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

Optimal. Leaf size=121 \[ -\frac{\sqrt{a d^2+c e^2} \tanh ^{-1}\left (\frac{a d-\frac{c e}{x}}{\sqrt{a+\frac{c}{x^2}} \sqrt{a d^2+c e^2}}\right )}{d e}-\frac{\sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c}}{x \sqrt{a+\frac{c}{x^2}}}\right )}{d}+\frac{\sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a+\frac{c}{x^2}}}{\sqrt{a}}\right )}{e} \]

[Out]

(Sqrt[a]*ArcTanh[Sqrt[a + c/x^2]/Sqrt[a]])/e - (Sqrt[a*d^2 + c*e^2]*ArcTanh[(a*d
- (c*e)/x)/(Sqrt[a*d^2 + c*e^2]*Sqrt[a + c/x^2])])/(d*e) - (Sqrt[c]*ArcTanh[Sqr
t[c]/(Sqrt[a + c/x^2]*x)])/d

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Rubi [A]  time = 0.466512, antiderivative size = 121, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 10, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.526 \[ -\frac{\sqrt{a d^2+c e^2} \tanh ^{-1}\left (\frac{a d-\frac{c e}{x}}{\sqrt{a+\frac{c}{x^2}} \sqrt{a d^2+c e^2}}\right )}{d e}-\frac{\sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c}}{x \sqrt{a+\frac{c}{x^2}}}\right )}{d}+\frac{\sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a+\frac{c}{x^2}}}{\sqrt{a}}\right )}{e} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[a + c/x^2]/(d + e*x),x]

[Out]

(Sqrt[a]*ArcTanh[Sqrt[a + c/x^2]/Sqrt[a]])/e - (Sqrt[a*d^2 + c*e^2]*ArcTanh[(a*d
- (c*e)/x)/(Sqrt[a*d^2 + c*e^2]*Sqrt[a + c/x^2])])/(d*e) - (Sqrt[c]*ArcTanh[Sqr
t[c]/(Sqrt[a + c/x^2]*x)])/d

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Rubi in Sympy [A]  time = 19.9262, size = 99, normalized size = 0.82 \[ \frac{\sqrt{a} \operatorname{atanh}{\left (\frac{\sqrt{a + \frac{c}{x^{2}}}}{\sqrt{a}} \right )}}{e} - \frac{\sqrt{c} \operatorname{atanh}{\left (\frac{\sqrt{c}}{x \sqrt{a + \frac{c}{x^{2}}}} \right )}}{d} - \frac{\sqrt{a d^{2} + c e^{2}} \operatorname{atanh}{\left (\frac{a d - \frac{c e}{x}}{\sqrt{a + \frac{c}{x^{2}}} \sqrt{a d^{2} + c e^{2}}} \right )}}{d e} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.163435, size = 173, normalized size = 1.43 \[ \frac{x \sqrt{a+\frac{c}{x^2}} \left (\sqrt{a d^2+c e^2} \log \left (\sqrt{a x^2+c} \sqrt{a d^2+c e^2}-a d x+c e\right )-\sqrt{a d^2+c e^2} \log (d+e x)+\sqrt{a} d \log \left (\sqrt{a} \sqrt{a x^2+c}+a x\right )-\sqrt{c} e \log \left (\sqrt{c} \sqrt{a x^2+c}+c\right )+\sqrt{c} e \log (x)\right )}{d e \sqrt{a x^2+c}} \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[a + c/x^2]/(d + e*x),x]

[Out]

(Sqrt[a + c/x^2]*x*(Sqrt[c]*e*Log[x] - Sqrt[a*d^2 + c*e^2]*Log[d + e*x] + Sqrt[a
]*d*Log[a*x + Sqrt[a]*Sqrt[c + a*x^2]] - Sqrt[c]*e*Log[c + Sqrt[c]*Sqrt[c + a*x^
2]] + Sqrt[a*d^2 + c*e^2]*Log[c*e - a*d*x + Sqrt[a*d^2 + c*e^2]*Sqrt[c + a*x^2]]
))/(d*e*Sqrt[c + a*x^2])

