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

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

[Out]

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

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Rubi [A]  time = 0.0384888, 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} \[ -\frac{2 a^2 (A b-a B)}{3 b^4 (a+b x)^{3/2}}+\frac{2 a (2 A b-3 a B)}{b^4 \sqrt{a+b x}}+\frac{2 \sqrt{a+b x} (A b-3 a B)}{b^4}+\frac{2 B (a+b x)^{3/2}}{3 b^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Mathematica [A]  time = 0.0459144, size = 63, normalized size = 0.69 \[ \frac{2 \left (8 a^2 b (A-3 B x)-16 a^3 B-6 a b^2 x (B x-2 A)+b^3 x^2 (3 A+B x)\right )}{3 b^4 (a+b x)^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.004, size = 70, normalized size = 0.8 \begin{align*}{\frac{2\,{b}^{3}B{x}^{3}+6\,A{x}^{2}{b}^{3}-12\,B{x}^{2}a{b}^{2}+24\,a{b}^{2}Ax-48\,{a}^{2}bBx+16\,Ab{a}^{2}-32\,B{a}^{3}}{3\,{b}^{4}} \left ( bx+a \right ) ^{-{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.07362, size = 109, normalized size = 1.2 \begin{align*} \frac{2 \,{\left (\frac{{\left (b x + a\right )}^{\frac{3}{2}} B - 3 \,{\left (3 \, B a - A b\right )} \sqrt{b x + a}}{b} + \frac{B a^{3} - A a^{2} b - 3 \,{\left (3 \, B a^{2} - 2 \, A a b\right )}{\left (b x + a\right )}}{{\left (b x + a\right )}^{\frac{3}{2}} b}\right )}}{3 \, b^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 2.30696, size = 193, normalized size = 2.12 \begin{align*} \frac{2 \,{\left (B b^{3} x^{3} - 16 \, B a^{3} + 8 \, A a^{2} b - 3 \,{\left (2 \, B a b^{2} - A b^{3}\right )} x^{2} - 12 \,{\left (2 \, B a^{2} b - A a b^{2}\right )} x\right )} \sqrt{b x + a}}{3 \,{\left (b^{6} x^{2} + 2 \, a b^{5} x + a^{2} 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)^(5/2),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 1.83897, size = 299, normalized size = 3.29 \begin{align*} \begin{cases} \frac{16 A a^{2} b}{3 a b^{4} \sqrt{a + b x} + 3 b^{5} x \sqrt{a + b x}} + \frac{24 A a b^{2} x}{3 a b^{4} \sqrt{a + b x} + 3 b^{5} x \sqrt{a + b x}} + \frac{6 A b^{3} x^{2}}{3 a b^{4} \sqrt{a + b x} + 3 b^{5} x \sqrt{a + b x}} - \frac{32 B a^{3}}{3 a b^{4} \sqrt{a + b x} + 3 b^{5} x \sqrt{a + b x}} - \frac{48 B a^{2} b x}{3 a b^{4} \sqrt{a + b x} + 3 b^{5} x \sqrt{a + b x}} - \frac{12 B a b^{2} x^{2}}{3 a b^{4} \sqrt{a + b x} + 3 b^{5} x \sqrt{a + b x}} + \frac{2 B b^{3} x^{3}}{3 a b^{4} \sqrt{a + b x} + 3 b^{5} x \sqrt{a + b x}} & \text{for}\: b \neq 0 \\\frac{\frac{A x^{3}}{3} + \frac{B x^{4}}{4}}{a^{\frac{5}{2}}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((16*A*a**2*b/(3*a*b**4*sqrt(a + b*x) + 3*b**5*x*sqrt(a + b*x)) + 24*A*a*b**2*x/(3*a*b**4*sqrt(a + b*
x) + 3*b**5*x*sqrt(a + b*x)) + 6*A*b**3*x**2/(3*a*b**4*sqrt(a + b*x) + 3*b**5*x*sqrt(a + b*x)) - 32*B*a**3/(3*
a*b**4*sqrt(a + b*x) + 3*b**5*x*sqrt(a + b*x)) - 48*B*a**2*b*x/(3*a*b**4*sqrt(a + b*x) + 3*b**5*x*sqrt(a + b*x
)) - 12*B*a*b**2*x**2/(3*a*b**4*sqrt(a + b*x) + 3*b**5*x*sqrt(a + b*x)) + 2*B*b**3*x**3/(3*a*b**4*sqrt(a + b*x
) + 3*b**5*x*sqrt(a + b*x)), Ne(b, 0)), ((A*x**3/3 + B*x**4/4)/a**(5/2), True))

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Giac [A]  time = 1.19464, size = 124, normalized size = 1.36 \begin{align*} -\frac{2 \,{\left (9 \,{\left (b x + a\right )} B a^{2} - B a^{3} - 6 \,{\left (b x + a\right )} A a b + A a^{2} b\right )}}{3 \,{\left (b x + a\right )}^{\frac{3}{2}} b^{4}} + \frac{2 \,{\left ({\left (b x + a\right )}^{\frac{3}{2}} B b^{8} - 9 \, \sqrt{b x + a} B a b^{8} + 3 \, \sqrt{b x + a} A b^{9}\right )}}{3 \, b^{12}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/3*(9*(b*x + a)*B*a^2 - B*a^3 - 6*(b*x + a)*A*a*b + A*a^2*b)/((b*x + a)^(3/2)*b^4) + 2/3*((b*x + a)^(3/2)*B*
b^8 - 9*sqrt(b*x + a)*B*a*b^8 + 3*sqrt(b*x + a)*A*b^9)/b^12