3.469 \(\int \frac{\sqrt{a+b \sqrt{c+d x}}}{x^2} \, dx\)

Optimal. Leaf size=137 \[ -\frac{\sqrt{a+b \sqrt{c+d x}}}{x}+\frac{b d \tanh ^{-1}\left (\frac{\sqrt{a+b \sqrt{c+d x}}}{\sqrt{a-b \sqrt{c}}}\right )}{2 \sqrt{c} \sqrt{a-b \sqrt{c}}}-\frac{b d \tanh ^{-1}\left (\frac{\sqrt{a+b \sqrt{c+d x}}}{\sqrt{a+b \sqrt{c}}}\right )}{2 \sqrt{c} \sqrt{a+b \sqrt{c}}} \]

[Out]

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

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Rubi [A]  time = 0.349087, antiderivative size = 137, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ -\frac{\sqrt{a+b \sqrt{c+d x}}}{x}+\frac{b d \tanh ^{-1}\left (\frac{\sqrt{a+b \sqrt{c+d x}}}{\sqrt{a-b \sqrt{c}}}\right )}{2 \sqrt{c} \sqrt{a-b \sqrt{c}}}-\frac{b d \tanh ^{-1}\left (\frac{\sqrt{a+b \sqrt{c+d x}}}{\sqrt{a+b \sqrt{c}}}\right )}{2 \sqrt{c} \sqrt{a+b \sqrt{c}}} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[a + b*Sqrt[c + d*x]]/x^2,x]

[Out]

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

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Rubi in Sympy [A]  time = 23.0225, size = 117, normalized size = 0.85 \[ - \frac{b d \operatorname{atanh}{\left (\frac{\sqrt{a + b \sqrt{c + d x}}}{\sqrt{a + b \sqrt{c}}} \right )}}{2 \sqrt{c} \sqrt{a + b \sqrt{c}}} + \frac{b d \operatorname{atanh}{\left (\frac{\sqrt{a + b \sqrt{c + d x}}}{\sqrt{a - b \sqrt{c}}} \right )}}{2 \sqrt{c} \sqrt{a - b \sqrt{c}}} - \frac{\sqrt{a + b \sqrt{c + d x}}}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.555435, size = 144, normalized size = 1.05 \[ \frac{1}{2} \left (-\frac{2 \sqrt{a+b \sqrt{c+d x}}}{x}+\frac{b d \tan ^{-1}\left (\frac{\sqrt{a+b \sqrt{c+d x}}}{\sqrt{-a-b \sqrt{c}}}\right )}{\sqrt{c} \sqrt{-a-b \sqrt{c}}}-\frac{b d \tan ^{-1}\left (\frac{\sqrt{a+b \sqrt{c+d x}}}{\sqrt{b \sqrt{c}-a}}\right )}{\sqrt{c} \sqrt{b \sqrt{c}-a}}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[a + b*Sqrt[c + d*x]]/x^2,x]

[Out]

((-2*Sqrt[a + b*Sqrt[c + d*x]])/x + (b*d*ArcTan[Sqrt[a + b*Sqrt[c + d*x]]/Sqrt[-
a - b*Sqrt[c]]])/(Sqrt[-a - b*Sqrt[c]]*Sqrt[c]) - (b*d*ArcTan[Sqrt[a + b*Sqrt[c
+ d*x]]/Sqrt[-a + b*Sqrt[c]]])/(Sqrt[-a + b*Sqrt[c]]*Sqrt[c]))/2

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Maple [A]  time = 0.03, size = 151, normalized size = 1.1 \[ -{\frac{{b}^{2}d}{{b}^{2} \left ( dx+c \right ) -{b}^{2}c}\sqrt{a+b\sqrt{dx+c}}}-{\frac{{b}^{2}d}{2}\arctan \left ({1\sqrt{a+b\sqrt{dx+c}}{\frac{1}{\sqrt{\sqrt{{b}^{2}c}-a}}}} \right ){\frac{1}{\sqrt{{b}^{2}c}}}{\frac{1}{\sqrt{\sqrt{{b}^{2}c}-a}}}}+{\frac{{b}^{2}d}{2}\arctan \left ({1\sqrt{a+b\sqrt{dx+c}}{\frac{1}{\sqrt{-\sqrt{{b}^{2}c}-a}}}} \right ){\frac{1}{\sqrt{{b}^{2}c}}}{\frac{1}{\sqrt{-\sqrt{{b}^{2}c}-a}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-b^2*d*(a+b*(d*x+c)^(1/2))^(1/2)/(b^2*(d*x+c)-b^2*c)-1/2*b^2*d/(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))+1
/2*b^2*d/(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))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{\sqrt{d x + c} b + a}}{x^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(sqrt(d*x + c)*b + a)/x^2,x, algorithm="maxima")

