3.5.75 \(\int \frac {(A+B x) (a^2+2 a b x+b^2 x^2)^3}{x^4} \, dx\)

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

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

Antiderivative was successfully verified.

[In]

Int[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^3)/x^4,x]

[Out]

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

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Mathematica [A]  time = 0.05, size = 127, normalized size = 0.95 \begin {gather*} -\frac {a^6 (2 A+3 B x)}{6 x^3}-\frac {3 a^5 b (A+2 B x)}{x^2}-\frac {15 a^4 A b^2}{x}+5 a^3 b^2 \log (x) (3 a B+4 A b)+20 a^3 b^3 B x+\frac {15}{2} a^2 b^4 x (2 A+B x)+a b^5 x^2 (3 A+2 B x)+\frac {1}{12} b^6 x^3 (4 A+3 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^4,x]

[Out]

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

[Out]

IntegrateAlgebraic[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^3)/x^4, x]

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fricas [A]  time = 0.41, size = 149, normalized size = 1.11 \begin {gather*} \frac {3 \, B b^{6} x^{7} - 4 \, A a^{6} + 4 \, {\left (6 \, B a b^{5} + A b^{6}\right )} x^{6} + 18 \, {\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} + 60 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} x^{3} \log \relax (x) - 36 \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} x^{2} - 6 \, {\left (B a^{6} + 6 \, A a^{5} b\right )} x}{12 \, x^{3}} \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^4,x, algorithm="fricas")

[Out]

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

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giac [A]  time = 0.17, size = 145, normalized size = 1.08 \begin {gather*} \frac {1}{4} \, B b^{6} x^{4} + 2 \, B a b^{5} x^{3} + \frac {1}{3} \, A b^{6} x^{3} + \frac {15}{2} \, B a^{2} b^{4} x^{2} + 3 \, A a b^{5} x^{2} + 20 \, B a^{3} b^{3} x + 15 \, A a^{2} b^{4} x + 5 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} \log \left ({\left | x \right |}\right ) - \frac {2 \, A a^{6} + 18 \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} x^{2} + 3 \, {\left (B a^{6} + 6 \, A a^{5} b\right )} x}{6 \, x^{3}} \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^4,x, algorithm="giac")

[Out]

1/4*B*b^6*x^4 + 2*B*a*b^5*x^3 + 1/3*A*b^6*x^3 + 15/2*B*a^2*b^4*x^2 + 3*A*a*b^5*x^2 + 20*B*a^3*b^3*x + 15*A*a^2
*b^4*x + 5*(3*B*a^4*b^2 + 4*A*a^3*b^3)*log(abs(x)) - 1/6*(2*A*a^6 + 18*(2*B*a^5*b + 5*A*a^4*b^2)*x^2 + 3*(B*a^
6 + 6*A*a^5*b)*x)/x^3

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maple [A]  time = 0.05, size = 144, normalized size = 1.07 \begin {gather*} \frac {B \,b^{6} x^{4}}{4}+\frac {A \,b^{6} x^{3}}{3}+2 B a \,b^{5} x^{3}+3 A a \,b^{5} x^{2}+\frac {15 B \,a^{2} b^{4} x^{2}}{2}+20 A \,a^{3} b^{3} \ln \relax (x )+15 A \,a^{2} b^{4} x +15 B \,a^{4} b^{2} \ln \relax (x )+20 B \,a^{3} b^{3} x -\frac {15 A \,a^{4} b^{2}}{x}-\frac {6 B \,a^{5} b}{x}-\frac {3 A \,a^{5} b}{x^{2}}-\frac {B \,a^{6}}{2 x^{2}}-\frac {A \,a^{6}}{3 x^{3}} \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^4,x)

[Out]

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

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maxima [A]  time = 0.52, size = 145, normalized size = 1.08 \begin {gather*} \frac {1}{4} \, B b^{6} x^{4} + \frac {1}{3} \, {\left (6 \, B a b^{5} + A b^{6}\right )} x^{3} + \frac {3}{2} \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} x^{2} + 5 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} x + 5 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} \log \relax (x) - \frac {2 \, A a^{6} + 18 \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} x^{2} + 3 \, {\left (B a^{6} + 6 \, A a^{5} b\right )} x}{6 \, x^{3}} \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^4,x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 0.06, size = 135, normalized size = 1.01 \begin {gather*} x^3\,\left (\frac {A\,b^6}{3}+2\,B\,a\,b^5\right )-\frac {x\,\left (\frac {B\,a^6}{2}+3\,A\,b\,a^5\right )+\frac {A\,a^6}{3}+x^2\,\left (6\,B\,a^5\,b+15\,A\,a^4\,b^2\right )}{x^3}+\ln \relax (x)\,\left (15\,B\,a^4\,b^2+20\,A\,a^3\,b^3\right )+\frac {B\,b^6\,x^4}{4}+5\,a^2\,b^3\,x\,\left (3\,A\,b+4\,B\,a\right )+\frac {3\,a\,b^4\,x^2\,\left (2\,A\,b+5\,B\,a\right )}{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^4,x)

[Out]

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

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sympy [A]  time = 0.83, size = 150, normalized size = 1.12 \begin {gather*} \frac {B b^{6} x^{4}}{4} + 5 a^{3} b^{2} \left (4 A b + 3 B a\right ) \log {\relax (x )} + x^{3} \left (\frac {A b^{6}}{3} + 2 B a b^{5}\right ) + x^{2} \left (3 A a b^{5} + \frac {15 B a^{2} b^{4}}{2}\right ) + x \left (15 A a^{2} b^{4} + 20 B a^{3} b^{3}\right ) + \frac {- 2 A a^{6} + x^{2} \left (- 90 A a^{4} b^{2} - 36 B a^{5} b\right ) + x \left (- 18 A a^{5} b - 3 B a^{6}\right )}{6 x^{3}} \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**4,x)

[Out]

B*b**6*x**4/4 + 5*a**3*b**2*(4*A*b + 3*B*a)*log(x) + x**3*(A*b**6/3 + 2*B*a*b**5) + x**2*(3*A*a*b**5 + 15*B*a*
*2*b**4/2) + x*(15*A*a**2*b**4 + 20*B*a**3*b**3) + (-2*A*a**6 + x**2*(-90*A*a**4*b**2 - 36*B*a**5*b) + x*(-18*
A*a**5*b - 3*B*a**6))/(6*x**3)

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