3.543 \(\int \frac{\sqrt{a+b x} \sqrt{c+d x}}{x^5} \, dx\)

Optimal. Leaf size=256 \[ \frac{\sqrt{a+b x} \sqrt{c+d x} \left (5 a^2 d^2-2 a b c d+5 b^2 c^2\right )}{96 a^2 c^2 x^2}+\frac{\left (5 a^2 d^2+6 a b c d+5 b^2 c^2\right ) (b c-a d)^2 \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{64 a^{7/2} c^{7/2}}-\frac{\sqrt{a+b x} \sqrt{c+d x} (a d+b c) \left (15 a^2 d^2-22 a b c d+15 b^2 c^2\right )}{192 a^3 c^3 x}-\frac{\sqrt{a+b x} \sqrt{c+d x}}{4 x^4}-\frac{\sqrt{a+b x} \sqrt{c+d x} (a d+b c)}{24 a c x^3} \]

[Out]

-(Sqrt[a + b*x]*Sqrt[c + d*x])/(4*x^4) - ((b*c + a*d)*Sqrt[a + b*x]*Sqrt[c + d*x
])/(24*a*c*x^3) + ((5*b^2*c^2 - 2*a*b*c*d + 5*a^2*d^2)*Sqrt[a + b*x]*Sqrt[c + d*
x])/(96*a^2*c^2*x^2) - ((b*c + a*d)*(15*b^2*c^2 - 22*a*b*c*d + 15*a^2*d^2)*Sqrt[
a + b*x]*Sqrt[c + d*x])/(192*a^3*c^3*x) + ((b*c - a*d)^2*(5*b^2*c^2 + 6*a*b*c*d
+ 5*a^2*d^2)*ArcTanh[(Sqrt[c]*Sqrt[a + b*x])/(Sqrt[a]*Sqrt[c + d*x])])/(64*a^(7/
2)*c^(7/2))

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Rubi [A]  time = 0.710303, antiderivative size = 256, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227 \[ \frac{\sqrt{a+b x} \sqrt{c+d x} \left (5 a^2 d^2-2 a b c d+5 b^2 c^2\right )}{96 a^2 c^2 x^2}+\frac{\left (5 a^2 d^2+6 a b c d+5 b^2 c^2\right ) (b c-a d)^2 \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{64 a^{7/2} c^{7/2}}-\frac{\sqrt{a+b x} \sqrt{c+d x} (a d+b c) \left (15 a^2 d^2-22 a b c d+15 b^2 c^2\right )}{192 a^3 c^3 x}-\frac{\sqrt{a+b x} \sqrt{c+d x}}{4 x^4}-\frac{\sqrt{a+b x} \sqrt{c+d x} (a d+b c)}{24 a c x^3} \]

Antiderivative was successfully verified.

[In]  Int[(Sqrt[a + b*x]*Sqrt[c + d*x])/x^5,x]

[Out]

-(Sqrt[a + b*x]*Sqrt[c + d*x])/(4*x^4) - ((b*c + a*d)*Sqrt[a + b*x]*Sqrt[c + d*x
])/(24*a*c*x^3) + ((5*b^2*c^2 - 2*a*b*c*d + 5*a^2*d^2)*Sqrt[a + b*x]*Sqrt[c + d*
x])/(96*a^2*c^2*x^2) - ((b*c + a*d)*(15*b^2*c^2 - 22*a*b*c*d + 15*a^2*d^2)*Sqrt[
a + b*x]*Sqrt[c + d*x])/(192*a^3*c^3*x) + ((b*c - a*d)^2*(5*b^2*c^2 + 6*a*b*c*d
+ 5*a^2*d^2)*ArcTanh[(Sqrt[c]*Sqrt[a + b*x])/(Sqrt[a]*Sqrt[c + d*x])])/(64*a^(7/
2)*c^(7/2))

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Rubi in Sympy [A]  time = 96.5852, size = 228, normalized size = 0.89 \[ - \frac{\sqrt{a + b x} \sqrt{c + d x}}{4 x^{4}} - \frac{\sqrt{a + b x} \sqrt{c + d x} \left (a d + b c\right )}{24 a c x^{3}} - \frac{\sqrt{a + b x} \sqrt{c + d x} \left (12 a b c d - 5 \left (a d + b c\right )^{2}\right )}{96 a^{2} c^{2} x^{2}} + \frac{\sqrt{a + b x} \sqrt{c + d x} \left (a d + b c\right ) \left (52 a b c d - 15 \left (a d + b c\right )^{2}\right )}{192 a^{3} c^{3} x} + \frac{\left (a d - b c\right )^{2} \left (5 a^{2} d^{2} + 6 a b c d + 5 b^{2} c^{2}\right ) \operatorname{atanh}{\left (\frac{\sqrt{c} \sqrt{a + b x}}{\sqrt{a} \sqrt{c + d x}} \right )}}{64 a^{\frac{7}{2}} c^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**(1/2)*(d*x+c)**(1/2)/x**5,x)

[Out]

