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

Optimal. Leaf size=121 \[ -\frac {A b-2 a B}{3 b^3 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {a (A b-a B)}{4 b^3 (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {B}{2 b^3 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}} \]

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Rubi [A]  time = 0.07, antiderivative size = 121, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.074, Rules used = {770, 77} \begin {gather*} -\frac {A b-2 a B}{3 b^3 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {a (A b-a B)}{4 b^3 (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {B}{2 b^3 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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]))))

Rule 770

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

Rubi steps

\begin {align*} \int \frac {x (A+B x)}{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx &=\frac {\left (b^4 \left (a b+b^2 x\right )\right ) \int \frac {x (A+B x)}{\left (a b+b^2 x\right )^5} \, dx}{\sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {\left (b^4 \left (a b+b^2 x\right )\right ) \int \left (\frac {a (-A b+a B)}{b^7 (a+b x)^5}+\frac {A b-2 a B}{b^7 (a+b x)^4}+\frac {B}{b^7 (a+b x)^3}\right ) \, dx}{\sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {a (A b-a B)}{4 b^3 (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {A b-2 a B}{3 b^3 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {B}{2 b^3 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}\\ \end {align*}

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Mathematica [A]  time = 0.02, size = 56, normalized size = 0.46 \begin {gather*} \frac {a^2 (-B)-a b (A+4 B x)-2 b^2 x (2 A+3 B x)}{12 b^3 (a+b x)^3 \sqrt {(a+b x)^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-(a^2*B) - 2*b^2*x*(2*A + 3*B*x) - a*b*(A + 4*B*x))/(12*b^3*(a + b*x)^3*Sqrt[(a + b*x)^2])

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IntegrateAlgebraic [B]  time = 1.07, size = 333, normalized size = 2.75 \begin {gather*} \frac {-2 \left (-3 a^6 b B+3 a^5 A b^2-a^2 b^5 B x^4-a A b^6 x^4-4 a b^6 B x^5-4 A b^7 x^5-6 b^7 B x^6\right )-2 \sqrt {b^2} \sqrt {a^2+2 a b x+b^2 x^2} \left (-3 a^5 B+3 a^4 A b+3 a^4 b B x-3 a^3 A b^2 x-3 a^3 b^2 B x^2+3 a^2 A b^3 x^2+3 a^2 b^3 B x^3-3 a A b^4 x^3-2 a b^4 B x^4+4 A b^5 x^4+6 b^5 B x^5\right )}{3 b^3 x^4 \sqrt {a^2+2 a b x+b^2 x^2} \left (-8 a^3 b^5-24 a^2 b^6 x-24 a b^7 x^2-8 b^8 x^3\right )+3 b^3 \sqrt {b^2} x^4 \left (8 a^4 b^4+32 a^3 b^5 x+48 a^2 b^6 x^2+32 a b^7 x^3+8 b^8 x^4\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*Sqrt[b^2]*Sqrt[a^2 + 2*a*b*x + b^2*x^2]*(3*a^4*A*b - 3*a^5*B - 3*a^3*A*b^2*x + 3*a^4*b*B*x + 3*a^2*A*b^3*x
^2 - 3*a^3*b^2*B*x^2 - 3*a*A*b^4*x^3 + 3*a^2*b^3*B*x^3 + 4*A*b^5*x^4 - 2*a*b^4*B*x^4 + 6*b^5*B*x^5) - 2*(3*a^5
*A*b^2 - 3*a^6*b*B - a*A*b^6*x^4 - a^2*b^5*B*x^4 - 4*A*b^7*x^5 - 4*a*b^6*B*x^5 - 6*b^7*B*x^6))/(3*b^3*x^4*Sqrt
[a^2 + 2*a*b*x + b^2*x^2]*(-8*a^3*b^5 - 24*a^2*b^6*x - 24*a*b^7*x^2 - 8*b^8*x^3) + 3*b^3*Sqrt[b^2]*x^4*(8*a^4*
b^4 + 32*a^3*b^5*x + 48*a^2*b^6*x^2 + 32*a*b^7*x^3 + 8*b^8*x^4))

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fricas [A]  time = 0.44, size = 80, normalized size = 0.66 \begin {gather*} -\frac {6 \, B b^{2} x^{2} + B a^{2} + A a b + 4 \, {\left (B a b + A b^{2}\right )} x}{12 \, {\left (b^{7} x^{4} + 4 \, a b^{6} x^{3} + 6 \, a^{2} b^{5} x^{2} + 4 \, a^{3} b^{4} x + a^{4} b^{3}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/12*(6*B*b^2*x^2 + B*a^2 + A*a*b + 4*(B*a*b + A*b^2)*x)/(b^7*x^4 + 4*a*b^6*x^3 + 6*a^2*b^5*x^2 + 4*a^3*b^4*x
+ a^4*b^3)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \mathit {sage}_{0} x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sage0*x

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maple [A]  time = 0.05, size = 52, normalized size = 0.43 \begin {gather*} -\frac {\left (b x +a \right ) \left (6 B \,b^{2} x^{2}+4 A \,b^{2} x +4 B a b x +A a b +B \,a^{2}\right )}{12 \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}} b^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/12*(b*x+a)/b^3*(6*B*b^2*x^2+4*A*b^2*x+4*B*a*b*x+A*a*b+B*a^2)/((b*x+a)^2)^(5/2)

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maxima [A]  time = 0.55, size = 90, normalized size = 0.74 \begin {gather*} -\frac {A}{3 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} b^{2}} - \frac {B}{2 \, b^{5} {\left (x + \frac {a}{b}\right )}^{2}} + \frac {2 \, B a}{3 \, b^{6} {\left (x + \frac {a}{b}\right )}^{3}} - \frac {B a^{2}}{4 \, b^{7} {\left (x + \frac {a}{b}\right )}^{4}} + \frac {A a}{4 \, b^{6} {\left (x + \frac {a}{b}\right )}^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*A/((b^2*x^2 + 2*a*b*x + a^2)^(3/2)*b^2) - 1/2*B/(b^5*(x + a/b)^2) + 2/3*B*a/(b^6*(x + a/b)^3) - 1/4*B*a^2
/(b^7*(x + a/b)^4) + 1/4*A*a/(b^6*(x + a/b)^4)

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mupad [B]  time = 1.25, size = 62, normalized size = 0.51 \begin {gather*} -\frac {\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}\,\left (B\,a^2+4\,B\,a\,b\,x+A\,a\,b+6\,B\,b^2\,x^2+4\,A\,b^2\,x\right )}{12\,b^3\,{\left (a+b\,x\right )}^5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-((a^2 + b^2*x^2 + 2*a*b*x)^(1/2)*(B*a^2 + 6*B*b^2*x^2 + A*a*b + 4*A*b^2*x + 4*B*a*b*x))/(12*b^3*(a + b*x)^5)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x \left (A + B x\right )}{\left (\left (a + b x\right )^{2}\right )^{\frac {5}{2}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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