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

Optimal. Leaf size=131 \[ -\frac{128 c^2 (b+2 c x) (7 b B-10 A c)}{105 b^6 \sqrt{b x+c x^2}}+\frac{16 c (b+2 c x) (7 b B-10 A c)}{105 b^4 \left (b x+c x^2\right )^{3/2}}-\frac{2 (7 b B-10 A c)}{35 b^2 x \left (b x+c x^2\right )^{3/2}}-\frac{2 A}{7 b x^2 \left (b x+c x^2\right )^{3/2}} \]

[Out]

(-2*A)/(7*b*x^2*(b*x + c*x^2)^(3/2)) - (2*(7*b*B - 10*A*c))/(35*b^2*x*(b*x + c*x
^2)^(3/2)) + (16*c*(7*b*B - 10*A*c)*(b + 2*c*x))/(105*b^4*(b*x + c*x^2)^(3/2)) -
(128*c^2*(7*b*B - 10*A*c)*(b + 2*c*x))/(105*b^6*Sqrt[b*x + c*x^2])

_______________________________________________________________________________________

Rubi [A]  time = 0.273628, antiderivative size = 131, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ -\frac{128 c^2 (b+2 c x) (7 b B-10 A c)}{105 b^6 \sqrt{b x+c x^2}}+\frac{16 c (b+2 c x) (7 b B-10 A c)}{105 b^4 \left (b x+c x^2\right )^{3/2}}-\frac{2 (7 b B-10 A c)}{35 b^2 x \left (b x+c x^2\right )^{3/2}}-\frac{2 A}{7 b x^2 \left (b x+c x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

(-2*A)/(7*b*x^2*(b*x + c*x^2)^(3/2)) - (2*(7*b*B - 10*A*c))/(35*b^2*x*(b*x + c*x
^2)^(3/2)) + (16*c*(7*b*B - 10*A*c)*(b + 2*c*x))/(105*b^4*(b*x + c*x^2)^(3/2)) -
(128*c^2*(7*b*B - 10*A*c)*(b + 2*c*x))/(105*b^6*Sqrt[b*x + c*x^2])

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 14.6352, size = 128, normalized size = 0.98 \[ - \frac{2 A}{7 b x^{2} \left (b x + c x^{2}\right )^{\frac{3}{2}}} + \frac{2 \left (10 A c - 7 B b\right )}{35 b^{2} x \left (b x + c x^{2}\right )^{\frac{3}{2}}} - \frac{16 c \left (b + 2 c x\right ) \left (10 A c - 7 B b\right )}{105 b^{4} \left (b x + c x^{2}\right )^{\frac{3}{2}}} + \frac{64 c^{2} \left (2 b + 4 c x\right ) \left (10 A c - 7 B b\right )}{105 b^{6} \sqrt{b x + c x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*A/(7*b*x**2*(b*x + c*x**2)**(3/2)) + 2*(10*A*c - 7*B*b)/(35*b**2*x*(b*x + c*x
**2)**(3/2)) - 16*c*(b + 2*c*x)*(10*A*c - 7*B*b)/(105*b**4*(b*x + c*x**2)**(3/2)
) + 64*c**2*(2*b + 4*c*x)*(10*A*c - 7*B*b)/(105*b**6*sqrt(b*x + c*x**2))

_______________________________________________________________________________________

Mathematica [A]  time = 0.17101, size = 123, normalized size = 0.94 \[ -\frac{2 \left (5 A \left (3 b^5-6 b^4 c x+16 b^3 c^2 x^2-96 b^2 c^3 x^3-384 b c^4 x^4-256 c^5 x^5\right )+7 b B x \left (3 b^4-8 b^3 c x+48 b^2 c^2 x^2+192 b c^3 x^3+128 c^4 x^4\right )\right )}{105 b^6 x^2 (x (b+c x))^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

(-2*(7*b*B*x*(3*b^4 - 8*b^3*c*x + 48*b^2*c^2*x^2 + 192*b*c^3*x^3 + 128*c^4*x^4)
+ 5*A*(3*b^5 - 6*b^4*c*x + 16*b^3*c^2*x^2 - 96*b^2*c^3*x^3 - 384*b*c^4*x^4 - 256
*c^5*x^5)))/(105*b^6*x^2*(x*(b + c*x))^(3/2))

_______________________________________________________________________________________

Maple [A]  time = 0.009, size = 134, normalized size = 1. \[ -{\frac{ \left ( 2\,cx+2\,b \right ) \left ( -1280\,A{c}^{5}{x}^{5}+896\,Bb{c}^{4}{x}^{5}-1920\,Ab{c}^{4}{x}^{4}+1344\,B{b}^{2}{c}^{3}{x}^{4}-480\,A{b}^{2}{c}^{3}{x}^{3}+336\,B{b}^{3}{c}^{2}{x}^{3}+80\,A{b}^{3}{c}^{2}{x}^{2}-56\,B{b}^{4}c{x}^{2}-30\,A{b}^{4}cx+21\,B{b}^{5}x+15\,A{b}^{5} \right ) }{105\,x{b}^{6}} \left ( c{x}^{2}+bx \right ) ^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/105*(c*x+b)*(-1280*A*c^5*x^5+896*B*b*c^4*x^5-1920*A*b*c^4*x^4+1344*B*b^2*c^3*
x^4-480*A*b^2*c^3*x^3+336*B*b^3*c^2*x^3+80*A*b^3*c^2*x^2-56*B*b^4*c*x^2-30*A*b^4
*c*x+21*B*b^5*x+15*A*b^5)/x/b^6/(c*x^2+b*x)^(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)^(5/2)*x^2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.294911, size = 192, normalized size = 1.47 \[ -\frac{2 \,{\left (15 \, A b^{5} + 128 \,{\left (7 \, B b c^{4} - 10 \, A c^{5}\right )} x^{5} + 192 \,{\left (7 \, B b^{2} c^{3} - 10 \, A b c^{4}\right )} x^{4} + 48 \,{\left (7 \, B b^{3} c^{2} - 10 \, A b^{2} c^{3}\right )} x^{3} - 8 \,{\left (7 \, B b^{4} c - 10 \, A b^{3} c^{2}\right )} x^{2} + 3 \,{\left (7 \, B b^{5} - 10 \, A b^{4} c\right )} x\right )}}{105 \,{\left (b^{6} c x^{4} + b^{7} x^{3}\right )} \sqrt{c x^{2} + b x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/105*(15*A*b^5 + 128*(7*B*b*c^4 - 10*A*c^5)*x^5 + 192*(7*B*b^2*c^3 - 10*A*b*c^
4)*x^4 + 48*(7*B*b^3*c^2 - 10*A*b^2*c^3)*x^3 - 8*(7*B*b^4*c - 10*A*b^3*c^2)*x^2
+ 3*(7*B*b^5 - 10*A*b^4*c)*x)/((b^6*c*x^4 + b^7*x^3)*sqrt(c*x^2 + b*x))

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((B*x + A)/((c*x^2 + b*x)^(5/2)*x^2), x)