3.5.77 \(\int \frac {(A+B x) (a^2+2 a b x+b^2 x^2)^3}{x^6} \, dx\)

Optimal. Leaf size=131 \[ -\frac {a^6 A}{5 x^5}-\frac {a^5 (a B+6 A b)}{4 x^4}-\frac {a^4 b (2 a B+5 A b)}{x^3}-\frac {5 a^3 b^2 (3 a B+4 A b)}{2 x^2}-\frac {5 a^2 b^3 (4 a B+3 A b)}{x}+b^5 x (6 a B+A b)+3 a b^4 \log (x) (5 a B+2 A b)+\frac {1}{2} b^6 B x^2 \]

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Rubi [A]  time = 0.08, antiderivative size = 131, 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, 76} \begin {gather*} -\frac {5 a^3 b^2 (3 a B+4 A b)}{2 x^2}-\frac {5 a^2 b^3 (4 a B+3 A b)}{x}-\frac {a^5 (a B+6 A b)}{4 x^4}-\frac {a^4 b (2 a B+5 A b)}{x^3}-\frac {a^6 A}{5 x^5}+b^5 x (6 a B+A b)+3 a b^4 \log (x) (5 a B+2 A b)+\frac {1}{2} b^6 B x^2 \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^3)/x^6,x]

[Out]

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

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 76

Int[((d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_))*((e_) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*
x)*(d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, d, e, f, n}, x] && IGtQ[p, 0] && (NeQ[n, -1] || EqQ[p, 1]) && N
eQ[b*e + a*f, 0] && ( !IntegerQ[n] || LtQ[9*p + 5*n, 0] || GeQ[n + p + 1, 0] || (GeQ[n + p + 2, 0] && Rational
Q[a, b, d, e, f])) && (NeQ[n + p + 3, 0] || EqQ[p, 1])

Rubi steps

\begin {align*} \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^3}{x^6} \, dx &=\int \frac {(a+b x)^6 (A+B x)}{x^6} \, dx\\ &=\int \left (b^5 (A b+6 a B)+\frac {a^6 A}{x^6}+\frac {a^5 (6 A b+a B)}{x^5}+\frac {3 a^4 b (5 A b+2 a B)}{x^4}+\frac {5 a^3 b^2 (4 A b+3 a B)}{x^3}+\frac {5 a^2 b^3 (3 A b+4 a B)}{x^2}+\frac {3 a b^4 (2 A b+5 a B)}{x}+b^6 B x\right ) \, dx\\ &=-\frac {a^6 A}{5 x^5}-\frac {a^5 (6 A b+a B)}{4 x^4}-\frac {a^4 b (5 A b+2 a B)}{x^3}-\frac {5 a^3 b^2 (4 A b+3 a B)}{2 x^2}-\frac {5 a^2 b^3 (3 A b+4 a B)}{x}+b^5 (A b+6 a B) x+\frac {1}{2} b^6 B x^2+3 a b^4 (2 A b+5 a B) \log (x)\\ \end {align*}

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Mathematica [A]  time = 0.05, size = 128, normalized size = 0.98 \begin {gather*} -\frac {a^6 (4 A+5 B x)}{20 x^5}-\frac {a^5 b (3 A+4 B x)}{2 x^4}-\frac {5 a^4 b^2 (2 A+3 B x)}{2 x^3}-\frac {10 a^3 b^3 (A+2 B x)}{x^2}-\frac {15 a^2 A b^4}{x}+3 a b^4 \log (x) (5 a B+2 A b)+6 a b^5 B x+\frac {1}{2} b^6 x (2 A+B x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^3)/x^6,x]

[Out]

(-15*a^2*A*b^4)/x + 6*a*b^5*B*x + (b^6*x*(2*A + B*x))/2 - (10*a^3*b^3*(A + 2*B*x))/x^2 - (5*a^4*b^2*(2*A + 3*B
*x))/(2*x^3) - (a^5*b*(3*A + 4*B*x))/(2*x^4) - (a^6*(4*A + 5*B*x))/(20*x^5) + 3*a*b^4*(2*A*b + 5*a*B)*Log[x]

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^3}{x^6} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^3)/x^6,x]

[Out]

IntegrateAlgebraic[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^3)/x^6, x]

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fricas [A]  time = 0.41, size = 149, normalized size = 1.14 \begin {gather*} \frac {10 \, B b^{6} x^{7} - 4 \, A a^{6} + 20 \, {\left (6 \, B a b^{5} + A b^{6}\right )} x^{6} + 60 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} x^{5} \log \relax (x) - 100 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} x^{4} - 50 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} x^{3} - 20 \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} x^{2} - 5 \, {\left (B a^{6} + 6 \, A a^{5} b\right )} x}{20 \, x^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(b^2*x^2+2*a*b*x+a^2)^3/x^6,x, algorithm="fricas")

[Out]

1/20*(10*B*b^6*x^7 - 4*A*a^6 + 20*(6*B*a*b^5 + A*b^6)*x^6 + 60*(5*B*a^2*b^4 + 2*A*a*b^5)*x^5*log(x) - 100*(4*B
*a^3*b^3 + 3*A*a^2*b^4)*x^4 - 50*(3*B*a^4*b^2 + 4*A*a^3*b^3)*x^3 - 20*(2*B*a^5*b + 5*A*a^4*b^2)*x^2 - 5*(B*a^6
+ 6*A*a^5*b)*x)/x^5

