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

Optimal. Leaf size=66 \[ \frac{2 \sqrt{a+b x}}{c \sqrt{c+d x}}-\frac{2 \sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{c^{3/2}} \]

[Out]

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

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Rubi [A]  time = 0.119295, antiderivative size = 66, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136 \[ \frac{2 \sqrt{a+b x}}{c \sqrt{c+d x}}-\frac{2 \sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{c^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 9.96553, size = 60, normalized size = 0.91 \[ - \frac{2 \sqrt{a} \operatorname{atanh}{\left (\frac{\sqrt{c} \sqrt{a + b x}}{\sqrt{a} \sqrt{c + d x}} \right )}}{c^{\frac{3}{2}}} + \frac{2 \sqrt{a + b x}}{c \sqrt{c + d x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.121194, size = 93, normalized size = 1.41 \[ -\frac{\sqrt{a} \log \left (2 \sqrt{a} \sqrt{c} \sqrt{a+b x} \sqrt{c+d x}+2 a c+a d x+b c x\right )}{c^{3/2}}+\frac{2 \sqrt{a+b x}}{c \sqrt{c+d x}}+\frac{\sqrt{a} \log (x)}{c^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.035, size = 143, normalized size = 2.2 \[{\frac{1}{c} \left ( -\ln \left ({\frac{1}{x} \left ( adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac \right ) } \right ) xad-\ln \left ({\frac{1}{x} \left ( adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac \right ) } \right ) ac+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) } \right ) \sqrt{bx+a}{\frac{1}{\sqrt{ac}}}{\frac{1}{\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }}}{\frac{1}{\sqrt{dx+c}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.283291, size = 1, normalized size = 0.02 \[ \left [\frac{{\left (d x + c\right )} \sqrt{\frac{a}{c}} \log \left (\frac{8 \, a^{2} c^{2} +{\left (b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2}\right )} x^{2} - 4 \,{\left (2 \, a c^{2} +{\left (b c^{2} + a c d\right )} x\right )} \sqrt{b x + a} \sqrt{d x + c} \sqrt{\frac{a}{c}} + 8 \,{\left (a b c^{2} + a^{2} c d\right )} x}{x^{2}}\right ) + 4 \, \sqrt{b x + a} \sqrt{d x + c}}{2 \,{\left (c d x + c^{2}\right )}}, -\frac{{\left (d x + c\right )} \sqrt{-\frac{a}{c}} \arctan \left (\frac{2 \, a c +{\left (b c + a d\right )} x}{2 \, \sqrt{b x + a} \sqrt{d x + c} c \sqrt{-\frac{a}{c}}}\right ) - 2 \, \sqrt{b x + a} \sqrt{d x + c}}{c d x + c^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/2*((d*x + c)*sqrt(a/c)*log((8*a^2*c^2 + (b^2*c^2 + 6*a*b*c*d + a^2*d^2)*x^2 -
4*(2*a*c^2 + (b*c^2 + a*c*d)*x)*sqrt(b*x + a)*sqrt(d*x + c)*sqrt(a/c) + 8*(a*b*
c^2 + a^2*c*d)*x)/x^2) + 4*sqrt(b*x + a)*sqrt(d*x + c))/(c*d*x + c^2), -((d*x +
c)*sqrt(-a/c)*arctan(1/2*(2*a*c + (b*c + a*d)*x)/(sqrt(b*x + a)*sqrt(d*x + c)*c*
sqrt(-a/c))) - 2*sqrt(b*x + a)*sqrt(d*x + c))/(c*d*x + c^2)]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{a + b x}}{x \left (c + d x\right )^{\frac{3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(sqrt(a + b*x)/(x*(c + d*x)**(3/2)), x)

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GIAC/XCAS [A]  time = 0.235624, size = 176, normalized size = 2.67 \[ -\frac{2 \, \sqrt{b d} a b \arctan \left (-\frac{b^{2} c + a b d -{\left (\sqrt{b d} \sqrt{b x + a} - \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d}\right )}^{2}}{2 \, \sqrt{-a b c d} b}\right )}{\sqrt{-a b c d} c{\left | b \right |}} + \frac{2 \, \sqrt{b x + a} b^{2}}{\sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d} c{\left | b \right |}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*sqrt(b*d)*a*b*arctan(-1/2*(b^2*c + a*b*d - (sqrt(b*d)*sqrt(b*x + a) - sqrt(b^
2*c + (b*x + a)*b*d - a*b*d))^2)/(sqrt(-a*b*c*d)*b))/(sqrt(-a*b*c*d)*c*abs(b)) +
2*sqrt(b*x + a)*b^2/(sqrt(b^2*c + (b*x + a)*b*d - a*b*d)*c*abs(b))