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

Optimal. Leaf size=143 \[ \frac{a^5 (A b-a B)}{3 b^7 (a+b x)^3}-\frac{a^4 (5 A b-6 a B)}{2 b^7 (a+b x)^2}+\frac{5 a^3 (2 A b-3 a B)}{b^7 (a+b x)}+\frac{10 a^2 (A b-2 a B) \log (a+b x)}{b^7}+\frac{x^2 (A b-4 a B)}{2 b^5}-\frac{2 a x (2 A b-5 a B)}{b^6}+\frac{B x^3}{3 b^4} \]

[Out]

(-2*a*(2*A*b - 5*a*B)*x)/b^6 + ((A*b - 4*a*B)*x^2)/(2*b^5) + (B*x^3)/(3*b^4) + (a^5*(A*b - a*B))/(3*b^7*(a + b
*x)^3) - (a^4*(5*A*b - 6*a*B))/(2*b^7*(a + b*x)^2) + (5*a^3*(2*A*b - 3*a*B))/(b^7*(a + b*x)) + (10*a^2*(A*b -
2*a*B)*Log[a + b*x])/b^7

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Rubi [A]  time = 0.159865, antiderivative size = 143, normalized size of antiderivative = 1., 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} \[ \frac{a^5 (A b-a B)}{3 b^7 (a+b x)^3}-\frac{a^4 (5 A b-6 a B)}{2 b^7 (a+b x)^2}+\frac{5 a^3 (2 A b-3 a B)}{b^7 (a+b x)}+\frac{10 a^2 (A b-2 a B) \log (a+b x)}{b^7}+\frac{x^2 (A b-4 a B)}{2 b^5}-\frac{2 a x (2 A b-5 a B)}{b^6}+\frac{B x^3}{3 b^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*a*(2*A*b - 5*a*B)*x)/b^6 + ((A*b - 4*a*B)*x^2)/(2*b^5) + (B*x^3)/(3*b^4) + (a^5*(A*b - a*B))/(3*b^7*(a + b
*x)^3) - (a^4*(5*A*b - 6*a*B))/(2*b^7*(a + b*x)^2) + (5*a^3*(2*A*b - 3*a*B))/(b^7*(a + b*x)) + (10*a^2*(A*b -
2*a*B)*Log[a + b*x])/b^7

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^5 (A+B x)}{\left (a^2+2 a b x+b^2 x^2\right )^2} \, dx &=\int \frac{x^5 (A+B x)}{(a+b x)^4} \, dx\\ &=\int \left (\frac{2 a (-2 A b+5 a B)}{b^6}+\frac{(A b-4 a B) x}{b^5}+\frac{B x^2}{b^4}+\frac{a^5 (-A b+a B)}{b^6 (a+b x)^4}-\frac{a^4 (-5 A b+6 a B)}{b^6 (a+b x)^3}+\frac{5 a^3 (-2 A b+3 a B)}{b^6 (a+b x)^2}-\frac{10 a^2 (-A b+2 a B)}{b^6 (a+b x)}\right ) \, dx\\ &=-\frac{2 a (2 A b-5 a B) x}{b^6}+\frac{(A b-4 a B) x^2}{2 b^5}+\frac{B x^3}{3 b^4}+\frac{a^5 (A b-a B)}{3 b^7 (a+b x)^3}-\frac{a^4 (5 A b-6 a B)}{2 b^7 (a+b x)^2}+\frac{5 a^3 (2 A b-3 a B)}{b^7 (a+b x)}+\frac{10 a^2 (A b-2 a B) \log (a+b x)}{b^7}\\ \end{align*}

Mathematica [A]  time = 0.0540591, size = 147, normalized size = 1.03 \[ \frac{a^3 b^3 x^2 (146 B x-9 A)+3 a^2 b^4 x^3 (10 B x-21 A)+3 a^4 b^2 x (27 A+26 B x)+a^5 b (47 A-102 B x)-60 a^2 (a+b x)^3 (2 a B-A b) \log (a+b x)-74 a^6 B-3 a b^5 x^4 (5 A+2 B x)+b^6 x^5 (3 A+2 B x)}{6 b^7 (a+b x)^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-74*a^6*B + a^5*b*(47*A - 102*B*x) + b^6*x^5*(3*A + 2*B*x) - 3*a*b^5*x^4*(5*A + 2*B*x) + 3*a^2*b^4*x^3*(-21*A
+ 10*B*x) + 3*a^4*b^2*x*(27*A + 26*B*x) + a^3*b^3*x^2*(-9*A + 146*B*x) - 60*a^2*(-(A*b) + 2*a*B)*(a + b*x)^3*
Log[a + b*x])/(6*b^7*(a + b*x)^3)

