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

Optimal. Leaf size=110 \[ \sqrt{a+b x} \sqrt{c+d x}-2 \sqrt{a} \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )+\frac{(a d+b c) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{\sqrt{b} \sqrt{d}} \]

[Out]

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

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Rubi [A]  time = 0.265255, antiderivative size = 110, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227 \[ \sqrt{a+b x} \sqrt{c+d x}-2 \sqrt{a} \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )+\frac{(a d+b c) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{\sqrt{b} \sqrt{d}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 22.7546, size = 102, normalized size = 0.93 \[ - 2 \sqrt{a} \sqrt{c} \operatorname{atanh}{\left (\frac{\sqrt{c} \sqrt{a + b x}}{\sqrt{a} \sqrt{c + d x}} \right )} + \sqrt{a + b x} \sqrt{c + d x} + \frac{\left (a d + b c\right ) \operatorname{atanh}{\left (\frac{\sqrt{b} \sqrt{c + d x}}{\sqrt{d} \sqrt{a + b x}} \right )}}{\sqrt{b} \sqrt{d}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.167211, size = 153, normalized size = 1.39 \[ \sqrt{a+b x} \sqrt{c+d x}-\sqrt{a} \sqrt{c} \log \left (2 \sqrt{a} \sqrt{c} \sqrt{a+b x} \sqrt{c+d x}+2 a c+a d x+b c x\right )+\frac{(a d+b c) \log \left (2 \sqrt{b} \sqrt{d} \sqrt{a+b x} \sqrt{c+d x}+a d+b c+2 b d x\right )}{2 \sqrt{b} \sqrt{d}}+\sqrt{a} \sqrt{c} \log (x) \]

Antiderivative was successfully verified.

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

[Out]

Sqrt[a + b*x]*Sqrt[c + d*x] + Sqrt[a]*Sqrt[c]*Log[x] - Sqrt[a]*Sqrt[c]*Log[2*a*c
+ b*c*x + a*d*x + 2*Sqrt[a]*Sqrt[c]*Sqrt[a + b*x]*Sqrt[c + d*x]] + ((b*c + a*d)
*Log[b*c + a*d + 2*b*d*x + 2*Sqrt[b]*Sqrt[d]*Sqrt[a + b*x]*Sqrt[c + d*x]])/(2*Sq
rt[b]*Sqrt[d])

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Maple [B]  time = 0.016, size = 244, normalized size = 2.2 \[{\frac{1}{2}\sqrt{bx+a}\sqrt{dx+c} \left ( \sqrt{ac}\ln \left ({\frac{1}{2} \left ( 2\,bdx+2\,\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}+ad+bc \right ){\frac{1}{\sqrt{bd}}}} \right ) ad+\sqrt{ac}\ln \left ({\frac{1}{2} \left ( 2\,bdx+2\,\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}+ad+bc \right ){\frac{1}{\sqrt{bd}}}} \right ) bc-2\,ac\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}+2\,ac}{x}} \right ) \sqrt{bd}+2\,\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}\sqrt{ac} \right ){\frac{1}{\sqrt{d{x}^{2}b+adx+bcx+ac}}}{\frac{1}{\sqrt{bd}}}{\frac{1}{\sqrt{ac}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.5504, 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(b*x + a)*sqrt(d*x + c)/x,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)) + 2*sqrt(a*c)*sqrt(b*d)*log((8*a^2*c^2 + (b^2*c^2 + 6*a*b*c*d + a^2
*d^2)*x^2 - 4*(2*a*c + (b*c + a*d)*x)*sqrt(a*c)*sqrt(b*x + a)*sqrt(d*x + c) + 8*
(a*b*c^2 + a^2*c*d)*x)/x^2) + 4*sqrt(b*d)*sqrt(b*x + a)*sqrt(d*x + c))/sqrt(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)*sq
rt(d*x + c)*b*d)) + sqrt(a*c)*sqrt(-b*d)*log((8*a^2*c^2 + (b^2*c^2 + 6*a*b*c*d +
a^2*d^2)*x^2 - 4*(2*a*c + (b*c + a*d)*x)*sqrt(a*c)*sqrt(b*x + a)*sqrt(d*x + c)
+ 8*(a*b*c^2 + a^2*c*d)*x)/x^2) + 2*sqrt(-b*d)*sqrt(b*x + a)*sqrt(d*x + c))/sqrt
(-b*d), -1/4*(4*sqrt(-a*c)*sqrt(b*d)*arctan(1/2*(2*a*c + (b*c + a*d)*x)/(sqrt(-a
*c)*sqrt(b*x + a)*sqrt(d*x + c))) - (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), -1/2*(2*sqrt(-a*c)*sqrt(-b*d)*arctan(1/2*(2*a*c + (b*c + a*d)*x
)/(sqrt(-a*c)*sqrt(b*x + a)*sqrt(d*x + c))) - (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)]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.235657, size = 246, normalized size = 2.24 \[ -\frac{{\left (\frac{4 \, \sqrt{b d} a b c \arctan \left (-\frac{b^{2} c + a b d -{\left (\sqrt{b d} \sqrt{b x + a} - \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d}\right )}^{2}}{2 \, \sqrt{-a b c d} b}\right )}{\sqrt{-a b c d}} - 2 \, \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d} \sqrt{b x + a} + \frac{{\left (\sqrt{b d} b c + \sqrt{b d} a d\right )}{\rm ln}\left ({\left (\sqrt{b d} \sqrt{b x + a} - \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d}\right )}^{2}\right )}{d}\right )}{\left | b \right |}}{2 \, b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2*(4*sqrt(b*d)*a*b*c*arctan(-1/2*(b^2*c + a*b*d - (sqrt(b*d)*sqrt(b*x + a) -
sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2)/(sqrt(-a*b*c*d)*b))/sqrt(-a*b*c*d) - 2*s
qrt(b^2*c + (b*x + a)*b*d - a*b*d)*sqrt(b*x + a) + (sqrt(b*d)*b*c + sqrt(b*d)*a*
d)*ln((sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2)/d)*abs(
b)/b^2