3.544 \(\int \frac{\sqrt{a+b x} \sqrt{c+d x}}{x^6} \, dx\)

Optimal. Leaf size=345 \[ \frac{\sqrt{a+b x} \sqrt{c+d x} \left (7 a^2 d^2-2 a b c d+7 b^2 c^2\right )}{240 a^2 c^2 x^3}-\frac{(a d+b c) \left (7 a^2 d^2+2 a b c d+7 b^2 c^2\right ) (b c-a d)^2 \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{128 a^{9/2} c^{9/2}}-\frac{\sqrt{a+b x} \sqrt{c+d x} (a d+b c) \left (35 a^2 d^2-46 a b c d+35 b^2 c^2\right )}{960 a^3 c^3 x^2}+\frac{\sqrt{a+b x} \sqrt{c+d x} \left (105 a^4 d^4-40 a^3 b c d^3-34 a^2 b^2 c^2 d^2-40 a b^3 c^3 d+105 b^4 c^4\right )}{1920 a^4 c^4 x}-\frac{\sqrt{a+b x} \sqrt{c+d x}}{5 x^5}-\frac{\sqrt{a+b x} \sqrt{c+d x} (a d+b c)}{40 a c x^4} \]

[Out]

-(Sqrt[a + b*x]*Sqrt[c + d*x])/(5*x^5) - ((b*c + a*d)*Sqrt[a + b*x]*Sqrt[c + d*x
])/(40*a*c*x^4) + ((7*b^2*c^2 - 2*a*b*c*d + 7*a^2*d^2)*Sqrt[a + b*x]*Sqrt[c + d*
x])/(240*a^2*c^2*x^3) - ((b*c + a*d)*(35*b^2*c^2 - 46*a*b*c*d + 35*a^2*d^2)*Sqrt
[a + b*x]*Sqrt[c + d*x])/(960*a^3*c^3*x^2) + ((105*b^4*c^4 - 40*a*b^3*c^3*d - 34
*a^2*b^2*c^2*d^2 - 40*a^3*b*c*d^3 + 105*a^4*d^4)*Sqrt[a + b*x]*Sqrt[c + d*x])/(1
920*a^4*c^4*x) - ((b*c - a*d)^2*(b*c + a*d)*(7*b^2*c^2 + 2*a*b*c*d + 7*a^2*d^2)*
ArcTanh[(Sqrt[c]*Sqrt[a + b*x])/(Sqrt[a]*Sqrt[c + d*x])])/(128*a^(9/2)*c^(9/2))

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Rubi [A]  time = 1.055, antiderivative size = 345, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227 \[ \frac{\sqrt{a+b x} \sqrt{c+d x} \left (7 a^2 d^2-2 a b c d+7 b^2 c^2\right )}{240 a^2 c^2 x^3}-\frac{(a d+b c) \left (7 a^2 d^2+2 a b c d+7 b^2 c^2\right ) (b c-a d)^2 \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{128 a^{9/2} c^{9/2}}-\frac{\sqrt{a+b x} \sqrt{c+d x} (a d+b c) \left (35 a^2 d^2-46 a b c d+35 b^2 c^2\right )}{960 a^3 c^3 x^2}+\frac{\sqrt{a+b x} \sqrt{c+d x} \left (105 a^4 d^4-40 a^3 b c d^3-34 a^2 b^2 c^2 d^2-40 a b^3 c^3 d+105 b^4 c^4\right )}{1920 a^4 c^4 x}-\frac{\sqrt{a+b x} \sqrt{c+d x}}{5 x^5}-\frac{\sqrt{a+b x} \sqrt{c+d x} (a d+b c)}{40 a c x^4} \]

Antiderivative was successfully verified.