-sqrt(a + b*x)*sqrt(c + d*x)/(4*x**4) - sqrt(a + b*x)*sqrt(c + d*x)*(a*d + b*c)/
(24*a*c*x**3) - sqrt(a + b*x)*sqrt(c + d*x)*(12*a*b*c*d - 5*(a*d + b*c)**2)/(96*
a**2*c**2*x**2) + sqrt(a + b*x)*sqrt(c + d*x)*(a*d + b*c)*(52*a*b*c*d - 15*(a*d
+ b*c)**2)/(192*a**3*c**3*x) + (a*d - b*c)**2*(5*a**2*d**2 + 6*a*b*c*d + 5*b**2*
c**2)*atanh(sqrt(c)*sqrt(a + b*x)/(sqrt(a)*sqrt(c + d*x)))/(64*a**(7/2)*c**(7/2)
)

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Mathematica [A]  time = 0.249008, size = 262, normalized size = 1.02 \[ \frac{-3 x^4 \log (x) (b c-a d)^2 \left (5 a^2 d^2+6 a b c d+5 b^2 c^2\right )+3 x^4 (b c-a d)^2 \left (5 a^2 d^2+6 a b c d+5 b^2 c^2\right ) \log \left (2 \sqrt{a} \sqrt{c} \sqrt{a+b x} \sqrt{c+d x}+2 a c+a d x+b c x\right )-2 \sqrt{a} \sqrt{c} \sqrt{a+b x} \sqrt{c+d x} \left (a^3 \left (48 c^3+8 c^2 d x-10 c d^2 x^2+15 d^3 x^3\right )+a^2 b c x \left (8 c^2+4 c d x-7 d^2 x^2\right )-a b^2 c^2 x^2 (10 c+7 d x)+15 b^3 c^3 x^3\right )}{384 a^{7/2} c^{7/2} x^4} \]

Antiderivative was successfully verified.

[In]  Integrate[(Sqrt[a + b*x]*Sqrt[c + d*x])/x^5,x]

[Out]

(-2*Sqrt[a]*Sqrt[c]*Sqrt[a + b*x]*Sqrt[c + d*x]*(15*b^3*c^3*x^3 - a*b^2*c^2*x^2*
(10*c + 7*d*x) + a^2*b*c*x*(8*c^2 + 4*c*d*x - 7*d^2*x^2) + a^3*(48*c^3 + 8*c^2*d
*x - 10*c*d^2*x^2 + 15*d^3*x^3)) - 3*(b*c - a*d)^2*(5*b^2*c^2 + 6*a*b*c*d + 5*a^
2*d^2)*x^4*Log[x] + 3*(b*c - a*d)^2*(5*b^2*c^2 + 6*a*b*c*d + 5*a^2*d^2)*x^4*Log[
2*a*c + b*c*x + a*d*x + 2*Sqrt[a]*Sqrt[c]*Sqrt[a + b*x]*Sqrt[c + d*x]])/(384*a^(
7/2)*c^(7/2)*x^4)

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Maple [B]  time = 0.023, size = 705, normalized size = 2.8 \[{\frac{1}{384\,{a}^{3}{c}^{3}{x}^{4}}\sqrt{bx+a}\sqrt{dx+c} \left ( 15\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}+2\,ac}{x}} \right ){x}^{4}{a}^{4}{d}^{4}-12\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}+2\,ac}{x}} \right ){x}^{4}{a}^{3}bc{d}^{3}-6\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}+2\,ac}{x}} \right ){x}^{4}{a}^{2}{b}^{2}{c}^{2}{d}^{2}-12\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}+2\,ac}{x}} \right ){x}^{4}a{b}^{3}{c}^{3}d+15\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}+2\,ac}{x}} \right ){x}^{4}{b}^{4}{c}^{4}-30\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}{x}^{3}{a}^{3}{d}^{3}+14\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}{x}^{3}{a}^{2}bc{d}^{2}+14\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}{x}^{3}a{b}^{2}{c}^{2}d-30\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}{x}^{3}{b}^{3}{c}^{3}+20\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}{x}^{2}{a}^{3}c{d}^{2}-8\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}{x}^{2}{a}^{2}b{c}^{2}d+20\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}{x}^{2}a{b}^{2}{c}^{3}-16\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}x{a}^{3}{c}^{2}d-16\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}x{a}^{2}b{c}^{3}-96\,\sqrt{d{x}^{2}b+adx+bcx+ac}{a}^{3}{c}^{3}\sqrt{ac} \right ){\frac{1}{\sqrt{ac}}}{\frac{1}{\sqrt{d{x}^{2}b+adx+bcx+ac}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^(1/2)*(d*x+c)^(1/2)/x^5,x)

[Out]