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Maple [B]  time = 0.042, size = 247, normalized size = 2. \[ -{\frac{x}{{e}^{2}d}\sqrt{{\frac{a{x}^{2}+c}{{x}^{2}}}} \left ( \sqrt{c}\ln \left ( 2\,{\frac{\sqrt{c}\sqrt{a{x}^{2}+c}+c}{x}} \right ){e}^{2}\sqrt{{\frac{a{d}^{2}+{e}^{2}c}{{e}^{2}}}}-\sqrt{a}\ln \left ({1 \left ( \sqrt{a{x}^{2}+c}\sqrt{a}+ax \right ){\frac{1}{\sqrt{a}}}} \right ) de\sqrt{{\frac{a{d}^{2}+{e}^{2}c}{{e}^{2}}}}-{d}^{2}\ln \left ( 2\,{\frac{1}{ex+d} \left ( \sqrt{a{x}^{2}+c}\sqrt{{\frac{a{d}^{2}+{e}^{2}c}{{e}^{2}}}}e-adx+ce \right ) } \right ) a-\ln \left ( 2\,{\frac{1}{ex+d} \left ( \sqrt{a{x}^{2}+c}\sqrt{{\frac{a{d}^{2}+{e}^{2}c}{{e}^{2}}}}e-adx+ce \right ) } \right ) c{e}^{2} \right ){\frac{1}{\sqrt{a{x}^{2}+c}}}{\frac{1}{\sqrt{{\frac{a{d}^{2}+{e}^{2}c}{{e}^{2}}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+c/x^2)^(1/2)/(e*x+d),x)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.03245, size = 1, normalized size = 0.01 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/2*(sqrt(a)*d*log(-2*a*x^2 - 2*sqrt(a)*x^2*sqrt((a*x^2 + c)/x^2) - c) + sqrt(c
)*e*log(-(a*x^2 - 2*sqrt(c)*x*sqrt((a*x^2 + c)/x^2) + 2*c)/x^2) + sqrt(a*d^2 + c
*e^2)*log((2*a*c*d*e*x - a*c*d^2 - 2*c^2*e^2 - (2*a^2*d^2 + a*c*e^2)*x^2 + 2*(a*
d*x^2 - c*e*x)*sqrt(a*d^2 + c*e^2)*sqrt((a*x^2 + c)/x^2))/(e^2*x^2 + 2*d*e*x + d
^2)))/(d*e), 1/2*(2*sqrt(-a)*d*arctan(a/(sqrt(-a)*sqrt((a*x^2 + c)/x^2))) + sqrt
(c)*e*log(-(a*x^2 - 2*sqrt(c)*x*sqrt((a*x^2 + c)/x^2) + 2*c)/x^2) + sqrt(a*d^2 +
c*e^2)*log((2*a*c*d*e*x - a*c*d^2 - 2*c^2*e^2 - (2*a^2*d^2 + a*c*e^2)*x^2 + 2*(
a*d*x^2 - c*e*x)*sqrt(a*d^2 + c*e^2)*sqrt((a*x^2 + c)/x^2))/(e^2*x^2 + 2*d*e*x +
d^2)))/(d*e), 1/2*(sqrt(a)*d*log(-2*a*x^2 - 2*sqrt(a)*x^2*sqrt((a*x^2 + c)/x^2)
- c) + sqrt(c)*e*log(-(a*x^2 - 2*sqrt(c)*x*sqrt((a*x^2 + c)/x^2) + 2*c)/x^2) -
2*sqrt(-a*d^2 - c*e^2)*arctan((a*d*x - c*e)/(sqrt(-a*d^2 - c*e^2)*x*sqrt((a*x^2
+ c)/x^2))))/(d*e), 1/2*(2*sqrt(-a)*d*arctan(a/(sqrt(-a)*sqrt((a*x^2 + c)/x^2)))
+ sqrt(c)*e*log(-(a*x^2 - 2*sqrt(c)*x*sqrt((a*x^2 + c)/x^2) + 2*c)/x^2) - 2*sqr
t(-a*d^2 - c*e^2)*arctan((a*d*x - c*e)/(sqrt(-a*d^2 - c*e^2)*x*sqrt((a*x^2 + c)/
x^2))))/(d*e), -1/2*(2*sqrt(-c)*e*arctan(c/(sqrt(-c)*x*sqrt((a*x^2 + c)/x^2))) -
sqrt(a)*d*log(-2*a*x^2 - 2*sqrt(a)*x^2*sqrt((a*x^2 + c)/x^2) - c) - sqrt(a*d^2
+ c*e^2)*log((2*a*c*d*e*x - a*c*d^2 - 2*c^2*e^2 - (2*a^2*d^2 + a*c*e^2)*x^2 + 2*
(a*d*x^2 - c*e*x)*sqrt(a*d^2 + c*e^2)*sqrt((a*x^2 + c)/x^2))/(e^2*x^2 + 2*d*e*x
+ d^2)))/(d*e), 1/2*(2*sqrt(-a)*d*arctan(a/(sqrt(-a)*sqrt((a*x^2 + c)/x^2))) - 2
*sqrt(-c)*e*arctan(c/(sqrt(-c)*x*sqrt((a*x^2 + c)/x^2))) + sqrt(a*d^2 + c*e^2)*l
og((2*a*c*d*e*x - a*c*d^2 - 2*c^2*e^2 - (2*a^2*d^2 + a*c*e^2)*x^2 + 2*(a*d*x^2 -
c*e*x)*sqrt(a*d^2 + c*e^2)*sqrt((a*x^2 + c)/x^2))/(e^2*x^2 + 2*d*e*x + d^2)))/(
d*e), -1/2*(2*sqrt(-c)*e*arctan(c/(sqrt(-c)*x*sqrt((a*x^2 + c)/x^2))) - sqrt(a)*
d*log(-2*a*x^2 - 2*sqrt(a)*x^2*sqrt((a*x^2 + c)/x^2) - c) + 2*sqrt(-a*d^2 - c*e^
2)*arctan((a*d*x - c*e)/(sqrt(-a*d^2 - c*e^2)*x*sqrt((a*x^2 + c)/x^2))))/(d*e),
(sqrt(-a)*d*arctan(a/(sqrt(-a)*sqrt((a*x^2 + c)/x^2))) - sqrt(-c)*e*arctan(c/(sq
rt(-c)*x*sqrt((a*x^2 + c)/x^2))) - sqrt(-a*d^2 - c*e^2)*arctan((a*d*x - c*e)/(sq
rt(-a*d^2 - c*e^2)*x*sqrt((a*x^2 + c)/x^2))))/(d*e)]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: TypeError