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

Optimal. Leaf size=70 \[ -\frac{8 (b+2 c x) (b B-2 A c)}{3 b^4 \sqrt{b x+c x^2}}-\frac{2 (A b-x (b B-2 A c))}{3 b^2 \left (b x+c x^2\right )^{3/2}} \]

[Out]

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

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Rubi [A]  time = 0.0619375, antiderivative size = 70, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105 \[ -\frac{8 (b+2 c x) (b B-2 A c)}{3 b^4 \sqrt{b x+c x^2}}-\frac{2 (A b-x (b B-2 A c))}{3 b^2 \left (b x+c x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 7.26416, size = 68, normalized size = 0.97 \[ - \frac{2 \left (A b + x \left (2 A c - B b\right )\right )}{3 b^{2} \left (b x + c x^{2}\right )^{\frac{3}{2}}} + \frac{4 \left (2 b + 4 c x\right ) \left (2 A c - B b\right )}{3 b^{4} \sqrt{b x + c x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*(A*b + x*(2*A*c - B*b))/(3*b**2*(b*x + c*x**2)**(3/2)) + 4*(2*b + 4*c*x)*(2*A
*c - B*b)/(3*b**4*sqrt(b*x + c*x**2))

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Mathematica [A]  time = 0.0960291, size = 72, normalized size = 1.03 \[ -\frac{2 \left (A \left (b^3-6 b^2 c x-24 b c^2 x^2-16 c^3 x^3\right )+b B x \left (3 b^2+12 b c x+8 c^2 x^2\right )\right )}{3 b^4 (x (b+c x))^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

(-2*(b*B*x*(3*b^2 + 12*b*c*x + 8*c^2*x^2) + A*(b^3 - 6*b^2*c*x - 24*b*c^2*x^2 -
16*c^3*x^3)))/(3*b^4*(x*(b + c*x))^(3/2))

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Maple [A]  time = 0.006, size = 83, normalized size = 1.2 \[ -{\frac{2\,x \left ( cx+b \right ) \left ( -16\,A{c}^{3}{x}^{3}+8\,B{x}^{3}b{c}^{2}-24\,Ab{c}^{2}{x}^{2}+12\,B{x}^{2}{b}^{2}c-6\,A{b}^{2}cx+3\,Bx{b}^{3}+A{b}^{3} \right ) }{3\,{b}^{4}} \left ( c{x}^{2}+bx \right ) ^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/3*x*(c*x+b)*(-16*A*c^3*x^3+8*B*b*c^2*x^3-24*A*b*c^2*x^2+12*B*b^2*c*x^2-6*A*b^
2*c*x+3*B*b^3*x+A*b^3)/b^4/(c*x^2+b*x)^(5/2)

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Maxima [A]  time = 0.6899, size = 176, normalized size = 2.51 \[ \frac{2 \, B x}{3 \,{\left (c x^{2} + b x\right )}^{\frac{3}{2}} b} - \frac{16 \, B c x}{3 \, \sqrt{c x^{2} + b x} b^{3}} - \frac{4 \, A c x}{3 \,{\left (c x^{2} + b x\right )}^{\frac{3}{2}} b^{2}} + \frac{32 \, A c^{2} x}{3 \, \sqrt{c x^{2} + b x} b^{4}} - \frac{8 \, B}{3 \, \sqrt{c x^{2} + b x} b^{2}} - \frac{2 \, A}{3 \,{\left (c x^{2} + b x\right )}^{\frac{3}{2}} b} + \frac{16 \, A c}{3 \, \sqrt{c x^{2} + b x} b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/3*B*x/((c*x^2 + b*x)^(3/2)*b) - 16/3*B*c*x/(sqrt(c*x^2 + b*x)*b^3) - 4/3*A*c*x
/((c*x^2 + b*x)^(3/2)*b^2) + 32/3*A*c^2*x/(sqrt(c*x^2 + b*x)*b^4) - 8/3*B/(sqrt(
c*x^2 + b*x)*b^2) - 2/3*A/((c*x^2 + b*x)^(3/2)*b) + 16/3*A*c/(sqrt(c*x^2 + b*x)*
b^3)

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Fricas [A]  time = 0.302678, size = 119, normalized size = 1.7 \[ -\frac{2 \,{\left (A b^{3} + 8 \,{\left (B b c^{2} - 2 \, A c^{3}\right )} x^{3} + 12 \,{\left (B b^{2} c - 2 \, A b c^{2}\right )} x^{2} + 3 \,{\left (B b^{3} - 2 \, A b^{2} c\right )} x\right )}}{3 \,{\left (b^{4} c x^{2} + b^{5} x\right )} \sqrt{c x^{2} + b x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/3*(A*b^3 + 8*(B*b*c^2 - 2*A*c^3)*x^3 + 12*(B*b^2*c - 2*A*b*c^2)*x^2 + 3*(B*b^
3 - 2*A*b^2*c)*x)/((b^4*c*x^2 + b^5*x)*sqrt(c*x^2 + b*x))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.279963, size = 127, normalized size = 1.81 \[ -\frac{{\left (4 \, x{\left (\frac{2 \,{\left (B b c^{2} - 2 \, A c^{3}\right )} x}{b^{4} c^{2}} + \frac{3 \,{\left (B b^{2} c - 2 \, A b c^{2}\right )}}{b^{4} c^{2}}\right )} + \frac{3 \,{\left (B b^{3} - 2 \, A b^{2} c\right )}}{b^{4} c^{2}}\right )} x + \frac{A}{b c^{2}}}{3 \,{\left (c x^{2} + b x\right )}^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/3*((4*x*(2*(B*b*c^2 - 2*A*c^3)*x/(b^4*c^2) + 3*(B*b^2*c - 2*A*b*c^2)/(b^4*c^2
)) + 3*(B*b^3 - 2*A*b^2*c)/(b^4*c^2))*x + A/(b*c^2))/(c*x^2 + b*x)^(3/2)