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

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]

-(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)*Log[x]

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Rubi [A]  time = 0.208044, antiderivative size = 132, 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 \[ -\frac{a^6 A}{6 x^6}-\frac{a^5 (a B+6 A b)}{5 x^5}-\frac{3 a^4 b (2 a B+5 A b)}{4 x^4}-\frac{5 a^3 b^2 (3 a B+4 A b)}{3 x^3}-\frac{5 a^2 b^3 (4 a B+3 A b)}{2 x^2}+b^5 \log (x) (6 a B+A b)-\frac{3 a b^4 (5 a B+2 A b)}{x}+b^6 B x \]

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)*Log[x]

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Rubi in Sympy [A]  time = 45.1704, size = 136, normalized size = 1.03 \[ - \frac{A a^{6}}{6 x^{6}} + B b^{6} x - \frac{a^{5} \left (6 A b + B a\right )}{5 x^{5}} - \frac{3 a^{4} b \left (5 A b + 2 B a\right )}{4 x^{4}} - \frac{5 a^{3} b^{2} \left (4 A b + 3 B a\right )}{3 x^{3}} - \frac{5 a^{2} b^{3} \left (3 A b + 4 B a\right )}{2 x^{2}} - \frac{3 a b^{4} \left (2 A b + 5 B a\right )}{x} + b^{5} \left (A b + 6 B a\right ) \log{\left (x \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.134665, 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]

-(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))/(60*x^6) + b^5*(A*b + 6*a*B)*Log[x]

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Maple [A]  time = 0.012, size = 144, normalized size = 1.1 \[{b}^{6}Bx+A\ln \left ( x \right ){b}^{6}+6\,B\ln \left ( x \right ) a{b}^{5}-{\frac{A{a}^{6}}{6\,{x}^{6}}}-{\frac{15\,A{b}^{2}{a}^{4}}{4\,{x}^{4}}}-{\frac{3\,B{a}^{5}b}{2\,{x}^{4}}}-{\frac{20\,A{a}^{3}{b}^{3}}{3\,{x}^{3}}}-5\,{\frac{B{b}^{2}{a}^{4}}{{x}^{3}}}-{\frac{15\,A{a}^{2}{b}^{4}}{2\,{x}^{2}}}-10\,{\frac{B{a}^{3}{b}^{3}}{{x}^{2}}}-{\frac{6\,A{a}^{5}b}{5\,{x}^{5}}}-{\frac{B{a}^{6}}{5\,{x}^{5}}}-6\,{\frac{Aa{b}^{5}}{x}}-15\,{\frac{B{a}^{2}{b}^{4}}{x}} \]

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

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Maxima [A]  time = 0.687829, size = 193, normalized size = 1.46 \[ B b^{6} x +{\left (6 \, B a b^{5} + A b^{6}\right )} \log \left (x\right ) - \frac{10 \, A a^{6} + 180 \,{\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} x^{5} + 150 \,{\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} x^{4} + 100 \,{\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} x^{3} + 45 \,{\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} x^{2} + 12 \,{\left (B a^{6} + 6 \, A a^{5} b\right )} x}{60 \, x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^2 + 2*a*b*x + a^2)^3*(B*x + A)/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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Fricas [A]  time = 0.269809, size = 201, normalized size = 1.52 \[ \frac{60 \, B b^{6} x^{7} + 60 \,{\left (6 \, B a b^{5} + A b^{6}\right )} x^{6} \log \left (x\right ) - 10 \, A a^{6} - 180 \,{\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} x^{5} - 150 \,{\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} x^{4} - 100 \,{\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} x^{3} - 45 \,{\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} x^{2} - 12 \,{\left (B a^{6} + 6 \, A a^{5} b\right )} x}{60 \, x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^2 + 2*a*b*x + a^2)^3*(B*x + A)/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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Sympy [A]  time = 12.946, size = 141, normalized size = 1.07 \[ B b^{6} x + b^{5} \left (A b + 6 B a\right ) \log{\left (x \right )} - \frac{10 A a^{6} + x^{5} \left (360 A a b^{5} + 900 B a^{2} b^{4}\right ) + x^{4} \left (450 A a^{2} b^{4} + 600 B a^{3} b^{3}\right ) + x^{3} \left (400 A a^{3} b^{3} + 300 B a^{4} b^{2}\right ) + x^{2} \left (225 A a^{4} b^{2} + 90 B a^{5} b\right ) + x \left (72 A a^{5} b + 12 B a^{6}\right )}{60 x^{6}} \]

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 + 3
00*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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GIAC/XCAS [A]  time = 0.270683, size = 194, normalized size = 1.47 \[ B b^{6} x +{\left (6 \, B a b^{5} + A b^{6}\right )}{\rm ln}\left ({\left | x \right |}\right ) - \frac{10 \, A a^{6} + 180 \,{\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} x^{5} + 150 \,{\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} x^{4} + 100 \,{\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} x^{3} + 45 \,{\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} x^{2} + 12 \,{\left (B a^{6} + 6 \, A a^{5} b\right )} x}{60 \, x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^2 + 2*a*b*x + a^2)^3*(B*x + A)/x^7,x, algorithm="giac")

[Out]

B*b^6*x + (6*B*a*b^5 + A*b^6)*ln(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