Optimal. Leaf size=106 \[ 2 \sqrt {a+b \sqrt {x}+c x}-2 \sqrt {a} \tanh ^{-1}\left (\frac {2 a+b \sqrt {x}}{2 \sqrt {a} \sqrt {a+b \sqrt {x}+c x}}\right )+\frac {b \tanh ^{-1}\left (\frac {b+2 c \sqrt {x}}{2 \sqrt {c} \sqrt {a+b \sqrt {x}+c x}}\right )}{\sqrt {c}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.09, antiderivative size = 106, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {1357, 734, 843, 621, 206, 724} \[ 2 \sqrt {a+b \sqrt {x}+c x}-2 \sqrt {a} \tanh ^{-1}\left (\frac {2 a+b \sqrt {x}}{2 \sqrt {a} \sqrt {a+b \sqrt {x}+c x}}\right )+\frac {b \tanh ^{-1}\left (\frac {b+2 c \sqrt {x}}{2 \sqrt {c} \sqrt {a+b \sqrt {x}+c x}}\right )}{\sqrt {c}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 206
Rule 621
Rule 724
Rule 734
Rule 843
Rule 1357
Rubi steps
\begin {align*} \int \frac {\sqrt {a+b \sqrt {x}+c x}}{x} \, dx &=2 \operatorname {Subst}\left (\int \frac {\sqrt {a+b x+c x^2}}{x} \, dx,x,\sqrt {x}\right )\\ &=2 \sqrt {a+b \sqrt {x}+c x}-\operatorname {Subst}\left (\int \frac {-2 a-b x}{x \sqrt {a+b x+c x^2}} \, dx,x,\sqrt {x}\right )\\ &=2 \sqrt {a+b \sqrt {x}+c x}+(2 a) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {a+b x+c x^2}} \, dx,x,\sqrt {x}\right )+b \operatorname {Subst}\left (\int \frac {1}{\sqrt {a+b x+c x^2}} \, dx,x,\sqrt {x}\right )\\ &=2 \sqrt {a+b \sqrt {x}+c x}-(4 a) \operatorname {Subst}\left (\int \frac {1}{4 a-x^2} \, dx,x,\frac {2 a+b \sqrt {x}}{\sqrt {a+b \sqrt {x}+c x}}\right )+(2 b) \operatorname {Subst}\left (\int \frac {1}{4 c-x^2} \, dx,x,\frac {b+2 c \sqrt {x}}{\sqrt {a+b \sqrt {x}+c x}}\right )\\ &=2 \sqrt {a+b \sqrt {x}+c x}-2 \sqrt {a} \tanh ^{-1}\left (\frac {2 a+b \sqrt {x}}{2 \sqrt {a} \sqrt {a+b \sqrt {x}+c x}}\right )+\frac {b \tanh ^{-1}\left (\frac {b+2 c \sqrt {x}}{2 \sqrt {c} \sqrt {a+b \sqrt {x}+c x}}\right )}{\sqrt {c}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.06, size = 106, normalized size = 1.00 \[ 2 \sqrt {a+b \sqrt {x}+c x}-2 \sqrt {a} \tanh ^{-1}\left (\frac {2 a+b \sqrt {x}}{2 \sqrt {a} \sqrt {a+b \sqrt {x}+c x}}\right )+\frac {b \tanh ^{-1}\left (\frac {b+2 c \sqrt {x}}{2 \sqrt {c} \sqrt {a+b \sqrt {x}+c x}}\right )}{\sqrt {c}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.01, size = 84, normalized size = 0.79 \[ -2 \sqrt {a}\, \ln \left (\frac {b \sqrt {x}+2 a +2 \sqrt {c x +b \sqrt {x}+a}\, \sqrt {a}}{\sqrt {x}}\right )+\frac {b \ln \left (\frac {c \sqrt {x}+\frac {b}{2}}{\sqrt {c}}+\sqrt {c x +b \sqrt {x}+a}\right )}{\sqrt {c}}+2 \sqrt {c x +b \sqrt {x}+a} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {c x + b \sqrt {x} + a}}{x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\sqrt {a+c\,x+b\,\sqrt {x}}}{x} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {a + b \sqrt {x} + c x}}{x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________