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

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

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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}{2} a^2 b^3 x^2 (4 a B+3 A b)+5 a^3 b^2 x (3 a B+4 A b)-\frac {a^5 (a B+6 A b)}{x}+3 a^4 b \log (x) (2 a B+5 A b)-\frac {a^6 A}{2 x^2}+a b^4 x^3 (5 a B+2 A b)+\frac {1}{4} b^5 x^4 (6 a B+A b)+\frac {1}{5} b^6 B x^5 \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Mathematica [A]  time = 0.06, size = 128, normalized size = 0.98 \begin {gather*} -\frac {a^6 (A+2 B x)}{2 x^2}-\frac {6 a^5 A b}{x}+3 a^4 b \log (x) (2 a B+5 A b)+15 a^4 b^2 B x+10 a^3 b^3 x (2 A+B x)+\frac {5}{2} a^2 b^4 x^2 (3 A+2 B x)+\frac {1}{2} a b^5 x^3 (4 A+3 B x)+\frac {1}{20} b^6 x^4 (5 A+4 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^3,x]

[Out]

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

[Out]

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

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fricas [A]  time = 0.42, size = 149, normalized size = 1.14 \begin {gather*} \frac {4 \, B b^{6} x^{7} - 10 \, A a^{6} + 5 \, {\left (6 \, B a b^{5} + A b^{6}\right )} x^{6} + 20 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} x^{5} + 50 \, {\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} + 60 \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} x^{2} \log \relax (x) - 20 \, {\left (B a^{6} + 6 \, A a^{5} b\right )} x}{20 \, x^{2}} \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^3,x, algorithm="fricas")

[Out]

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

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giac [A]  time = 0.17, size = 144, normalized size = 1.10 \begin {gather*} \frac {1}{5} \, B b^{6} x^{5} + \frac {3}{2} \, B a b^{5} x^{4} + \frac {1}{4} \, A b^{6} x^{4} + 5 \, B a^{2} b^{4} x^{3} + 2 \, A a b^{5} x^{3} + 10 \, B a^{3} b^{3} x^{2} + \frac {15}{2} \, A a^{2} b^{4} x^{2} + 15 \, B a^{4} b^{2} x + 20 \, A a^{3} b^{3} x + 3 \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} \log \left ({\left | x \right |}\right ) - \frac {A a^{6} + 2 \, {\left (B a^{6} + 6 \, A a^{5} b\right )} x}{2 \, x^{2}} \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^3,x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.06, size = 144, normalized size = 1.10 \begin {gather*} \frac {B \,b^{6} x^{5}}{5}+\frac {A \,b^{6} x^{4}}{4}+\frac {3 B a \,b^{5} x^{4}}{2}+2 A a \,b^{5} x^{3}+5 B \,a^{2} b^{4} x^{3}+\frac {15 A \,a^{2} b^{4} x^{2}}{2}+10 B \,a^{3} b^{3} x^{2}+15 A \,a^{4} b^{2} \ln \relax (x )+20 A \,a^{3} b^{3} x +6 B \,a^{5} b \ln \relax (x )+15 B \,a^{4} b^{2} x -\frac {6 A \,a^{5} b}{x}-\frac {B \,a^{6}}{x}-\frac {A \,a^{6}}{2 x^{2}} \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^3,x)

[Out]

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

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maxima [A]  time = 0.56, size = 143, normalized size = 1.09 \begin {gather*} \frac {1}{5} \, B b^{6} x^{5} + \frac {1}{4} \, {\left (6 \, B a b^{5} + A b^{6}\right )} x^{4} + {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} x^{3} + \frac {5}{2} \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} x^{2} + 5 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} x + 3 \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} \log \relax (x) - \frac {A a^{6} + 2 \, {\left (B a^{6} + 6 \, A a^{5} b\right )} x}{2 \, x^{2}} \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^3,x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 1.08, size = 130, normalized size = 0.99 \begin {gather*} \ln \relax (x)\,\left (6\,B\,a^5\,b+15\,A\,a^4\,b^2\right )-\frac {x\,\left (B\,a^6+6\,A\,b\,a^5\right )+\frac {A\,a^6}{2}}{x^2}+x^4\,\left (\frac {A\,b^6}{4}+\frac {3\,B\,a\,b^5}{2}\right )+\frac {B\,b^6\,x^5}{5}+\frac {5\,a^2\,b^3\,x^2\,\left (3\,A\,b+4\,B\,a\right )}{2}+5\,a^3\,b^2\,x\,\left (4\,A\,b+3\,B\,a\right )+a\,b^4\,x^3\,\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^3,x)

[Out]

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

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sympy [A]  time = 0.51, size = 148, normalized size = 1.13 \begin {gather*} \frac {B b^{6} x^{5}}{5} + 3 a^{4} b \left (5 A b + 2 B a\right ) \log {\relax (x )} + x^{4} \left (\frac {A b^{6}}{4} + \frac {3 B a b^{5}}{2}\right ) + x^{3} \left (2 A a b^{5} + 5 B a^{2} b^{4}\right ) + x^{2} \left (\frac {15 A a^{2} b^{4}}{2} + 10 B a^{3} b^{3}\right ) + x \left (20 A a^{3} b^{3} + 15 B a^{4} b^{2}\right ) + \frac {- A a^{6} + x \left (- 12 A a^{5} b - 2 B a^{6}\right )}{2 x^{2}} \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**3,x)

[Out]

B*b**6*x**5/5 + 3*a**4*b*(5*A*b + 2*B*a)*log(x) + x**4*(A*b**6/4 + 3*B*a*b**5/2) + x**3*(2*A*a*b**5 + 5*B*a**2
*b**4) + x**2*(15*A*a**2*b**4/2 + 10*B*a**3*b**3) + x*(20*A*a**3*b**3 + 15*B*a**4*b**2) + (-A*a**6 + x*(-12*A*
a**5*b - 2*B*a**6))/(2*x**2)

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