3.542 \(\int \frac{\sqrt{a+b x} \sqrt{c+d x}}{x^4} \, dx\)

Optimal. Leaf size=172 \[ -\frac{(a d+b c) (b c-a d)^2 \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{8 a^{5/2} c^{5/2}}+\frac{\sqrt{a+b x} \sqrt{c+d x} \left (\frac{b^2}{a^2}-\frac{d^2}{c^2}\right )}{8 x}+\frac{\sqrt{a+b x} (c+d x)^{3/2} (a d+b c)}{4 a c^2 x^2}-\frac{(a+b x)^{3/2} (c+d x)^{3/2}}{3 a c x^3} \]

[Out]

((b^2/a^2 - d^2/c^2)*Sqrt[a + b*x]*Sqrt[c + d*x])/(8*x) + ((b*c + a*d)*Sqrt[a +
b*x]*(c + d*x)^(3/2))/(4*a*c^2*x^2) - ((a + b*x)^(3/2)*(c + d*x)^(3/2))/(3*a*c*x
^3) - ((b*c - a*d)^2*(b*c + a*d)*ArcTanh[(Sqrt[c]*Sqrt[a + b*x])/(Sqrt[a]*Sqrt[c
+ d*x])])/(8*a^(5/2)*c^(5/2))

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Rubi [A]  time = 0.319927, antiderivative size = 172, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ -\frac{(a d+b c) (b c-a d)^2 \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{8 a^{5/2} c^{5/2}}+\frac{\sqrt{a+b x} \sqrt{c+d x} \left (\frac{b^2}{a^2}-\frac{d^2}{c^2}\right )}{8 x}+\frac{\sqrt{a+b x} (c+d x)^{3/2} (a d+b c)}{4 a c^2 x^2}-\frac{(a+b x)^{3/2} (c+d x)^{3/2}}{3 a c x^3} \]

Antiderivative was successfully verified.

[In]  Int[(Sqrt[a + b*x]*Sqrt[c + d*x])/x^4,x]

[Out]

((b^2/a^2 - d^2/c^2)*Sqrt[a + b*x]*Sqrt[c + d*x])/(8*x) + ((b*c + a*d)*Sqrt[a +
b*x]*(c + d*x)^(3/2))/(4*a*c^2*x^2) - ((a + b*x)^(3/2)*(c + d*x)^(3/2))/(3*a*c*x
^3) - ((b*c - a*d)^2*(b*c + a*d)*ArcTanh[(Sqrt[c]*Sqrt[a + b*x])/(Sqrt[a]*Sqrt[c
+ d*x])])/(8*a^(5/2)*c^(5/2))

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Rubi in Sympy [A]  time = 25.5729, size = 150, normalized size = 0.87 \[ - \frac{\sqrt{a + b x} \sqrt{c + d x} \left (- \frac{d^{2}}{8 c^{2}} + \frac{b^{2}}{8 a^{2}}\right )}{x} - \frac{\left (a + b x\right )^{\frac{3}{2}} \left (c + d x\right )^{\frac{3}{2}}}{3 a c x^{3}} + \frac{\left (a + b x\right )^{\frac{3}{2}} \sqrt{c + d x} \left (a d + b c\right )}{4 a^{2} c x^{2}} - \frac{\left (a d - b c\right )^{2} \left (a d + b c\right ) \operatorname{atanh}{\left (\frac{\sqrt{c} \sqrt{a + b x}}{\sqrt{a} \sqrt{c + d x}} \right )}}{8 a^{\frac{5}{2}} c^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**(1/2)*(d*x+c)**(1/2)/x**4,x)

[Out]

-sqrt(a + b*x)*sqrt(c + d*x)*(-d**2/(8*c**2) + b**2/(8*a**2))/x - (a + b*x)**(3/
2)*(c + d*x)**(3/2)/(3*a*c*x**3) + (a + b*x)**(3/2)*sqrt(c + d*x)*(a*d + b*c)/(4
*a**2*c*x**2) - (a*d - b*c)**2*(a*d + b*c)*atanh(sqrt(c)*sqrt(a + b*x)/(sqrt(a)*
sqrt(c + d*x)))/(8*a**(5/2)*c**(5/2))

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Mathematica [A]  time = 0.180227, size = 184, normalized size = 1.07 \[ \frac{-2 \sqrt{a} \sqrt{c} \sqrt{a+b x} \sqrt{c+d x} \left (a^2 \left (8 c^2+2 c d x-3 d^2 x^2\right )+2 a b c x (c+d x)-3 b^2 c^2 x^2\right )+3 x^3 \log (x) (b c-a d)^2 (a d+b c)-3 x^3 (b c-a d)^2 (a d+b c) \log \left (2 \sqrt{a} \sqrt{c} \sqrt{a+b x} \sqrt{c+d x}+2 a c+a d x+b c x\right )}{48 a^{5/2} c^{5/2} x^3} \]

Antiderivative was successfully verified.

