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

Optimal. Leaf size=18 \[ -\frac {c^3 (a-b x)^4}{2 x^2} \]

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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} \begin {gather*} -\frac {c^3 (a-b x)^4}{2 x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-(c^3*(a - b*x)^4)/(2*x^2)

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)^3}{x^3} \, dx &=-\frac {c^3 (a-b x)^4}{2 x^2}\\ \end {align*}

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Mathematica [B]  time = 0.01, size = 41, normalized size = 2.28 \begin {gather*} c^3 \left (-\frac {a^4}{2 x^2}+\frac {2 a^3 b}{x}+2 a b^3 x-\frac {b^4 x^2}{2}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

c^3*(-1/2*a^4/x^2 + (2*a^3*b)/x + 2*a*b^3*x - (b^4*x^2)/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^3} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [B]  time = 1.16, size = 45, normalized size = 2.50 \begin {gather*} -\frac {b^{4} c^{3} x^{4} - 4 \, a b^{3} c^{3} x^{3} - 4 \, a^{3} b c^{3} x + a^{4} c^{3}}{2 \, x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [B]  time = 0.88, size = 46, normalized size = 2.56 \begin {gather*} -\frac {1}{2} \, b^{4} c^{3} x^{2} + 2 \, a b^{3} c^{3} x + \frac {4 \, a^{3} b c^{3} x - a^{4} c^{3}}{2 \, x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [B]  time = 0.00, size = 38, normalized size = 2.11 \begin {gather*} \left (-\frac {b^{4} x^{2}}{2}+2 a \,b^{3} x +\frac {2 a^{3} b}{x}-\frac {a^{4}}{2 x^{2}}\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^3,x)

[Out]

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

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maxima [B]  time = 1.08, size = 46, normalized size = 2.56 \begin {gather*} -\frac {1}{2} \, b^{4} c^{3} x^{2} + 2 \, a b^{3} c^{3} x + \frac {4 \, a^{3} b c^{3} x - a^{4} c^{3}}{2 \, x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.29, size = 35, normalized size = 1.94 \begin {gather*} -\frac {c^3\,\left (a^4-4\,a^3\,b\,x-4\,a\,b^3\,x^3+b^4\,x^4\right )}{2\,x^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [B]  time = 0.16, size = 46, normalized size = 2.56 \begin {gather*} 2 a b^{3} c^{3} x - \frac {b^{4} c^{3} x^{2}}{2} - \frac {a^{4} c^{3} - 4 a^{3} b c^{3} x}{2 x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*a*b**3*c**3*x - b**4*c**3*x**2/2 - (a**4*c**3 - 4*a**3*b*c**3*x)/(2*x**2)

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