3.35 \(\int \frac {(a+b x) (a c-b c x)^5}{x^4} \, dx\)

Optimal. Leaf size=18 \[ -\frac {c^5 (a-b x)^6}{3 x^3} \]

[Out]

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

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Rubi [A]  time = 0.00, antiderivative size = 18, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {74} \[ -\frac {c^5 (a-b x)^6}{3 x^3} \]

Antiderivative was successfully verified.

[In]

Int[((a + b*x)*(a*c - b*c*x)^5)/x^4,x]

[Out]

-(c^5*(a - b*x)^6)/(3*x^3)

Rule 74

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*(c + d*x)
^(n + 1)*(e + f*x)^(p + 1))/(d*f*(n + p + 2)), x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 2, 0] &
& EqQ[a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)), 0]

Rubi steps

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

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Mathematica [B]  time = 0.02, size = 63, normalized size = 3.50 \[ c^5 \left (-\frac {a^6}{3 x^3}+\frac {2 a^5 b}{x^2}-\frac {5 a^4 b^2}{x}-5 a^2 b^4 x+2 a b^5 x^2-\frac {b^6 x^3}{3}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x)*(a*c - b*c*x)^5)/x^4,x]

[Out]

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

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fricas [B]  time = 0.48, size = 73, normalized size = 4.06 \[ -\frac {b^{6} c^{5} x^{6} - 6 \, a b^{5} c^{5} x^{5} + 15 \, a^{2} b^{4} c^{5} x^{4} + 15 \, a^{4} b^{2} c^{5} x^{2} - 6 \, a^{5} b c^{5} x + a^{6} c^{5}}{3 \, x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(-b*c*x+a*c)^5/x^4,x, algorithm="fricas")

[Out]

-1/3*(b^6*c^5*x^6 - 6*a*b^5*c^5*x^5 + 15*a^2*b^4*c^5*x^4 + 15*a^4*b^2*c^5*x^2 - 6*a^5*b*c^5*x + a^6*c^5)/x^3

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giac [B]  time = 1.04, size = 73, normalized size = 4.06 \[ -\frac {1}{3} \, b^{6} c^{5} x^{3} + 2 \, a b^{5} c^{5} x^{2} - 5 \, a^{2} b^{4} c^{5} x - \frac {15 \, a^{4} b^{2} c^{5} x^{2} - 6 \, a^{5} b c^{5} x + a^{6} c^{5}}{3 \, x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(-b*c*x+a*c)^5/x^4,x, algorithm="giac")

[Out]

-1/3*b^6*c^5*x^3 + 2*a*b^5*c^5*x^2 - 5*a^2*b^4*c^5*x - 1/3*(15*a^4*b^2*c^5*x^2 - 6*a^5*b*c^5*x + a^6*c^5)/x^3

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maple [B]  time = 0.01, size = 60, normalized size = 3.33 \[ \left (-\frac {b^{6} x^{3}}{3}+2 a \,b^{5} x^{2}-5 a^{2} b^{4} x -\frac {5 a^{4} b^{2}}{x}+\frac {2 a^{5} b}{x^{2}}-\frac {a^{6}}{3 x^{3}}\right ) c^{5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)*(-b*c*x+a*c)^5/x^4,x)

[Out]

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

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maxima [B]  time = 1.04, size = 73, normalized size = 4.06 \[ -\frac {1}{3} \, b^{6} c^{5} x^{3} + 2 \, a b^{5} c^{5} x^{2} - 5 \, a^{2} b^{4} c^{5} x - \frac {15 \, a^{4} b^{2} c^{5} x^{2} - 6 \, a^{5} b c^{5} x + a^{6} c^{5}}{3 \, x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(-b*c*x+a*c)^5/x^4,x, algorithm="maxima")

[Out]

-1/3*b^6*c^5*x^3 + 2*a*b^5*c^5*x^2 - 5*a^2*b^4*c^5*x - 1/3*(15*a^4*b^2*c^5*x^2 - 6*a^5*b*c^5*x + a^6*c^5)/x^3

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mupad [B]  time = 0.06, size = 74, normalized size = 4.11 \[ 2\,a\,b^5\,c^5\,x^2-\frac {b^6\,c^5\,x^3}{3}-5\,a^2\,b^4\,c^5\,x-\frac {\frac {a^6\,c^5}{3}-2\,a^5\,b\,c^5\,x+5\,a^4\,b^2\,c^5\,x^2}{x^3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((a*c - b*c*x)^5*(a + b*x))/x^4,x)

[Out]

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

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sympy [B]  time = 0.24, size = 76, normalized size = 4.22 \[ - 5 a^{2} b^{4} c^{5} x + 2 a b^{5} c^{5} x^{2} - \frac {b^{6} c^{5} x^{3}}{3} - \frac {a^{6} c^{5} - 6 a^{5} b c^{5} x + 15 a^{4} b^{2} c^{5} x^{2}}{3 x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(-b*c*x+a*c)**5/x**4,x)

[Out]

-5*a**2*b**4*c**5*x + 2*a*b**5*c**5*x**2 - b**6*c**5*x**3/3 - (a**6*c**5 - 6*a**5*b*c**5*x + 15*a**4*b**2*c**5
*x**2)/(3*x**3)

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