3.5.76 \(\int \frac {(A+B x) (a^2+2 a b x+b^2 x^2)^3}{x^5} \, dx\)

Optimal. Leaf size=134 \[ -\frac {a^6 A}{4 x^4}-\frac {a^5 (a B+6 A b)}{3 x^3}-\frac {3 a^4 b (2 a B+5 A b)}{2 x^2}-\frac {5 a^3 b^2 (3 a B+4 A b)}{x}+5 a^2 b^3 \log (x) (4 a B+3 A b)+\frac {1}{2} b^5 x^2 (6 a B+A b)+3 a b^4 x (5 a B+2 A b)+\frac {1}{3} b^6 B x^3 \]

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

Antiderivative was successfully verified.

[In]

Int[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^3)/x^5,x]

[Out]

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

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Mathematica [A]  time = 0.05, size = 128, normalized size = 0.96 \begin {gather*} -\frac {a^6 (3 A+4 B x)}{12 x^4}-\frac {a^5 b (2 A+3 B x)}{x^3}-\frac {15 a^4 b^2 (A+2 B x)}{2 x^2}-\frac {20 a^3 A b^3}{x}+5 a^2 b^3 \log (x) (4 a B+3 A b)+15 a^2 b^4 B x+3 a b^5 x (2 A+B x)+\frac {1}{6} b^6 x^2 (3 A+2 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^5,x]

[Out]

(-20*a^3*A*b^3)/x + 15*a^2*b^4*B*x + 3*a*b^5*x*(2*A + B*x) - (15*a^4*b^2*(A + 2*B*x))/(2*x^2) + (b^6*x^2*(3*A
+ 2*B*x))/6 - (a^5*b*(2*A + 3*B*x))/x^3 - (a^6*(3*A + 4*B*x))/(12*x^4) + 5*a^2*b^3*(3*A*b + 4*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^5} \, 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^5,x]

[Out]

IntegrateAlgebraic[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^3)/x^5, x]

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fricas [A]  time = 0.42, size = 149, normalized size = 1.11 \begin {gather*} \frac {4 \, B b^{6} x^{7} - 3 \, A a^{6} + 6 \, {\left (6 \, B a b^{5} + A b^{6}\right )} x^{6} + 36 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} x^{5} + 60 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} x^{4} \log \relax (x) - 60 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} x^{3} - 18 \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} x^{2} - 4 \, {\left (B a^{6} + 6 \, A a^{5} b\right )} x}{12 \, x^{4}} \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^5,x, algorithm="fricas")

[Out]

1/12*(4*B*b^6*x^7 - 3*A*a^6 + 6*(6*B*a*b^5 + A*b^6)*x^6 + 36*(5*B*a^2*b^4 + 2*A*a*b^5)*x^5 + 60*(4*B*a^3*b^3 +
3*A*a^2*b^4)*x^4*log(x) - 60*(3*B*a^4*b^2 + 4*A*a^3*b^3)*x^3 - 18*(2*B*a^5*b + 5*A*a^4*b^2)*x^2 - 4*(B*a^6 +
6*A*a^5*b)*x)/x^4

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giac [A]  time = 0.19, size = 145, normalized size = 1.08 \begin {gather*} \frac {1}{3} \, B b^{6} x^{3} + 3 \, B a b^{5} x^{2} + \frac {1}{2} \, A b^{6} x^{2} + 15 \, B a^{2} b^{4} x + 6 \, A a b^{5} x + 5 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} \log \left ({\left | x \right |}\right ) - \frac {3 \, A a^{6} + 60 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} x^{3} + 18 \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} x^{2} + 4 \, {\left (B a^{6} + 6 \, A a^{5} b\right )} x}{12 \, x^{4}} \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^5,x, algorithm="giac")

[Out]

1/3*B*b^6*x^3 + 3*B*a*b^5*x^2 + 1/2*A*b^6*x^2 + 15*B*a^2*b^4*x + 6*A*a*b^5*x + 5*(4*B*a^3*b^3 + 3*A*a^2*b^4)*l
og(abs(x)) - 1/12*(3*A*a^6 + 60*(3*B*a^4*b^2 + 4*A*a^3*b^3)*x^3 + 18*(2*B*a^5*b + 5*A*a^4*b^2)*x^2 + 4*(B*a^6
+ 6*A*a^5*b)*x)/x^4