[In]  Integrate[(Sqrt[a + b*x]*Sqrt[c + d*x])/x^4,x]

[Out]

(-2*Sqrt[a]*Sqrt[c]*Sqrt[a + b*x]*Sqrt[c + d*x]*(-3*b^2*c^2*x^2 + 2*a*b*c*x*(c +
d*x) + a^2*(8*c^2 + 2*c*d*x - 3*d^2*x^2)) + 3*(b*c - a*d)^2*(b*c + a*d)*x^3*Log
[x] - 3*(b*c - a*d)^2*(b*c + a*d)*x^3*Log[2*a*c + b*c*x + a*d*x + 2*Sqrt[a]*Sqrt
[c]*Sqrt[a + b*x]*Sqrt[c + d*x]])/(48*a^(5/2)*c^(5/2)*x^3)

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Maple [B]  time = 0.021, size = 485, normalized size = 2.8 \[ -{\frac{1}{48\,{a}^{2}{c}^{2}{x}^{3}}\sqrt{bx+a}\sqrt{dx+c} \left ( 3\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}+2\,ac}{x}} \right ){x}^{3}{a}^{3}{d}^{3}-3\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}+2\,ac}{x}} \right ){x}^{3}{a}^{2}bc{d}^{2}-3\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}+2\,ac}{x}} \right ){x}^{3}a{b}^{2}{c}^{2}d+3\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}+2\,ac}{x}} \right ){x}^{3}{b}^{3}{c}^{3}-6\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}{x}^{2}{a}^{2}{d}^{2}+4\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}{x}^{2}abcd-6\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}{x}^{2}{b}^{2}{c}^{2}+4\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}x{a}^{2}cd+4\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}xab{c}^{2}+16\,\sqrt{d{x}^{2}b+adx+bcx+ac}{a}^{2}{c}^{2}\sqrt{ac} \right ){\frac{1}{\sqrt{ac}}}{\frac{1}{\sqrt{d{x}^{2}b+adx+bcx+ac}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^(1/2)*(d*x+c)^(1/2)/x^4,x)

[Out]

-1/48*(b*x+a)^(1/2)*(d*x+c)^(1/2)/a^2/c^2*(3*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^3*a^3*d^3-3*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^3*a^2*b*c*d^2-3*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^3*a*b^2*c^2*d+3*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^3*b^3*c^3-6*(a*
c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*x^2*a^2*d^2+4*(a*c)^(1/2)*(b*d*x^2+a*d*
x+b*c*x+a*c)^(1/2)*x^2*a*b*c*d-6*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*x^2
*b^2*c^2+4*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*x*a^2*c*d+4*(a*c)^(1/2)*(
b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*x*a*b*c^2+16*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*a^2*
c^2*(a*c)^(1/2))/(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)/x^3/(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^4,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.409841, size = 1, normalized size = 0.01 \[ \left [\frac{3 \,{\left (b^{3} c^{3} - a b^{2} c^{2} d - a^{2} b c d^{2} + a^{3} d^{3}\right )} x^{3} \log \left (-\frac{4 \,{\left (2 \, a^{2} c^{2} +{\left (a b c^{2} + a^{2} c d\right )} x\right )} \sqrt{b x + a} \sqrt{d x + c} -{\left (8 \, a^{2} c^{2} +{\left (b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2}\right )} x^{2} + 8 \,{\left (a b c^{2} + a^{2} c d\right )} x\right )} \sqrt{a c}}{x^{2}}\right ) - 4 \,{\left (8 \, a^{2} c^{2} -{\left (3 \, b^{2} c^{2} - 2 \, a b c d + 3 \, a^{2} d^{2}\right )} x^{2} + 2 \,{\left (a b c^{2} + a^{2} c d\right )} x\right )} \sqrt{a c} \sqrt{b x + a} \sqrt{d x + c}}{96 \, \sqrt{a c} a^{2} c^{2} x^{3}}, -\frac{3 \,{\left (b^{3} c^{3} - a b^{2} c^{2} d - a^{2} b c d^{2} + a^{3} d^{3}\right )} x^{3} \arctan \left (\frac{{\left (2 \, a c +{\left (b c + a d\right )} x\right )} \sqrt{-a c}}{2 \, \sqrt{b x + a} \sqrt{d x + c} a c}\right ) + 2 \,{\left (8 \, a^{2} c^{2} -{\left (3 \, b^{2} c^{2} - 2 \, a b c d + 3 \, a^{2} d^{2}\right )} x^{2} + 2 \,{\left (a b c^{2} + a^{2} c d\right )} x\right )} \sqrt{-a c} \sqrt{b x + a} \sqrt{d x + c}}{48 \, \sqrt{-a c} a^{2} c^{2} x^{3}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(b*x + a)*sqrt(d*x + c)/x^4,x, algorithm="fricas")

[Out]

[1/96*(3*(b^3*c^3 - a*b^2*c^2*d - a^2*b*c*d^2 + a^3*d^3)*x^3*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*(8*a^2*c^
2 - (3*b^2*c^2 - 2*a*b*c*d + 3*a^2*d^2)*x^2 + 2*(a*b*c^2 + a^2*c*d)*x)*sqrt(a*c)
*sqrt(b*x + a)*sqrt(d*x + c))/(sqrt(a*c)*a^2*c^2*x^3), -1/48*(3*(b^3*c^3 - a*b^2
*c^2*d - a^2*b*c*d^2 + a^3*d^3)*x^3*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*(8*a^2*c^2 - (3*b^2*c^2 - 2*a*b*c*d + 3
*a^2*d^2)*x^2 + 2*(a*b*c^2 + a^2*c*d)*x)*sqrt(-a*c)*sqrt(b*x + a)*sqrt(d*x + c))
/(sqrt(-a*c)*a^2*c^2*x^3)]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{a + b x} \sqrt{c + d x}}{x^{4}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x+a)**(1/2)*(d*x+c)**(1/2)/x**4,x)

[Out]

Integral(sqrt(a + b*x)*sqrt(c + d*x)/x**4, 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^4,x, algorithm="giac")

[Out]

Exception raised: TypeError