Optimal. Leaf size=215 \[ \frac {2 \tan ^{-1}\left (\frac {\sqrt {c} \sqrt {\sqrt {a^2 x^2+b^2}+a x}}{\sqrt {a d-\sqrt {a^2 d^2+b^2 c^2}}}\right )}{\sqrt {c} \sqrt {a d-\sqrt {a^2 d^2+b^2 c^2}}}+\frac {2 \tan ^{-1}\left (\frac {\sqrt {c} \sqrt {\sqrt {a^2 x^2+b^2}+a x}}{\sqrt {\sqrt {a^2 d^2+b^2 c^2}+a d}}\right )}{\sqrt {c} \sqrt {\sqrt {a^2 d^2+b^2 c^2}+a d}}+\frac {2}{c \sqrt {\sqrt {a^2 x^2+b^2}+a x}} \]
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Rubi [A] time = 0.50, antiderivative size = 215, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.194, Rules used = {2119, 1628, 828, 826, 1166, 205} \begin {gather*} \frac {2 \tan ^{-1}\left (\frac {\sqrt {c} \sqrt {\sqrt {a^2 x^2+b^2}+a x}}{\sqrt {a d-\sqrt {a^2 d^2+b^2 c^2}}}\right )}{\sqrt {c} \sqrt {a d-\sqrt {a^2 d^2+b^2 c^2}}}+\frac {2 \tan ^{-1}\left (\frac {\sqrt {c} \sqrt {\sqrt {a^2 x^2+b^2}+a x}}{\sqrt {\sqrt {a^2 d^2+b^2 c^2}+a d}}\right )}{\sqrt {c} \sqrt {\sqrt {a^2 d^2+b^2 c^2}+a d}}+\frac {2}{c \sqrt {\sqrt {a^2 x^2+b^2}+a x}} \end {gather*}
Antiderivative was successfully verified.
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Rule 205
Rule 826
Rule 828
Rule 1166
Rule 1628
Rule 2119
Rubi steps
\begin {align*} \int \frac {1}{(d+c x) \sqrt {a x+\sqrt {b^2+a^2 x^2}}} \, dx &=\operatorname {Subst}\left (\int \frac {b^2+x^2}{x^{3/2} \left (-b^2 c+2 a d x+c x^2\right )} \, dx,x,a x+\sqrt {b^2+a^2 x^2}\right )\\ &=\operatorname {Subst}\left (\int \left (\frac {1}{c x^{3/2}}+\frac {2 \left (b^2 c-a d x\right )}{c x^{3/2} \left (-b^2 c+2 a d x+c x^2\right )}\right ) \, dx,x,a x+\sqrt {b^2+a^2 x^2}\right )\\ &=-\frac {2}{c \sqrt {a x+\sqrt {b^2+a^2 x^2}}}+\frac {2 \operatorname {Subst}\left (\int \frac {b^2 c-a d x}{x^{3/2} \left (-b^2 c+2 a d x+c x^2\right )} \, dx,x,a x+\sqrt {b^2+a^2 x^2}\right )}{c}\\ &=\frac {2}{c \sqrt {a x+\sqrt {b^2+a^2 x^2}}}-\frac {2 \operatorname {Subst}\left (\int \frac {-a b^2 c d-b^2 c^2 x}{\sqrt {x} \left (-b^2 c+2 a d x+c x^2\right )} \, dx,x,a x+\sqrt {b^2+a^2 x^2}\right )}{b^2 c^2}\\ &=\frac {2}{c \sqrt {a x+\sqrt {b^2+a^2 x^2}}}-\frac {4 \operatorname {Subst}\left (\int \frac {-a b^2 c d-b^2 c^2 x^2}{-b^2 c+2 a d x^2+c x^4} \, dx,x,\sqrt {a x+\sqrt {b^2+a^2 x^2}}\right )}{b^2 c^2}\\ &=\frac {2}{c \sqrt {a x+\sqrt {b^2+a^2 x^2}}}+2 \operatorname {Subst}\left (\int \frac {1}{a d-\sqrt {b^2 c^2+a^2 d^2}+c x^2} \, dx,x,\sqrt {a x+\sqrt {b^2+a^2 x^2}}\right )+2 \operatorname {Subst}\left (\int \frac {1}{a d+\sqrt {b^2 c^2+a^2 d^2}+c x^2} \, dx,x,\sqrt {a x+\sqrt {b^2+a^2 x^2}}\right )\\ &=\frac {2}{c \sqrt {a x+\sqrt {b^2+a^2 x^2}}}+\frac {2 \tan ^{-1}\left (\frac {\sqrt {c} \sqrt {a x+\sqrt {b^2+a^2 x^2}}}{\sqrt {a d-\sqrt {b^2 c^2+a^2 d^2}}}\right )}{\sqrt {c} \sqrt {a d-\sqrt {b^2 c^2+a^2 d^2}}}+\frac {2 \tan ^{-1}\left (\frac {\sqrt {c} \sqrt {a x+\sqrt {b^2+a^2 x^2}}}{\sqrt {a d+\sqrt {b^2 c^2+a^2 d^2}}}\right )}{\sqrt {c} \sqrt {a d+\sqrt {b^2 c^2+a^2 d^2}}}\\ \end {align*}
