3.938 \(\int \frac{\sqrt{a+b x^2}}{x^3 \sqrt{c+d x^2}} \, dx\)

Optimal. Leaf size=89 \[ -\frac{(b c-a d) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x^2}}{\sqrt{a} \sqrt{c+d x^2}}\right )}{2 \sqrt{a} c^{3/2}}-\frac{\sqrt{a+b x^2} \sqrt{c+d x^2}}{2 c x^2} \]

[Out]

-(Sqrt[a + b*x^2]*Sqrt[c + d*x^2])/(2*c*x^2) - ((b*c - a*d)*ArcTanh[(Sqrt[c]*Sqr
t[a + b*x^2])/(Sqrt[a]*Sqrt[c + d*x^2])])/(2*Sqrt[a]*c^(3/2))

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Rubi [A]  time = 0.251465, antiderivative size = 89, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ -\frac{(b c-a d) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x^2}}{\sqrt{a} \sqrt{c+d x^2}}\right )}{2 \sqrt{a} c^{3/2}}-\frac{\sqrt{a+b x^2} \sqrt{c+d x^2}}{2 c x^2} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[a + b*x^2]/(x^3*Sqrt[c + d*x^2]),x]

[Out]

-(Sqrt[a + b*x^2]*Sqrt[c + d*x^2])/(2*c*x^2) - ((b*c - a*d)*ArcTanh[(Sqrt[c]*Sqr
t[a + b*x^2])/(Sqrt[a]*Sqrt[c + d*x^2])])/(2*Sqrt[a]*c^(3/2))

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Rubi in Sympy [A]  time = 20.945, size = 76, normalized size = 0.85 \[ - \frac{\sqrt{a + b x^{2}} \sqrt{c + d x^{2}}}{2 c x^{2}} + \frac{\left (a d - b c\right ) \operatorname{atanh}{\left (\frac{\sqrt{c} \sqrt{a + b x^{2}}}{\sqrt{a} \sqrt{c + d x^{2}}} \right )}}{2 \sqrt{a} c^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**2+a)**(1/2)/x**3/(d*x**2+c)**(1/2),x)

[Out]

-sqrt(a + b*x**2)*sqrt(c + d*x**2)/(2*c*x**2) + (a*d - b*c)*atanh(sqrt(c)*sqrt(a
+ b*x**2)/(sqrt(a)*sqrt(c + d*x**2)))/(2*sqrt(a)*c**(3/2))

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Mathematica [C]  time = 0.420173, size = 188, normalized size = 2.11 \[ \frac{\frac{2 b d x^4 (b c-a d) F_1\left (1;\frac{1}{2},\frac{1}{2};2;-\frac{a}{b x^2},-\frac{c}{d x^2}\right )}{-4 b d x^2 F_1\left (1;\frac{1}{2},\frac{1}{2};2;-\frac{a}{b x^2},-\frac{c}{d x^2}\right )+b c F_1\left (2;\frac{1}{2},\frac{3}{2};3;-\frac{a}{b x^2},-\frac{c}{d x^2}\right )+a d F_1\left (2;\frac{3}{2},\frac{1}{2};3;-\frac{a}{b x^2},-\frac{c}{d x^2}\right )}-\left (a+b x^2\right ) \left (c+d x^2\right )}{2 c x^2 \sqrt{a+b x^2} \sqrt{c+d x^2}} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[Sqrt[a + b*x^2]/(x^3*Sqrt[c + d*x^2]),x]

[Out]

(-((a + b*x^2)*(c + d*x^2)) + (2*b*d*(b*c - a*d)*x^4*AppellF1[1, 1/2, 1/2, 2, -(
a/(b*x^2)), -(c/(d*x^2))])/(-4*b*d*x^2*AppellF1[1, 1/2, 1/2, 2, -(a/(b*x^2)), -(
c/(d*x^2))] + b*c*AppellF1[2, 1/2, 3/2, 3, -(a/(b*x^2)), -(c/(d*x^2))] + a*d*App
ellF1[2, 3/2, 1/2, 3, -(a/(b*x^2)), -(c/(d*x^2))]))/(2*c*x^2*Sqrt[a + b*x^2]*Sqr
t[c + d*x^2])

