3.238 \(\int \frac{A+B x}{\sqrt{x} \left (b x+c x^2\right )^{3/2}} \, dx\)

Optimal. Leaf size=97 \[ -\frac{(2 b B-3 A c) \tanh ^{-1}\left (\frac{\sqrt{b x+c x^2}}{\sqrt{b} \sqrt{x}}\right )}{b^{5/2}}+\frac{\sqrt{x} (2 b B-3 A c)}{b^2 \sqrt{b x+c x^2}}-\frac{A}{b \sqrt{x} \sqrt{b x+c x^2}} \]

[Out]

-(A/(b*Sqrt[x]*Sqrt[b*x + c*x^2])) + ((2*b*B - 3*A*c)*Sqrt[x])/(b^2*Sqrt[b*x + c
*x^2]) - ((2*b*B - 3*A*c)*ArcTanh[Sqrt[b*x + c*x^2]/(Sqrt[b]*Sqrt[x])])/b^(5/2)

_______________________________________________________________________________________

Rubi [A]  time = 0.189481, antiderivative size = 97, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ -\frac{(2 b B-3 A c) \tanh ^{-1}\left (\frac{\sqrt{b x+c x^2}}{\sqrt{b} \sqrt{x}}\right )}{b^{5/2}}+\frac{\sqrt{x} (2 b B-3 A c)}{b^2 \sqrt{b x+c x^2}}-\frac{A}{b \sqrt{x} \sqrt{b x+c x^2}} \]

Antiderivative was successfully verified.

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

[Out]

-(A/(b*Sqrt[x]*Sqrt[b*x + c*x^2])) + ((2*b*B - 3*A*c)*Sqrt[x])/(b^2*Sqrt[b*x + c
*x^2]) - ((2*b*B - 3*A*c)*ArcTanh[Sqrt[b*x + c*x^2]/(Sqrt[b]*Sqrt[x])])/b^(5/2)

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 13.9486, size = 92, normalized size = 0.95 \[ - \frac{A}{b \sqrt{x} \sqrt{b x + c x^{2}}} - \frac{2 \sqrt{x} \left (\frac{3 A c}{2} - B b\right )}{b^{2} \sqrt{b x + c x^{2}}} + \frac{2 \left (\frac{3 A c}{2} - B b\right ) \operatorname{atanh}{\left (\frac{\sqrt{b x + c x^{2}}}{\sqrt{b} \sqrt{x}} \right )}}{b^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-A/(b*sqrt(x)*sqrt(b*x + c*x**2)) - 2*sqrt(x)*(3*A*c/2 - B*b)/(b**2*sqrt(b*x + c
*x**2)) + 2*(3*A*c/2 - B*b)*atanh(sqrt(b*x + c*x**2)/(sqrt(b)*sqrt(x)))/b**(5/2)

_______________________________________________________________________________________

Mathematica [A]  time = 0.114557, size = 81, normalized size = 0.84 \[ \frac{\sqrt{b} (2 b B x-A (b+3 c x))-x \sqrt{b+c x} (2 b B-3 A c) \tanh ^{-1}\left (\frac{\sqrt{b+c x}}{\sqrt{b}}\right )}{b^{5/2} \sqrt{x} \sqrt{x (b+c x)}} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Maple [A]  time = 0.032, size = 94, normalized size = 1. \[{\frac{1}{cx+b}\sqrt{x \left ( cx+b \right ) } \left ( 3\,A{\it Artanh} \left ({\frac{\sqrt{cx+b}}{\sqrt{b}}} \right ) \sqrt{cx+b}xc-2\,B{\it Artanh} \left ({\frac{\sqrt{cx+b}}{\sqrt{b}}} \right ) \sqrt{cx+b}xb-3\,A\sqrt{b}xc+2\,B{b}^{3/2}x-A{b}^{{\frac{3}{2}}} \right ){x}^{-{\frac{3}{2}}}{b}^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(x*(c*x+b))^(1/2)*(3*A*arctanh((c*x+b)^(1/2)/b^(1/2))*(c*x+b)^(1/2)*x*c-2*B*arct
anh((c*x+b)^(1/2)/b^(1/2))*(c*x+b)^(1/2)*x*b-3*A*b^(1/2)*x*c+2*B*b^(3/2)*x-A*b^(
3/2))/x^(3/2)/(c*x+b)/b^(5/2)

