3.728 \(\int \frac{1}{x^4 \sqrt{a+b x} \sqrt{c+d x}} \, dx\)

Optimal. Leaf size=198 \[ \frac{5 \sqrt{a+b x} \sqrt{c+d x} (a d+b c)}{12 a^2 c^2 x^2}+\frac{\left (5 a^2 d^2-2 a b c d+5 b^2 c^2\right ) (a d+b c) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{8 a^{7/2} c^{7/2}}-\frac{\sqrt{a+b x} \sqrt{c+d x} \left (15 a^2 d^2+14 a b c d+15 b^2 c^2\right )}{24 a^3 c^3 x}-\frac{\sqrt{a+b x} \sqrt{c+d x}}{3 a c x^3} \]

[Out]

-(Sqrt[a + b*x]*Sqrt[c + d*x])/(3*a*c*x^3) + (5*(b*c + a*d)*Sqrt[a + b*x]*Sqrt[c
+ d*x])/(12*a^2*c^2*x^2) - ((15*b^2*c^2 + 14*a*b*c*d + 15*a^2*d^2)*Sqrt[a + b*x
]*Sqrt[c + d*x])/(24*a^3*c^3*x) + ((b*c + a*d)*(5*b^2*c^2 - 2*a*b*c*d + 5*a^2*d^
2)*ArcTanh[(Sqrt[c]*Sqrt[a + b*x])/(Sqrt[a]*Sqrt[c + d*x])])/(8*a^(7/2)*c^(7/2))

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Rubi [A]  time = 0.493566, antiderivative size = 198, 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 \[ \frac{5 \sqrt{a+b x} \sqrt{c+d x} (a d+b c)}{12 a^2 c^2 x^2}+\frac{\left (5 a^2 d^2-2 a b c d+5 b^2 c^2\right ) (a d+b c) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{8 a^{7/2} c^{7/2}}-\frac{\sqrt{a+b x} \sqrt{c+d x} \left (15 a^2 d^2+14 a b c d+15 b^2 c^2\right )}{24 a^3 c^3 x}-\frac{\sqrt{a+b x} \sqrt{c+d x}}{3 a c x^3} \]

Antiderivative was successfully verified.

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

[Out]

-(Sqrt[a + b*x]*Sqrt[c + d*x])/(3*a*c*x^3) + (5*(b*c + a*d)*Sqrt[a + b*x]*Sqrt[c
+ d*x])/(12*a^2*c^2*x^2) - ((15*b^2*c^2 + 14*a*b*c*d + 15*a^2*d^2)*Sqrt[a + b*x
]*Sqrt[c + d*x])/(24*a^3*c^3*x) + ((b*c + a*d)*(5*b^2*c^2 - 2*a*b*c*d + 5*a^2*d^
2)*ArcTanh[(Sqrt[c]*Sqrt[a + b*x])/(Sqrt[a]*Sqrt[c + d*x])])/(8*a^(7/2)*c^(7/2))

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Rubi in Sympy [A]  time = 65.2013, size = 173, normalized size = 0.87 \[ - \frac{\sqrt{a + b x} \sqrt{c + d x}}{3 a c x^{3}} + \frac{5 \sqrt{a + b x} \sqrt{c + d x} \left (a d + b c\right )}{12 a^{2} c^{2} x^{2}} + \frac{\sqrt{a + b x} \sqrt{c + d x} \left (16 a b c d - 15 \left (a d + b c\right )^{2}\right )}{24 a^{3} c^{3} x} - \frac{\left (a d + b c\right ) \left (12 a b c d - 5 \left (a d + b c\right )^{2}\right ) \operatorname{atanh}{\left (\frac{\sqrt{c} \sqrt{a + b x}}{\sqrt{a} \sqrt{c + d x}} \right )}}{8 a^{\frac{7}{2}} c^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(a + b*x)*sqrt(c + d*x)/(3*a*c*x**3) + 5*sqrt(a + b*x)*sqrt(c + d*x)*(a*d +
b*c)/(12*a**2*c**2*x**2) + sqrt(a + b*x)*sqrt(c + d*x)*(16*a*b*c*d - 15*(a*d +
b*c)**2)/(24*a**3*c**3*x) - (a*d + b*c)*(12*a*b*c*d - 5*(a*d + b*c)**2)*atanh(sq
rt(c)*sqrt(a + b*x)/(sqrt(a)*sqrt(c + d*x)))/(8*a**(7/2)*c**(7/2))

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Mathematica [A]  time = 0.17325, size = 212, normalized size = 1.07 \[ \frac{-3 x^3 \log (x) (a d+b c) \left (5 a^2 d^2-2 a b c d+5 b^2 c^2\right )+3 x^3 (a d+b c) \left (5 a^2 d^2-2 a b c d+5 b^2 c^2\right ) \log \left (2 \sqrt{a} \sqrt{c} \sqrt{a+b x} \sqrt{c+d x}+2 a c+a d x+b c x\right )-2 \sqrt{a} \sqrt{c} \sqrt{a+b x} \sqrt{c+d x} \left (x^2 \left (15 a^2 d^2+14 a b c d+15 b^2 c^2\right )+8 a^2 c^2-10 a c x (a d+b c)\right )}{48 a^{7/2} c^{7/2} x^3} \]

Antiderivative was successfully verified.

