3.468 \(\int \frac{\sqrt{a+b \sqrt{c+d x}}}{x} \, dx\)

Optimal. Leaf size=116 \[ 4 \sqrt{a+b \sqrt{c+d x}}-2 \sqrt{a-b \sqrt{c}} \tanh ^{-1}\left (\frac{\sqrt{a+b \sqrt{c+d x}}}{\sqrt{a-b \sqrt{c}}}\right )-2 \sqrt{a+b \sqrt{c}} \tanh ^{-1}\left (\frac{\sqrt{a+b \sqrt{c+d x}}}{\sqrt{a+b \sqrt{c}}}\right ) \]

[Out]

4*Sqrt[a + b*Sqrt[c + d*x]] - 2*Sqrt[a - b*Sqrt[c]]*ArcTanh[Sqrt[a + b*Sqrt[c +
d*x]]/Sqrt[a - b*Sqrt[c]]] - 2*Sqrt[a + b*Sqrt[c]]*ArcTanh[Sqrt[a + b*Sqrt[c + d
*x]]/Sqrt[a + b*Sqrt[c]]]

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Rubi [A]  time = 0.349722, antiderivative size = 116, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286 \[ 4 \sqrt{a+b \sqrt{c+d x}}-2 \sqrt{a-b \sqrt{c}} \tanh ^{-1}\left (\frac{\sqrt{a+b \sqrt{c+d x}}}{\sqrt{a-b \sqrt{c}}}\right )-2 \sqrt{a+b \sqrt{c}} \tanh ^{-1}\left (\frac{\sqrt{a+b \sqrt{c+d x}}}{\sqrt{a+b \sqrt{c}}}\right ) \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[a + b*Sqrt[c + d*x]]/x,x]

[Out]

4*Sqrt[a + b*Sqrt[c + d*x]] - 2*Sqrt[a - b*Sqrt[c]]*ArcTanh[Sqrt[a + b*Sqrt[c +
d*x]]/Sqrt[a - b*Sqrt[c]]] - 2*Sqrt[a + b*Sqrt[c]]*ArcTanh[Sqrt[a + b*Sqrt[c + d
*x]]/Sqrt[a + b*Sqrt[c]]]

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Rubi in Sympy [A]  time = 27.8602, size = 100, normalized size = 0.86 \[ - 2 \sqrt{a - b \sqrt{c}} \operatorname{atanh}{\left (\frac{\sqrt{a + b \sqrt{c + d x}}}{\sqrt{a - b \sqrt{c}}} \right )} - 2 \sqrt{a + b \sqrt{c}} \operatorname{atanh}{\left (\frac{\sqrt{a + b \sqrt{c + d x}}}{\sqrt{a + b \sqrt{c}}} \right )} + 4 \sqrt{a + b \sqrt{c + d x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((a+b*(d*x+c)**(1/2))**(1/2)/x,x)

[Out]

-2*sqrt(a - b*sqrt(c))*atanh(sqrt(a + b*sqrt(c + d*x))/sqrt(a - b*sqrt(c))) - 2*
sqrt(a + b*sqrt(c))*atanh(sqrt(a + b*sqrt(c + d*x))/sqrt(a + b*sqrt(c))) + 4*sqr
t(a + b*sqrt(c + d*x))

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Mathematica [A]  time = 0.235681, size = 116, normalized size = 1. \[ 4 \sqrt{a+b \sqrt{c+d x}}-2 \sqrt{a-b \sqrt{c}} \tanh ^{-1}\left (\frac{\sqrt{a+b \sqrt{c+d x}}}{\sqrt{a-b \sqrt{c}}}\right )-2 \sqrt{a+b \sqrt{c}} \tanh ^{-1}\left (\frac{\sqrt{a+b \sqrt{c+d x}}}{\sqrt{a+b \sqrt{c}}}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[a + b*Sqrt[c + d*x]]/x,x]

[Out]

4*Sqrt[a + b*Sqrt[c + d*x]] - 2*Sqrt[a - b*Sqrt[c]]*ArcTanh[Sqrt[a + b*Sqrt[c +
d*x]]/Sqrt[a - b*Sqrt[c]]] - 2*Sqrt[a + b*Sqrt[c]]*ArcTanh[Sqrt[a + b*Sqrt[c + d
*x]]/Sqrt[a + b*Sqrt[c]]]