1/384*(b*x+a)^(1/2)*(d*x+c)^(1/2)/a^3/c^3*(15*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^4*a^4*d^4-12*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^4*a^3*b*c*d^3-6*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^4*a^2*b^2*c^2*d^2-12*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^4*a*b^3*
c^3*d+15*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^4*b^4*c^4-30*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*x^3*a^3*d^3+14*(a*c)
^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*x^3*a^2*b*c*d^2+14*(a*c)^(1/2)*(b*d*x^2+a
*d*x+b*c*x+a*c)^(1/2)*x^3*a*b^2*c^2*d-30*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(
1/2)*x^3*b^3*c^3+20*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*x^2*a^3*c*d^2-8*
(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*x^2*a^2*b*c^2*d+20*(a*c)^(1/2)*(b*d*
x^2+a*d*x+b*c*x+a*c)^(1/2)*x^2*a*b^2*c^3-16*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c
)^(1/2)*x*a^3*c^2*d-16*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*x*a^2*b*c^3-9
6*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*a^3*c^3*(a*c)^(1/2))/(b*d*x^2+a*d*x+b*c*x+a*c)
^(1/2)/x^4/(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^5,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.771619, size = 1, normalized size = 0. \[ \left [\frac{3 \,{\left (5 \, b^{4} c^{4} - 4 \, a b^{3} c^{3} d - 2 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + 5 \, a^{4} d^{4}\right )} x^{4} \log \left (\frac{4 \,{\left (2 \, a^{2} c^{2} +{\left (a b c^{2} + a^{2} c d\right )} x\right )} \sqrt{b x + a} \sqrt{d x + c} +{\left (8 \, a^{2} c^{2} +{\left (b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2}\right )} x^{2} + 8 \,{\left (a b c^{2} + a^{2} c d\right )} x\right )} \sqrt{a c}}{x^{2}}\right ) - 4 \,{\left (48 \, a^{3} c^{3} +{\left (15 \, b^{3} c^{3} - 7 \, a b^{2} c^{2} d - 7 \, a^{2} b c d^{2} + 15 \, a^{3} d^{3}\right )} x^{3} - 2 \,{\left (5 \, a b^{2} c^{3} - 2 \, a^{2} b c^{2} d + 5 \, a^{3} c d^{2}\right )} x^{2} + 8 \,{\left (a^{2} b c^{3} + a^{3} c^{2} d\right )} x\right )} \sqrt{a c} \sqrt{b x + a} \sqrt{d x + c}}{768 \, \sqrt{a c} a^{3} c^{3} x^{4}}, \frac{3 \,{\left (5 \, b^{4} c^{4} - 4 \, a b^{3} c^{3} d - 2 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + 5 \, a^{4} d^{4}\right )} x^{4} \arctan \left (\frac{{\left (2 \, a c +{\left (b c + a d\right )} x\right )} \sqrt{-a c}}{2 \, \sqrt{b x + a} \sqrt{d x + c} a c}\right ) - 2 \,{\left (48 \, a^{3} c^{3} +{\left (15 \, b^{3} c^{3} - 7 \, a b^{2} c^{2} d - 7 \, a^{2} b c d^{2} + 15 \, a^{3} d^{3}\right )} x^{3} - 2 \,{\left (5 \, a b^{2} c^{3} - 2 \, a^{2} b c^{2} d + 5 \, a^{3} c d^{2}\right )} x^{2} + 8 \,{\left (a^{2} b c^{3} + a^{3} c^{2} d\right )} x\right )} \sqrt{-a c} \sqrt{b x + a} \sqrt{d x + c}}{384 \, \sqrt{-a c} a^{3} c^{3} x^{4}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(b*x + a)*sqrt(d*x + c)/x^5,x, algorithm="fricas")

[Out]

[1/768*(3*(5*b^4*c^4 - 4*a*b^3*c^3*d - 2*a^2*b^2*c^2*d^2 - 4*a^3*b*c*d^3 + 5*a^4
*d^4)*x^4*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)*s
qrt(a*c))/x^2) - 4*(48*a^3*c^3 + (15*b^3*c^3 - 7*a*b^2*c^2*d - 7*a^2*b*c*d^2 + 1
5*a^3*d^3)*x^3 - 2*(5*a*b^2*c^3 - 2*a^2*b*c^2*d + 5*a^3*c*d^2)*x^2 + 8*(a^2*b*c^
3 + a^3*c^2*d)*x)*sqrt(a*c)*sqrt(b*x + a)*sqrt(d*x + c))/(sqrt(a*c)*a^3*c^3*x^4)
, 1/384*(3*(5*b^4*c^4 - 4*a*b^3*c^3*d - 2*a^2*b^2*c^2*d^2 - 4*a^3*b*c*d^3 + 5*a^
4*d^4)*x^4*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*(48*a^3*c^3 + (15*b^3*c^3 - 7*a*b^2*c^2*d - 7*a^2*b*c*d^2 + 15*a
^3*d^3)*x^3 - 2*(5*a*b^2*c^3 - 2*a^2*b*c^2*d + 5*a^3*c*d^2)*x^2 + 8*(a^2*b*c^3 +
a^3*c^2*d)*x)*sqrt(-a*c)*sqrt(b*x + a)*sqrt(d*x + c))/(sqrt(-a*c)*a^3*c^3*x^4)]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{a + b x} \sqrt{c + d x}}{x^{5}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x+a)**(1/2)*(d*x+c)**(1/2)/x**5,x)

[Out]

Integral(sqrt(a + b*x)*sqrt(c + d*x)/x**5, 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^5,x, algorithm="giac")

[Out]

Exception raised: TypeError