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

Optimal. Leaf size=146 \[ -\frac{(2 b B-5 A c) \tanh ^{-1}\left (\frac{\sqrt{b x+c x^2}}{\sqrt{b} \sqrt{x}}\right )}{b^{7/2}}+\frac{\sqrt{x} (2 b B-5 A c)}{b^3 \sqrt{b x+c x^2}}+\frac{2 b B-5 A c}{3 b^2 c \sqrt{x} \sqrt{b x+c x^2}}-\frac{2 \sqrt{x} (b B-A c)}{3 b c \left (b x+c x^2\right )^{3/2}} \]

[Out]

(-2*(b*B - A*c)*Sqrt[x])/(3*b*c*(b*x + c*x^2)^(3/2)) + (2*b*B - 5*A*c)/(3*b^2*c*
Sqrt[x]*Sqrt[b*x + c*x^2]) + ((2*b*B - 5*A*c)*Sqrt[x])/(b^3*Sqrt[b*x + c*x^2]) -
((2*b*B - 5*A*c)*ArcTanh[Sqrt[b*x + c*x^2]/(Sqrt[b]*Sqrt[x])])/b^(7/2)

_______________________________________________________________________________________

Rubi [A]  time = 0.279091, antiderivative size = 146, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208 \[ -\frac{(2 b B-5 A c) \tanh ^{-1}\left (\frac{\sqrt{b x+c x^2}}{\sqrt{b} \sqrt{x}}\right )}{b^{7/2}}+\frac{\sqrt{x} (2 b B-5 A c)}{b^3 \sqrt{b x+c x^2}}+\frac{2 b B-5 A c}{3 b^2 c \sqrt{x} \sqrt{b x+c x^2}}-\frac{2 \sqrt{x} (b B-A c)}{3 b c \left (b x+c x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

(-2*(b*B - A*c)*Sqrt[x])/(3*b*c*(b*x + c*x^2)^(3/2)) + (2*b*B - 5*A*c)/(3*b^2*c*
Sqrt[x]*Sqrt[b*x + c*x^2]) + ((2*b*B - 5*A*c)*Sqrt[x])/(b^3*Sqrt[b*x + c*x^2]) -
((2*b*B - 5*A*c)*ArcTanh[Sqrt[b*x + c*x^2]/(Sqrt[b]*Sqrt[x])])/b^(7/2)

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 20.5402, size = 136, normalized size = 0.93 \[ \frac{2 \sqrt{x} \left (A c - B b\right )}{3 b c \left (b x + c x^{2}\right )^{\frac{3}{2}}} - \frac{2 \left (\frac{5 A c}{2} - B b\right )}{3 b^{2} c \sqrt{x} \sqrt{b x + c x^{2}}} - \frac{2 \sqrt{x} \left (\frac{5 A c}{2} - B b\right )}{b^{3} \sqrt{b x + c x^{2}}} + \frac{2 \left (\frac{5 A c}{2} - B b\right ) \operatorname{atanh}{\left (\frac{\sqrt{b x + c x^{2}}}{\sqrt{b} \sqrt{x}} \right )}}{b^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*sqrt(x)*(A*c - B*b)/(3*b*c*(b*x + c*x**2)**(3/2)) - 2*(5*A*c/2 - B*b)/(3*b**2*
c*sqrt(x)*sqrt(b*x + c*x**2)) - 2*sqrt(x)*(5*A*c/2 - B*b)/(b**3*sqrt(b*x + c*x**
2)) + 2*(5*A*c/2 - B*b)*atanh(sqrt(b*x + c*x**2)/(sqrt(b)*sqrt(x)))/b**(7/2)

_______________________________________________________________________________________

Mathematica [A]  time = 0.175437, size = 105, normalized size = 0.72 \[ \frac{\sqrt{x} \left (\sqrt{b} \left (2 b B x (4 b+3 c x)-A \left (3 b^2+20 b c x+15 c^2 x^2\right )\right )-3 x (b+c x)^{3/2} (2 b B-5 A c) \tanh ^{-1}\left (\frac{\sqrt{b+c x}}{\sqrt{b}}\right )\right )}{3 b^{7/2} (x (b+c x))^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[x]*(Sqrt[b]*(2*b*B*x*(4*b + 3*c*x) - A*(3*b^2 + 20*b*c*x + 15*c^2*x^2)) -
3*(2*b*B - 5*A*c)*x*(b + c*x)^(3/2)*ArcTanh[Sqrt[b + c*x]/Sqrt[b]]))/(3*b^(7/2)*
(x*(b + c*x))^(3/2))

