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

Optimal. Leaf size=279 \[ \frac{\sqrt{a+b x} \sqrt{c+d x} \left (-35 a^2 d^2+6 a b c d+5 b^2 c^2\right )}{96 a^2 c^3 x^2}-\frac{\sqrt{a+b x} \sqrt{c+d x} \left (-105 a^3 d^3+25 a^2 b c d^2+17 a b^2 c^2 d+15 b^3 c^3\right )}{192 a^3 c^4 x}+\frac{(b c-a d) \left (35 a^3 d^3+15 a^2 b c d^2+9 a b^2 c^2 d+5 b^3 c^3\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{64 a^{7/2} c^{9/2}}-\frac{\sqrt{a+b x} \sqrt{c+d x} (b c-7 a d)}{24 a c^2 x^3}-\frac{\sqrt{a+b x} \sqrt{c+d x}}{4 c x^4} \]

[Out]

-(Sqrt[a + b*x]*Sqrt[c + d*x])/(4*c*x^4) - ((b*c - 7*a*d)*Sqrt[a + b*x]*Sqrt[c +
d*x])/(24*a*c^2*x^3) + ((5*b^2*c^2 + 6*a*b*c*d - 35*a^2*d^2)*Sqrt[a + b*x]*Sqrt
[c + d*x])/(96*a^2*c^3*x^2) - ((15*b^3*c^3 + 17*a*b^2*c^2*d + 25*a^2*b*c*d^2 - 1
05*a^3*d^3)*Sqrt[a + b*x]*Sqrt[c + d*x])/(192*a^3*c^4*x) + ((b*c - a*d)*(5*b^3*c
^3 + 9*a*b^2*c^2*d + 15*a^2*b*c*d^2 + 35*a^3*d^3)*ArcTanh[(Sqrt[c]*Sqrt[a + b*x]
)/(Sqrt[a]*Sqrt[c + d*x])])/(64*a^(7/2)*c^(9/2))

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Rubi [A]  time = 0.778308, antiderivative size = 279, 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 \[ \frac{\sqrt{a+b x} \sqrt{c+d x} \left (-35 a^2 d^2+6 a b c d+5 b^2 c^2\right )}{96 a^2 c^3 x^2}-\frac{\sqrt{a+b x} \sqrt{c+d x} \left (-105 a^3 d^3+25 a^2 b c d^2+17 a b^2 c^2 d+15 b^3 c^3\right )}{192 a^3 c^4 x}+\frac{(b c-a d) \left (35 a^3 d^3+15 a^2 b c d^2+9 a b^2 c^2 d+5 b^3 c^3\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{64 a^{7/2} c^{9/2}}-\frac{\sqrt{a+b x} \sqrt{c+d x} (b c-7 a d)}{24 a c^2 x^3}-\frac{\sqrt{a+b x} \sqrt{c+d x}}{4 c x^4} \]

Antiderivative was successfully verified.

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

[Out]

-(Sqrt[a + b*x]*Sqrt[c + d*x])/(4*c*x^4) - ((b*c - 7*a*d)*Sqrt[a + b*x]*Sqrt[c +
d*x])/(24*a*c^2*x^3) + ((5*b^2*c^2 + 6*a*b*c*d - 35*a^2*d^2)*Sqrt[a + b*x]*Sqrt
[c + d*x])/(96*a^2*c^3*x^2) - ((15*b^3*c^3 + 17*a*b^2*c^2*d + 25*a^2*b*c*d^2 - 1
05*a^3*d^3)*Sqrt[a + b*x]*Sqrt[c + d*x])/(192*a^3*c^4*x) + ((b*c - a*d)*(5*b^3*c
^3 + 9*a*b^2*c^2*d + 15*a^2*b*c*d^2 + 35*a^3*d^3)*ArcTanh[(Sqrt[c]*Sqrt[a + b*x]
)/(Sqrt[a]*Sqrt[c + d*x])])/(64*a^(7/2)*c^(9/2))

