3.577 \(\int \frac{\sqrt{a+b x}}{x^2 (c+d x)^{3/2}} \, dx\)

Optimal. Leaf size=113 \[ -\frac{(b c-3 a d) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{\sqrt{a} c^{5/2}}+\frac{\sqrt{a+b x} (b c-3 a d)}{a c^2 \sqrt{c+d x}}-\frac{(a+b x)^{3/2}}{a c x \sqrt{c+d x}} \]

[Out]

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

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Rubi [A]  time = 0.215937, antiderivative size = 113, 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 \[ -\frac{(b c-3 a d) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{\sqrt{a} c^{5/2}}+\frac{\sqrt{a+b x} (b c-3 a d)}{a c^2 \sqrt{c+d x}}-\frac{(a+b x)^{3/2}}{a c x \sqrt{c+d x}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 18.9761, size = 114, normalized size = 1.01 \[ \frac{2 d \left (a + b x\right )^{\frac{3}{2}}}{c x \sqrt{c + d x} \left (a d - b c\right )} - \frac{\sqrt{a + b x} \sqrt{c + d x} \left (3 a d - b c\right )}{c^{2} x \left (a d - b c\right )} + \frac{\left (3 a d - b c\right ) \operatorname{atanh}{\left (\frac{\sqrt{c} \sqrt{a + b x}}{\sqrt{a} \sqrt{c + d x}} \right )}}{\sqrt{a} c^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.242354, size = 119, normalized size = 1.05 \[ \frac{-\frac{2 \sqrt{c} \sqrt{a+b x} (c+3 d x)}{x \sqrt{c+d x}}+\frac{\log (x) (b c-3 a d)}{\sqrt{a}}+\frac{(3 a d-b c) \log \left (2 \sqrt{a} \sqrt{c} \sqrt{a+b x} \sqrt{c+d x}+2 a c+a d x+b c x\right )}{\sqrt{a}}}{2 c^{5/2}} \]

Antiderivative was successfully verified.

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

[Out]

((-2*Sqrt[c]*Sqrt[a + b*x]*(c + 3*d*x))/(x*Sqrt[c + d*x]) + ((b*c - 3*a*d)*Log[x
])/Sqrt[a] + ((-(b*c) + 3*a*d)*Log[2*a*c + b*c*x + a*d*x + 2*Sqrt[a]*Sqrt[c]*Sqr
t[a + b*x]*Sqrt[c + d*x]])/Sqrt[a])/(2*c^(5/2))

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Maple [B]  time = 0.038, size = 267, normalized size = 2.4 \[{\frac{1}{2\,{c}^{2}x}\sqrt{bx+a} \left ( 3\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac}{x}} \right ){x}^{2}a{d}^{2}-\ln \left ({\frac{1}{x} \left ( adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac \right ) } \right ){x}^{2}bcd+3\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac}{x}} \right ) xacd-\ln \left ({\frac{1}{x} \left ( adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac \right ) } \right ) xb{c}^{2}-6\,xd\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }-2\,c\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) } \right ){\frac{1}{\sqrt{ac}}}{\frac{1}{\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }}}{\frac{1}{\sqrt{dx+c}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.304105, size = 1, normalized size = 0.01 \[ \left [-\frac{4 \, \sqrt{a c} \sqrt{b x + a}{\left (3 \, d x + c\right )} \sqrt{d x + c} +{\left ({\left (b c d - 3 \, a d^{2}\right )} x^{2} +{\left (b c^{2} - 3 \, a c d\right )} x\right )} \log \left (\frac{4 \,{\left (2 \, a^{2} c^{2} +{\left (a b c^{2} + a^{2} c d\right )} x\right )} \sqrt{b x + a} \sqrt{d x + c} +{\left (8 \, a^{2} c^{2} +{\left (b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2}\right )} x^{2} + 8 \,{\left (a b c^{2} + a^{2} c d\right )} x\right )} \sqrt{a c}}{x^{2}}\right )}{4 \,{\left (c^{2} d x^{2} + c^{3} x\right )} \sqrt{a c}}, -\frac{2 \, \sqrt{-a c} \sqrt{b x + a}{\left (3 \, d x + c\right )} \sqrt{d x + c} +{\left ({\left (b c d - 3 \, a d^{2}\right )} x^{2} +{\left (b c^{2} - 3 \, a c d\right )} x\right )} \arctan \left (\frac{{\left (2 \, a c +{\left (b c + a d\right )} x\right )} \sqrt{-a c}}{2 \, \sqrt{b x + a} \sqrt{d x + c} a c}\right )}{2 \,{\left (c^{2} d x^{2} + c^{3} x\right )} \sqrt{-a c}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(b*x + a)/((d*x + c)^(3/2)*x^2),x, algorithm="fricas")

[Out]

[-1/4*(4*sqrt(a*c)*sqrt(b*x + a)*(3*d*x + c)*sqrt(d*x + c) + ((b*c*d - 3*a*d^2)*
x^2 + (b*c^2 - 3*a*c*d)*x)*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))/((c^2*d*x^2 + c^3*x)*sqrt(a*c)), -1/2*(2*sqrt(-
a*c)*sqrt(b*x + a)*(3*d*x + c)*sqrt(d*x + c) + ((b*c*d - 3*a*d^2)*x^2 + (b*c^2 -
3*a*c*d)*x)*arctan(1/2*(2*a*c + (b*c + a*d)*x)*sqrt(-a*c)/(sqrt(b*x + a)*sqrt(d
*x + c)*a*c)))/((c^2*d*x^2 + c^3*x)*sqrt(-a*c))]

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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**2/(d*x+c)**(3/2),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)/((d*x + c)^(3/2)*x^2),x, algorithm="giac")

[Out]

Exception raised: TypeError