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maple [A]  time = 0.06, size = 144, normalized size = 1.07 \begin {gather*} \frac {B \,b^{6} x^{3}}{3}+\frac {A \,b^{6} x^{2}}{2}+3 B a \,b^{5} x^{2}+15 A \,a^{2} b^{4} \ln \relax (x )+6 A a \,b^{5} x +20 B \,a^{3} b^{3} \ln \relax (x )+15 B \,a^{2} b^{4} x -\frac {20 A \,a^{3} b^{3}}{x}-\frac {15 B \,a^{4} b^{2}}{x}-\frac {15 A \,a^{4} b^{2}}{2 x^{2}}-\frac {3 B \,a^{5} b}{x^{2}}-\frac {2 A \,a^{5} b}{x^{3}}-\frac {B \,a^{6}}{3 x^{3}}-\frac {A \,a^{6}}{4 x^{4}} \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^5,x)

[Out]

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

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maxima [A]  time = 0.55, size = 145, normalized size = 1.08 \begin {gather*} \frac {1}{3} \, B b^{6} x^{3} + \frac {1}{2} \, {\left (6 \, B a b^{5} + A b^{6}\right )} x^{2} + 3 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} x + 5 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} \log \relax (x) - \frac {3 \, A a^{6} + 60 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} x^{3} + 18 \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} x^{2} + 4 \, {\left (B a^{6} + 6 \, A a^{5} b\right )} x}{12 \, x^{4}} \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^5,x, algorithm="maxima")

[Out]

1/3*B*b^6*x^3 + 1/2*(6*B*a*b^5 + A*b^6)*x^2 + 3*(5*B*a^2*b^4 + 2*A*a*b^5)*x + 5*(4*B*a^3*b^3 + 3*A*a^2*b^4)*lo
g(x) - 1/12*(3*A*a^6 + 60*(3*B*a^4*b^2 + 4*A*a^3*b^3)*x^3 + 18*(2*B*a^5*b + 5*A*a^4*b^2)*x^2 + 4*(B*a^6 + 6*A*
a^5*b)*x)/x^4

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mupad [B]  time = 0.06, size = 138, normalized size = 1.03 \begin {gather*} x^2\,\left (\frac {A\,b^6}{2}+3\,B\,a\,b^5\right )-\frac {x\,\left (\frac {B\,a^6}{3}+2\,A\,b\,a^5\right )+\frac {A\,a^6}{4}+x^2\,\left (3\,B\,a^5\,b+\frac {15\,A\,a^4\,b^2}{2}\right )+x^3\,\left (15\,B\,a^4\,b^2+20\,A\,a^3\,b^3\right )}{x^4}+\ln \relax (x)\,\left (20\,B\,a^3\,b^3+15\,A\,a^2\,b^4\right )+\frac {B\,b^6\,x^3}{3}+3\,a\,b^4\,x\,\left (2\,A\,b+5\,B\,a\right ) \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^5,x)

[Out]

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

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sympy [A]  time = 1.35, size = 150, normalized size = 1.12 \begin {gather*} \frac {B b^{6} x^{3}}{3} + 5 a^{2} b^{3} \left (3 A b + 4 B a\right ) \log {\relax (x )} + x^{2} \left (\frac {A b^{6}}{2} + 3 B a b^{5}\right ) + x \left (6 A a b^{5} + 15 B a^{2} b^{4}\right ) + \frac {- 3 A a^{6} + x^{3} \left (- 240 A a^{3} b^{3} - 180 B a^{4} b^{2}\right ) + x^{2} \left (- 90 A a^{4} b^{2} - 36 B a^{5} b\right ) + x \left (- 24 A a^{5} b - 4 B a^{6}\right )}{12 x^{4}} \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**5,x)

[Out]

B*b**6*x**3/3 + 5*a**2*b**3*(3*A*b + 4*B*a)*log(x) + x**2*(A*b**6/2 + 3*B*a*b**5) + x*(6*A*a*b**5 + 15*B*a**2*
b**4) + (-3*A*a**6 + x**3*(-240*A*a**3*b**3 - 180*B*a**4*b**2) + x**2*(-90*A*a**4*b**2 - 36*B*a**5*b) + x*(-24
*A*a**5*b - 4*B*a**6))/(12*x**4)

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