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Maple [A]  time = 0.01, size = 174, normalized size = 1.2 \begin{align*}{\frac{B{x}^{3}}{3\,{b}^{4}}}+{\frac{A{x}^{2}}{2\,{b}^{4}}}-2\,{\frac{B{x}^{2}a}{{b}^{5}}}-4\,{\frac{aAx}{{b}^{5}}}+10\,{\frac{{a}^{2}Bx}{{b}^{6}}}+10\,{\frac{A{a}^{3}}{{b}^{6} \left ( bx+a \right ) }}-15\,{\frac{B{a}^{4}}{{b}^{7} \left ( bx+a \right ) }}+10\,{\frac{{a}^{2}\ln \left ( bx+a \right ) A}{{b}^{6}}}-20\,{\frac{{a}^{3}\ln \left ( bx+a \right ) B}{{b}^{7}}}-{\frac{5\,A{a}^{4}}{2\,{b}^{6} \left ( bx+a \right ) ^{2}}}+3\,{\frac{B{a}^{5}}{{b}^{7} \left ( bx+a \right ) ^{2}}}+{\frac{{a}^{5}A}{3\,{b}^{6} \left ( bx+a \right ) ^{3}}}-{\frac{B{a}^{6}}{3\,{b}^{7} \left ( bx+a \right ) ^{3}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*B*x^3/b^4+1/2/b^4*A*x^2-2/b^5*B*x^2*a-4/b^5*A*a*x+10/b^6*B*a^2*x+10*a^3/b^6/(b*x+a)*A-15*a^4/b^7/(b*x+a)*B
+10*a^2/b^6*ln(b*x+a)*A-20*a^3/b^7*ln(b*x+a)*B-5/2*a^4/b^6/(b*x+a)^2*A+3*a^5/b^7/(b*x+a)^2*B+1/3*a^5/b^6/(b*x+
a)^3*A-1/3*a^6/b^7/(b*x+a)^3*B

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Maxima [A]  time = 1.00943, size = 227, normalized size = 1.59 \begin{align*} -\frac{74 \, B a^{6} - 47 \, A a^{5} b + 30 \,{\left (3 \, B a^{4} b^{2} - 2 \, A a^{3} b^{3}\right )} x^{2} + 3 \,{\left (54 \, B a^{5} b - 35 \, A a^{4} b^{2}\right )} x}{6 \,{\left (b^{10} x^{3} + 3 \, a b^{9} x^{2} + 3 \, a^{2} b^{8} x + a^{3} b^{7}\right )}} + \frac{2 \, B b^{2} x^{3} - 3 \,{\left (4 \, B a b - A b^{2}\right )} x^{2} + 12 \,{\left (5 \, B a^{2} - 2 \, A a b\right )} x}{6 \, b^{6}} - \frac{10 \,{\left (2 \, B a^{3} - A a^{2} b\right )} \log \left (b x + a\right )}{b^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/6*(74*B*a^6 - 47*A*a^5*b + 30*(3*B*a^4*b^2 - 2*A*a^3*b^3)*x^2 + 3*(54*B*a^5*b - 35*A*a^4*b^2)*x)/(b^10*x^3
+ 3*a*b^9*x^2 + 3*a^2*b^8*x + a^3*b^7) + 1/6*(2*B*b^2*x^3 - 3*(4*B*a*b - A*b^2)*x^2 + 12*(5*B*a^2 - 2*A*a*b)*x
)/b^6 - 10*(2*B*a^3 - A*a^2*b)*log(b*x + a)/b^7