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giac [A]  time = 0.18, size = 144, normalized size = 1.10 \begin {gather*} \frac {1}{2} \, B b^{6} x^{2} + 6 \, B a b^{5} x + A b^{6} x + 3 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} \log \left ({\left | x \right |}\right ) - \frac {4 \, A a^{6} + 100 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} x^{4} + 50 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} x^{3} + 20 \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} x^{2} + 5 \, {\left (B a^{6} + 6 \, A a^{5} b\right )} x}{20 \, x^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(b^2*x^2+2*a*b*x+a^2)^3/x^6,x, algorithm="giac")

[Out]

1/2*B*b^6*x^2 + 6*B*a*b^5*x + A*b^6*x + 3*(5*B*a^2*b^4 + 2*A*a*b^5)*log(abs(x)) - 1/20*(4*A*a^6 + 100*(4*B*a^3
*b^3 + 3*A*a^2*b^4)*x^4 + 50*(3*B*a^4*b^2 + 4*A*a^3*b^3)*x^3 + 20*(2*B*a^5*b + 5*A*a^4*b^2)*x^2 + 5*(B*a^6 + 6
*A*a^5*b)*x)/x^5

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maple [A]  time = 0.06, size = 143, normalized size = 1.09 \begin {gather*} \frac {B \,b^{6} x^{2}}{2}+6 A a \,b^{5} \ln \relax (x )+A \,b^{6} x +15 B \,a^{2} b^{4} \ln \relax (x )+6 B a \,b^{5} x -\frac {15 A \,a^{2} b^{4}}{x}-\frac {20 B \,a^{3} b^{3}}{x}-\frac {10 A \,a^{3} b^{3}}{x^{2}}-\frac {15 B \,a^{4} b^{2}}{2 x^{2}}-\frac {5 A \,a^{4} b^{2}}{x^{3}}-\frac {2 B \,a^{5} b}{x^{3}}-\frac {3 A \,a^{5} b}{2 x^{4}}-\frac {B \,a^{6}}{4 x^{4}}-\frac {A \,a^{6}}{5 x^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)*(b^2*x^2+2*a*b*x+a^2)^3/x^6,x)

[Out]

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

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maxima [A]  time = 0.61, size = 144, normalized size = 1.10 \begin {gather*} \frac {1}{2} \, B b^{6} x^{2} + {\left (6 \, B a b^{5} + A b^{6}\right )} x + 3 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} \log \relax (x) - \frac {4 \, A a^{6} + 100 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} x^{4} + 50 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} x^{3} + 20 \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} x^{2} + 5 \, {\left (B a^{6} + 6 \, A a^{5} b\right )} x}{20 \, x^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(b^2*x^2+2*a*b*x+a^2)^3/x^6,x, algorithm="maxima")

[Out]

1/2*B*b^6*x^2 + (6*B*a*b^5 + A*b^6)*x + 3*(5*B*a^2*b^4 + 2*A*a*b^5)*log(x) - 1/20*(4*A*a^6 + 100*(4*B*a^3*b^3
+ 3*A*a^2*b^4)*x^4 + 50*(3*B*a^4*b^2 + 4*A*a^3*b^3)*x^3 + 20*(2*B*a^5*b + 5*A*a^4*b^2)*x^2 + 5*(B*a^6 + 6*A*a^
5*b)*x)/x^5

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mupad [B]  time = 1.09, size = 140, normalized size = 1.07 \begin {gather*} x\,\left (A\,b^6+6\,B\,a\,b^5\right )-\frac {x\,\left (\frac {B\,a^6}{4}+\frac {3\,A\,b\,a^5}{2}\right )+\frac {A\,a^6}{5}+x^2\,\left (2\,B\,a^5\,b+5\,A\,a^4\,b^2\right )+x^3\,\left (\frac {15\,B\,a^4\,b^2}{2}+10\,A\,a^3\,b^3\right )+x^4\,\left (20\,B\,a^3\,b^3+15\,A\,a^2\,b^4\right )}{x^5}+\ln \relax (x)\,\left (15\,B\,a^2\,b^4+6\,A\,a\,b^5\right )+\frac {B\,b^6\,x^2}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((A + B*x)*(a^2 + b^2*x^2 + 2*a*b*x)^3)/x^6,x)

[Out]

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

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sympy [A]  time = 2.09, size = 150, normalized size = 1.15 \begin {gather*} \frac {B b^{6} x^{2}}{2} + 3 a b^{4} \left (2 A b + 5 B a\right ) \log {\relax (x )} + x \left (A b^{6} + 6 B a b^{5}\right ) + \frac {- 4 A a^{6} + x^{4} \left (- 300 A a^{2} b^{4} - 400 B a^{3} b^{3}\right ) + x^{3} \left (- 200 A a^{3} b^{3} - 150 B a^{4} b^{2}\right ) + x^{2} \left (- 100 A a^{4} b^{2} - 40 B a^{5} b\right ) + x \left (- 30 A a^{5} b - 5 B a^{6}\right )}{20 x^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(b**2*x**2+2*a*b*x+a**2)**3/x**6,x)

[Out]

B*b**6*x**2/2 + 3*a*b**4*(2*A*b + 5*B*a)*log(x) + x*(A*b**6 + 6*B*a*b**5) + (-4*A*a**6 + x**4*(-300*A*a**2*b**
4 - 400*B*a**3*b**3) + x**3*(-200*A*a**3*b**3 - 150*B*a**4*b**2) + x**2*(-100*A*a**4*b**2 - 40*B*a**5*b) + x*(
-30*A*a**5*b - 5*B*a**6))/(20*x**5)

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