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

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

________________________________________________________________________________________

Rubi [A]  time = 0.02, antiderivative size = 63, normalized size of antiderivative = 1.00, 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} \begin {gather*} \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} \end {gather*}

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 = 0.04, size = 47, normalized size = 0.75 \begin {gather*} \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}} \end {gather*}

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])

________________________________________________________________________________________

IntegrateAlgebraic [A]  time = 0.04, size = 55, normalized size = 0.87 \begin {gather*} \frac {2 \left (-3 a^2 B+3 A b (a+b x)+3 a A b-6 a B (a+b x)+B (a+b x)^2\right )}{3 b^3 \sqrt {a+b x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

fricas [A]  time = 1.12, size = 57, normalized size = 0.90 \begin {gather*} \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 {gather*}

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)

________________________________________________________________________________________

giac [A]  time = 1.30, size = 69, normalized size = 1.10 \begin {gather*} -\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 {gather*}

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

________________________________________________________________________________________

maple [A]  time = 0.00, size = 46, normalized size = 0.73 \begin {gather*} \frac {\frac {2}{3} B \,b^{2} x^{2}+2 A \,b^{2} x -\frac {8}{3} B a b x +4 A a b -\frac {16}{3} B \,a^{2}}{\sqrt {b x +a}\, b^{3}} \end {gather*}

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

________________________________________________________________________________________

maxima [A]  time = 0.89, size = 61, normalized size = 0.97 \begin {gather*} \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 {gather*}

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

________________________________________________________________________________________

mupad [B]  time = 0.06, size = 52, normalized size = 0.83 \begin {gather*} \frac {2\,B\,{\left (a+b\,x\right )}^2-6\,B\,a^2+6\,A\,a\,b+6\,A\,b\,\left (a+b\,x\right )-12\,B\,a\,\left (a+b\,x\right )}{3\,b^3\,\sqrt {a+b\,x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x*(A + B*x))/(a + b*x)^(3/2),x)

[Out]

(2*B*(a + b*x)^2 - 6*B*a^2 + 6*A*a*b + 6*A*b*(a + b*x) - 12*B*a*(a + b*x))/(3*b^3*(a + b*x)^(1/2))

________________________________________________________________________________________

sympy [A]  time = 10.13, size = 60, normalized size = 0.95 \begin {gather*} \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 {gather*}

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

________________________________________________________________________________________