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

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

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Rubi [A]  time = 0.00, antiderivative size = 36, normalized size of antiderivative = 1.00, 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} \begin {gather*} \frac {b (a+b x)^6}{42 a^2 x^6}-\frac {(a+b x)^6}{7 a x^7} \end {gather*}

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 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]

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])

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*}

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Mathematica [A]  time = 0.00, size = 67, normalized size = 1.86 \begin {gather*} -\frac {a^5}{7 x^7}-\frac {5 a^4 b}{6 x^6}-\frac {2 a^3 b^2}{x^5}-\frac {5 a^2 b^3}{2 x^4}-\frac {5 a b^4}{3 x^3}-\frac {b^5}{2 x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^5/x^8,x]

[Out]

-1/7*a^5/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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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(a+b x)^5}{x^8} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[(a + b*x)^5/x^8,x]

[Out]

IntegrateAlgebraic[(a + b*x)^5/x^8, x]

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fricas [A]  time = 1.37, size = 57, normalized size = 1.58 \begin {gather*} -\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 {gather*}

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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giac [A]  time = 1.12, size = 57, normalized size = 1.58 \begin {gather*} -\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 {gather*}

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

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maple [A]  time = 0.01, size = 58, normalized size = 1.61 \begin {gather*} -\frac {b^{5}}{2 x^{2}}-\frac {5 a \,b^{4}}{3 x^{3}}-\frac {5 a^{2} b^{3}}{2 x^{4}}-\frac {2 a^{3} b^{2}}{x^{5}}-\frac {5 a^{4} b}{6 x^{6}}-\frac {a^{5}}{7 x^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 1.36, size = 57, normalized size = 1.58 \begin {gather*} -\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 {gather*}

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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mupad [B]  time = 0.07, size = 57, normalized size = 1.58 \begin {gather*} -\frac {\frac {a^5}{7}+\frac {5\,a^4\,b\,x}{6}+2\,a^3\,b^2\,x^2+\frac {5\,a^2\,b^3\,x^3}{2}+\frac {5\,a\,b^4\,x^4}{3}+\frac {b^5\,x^5}{2}}{x^7} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x)^5/x^8,x)

[Out]

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

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sympy [B]  time = 0.41, size = 61, normalized size = 1.69 \begin {gather*} \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 {gather*}

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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