[Out]

integrate(sqrt(sqrt(d*x + c)*b + a)/x^2, x)

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Fricas [A]  time = 0.366363, size = 1354, normalized size = 9.88 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(sqrt(d*x + c)*b + a)/x^2,x, algorithm="fricas")

[Out]

-1/4*(x*sqrt(-(a*b^2*d^2 + sqrt(b^6*d^4/(b^4*c^3 - 2*a^2*b^2*c^2 + a^4*c))*(b^2*
c^2 - a^2*c))/(b^2*c^2 - a^2*c))*log(sqrt(sqrt(d*x + c)*b + a)*b^4*d^3 + (b^4*c*
d^2 - sqrt(b^6*d^4/(b^4*c^3 - 2*a^2*b^2*c^2 + a^4*c))*(a*b^2*c^2 - a^3*c))*sqrt(
-(a*b^2*d^2 + sqrt(b^6*d^4/(b^4*c^3 - 2*a^2*b^2*c^2 + a^4*c))*(b^2*c^2 - a^2*c))
/(b^2*c^2 - a^2*c))) - x*sqrt(-(a*b^2*d^2 + sqrt(b^6*d^4/(b^4*c^3 - 2*a^2*b^2*c^
2 + a^4*c))*(b^2*c^2 - a^2*c))/(b^2*c^2 - a^2*c))*log(sqrt(sqrt(d*x + c)*b + a)*
b^4*d^3 - (b^4*c*d^2 - sqrt(b^6*d^4/(b^4*c^3 - 2*a^2*b^2*c^2 + a^4*c))*(a*b^2*c^
2 - a^3*c))*sqrt(-(a*b^2*d^2 + sqrt(b^6*d^4/(b^4*c^3 - 2*a^2*b^2*c^2 + a^4*c))*(
b^2*c^2 - a^2*c))/(b^2*c^2 - a^2*c))) + x*sqrt(-(a*b^2*d^2 - sqrt(b^6*d^4/(b^4*c
^3 - 2*a^2*b^2*c^2 + a^4*c))*(b^2*c^2 - a^2*c))/(b^2*c^2 - a^2*c))*log(sqrt(sqrt
(d*x + c)*b + a)*b^4*d^3 + (b^4*c*d^2 + sqrt(b^6*d^4/(b^4*c^3 - 2*a^2*b^2*c^2 +
a^4*c))*(a*b^2*c^2 - a^3*c))*sqrt(-(a*b^2*d^2 - sqrt(b^6*d^4/(b^4*c^3 - 2*a^2*b^
2*c^2 + a^4*c))*(b^2*c^2 - a^2*c))/(b^2*c^2 - a^2*c))) - x*sqrt(-(a*b^2*d^2 - sq
rt(b^6*d^4/(b^4*c^3 - 2*a^2*b^2*c^2 + a^4*c))*(b^2*c^2 - a^2*c))/(b^2*c^2 - a^2*
c))*log(sqrt(sqrt(d*x + c)*b + a)*b^4*d^3 - (b^4*c*d^2 + sqrt(b^6*d^4/(b^4*c^3 -
2*a^2*b^2*c^2 + a^4*c))*(a*b^2*c^2 - a^3*c))*sqrt(-(a*b^2*d^2 - sqrt(b^6*d^4/(b
^4*c^3 - 2*a^2*b^2*c^2 + a^4*c))*(b^2*c^2 - a^2*c))/(b^2*c^2 - a^2*c))) + 4*sqrt
(sqrt(d*x + c)*b + a))/x

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{a + b \sqrt{c + d x}}}{x^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(sqrt(a + b*sqrt(c + d*x))/x**2, 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^2,x, algorithm="giac")

[Out]

Exception raised: TypeError