_______________________________________________________________________________________

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((B*x + A)/((c*x^2 + b*x)^(3/2)*sqrt(x)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.30973, size = 1, normalized size = 0.01 \[ \left [-\frac{2 \, \sqrt{c x^{2} + b x}{\left (A b -{\left (2 \, B b - 3 \, A c\right )} x\right )} \sqrt{b} \sqrt{x} +{\left ({\left (2 \, B b c - 3 \, A c^{2}\right )} x^{3} +{\left (2 \, B b^{2} - 3 \, A b c\right )} x^{2}\right )} \log \left (-\frac{2 \, \sqrt{c x^{2} + b x} b \sqrt{x} +{\left (c x^{2} + 2 \, b x\right )} \sqrt{b}}{x^{2}}\right )}{2 \,{\left (b^{2} c x^{3} + b^{3} x^{2}\right )} \sqrt{b}}, -\frac{\sqrt{c x^{2} + b x}{\left (A b -{\left (2 \, B b - 3 \, A c\right )} x\right )} \sqrt{-b} \sqrt{x} +{\left ({\left (2 \, B b c - 3 \, A c^{2}\right )} x^{3} +{\left (2 \, B b^{2} - 3 \, A b c\right )} x^{2}\right )} \arctan \left (\frac{\sqrt{-b} \sqrt{x}}{\sqrt{c x^{2} + b x}}\right )}{{\left (b^{2} c x^{3} + b^{3} x^{2}\right )} \sqrt{-b}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[-1/2*(2*sqrt(c*x^2 + b*x)*(A*b - (2*B*b - 3*A*c)*x)*sqrt(b)*sqrt(x) + ((2*B*b*c
- 3*A*c^2)*x^3 + (2*B*b^2 - 3*A*b*c)*x^2)*log(-(2*sqrt(c*x^2 + b*x)*b*sqrt(x) +
(c*x^2 + 2*b*x)*sqrt(b))/x^2))/((b^2*c*x^3 + b^3*x^2)*sqrt(b)), -(sqrt(c*x^2 +
b*x)*(A*b - (2*B*b - 3*A*c)*x)*sqrt(-b)*sqrt(x) + ((2*B*b*c - 3*A*c^2)*x^3 + (2*
B*b^2 - 3*A*b*c)*x^2)*arctan(sqrt(-b)*sqrt(x)/sqrt(c*x^2 + b*x)))/((b^2*c*x^3 +
b^3*x^2)*sqrt(-b))]

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral((A + B*x)/(sqrt(x)*(x*(b + c*x))**(3/2)), x)

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.30415, size = 117, normalized size = 1.21 \[ \frac{{\left (2 \, B b - 3 \, A c\right )} \arctan \left (\frac{\sqrt{c x + b}}{\sqrt{-b}}\right )}{\sqrt{-b} b^{2}} + \frac{2 \,{\left (c x + b\right )} B b - 2 \, B b^{2} - 3 \,{\left (c x + b\right )} A c + 2 \, A b c}{{\left ({\left (c x + b\right )}^{\frac{3}{2}} - \sqrt{c x + b} b\right )} b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(2*B*b - 3*A*c)*arctan(sqrt(c*x + b)/sqrt(-b))/(sqrt(-b)*b^2) + (2*(c*x + b)*B*b
- 2*B*b^2 - 3*(c*x + b)*A*c + 2*A*b*c)/(((c*x + b)^(3/2) - sqrt(c*x + b)*b)*b^2
)