3.570 \(\int \frac{\sqrt{a+b x}}{x^3 \sqrt{c+d x}} \, dx\)

Optimal. Leaf size=131 \[ \frac{(b c-a d) (3 a d+b c) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{4 a^{3/2} c^{5/2}}+\frac{\sqrt{a+b x} \sqrt{c+d x} (3 a d+b c)}{4 a c^2 x}-\frac{(a+b x)^{3/2} \sqrt{c+d x}}{2 a c x^2} \]

[Out]

((b*c + 3*a*d)*Sqrt[a + b*x]*Sqrt[c + d*x])/(4*a*c^2*x) - ((a + b*x)^(3/2)*Sqrt[
c + d*x])/(2*a*c*x^2) + ((b*c - a*d)*(b*c + 3*a*d)*ArcTanh[(Sqrt[c]*Sqrt[a + b*x
])/(Sqrt[a]*Sqrt[c + d*x])])/(4*a^(3/2)*c^(5/2))

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Rubi [A]  time = 0.223305, antiderivative size = 131, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ \frac{(b c-a d) (3 a d+b c) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{4 a^{3/2} c^{5/2}}+\frac{\sqrt{a+b x} \sqrt{c+d x} (3 a d+b c)}{4 a c^2 x}-\frac{(a+b x)^{3/2} \sqrt{c+d x}}{2 a c x^2} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[a + b*x]/(x^3*Sqrt[c + d*x]),x]

[Out]

((b*c + 3*a*d)*Sqrt[a + b*x]*Sqrt[c + d*x])/(4*a*c^2*x) - ((a + b*x)^(3/2)*Sqrt[
c + d*x])/(2*a*c*x^2) + ((b*c - a*d)*(b*c + 3*a*d)*ArcTanh[(Sqrt[c]*Sqrt[a + b*x
])/(Sqrt[a]*Sqrt[c + d*x])])/(4*a^(3/2)*c^(5/2))

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Rubi in Sympy [A]  time = 17.0343, size = 114, normalized size = 0.87 \[ - \frac{\left (a + b x\right )^{\frac{3}{2}} \sqrt{c + d x}}{2 a c x^{2}} + \frac{\sqrt{a + b x} \sqrt{c + d x} \left (3 a d + b c\right )}{4 a c^{2} x} - \frac{\left (a d - b c\right ) \left (3 a d + b c\right ) \operatorname{atanh}{\left (\frac{\sqrt{c} \sqrt{a + b x}}{\sqrt{a} \sqrt{c + d x}} \right )}}{4 a^{\frac{3}{2}} c^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**(1/2)/x**3/(d*x+c)**(1/2),x)

[Out]

-(a + b*x)**(3/2)*sqrt(c + d*x)/(2*a*c*x**2) + sqrt(a + b*x)*sqrt(c + d*x)*(3*a*
d + b*c)/(4*a*c**2*x) - (a*d - b*c)*(3*a*d + b*c)*atanh(sqrt(c)*sqrt(a + b*x)/(s
qrt(a)*sqrt(c + d*x)))/(4*a**(3/2)*c**(5/2))

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Mathematica [A]  time = 0.19229, size = 159, normalized size = 1.21 \[ -\frac{\log (x) (b c-a d) (3 a d+b c)}{8 a^{3/2} c^{5/2}}+\frac{(b c-a d) (3 a d+b c) \log \left (2 \sqrt{a} \sqrt{c} \sqrt{a+b x} \sqrt{c+d x}+2 a c+a d x+b c x\right )}{8 a^{3/2} c^{5/2}}+\sqrt{a+b x} \sqrt{c+d x} \left (\frac{3 a d-b c}{4 a c^2 x}-\frac{1}{2 c x^2}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[a + b*x]/(x^3*Sqrt[c + d*x]),x]

[Out]

(-1/(2*c*x^2) + (-(b*c) + 3*a*d)/(4*a*c^2*x))*Sqrt[a + b*x]*Sqrt[c + d*x] - ((b*
c - a*d)*(b*c + 3*a*d)*Log[x])/(8*a^(3/2)*c^(5/2)) + ((b*c - a*d)*(b*c + 3*a*d)*
Log[2*a*c + b*c*x + a*d*x + 2*Sqrt[a]*Sqrt[c]*Sqrt[a + b*x]*Sqrt[c + d*x]])/(8*a
^(3/2)*c^(5/2))

