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

Optimal. Leaf size=63 \[ \frac{2 \sqrt{a+b x} (A b-2 a B)}{b^3}+\frac{2 a (A b-a B)}{b^3 \sqrt{a+b x}}+\frac{2 B (a+b x)^{3/2}}{3 b^3} \]

[Out]

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

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Rubi [A]  time = 0.0238114, antiderivative size = 63, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.062, Rules used = {77} \[ \frac{2 \sqrt{a+b x} (A b-2 a B)}{b^3}+\frac{2 a (A b-a B)}{b^3 \sqrt{a+b x}}+\frac{2 B (a+b x)^{3/2}}{3 b^3} \]

Antiderivative was successfully verified.

[In]

Int[(x*(A + B*x))/(a + b*x)^(3/2),x]

[Out]

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

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

Mathematica [A]  time = 0.0289568, size = 47, normalized size = 0.75 \[ \frac{2 \left (-8 a^2 B+a (6 A b-4 b B x)+b^2 x (3 A+B x)\right )}{3 b^3 \sqrt{a+b x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.002, size = 46, normalized size = 0.7 \begin{align*}{\frac{2\,{b}^{2}B{x}^{2}+6\,{b}^{2}Ax-8\,abBx+12\,Aba-16\,B{a}^{2}}{3\,{b}^{3}}{\frac{1}{\sqrt{bx+a}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3/(b*x+a)^(1/2)*(B*b^2*x^2+3*A*b^2*x-4*B*a*b*x+6*A*a*b-8*B*a^2)/b^3

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Maxima [A]  time = 1.08214, size = 82, normalized size = 1.3 \begin{align*} \frac{2 \,{\left (\frac{{\left (b x + a\right )}^{\frac{3}{2}} B - 3 \,{\left (2 \, B a - A b\right )} \sqrt{b x + a}}{b} - \frac{3 \,{\left (B a^{2} - A a b\right )}}{\sqrt{b x + a} b}\right )}}{3 \, b^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 2.38879, size = 123, normalized size = 1.95 \begin{align*} \frac{2 \,{\left (B b^{2} x^{2} - 8 \, B a^{2} + 6 \, A a b -{\left (4 \, B a b - 3 \, A b^{2}\right )} x\right )} \sqrt{b x + a}}{3 \,{\left (b^{4} x + a b^{3}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 11.2369, size = 60, normalized size = 0.95 \begin{align*} \frac{2 B \left (a + b x\right )^{\frac{3}{2}}}{3 b^{3}} - \frac{2 a \left (- A b + B a\right )}{b^{3} \sqrt{a + b x}} + \frac{\sqrt{a + b x} \left (2 A b - 4 B a\right )}{b^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.14737, size = 93, normalized size = 1.48 \begin{align*} -\frac{2 \,{\left (B a^{2} - A a b\right )}}{\sqrt{b x + a} b^{3}} + \frac{2 \,{\left ({\left (b x + a\right )}^{\frac{3}{2}} B b^{6} - 6 \, \sqrt{b x + a} B a b^{6} + 3 \, \sqrt{b x + a} A b^{7}\right )}}{3 \, b^{9}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*(B*a^2 - A*a*b)/(sqrt(b*x + a)*b^3) + 2/3*((b*x + a)^(3/2)*B*b^6 - 6*sqrt(b*x + a)*B*a*b^6 + 3*sqrt(b*x + a
)*A*b^7)/b^9