3.551 \(\int \frac {(A+B x) (a^2+2 a b x+b^2 x^2)^3}{x^7} \, dx\)

Optimal. Leaf size=132 \[ -\frac {a^6 A}{6 x^6}-\frac {a^5 (a B+6 A b)}{5 x^5}-\frac {3 a^4 b (2 a B+5 A b)}{4 x^4}-\frac {5 a^3 b^2 (3 a B+4 A b)}{3 x^3}-\frac {5 a^2 b^3 (4 a B+3 A b)}{2 x^2}+b^5 \log (x) (6 a B+A b)-\frac {3 a b^4 (5 a B+2 A b)}{x}+b^6 B x \]

[Out]

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

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Rubi [A]  time = 0.08, antiderivative size = 132, 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} \[ -\frac {5 a^3 b^2 (3 a B+4 A b)}{3 x^3}-\frac {5 a^2 b^3 (4 a B+3 A b)}{2 x^2}-\frac {a^5 (a B+6 A b)}{5 x^5}-\frac {3 a^4 b (2 a B+5 A b)}{4 x^4}-\frac {a^6 A}{6 x^6}-\frac {3 a b^4 (5 a B+2 A b)}{x}+b^5 \log (x) (6 a B+A b)+b^6 B x \]

Antiderivative was successfully verified.

[In]

Int[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^3)/x^7,x]

[Out]

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

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Mathematica [A]  time = 0.07, size = 125, normalized size = 0.95 \[ b^5 \log (x) (6 a B+A b)-\frac {2 a^6 (5 A+6 B x)+18 a^5 b x (4 A+5 B x)+75 a^4 b^2 x^2 (3 A+4 B x)+200 a^3 b^3 x^3 (2 A+3 B x)+450 a^2 b^4 x^4 (A+2 B x)+360 a A b^5 x^5-60 b^6 B x^7}{60 x^6} \]

Antiderivative was successfully verified.

[In]

Integrate[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^3)/x^7,x]

[Out]

-1/60*(360*a*A*b^5*x^5 - 60*b^6*B*x^7 + 450*a^2*b^4*x^4*(A + 2*B*x) + 200*a^3*b^3*x^3*(2*A + 3*B*x) + 75*a^4*b
^2*x^2*(3*A + 4*B*x) + 18*a^5*b*x*(4*A + 5*B*x) + 2*a^6*(5*A + 6*B*x))/x^6 + b^5*(A*b + 6*a*B)*Log[x]

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fricas [A]  time = 1.03, size = 149, normalized size = 1.13 \[ \frac {60 \, B b^{6} x^{7} + 60 \, {\left (6 \, B a b^{5} + A b^{6}\right )} x^{6} \log \relax (x) - 10 \, A a^{6} - 180 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} x^{5} - 150 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} x^{4} - 100 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} x^{3} - 45 \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} x^{2} - 12 \, {\left (B a^{6} + 6 \, A a^{5} b\right )} x}{60 \, x^{6}} \]

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^7,x, algorithm="fricas")

[Out]

1/60*(60*B*b^6*x^7 + 60*(6*B*a*b^5 + A*b^6)*x^6*log(x) - 10*A*a^6 - 180*(5*B*a^2*b^4 + 2*A*a*b^5)*x^5 - 150*(4
*B*a^3*b^3 + 3*A*a^2*b^4)*x^4 - 100*(3*B*a^4*b^2 + 4*A*a^3*b^3)*x^3 - 45*(2*B*a^5*b + 5*A*a^4*b^2)*x^2 - 12*(B
*a^6 + 6*A*a^5*b)*x)/x^6

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giac [A]  time = 0.16, size = 144, normalized size = 1.09 \[ B b^{6} x + {\left (6 \, B a b^{5} + A b^{6}\right )} \log \left ({\left | x \right |}\right ) - \frac {10 \, A a^{6} + 180 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} x^{5} + 150 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} x^{4} + 100 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} x^{3} + 45 \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} x^{2} + 12 \, {\left (B a^{6} + 6 \, A a^{5} b\right )} x}{60 \, x^{6}} \]

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^7,x, algorithm="giac")

[Out]

B*b^6*x + (6*B*a*b^5 + A*b^6)*log(abs(x)) - 1/60*(10*A*a^6 + 180*(5*B*a^2*b^4 + 2*A*a*b^5)*x^5 + 150*(4*B*a^3*
b^3 + 3*A*a^2*b^4)*x^4 + 100*(3*B*a^4*b^2 + 4*A*a^3*b^3)*x^3 + 45*(2*B*a^5*b + 5*A*a^4*b^2)*x^2 + 12*(B*a^6 +
6*A*a^5*b)*x)/x^6

