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

Optimal. Leaf size=92 \[ \frac{\sqrt{b} \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x^2}}{\sqrt{b} \sqrt{c+d x^2}}\right )}{\sqrt{d}}-\frac{\sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x^2}}{\sqrt{a} \sqrt{c+d x^2}}\right )}{\sqrt{c}} \]

[Out]

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

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Rubi [A]  time = 0.27373, antiderivative size = 92, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.192 \[ \frac{\sqrt{b} \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x^2}}{\sqrt{b} \sqrt{c+d x^2}}\right )}{\sqrt{d}}-\frac{\sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x^2}}{\sqrt{a} \sqrt{c+d x^2}}\right )}{\sqrt{c}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 26.6491, size = 83, normalized size = 0.9 \[ - \frac{\sqrt{a} \operatorname{atanh}{\left (\frac{\sqrt{c} \sqrt{a + b x^{2}}}{\sqrt{a} \sqrt{c + d x^{2}}} \right )}}{\sqrt{c}} + \frac{\sqrt{b} \operatorname{atanh}{\left (\frac{\sqrt{b} \sqrt{c + d x^{2}}}{\sqrt{d} \sqrt{a + b x^{2}}} \right )}}{\sqrt{d}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [C]  time = 0.554556, size = 238, normalized size = 2.59 \[ \frac{5 a \left (a+b x^2\right )^{3/2} (b c-a d) F_1\left (\frac{3}{2};\frac{1}{2},1;\frac{5}{2};\frac{d \left (b x^2+a\right )}{a d-b c},\frac{b x^2}{a}+1\right )}{3 b x^2 \sqrt{c+d x^2} \left (5 a (b c-a d) F_1\left (\frac{3}{2};\frac{1}{2},1;\frac{5}{2};\frac{d \left (b x^2+a\right )}{a d-b c},\frac{b x^2}{a}+1\right )-\left (a+b x^2\right ) \left ((2 a d-2 b c) F_1\left (\frac{5}{2};\frac{1}{2},2;\frac{7}{2};\frac{d \left (b x^2+a\right )}{a d-b c},\frac{b x^2}{a}+1\right )+a d F_1\left (\frac{5}{2};\frac{3}{2},1;\frac{7}{2};\frac{d \left (b x^2+a\right )}{a d-b c},\frac{b x^2}{a}+1\right )\right )\right )} \]

Warning: Unable to verify antiderivative.

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

[Out]

(5*a*(b*c - a*d)*(a + b*x^2)^(3/2)*AppellF1[3/2, 1/2, 1, 5/2, (d*(a + b*x^2))/(-
(b*c) + a*d), 1 + (b*x^2)/a])/(3*b*x^2*Sqrt[c + d*x^2]*(5*a*(b*c - a*d)*AppellF1
[3/2, 1/2, 1, 5/2, (d*(a + b*x^2))/(-(b*c) + a*d), 1 + (b*x^2)/a] - (a + b*x^2)*
((-2*b*c + 2*a*d)*AppellF1[5/2, 1/2, 2, 7/2, (d*(a + b*x^2))/(-(b*c) + a*d), 1 +
(b*x^2)/a] + a*d*AppellF1[5/2, 3/2, 1, 7/2, (d*(a + b*x^2))/(-(b*c) + a*d), 1 +
(b*x^2)/a])))

