3.24 \(\int \frac{(a+b x) (a c-b c x)^4}{x^7} \, dx\)

Optimal. Leaf size=41 \[ -\frac{7 b c^4 (a-b x)^5}{30 a x^5}-\frac{c^4 (a-b x)^5}{6 x^6} \]

[Out]

-(c^4*(a - b*x)^5)/(6*x^6) - (7*b*c^4*(a - b*x)^5)/(30*a*x^5)

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Rubi [A]  time = 0.0084508, antiderivative size = 41, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {78, 37} \[ -\frac{7 b c^4 (a-b x)^5}{30 a x^5}-\frac{c^4 (a-b x)^5}{6 x^6} \]

Antiderivative was successfully verified.

[In]

Int[((a + b*x)*(a*c - b*c*x)^4)/x^7,x]

[Out]

-(c^4*(a - b*x)^5)/(6*x^6) - (7*b*c^4*(a - b*x)^5)/(30*a*x^5)

Rule 78

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

Rule 37

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

Rubi steps

\begin{align*} \int \frac{(a+b x) (a c-b c x)^4}{x^7} \, dx &=-\frac{c^4 (a-b x)^5}{6 x^6}+\frac{1}{6} (7 b) \int \frac{(a c-b c x)^4}{x^6} \, dx\\ &=-\frac{c^4 (a-b x)^5}{6 x^6}-\frac{7 b c^4 (a-b x)^5}{30 a x^5}\\ \end{align*}

Mathematica [B]  time = 0.0076665, size = 85, normalized size = 2.07 \[ -\frac{a^3 b^2 c^4}{2 x^4}-\frac{2 a^2 b^3 c^4}{3 x^3}+\frac{3 a^4 b c^4}{5 x^5}-\frac{a^5 c^4}{6 x^6}+\frac{3 a b^4 c^4}{2 x^2}-\frac{b^5 c^4}{x} \]

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x)*(a*c - b*c*x)^4)/x^7,x]

[Out]

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

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Maple [A]  time = 0.005, size = 62, normalized size = 1.5 \begin{align*}{c}^{4} \left ( -{\frac{2\,{a}^{2}{b}^{3}}{3\,{x}^{3}}}+{\frac{3\,{a}^{4}b}{5\,{x}^{5}}}-{\frac{{a}^{3}{b}^{2}}{2\,{x}^{4}}}+{\frac{3\,a{b}^{4}}{2\,{x}^{2}}}-{\frac{{a}^{5}}{6\,{x}^{6}}}-{\frac{{b}^{5}}{x}} \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)*(-b*c*x+a*c)^4/x^7,x)

[Out]

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

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Maxima [A]  time = 1.07027, size = 101, normalized size = 2.46 \begin{align*} -\frac{30 \, b^{5} c^{4} x^{5} - 45 \, a b^{4} c^{4} x^{4} + 20 \, a^{2} b^{3} c^{4} x^{3} + 15 \, a^{3} b^{2} c^{4} x^{2} - 18 \, a^{4} b c^{4} x + 5 \, a^{5} c^{4}}{30 \, x^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(-b*c*x+a*c)^4/x^7,x, algorithm="maxima")

[Out]

-1/30*(30*b^5*c^4*x^5 - 45*a*b^4*c^4*x^4 + 20*a^2*b^3*c^4*x^3 + 15*a^3*b^2*c^4*x^2 - 18*a^4*b*c^4*x + 5*a^5*c^
4)/x^6

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Fricas [A]  time = 1.86211, size = 159, normalized size = 3.88 \begin{align*} -\frac{30 \, b^{5} c^{4} x^{5} - 45 \, a b^{4} c^{4} x^{4} + 20 \, a^{2} b^{3} c^{4} x^{3} + 15 \, a^{3} b^{2} c^{4} x^{2} - 18 \, a^{4} b c^{4} x + 5 \, a^{5} c^{4}}{30 \, x^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(-b*c*x+a*c)^4/x^7,x, algorithm="fricas")

[Out]

-1/30*(30*b^5*c^4*x^5 - 45*a*b^4*c^4*x^4 + 20*a^2*b^3*c^4*x^3 + 15*a^3*b^2*c^4*x^2 - 18*a^4*b*c^4*x + 5*a^5*c^
4)/x^6

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Sympy [B]  time = 0.592354, size = 82, normalized size = 2. \begin{align*} - \frac{5 a^{5} c^{4} - 18 a^{4} b c^{4} x + 15 a^{3} b^{2} c^{4} x^{2} + 20 a^{2} b^{3} c^{4} x^{3} - 45 a b^{4} c^{4} x^{4} + 30 b^{5} c^{4} x^{5}}{30 x^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(-b*c*x+a*c)**4/x**7,x)

[Out]

-(5*a**5*c**4 - 18*a**4*b*c**4*x + 15*a**3*b**2*c**4*x**2 + 20*a**2*b**3*c**4*x**3 - 45*a*b**4*c**4*x**4 + 30*
b**5*c**4*x**5)/(30*x**6)

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Giac [A]  time = 1.24304, size = 101, normalized size = 2.46 \begin{align*} -\frac{30 \, b^{5} c^{4} x^{5} - 45 \, a b^{4} c^{4} x^{4} + 20 \, a^{2} b^{3} c^{4} x^{3} + 15 \, a^{3} b^{2} c^{4} x^{2} - 18 \, a^{4} b c^{4} x + 5 \, a^{5} c^{4}}{30 \, x^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(-b*c*x+a*c)^4/x^7,x, algorithm="giac")

[Out]

-1/30*(30*b^5*c^4*x^5 - 45*a*b^4*c^4*x^4 + 20*a^2*b^3*c^4*x^3 + 15*a^3*b^2*c^4*x^2 - 18*a^4*b*c^4*x + 5*a^5*c^
4)/x^6