[In]  Int[(Sqrt[a + b*x]*Sqrt[c + d*x])/x^6,x]

[Out]

-(Sqrt[a + b*x]*Sqrt[c + d*x])/(5*x^5) - ((b*c + a*d)*Sqrt[a + b*x]*Sqrt[c + d*x
])/(40*a*c*x^4) + ((7*b^2*c^2 - 2*a*b*c*d + 7*a^2*d^2)*Sqrt[a + b*x]*Sqrt[c + d*
x])/(240*a^2*c^2*x^3) - ((b*c + a*d)*(35*b^2*c^2 - 46*a*b*c*d + 35*a^2*d^2)*Sqrt
[a + b*x]*Sqrt[c + d*x])/(960*a^3*c^3*x^2) + ((105*b^4*c^4 - 40*a*b^3*c^3*d - 34
*a^2*b^2*c^2*d^2 - 40*a^3*b*c*d^3 + 105*a^4*d^4)*Sqrt[a + b*x]*Sqrt[c + d*x])/(1
920*a^4*c^4*x) - ((b*c - a*d)^2*(b*c + a*d)*(7*b^2*c^2 + 2*a*b*c*d + 7*a^2*d^2)*
ArcTanh[(Sqrt[c]*Sqrt[a + b*x])/(Sqrt[a]*Sqrt[c + d*x])])/(128*a^(9/2)*c^(9/2))

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Rubi in Sympy [A]  time = 171.808, size = 306, normalized size = 0.89 \[ - \frac{\sqrt{a + b x} \sqrt{c + d x}}{5 x^{5}} - \frac{\sqrt{a + b x} \sqrt{c + d x} \left (a d + b c\right )}{40 a c x^{4}} - \frac{\sqrt{a + b x} \sqrt{c + d x} \left (16 a b c d - 7 \left (a d + b c\right )^{2}\right )}{240 a^{2} c^{2} x^{3}} + \frac{\sqrt{a + b x} \sqrt{c + d x} \left (a d + b c\right ) \left (116 a b c d - 35 \left (a d + b c\right )^{2}\right )}{960 a^{3} c^{3} x^{2}} + \frac{\sqrt{a + b x} \sqrt{c + d x} \left (256 a^{2} b^{2} c^{2} d^{2} - 460 a b c d \left (a d + b c\right )^{2} + 105 \left (a d + b c\right )^{4}\right )}{1920 a^{4} c^{4} x} - \frac{\left (a d - b c\right )^{2} \left (a d + b c\right ) \left (7 a^{2} d^{2} + 2 a b c d + 7 b^{2} c^{2}\right ) \operatorname{atanh}{\left (\frac{\sqrt{c} \sqrt{a + b x}}{\sqrt{a} \sqrt{c + d x}} \right )}}{128 a^{\frac{9}{2}} c^{\frac{9}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**(1/2)*(d*x+c)**(1/2)/x**6,x)

[Out]

-sqrt(a + b*x)*sqrt(c + d*x)/(5*x**5) - sqrt(a + b*x)*sqrt(c + d*x)*(a*d + b*c)/
(40*a*c*x**4) - sqrt(a + b*x)*sqrt(c + d*x)*(16*a*b*c*d - 7*(a*d + b*c)**2)/(240
*a**2*c**2*x**3) + sqrt(a + b*x)*sqrt(c + d*x)*(a*d + b*c)*(116*a*b*c*d - 35*(a*
d + b*c)**2)/(960*a**3*c**3*x**2) + sqrt(a + b*x)*sqrt(c + d*x)*(256*a**2*b**2*c
**2*d**2 - 460*a*b*c*d*(a*d + b*c)**2 + 105*(a*d + b*c)**4)/(1920*a**4*c**4*x) -
(a*d - b*c)**2*(a*d + b*c)*(7*a**2*d**2 + 2*a*b*c*d + 7*b**2*c**2)*atanh(sqrt(c
)*sqrt(a + b*x)/(sqrt(a)*sqrt(c + d*x)))/(128*a**(9/2)*c**(9/2))

