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

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

[Out]

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

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Rubi [A]  time = 0.0313752, antiderivative size = 70, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {777, 613} \[ \frac{2 (b+2 c x) (b B-4 A c)}{3 b^3 c \sqrt{b x+c x^2}}-\frac{2 x (b B-A c)}{3 b c \left (b x+c x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 777

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

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

Mathematica [A]  time = 0.0270668, size = 55, normalized size = 0.79 \[ \frac{x \left (2 b B x (3 b+2 c x)-2 A \left (3 b^2+12 b c x+8 c^2 x^2\right )\right )}{3 b^3 (x (b+c x))^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.006, size = 62, normalized size = 0.9 \begin{align*} -{\frac{2\,{x}^{2} \left ( cx+b \right ) \left ( 8\,A{c}^{2}{x}^{2}-2\,B{x}^{2}bc+12\,Abcx-3\,{b}^{2}Bx+3\,A{b}^{2} \right ) }{3\,{b}^{3}} \left ( c{x}^{2}+bx \right ) ^{-{\frac{5}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.02475, size = 150, normalized size = 2.14 \begin{align*} \frac{4 \, B x}{3 \, \sqrt{c x^{2} + b x} b^{2}} + \frac{2 \, A x}{3 \,{\left (c x^{2} + b x\right )}^{\frac{3}{2}} b} - \frac{2 \, B x}{3 \,{\left (c x^{2} + b x\right )}^{\frac{3}{2}} c} - \frac{16 \, A c x}{3 \, \sqrt{c x^{2} + b x} b^{3}} - \frac{8 \, A}{3 \, \sqrt{c x^{2} + b x} b^{2}} + \frac{2 \, B}{3 \, \sqrt{c x^{2} + b x} b c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.96564, size = 162, normalized size = 2.31 \begin{align*} -\frac{2 \,{\left (3 \, A b^{2} - 2 \,{\left (B b c - 4 \, A c^{2}\right )} x^{2} - 3 \,{\left (B b^{2} - 4 \, A b c\right )} x\right )} \sqrt{c x^{2} + b x}}{3 \,{\left (b^{3} c^{2} x^{3} + 2 \, b^{4} c x^{2} + b^{5} x\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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