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

Optimal. Leaf size=91 \[ -\frac{2 a^2 (A b-a B)}{b^4 \sqrt{a+b x}}+\frac{2 (a+b x)^{3/2} (A b-3 a B)}{3 b^4}-\frac{2 a \sqrt{a+b x} (2 A b-3 a B)}{b^4}+\frac{2 B (a+b x)^{5/2}}{5 b^4} \]

[Out]

(-2*a^2*(A*b - a*B))/(b^4*Sqrt[a + b*x]) - (2*a*(2*A*b - 3*a*B)*Sqrt[a + b*x])/b^4 + (2*(A*b - 3*a*B)*(a + b*x
)^(3/2))/(3*b^4) + (2*B*(a + b*x)^(5/2))/(5*b^4)

________________________________________________________________________________________

Rubi [A]  time = 0.0342253, antiderivative size = 91, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.056, Rules used = {77} \[ -\frac{2 a^2 (A b-a B)}{b^4 \sqrt{a+b x}}+\frac{2 (a+b x)^{3/2} (A b-3 a B)}{3 b^4}-\frac{2 a \sqrt{a+b x} (2 A b-3 a B)}{b^4}+\frac{2 B (a+b x)^{5/2}}{5 b^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*a^2*(A*b - a*B))/(b^4*Sqrt[a + b*x]) - (2*a*(2*A*b - 3*a*B)*Sqrt[a + b*x])/b^4 + (2*(A*b - 3*a*B)*(a + b*x
)^(3/2))/(3*b^4) + (2*B*(a + b*x)^(5/2))/(5*b^4)

Rule 77

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))

Rubi steps

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

Mathematica [A]  time = 0.0421447, size = 67, normalized size = 0.74 \[ \frac{2 \left (-8 a^2 b (5 A-3 B x)+48 a^3 B-2 a b^2 x (10 A+3 B x)+b^3 x^2 (5 A+3 B x)\right )}{15 b^4 \sqrt{a+b x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(48*a^3*B - 8*a^2*b*(5*A - 3*B*x) + b^3*x^2*(5*A + 3*B*x) - 2*a*b^2*x*(10*A + 3*B*x)))/(15*b^4*Sqrt[a + b*x
])

________________________________________________________________________________________

Maple [A]  time = 0.004, size = 71, normalized size = 0.8 \begin{align*} -{\frac{-6\,{b}^{3}B{x}^{3}-10\,A{x}^{2}{b}^{3}+12\,B{x}^{2}a{b}^{2}+40\,a{b}^{2}Ax-48\,{a}^{2}bBx+80\,Ab{a}^{2}-96\,B{a}^{3}}{15\,{b}^{4}}{\frac{1}{\sqrt{bx+a}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/15/(b*x+a)^(1/2)*(-3*B*b^3*x^3-5*A*b^3*x^2+6*B*a*b^2*x^2+20*A*a*b^2*x-24*B*a^2*b*x+40*A*a^2*b-48*B*a^3)/b^4

________________________________________________________________________________________

Maxima [A]  time = 1.17494, size = 115, normalized size = 1.26 \begin{align*} \frac{2 \,{\left (\frac{3 \,{\left (b x + a\right )}^{\frac{5}{2}} B - 5 \,{\left (3 \, B a - A b\right )}{\left (b x + a\right )}^{\frac{3}{2}} + 15 \,{\left (3 \, B a^{2} - 2 \, A a b\right )} \sqrt{b x + a}}{b} + \frac{15 \,{\left (B a^{3} - A a^{2} b\right )}}{\sqrt{b x + a} b}\right )}}{15 \, b^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/15*((3*(b*x + a)^(5/2)*B - 5*(3*B*a - A*b)*(b*x + a)^(3/2) + 15*(3*B*a^2 - 2*A*a*b)*sqrt(b*x + a))/b + 15*(B
*a^3 - A*a^2*b)/(sqrt(b*x + a)*b))/b^3

________________________________________________________________________________________

Fricas [A]  time = 2.21701, size = 178, normalized size = 1.96 \begin{align*} \frac{2 \,{\left (3 \, B b^{3} x^{3} + 48 \, B a^{3} - 40 \, A a^{2} b -{\left (6 \, B a b^{2} - 5 \, A b^{3}\right )} x^{2} + 4 \,{\left (6 \, B a^{2} b - 5 \, A a b^{2}\right )} x\right )} \sqrt{b x + a}}{15 \,{\left (b^{5} x + a b^{4}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/15*(3*B*b^3*x^3 + 48*B*a^3 - 40*A*a^2*b - (6*B*a*b^2 - 5*A*b^3)*x^2 + 4*(6*B*a^2*b - 5*A*a*b^2)*x)*sqrt(b*x
+ a)/(b^5*x + a*b^4)

________________________________________________________________________________________

Sympy [A]  time = 11.1527, size = 88, normalized size = 0.97 \begin{align*} \frac{2 B \left (a + b x\right )^{\frac{5}{2}}}{5 b^{4}} + \frac{2 a^{2} \left (- A b + B a\right )}{b^{4} \sqrt{a + b x}} + \frac{\left (a + b x\right )^{\frac{3}{2}} \left (2 A b - 6 B a\right )}{3 b^{4}} + \frac{\sqrt{a + b x} \left (- 4 A a b + 6 B a^{2}\right )}{b^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*B*(a + b*x)**(5/2)/(5*b**4) + 2*a**2*(-A*b + B*a)/(b**4*sqrt(a + b*x)) + (a + b*x)**(3/2)*(2*A*b - 6*B*a)/(3
*b**4) + sqrt(a + b*x)*(-4*A*a*b + 6*B*a**2)/b**4

________________________________________________________________________________________

Giac [A]  time = 1.205, size = 138, normalized size = 1.52 \begin{align*} \frac{2 \,{\left (B a^{3} - A a^{2} b\right )}}{\sqrt{b x + a} b^{4}} + \frac{2 \,{\left (3 \,{\left (b x + a\right )}^{\frac{5}{2}} B b^{16} - 15 \,{\left (b x + a\right )}^{\frac{3}{2}} B a b^{16} + 45 \, \sqrt{b x + a} B a^{2} b^{16} + 5 \,{\left (b x + a\right )}^{\frac{3}{2}} A b^{17} - 30 \, \sqrt{b x + a} A a b^{17}\right )}}{15 \, b^{20}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*(B*a^3 - A*a^2*b)/(sqrt(b*x + a)*b^4) + 2/15*(3*(b*x + a)^(5/2)*B*b^16 - 15*(b*x + a)^(3/2)*B*a*b^16 + 45*sq
rt(b*x + a)*B*a^2*b^16 + 5*(b*x + a)^(3/2)*A*b^17 - 30*sqrt(b*x + a)*A*a*b^17)/b^20