3.91 \(\int \frac{(a+b x)^5}{x^8} \, dx\)

Optimal. Leaf size=36 \[ \frac{b (a+b x)^6}{42 a^2 x^6}-\frac{(a+b x)^6}{7 a x^7} \]

[Out]

-(a + b*x)^6/(7*a*x^7) + (b*(a + b*x)^6)/(42*a^2*x^6)

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Rubi [A]  time = 0.0045536, antiderivative size = 36, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {45, 37} \[ \frac{b (a+b x)^6}{42 a^2 x^6}-\frac{(a+b x)^6}{7 a x^7} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^5/x^8,x]

[Out]

-(a + b*x)^6/(7*a*x^7) + (b*(a + b*x)^6)/(42*a^2*x^6)

Rule 45

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] - Dist[(d*Simplify[m + n + 2])/((b*c - a*d)*(m + 1)), Int[(a + b*x)^Simplify[m +
1]*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && ILtQ[Simplify[m + n + 2], 0] &&
NeQ[m, -1] &&  !(LtQ[m, -1] && LtQ[n, -1] && (EqQ[a, 0] || (NeQ[c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && (
SumSimplerQ[m, 1] ||  !SumSimplerQ[n, 1])

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)^5}{x^8} \, dx &=-\frac{(a+b x)^6}{7 a x^7}-\frac{b \int \frac{(a+b x)^5}{x^7} \, dx}{7 a}\\ &=-\frac{(a+b x)^6}{7 a x^7}+\frac{b (a+b x)^6}{42 a^2 x^6}\\ \end{align*}

Mathematica [A]  time = 0.0085953, size = 67, normalized size = 1.86 \[ -\frac{2 a^3 b^2}{x^5}-\frac{5 a^2 b^3}{2 x^4}-\frac{5 a^4 b}{6 x^6}-\frac{a^5}{7 x^7}-\frac{5 a b^4}{3 x^3}-\frac{b^5}{2 x^2} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^5/x^8,x]

[Out]

-a^5/(7*x^7) - (5*a^4*b)/(6*x^6) - (2*a^3*b^2)/x^5 - (5*a^2*b^3)/(2*x^4) - (5*a*b^4)/(3*x^3) - b^5/(2*x^2)

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Maple [A]  time = 0.006, size = 58, normalized size = 1.6 \begin{align*} -{\frac{5\,a{b}^{4}}{3\,{x}^{3}}}-2\,{\frac{{a}^{3}{b}^{2}}{{x}^{5}}}-{\frac{5\,{a}^{2}{b}^{3}}{2\,{x}^{4}}}-{\frac{5\,{a}^{4}b}{6\,{x}^{6}}}-{\frac{{b}^{5}}{2\,{x}^{2}}}-{\frac{{a}^{5}}{7\,{x}^{7}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^5/x^8,x)

[Out]

-5/3*a*b^4/x^3-2*a^3*b^2/x^5-5/2*a^2*b^3/x^4-5/6*a^4*b/x^6-1/2*b^5/x^2-1/7*a^5/x^7

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Maxima [A]  time = 1.03991, size = 77, normalized size = 2.14 \begin{align*} -\frac{21 \, b^{5} x^{5} + 70 \, a b^{4} x^{4} + 105 \, a^{2} b^{3} x^{3} + 84 \, a^{3} b^{2} x^{2} + 35 \, a^{4} b x + 6 \, a^{5}}{42 \, x^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^5/x^8,x, algorithm="maxima")

[Out]

-1/42*(21*b^5*x^5 + 70*a*b^4*x^4 + 105*a^2*b^3*x^3 + 84*a^3*b^2*x^2 + 35*a^4*b*x + 6*a^5)/x^7

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Fricas [A]  time = 1.39562, size = 128, normalized size = 3.56 \begin{align*} -\frac{21 \, b^{5} x^{5} + 70 \, a b^{4} x^{4} + 105 \, a^{2} b^{3} x^{3} + 84 \, a^{3} b^{2} x^{2} + 35 \, a^{4} b x + 6 \, a^{5}}{42 \, x^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^5/x^8,x, algorithm="fricas")

[Out]

-1/42*(21*b^5*x^5 + 70*a*b^4*x^4 + 105*a^2*b^3*x^3 + 84*a^3*b^2*x^2 + 35*a^4*b*x + 6*a^5)/x^7

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Sympy [B]  time = 0.742721, size = 61, normalized size = 1.69 \begin{align*} - \frac{6 a^{5} + 35 a^{4} b x + 84 a^{3} b^{2} x^{2} + 105 a^{2} b^{3} x^{3} + 70 a b^{4} x^{4} + 21 b^{5} x^{5}}{42 x^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**5/x**8,x)

[Out]

-(6*a**5 + 35*a**4*b*x + 84*a**3*b**2*x**2 + 105*a**2*b**3*x**3 + 70*a*b**4*x**4 + 21*b**5*x**5)/(42*x**7)

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Giac [A]  time = 1.19444, size = 77, normalized size = 2.14 \begin{align*} -\frac{21 \, b^{5} x^{5} + 70 \, a b^{4} x^{4} + 105 \, a^{2} b^{3} x^{3} + 84 \, a^{3} b^{2} x^{2} + 35 \, a^{4} b x + 6 \, a^{5}}{42 \, x^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^5/x^8,x, algorithm="giac")

[Out]

-1/42*(21*b^5*x^5 + 70*a*b^4*x^4 + 105*a^2*b^3*x^3 + 84*a^3*b^2*x^2 + 35*a^4*b*x + 6*a^5)/x^7