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Maple [B]  time = 0.031, size = 258, normalized size = 2. \[ -{\frac{1}{8\,{c}^{2}a{x}^{2}}\sqrt{bx+a}\sqrt{dx+c} \left ( 3\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac}{x}} \right ){x}^{2}{a}^{2}{d}^{2}-2\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac}{x}} \right ){x}^{2}abcd-\ln \left ({\frac{1}{x} \left ( adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac \right ) } \right ){x}^{2}{b}^{2}{c}^{2}-6\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }dax\sqrt{ac}+2\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }bcx\sqrt{ac}+4\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }ca\sqrt{ac} \right ){\frac{1}{\sqrt{ac}}}{\frac{1}{\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^(1/2)/x^3/(d*x+c)^(1/2),x)

[Out]

-1/8*(b*x+a)^(1/2)*(d*x+c)^(1/2)/a/c^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^2*a^2*d^2-2*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^2*a*b*c*d-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^2*b^2*c^2-6*((b*x+a)*(d*x+c))^(1/2)*d*a*x*(a*c)^(1/2)+2*((
b*x+a)*(d*x+c))^(1/2)*b*c*x*(a*c)^(1/2)+4*((b*x+a)*(d*x+c))^(1/2)*c*a*(a*c)^(1/2
))/((b*x+a)*(d*x+c))^(1/2)/x^2/(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^3),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.293353, size = 1, normalized size = 0.01 \[ \left [-\frac{{\left (b^{2} c^{2} + 2 \, a b c d - 3 \, a^{2} d^{2}\right )} x^{2} \log \left (-\frac{4 \,{\left (2 \, a^{2} c^{2} +{\left (a b c^{2} + a^{2} c d\right )} x\right )} \sqrt{b x + a} \sqrt{d x + c} -{\left (8 \, a^{2} c^{2} +{\left (b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2}\right )} x^{2} + 8 \,{\left (a b c^{2} + a^{2} c d\right )} x\right )} \sqrt{a c}}{x^{2}}\right ) + 4 \,{\left (2 \, a c +{\left (b c - 3 \, a d\right )} x\right )} \sqrt{a c} \sqrt{b x + a} \sqrt{d x + c}}{16 \, \sqrt{a c} a c^{2} x^{2}}, \frac{{\left (b^{2} c^{2} + 2 \, a b c d - 3 \, a^{2} d^{2}\right )} x^{2} \arctan \left (\frac{{\left (2 \, a c +{\left (b c + a d\right )} x\right )} \sqrt{-a c}}{2 \, \sqrt{b x + a} \sqrt{d x + c} a c}\right ) - 2 \,{\left (2 \, a c +{\left (b c - 3 \, a d\right )} x\right )} \sqrt{-a c} \sqrt{b x + a} \sqrt{d x + c}}{8 \, \sqrt{-a c} a c^{2} x^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(b*x + a)/(sqrt(d*x + c)*x^3),x, algorithm="fricas")

[Out]

[-1/16*((b^2*c^2 + 2*a*b*c*d - 3*a^2*d^2)*x^2*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*(2*a*c + (b*c - 3*a*d)*x
)*sqrt(a*c)*sqrt(b*x + a)*sqrt(d*x + c))/(sqrt(a*c)*a*c^2*x^2), 1/8*((b^2*c^2 +
2*a*b*c*d - 3*a^2*d^2)*x^2*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*(2*a*c + (b*c - 3*a*d)*x)*sqrt(-a*c)*sqrt(b*x +
a)*sqrt(d*x + c))/(sqrt(-a*c)*a*c^2*x^2)]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{a + b x}}{x^{3} \sqrt{c + d x}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x+a)**(1/2)/x**3/(d*x+c)**(1/2),x)

[Out]

Integral(sqrt(a + b*x)/(x**3*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(sqrt(b*x + a)/(sqrt(d*x + c)*x^3),x, algorithm="giac")

[Out]

Exception raised: TypeError