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Mathematica [A]  time = 0.330509, size = 332, normalized size = 0.96 \[ \frac{15 x^5 \log (x) (b c-a d)^2 (a d+b c) \left (7 a^2 d^2+2 a b c d+7 b^2 c^2\right )-15 x^5 (b c-a d)^2 (a d+b c) \left (7 a^2 d^2+2 a b c d+7 b^2 c^2\right ) \log \left (2 \sqrt{a} \sqrt{c} \sqrt{a+b x} \sqrt{c+d x}+2 a c+a d x+b c x\right )-2 \sqrt{a} \sqrt{c} \sqrt{a+b x} \sqrt{c+d x} \left (a^4 \left (384 c^4+48 c^3 d x-56 c^2 d^2 x^2+70 c d^3 x^3-105 d^4 x^4\right )+2 a^3 b c x \left (24 c^3+8 c^2 d x-11 c d^2 x^2+20 d^3 x^3\right )-2 a^2 b^2 c^2 x^2 \left (28 c^2+11 c d x-17 d^2 x^2\right )+10 a b^3 c^3 x^3 (7 c+4 d x)-105 b^4 c^4 x^4\right )}{3840 a^{9/2} c^{9/2} x^5} \]

Antiderivative was successfully verified.

[In]  Integrate[(Sqrt[a + b*x]*Sqrt[c + d*x])/x^6,x]

[Out]

(-2*Sqrt[a]*Sqrt[c]*Sqrt[a + b*x]*Sqrt[c + d*x]*(-105*b^4*c^4*x^4 + 10*a*b^3*c^3
*x^3*(7*c + 4*d*x) - 2*a^2*b^2*c^2*x^2*(28*c^2 + 11*c*d*x - 17*d^2*x^2) + 2*a^3*
b*c*x*(24*c^3 + 8*c^2*d*x - 11*c*d^2*x^2 + 20*d^3*x^3) + a^4*(384*c^4 + 48*c^3*d
*x - 56*c^2*d^2*x^2 + 70*c*d^3*x^3 - 105*d^4*x^4)) + 15*(b*c - a*d)^2*(b*c + a*d
)*(7*b^2*c^2 + 2*a*b*c*d + 7*a^2*d^2)*x^5*Log[x] - 15*(b*c - a*d)^2*(b*c + a*d)*
(7*b^2*c^2 + 2*a*b*c*d + 7*a^2*d^2)*x^5*Log[2*a*c + b*c*x + a*d*x + 2*Sqrt[a]*Sq
rt[c]*Sqrt[a + b*x]*Sqrt[c + d*x]])/(3840*a^(9/2)*c^(9/2)*x^5)

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Maple [B]  time = 0.029, size = 967, normalized size = 2.8 \[ -{\frac{1}{3840\,{a}^{4}{c}^{4}{x}^{5}}\sqrt{bx+a}\sqrt{dx+c} \left ( 105\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}+2\,ac}{x}} \right ){x}^{5}{a}^{5}{d}^{5}-75\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}+2\,ac}{x}} \right ){x}^{5}{a}^{4}bc{d}^{4}-30\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}+2\,ac}{x}} \right ){x}^{5}{a}^{3}{b}^{2}{c}^{2}{d}^{3}-30\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}+2\,ac}{x}} \right ){x}^{5}{a}^{2}{b}^{3}{c}^{3}{d}^{2}-75\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}+2\,ac}{x}} \right ){x}^{5}a{b}^{4}{c}^{4}d+105\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}+2\,ac}{x}} \right ){x}^{5}{b}^{5}{c}^{5}-210\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}{x}^{4}{a}^{4}{d}^{4}+80\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}{x}^{4}{a}^{3}bc{d}^{3}+68\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}{x}^{4}{a}^{2}{b}^{2}{c}^{2}{d}^{2}+80\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}{x}^{4}a{b}^{3}{c}^{3}d-210\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}{x}^{4}{b}^{4}{c}^{4}+140\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}{x}^{3}{a}^{4}c{d}^{3}-44\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}{x}^{3}{a}^{3}b{c}^{2}{d}^{2}-44\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}{x}^{3}{a}^{2}{b}^{2}{c}^{3}d+140\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}{x}^{3}a{b}^{3}{c}^{4}-112\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}{x}^{2}{a}^{4}{c}^{2}{d}^{2}+32\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}{x}^{2}{a}^{3}b{c}^{3}d-112\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}{x}^{2}{a}^{2}{b}^{2}{c}^{4}+96\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}x{a}^{4}{c}^{3}d+96\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}x{a}^{3}b{c}^{4}+768\,\sqrt{d{x}^{2}b+adx+bcx+ac}{a}^{4}{c}^{4}\sqrt{ac} \right ){\frac{1}{\sqrt{ac}}}{\frac{1}{\sqrt{d{x}^{2}b+adx+bcx+ac}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^(1/2)*(d*x+c)^(1/2)/x^6,x)

