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

Optimal. Leaf size=134 \[ \frac{a^2 x^2 (3 A b-4 a B)}{2 b^5}+\frac{a^5 (A b-a B)}{b^7 (a+b x)}-\frac{a^3 x (4 A b-5 a B)}{b^6}+\frac{a^4 (5 A b-6 a B) \log (a+b x)}{b^7}-\frac{a x^3 (2 A b-3 a B)}{3 b^4}+\frac{x^4 (A b-2 a B)}{4 b^3}+\frac{B x^5}{5 b^2} \]

[Out]

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

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Rubi [A]  time = 0.157538, antiderivative size = 134, 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} \[ \frac{a^2 x^2 (3 A b-4 a B)}{2 b^5}+\frac{a^5 (A b-a B)}{b^7 (a+b x)}-\frac{a^3 x (4 A b-5 a B)}{b^6}+\frac{a^4 (5 A b-6 a B) \log (a+b x)}{b^7}-\frac{a x^3 (2 A b-3 a B)}{3 b^4}+\frac{x^4 (A b-2 a B)}{4 b^3}+\frac{B x^5}{5 b^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Mathematica [A]  time = 0.0652993, size = 127, normalized size = 0.95 \[ \frac{-30 a^2 b^2 x^2 (4 a B-3 A b)+\frac{60 a^5 (A b-a B)}{a+b x}+60 a^3 b x (5 a B-4 A b)+60 a^4 (5 A b-6 a B) \log (a+b x)+20 a b^3 x^3 (3 a B-2 A b)+15 b^4 x^4 (A b-2 a B)+12 b^5 B x^5}{60 b^7} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(60*a^3*b*(-4*A*b + 5*a*B)*x - 30*a^2*b^2*(-3*A*b + 4*a*B)*x^2 + 20*a*b^3*(-2*A*b + 3*a*B)*x^3 + 15*b^4*(A*b -
2*a*B)*x^4 + 12*b^5*B*x^5 + (60*a^5*(A*b - a*B))/(a + b*x) + 60*a^4*(5*A*b - 6*a*B)*Log[a + b*x])/(60*b^7)

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Maple [A]  time = 0.007, size = 156, normalized size = 1.2 \begin{align*}{\frac{B{x}^{5}}{5\,{b}^{2}}}+{\frac{A{x}^{4}}{4\,{b}^{2}}}-{\frac{aB{x}^{4}}{2\,{b}^{3}}}-{\frac{2\,aA{x}^{3}}{3\,{b}^{3}}}+{\frac{{a}^{2}B{x}^{3}}{{b}^{4}}}+{\frac{3\,{a}^{2}A{x}^{2}}{2\,{b}^{4}}}-2\,{\frac{{a}^{3}B{x}^{2}}{{b}^{5}}}-4\,{\frac{{a}^{3}Ax}{{b}^{5}}}+5\,{\frac{{a}^{4}Bx}{{b}^{6}}}+{\frac{{a}^{5}A}{{b}^{6} \left ( bx+a \right ) }}-{\frac{B{a}^{6}}{{b}^{7} \left ( bx+a \right ) }}+5\,{\frac{{a}^{4}\ln \left ( bx+a \right ) A}{{b}^{6}}}-6\,{\frac{{a}^{5}\ln \left ( bx+a \right ) B}{{b}^{7}}} \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),x)

[Out]

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

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Maxima [A]  time = 0.99927, size = 201, normalized size = 1.5 \begin{align*} -\frac{B a^{6} - A a^{5} b}{b^{8} x + a b^{7}} + \frac{12 \, B b^{4} x^{5} - 15 \,{\left (2 \, B a b^{3} - A b^{4}\right )} x^{4} + 20 \,{\left (3 \, B a^{2} b^{2} - 2 \, A a b^{3}\right )} x^{3} - 30 \,{\left (4 \, B a^{3} b - 3 \, A a^{2} b^{2}\right )} x^{2} + 60 \,{\left (5 \, B a^{4} - 4 \, A a^{3} b\right )} x}{60 \, b^{6}} - \frac{{\left (6 \, B a^{5} - 5 \, A a^{4} 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),x, algorithm="maxima")

[Out]

-(B*a^6 - A*a^5*b)/(b^8*x + a*b^7) + 1/60*(12*B*b^4*x^5 - 15*(2*B*a*b^3 - A*b^4)*x^4 + 20*(3*B*a^2*b^2 - 2*A*a
*b^3)*x^3 - 30*(4*B*a^3*b - 3*A*a^2*b^2)*x^2 + 60*(5*B*a^4 - 4*A*a^3*b)*x)/b^6 - (6*B*a^5 - 5*A*a^4*b)*log(b*x
+ a)/b^7

