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

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

[Out]

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

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Rubi [A]  time = 0.227353, 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 \[ -\frac{(b c-a d) (a d+3 b c) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{4 a^{5/2} c^{3/2}}+\frac{\sqrt{a+b x} \sqrt{c+d x} (a d+3 b c)}{4 a^2 c x}-\frac{\sqrt{a+b x} (c+d x)^{3/2}}{2 a c x^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 16.8126, size = 114, normalized size = 0.87 \[ - \frac{\sqrt{a + b x} \left (c + d x\right )^{\frac{3}{2}}}{2 a c x^{2}} + \frac{\sqrt{a + b x} \sqrt{c + d x} \left (a d + 3 b c\right )}{4 a^{2} c x} + \frac{\left (a d - b c\right ) \left (a d + 3 b c\right ) \operatorname{atanh}{\left (\frac{\sqrt{c} \sqrt{a + b x}}{\sqrt{a} \sqrt{c + d x}} \right )}}{4 a^{\frac{5}{2}} c^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.194389, size = 159, normalized size = 1.21 \[ \frac{\log (x) (b c-a d) (a d+3 b c)}{8 a^{5/2} c^{3/2}}-\frac{(b c-a d) (a d+3 b c) \log \left (2 \sqrt{a} \sqrt{c} \sqrt{a+b x} \sqrt{c+d x}+2 a c+a d x+b c x\right )}{8 a^{5/2} c^{3/2}}+\sqrt{a+b x} \sqrt{c+d x} \left (\frac{3 b c-a d}{4 a^2 c x}-\frac{1}{2 a x^2}\right ) \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.033, size = 257, normalized size = 2. \[{\frac{1}{8\,{a}^{2}c{x}^{2}}\sqrt{bx+a}\sqrt{dx+c} \left ( \ln \left ({\frac{1}{x} \left ( adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac \right ) } \right ){x}^{2}{a}^{2}{d}^{2}+2\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac}{x}} \right ){x}^{2}abcd-3\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac}{x}} \right ){x}^{2}{b}^{2}{c}^{2}-2\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }dax\sqrt{ac}+6\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }bcx\sqrt{ac}-4\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }ca\sqrt{ac} \right ){\frac{1}{\sqrt{ac}}}{\frac{1}{\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.331572, size = 1, normalized size = 0.01 \[ \left [-\frac{{\left (3 \, b^{2} c^{2} - 2 \, a b c d - a^{2} d^{2}\right )} x^{2} \log \left (\frac{4 \,{\left (2 \, a^{2} c^{2} +{\left (a b c^{2} + a^{2} c d\right )} x\right )} \sqrt{b x + a} \sqrt{d x + c} +{\left (8 \, a^{2} c^{2} +{\left (b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2}\right )} x^{2} + 8 \,{\left (a b c^{2} + a^{2} c d\right )} x\right )} \sqrt{a c}}{x^{2}}\right ) + 4 \,{\left (2 \, a c -{\left (3 \, b c - a d\right )} x\right )} \sqrt{a c} \sqrt{b x + a} \sqrt{d x + c}}{16 \, \sqrt{a c} a^{2} c x^{2}}, -\frac{{\left (3 \, b^{2} c^{2} - 2 \, a b c d - a^{2} d^{2}\right )} x^{2} \arctan \left (\frac{{\left (2 \, a c +{\left (b c + a d\right )} x\right )} \sqrt{-a c}}{2 \, \sqrt{b x + a} \sqrt{d x + c} a c}\right ) + 2 \,{\left (2 \, a c -{\left (3 \, b c - a d\right )} x\right )} \sqrt{-a c} \sqrt{b x + a} \sqrt{d x + c}}{8 \, \sqrt{-a c} a^{2} c x^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: TypeError