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

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

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Rubi [A]  time = 0.16, antiderivative size = 134, 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 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} \end {gather*}

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*}

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Mathematica [A]  time = 0.06, size = 127, normalized size = 0.95 \begin {gather*} \frac {\frac {60 a^5 (A b-a B)}{a+b x}+60 a^4 (5 A b-6 a B) \log (a+b x)+60 a^3 b x (5 a B-4 A b)-30 a^2 b^2 x^2 (4 a B-3 A b)+15 b^4 x^4 (A b-2 a B)+20 a b^3 x^3 (3 a B-2 A b)+12 b^5 B x^5}{60 b^7} \end {gather*}

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

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.40, size = 188, normalized size = 1.40 \begin {gather*} \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 {gather*}

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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giac [A]  time = 0.16, size = 152, normalized size = 1.13 \begin {gather*} -\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 {gather*}

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

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maple [A]  time = 0.06, size = 156, normalized size = 1.16 \begin {gather*} \frac {B \,x^{5}}{5 b^{2}}+\frac {A \,x^{4}}{4 b^{2}}-\frac {B a \,x^{4}}{2 b^{3}}-\frac {2 A a \,x^{3}}{3 b^{3}}+\frac {B \,a^{2} x^{3}}{b^{4}}+\frac {3 A \,a^{2} x^{2}}{2 b^{4}}-\frac {2 B \,a^{3} x^{2}}{b^{5}}+\frac {A \,a^{5}}{\left (b x +a \right ) b^{6}}+\frac {5 A \,a^{4} \ln \left (b x +a \right )}{b^{6}}-\frac {4 A \,a^{3} x}{b^{5}}-\frac {B \,a^{6}}{\left (b x +a \right ) b^{7}}-\frac {6 B \,a^{5} \ln \left (b x +a \right )}{b^{7}}+\frac {5 B \,a^{4} x}{b^{6}} \end {gather*}

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

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maxima [A]  time = 0.55, size = 149, normalized size = 1.11 \begin {gather*} -\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 {gather*}

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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mupad [B]  time = 1.10, size = 279, normalized size = 2.08 \begin {gather*} x^2\,\left (\frac {a\,\left (\frac {2\,a\,\left (\frac {A}{b^2}-\frac {2\,B\,a}{b^3}\right )}{b}+\frac {B\,a^2}{b^4}\right )}{b}-\frac {a^2\,\left (\frac {A}{b^2}-\frac {2\,B\,a}{b^3}\right )}{2\,b^2}\right )+x^4\,\left (\frac {A}{4\,b^2}-\frac {B\,a}{2\,b^3}\right )-x^3\,\left (\frac {2\,a\,\left (\frac {A}{b^2}-\frac {2\,B\,a}{b^3}\right )}{3\,b}+\frac {B\,a^2}{3\,b^4}\right )-x\,\left (\frac {2\,a\,\left (\frac {2\,a\,\left (\frac {2\,a\,\left (\frac {A}{b^2}-\frac {2\,B\,a}{b^3}\right )}{b}+\frac {B\,a^2}{b^4}\right )}{b}-\frac {a^2\,\left (\frac {A}{b^2}-\frac {2\,B\,a}{b^3}\right )}{b^2}\right )}{b}-\frac {a^2\,\left (\frac {2\,a\,\left (\frac {A}{b^2}-\frac {2\,B\,a}{b^3}\right )}{b}+\frac {B\,a^2}{b^4}\right )}{b^2}\right )-\frac {\ln \left (a+b\,x\right )\,\left (6\,B\,a^5-5\,A\,a^4\,b\right )}{b^7}+\frac {B\,x^5}{5\,b^2}-\frac {B\,a^6-A\,a^5\,b}{b\,\left (x\,b^7+a\,b^6\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 0.51, size = 143, normalized size = 1.07 \begin {gather*} \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}} + x^{4} \left (\frac {A}{4 b^{2}} - \frac {B a}{2 b^{3}}\right ) + x^{3} \left (- \frac {2 A a}{3 b^{3}} + \frac {B a^{2}}{b^{4}}\right ) + x^{2} \left (\frac {3 A a^{2}}{2 b^{4}} - \frac {2 B a^{3}}{b^{5}}\right ) + x \left (- \frac {4 A a^{3}}{b^{5}} + \frac {5 B a^{4}}{b^{6}}\right ) + \frac {A a^{5} b - B a^{6}}{a b^{7} + b^{8} x} \end {gather*}

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

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