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

Optimal. Leaf size=86 \[ \frac{\sqrt{a+b x^2} \sqrt{c+d x^2}}{2 d}-\frac{(b c-a d) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x^2}}{\sqrt{b} \sqrt{c+d x^2}}\right )}{2 \sqrt{b} d^{3/2}} \]

[Out]

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

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Rubi [A]  time = 0.177337, antiderivative size = 86, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ \frac{\sqrt{a+b x^2} \sqrt{c+d x^2}}{2 d}-\frac{(b c-a d) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x^2}}{\sqrt{b} \sqrt{c+d x^2}}\right )}{2 \sqrt{b} d^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 18.1603, size = 73, normalized size = 0.85 \[ \frac{\sqrt{a + b x^{2}} \sqrt{c + d x^{2}}}{2 d} + \frac{\left (a d - b c\right ) \operatorname{atanh}{\left (\frac{\sqrt{b} \sqrt{c + d x^{2}}}{\sqrt{d} \sqrt{a + b x^{2}}} \right )}}{2 \sqrt{b} d^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.086989, size = 101, normalized size = 1.17 \[ \frac{(a d-b c) \log \left (2 \sqrt{b} \sqrt{d} \sqrt{a+b x^2} \sqrt{c+d x^2}+a d+b c+2 b d x^2\right )}{4 \sqrt{b} d^{3/2}}+\frac{\sqrt{a+b x^2} \sqrt{c+d x^2}}{2 d} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[a + b*x^2]*Sqrt[c + d*x^2])/(2*d) + ((-(b*c) + a*d)*Log[b*c + a*d + 2*b*d*
x^2 + 2*Sqrt[b]*Sqrt[d]*Sqrt[a + b*x^2]*Sqrt[c + d*x^2]])/(4*Sqrt[b]*d^(3/2))

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Maple [B]  time = 0.014, size = 198, normalized size = 2.3 \[{\frac{1}{4\,d}\sqrt{b{x}^{2}+a}\sqrt{d{x}^{2}+c} \left ( a\ln \left ({\frac{1}{2} \left ( 2\,bd{x}^{2}+2\,\sqrt{bd{x}^{4}+ad{x}^{2}+c{x}^{2}b+ac}\sqrt{bd}+ad+bc \right ){\frac{1}{\sqrt{bd}}}} \right ) d-b\ln \left ({\frac{1}{2} \left ( 2\,bd{x}^{2}+2\,\sqrt{bd{x}^{4}+ad{x}^{2}+c{x}^{2}b+ac}\sqrt{bd}+ad+bc \right ){\frac{1}{\sqrt{bd}}}} \right ) c+2\,\sqrt{bd{x}^{4}+ad{x}^{2}+c{x}^{2}b+ac}\sqrt{bd} \right ){\frac{1}{\sqrt{bd{x}^{4}+ad{x}^{2}+c{x}^{2}b+ac}}}{\frac{1}{\sqrt{bd}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.250387, size = 1, normalized size = 0.01 \[ \left [-\frac{{\left (b c - a d\right )} \log \left (4 \,{\left (2 \, b^{2} d^{2} x^{2} + b^{2} c d + a b d^{2}\right )} \sqrt{b x^{2} + a} \sqrt{d x^{2} + c} +{\left (8 \, b^{2} d^{2} x^{4} + b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2} + 8 \,{\left (b^{2} c d + a b d^{2}\right )} x^{2}\right )} \sqrt{b d}\right ) - 4 \, \sqrt{b x^{2} + a} \sqrt{d x^{2} + c} \sqrt{b d}}{8 \, \sqrt{b d} d}, -\frac{{\left (b c - a d\right )} \arctan \left (\frac{{\left (2 \, b d x^{2} + b c + a d\right )} \sqrt{-b d}}{2 \, \sqrt{b x^{2} + a} \sqrt{d x^{2} + c} b d}\right ) - 2 \, \sqrt{b x^{2} + a} \sqrt{d x^{2} + c} \sqrt{-b d}}{4 \, \sqrt{-b d} d}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.249572, size = 143, normalized size = 1.66 \[ \frac{b{\left (\frac{{\left (b c - a d\right )}{\rm ln}\left ({\left | -\sqrt{b x^{2} + a} \sqrt{b d} + \sqrt{b^{2} c +{\left (b x^{2} + a\right )} b d - a b d} \right |}\right )}{\sqrt{b d} d} + \frac{\sqrt{b^{2} c +{\left (b x^{2} + a\right )} b d - a b d} \sqrt{b x^{2} + a}}{b d}\right )}}{2 \,{\left | b \right |}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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