3.1210 \(\int \frac{A+B x}{(b x+c x^2)^{5/2}} \, dx\)

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 = 0.0215591, 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, Rules used = {638, 613} \[ -\frac{8 (b+2 c x) (b B-2 A c)}{3 b^4 \sqrt{b x+c x^2}}-\frac{2 (A b-x (b B-2 A c))}{3 b^2 \left (b x+c x^2\right )^{3/2}} \]

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])

Rule 638

Int[((d_.) + (e_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((b*d - 2*a*e + (2*c*d -
b*e)*x)*(a + b*x + c*x^2)^(p + 1))/((p + 1)*(b^2 - 4*a*c)), x] - Dist[((2*p + 3)*(2*c*d - b*e))/((p + 1)*(b^2
- 4*a*c)), Int[(a + b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[2*c*d - b*e, 0] && NeQ[b^
2 - 4*a*c, 0] && LtQ[p, -1] && NeQ[p, -3/2]

Rule 613

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-3/2), x_Symbol] :> Simp[(-2*(b + 2*c*x))/((b^2 - 4*a*c)*Sqrt[a + b*x
+ c*x^2]), x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rubi steps

\begin{align*} \int \frac{A+B x}{\left (b x+c x^2\right )^{5/2}} \, dx &=-\frac{2 (A b-(b B-2 A c) x)}{3 b^2 \left (b x+c x^2\right )^{3/2}}+\frac{(4 (b B-2 A c)) \int \frac{1}{\left (b x+c x^2\right )^{3/2}} \, dx}{3 b^2}\\ &=-\frac{2 (A b-(b B-2 A c) x)}{3 b^2 \left (b x+c x^2\right )^{3/2}}-\frac{8 (b B-2 A c) (b+2 c x)}{3 b^4 \sqrt{b x+c x^2}}\\ \end{align*}

Mathematica [A]  time = 0.0243835, size = 72, normalized size = 1.03 \[ -\frac{2 \left (A \left (-6 b^2 c x+b^3-24 b c^2 x^2-16 c^3 x^3\right )+b B x \left (3 b^2+12 b c x+8 c^2 x^2\right )\right )}{3 b^4 (x (b+c x))^{3/2}} \]

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 \begin{align*} -{\frac{2\,x \left ( cx+b \right ) \left ( -16\,A{x}^{3}{c}^{3}+8\,B{x}^{3}b{c}^{2}-24\,A{x}^{2}b{c}^{2}+12\,B{x}^{2}{b}^{2}c-6\,A{b}^{2}cx+3\,{b}^{3}Bx+A{b}^{3} \right ) }{3\,{b}^{4}} \left ( c{x}^{2}+bx \right ) ^{-{\frac{5}{2}}}} \end{align*}

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 [B]  time = 0.984022, size = 176, normalized size = 2.51 \begin{align*} \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}} \end{align*}

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) + 3
2/3*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 = 1.81878, size = 209, normalized size = 2.99 \begin{align*} -\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 )} \sqrt{c x^{2} + b x}}{3 \,{\left (b^{4} c^{2} x^{4} + 2 \, b^{5} c x^{3} + b^{6} x^{2}\right )}} \end{align*}

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)*sqrt(c*x^2 +
b*x)/(b^4*c^2*x^4 + 2*b^5*c*x^3 + b^6*x^2)

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

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 [A]  time = 1.3277, size = 127, normalized size = 1.81 \begin{align*} -\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}}} \end{align*}

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)