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

Optimal. Leaf size=87 \[ -\frac {a^2 (A b-a B)}{5 b^4 (a+b x)^5}+\frac {a (2 A b-3 a B)}{4 b^4 (a+b x)^4}-\frac {A b-3 a B}{3 b^4 (a+b x)^3}-\frac {B}{2 b^4 (a+b x)^2} \]

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Rubi [A]  time = 0.07, antiderivative size = 87, 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 = {27, 77} \begin {gather*} -\frac {a^2 (A b-a B)}{5 b^4 (a+b x)^5}+\frac {a (2 A b-3 a B)}{4 b^4 (a+b x)^4}-\frac {A b-3 a B}{3 b^4 (a+b x)^3}-\frac {B}{2 b^4 (a+b x)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 27

Int[(u_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[u*Cancel[(b/2 + c*x)^(2*p)/c^p], x] /; Fr
eeQ[{a, b, c}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]

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)}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx &=\int \frac {x^2 (A+B x)}{(a+b x)^6} \, dx\\ &=\int \left (-\frac {a^2 (-A b+a B)}{b^3 (a+b x)^6}+\frac {a (-2 A b+3 a B)}{b^3 (a+b x)^5}+\frac {A b-3 a B}{b^3 (a+b x)^4}+\frac {B}{b^3 (a+b x)^3}\right ) \, dx\\ &=-\frac {a^2 (A b-a B)}{5 b^4 (a+b x)^5}+\frac {a (2 A b-3 a B)}{4 b^4 (a+b x)^4}-\frac {A b-3 a B}{3 b^4 (a+b x)^3}-\frac {B}{2 b^4 (a+b x)^2}\\ \end {align*}

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Mathematica [A]  time = 0.02, size = 63, normalized size = 0.72 \begin {gather*} -\frac {3 a^3 B+a^2 b (2 A+15 B x)+10 a b^2 x (A+3 B x)+10 b^3 x^2 (2 A+3 B x)}{60 b^4 (a+b x)^5} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/60*(3*a^3*B + 10*a*b^2*x*(A + 3*B*x) + 10*b^3*x^2*(2*A + 3*B*x) + a^2*b*(2*A + 15*B*x))/(b^4*(a + b*x)^5)

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^2 (A+B x)}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.39, size = 119, normalized size = 1.37 \begin {gather*} -\frac {30 \, B b^{3} x^{3} + 3 \, B a^{3} + 2 \, A a^{2} b + 10 \, {\left (3 \, B a b^{2} + 2 \, A b^{3}\right )} x^{2} + 5 \, {\left (3 \, B a^{2} b + 2 \, A a b^{2}\right )} x}{60 \, {\left (b^{9} x^{5} + 5 \, a b^{8} x^{4} + 10 \, a^{2} b^{7} x^{3} + 10 \, a^{3} b^{6} x^{2} + 5 \, a^{4} b^{5} x + a^{5} b^{4}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/60*(30*B*b^3*x^3 + 3*B*a^3 + 2*A*a^2*b + 10*(3*B*a*b^2 + 2*A*b^3)*x^2 + 5*(3*B*a^2*b + 2*A*a*b^2)*x)/(b^9*x
^5 + 5*a*b^8*x^4 + 10*a^2*b^7*x^3 + 10*a^3*b^6*x^2 + 5*a^4*b^5*x + a^5*b^4)

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giac [A]  time = 0.16, size = 70, normalized size = 0.80 \begin {gather*} -\frac {30 \, B b^{3} x^{3} + 30 \, B a b^{2} x^{2} + 20 \, A b^{3} x^{2} + 15 \, B a^{2} b x + 10 \, A a b^{2} x + 3 \, B a^{3} + 2 \, A a^{2} b}{60 \, {\left (b x + a\right )}^{5} b^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/60*(30*B*b^3*x^3 + 30*B*a*b^2*x^2 + 20*A*b^3*x^2 + 15*B*a^2*b*x + 10*A*a*b^2*x + 3*B*a^3 + 2*A*a^2*b)/((b*x
+ a)^5*b^4)

