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

Optimal. Leaf size=75 \[ \frac{\sqrt{a+b x} \sqrt{c+d x}}{b d}-\frac{(a d+b c) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{b^{3/2} d^{3/2}} \]

[Out]

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

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Rubi [A]  time = 0.106902, antiderivative size = 75, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15 \[ \frac{\sqrt{a+b x} \sqrt{c+d x}}{b d}-\frac{(a d+b c) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{b^{3/2} d^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 10.13, size = 65, normalized size = 0.87 \[ \frac{\sqrt{a + b x} \sqrt{c + d x}}{b d} - \frac{\left (a d + b c\right ) \operatorname{atanh}{\left (\frac{\sqrt{d} \sqrt{a + b x}}{\sqrt{b} \sqrt{c + d x}} \right )}}{b^{\frac{3}{2}} d^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.075899, size = 90, normalized size = 1.2 \[ \frac{\sqrt{a+b x} \sqrt{c+d x}}{b d}-\frac{(a d+b c) \log \left (2 \sqrt{b} \sqrt{d} \sqrt{a+b x} \sqrt{c+d x}+a d+b c+2 b d x\right )}{2 b^{3/2} d^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.026, size = 148, normalized size = 2. \[ -{\frac{1}{2\,bd} \left ( \ln \left ({\frac{1}{2} \left ( 2\,bdx+2\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}+ad+bc \right ){\frac{1}{\sqrt{bd}}}} \right ) ad+\ln \left ({\frac{1}{2} \left ( 2\,bdx+2\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}+ad+bc \right ){\frac{1}{\sqrt{bd}}}} \right ) bc-2\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd} \right ) \sqrt{bx+a}\sqrt{dx+c}{\frac{1}{\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }}}{\frac{1}{\sqrt{bd}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.261238, size = 1, normalized size = 0.01 \[ \left [\frac{{\left (b c + a d\right )} \log \left (-4 \,{\left (2 \, b^{2} d^{2} x + b^{2} c d + a b d^{2}\right )} \sqrt{b x + a} \sqrt{d x + c} +{\left (8 \, b^{2} d^{2} x^{2} + b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2} + 8 \,{\left (b^{2} c d + a b d^{2}\right )} x\right )} \sqrt{b d}\right ) + 4 \, \sqrt{b d} \sqrt{b x + a} \sqrt{d x + c}}{4 \, \sqrt{b d} b d}, -\frac{{\left (b c + a d\right )} \arctan \left (\frac{{\left (2 \, b d x + b c + a d\right )} \sqrt{-b d}}{2 \, \sqrt{b x + a} \sqrt{d x + c} b d}\right ) - 2 \, \sqrt{-b d} \sqrt{b x + a} \sqrt{d x + c}}{2 \, \sqrt{-b d} b d}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.238393, size = 128, normalized size = 1.71 \[ \frac{\frac{{\left (b c + a d\right )}{\rm ln}\left ({\left | -\sqrt{b d} \sqrt{b x + a} + \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d} \right |}\right )}{\sqrt{b d} d} + \frac{\sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d} \sqrt{b x + a}}{b d}}{{\left | b \right |}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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