3.231 \(\int \frac{A+B x}{x^{5/2} \sqrt{b x+c x^2}} \, dx\)

Optimal. Leaf size=105 \[ \frac{c (4 b B-3 A c) \tanh ^{-1}\left (\frac{\sqrt{b x+c x^2}}{\sqrt{b} \sqrt{x}}\right )}{4 b^{5/2}}-\frac{\sqrt{b x+c x^2} (4 b B-3 A c)}{4 b^2 x^{3/2}}-\frac{A \sqrt{b x+c x^2}}{2 b x^{5/2}} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.212121, antiderivative size = 105, 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 \[ \frac{c (4 b B-3 A c) \tanh ^{-1}\left (\frac{\sqrt{b x+c x^2}}{\sqrt{b} \sqrt{x}}\right )}{4 b^{5/2}}-\frac{\sqrt{b x+c x^2} (4 b B-3 A c)}{4 b^2 x^{3/2}}-\frac{A \sqrt{b x+c x^2}}{2 b x^{5/2}} \]

Antiderivative was successfully verified.

[In]  Int[(A + B*x)/(x^(5/2)*Sqrt[b*x + c*x^2]),x]

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 13.7927, size = 95, normalized size = 0.9 \[ - \frac{A \sqrt{b x + c x^{2}}}{2 b x^{\frac{5}{2}}} + \frac{\left (3 A c - 4 B b\right ) \sqrt{b x + c x^{2}}}{4 b^{2} x^{\frac{3}{2}}} - \frac{c \left (3 A c - 4 B b\right ) \operatorname{atanh}{\left (\frac{\sqrt{b x + c x^{2}}}{\sqrt{b} \sqrt{x}} \right )}}{4 b^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x+A)/x**(5/2)/(c*x**2+b*x)**(1/2),x)

[Out]

-A*sqrt(b*x + c*x**2)/(2*b*x**(5/2)) + (3*A*c - 4*B*b)*sqrt(b*x + c*x**2)/(4*b**
2*x**(3/2)) - c*(3*A*c - 4*B*b)*atanh(sqrt(b*x + c*x**2)/(sqrt(b)*sqrt(x)))/(4*b
**(5/2))

_______________________________________________________________________________________

Mathematica [A]  time = 0.121206, size = 92, normalized size = 0.88 \[ \frac{c x^2 \sqrt{b+c x} (4 b B-3 A c) \tanh ^{-1}\left (\frac{\sqrt{b+c x}}{\sqrt{b}}\right )-\sqrt{b} (b+c x) (2 A b-3 A c x+4 b B x)}{4 b^{5/2} x^{3/2} \sqrt{x (b+c x)}} \]

Antiderivative was successfully verified.

[In]  Integrate[(A + B*x)/(x^(5/2)*Sqrt[b*x + c*x^2]),x]

[Out]

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

_______________________________________________________________________________________

Maple [A]  time = 0.028, size = 109, normalized size = 1. \[ -{\frac{1}{4}\sqrt{x \left ( cx+b \right ) } \left ( 3\,A{\it Artanh} \left ({\frac{\sqrt{cx+b}}{\sqrt{b}}} \right ){x}^{2}{c}^{2}-4\,B{\it Artanh} \left ({\frac{\sqrt{cx+b}}{\sqrt{b}}} \right ){x}^{2}bc-3\,Axc\sqrt{cx+b}\sqrt{b}+4\,Bx{b}^{3/2}\sqrt{cx+b}+2\,A{b}^{3/2}\sqrt{cx+b} \right ){b}^{-{\frac{5}{2}}}{x}^{-{\frac{5}{2}}}{\frac{1}{\sqrt{cx+b}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)/x^(5/2)/(c*x^2+b*x)^(1/2),x)

[Out]