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Maple [B]  time = 0.051, size = 221, normalized size = 1.9 \[ 4\,\sqrt{a+b\sqrt{dx+c}}-2\,{\frac{{b}^{2}c}{\sqrt{{b}^{2}c}\sqrt{\sqrt{{b}^{2}c}-a}}\arctan \left ({\frac{\sqrt{a+b\sqrt{dx+c}}}{\sqrt{\sqrt{{b}^{2}c}-a}}} \right ) }+2\,{\frac{a}{\sqrt{\sqrt{{b}^{2}c}-a}}\arctan \left ({\frac{\sqrt{a+b\sqrt{dx+c}}}{\sqrt{\sqrt{{b}^{2}c}-a}}} \right ) }+2\,{\frac{{b}^{2}c}{\sqrt{{b}^{2}c}\sqrt{-\sqrt{{b}^{2}c}-a}}\arctan \left ({\frac{\sqrt{a+b\sqrt{dx+c}}}{\sqrt{-\sqrt{{b}^{2}c}-a}}} \right ) }+2\,{\frac{a}{\sqrt{-\sqrt{{b}^{2}c}-a}}\arctan \left ({\frac{\sqrt{a+b\sqrt{dx+c}}}{\sqrt{-\sqrt{{b}^{2}c}-a}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b*(d*x+c)^(1/2))^(1/2)/x,x)

[Out]

4*(a+b*(d*x+c)^(1/2))^(1/2)-2/(b^2*c)^(1/2)/((b^2*c)^(1/2)-a)^(1/2)*arctan((a+b*
(d*x+c)^(1/2))^(1/2)/((b^2*c)^(1/2)-a)^(1/2))*b^2*c+2/((b^2*c)^(1/2)-a)^(1/2)*ar
ctan((a+b*(d*x+c)^(1/2))^(1/2)/((b^2*c)^(1/2)-a)^(1/2))*a+2/(b^2*c)^(1/2)/(-(b^2
*c)^(1/2)-a)^(1/2)*arctan((a+b*(d*x+c)^(1/2))^(1/2)/(-(b^2*c)^(1/2)-a)^(1/2))*b^
2*c+2/(-(b^2*c)^(1/2)-a)^(1/2)*arctan((a+b*(d*x+c)^(1/2))^(1/2)/(-(b^2*c)^(1/2)-
a)^(1/2))*a

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{\sqrt{d x + c} b + a}}{x}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(sqrt(d*x + c)*b + a)/x,x, algorithm="maxima")

[Out]

integrate(sqrt(sqrt(d*x + c)*b + a)/x, x)

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Fricas [A]  time = 0.349505, size = 262, normalized size = 2.26 \[ -\sqrt{a + \sqrt{b^{2} c}} \log \left (2 \, \sqrt{\sqrt{d x + c} b + a} + 2 \, \sqrt{a + \sqrt{b^{2} c}}\right ) + \sqrt{a + \sqrt{b^{2} c}} \log \left (2 \, \sqrt{\sqrt{d x + c} b + a} - 2 \, \sqrt{a + \sqrt{b^{2} c}}\right ) - \sqrt{a - \sqrt{b^{2} c}} \log \left (2 \, \sqrt{\sqrt{d x + c} b + a} + 2 \, \sqrt{a - \sqrt{b^{2} c}}\right ) + \sqrt{a - \sqrt{b^{2} c}} \log \left (2 \, \sqrt{\sqrt{d x + c} b + a} - 2 \, \sqrt{a - \sqrt{b^{2} c}}\right ) + 4 \, \sqrt{\sqrt{d x + c} b + a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(sqrt(d*x + c)*b + a)/x,x, algorithm="fricas")

[Out]

-sqrt(a + sqrt(b^2*c))*log(2*sqrt(sqrt(d*x + c)*b + a) + 2*sqrt(a + sqrt(b^2*c))
) + sqrt(a + sqrt(b^2*c))*log(2*sqrt(sqrt(d*x + c)*b + a) - 2*sqrt(a + sqrt(b^2*
c))) - sqrt(a - sqrt(b^2*c))*log(2*sqrt(sqrt(d*x + c)*b + a) + 2*sqrt(a - sqrt(b
^2*c))) + sqrt(a - sqrt(b^2*c))*log(2*sqrt(sqrt(d*x + c)*b + a) - 2*sqrt(a - sqr
t(b^2*c))) + 4*sqrt(sqrt(d*x + c)*b + a)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{a + b \sqrt{c + d x}}}{x}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a+b*(d*x+c)**(1/2))**(1/2)/x,x)

[Out]

Integral(sqrt(a + b*sqrt(c + d*x))/x, x)

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GIAC/XCAS [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(sqrt(d*x + c)*b + a)/x,x, algorithm="giac")

[Out]

Exception raised: TypeError