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Fricas [A]  time = 1.30951, size = 408, normalized size = 3.04 \begin{align*} \frac{12 \, B b^{6} x^{6} - 60 \, B a^{6} + 60 \, A a^{5} b - 3 \,{\left (6 \, B a b^{5} - 5 \, A b^{6}\right )} x^{5} + 5 \,{\left (6 \, B a^{2} b^{4} - 5 \, A a b^{5}\right )} x^{4} - 10 \,{\left (6 \, B a^{3} b^{3} - 5 \, A a^{2} b^{4}\right )} x^{3} + 30 \,{\left (6 \, B a^{4} b^{2} - 5 \, A a^{3} b^{3}\right )} x^{2} + 60 \,{\left (5 \, B a^{5} b - 4 \, A a^{4} b^{2}\right )} x - 60 \,{\left (6 \, B a^{6} - 5 \, A a^{5} b +{\left (6 \, B a^{5} b - 5 \, A a^{4} b^{2}\right )} x\right )} \log \left (b x + a\right )}{60 \,{\left (b^{8} x + a 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),x, algorithm="fricas")

[Out]

1/60*(12*B*b^6*x^6 - 60*B*a^6 + 60*A*a^5*b - 3*(6*B*a*b^5 - 5*A*b^6)*x^5 + 5*(6*B*a^2*b^4 - 5*A*a*b^5)*x^4 - 1
0*(6*B*a^3*b^3 - 5*A*a^2*b^4)*x^3 + 30*(6*B*a^4*b^2 - 5*A*a^3*b^3)*x^2 + 60*(5*B*a^5*b - 4*A*a^4*b^2)*x - 60*(
6*B*a^6 - 5*A*a^5*b + (6*B*a^5*b - 5*A*a^4*b^2)*x)*log(b*x + a))/(b^8*x + a*b^7)

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Sympy [A]  time = 0.674527, size = 138, normalized size = 1.03 \begin{align*} \frac{B x^{5}}{5 b^{2}} - \frac{a^{4} \left (- 5 A b + 6 B a\right ) \log{\left (a + b x \right )}}{b^{7}} - \frac{- A a^{5} b + B a^{6}}{a b^{7} + b^{8} x} - \frac{x^{4} \left (- A b + 2 B a\right )}{4 b^{3}} + \frac{x^{3} \left (- 2 A a b + 3 B a^{2}\right )}{3 b^{4}} - \frac{x^{2} \left (- 3 A a^{2} b + 4 B a^{3}\right )}{2 b^{5}} + \frac{x \left (- 4 A a^{3} b + 5 B a^{4}\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),x)

[Out]

B*x**5/(5*b**2) - a**4*(-5*A*b + 6*B*a)*log(a + b*x)/b**7 - (-A*a**5*b + B*a**6)/(a*b**7 + b**8*x) - x**4*(-A*
b + 2*B*a)/(4*b**3) + x**3*(-2*A*a*b + 3*B*a**2)/(3*b**4) - x**2*(-3*A*a**2*b + 4*B*a**3)/(2*b**5) + x*(-4*A*a
**3*b + 5*B*a**4)/b**6

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Giac [A]  time = 1.1328, size = 205, normalized size = 1.53 \begin{align*} -\frac{{\left (6 \, B a^{5} - 5 \, A a^{4} b\right )} \log \left ({\left | b x + a \right |}\right )}{b^{7}} - \frac{B a^{6} - A a^{5} b}{{\left (b x + a\right )} b^{7}} + \frac{12 \, B b^{8} x^{5} - 30 \, B a b^{7} x^{4} + 15 \, A b^{8} x^{4} + 60 \, B a^{2} b^{6} x^{3} - 40 \, A a b^{7} x^{3} - 120 \, B a^{3} b^{5} x^{2} + 90 \, A a^{2} b^{6} x^{2} + 300 \, B a^{4} b^{4} x - 240 \, A a^{3} b^{5} x}{60 \, b^{10}} \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),x, algorithm="giac")

[Out]

-(6*B*a^5 - 5*A*a^4*b)*log(abs(b*x + a))/b^7 - (B*a^6 - A*a^5*b)/((b*x + a)*b^7) + 1/60*(12*B*b^8*x^5 - 30*B*a
*b^7*x^4 + 15*A*b^8*x^4 + 60*B*a^2*b^6*x^3 - 40*A*a*b^7*x^3 - 120*B*a^3*b^5*x^2 + 90*A*a^2*b^6*x^2 + 300*B*a^4
*b^4*x - 240*A*a^3*b^5*x)/b^10