_______________________________________________________________________________________

Maple [A]  time = 0.033, size = 175, normalized size = 1.2 \[{\frac{1}{3\, \left ( cx+b \right ) ^{2}}\sqrt{x \left ( cx+b \right ) } \left ( 15\,A{\it Artanh} \left ({\frac{\sqrt{cx+b}}{\sqrt{b}}} \right ) \sqrt{cx+b}{x}^{2}{c}^{2}-6\,B{\it Artanh} \left ({\frac{\sqrt{cx+b}}{\sqrt{b}}} \right ) \sqrt{cx+b}{x}^{2}bc+15\,A{\it Artanh} \left ({\frac{\sqrt{cx+b}}{\sqrt{b}}} \right ) xbc\sqrt{cx+b}-15\,A\sqrt{b}{x}^{2}{c}^{2}-6\,B{\it Artanh} \left ({\frac{\sqrt{cx+b}}{\sqrt{b}}} \right ) x{b}^{2}\sqrt{cx+b}+6\,B{b}^{3/2}{x}^{2}c-20\,A{b}^{3/2}xc+8\,B{b}^{5/2}x-3\,A{b}^{5/2} \right ){x}^{-{\frac{3}{2}}}{b}^{-{\frac{7}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*(x*(c*x+b))^(1/2)*(15*A*arctanh((c*x+b)^(1/2)/b^(1/2))*(c*x+b)^(1/2)*x^2*c^2
-6*B*arctanh((c*x+b)^(1/2)/b^(1/2))*(c*x+b)^(1/2)*x^2*b*c+15*A*arctanh((c*x+b)^(
1/2)/b^(1/2))*x*b*c*(c*x+b)^(1/2)-15*A*b^(1/2)*x^2*c^2-6*B*arctanh((c*x+b)^(1/2)
/b^(1/2))*x*b^2*(c*x+b)^(1/2)+6*B*b^(3/2)*x^2*c-20*A*b^(3/2)*x*c+8*B*b^(5/2)*x-3
*A*b^(5/2))/x^(3/2)/(c*x+b)^2/b^(7/2)

_______________________________________________________________________________________

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

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.316316, size = 1, normalized size = 0.01 \[ \left [-\frac{2 \,{\left (3 \, A b^{2} - 3 \,{\left (2 \, B b c - 5 \, A c^{2}\right )} x^{2} - 4 \,{\left (2 \, B b^{2} - 5 \, A b c\right )} x\right )} \sqrt{c x^{2} + b x} \sqrt{b} \sqrt{x} + 3 \,{\left ({\left (2 \, B b c^{2} - 5 \, A c^{3}\right )} x^{4} + 2 \,{\left (2 \, B b^{2} c - 5 \, A b c^{2}\right )} x^{3} +{\left (2 \, B b^{3} - 5 \, A b^{2} c\right )} x^{2}\right )} \log \left (-\frac{2 \, \sqrt{c x^{2} + b x} b \sqrt{x} +{\left (c x^{2} + 2 \, b x\right )} \sqrt{b}}{x^{2}}\right )}{6 \,{\left (b^{3} c^{2} x^{4} + 2 \, b^{4} c x^{3} + b^{5} x^{2}\right )} \sqrt{b}}, -\frac{{\left (3 \, A b^{2} - 3 \,{\left (2 \, B b c - 5 \, A c^{2}\right )} x^{2} - 4 \,{\left (2 \, B b^{2} - 5 \, A b c\right )} x\right )} \sqrt{c x^{2} + b x} \sqrt{-b} \sqrt{x} + 3 \,{\left ({\left (2 \, B b c^{2} - 5 \, A c^{3}\right )} x^{4} + 2 \,{\left (2 \, B b^{2} c - 5 \, A b c^{2}\right )} x^{3} +{\left (2 \, B b^{3} - 5 \, A b^{2} c\right )} x^{2}\right )} \arctan \left (\frac{\sqrt{-b} \sqrt{x}}{\sqrt{c x^{2} + b x}}\right )}{3 \,{\left (b^{3} c^{2} x^{4} + 2 \, b^{4} c x^{3} + b^{5} x^{2}\right )} \sqrt{-b}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[-1/6*(2*(3*A*b^2 - 3*(2*B*b*c - 5*A*c^2)*x^2 - 4*(2*B*b^2 - 5*A*b*c)*x)*sqrt(c*
x^2 + b*x)*sqrt(b)*sqrt(x) + 3*((2*B*b*c^2 - 5*A*c^3)*x^4 + 2*(2*B*b^2*c - 5*A*b
*c^2)*x^3 + (2*B*b^3 - 5*A*b^2*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^3*c^2*x^4 + 2*b^4*c*x^3 + b^5*x^2)*sqrt(b)), -1/
3*((3*A*b^2 - 3*(2*B*b*c - 5*A*c^2)*x^2 - 4*(2*B*b^2 - 5*A*b*c)*x)*sqrt(c*x^2 +
b*x)*sqrt(-b)*sqrt(x) + 3*((2*B*b*c^2 - 5*A*c^3)*x^4 + 2*(2*B*b^2*c - 5*A*b*c^2)
*x^3 + (2*B*b^3 - 5*A*b^2*c)*x^2)*arctan(sqrt(-b)*sqrt(x)/sqrt(c*x^2 + b*x)))/((
b^3*c^2*x^4 + 2*b^4*c*x^3 + b^5*x^2)*sqrt(-b))]

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.327866, size = 122, normalized size = 0.84 \[ \frac{{\left (2 \, B b - 5 \, A c\right )} \arctan \left (\frac{\sqrt{c x + b}}{\sqrt{-b}}\right )}{\sqrt{-b} b^{3}} - \frac{\sqrt{c x + b} A}{b^{3} x} + \frac{2 \,{\left (3 \,{\left (c x + b\right )} B b + B b^{2} - 6 \,{\left (c x + b\right )} A c - A b c\right )}}{3 \,{\left (c x + b\right )}^{\frac{3}{2}} b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(2*B*b - 5*A*c)*arctan(sqrt(c*x + b)/sqrt(-b))/(sqrt(-b)*b^3) - sqrt(c*x + b)*A/
(b^3*x) + 2/3*(3*(c*x + b)*B*b + B*b^2 - 6*(c*x + b)*A*c - A*b*c)/((c*x + b)^(3/
2)*b^3)