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

Optimal. Leaf size=115 \[ -\frac{\sqrt{a+b x} \sqrt{c+d x}}{x}-\frac{(a d+b c) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{\sqrt{a} \sqrt{c}}+2 \sqrt{b} \sqrt{d} \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right ) \]

[Out]

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

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Rubi [A]  time = 0.252887, antiderivative size = 115, 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 \[ -\frac{\sqrt{a+b x} \sqrt{c+d x}}{x}-\frac{(a d+b c) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{\sqrt{a} \sqrt{c}}+2 \sqrt{b} \sqrt{d} \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right ) \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 25.9541, size = 104, normalized size = 0.9 \[ 2 \sqrt{b} \sqrt{d} \operatorname{atanh}{\left (\frac{\sqrt{d} \sqrt{a + b x}}{\sqrt{b} \sqrt{c + d x}} \right )} - \frac{\sqrt{a + b x} \sqrt{c + d x}}{x} - \frac{\left (a d + b c\right ) \operatorname{atanh}{\left (\frac{\sqrt{c} \sqrt{a + b x}}{\sqrt{a} \sqrt{c + d x}} \right )}}{\sqrt{a} \sqrt{c}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.128864, size = 170, normalized size = 1.48 \[ -\frac{\sqrt{a+b x} \sqrt{c+d x}}{x}-\frac{\log (x) (-a d-b c)}{2 \sqrt{a} \sqrt{c}}+\frac{(-a d-b c) \log \left (2 \sqrt{a} \sqrt{c} \sqrt{a+b x} \sqrt{c+d x}+2 a c+a d x+b c x\right )}{2 \sqrt{a} \sqrt{c}}+\sqrt{b} \sqrt{d} \log \left (2 \sqrt{b} \sqrt{d} \sqrt{a+b x} \sqrt{c+d x}+a d+b c+2 b d x\right ) \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.017, size = 250, normalized size = 2.2 \[{\frac{1}{2\,x}\sqrt{bx+a}\sqrt{dx+c} \left ( 2\,\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ) xbd\sqrt{ac}-\ln \left ({\frac{1}{x} \left ( adx+bcx+2\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}+2\,ac \right ) } \right ) xad\sqrt{bd}-\ln \left ({\frac{1}{x} \left ( adx+bcx+2\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}+2\,ac \right ) } \right ) xbc\sqrt{bd}-2\,\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}\sqrt{ac} \right ){\frac{1}{\sqrt{d{x}^{2}b+adx+bcx+ac}}}{\frac{1}{\sqrt{bd}}}{\frac{1}{\sqrt{ac}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.392501, 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^2,x, algorithm="fricas")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.537708, size = 4, normalized size = 0.03 \[ \mathit{sage}_{0} x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sage0*x