[Out]

-1/3840*(b*x+a)^(1/2)*(d*x+c)^(1/2)/a^4/c^4*(105*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^5*a^5*d^5-75*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^5*a^4*b*c*d^4-30*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^5*a^3*b^2*c^2*d^3-3
0*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^5*a^
2*b^3*c^3*d^2-75*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^5*a*b^4*c^4*d+105*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^5*b^5*c^5-210*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)
*x^4*a^4*d^4+80*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*x^4*a^3*b*c*d^3+68*(
a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*x^4*a^2*b^2*c^2*d^2+80*(a*c)^(1/2)*(b
*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*x^4*a*b^3*c^3*d-210*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c
*x+a*c)^(1/2)*x^4*b^4*c^4+140*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*x^3*a^
4*c*d^3-44*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*x^3*a^3*b*c^2*d^2-44*(a*c
)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*x^3*a^2*b^2*c^3*d+140*(a*c)^(1/2)*(b*d*x
^2+a*d*x+b*c*x+a*c)^(1/2)*x^3*a*b^3*c^4-112*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c
)^(1/2)*x^2*a^4*c^2*d^2+32*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*x^2*a^3*b
*c^3*d-112*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*x^2*a^2*b^2*c^4+96*(a*c)^
(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*x*a^4*c^3*d+96*(a*c)^(1/2)*(b*d*x^2+a*d*x+
b*c*x+a*c)^(1/2)*x*a^3*b*c^4+768*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*a^4*c^4*(a*c)^(
1/2))/(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)/x^5/(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^6,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 2.87504, size = 1, normalized size = 0. \[ \left [\frac{15 \,{\left (7 \, b^{5} c^{5} - 5 \, a b^{4} c^{4} d - 2 \, a^{2} b^{3} c^{3} d^{2} - 2 \, a^{3} b^{2} c^{2} d^{3} - 5 \, a^{4} b c d^{4} + 7 \, a^{5} d^{5}\right )} x^{5} \log \left (-\frac{4 \,{\left (2 \, a^{2} c^{2} +{\left (a b c^{2} + a^{2} c d\right )} x\right )} \sqrt{b x + a} \sqrt{d x + c} -{\left (8 \, a^{2} c^{2} +{\left (b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2}\right )} x^{2} + 8 \,{\left (a b c^{2} + a^{2} c d\right )} x\right )} \sqrt{a c}}{x^{2}}\right ) - 4 \,{\left (384 \, a^{4} c^{4} -{\left (105 \, b^{4} c^{4} - 40 \, a b^{3} c^{3} d - 34 \, a^{2} b^{2} c^{2} d^{2} - 40 \, a^{3} b c d^{3} + 105 \, a^{4} d^{4}\right )} x^{4} + 2 \,{\left (35 \, a b^{3} c^{4} - 11 \, a^{2} b^{2} c^{3} d - 11 \, a^{3} b c^{2} d^{2} + 35 \, a^{4} c d^{3}\right )} x^{3} - 8 \,{\left (7 \, a^{2} b^{2} c^{4} - 2 \, a^{3} b c^{3} d + 7 \, a^{4} c^{2} d^{2}\right )} x^{2} + 48 \,{\left (a^{3} b c^{4} + a^{4} c^{3} d\right )} x\right )} \sqrt{a c} \sqrt{b x + a} \sqrt{d x + c}}{7680 \, \sqrt{a c} a^{4} c^{4} x^{5}}, -\frac{15 \,{\left (7 \, b^{5} c^{5} - 5 \, a b^{4} c^{4} d - 2 \, a^{2} b^{3} c^{3} d^{2} - 2 \, a^{3} b^{2} c^{2} d^{3} - 5 \, a^{4} b c d^{4} + 7 \, a^{5} d^{5}\right )} x^{5} \arctan \left (\frac{{\left (2 \, a c +{\left (b c + a d\right )} x\right )} \sqrt{-a c}}{2 \, \sqrt{b x + a} \sqrt{d x + c} a c}\right ) + 2 \,{\left (384 \, a^{4} c^{4} -{\left (105 \, b^{4} c^{4} - 40 \, a b^{3} c^{3} d - 34 \, a^{2} b^{2} c^{2} d^{2} - 40 \, a^{3} b c d^{3} + 105 \, a^{4} d^{4}\right )} x^{4} + 2 \,{\left (35 \, a b^{3} c^{4} - 11 \, a^{2} b^{2} c^{3} d - 11 \, a^{3} b c^{2} d^{2} + 35 \, a^{4} c d^{3}\right )} x^{3} - 8 \,{\left (7 \, a^{2} b^{2} c^{4} - 2 \, a^{3} b c^{3} d + 7 \, a^{4} c^{2} d^{2}\right )} x^{2} + 48 \,{\left (a^{3} b c^{4} + a^{4} c^{3} d\right )} x\right )} \sqrt{-a c} \sqrt{b x + a} \sqrt{d x + c}}{3840 \, \sqrt{-a c} a^{4} c^{4} x^{5}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(b*x + a)*sqrt(d*x + c)/x^6,x, algorithm="fricas")

