3.541 \(\int \frac{\sqrt{a+b x} \sqrt{c+d x}}{x^3} \, dx\)

Optimal. Leaf size=122 \[ \frac{(b c-a d)^2 \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{4 a^{3/2} c^{3/2}}-\frac{\sqrt{a+b x} (c+d x)^{3/2}}{2 c x^2}-\frac{\sqrt{a+b x} \sqrt{c+d x} (b c-a d)}{4 a c x} \]

[Out]

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

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Rubi [A]  time = 0.208999, antiderivative size = 122, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136 \[ \frac{(b c-a d)^2 \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{4 a^{3/2} c^{3/2}}-\frac{\sqrt{a+b x} (c+d x)^{3/2}}{2 c x^2}-\frac{\sqrt{a+b x} \sqrt{c+d x} (b c-a d)}{4 a c x} \]

Antiderivative was successfully verified.

[In]  Int[(Sqrt[a + b*x]*Sqrt[c + d*x])/x^3,x]

[Out]

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

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Rubi in Sympy [A]  time = 16.6825, size = 102, normalized size = 0.84 \[ - \frac{\sqrt{a + b x} \left (c + d x\right )^{\frac{3}{2}}}{2 c x^{2}} + \frac{\sqrt{a + b x} \sqrt{c + d x} \left (a d - b c\right )}{4 a c x} + \frac{\left (a d - b c\right )^{2} \operatorname{atanh}{\left (\frac{\sqrt{c} \sqrt{a + b x}}{\sqrt{a} \sqrt{c + d x}} \right )}}{4 a^{\frac{3}{2}} c^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**(1/2)*(d*x+c)**(1/2)/x**3,x)

[Out]

-sqrt(a + b*x)*(c + d*x)**(3/2)/(2*c*x**2) + sqrt(a + b*x)*sqrt(c + d*x)*(a*d -
b*c)/(4*a*c*x) + (a*d - b*c)**2*atanh(sqrt(c)*sqrt(a + b*x)/(sqrt(a)*sqrt(c + d*
x)))/(4*a**(3/2)*c**(3/2))

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Mathematica [A]  time = 0.106433, size = 136, normalized size = 1.11 \[ \frac{x^2 \log (x) \left (-(b c-a d)^2\right )+x^2 (b c-a d)^2 \log \left (2 \sqrt{a} \sqrt{c} \sqrt{a+b x} \sqrt{c+d x}+2 a c+a d x+b c x\right )-2 \sqrt{a} \sqrt{c} \sqrt{a+b x} \sqrt{c+d x} (2 a c+a d x+b c x)}{8 a^{3/2} c^{3/2} x^2} \]

Antiderivative was successfully verified.

[In]  Integrate[(Sqrt[a + b*x]*Sqrt[c + d*x])/x^3,x]

[Out]

(-2*Sqrt[a]*Sqrt[c]*Sqrt[a + b*x]*Sqrt[c + d*x]*(2*a*c + b*c*x + a*d*x) - (b*c -
a*d)^2*x^2*Log[x] + (b*c - a*d)^2*x^2*Log[2*a*c + b*c*x + a*d*x + 2*Sqrt[a]*Sqr
t[c]*Sqrt[a + b*x]*Sqrt[c + d*x]])/(8*a^(3/2)*c^(3/2)*x^2)

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Maple [B]  time = 0.018, size = 305, normalized size = 2.5 \[{\frac{1}{8\,ac{x}^{2}}\sqrt{bx+a}\sqrt{dx+c} \left ( \ln \left ({\frac{1}{x} \left ( adx+bcx+2\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}+2\,ac \right ) } \right ){x}^{2}{a}^{2}{d}^{2}-2\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}+2\,ac}{x}} \right ){x}^{2}abcd+\ln \left ({\frac{1}{x} \left ( adx+bcx+2\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}+2\,ac \right ) } \right ){x}^{2}{b}^{2}{c}^{2}-2\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}xad-2\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}xbc-4\,\sqrt{d{x}^{2}b+adx+bcx+ac}ac\sqrt{ac} \right ){\frac{1}{\sqrt{ac}}}{\frac{1}{\sqrt{d{x}^{2}b+adx+bcx+ac}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^(1/2)*(d*x+c)^(1/2)/x^3,x)

[Out]

1/8*(b*x+a)^(1/2)*(d*x+c)^(1/2)/a/c*(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^2*a^2*d^2-2*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^2*a*b*c*d+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^2*b^2*c^2-2*(a*c)^(1/2)*(b*d*x^2+a*d*x+
b*c*x+a*c)^(1/2)*x*a*d-2*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*x*b*c-4*(b*
d*x^2+a*d*x+b*c*x+a*c)^(1/2)*a*c*(a*c)^(1/2))/(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)/x^
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^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.311441, size = 1, normalized size = 0.01 \[ \left [\frac{{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} x^{2} \log \left (\frac{4 \,{\left (2 \, a^{2} c^{2} +{\left (a b c^{2} + a^{2} c d\right )} x\right )} \sqrt{b x + a} \sqrt{d x + c} +{\left (8 \, a^{2} c^{2} +{\left (b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2}\right )} x^{2} + 8 \,{\left (a b c^{2} + a^{2} c d\right )} x\right )} \sqrt{a c}}{x^{2}}\right ) - 4 \,{\left (2 \, a c +{\left (b c + a d\right )} x\right )} \sqrt{a c} \sqrt{b x + a} \sqrt{d x + c}}{16 \, \sqrt{a c} a c x^{2}}, \frac{{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} x^{2} \arctan \left (\frac{{\left (2 \, a c +{\left (b c + a d\right )} x\right )} \sqrt{-a c}}{2 \, \sqrt{b x + a} \sqrt{d x + c} a c}\right ) - 2 \,{\left (2 \, a c +{\left (b c + a d\right )} x\right )} \sqrt{-a c} \sqrt{b x + a} \sqrt{d x + c}}{8 \, \sqrt{-a c} a c x^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(b*x + a)*sqrt(d*x + c)/x^3,x, algorithm="fricas")

[Out]

[1/16*((b^2*c^2 - 2*a*b*c*d + a^2*d^2)*x^2*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*(2*a*c + (b*c + a*d)*x)*sqrt
(a*c)*sqrt(b*x + a)*sqrt(d*x + c))/(sqrt(a*c)*a*c*x^2), 1/8*((b^2*c^2 - 2*a*b*c*
d + a^2*d^2)*x^2*arctan(1/2*(2*a*c + (b*c + a*d)*x)*sqrt(-a*c)/(sqrt(b*x + a)*sq
rt(d*x + c)*a*c)) - 2*(2*a*c + (b*c + a*d)*x)*sqrt(-a*c)*sqrt(b*x + a)*sqrt(d*x
+ c))/(sqrt(-a*c)*a*c*x^2)]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{a + b x} \sqrt{c + d x}}{x^{3}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x+a)**(1/2)*(d*x+c)**(1/2)/x**3,x)

[Out]

Integral(sqrt(a + b*x)*sqrt(c + d*x)/x**3, x)

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GIAC/XCAS [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(b*x + a)*sqrt(d*x + c)/x^3,x, algorithm="giac")

[Out]

Exception raised: TypeError