3.1.21 \(\int \frac {(a+b x) (a c-b c x)^4}{x^4} \, dx\)

Optimal. Leaf size=78 \[ -\frac {a^5 c^4}{3 x^3}+\frac {3 a^4 b c^4}{2 x^2}-\frac {2 a^3 b^2 c^4}{x}+2 a^2 b^3 c^4 \log (x)-3 a b^4 c^4 x+\frac {1}{2} b^5 c^4 x^2 \]

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Rubi [A]  time = 0.04, antiderivative size = 78, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {75} \begin {gather*} -\frac {2 a^3 b^2 c^4}{x}+2 a^2 b^3 c^4 \log (x)+\frac {3 a^4 b c^4}{2 x^2}-\frac {a^5 c^4}{3 x^3}-3 a b^4 c^4 x+\frac {1}{2} b^5 c^4 x^2 \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((a + b*x)*(a*c - b*c*x)^4)/x^4,x]

[Out]

-(a^5*c^4)/(3*x^3) + (3*a^4*b*c^4)/(2*x^2) - (2*a^3*b^2*c^4)/x - 3*a*b^4*c^4*x + (b^5*c^4*x^2)/2 + 2*a^2*b^3*c
^4*Log[x]

Rule 75

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] && EqQ[b*e + a*f, 0] &&  !(ILtQ[n
+ p + 2, 0] && GtQ[n + 2*p, 0])

Rubi steps

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

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Mathematica [A]  time = 0.01, size = 78, normalized size = 1.00 \begin {gather*} -\frac {a^5 c^4}{3 x^3}+\frac {3 a^4 b c^4}{2 x^2}-\frac {2 a^3 b^2 c^4}{x}+2 a^2 b^3 c^4 \log (x)-3 a b^4 c^4 x+\frac {1}{2} b^5 c^4 x^2 \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x)*(a*c - b*c*x)^4)/x^4,x]

[Out]

-1/3*(a^5*c^4)/x^3 + (3*a^4*b*c^4)/(2*x^2) - (2*a^3*b^2*c^4)/x - 3*a*b^4*c^4*x + (b^5*c^4*x^2)/2 + 2*a^2*b^3*c
^4*Log[x]

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(a+b x) (a c-b c x)^4}{x^4} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[((a + b*x)*(a*c - b*c*x)^4)/x^4,x]

[Out]

IntegrateAlgebraic[((a + b*x)*(a*c - b*c*x)^4)/x^4, x]

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fricas [A]  time = 1.32, size = 77, normalized size = 0.99 \begin {gather*} \frac {3 \, b^{5} c^{4} x^{5} - 18 \, a b^{4} c^{4} x^{4} + 12 \, a^{2} b^{3} c^{4} x^{3} \log \relax (x) - 12 \, a^{3} b^{2} c^{4} x^{2} + 9 \, a^{4} b c^{4} x - 2 \, a^{5} c^{4}}{6 \, x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(-b*c*x+a*c)^4/x^4,x, algorithm="fricas")

[Out]

1/6*(3*b^5*c^4*x^5 - 18*a*b^4*c^4*x^4 + 12*a^2*b^3*c^4*x^3*log(x) - 12*a^3*b^2*c^4*x^2 + 9*a^4*b*c^4*x - 2*a^5
*c^4)/x^3

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giac [A]  time = 1.09, size = 74, normalized size = 0.95 \begin {gather*} \frac {1}{2} \, b^{5} c^{4} x^{2} - 3 \, a b^{4} c^{4} x + 2 \, a^{2} b^{3} c^{4} \log \left ({\left | x \right |}\right ) - \frac {12 \, a^{3} b^{2} c^{4} x^{2} - 9 \, a^{4} b c^{4} x + 2 \, a^{5} c^{4}}{6 \, x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(-b*c*x+a*c)^4/x^4,x, algorithm="giac")

[Out]

1/2*b^5*c^4*x^2 - 3*a*b^4*c^4*x + 2*a^2*b^3*c^4*log(abs(x)) - 1/6*(12*a^3*b^2*c^4*x^2 - 9*a^4*b*c^4*x + 2*a^5*
c^4)/x^3

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maple [A]  time = 0.01, size = 73, normalized size = 0.94 \begin {gather*} \frac {b^{5} c^{4} x^{2}}{2}+2 a^{2} b^{3} c^{4} \ln \relax (x )-3 a \,b^{4} c^{4} x -\frac {2 a^{3} b^{2} c^{4}}{x}+\frac {3 a^{4} b \,c^{4}}{2 x^{2}}-\frac {a^{5} c^{4}}{3 x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)*(-b*c*x+a*c)^4/x^4,x)

[Out]

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

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maxima [A]  time = 1.04, size = 73, normalized size = 0.94 \begin {gather*} \frac {1}{2} \, b^{5} c^{4} x^{2} - 3 \, a b^{4} c^{4} x + 2 \, a^{2} b^{3} c^{4} \log \relax (x) - \frac {12 \, a^{3} b^{2} c^{4} x^{2} - 9 \, a^{4} b c^{4} x + 2 \, a^{5} c^{4}}{6 \, x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(-b*c*x+a*c)^4/x^4,x, algorithm="maxima")

[Out]

1/2*b^5*c^4*x^2 - 3*a*b^4*c^4*x + 2*a^2*b^3*c^4*log(x) - 1/6*(12*a^3*b^2*c^4*x^2 - 9*a^4*b*c^4*x + 2*a^5*c^4)/
x^3

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mupad [B]  time = 0.05, size = 73, normalized size = 0.94 \begin {gather*} \frac {b^5\,c^4\,x^2}{2}-\frac {\frac {a^5\,c^4}{3}-\frac {3\,a^4\,b\,c^4\,x}{2}+2\,a^3\,b^2\,c^4\,x^2}{x^3}+2\,a^2\,b^3\,c^4\,\ln \relax (x)-3\,a\,b^4\,c^4\,x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((a*c - b*c*x)^4*(a + b*x))/x^4,x)

[Out]

(b^5*c^4*x^2)/2 - ((a^5*c^4)/3 + 2*a^3*b^2*c^4*x^2 - (3*a^4*b*c^4*x)/2)/x^3 + 2*a^2*b^3*c^4*log(x) - 3*a*b^4*c
^4*x

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sympy [A]  time = 0.27, size = 78, normalized size = 1.00 \begin {gather*} 2 a^{2} b^{3} c^{4} \log {\relax (x )} - 3 a b^{4} c^{4} x + \frac {b^{5} c^{4} x^{2}}{2} + \frac {- 2 a^{5} c^{4} + 9 a^{4} b c^{4} x - 12 a^{3} b^{2} c^{4} x^{2}}{6 x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(-b*c*x+a*c)**4/x**4,x)

[Out]

2*a**2*b**3*c**4*log(x) - 3*a*b**4*c**4*x + b**5*c**4*x**2/2 + (-2*a**5*c**4 + 9*a**4*b*c**4*x - 12*a**3*b**2*
c**4*x**2)/(6*x**3)

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