[Out]

[1/7680*(15*(7*b^5*c^5 - 5*a*b^4*c^4*d - 2*a^2*b^3*c^3*d^2 - 2*a^3*b^2*c^2*d^3 -
5*a^4*b*c*d^4 + 7*a^5*d^5)*x^5*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*(384*a^4*c^4 - (105*b^4*c^4 - 40*a*b^3
*c^3*d - 34*a^2*b^2*c^2*d^2 - 40*a^3*b*c*d^3 + 105*a^4*d^4)*x^4 + 2*(35*a*b^3*c^
4 - 11*a^2*b^2*c^3*d - 11*a^3*b*c^2*d^2 + 35*a^4*c*d^3)*x^3 - 8*(7*a^2*b^2*c^4 -
2*a^3*b*c^3*d + 7*a^4*c^2*d^2)*x^2 + 48*(a^3*b*c^4 + a^4*c^3*d)*x)*sqrt(a*c)*sq
rt(b*x + a)*sqrt(d*x + c))/(sqrt(a*c)*a^4*c^4*x^5), -1/3840*(15*(7*b^5*c^5 - 5*a
*b^4*c^4*d - 2*a^2*b^3*c^3*d^2 - 2*a^3*b^2*c^2*d^3 - 5*a^4*b*c*d^4 + 7*a^5*d^5)*
x^5*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*(384*a^4*c^4 - (105*b^4*c^4 - 40*a*b^3*c^3*d - 34*a^2*b^2*c^2*d^2 - 40*
a^3*b*c*d^3 + 105*a^4*d^4)*x^4 + 2*(35*a*b^3*c^4 - 11*a^2*b^2*c^3*d - 11*a^3*b*c
^2*d^2 + 35*a^4*c*d^3)*x^3 - 8*(7*a^2*b^2*c^4 - 2*a^3*b*c^3*d + 7*a^4*c^2*d^2)*x
^2 + 48*(a^3*b*c^4 + a^4*c^3*d)*x)*sqrt(-a*c)*sqrt(b*x + a)*sqrt(d*x + c))/(sqrt
(-a*c)*a^4*c^4*x^5)]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{a + b x} \sqrt{c + d x}}{x^{6}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x+a)**(1/2)*(d*x+c)**(1/2)/x**6,x)

[Out]

Integral(sqrt(a + b*x)*sqrt(c + d*x)/x**6, 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^6,x, algorithm="giac")

[Out]

Exception raised: TypeError