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

Optimal. Leaf size=190 \[ \frac{\sqrt{a+b x} \sqrt{c+d x} (3 b c-5 a d) (3 a d+b c)}{24 a^2 c^3 x}-\frac{(b c-a d) \left (5 a^2 d^2+2 a b c d+b^2 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{8 a^{5/2} c^{7/2}}-\frac{\sqrt{a+b x} \sqrt{c+d x} (b c-5 a d)}{12 a c^2 x^2}-\frac{\sqrt{a+b x} \sqrt{c+d x}}{3 c x^3} \]

[Out]

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

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Rubi [A]  time = 0.498191, antiderivative size = 190, 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 \[ \frac{\sqrt{a+b x} \sqrt{c+d x} (3 b c-5 a d) (3 a d+b c)}{24 a^2 c^3 x}-\frac{(b c-a d) \left (5 a^2 d^2+2 a b c d+b^2 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{8 a^{5/2} c^{7/2}}-\frac{\sqrt{a+b x} \sqrt{c+d x} (b c-5 a d)}{12 a c^2 x^2}-\frac{\sqrt{a+b x} \sqrt{c+d x}}{3 c x^3} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 59.5199, size = 175, normalized size = 0.92 \[ - \frac{\sqrt{a + b x} \sqrt{c + d x}}{3 c x^{3}} + \frac{\sqrt{a + b x} \sqrt{c + d x} \left (5 a d - b c\right )}{12 a c^{2} x^{2}} - \frac{\sqrt{a + b x} \sqrt{c + d x} \left (3 a d + b c\right ) \left (5 a d - 3 b c\right )}{24 a^{2} c^{3} x} + \frac{\left (a d - b c\right ) \left (5 a^{2} d^{2} + 2 a b c d + b^{2} c^{2}\right ) \operatorname{atanh}{\left (\frac{\sqrt{c} \sqrt{a + b x}}{\sqrt{a} \sqrt{c + d x}} \right )}}{8 a^{\frac{5}{2}} c^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.18498, size = 211, normalized size = 1.11 \[ \frac{3 x^3 \log (x) (b c-a d) \left (5 a^2 d^2+2 a b c d+b^2 c^2\right )-3 x^3 (b c-a d) \left (5 a^2 d^2+2 a b c d+b^2 c^2\right ) \log \left (2 \sqrt{a} \sqrt{c} \sqrt{a+b x} \sqrt{c+d x}+2 a c+a d x+b c x\right )-2 \sqrt{a} \sqrt{c} \sqrt{a+b x} \sqrt{c+d x} \left (a^2 \left (8 c^2-10 c d x+15 d^2 x^2\right )+2 a b c x (c-2 d x)-3 b^2 c^2 x^2\right )}{48 a^{5/2} c^{7/2} x^3} \]

Antiderivative was successfully verified.

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

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Maple [B]  time = 0.036, size = 408, normalized size = 2.2 \[{\frac{1}{48\,{c}^{3}{a}^{2}{x}^{3}}\sqrt{bx+a}\sqrt{dx+c} \left ( 15\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac}{x}} \right ){x}^{3}{a}^{3}{d}^{3}-9\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac}{x}} \right ){x}^{3}{a}^{2}bc{d}^{2}-3\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac}{x}} \right ){x}^{3}a{b}^{2}{c}^{2}d-3\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac}{x}} \right ){x}^{3}{b}^{3}{c}^{3}-30\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }{d}^{2}{a}^{2}{x}^{2}\sqrt{ac}+8\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }dbca{x}^{2}\sqrt{ac}+6\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }{b}^{2}{c}^{2}{x}^{2}\sqrt{ac}+20\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }dc{a}^{2}x\sqrt{ac}-4\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }b{c}^{2}ax\sqrt{ac}-16\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }{c}^{2}{a}^{2}\sqrt{ac} \right ){\frac{1}{\sqrt{ac}}}{\frac{1}{\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/48*(b*x+a)^(1/2)*(d*x+c)^(1/2)/a^2/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^2-3*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-3*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)+8*((b*x+a)*(d*x+c))^(1/2)*d*b*c*a*x^2*(a*c)^(1/2)+6*((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)-4*((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))/((b*x+a)*(d*x+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 = 0.418214, size = 1, normalized size = 0.01 \[ \left [-\frac{3 \,{\left (b^{3} c^{3} + a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - 5 \, a^{3} d^{3}\right )} x^{3} \log \left (\frac{4 \,{\left (2 \, a^{2} c^{2} +{\left (a b c^{2} + a^{2} c d\right )} x\right )} \sqrt{b x + a} \sqrt{d x + c} +{\left (8 \, a^{2} c^{2} +{\left (b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2}\right )} x^{2} + 8 \,{\left (a b c^{2} + a^{2} c d\right )} x\right )} \sqrt{a c}}{x^{2}}\right ) + 4 \,{\left (8 \, a^{2} c^{2} -{\left (3 \, b^{2} c^{2} + 4 \, a b c d - 15 \, a^{2} d^{2}\right )} x^{2} + 2 \,{\left (a b c^{2} - 5 \, a^{2} c d\right )} x\right )} \sqrt{a c} \sqrt{b x + a} \sqrt{d x + c}}{96 \, \sqrt{a c} a^{2} c^{3} x^{3}}, -\frac{3 \,{\left (b^{3} c^{3} + a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - 5 \, a^{3} d^{3}\right )} x^{3} \arctan \left (\frac{{\left (2 \, a c +{\left (b c + a d\right )} x\right )} \sqrt{-a c}}{2 \, \sqrt{b x + a} \sqrt{d x + c} a c}\right ) + 2 \,{\left (8 \, a^{2} c^{2} -{\left (3 \, b^{2} c^{2} + 4 \, a b c d - 15 \, a^{2} d^{2}\right )} x^{2} + 2 \,{\left (a b c^{2} - 5 \, a^{2} c d\right )} x\right )} \sqrt{-a c} \sqrt{b x + a} \sqrt{d x + c}}{48 \, \sqrt{-a c} a^{2} c^{3} x^{3}}\right ] \]

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 + 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 - (3*b^2*c^2 + 4*a*b*c*d - 15*a^2*d^2)*x^2 + 2*(a*b*c^2 - 5*a^2*c*d)*x)*sq
rt(a*c)*sqrt(b*x + a)*sqrt(d*x + c))/(sqrt(a*c)*a^2*c^3*x^3), -1/48*(3*(b^3*c^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 - (3*b^2*c^2 + 4*
a*b*c*d - 15*a^2*d^2)*x^2 + 2*(a*b*c^2 - 5*a^2*c*d)*x)*sqrt(-a*c)*sqrt(b*x + a)*
sqrt(d*x + c))/(sqrt(-a*c)*a^2*c^3*x^3)]

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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**4/(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^4),x, algorithm="giac")

[Out]

Exception raised: TypeError