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Mathematica [A] time = 0.64, size = 333, normalized size = 1.55 \begin {gather*} -\frac {2 \left (-\frac {\left (a d \left (\sqrt {a^2 d^2+b^2 c^2}+a d\right )+b^2 c^2\right ) \tan ^{-1}\left (\frac {b \sqrt {c}}{\sqrt {\sqrt {a^2 x^2+b^2}+a x} \sqrt {-\sqrt {a^2 d^2+b^2 c^2}-a d}}\right )}{b \sqrt {a^2 d^2+b^2 c^2} \sqrt {-\sqrt {a^2 d^2+b^2 c^2}-a d}}+\frac {\left (a d \left (a d-\sqrt {a^2 d^2+b^2 c^2}\right )+b^2 c^2\right ) \tan ^{-1}\left (\frac {b \sqrt {c}}{\sqrt {\sqrt {a^2 x^2+b^2}+a x} \sqrt {\sqrt {a^2 d^2+b^2 c^2}-a d}}\right )}{b \sqrt {a^2 d^2+b^2 c^2} \sqrt {\sqrt {a^2 d^2+b^2 c^2}-a d}}-\frac {\sqrt {c}}{\sqrt {\sqrt {a^2 x^2+b^2}+a x}}\right )}{c^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.66, size = 215, normalized size = 1.00 \begin {gather*} \frac {2}{c \sqrt {a x+\sqrt {b^2+a^2 x^2}}}+\frac {2 \tan ^{-1}\left (\frac {\sqrt {c} \sqrt {a x+\sqrt {b^2+a^2 x^2}}}{\sqrt {a d-\sqrt {b^2 c^2+a^2 d^2}}}\right )}{\sqrt {c} \sqrt {a d-\sqrt {b^2 c^2+a^2 d^2}}}+\frac {2 \tan ^{-1}\left (\frac {\sqrt {c} \sqrt {a x+\sqrt {b^2+a^2 x^2}}}{\sqrt {a d+\sqrt {b^2 c^2+a^2 d^2}}}\right )}{\sqrt {c} \sqrt {a d+\sqrt {b^2 c^2+a^2 d^2}}} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.65, size = 673, normalized size = 3.13 \begin {gather*} -\frac {b^{2} c \sqrt {\frac {b^{2} c^{3} \sqrt {\frac {b^{2} c^{2} + a^{2} d^{2}}{b^{4} c^{6}}} + a d}{b^{2} c^{3}}} \log \left (4 \, {\left (b^{2} c^{3} \sqrt {\frac {b^{2} c^{2} + a^{2} d^{2}}{b^{4} c^{6}}} - a d\right )} \sqrt {\frac {b^{2} c^{3} \sqrt {\frac {b^{2} c^{2} + a^{2} d^{2}}{b^{4} c^{6}}} + a d}{b^{2} c^{3}}} + 4 \, \sqrt {a x + \sqrt {a^{2} x^{2} + b^{2}}}\right ) - b^{2} c \sqrt {\frac {b^{2} c^{3} \sqrt {\frac {b^{2} c^{2} + a^{2} d^{2}}{b^{4} c^{6}}} + a d}{b^{2} c^{3}}} \log \left (-4 \, {\left (b^{2} c^{3} \sqrt {\frac {b^{2} c^{2} + a^{2} d^{2}}{b^{4} c^{6}}} - a d\right )} \sqrt {\frac {b^{2} c^{3} \sqrt {\frac {b^{2} c^{2} + a^{2} d^{2}}{b^{4} c^{6}}} + a d}{b^{2} c^{3}}} + 4 \, \sqrt {a x + \sqrt {a^{2} x^{2} + b^{2}}}\right ) - b^{2} c \sqrt {-\frac {b^{2} c^{3} \sqrt {\frac {b^{2} c^{2} + a^{2} d^{2}}{b^{4} c^{6}}} - a d}{b^{2} c^{3}}} \log \left (4 \, {\left (b^{2} c^{3} \sqrt {\frac {b^{2} c^{2} + a^{2} d^{2}}{b^{4} c^{6}}} + a d\right )} \sqrt {-\frac {b^{2} c^{3} \sqrt {\frac {b^{2} c^{2} + a^{2} d^{2}}{b^{4} c^{6}}} - a d}{b^{2} c^{3}}} + 4 \, \sqrt {a x + \sqrt {a^{2} x^{2} + b^{2}}}\right ) + b^{2} c \sqrt {-\frac {b^{2} c^{3} \sqrt {\frac {b^{2} c^{2} + a^{2} d^{2}}{b^{4} c^{6}}} - a d}{b^{2} c^{3}}} \log \left (-4 \, {\left (b^{2} c^{3} \sqrt {\frac {b^{2} c^{2} + a^{2} d^{2}}{b^{4} c^{6}}} + a d\right )} \sqrt {-\frac {b^{2} c^{3} \sqrt {\frac {b^{2} c^{2} + a^{2} d^{2}}{b^{4} c^{6}}} - a d}{b^{2} c^{3}}} + 4 \, \sqrt {a x + \sqrt {a^{2} x^{2} + b^{2}}}\right ) + 2 \, \sqrt {a x + \sqrt {a^{2} x^{2} + b^{2}}} {\left (a x - \sqrt {a^{2} x^{2} + b^{2}}\right )}}{b^{2} c} \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}{\sqrt {a x + \sqrt {a^{2} x^{2} + b^{2}}} {\left (c x + d\right )}}\,{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 \[\int \frac {1}{\left (c x +d \right ) \sqrt {a x +\sqrt {a^{2} x^{2}+b^{2}}}}\, dx\]
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}{\sqrt {a x + \sqrt {a^{2} x^{2} + b^{2}}} {\left (c x + d\right )}}\,{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}{\sqrt {a\,x+\sqrt {a^2\,x^2+b^2}}\,\left (d+c\,x\right )} \,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}{\sqrt {a x + \sqrt {a^{2} x^{2} + b^{2}}} \left (c x + d\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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