3.128 \(\int \frac{A+B x}{x^4 \left (b x+c x^2\right )^{3/2}} \, dx\)

Optimal. Leaf size=163 \[ \frac{128 c^3 (b+2 c x) (9 b B-10 A c)}{315 b^6 \sqrt{b x+c x^2}}-\frac{32 c^2 (9 b B-10 A c)}{315 b^4 x \sqrt{b x+c x^2}}+\frac{16 c (9 b B-10 A c)}{315 b^3 x^2 \sqrt{b x+c x^2}}-\frac{2 (9 b B-10 A c)}{63 b^2 x^3 \sqrt{b x+c x^2}}-\frac{2 A}{9 b x^4 \sqrt{b x+c x^2}} \]

[Out]

(-2*A)/(9*b*x^4*Sqrt[b*x + c*x^2]) - (2*(9*b*B - 10*A*c))/(63*b^2*x^3*Sqrt[b*x +
c*x^2]) + (16*c*(9*b*B - 10*A*c))/(315*b^3*x^2*Sqrt[b*x + c*x^2]) - (32*c^2*(9*
b*B - 10*A*c))/(315*b^4*x*Sqrt[b*x + c*x^2]) + (128*c^3*(9*b*B - 10*A*c)*(b + 2*
c*x))/(315*b^6*Sqrt[b*x + c*x^2])

_______________________________________________________________________________________

Rubi [A]  time = 0.337956, antiderivative size = 163, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136 \[ \frac{128 c^3 (b+2 c x) (9 b B-10 A c)}{315 b^6 \sqrt{b x+c x^2}}-\frac{32 c^2 (9 b B-10 A c)}{315 b^4 x \sqrt{b x+c x^2}}+\frac{16 c (9 b B-10 A c)}{315 b^3 x^2 \sqrt{b x+c x^2}}-\frac{2 (9 b B-10 A c)}{63 b^2 x^3 \sqrt{b x+c x^2}}-\frac{2 A}{9 b x^4 \sqrt{b x+c x^2}} \]

Antiderivative was successfully verified.

[In]  Int[(A + B*x)/(x^4*(b*x + c*x^2)^(3/2)),x]

[Out]

(-2*A)/(9*b*x^4*Sqrt[b*x + c*x^2]) - (2*(9*b*B - 10*A*c))/(63*b^2*x^3*Sqrt[b*x +
c*x^2]) + (16*c*(9*b*B - 10*A*c))/(315*b^3*x^2*Sqrt[b*x + c*x^2]) - (32*c^2*(9*
b*B - 10*A*c))/(315*b^4*x*Sqrt[b*x + c*x^2]) + (128*c^3*(9*b*B - 10*A*c)*(b + 2*
c*x))/(315*b^6*Sqrt[b*x + c*x^2])

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 21.4323, size = 160, normalized size = 0.98 \[ - \frac{2 A}{9 b x^{4} \sqrt{b x + c x^{2}}} + \frac{2 \left (10 A c - 9 B b\right )}{63 b^{2} x^{3} \sqrt{b x + c x^{2}}} - \frac{16 c \left (10 A c - 9 B b\right )}{315 b^{3} x^{2} \sqrt{b x + c x^{2}}} + \frac{32 c^{2} \left (10 A c - 9 B b\right )}{315 b^{4} x \sqrt{b x + c x^{2}}} - \frac{64 c^{3} \left (2 b + 4 c x\right ) \left (10 A c - 9 B b\right )}{315 b^{6} \sqrt{b x + c x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x+A)/x**4/(c*x**2+b*x)**(3/2),x)

[Out]

-2*A/(9*b*x**4*sqrt(b*x + c*x**2)) + 2*(10*A*c - 9*B*b)/(63*b**2*x**3*sqrt(b*x +
c*x**2)) - 16*c*(10*A*c - 9*B*b)/(315*b**3*x**2*sqrt(b*x + c*x**2)) + 32*c**2*(
10*A*c - 9*B*b)/(315*b**4*x*sqrt(b*x + c*x**2)) - 64*c**3*(2*b + 4*c*x)*(10*A*c
- 9*B*b)/(315*b**6*sqrt(b*x + c*x**2))

_______________________________________________________________________________________

Mathematica [A]  time = 0.150643, size = 123, normalized size = 0.75 \[ -\frac{2 \left (5 A \left (7 b^5-10 b^4 c x+16 b^3 c^2 x^2-32 b^2 c^3 x^3+128 b c^4 x^4+256 c^5 x^5\right )+9 b B x \left (5 b^4-8 b^3 c x+16 b^2 c^2 x^2-64 b c^3 x^3-128 c^4 x^4\right )\right )}{315 b^6 x^4 \sqrt{x (b+c x)}} \]

Antiderivative was successfully verified.