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Rubi in Sympy [A]  time = 129.182, size = 269, normalized size = 0.96 \[ - \frac{\sqrt{a + b x} \sqrt{c + d x}}{4 c x^{4}} + \frac{\sqrt{a + b x} \sqrt{c + d x} \left (7 a d - b c\right )}{24 a c^{2} x^{3}} - \frac{\sqrt{a + b x} \sqrt{c + d x} \left (35 a^{2} d^{2} - 6 a b c d - 5 b^{2} c^{2}\right )}{96 a^{2} c^{3} x^{2}} + \frac{\sqrt{a + b x} \sqrt{c + d x} \left (105 a^{3} d^{3} - 25 a^{2} b c d^{2} - 17 a b^{2} c^{2} d - 15 b^{3} c^{3}\right )}{192 a^{3} c^{4} x} - \frac{\left (a d - b c\right ) \left (35 a^{3} d^{3} + 15 a^{2} b c d^{2} + 9 a b^{2} c^{2} d + 5 b^{3} c^{3}\right ) \operatorname{atanh}{\left (\frac{\sqrt{c} \sqrt{a + b x}}{\sqrt{a} \sqrt{c + d x}} \right )}}{64 a^{\frac{7}{2}} c^{\frac{9}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(a + b*x)*sqrt(c + d*x)/(4*c*x**4) + sqrt(a + b*x)*sqrt(c + d*x)*(7*a*d - b
*c)/(24*a*c**2*x**3) - sqrt(a + b*x)*sqrt(c + d*x)*(35*a**2*d**2 - 6*a*b*c*d - 5
*b**2*c**2)/(96*a**2*c**3*x**2) + sqrt(a + b*x)*sqrt(c + d*x)*(105*a**3*d**3 - 2
5*a**2*b*c*d**2 - 17*a*b**2*c**2*d - 15*b**3*c**3)/(192*a**3*c**4*x) - (a*d - b*
c)*(35*a**3*d**3 + 15*a**2*b*c*d**2 + 9*a*b**2*c**2*d + 5*b**3*c**3)*atanh(sqrt(
c)*sqrt(a + b*x)/(sqrt(a)*sqrt(c + d*x)))/(64*a**(7/2)*c**(9/2))

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Mathematica [A]  time = 0.253508, size = 285, normalized size = 1.02 \[ \frac{-3 x^4 \log (x) (b c-a d) \left (35 a^3 d^3+15 a^2 b c d^2+9 a b^2 c^2 d+5 b^3 c^3\right )+3 x^4 (b c-a d) \left (35 a^3 d^3+15 a^2 b c d^2+9 a b^2 c^2 d+5 b^3 c^3\right ) \log \left (2 \sqrt{a} \sqrt{c} \sqrt{a+b x} \sqrt{c+d x}+2 a c+a d x+b c x\right )-2 \sqrt{a} \sqrt{c} \sqrt{a+b x} \sqrt{c+d x} \left (a^3 \left (48 c^3-56 c^2 d x+70 c d^2 x^2-105 d^3 x^3\right )+a^2 b c x \left (8 c^2-12 c d x+25 d^2 x^2\right )+a b^2 c^2 x^2 (17 d x-10 c)+15 b^3 c^3 x^3\right )}{384 a^{7/2} c^{9/2} x^4} \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[a + b*x]/(x^5*Sqrt[c + d*x]),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 + 17*d*x) + a^2*b*c*x*(8*c^2 - 12*c*d*x + 25*d^2*x^2) + a^3*(48*c^3 - 56*
c^2*d*x + 70*c*d^2*x^2 - 105*d^3*x^3)) - 3*(b*c - a*d)*(5*b^3*c^3 + 9*a*b^2*c^2*
d + 15*a^2*b*c*d^2 + 35*a^3*d^3)*x^4*Log[x] + 3*(b*c - a*d)*(5*b^3*c^3 + 9*a*b^2
*c^2*d + 15*a^2*b*c*d^2 + 35*a^3*d^3)*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^(9/2)*x^4)