-1/4*(x*(c*x+b))^(1/2)/b^(5/2)*(3*A*arctanh((c*x+b)^(1/2)/b^(1/2))*x^2*c^2-4*B*a
rctanh((c*x+b)^(1/2)/b^(1/2))*x^2*b*c-3*A*x*c*(c*x+b)^(1/2)*b^(1/2)+4*B*x*b^(3/2
)*(c*x+b)^(1/2)+2*A*b^(3/2)*(c*x+b)^(1/2))/x^(5/2)/(c*x+b)^(1/2)

_______________________________________________________________________________________

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)/(sqrt(c*x^2 + b*x)*x^(5/2)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.294539, size = 1, normalized size = 0.01 \[ \left [-\frac{{\left (4 \, B b c - 3 \, A c^{2}\right )} x^{3} \log \left (\frac{2 \, \sqrt{c x^{2} + b x} b \sqrt{x} -{\left (c x^{2} + 2 \, b x\right )} \sqrt{b}}{x^{2}}\right ) + 2 \, \sqrt{c x^{2} + b x}{\left (2 \, A b +{\left (4 \, B b - 3 \, A c\right )} x\right )} \sqrt{b} \sqrt{x}}{8 \, b^{\frac{5}{2}} x^{3}}, \frac{{\left (4 \, B b c - 3 \, A c^{2}\right )} x^{3} \arctan \left (\frac{\sqrt{-b} \sqrt{x}}{\sqrt{c x^{2} + b x}}\right ) - \sqrt{c x^{2} + b x}{\left (2 \, A b +{\left (4 \, B b - 3 \, A c\right )} x\right )} \sqrt{-b} \sqrt{x}}{4 \, \sqrt{-b} b^{2} x^{3}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)/(sqrt(c*x^2 + b*x)*x^(5/2)),x, algorithm="fricas")

[Out]

[-1/8*((4*B*b*c - 3*A*c^2)*x^3*log((2*sqrt(c*x^2 + b*x)*b*sqrt(x) - (c*x^2 + 2*b
*x)*sqrt(b))/x^2) + 2*sqrt(c*x^2 + b*x)*(2*A*b + (4*B*b - 3*A*c)*x)*sqrt(b)*sqrt
(x))/(b^(5/2)*x^3), 1/4*((4*B*b*c - 3*A*c^2)*x^3*arctan(sqrt(-b)*sqrt(x)/sqrt(c*
x^2 + b*x)) - sqrt(c*x^2 + b*x)*(2*A*b + (4*B*b - 3*A*c)*x)*sqrt(-b)*sqrt(x))/(s
qrt(-b)*b^2*x^3)]

_______________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{A + B x}{x^{\frac{5}{2}} \sqrt{x \left (b + c x\right )}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x+A)/x**(5/2)/(c*x**2+b*x)**(1/2),x)

[Out]

Integral((A + B*x)/(x**(5/2)*sqrt(x*(b + c*x))), x)

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.309137, size = 150, normalized size = 1.43 \[ -\frac{\frac{{\left (4 \, B b c^{2} - 3 \, A c^{3}\right )} \arctan \left (\frac{\sqrt{c x + b}}{\sqrt{-b}}\right )}{\sqrt{-b} b^{2}} + \frac{4 \,{\left (c x + b\right )}^{\frac{3}{2}} B b c^{2} - 4 \, \sqrt{c x + b} B b^{2} c^{2} - 3 \,{\left (c x + b\right )}^{\frac{3}{2}} A c^{3} + 5 \, \sqrt{c x + b} A b c^{3}}{b^{2} c^{2} x^{2}}}{4 \, c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)/(sqrt(c*x^2 + b*x)*x^(5/2)),x, algorithm="giac")

[Out]

-1/4*((4*B*b*c^2 - 3*A*c^3)*arctan(sqrt(c*x + b)/sqrt(-b))/(sqrt(-b)*b^2) + (4*(
c*x + b)^(3/2)*B*b*c^2 - 4*sqrt(c*x + b)*B*b^2*c^2 - 3*(c*x + b)^(3/2)*A*c^3 + 5
*sqrt(c*x + b)*A*b*c^3)/(b^2*c^2*x^2))/c