[In]  Integrate[(A + B*x)/(x^4*(b*x + c*x^2)^(3/2)),x]

[Out]

(-2*(9*b*B*x*(5*b^4 - 8*b^3*c*x + 16*b^2*c^2*x^2 - 64*b*c^3*x^3 - 128*c^4*x^4) +
5*A*(7*b^5 - 10*b^4*c*x + 16*b^3*c^2*x^2 - 32*b^2*c^3*x^3 + 128*b*c^4*x^4 + 256
*c^5*x^5)))/(315*b^6*x^4*Sqrt[x*(b + c*x)])

_______________________________________________________________________________________

Maple [A]  time = 0.009, size = 134, normalized size = 0.8 \[ -{\frac{ \left ( 2\,cx+2\,b \right ) \left ( 1280\,A{c}^{5}{x}^{5}-1152\,Bb{c}^{4}{x}^{5}+640\,Ab{c}^{4}{x}^{4}-576\,B{b}^{2}{c}^{3}{x}^{4}-160\,A{b}^{2}{c}^{3}{x}^{3}+144\,B{b}^{3}{c}^{2}{x}^{3}+80\,A{b}^{3}{c}^{2}{x}^{2}-72\,B{b}^{4}c{x}^{2}-50\,A{b}^{4}cx+45\,B{b}^{5}x+35\,A{b}^{5} \right ) }{315\,{x}^{3}{b}^{6}} \left ( c{x}^{2}+bx \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)/x^4/(c*x^2+b*x)^(3/2),x)

[Out]

-2/315*(c*x+b)*(1280*A*c^5*x^5-1152*B*b*c^4*x^5+640*A*b*c^4*x^4-576*B*b^2*c^3*x^
4-160*A*b^2*c^3*x^3+144*B*b^3*c^2*x^3+80*A*b^3*c^2*x^2-72*B*b^4*c*x^2-50*A*b^4*c
*x+45*B*b^5*x+35*A*b^5)/x^3/b^6/(c*x^2+b*x)^(3/2)

_______________________________________________________________________________________

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)*x^4),x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.280147, size = 176, normalized size = 1.08 \[ -\frac{2 \,{\left (35 \, A b^{5} - 128 \,{\left (9 \, B b c^{4} - 10 \, A c^{5}\right )} x^{5} - 64 \,{\left (9 \, B b^{2} c^{3} - 10 \, A b c^{4}\right )} x^{4} + 16 \,{\left (9 \, B b^{3} c^{2} - 10 \, A b^{2} c^{3}\right )} x^{3} - 8 \,{\left (9 \, B b^{4} c - 10 \, A b^{3} c^{2}\right )} x^{2} + 5 \,{\left (9 \, B b^{5} - 10 \, A b^{4} c\right )} x\right )}}{315 \, \sqrt{c x^{2} + b x} b^{6} x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)/((c*x^2 + b*x)^(3/2)*x^4),x, algorithm="fricas")

[Out]

-2/315*(35*A*b^5 - 128*(9*B*b*c^4 - 10*A*c^5)*x^5 - 64*(9*B*b^2*c^3 - 10*A*b*c^4
)*x^4 + 16*(9*B*b^3*c^2 - 10*A*b^2*c^3)*x^3 - 8*(9*B*b^4*c - 10*A*b^3*c^2)*x^2 +
5*(9*B*b^5 - 10*A*b^4*c)*x)/(sqrt(c*x^2 + b*x)*b^6*x^4)

_______________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{A + B x}{x^{4} \left (x \left (b + c x\right )\right )^{\frac{3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x+A)/x**4/(c*x**2+b*x)**(3/2),x)

[Out]

Integral((A + B*x)/(x**4*(x*(b + c*x))**(3/2)), x)

_______________________________________________________________________________________

GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{B x + A}{{\left (c x^{2} + b x\right )}^{\frac{3}{2}} x^{4}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)/((c*x^2 + b*x)^(3/2)*x^4),x, algorithm="giac")

[Out]

integrate((B*x + A)/((c*x^2 + b*x)^(3/2)*x^4), x)