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

Optimal. Leaf size=90 \[ \frac{4 c \left (b x+c x^2\right )^{3/2} (7 b B-4 A c)}{105 b^3 x^3}-\frac{2 \left (b x+c x^2\right )^{3/2} (7 b B-4 A c)}{35 b^2 x^4}-\frac{2 A \left (b x+c x^2\right )^{3/2}}{7 b x^5} \]

[Out]

(-2*A*(b*x + c*x^2)^(3/2))/(7*b*x^5) - (2*(7*b*B - 4*A*c)*(b*x + c*x^2)^(3/2))/(
35*b^2*x^4) + (4*c*(7*b*B - 4*A*c)*(b*x + c*x^2)^(3/2))/(105*b^3*x^3)

_______________________________________________________________________________________

Rubi [A]  time = 0.210152, antiderivative size = 90, 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 \[ \frac{4 c \left (b x+c x^2\right )^{3/2} (7 b B-4 A c)}{105 b^3 x^3}-\frac{2 \left (b x+c x^2\right )^{3/2} (7 b B-4 A c)}{35 b^2 x^4}-\frac{2 A \left (b x+c x^2\right )^{3/2}}{7 b x^5} \]

Antiderivative was successfully verified.

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

[Out]

(-2*A*(b*x + c*x^2)^(3/2))/(7*b*x^5) - (2*(7*b*B - 4*A*c)*(b*x + c*x^2)^(3/2))/(
35*b^2*x^4) + (4*c*(7*b*B - 4*A*c)*(b*x + c*x^2)^(3/2))/(105*b^3*x^3)

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 12.0737, size = 87, normalized size = 0.97 \[ - \frac{2 A \left (b x + c x^{2}\right )^{\frac{3}{2}}}{7 b x^{5}} + \frac{2 \left (4 A c - 7 B b\right ) \left (b x + c x^{2}\right )^{\frac{3}{2}}}{35 b^{2} x^{4}} - \frac{4 c \left (4 A c - 7 B b\right ) \left (b x + c x^{2}\right )^{\frac{3}{2}}}{105 b^{3} x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*A*(b*x + c*x**2)**(3/2)/(7*b*x**5) + 2*(4*A*c - 7*B*b)*(b*x + c*x**2)**(3/2)/
(35*b**2*x**4) - 4*c*(4*A*c - 7*B*b)*(b*x + c*x**2)**(3/2)/(105*b**3*x**3)

_______________________________________________________________________________________

Mathematica [A]  time = 0.0862873, size = 56, normalized size = 0.62 \[ -\frac{2 (x (b+c x))^{3/2} \left (A \left (15 b^2-12 b c x+8 c^2 x^2\right )+7 b B x (3 b-2 c x)\right )}{105 b^3 x^5} \]

Antiderivative was successfully verified.

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

[Out]

(-2*(x*(b + c*x))^(3/2)*(7*b*B*x*(3*b - 2*c*x) + A*(15*b^2 - 12*b*c*x + 8*c^2*x^
2)))/(105*b^3*x^5)

_______________________________________________________________________________________

Maple [A]  time = 0.008, size = 62, normalized size = 0.7 \[ -{\frac{ \left ( 2\,cx+2\,b \right ) \left ( 8\,A{c}^{2}{x}^{2}-14\,B{x}^{2}bc-12\,Abcx+21\,{b}^{2}Bx+15\,{b}^{2}A \right ) }{105\,{x}^{4}{b}^{3}}\sqrt{c{x}^{2}+bx}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/105*(c*x+b)*(8*A*c^2*x^2-14*B*b*c*x^2-12*A*b*c*x+21*B*b^2*x+15*A*b^2)*(c*x^2+
b*x)^(1/2)/x^4/b^3

_______________________________________________________________________________________

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

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.266915, size = 108, normalized size = 1.2 \[ -\frac{2 \,{\left (15 \, A b^{3} - 2 \,{\left (7 \, B b c^{2} - 4 \, A c^{3}\right )} x^{3} +{\left (7 \, B b^{2} c - 4 \, A b c^{2}\right )} x^{2} + 3 \,{\left (7 \, B b^{3} + A b^{2} c\right )} x\right )} \sqrt{c x^{2} + b x}}{105 \, b^{3} x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/105*(15*A*b^3 - 2*(7*B*b*c^2 - 4*A*c^3)*x^3 + (7*B*b^2*c - 4*A*b*c^2)*x^2 + 3
*(7*B*b^3 + A*b^2*c)*x)*sqrt(c*x^2 + b*x)/(b^3*x^4)

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.27703, size = 339, normalized size = 3.77 \[ \frac{2 \,{\left (105 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{5} B c^{\frac{3}{2}} + 175 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{4} B b c + 140 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{4} A c^{2} + 105 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{3} B b^{2} \sqrt{c} + 315 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{3} A b c^{\frac{3}{2}} + 21 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{2} B b^{3} + 273 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{2} A b^{2} c + 105 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )} A b^{3} \sqrt{c} + 15 \, A b^{4}\right )}}{105 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/105*(105*(sqrt(c)*x - sqrt(c*x^2 + b*x))^5*B*c^(3/2) + 175*(sqrt(c)*x - sqrt(c
*x^2 + b*x))^4*B*b*c + 140*(sqrt(c)*x - sqrt(c*x^2 + b*x))^4*A*c^2 + 105*(sqrt(c
)*x - sqrt(c*x^2 + b*x))^3*B*b^2*sqrt(c) + 315*(sqrt(c)*x - sqrt(c*x^2 + b*x))^3
*A*b*c^(3/2) + 21*(sqrt(c)*x - sqrt(c*x^2 + b*x))^2*B*b^3 + 273*(sqrt(c)*x - sqr
t(c*x^2 + b*x))^2*A*b^2*c + 105*(sqrt(c)*x - sqrt(c*x^2 + b*x))*A*b^3*sqrt(c) +
15*A*b^4)/(sqrt(c)*x - sqrt(c*x^2 + b*x))^7