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Fricas [A]  time = 1.32878, size = 533, normalized size = 3.73 \begin{align*} \frac{2 \, B b^{6} x^{6} - 74 \, B a^{6} + 47 \, A a^{5} b - 3 \,{\left (2 \, B a b^{5} - A b^{6}\right )} x^{5} + 15 \,{\left (2 \, B a^{2} b^{4} - A a b^{5}\right )} x^{4} +{\left (146 \, B a^{3} b^{3} - 63 \, A a^{2} b^{4}\right )} x^{3} + 3 \,{\left (26 \, B a^{4} b^{2} - 3 \, A a^{3} b^{3}\right )} x^{2} - 3 \,{\left (34 \, B a^{5} b - 27 \, A a^{4} b^{2}\right )} x - 60 \,{\left (2 \, B a^{6} - A a^{5} b +{\left (2 \, B a^{3} b^{3} - A a^{2} b^{4}\right )} x^{3} + 3 \,{\left (2 \, B a^{4} b^{2} - A a^{3} b^{3}\right )} x^{2} + 3 \,{\left (2 \, B a^{5} b - A a^{4} b^{2}\right )} x\right )} \log \left (b x + a\right )}{6 \,{\left (b^{10} x^{3} + 3 \, a b^{9} x^{2} + 3 \, a^{2} b^{8} x + a^{3} b^{7}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*(2*B*b^6*x^6 - 74*B*a^6 + 47*A*a^5*b - 3*(2*B*a*b^5 - A*b^6)*x^5 + 15*(2*B*a^2*b^4 - A*a*b^5)*x^4 + (146*B
*a^3*b^3 - 63*A*a^2*b^4)*x^3 + 3*(26*B*a^4*b^2 - 3*A*a^3*b^3)*x^2 - 3*(34*B*a^5*b - 27*A*a^4*b^2)*x - 60*(2*B*
a^6 - A*a^5*b + (2*B*a^3*b^3 - A*a^2*b^4)*x^3 + 3*(2*B*a^4*b^2 - A*a^3*b^3)*x^2 + 3*(2*B*a^5*b - A*a^4*b^2)*x)
*log(b*x + a))/(b^10*x^3 + 3*a*b^9*x^2 + 3*a^2*b^8*x + a^3*b^7)

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Sympy [A]  time = 1.6348, size = 165, normalized size = 1.15 \begin{align*} \frac{B x^{3}}{3 b^{4}} - \frac{10 a^{2} \left (- A b + 2 B a\right ) \log{\left (a + b x \right )}}{b^{7}} - \frac{- 47 A a^{5} b + 74 B a^{6} + x^{2} \left (- 60 A a^{3} b^{3} + 90 B a^{4} b^{2}\right ) + x \left (- 105 A a^{4} b^{2} + 162 B a^{5} b\right )}{6 a^{3} b^{7} + 18 a^{2} b^{8} x + 18 a b^{9} x^{2} + 6 b^{10} x^{3}} - \frac{x^{2} \left (- A b + 4 B a\right )}{2 b^{5}} + \frac{x \left (- 4 A a b + 10 B a^{2}\right )}{b^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

B*x**3/(3*b**4) - 10*a**2*(-A*b + 2*B*a)*log(a + b*x)/b**7 - (-47*A*a**5*b + 74*B*a**6 + x**2*(-60*A*a**3*b**3
+ 90*B*a**4*b**2) + x*(-105*A*a**4*b**2 + 162*B*a**5*b))/(6*a**3*b**7 + 18*a**2*b**8*x + 18*a*b**9*x**2 + 6*b
**10*x**3) - x**2*(-A*b + 4*B*a)/(2*b**5) + x*(-4*A*a*b + 10*B*a**2)/b**6

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Giac [A]  time = 1.11062, size = 201, normalized size = 1.41 \begin{align*} -\frac{10 \,{\left (2 \, B a^{3} - A a^{2} b\right )} \log \left ({\left | b x + a \right |}\right )}{b^{7}} - \frac{74 \, B a^{6} - 47 \, A a^{5} b + 30 \,{\left (3 \, B a^{4} b^{2} - 2 \, A a^{3} b^{3}\right )} x^{2} + 3 \,{\left (54 \, B a^{5} b - 35 \, A a^{4} b^{2}\right )} x}{6 \,{\left (b x + a\right )}^{3} b^{7}} + \frac{2 \, B b^{8} x^{3} - 12 \, B a b^{7} x^{2} + 3 \, A b^{8} x^{2} + 60 \, B a^{2} b^{6} x - 24 \, A a b^{7} x}{6 \, b^{12}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-10*(2*B*a^3 - A*a^2*b)*log(abs(b*x + a))/b^7 - 1/6*(74*B*a^6 - 47*A*a^5*b + 30*(3*B*a^4*b^2 - 2*A*a^3*b^3)*x^
2 + 3*(54*B*a^5*b - 35*A*a^4*b^2)*x)/((b*x + a)^3*b^7) + 1/6*(2*B*b^8*x^3 - 12*B*a*b^7*x^2 + 3*A*b^8*x^2 + 60*
B*a^2*b^6*x - 24*A*a*b^7*x)/b^12