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Maple [B]  time = 0.043, size = 207, normalized size = 2.3 \[{\frac{1}{4\,c{x}^{2}}\sqrt{b{x}^{2}+a}\sqrt{d{x}^{2}+c} \left ( \ln \left ({\frac{1}{{x}^{2}} \left ( ad{x}^{2}+c{x}^{2}b+2\,\sqrt{ac}\sqrt{bd{x}^{4}+ad{x}^{2}+c{x}^{2}b+ac}+2\,ac \right ) } \right ){x}^{2}ad-\ln \left ({\frac{1}{{x}^{2}} \left ( ad{x}^{2}+c{x}^{2}b+2\,\sqrt{ac}\sqrt{bd{x}^{4}+ad{x}^{2}+c{x}^{2}b+ac}+2\,ac \right ) } \right ){x}^{2}bc-2\,\sqrt{ac}\sqrt{bd{x}^{4}+ad{x}^{2}+c{x}^{2}b+ac} \right ){\frac{1}{\sqrt{ac}}}{\frac{1}{\sqrt{bd{x}^{4}+ad{x}^{2}+c{x}^{2}b+ac}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^2+a)^(1/2)/x^3/(d*x^2+c)^(1/2),x)

[Out]

1/4*(b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)/c*(ln((a*d*x^2+c*x^2*b+2*(a*c)^(1/2)*(b*d*x^
4+a*d*x^2+b*c*x^2+a*c)^(1/2)+2*a*c)/x^2)*x^2*a*d-ln((a*d*x^2+c*x^2*b+2*(a*c)^(1/
2)*(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)+2*a*c)/x^2)*x^2*b*c-2*(a*c)^(1/2)*(b*d*x^
4+a*d*x^2+b*c*x^2+a*c)^(1/2))/(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)/x^2/(a*c)^(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(b*x^2 + a)/(sqrt(d*x^2 + c)*x^3),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.278143, size = 1, normalized size = 0.01 \[ \left [-\frac{{\left (b c - a d\right )} x^{2} \log \left (\frac{4 \,{\left (2 \, a^{2} c^{2} +{\left (a b c^{2} + a^{2} c d\right )} x^{2}\right )} \sqrt{b x^{2} + a} \sqrt{d x^{2} + c} +{\left ({\left (b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2}\right )} x^{4} + 8 \, a^{2} c^{2} + 8 \,{\left (a b c^{2} + a^{2} c d\right )} x^{2}\right )} \sqrt{a c}}{x^{4}}\right ) + 4 \, \sqrt{b x^{2} + a} \sqrt{d x^{2} + c} \sqrt{a c}}{8 \, \sqrt{a c} c x^{2}}, -\frac{{\left (b c - a d\right )} x^{2} \arctan \left (\frac{{\left ({\left (b c + a d\right )} x^{2} + 2 \, a c\right )} \sqrt{-a c}}{2 \, \sqrt{b x^{2} + a} \sqrt{d x^{2} + c} a c}\right ) + 2 \, \sqrt{b x^{2} + a} \sqrt{d x^{2} + c} \sqrt{-a c}}{4 \, \sqrt{-a c} c x^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(b*x^2 + a)/(sqrt(d*x^2 + c)*x^3),x, algorithm="fricas")

[Out]

[-1/8*((b*c - a*d)*x^2*log((4*(2*a^2*c^2 + (a*b*c^2 + a^2*c*d)*x^2)*sqrt(b*x^2 +
a)*sqrt(d*x^2 + c) + ((b^2*c^2 + 6*a*b*c*d + a^2*d^2)*x^4 + 8*a^2*c^2 + 8*(a*b*
c^2 + a^2*c*d)*x^2)*sqrt(a*c))/x^4) + 4*sqrt(b*x^2 + a)*sqrt(d*x^2 + c)*sqrt(a*c
))/(sqrt(a*c)*c*x^2), -1/4*((b*c - a*d)*x^2*arctan(1/2*((b*c + a*d)*x^2 + 2*a*c)
*sqrt(-a*c)/(sqrt(b*x^2 + a)*sqrt(d*x^2 + c)*a*c)) + 2*sqrt(b*x^2 + a)*sqrt(d*x^
2 + c)*sqrt(-a*c))/(sqrt(-a*c)*c*x^2)]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{a + b x^{2}}}{x^{3} \sqrt{c + d x^{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x**2+a)**(1/2)/x**3/(d*x**2+c)**(1/2),x)

[Out]

Integral(sqrt(a + b*x**2)/(x**3*sqrt(c + d*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(b*x^2 + a)/(sqrt(d*x^2 + c)*x^3),x, algorithm="giac")

[Out]

Exception raised: TypeError