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Maple [B]  time = 0.04, size = 593, normalized size = 2.1 \[ -{\frac{1}{384\,{a}^{3}{c}^{4}{x}^{4}}\sqrt{bx+a}\sqrt{dx+c} \left ( 105\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac}{x}} \right ){x}^{4}{a}^{4}{d}^{4}-60\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac}{x}} \right ){x}^{4}{a}^{3}bc{d}^{3}-18\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac}{x}} \right ){x}^{4}{a}^{2}{b}^{2}{c}^{2}{d}^{2}-12\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac}{x}} \right ){x}^{4}a{b}^{3}{c}^{3}d-15\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac}{x}} \right ){x}^{4}{b}^{4}{c}^{4}-210\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }{d}^{3}{a}^{3}{x}^{3}\sqrt{ac}+50\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }{d}^{2}bc{a}^{2}{x}^{3}\sqrt{ac}+34\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }d{b}^{2}{c}^{2}a{x}^{3}\sqrt{ac}+30\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }{b}^{3}{c}^{3}{x}^{3}\sqrt{ac}+140\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }{d}^{2}c{a}^{3}{x}^{2}\sqrt{ac}-24\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }db{c}^{2}{a}^{2}{x}^{2}\sqrt{ac}-20\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }{b}^{2}{c}^{3}a{x}^{2}\sqrt{ac}-112\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }d{c}^{2}{a}^{3}x\sqrt{ac}+16\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }b{c}^{3}{a}^{2}x\sqrt{ac}+96\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }{c}^{3}{a}^{3}\sqrt{ac} \right ){\frac{1}{\sqrt{ac}}}{\frac{1}{\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/384*(b*x+a)^(1/2)*(d*x+c)^(1/2)/a^3/c^4*(105*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^4*a^4*d^4-60*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^4*a^3*b*c*d^3-18*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^4*a^2*b^2*c^2*d^2-12*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^4*a*b^3*c^3*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^4*b^4*c^4-210*((b*x+a)*(d*x+c))^(1
/2)*d^3*a^3*x^3*(a*c)^(1/2)+50*((b*x+a)*(d*x+c))^(1/2)*d^2*b*c*a^2*x^3*(a*c)^(1/
2)+34*((b*x+a)*(d*x+c))^(1/2)*d*b^2*c^2*a*x^3*(a*c)^(1/2)+30*((b*x+a)*(d*x+c))^(
1/2)*b^3*c^3*x^3*(a*c)^(1/2)+140*((b*x+a)*(d*x+c))^(1/2)*d^2*c*a^3*x^2*(a*c)^(1/
2)-24*((b*x+a)*(d*x+c))^(1/2)*d*b*c^2*a^2*x^2*(a*c)^(1/2)-20*((b*x+a)*(d*x+c))^(
1/2)*b^2*c^3*a*x^2*(a*c)^(1/2)-112*((b*x+a)*(d*x+c))^(1/2)*d*c^2*a^3*x*(a*c)^(1/
2)+16*((b*x+a)*(d*x+c))^(1/2)*b*c^3*a^2*x*(a*c)^(1/2)+96*((b*x+a)*(d*x+c))^(1/2)
*c^3*a^3*(a*c)^(1/2))/((b*x+a)*(d*x+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 = 0.739525, size = 1, normalized size = 0. \[ \left [-\frac{3 \,{\left (5 \, b^{4} c^{4} + 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} + 20 \, a^{3} b c d^{3} - 35 \, a^{4} d^{4}\right )} x^{4} \log \left (-\frac{4 \,{\left (2 \, a^{2} c^{2} +{\left (a b c^{2} + a^{2} c d\right )} x\right )} \sqrt{b x + a} \sqrt{d x + c} -{\left (8 \, a^{2} c^{2} +{\left (b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2}\right )} x^{2} + 8 \,{\left (a b c^{2} + a^{2} c d\right )} x\right )} \sqrt{a c}}{x^{2}}\right ) + 4 \,{\left (48 \, a^{3} c^{3} +{\left (15 \, b^{3} c^{3} + 17 \, a b^{2} c^{2} d + 25 \, a^{2} b c d^{2} - 105 \, a^{3} d^{3}\right )} x^{3} - 2 \,{\left (5 \, a b^{2} c^{3} + 6 \, a^{2} b c^{2} d - 35 \, a^{3} c d^{2}\right )} x^{2} + 8 \,{\left (a^{2} b c^{3} - 7 \, a^{3} c^{2} d\right )} x\right )} \sqrt{a c} \sqrt{b x + a} \sqrt{d x + c}}{768 \, \sqrt{a c} a^{3} c^{4} x^{4}}, \frac{3 \,{\left (5 \, b^{4} c^{4} + 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} + 20 \, a^{3} b c d^{3} - 35 \, a^{4} d^{4}\right )} x^{4} \arctan \left (\frac{{\left (2 \, a c +{\left (b c + a d\right )} x\right )} \sqrt{-a c}}{2 \, \sqrt{b x + a} \sqrt{d x + c} a c}\right ) - 2 \,{\left (48 \, a^{3} c^{3} +{\left (15 \, b^{3} c^{3} + 17 \, a b^{2} c^{2} d + 25 \, a^{2} b c d^{2} - 105 \, a^{3} d^{3}\right )} x^{3} - 2 \,{\left (5 \, a b^{2} c^{3} + 6 \, a^{2} b c^{2} d - 35 \, a^{3} c d^{2}\right )} x^{2} + 8 \,{\left (a^{2} b c^{3} - 7 \, a^{3} c^{2} d\right )} x\right )} \sqrt{-a c} \sqrt{b x + a} \sqrt{d x + c}}{384 \, \sqrt{-a c} a^{3} c^{4} x^{4}}\right ] \]

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 + 6*a^2*b^2*c^2*d^2 + 20*a^3*b*c*d^3 - 35*
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)*sqrt(a*c))/x^2) + 4*(48*a^3*c^3 + (15*b^3*c^3 + 17*a*b^2*c^2*d + 25*a^2*b*c*d
^2 - 105*a^3*d^3)*x^3 - 2*(5*a*b^2*c^3 + 6*a^2*b*c^2*d - 35*a^3*c*d^2)*x^2 + 8*(
a^2*b*c^3 - 7*a^3*c^2*d)*x)*sqrt(a*c)*sqrt(b*x + a)*sqrt(d*x + c))/(sqrt(a*c)*a^
3*c^4*x^4), 1/384*(3*(5*b^4*c^4 + 4*a*b^3*c^3*d + 6*a^2*b^2*c^2*d^2 + 20*a^3*b*c
*d^3 - 35*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 + 17*a*b^2*c^2*d + 25*a^2*
b*c*d^2 - 105*a^3*d^3)*x^3 - 2*(5*a*b^2*c^3 + 6*a^2*b*c^2*d - 35*a^3*c*d^2)*x^2
+ 8*(a^2*b*c^3 - 7*a^3*c^2*d)*x)*sqrt(-a*c)*sqrt(b*x + a)*sqrt(d*x + c))/(sqrt(-
a*c)*a^3*c^4*x^4)]

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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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