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Maple [B]  time = 0.036, size = 177, normalized size = 1.9 \[{\frac{1}{2}\sqrt{b{x}^{2}+a}\sqrt{d{x}^{2}+c} \left ( \ln \left ({\frac{1}{2} \left ( 2\,bd{x}^{2}+2\,\sqrt{bd{x}^{4}+ad{x}^{2}+c{x}^{2}b+ac}\sqrt{bd}+ad+bc \right ){\frac{1}{\sqrt{bd}}}} \right ) b\sqrt{ac}-a\ln \left ({\frac{1}{{x}^{2}} \left ( ad{x}^{2}+c{x}^{2}b+2\,\sqrt{ac}\sqrt{bd{x}^{4}+ad{x}^{2}+c{x}^{2}b+ac}+2\,ac \right ) } \right ) \sqrt{bd} \right ){\frac{1}{\sqrt{bd{x}^{4}+ad{x}^{2}+c{x}^{2}b+ac}}}{\frac{1}{\sqrt{bd}}}{\frac{1}{\sqrt{ac}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.356338, size = 1, normalized size = 0.01 \[ \left [\frac{1}{4} \, \sqrt{\frac{b}{d}} \log \left (8 \, b^{2} d^{2} x^{4} + b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2} + 8 \,{\left (b^{2} c d + a b d^{2}\right )} x^{2} + 4 \,{\left (2 \, b d^{2} x^{2} + b c d + a d^{2}\right )} \sqrt{b x^{2} + a} \sqrt{d x^{2} + c} \sqrt{\frac{b}{d}}\right ) + \frac{1}{4} \, \sqrt{\frac{a}{c}} \log \left (\frac{{\left (b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2}\right )} x^{4} + 8 \, a^{2} c^{2} + 8 \,{\left (a b c^{2} + a^{2} c d\right )} x^{2} - 4 \,{\left (2 \, a c^{2} +{\left (b c^{2} + a c d\right )} x^{2}\right )} \sqrt{b x^{2} + a} \sqrt{d x^{2} + c} \sqrt{\frac{a}{c}}}{x^{4}}\right ), \frac{1}{2} \, \sqrt{-\frac{b}{d}} \arctan \left (\frac{2 \, b d x^{2} + b c + a d}{2 \, \sqrt{b x^{2} + a} \sqrt{d x^{2} + c} d \sqrt{-\frac{b}{d}}}\right ) + \frac{1}{4} \, \sqrt{\frac{a}{c}} \log \left (\frac{{\left (b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2}\right )} x^{4} + 8 \, a^{2} c^{2} + 8 \,{\left (a b c^{2} + a^{2} c d\right )} x^{2} - 4 \,{\left (2 \, a c^{2} +{\left (b c^{2} + a c d\right )} x^{2}\right )} \sqrt{b x^{2} + a} \sqrt{d x^{2} + c} \sqrt{\frac{a}{c}}}{x^{4}}\right ), -\frac{1}{2} \, \sqrt{-\frac{a}{c}} \arctan \left (\frac{{\left (b c + a d\right )} x^{2} + 2 \, a c}{2 \, \sqrt{b x^{2} + a} \sqrt{d x^{2} + c} c \sqrt{-\frac{a}{c}}}\right ) + \frac{1}{4} \, \sqrt{\frac{b}{d}} \log \left (8 \, b^{2} d^{2} x^{4} + b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2} + 8 \,{\left (b^{2} c d + a b d^{2}\right )} x^{2} + 4 \,{\left (2 \, b d^{2} x^{2} + b c d + a d^{2}\right )} \sqrt{b x^{2} + a} \sqrt{d x^{2} + c} \sqrt{\frac{b}{d}}\right ), -\frac{1}{2} \, \sqrt{-\frac{a}{c}} \arctan \left (\frac{{\left (b c + a d\right )} x^{2} + 2 \, a c}{2 \, \sqrt{b x^{2} + a} \sqrt{d x^{2} + c} c \sqrt{-\frac{a}{c}}}\right ) + \frac{1}{2} \, \sqrt{-\frac{b}{d}} \arctan \left (\frac{2 \, b d x^{2} + b c + a d}{2 \, \sqrt{b x^{2} + a} \sqrt{d x^{2} + c} d \sqrt{-\frac{b}{d}}}\right )\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.249875, size = 208, normalized size = 2.26 \[ -\frac{b^{2}{\left (\frac{2 \, \sqrt{b d} a \arctan \left (-\frac{b^{2} c + a b d -{\left (\sqrt{b x^{2} + a} \sqrt{b d} - \sqrt{b^{2} c +{\left (b x^{2} + a\right )} b d - a b d}\right )}^{2}}{2 \, \sqrt{-a b c d} b}\right )}{\sqrt{-a b c d} b} + \frac{\sqrt{b d}{\rm ln}\left ({\left (\sqrt{b x^{2} + a} \sqrt{b d} - \sqrt{b^{2} c +{\left (b x^{2} + a\right )} b d - a b d}\right )}^{2}\right )}{b d}\right )}}{2 \,{\left | b \right |}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2*b^2*(2*sqrt(b*d)*a*arctan(-1/2*(b^2*c + a*b*d - (sqrt(b*x^2 + a)*sqrt(b*d)
- sqrt(b^2*c + (b*x^2 + a)*b*d - a*b*d))^2)/(sqrt(-a*b*c*d)*b))/(sqrt(-a*b*c*d)*
b) + sqrt(b*d)*ln((sqrt(b*x^2 + a)*sqrt(b*d) - sqrt(b^2*c + (b*x^2 + a)*b*d - a*
b*d))^2)/(b*d))/abs(b)