{2} x^{2} + b^{2}}}}{15 \, a^{2} b^{2}} \]
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\[ \int \frac {d+c x}{\sqrt {a x+\sqrt {b^2+a^2 x^2}}} \, dx=\int \frac {c x + d}{\sqrt {a x + \sqrt {a^{2} x^{2} + b^{2}}}}\, dx \]
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\[ \int \frac {d+c x}{\sqrt {a x+\sqrt {b^2+a^2 x^2}}} \, dx=\int { \frac {c x + d}{\sqrt {a x + \sqrt {a^{2} x^{2} + b^{2}}}} \,d x } \]
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\[ \int \frac {d+c x}{\sqrt {a x+\sqrt {b^2+a^2 x^2}}} \, dx=\int { \frac {c x + d}{\sqrt {a x + \sqrt {a^{2} x^{2} + b^{2}}}} \,d x } \]
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Timed out. \[ \int \frac {d+c x}{\sqrt {a x+\sqrt {b^2+a^2 x^2}}} \, dx=\int \frac {d+c\,x}{\sqrt {a\,x+\sqrt {a^2\,x^2+b^2}}} \,d x \]
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