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maple [A]  time = 0.05, size = 80, normalized size = 0.92 \begin {gather*} -\frac {\left (A b -B a \right ) a^{2}}{5 \left (b x +a \right )^{5} b^{4}}-\frac {B}{2 \left (b x +a \right )^{2} b^{4}}+\frac {\left (2 A b -3 B a \right ) a}{4 \left (b x +a \right )^{4} b^{4}}-\frac {A b -3 B a}{3 \left (b x +a \right )^{3} b^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.61, size = 119, normalized size = 1.37 \begin {gather*} -\frac {30 \, B b^{3} x^{3} + 3 \, B a^{3} + 2 \, A a^{2} b + 10 \, {\left (3 \, B a b^{2} + 2 \, A b^{3}\right )} x^{2} + 5 \, {\left (3 \, B a^{2} b + 2 \, A a b^{2}\right )} x}{60 \, {\left (b^{9} x^{5} + 5 \, a b^{8} x^{4} + 10 \, a^{2} b^{7} x^{3} + 10 \, a^{3} b^{6} x^{2} + 5 \, a^{4} b^{5} x + a^{5} b^{4}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/60*(30*B*b^3*x^3 + 3*B*a^3 + 2*A*a^2*b + 10*(3*B*a*b^2 + 2*A*b^3)*x^2 + 5*(3*B*a^2*b + 2*A*a*b^2)*x)/(b^9*x
^5 + 5*a*b^8*x^4 + 10*a^2*b^7*x^3 + 10*a^3*b^6*x^2 + 5*a^4*b^5*x + a^5*b^4)

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mupad [B]  time = 0.04, size = 113, normalized size = 1.30 \begin {gather*} -\frac {\frac {B\,x^3}{2\,b}+\frac {a^2\,\left (2\,A\,b+3\,B\,a\right )}{60\,b^4}+\frac {x^2\,\left (2\,A\,b+3\,B\,a\right )}{6\,b^2}+\frac {a\,x\,\left (2\,A\,b+3\,B\,a\right )}{12\,b^3}}{a^5+5\,a^4\,b\,x+10\,a^3\,b^2\,x^2+10\,a^2\,b^3\,x^3+5\,a\,b^4\,x^4+b^5\,x^5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-((B*x^3)/(2*b) + (a^2*(2*A*b + 3*B*a))/(60*b^4) + (x^2*(2*A*b + 3*B*a))/(6*b^2) + (a*x*(2*A*b + 3*B*a))/(12*b
^3))/(a^5 + b^5*x^5 + 5*a*b^4*x^4 + 10*a^3*b^2*x^2 + 10*a^2*b^3*x^3 + 5*a^4*b*x)

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sympy [A]  time = 1.01, size = 126, normalized size = 1.45 \begin {gather*} \frac {- 2 A a^{2} b - 3 B a^{3} - 30 B b^{3} x^{3} + x^{2} \left (- 20 A b^{3} - 30 B a b^{2}\right ) + x \left (- 10 A a b^{2} - 15 B a^{2} b\right )}{60 a^{5} b^{4} + 300 a^{4} b^{5} x + 600 a^{3} b^{6} x^{2} + 600 a^{2} b^{7} x^{3} + 300 a b^{8} x^{4} + 60 b^{9} x^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(-2*A*a**2*b - 3*B*a**3 - 30*B*b**3*x**3 + x**2*(-20*A*b**3 - 30*B*a*b**2) + x*(-10*A*a*b**2 - 15*B*a**2*b))/(
60*a**5*b**4 + 300*a**4*b**5*x + 600*a**3*b**6*x**2 + 600*a**2*b**7*x**3 + 300*a*b**8*x**4 + 60*b**9*x**5)

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