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

Optimal. Leaf size=128 \[ -\frac{16 c^2 (b+2 c x) (7 b B-8 A c)}{35 b^5 \sqrt{b x+c x^2}}+\frac{4 c (7 b B-8 A c)}{35 b^3 x \sqrt{b x+c x^2}}-\frac{2 (7 b B-8 A c)}{35 b^2 x^2 \sqrt{b x+c x^2}}-\frac{2 A}{7 b x^3 \sqrt{b x+c x^2}} \]

[Out]

(-2*A)/(7*b*x^3*Sqrt[b*x + c*x^2]) - (2*(7*b*B - 8*A*c))/(35*b^2*x^2*Sqrt[b*x +
c*x^2]) + (4*c*(7*b*B - 8*A*c))/(35*b^3*x*Sqrt[b*x + c*x^2]) - (16*c^2*(7*b*B -
8*A*c)*(b + 2*c*x))/(35*b^5*Sqrt[b*x + c*x^2])

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Rubi [A]  time = 0.260653, antiderivative size = 128, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136 \[ -\frac{16 c^2 (b+2 c x) (7 b B-8 A c)}{35 b^5 \sqrt{b x+c x^2}}+\frac{4 c (7 b B-8 A c)}{35 b^3 x \sqrt{b x+c x^2}}-\frac{2 (7 b B-8 A c)}{35 b^2 x^2 \sqrt{b x+c x^2}}-\frac{2 A}{7 b x^3 \sqrt{b x+c x^2}} \]

Antiderivative was successfully verified.

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

[Out]

(-2*A)/(7*b*x^3*Sqrt[b*x + c*x^2]) - (2*(7*b*B - 8*A*c))/(35*b^2*x^2*Sqrt[b*x +
c*x^2]) + (4*c*(7*b*B - 8*A*c))/(35*b^3*x*Sqrt[b*x + c*x^2]) - (16*c^2*(7*b*B -
8*A*c)*(b + 2*c*x))/(35*b^5*Sqrt[b*x + c*x^2])

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Rubi in Sympy [A]  time = 16.0742, size = 124, normalized size = 0.97 \[ - \frac{2 A}{7 b x^{3} \sqrt{b x + c x^{2}}} + \frac{2 \left (8 A c - 7 B b\right )}{35 b^{2} x^{2} \sqrt{b x + c x^{2}}} - \frac{4 c \left (8 A c - 7 B b\right )}{35 b^{3} x \sqrt{b x + c x^{2}}} + \frac{8 c^{2} \left (2 b + 4 c x\right ) \left (8 A c - 7 B b\right )}{35 b^{5} \sqrt{b x + c x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*A/(7*b*x**3*sqrt(b*x + c*x**2)) + 2*(8*A*c - 7*B*b)/(35*b**2*x**2*sqrt(b*x +
c*x**2)) - 4*c*(8*A*c - 7*B*b)/(35*b**3*x*sqrt(b*x + c*x**2)) + 8*c**2*(2*b + 4*
c*x)*(8*A*c - 7*B*b)/(35*b**5*sqrt(b*x + c*x**2))

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Mathematica [A]  time = 0.127395, size = 98, normalized size = 0.77 \[ -\frac{2 \left (A \left (5 b^4-8 b^3 c x+16 b^2 c^2 x^2-64 b c^3 x^3-128 c^4 x^4\right )+7 b B x \left (b^3-2 b^2 c x+8 b c^2 x^2+16 c^3 x^3\right )\right )}{35 b^5 x^3 \sqrt{x (b+c x)}} \]

Antiderivative was successfully verified.

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

[Out]

(-2*(7*b*B*x*(b^3 - 2*b^2*c*x + 8*b*c^2*x^2 + 16*c^3*x^3) + A*(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)))/(35*b^5*x^3*Sqrt[x*(b + c*x)])

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Maple [A]  time = 0.009, size = 110, normalized size = 0.9 \[ -{\frac{ \left ( 2\,cx+2\,b \right ) \left ( -128\,A{c}^{4}{x}^{4}+112\,Bb{c}^{3}{x}^{4}-64\,Ab{c}^{3}{x}^{3}+56\,B{b}^{2}{c}^{2}{x}^{3}+16\,A{b}^{2}{c}^{2}{x}^{2}-14\,B{b}^{3}c{x}^{2}-8\,A{b}^{3}cx+7\,{b}^{4}Bx+5\,A{b}^{4} \right ) }{35\,{x}^{2}{b}^{5}} \left ( c{x}^{2}+bx \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/35*(c*x+b)*(-128*A*c^4*x^4+112*B*b*c^3*x^4-64*A*b*c^3*x^3+56*B*b^2*c^2*x^3+16
*A*b^2*c^2*x^2-14*B*b^3*c*x^2-8*A*b^3*c*x+7*B*b^4*x+5*A*b^4)/x^2/b^5/(c*x^2+b*x)
^(3/2)

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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^3),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.281456, size = 142, normalized size = 1.11 \[ -\frac{2 \,{\left (5 \, A b^{4} + 16 \,{\left (7 \, B b c^{3} - 8 \, A c^{4}\right )} x^{4} + 8 \,{\left (7 \, B b^{2} c^{2} - 8 \, A b c^{3}\right )} x^{3} - 2 \,{\left (7 \, B b^{3} c - 8 \, A b^{2} c^{2}\right )} x^{2} +{\left (7 \, B b^{4} - 8 \, A b^{3} c\right )} x\right )}}{35 \, \sqrt{c x^{2} + b x} b^{5} x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/35*(5*A*b^4 + 16*(7*B*b*c^3 - 8*A*c^4)*x^4 + 8*(7*B*b^2*c^2 - 8*A*b*c^3)*x^3
- 2*(7*B*b^3*c - 8*A*b^2*c^2)*x^2 + (7*B*b^4 - 8*A*b^3*c)*x)/(sqrt(c*x^2 + b*x)*
b^5*x^3)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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