3.1.11 \(\int \frac {(a+b x) (a c-b c x)^3}{x^7} \, dx\)

Optimal. Leaf size=55 \[ -\frac {a^4 c^3}{6 x^6}+\frac {2 a^3 b c^3}{5 x^5}-\frac {2 a b^3 c^3}{3 x^3}+\frac {b^4 c^3}{2 x^2} \]

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Rubi [A]  time = 0.02, antiderivative size = 55, 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 c^3}{5 x^5}-\frac {a^4 c^3}{6 x^6}-\frac {2 a b^3 c^3}{3 x^3}+\frac {b^4 c^3}{2 x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((a + b*x)*(a*c - b*c*x)^3)/x^7,x]

[Out]

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

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)^3}{x^7} \, dx &=\int \left (\frac {a^4 c^3}{x^7}-\frac {2 a^3 b c^3}{x^6}+\frac {2 a b^3 c^3}{x^4}-\frac {b^4 c^3}{x^3}\right ) \, dx\\ &=-\frac {a^4 c^3}{6 x^6}+\frac {2 a^3 b c^3}{5 x^5}-\frac {2 a b^3 c^3}{3 x^3}+\frac {b^4 c^3}{2 x^2}\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 47, normalized size = 0.85 \begin {gather*} c^3 \left (-\frac {a^4}{6 x^6}+\frac {2 a^3 b}{5 x^5}-\frac {2 a b^3}{3 x^3}+\frac {b^4}{2 x^2}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x)*(a*c - b*c*x)^3)/x^7,x]

[Out]

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

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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)^3}{x^7} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[((a + b*x)*(a*c - b*c*x)^3)/x^7,x]

[Out]

IntegrateAlgebraic[((a + b*x)*(a*c - b*c*x)^3)/x^7, x]

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fricas [A]  time = 1.11, size = 47, normalized size = 0.85 \begin {gather*} \frac {15 \, b^{4} c^{3} x^{4} - 20 \, a b^{3} c^{3} x^{3} + 12 \, a^{3} b c^{3} x - 5 \, a^{4} c^{3}}{30 \, x^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(-b*c*x+a*c)^3/x^7,x, algorithm="fricas")

[Out]

1/30*(15*b^4*c^3*x^4 - 20*a*b^3*c^3*x^3 + 12*a^3*b*c^3*x - 5*a^4*c^3)/x^6

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giac [A]  time = 1.04, size = 47, normalized size = 0.85 \begin {gather*} \frac {15 \, b^{4} c^{3} x^{4} - 20 \, a b^{3} c^{3} x^{3} + 12 \, a^{3} b c^{3} x - 5 \, a^{4} c^{3}}{30 \, x^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(-b*c*x+a*c)^3/x^7,x, algorithm="giac")

[Out]

1/30*(15*b^4*c^3*x^4 - 20*a*b^3*c^3*x^3 + 12*a^3*b*c^3*x - 5*a^4*c^3)/x^6

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maple [A]  time = 0.01, size = 40, normalized size = 0.73 \begin {gather*} \left (\frac {b^{4}}{2 x^{2}}-\frac {2 a \,b^{3}}{3 x^{3}}+\frac {2 a^{3} b}{5 x^{5}}-\frac {a^{4}}{6 x^{6}}\right ) c^{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)*(-b*c*x+a*c)^3/x^7,x)

[Out]

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

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maxima [A]  time = 1.07, size = 47, normalized size = 0.85 \begin {gather*} \frac {15 \, b^{4} c^{3} x^{4} - 20 \, a b^{3} c^{3} x^{3} + 12 \, a^{3} b c^{3} x - 5 \, a^{4} c^{3}}{30 \, x^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(-b*c*x+a*c)^3/x^7,x, algorithm="maxima")

[Out]

1/30*(15*b^4*c^3*x^4 - 20*a*b^3*c^3*x^3 + 12*a^3*b*c^3*x - 5*a^4*c^3)/x^6

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mupad [B]  time = 0.29, size = 47, normalized size = 0.85 \begin {gather*} -\frac {\frac {a^4\,c^3}{6}-\frac {2\,a^3\,b\,c^3\,x}{5}+\frac {2\,a\,b^3\,c^3\,x^3}{3}-\frac {b^4\,c^3\,x^4}{2}}{x^6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((a*c - b*c*x)^3*(a + b*x))/x^7,x)

[Out]

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

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sympy [A]  time = 0.30, size = 51, normalized size = 0.93 \begin {gather*} - \frac {5 a^{4} c^{3} - 12 a^{3} b c^{3} x + 20 a b^{3} c^{3} x^{3} - 15 b^{4} c^{3} x^{4}}{30 x^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(-b*c*x+a*c)**3/x**7,x)

[Out]

-(5*a**4*c**3 - 12*a**3*b*c**3*x + 20*a*b**3*c**3*x**3 - 15*b**4*c**3*x**4)/(30*x**6)

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