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

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

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0450948, antiderivative size = 70, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083, Rules used = {788, 648} \[ \frac{2 \sqrt{b x+c x^2} (2 b B-A c)}{b c^2 \sqrt{x}}-\frac{2 x^{3/2} (b B-A c)}{b c \sqrt{b x+c x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 788

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

Rule 648

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

Rubi steps

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

Mathematica [A]  time = 0.0208443, size = 34, normalized size = 0.49 \[ \frac{2 \sqrt{x} (-A c+2 b B+B c x)}{c^2 \sqrt{x (b+c x)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 0.004, size = 38, normalized size = 0.5 \begin{align*} -2\,{\frac{ \left ( cx+b \right ) \left ( -Bcx+Ac-2\,bB \right ){x}^{3/2}}{{c}^{2} \left ( c{x}^{2}+bx \right ) ^{3/2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (B x + A\right )} x^{\frac{3}{2}}}{{\left (c x^{2} + b x\right )}^{\frac{3}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(3/2)*(B*x+A)/(c*x^2+b*x)^(3/2),x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

Fricas [A]  time = 1.82694, size = 96, normalized size = 1.37 \begin{align*} \frac{2 \,{\left (B c x + 2 \, B b - A c\right )} \sqrt{c x^{2} + b x} \sqrt{x}}{c^{3} x^{2} + b c^{2} x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Giac [A]  time = 1.16047, size = 66, normalized size = 0.94 \begin{align*} \frac{2 \,{\left (\sqrt{c x + b} B + \frac{B b - A c}{\sqrt{c x + b}}\right )}}{c^{2}} - \frac{2 \,{\left (2 \, B b - A c\right )}}{\sqrt{b} c^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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