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maple [A]  time = 0.05, size = 144, normalized size = 1.09 \[ A \,b^{6} \ln \relax (x )+6 B a \,b^{5} \ln \relax (x )+B \,b^{6} x -\frac {6 A a \,b^{5}}{x}-\frac {15 B \,a^{2} b^{4}}{x}-\frac {15 A \,a^{2} b^{4}}{2 x^{2}}-\frac {10 B \,a^{3} b^{3}}{x^{2}}-\frac {20 A \,a^{3} b^{3}}{3 x^{3}}-\frac {5 B \,a^{4} b^{2}}{x^{3}}-\frac {15 A \,a^{4} b^{2}}{4 x^{4}}-\frac {3 B \,a^{5} b}{2 x^{4}}-\frac {6 A \,a^{5} b}{5 x^{5}}-\frac {B \,a^{6}}{5 x^{5}}-\frac {A \,a^{6}}{6 x^{6}} \]

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^7,x)

[Out]

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

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maxima [A]  time = 0.47, size = 143, normalized size = 1.08 \[ B b^{6} x + {\left (6 \, B a b^{5} + A b^{6}\right )} \log \relax (x) - \frac {10 \, A a^{6} + 180 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} x^{5} + 150 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} x^{4} + 100 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} x^{3} + 45 \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} x^{2} + 12 \, {\left (B a^{6} + 6 \, A a^{5} b\right )} x}{60 \, x^{6}} \]

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^7,x, algorithm="maxima")

[Out]

B*b^6*x + (6*B*a*b^5 + A*b^6)*log(x) - 1/60*(10*A*a^6 + 180*(5*B*a^2*b^4 + 2*A*a*b^5)*x^5 + 150*(4*B*a^3*b^3 +
3*A*a^2*b^4)*x^4 + 100*(3*B*a^4*b^2 + 4*A*a^3*b^3)*x^3 + 45*(2*B*a^5*b + 5*A*a^4*b^2)*x^2 + 12*(B*a^6 + 6*A*a
^5*b)*x)/x^6

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mupad [B]  time = 0.08, size = 139, normalized size = 1.05 \[ \ln \relax (x)\,\left (A\,b^6+6\,B\,a\,b^5\right )-\frac {x\,\left (\frac {B\,a^6}{5}+\frac {6\,A\,b\,a^5}{5}\right )+\frac {A\,a^6}{6}+x^2\,\left (\frac {3\,B\,a^5\,b}{2}+\frac {15\,A\,a^4\,b^2}{4}\right )+x^5\,\left (15\,B\,a^2\,b^4+6\,A\,a\,b^5\right )+x^3\,\left (5\,B\,a^4\,b^2+\frac {20\,A\,a^3\,b^3}{3}\right )+x^4\,\left (10\,B\,a^3\,b^3+\frac {15\,A\,a^2\,b^4}{2}\right )}{x^6}+B\,b^6\,x \]

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^7,x)

[Out]

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

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sympy [A]  time = 3.16, size = 150, normalized size = 1.14 \[ B b^{6} x + b^{5} \left (A b + 6 B a\right ) \log {\relax (x )} + \frac {- 10 A a^{6} + x^{5} \left (- 360 A a b^{5} - 900 B a^{2} b^{4}\right ) + x^{4} \left (- 450 A a^{2} b^{4} - 600 B a^{3} b^{3}\right ) + x^{3} \left (- 400 A a^{3} b^{3} - 300 B a^{4} b^{2}\right ) + x^{2} \left (- 225 A a^{4} b^{2} - 90 B a^{5} b\right ) + x \left (- 72 A a^{5} b - 12 B a^{6}\right )}{60 x^{6}} \]

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**7,x)

[Out]

B*b**6*x + b**5*(A*b + 6*B*a)*log(x) + (-10*A*a**6 + x**5*(-360*A*a*b**5 - 900*B*a**2*b**4) + x**4*(-450*A*a**
2*b**4 - 600*B*a**3*b**3) + x**3*(-400*A*a**3*b**3 - 300*B*a**4*b**2) + x**2*(-225*A*a**4*b**2 - 90*B*a**5*b)
+ x*(-72*A*a**5*b - 12*B*a**6))/(60*x**6)

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