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

[Out]

(-2*Sqrt[a]*Sqrt[c]*Sqrt[a + b*x]*Sqrt[c + d*x]*(8*a^2*c^2 - 10*a*c*(b*c + a*d)*
x + (15*b^2*c^2 + 14*a*b*c*d + 15*a^2*d^2)*x^2) - 3*(b*c + a*d)*(5*b^2*c^2 - 2*a
*b*c*d + 5*a^2*d^2)*x^3*Log[x] + 3*(b*c + a*d)*(5*b^2*c^2 - 2*a*b*c*d + 5*a^2*d^
2)*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^(7/2)*c^(7/2)*x^3)

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Maple [B]  time = 0.039, size = 408, normalized size = 2.1 \[{\frac{1}{48\,{a}^{3}{c}^{3}{x}^{3}} \left ( 15\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac}{x}} \right ){x}^{3}{a}^{3}{d}^{3}+9\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac}{x}} \right ){x}^{3}{a}^{2}bc{d}^{2}+9\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac}{x}} \right ){x}^{3}a{b}^{2}{c}^{2}d+15\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac}{x}} \right ){x}^{3}{b}^{3}{c}^{3}-30\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }{d}^{2}{a}^{2}{x}^{2}\sqrt{ac}-28\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }dbca{x}^{2}\sqrt{ac}-30\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }{b}^{2}{c}^{2}{x}^{2}\sqrt{ac}+20\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }dc{a}^{2}x\sqrt{ac}+20\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }b{c}^{2}ax\sqrt{ac}-16\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }{c}^{2}{a}^{2}\sqrt{ac} \right ) \sqrt{dx+c}\sqrt{bx+a}{\frac{1}{\sqrt{ac}}}{\frac{1}{\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/48/a^3/c^3*(15*ln((a*d*x+b*c*x+2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+2*a*c)/x)
*x^3*a^3*d^3+9*ln((a*d*x+b*c*x+2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+2*a*c)/x)*x
^3*a^2*b*c*d^2+9*ln((a*d*x+b*c*x+2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+2*a*c)/x)
*x^3*a*b^2*c^2*d+15*ln((a*d*x+b*c*x+2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+2*a*c)
/x)*x^3*b^3*c^3-30*((b*x+a)*(d*x+c))^(1/2)*d^2*a^2*x^2*(a*c)^(1/2)-28*((b*x+a)*(
d*x+c))^(1/2)*d*b*c*a*x^2*(a*c)^(1/2)-30*((b*x+a)*(d*x+c))^(1/2)*b^2*c^2*x^2*(a*
c)^(1/2)+20*((b*x+a)*(d*x+c))^(1/2)*d*c*a^2*x*(a*c)^(1/2)+20*((b*x+a)*(d*x+c))^(
1/2)*b*c^2*a*x*(a*c)^(1/2)-16*((b*x+a)*(d*x+c))^(1/2)*c^2*a^2*(a*c)^(1/2))*(d*x+
c)^(1/2)*(b*x+a)^(1/2)/(a*c)^(1/2)/x^3/((b*x+a)*(d*x+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(1/(sqrt(b*x + a)*sqrt(d*x + c)*x^4),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.436547, size = 1, normalized size = 0.01 \[ \left [\frac{3 \,{\left (5 \, b^{3} c^{3} + 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} + 5 \, a^{3} d^{3}\right )} x^{3} \log \left (\frac{4 \,{\left (2 \, a^{2} c^{2} +{\left (a b c^{2} + a^{2} c d\right )} x\right )} \sqrt{b x + a} \sqrt{d x + c} +{\left (8 \, a^{2} c^{2} +{\left (b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2}\right )} x^{2} + 8 \,{\left (a b c^{2} + a^{2} c d\right )} x\right )} \sqrt{a c}}{x^{2}}\right ) - 4 \,{\left (8 \, a^{2} c^{2} +{\left (15 \, b^{2} c^{2} + 14 \, a b c d + 15 \, a^{2} d^{2}\right )} x^{2} - 10 \,{\left (a b c^{2} + a^{2} c d\right )} x\right )} \sqrt{a c} \sqrt{b x + a} \sqrt{d x + c}}{96 \, \sqrt{a c} a^{3} c^{3} x^{3}}, \frac{3 \,{\left (5 \, b^{3} c^{3} + 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} + 5 \, a^{3} d^{3}\right )} x^{3} \arctan \left (\frac{{\left (2 \, a c +{\left (b c + a d\right )} x\right )} \sqrt{-a c}}{2 \, \sqrt{b x + a} \sqrt{d x + c} a c}\right ) - 2 \,{\left (8 \, a^{2} c^{2} +{\left (15 \, b^{2} c^{2} + 14 \, a b c d + 15 \, a^{2} d^{2}\right )} x^{2} - 10 \,{\left (a b c^{2} + a^{2} c d\right )} x\right )} \sqrt{-a c} \sqrt{b x + a} \sqrt{d x + c}}{48 \, \sqrt{-a c} a^{3} c^{3} x^{3}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(b*x + a)*sqrt(d*x + c)*x^4),x, algorithm="fricas")

[Out]

[1/96*(3*(5*b^3*c^3 + 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 + 5*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 + (15*b^2*c^2 + 14*a*b*c*d + 15*a^2*d^2)*x^2 - 10*(a*b*c^2 + a^2*c*d)*x
)*sqrt(a*c)*sqrt(b*x + a)*sqrt(d*x + c))/(sqrt(a*c)*a^3*c^3*x^3), 1/48*(3*(5*b^3
*c^3 + 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 + 5*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 + (15*b^2*
c^2 + 14*a*b*c*d + 15*a^2*d^2)*x^2 - 10*(a*b*c^2 + a^2*c*d)*x)*sqrt(-a*c)*sqrt(b
*x + a)*sqrt(d*x + c))/(sqrt(-a*c)*a^3*c^3*x^3)]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{x^{4} \sqrt{a + b x} \sqrt{c + d x}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(1/(x**4*sqrt(a + b*x)*sqrt(c + d*x)), 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(1/(sqrt(b*x + a)*sqrt(d*x + c)*x^4),x, algorithm="giac")

[Out]

Exception raised: TypeError