3.2.16 \(\int \frac {(a+b x)^7}{x^{10}} \, dx\)

Optimal. Leaf size=36 \[ \frac {b (a+b x)^8}{72 a^2 x^8}-\frac {(a+b x)^8}{9 a x^9} \]

________________________________________________________________________________________

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)^8}{72 a^2 x^8}-\frac {(a+b x)^8}{9 a x^9} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^7/x^10,x]

[Out]

-(a + b*x)^8/(9*a*x^9) + (b*(a + b*x)^8)/(72*a^2*x^8)

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)^7}{x^{10}} \, dx &=-\frac {(a+b x)^8}{9 a x^9}-\frac {b \int \frac {(a+b x)^7}{x^9} \, dx}{9 a}\\ &=-\frac {(a+b x)^8}{9 a x^9}+\frac {b (a+b x)^8}{72 a^2 x^8}\\ \end {align*}

________________________________________________________________________________________

Mathematica [B]  time = 0.00, size = 91, normalized size = 2.53 \begin {gather*} -\frac {a^7}{9 x^9}-\frac {7 a^6 b}{8 x^8}-\frac {3 a^5 b^2}{x^7}-\frac {35 a^4 b^3}{6 x^6}-\frac {7 a^3 b^4}{x^5}-\frac {21 a^2 b^5}{4 x^4}-\frac {7 a b^6}{3 x^3}-\frac {b^7}{2 x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^7/x^10,x]

[Out]

-1/9*a^7/x^9 - (7*a^6*b)/(8*x^8) - (3*a^5*b^2)/x^7 - (35*a^4*b^3)/(6*x^6) - (7*a^3*b^4)/x^5 - (21*a^2*b^5)/(4*
x^4) - (7*a*b^6)/(3*x^3) - b^7/(2*x^2)

________________________________________________________________________________________

IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(a+b x)^7}{x^{10}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[(a + b*x)^7/x^10,x]

[Out]

IntegrateAlgebraic[(a + b*x)^7/x^10, x]

________________________________________________________________________________________

fricas [B]  time = 1.21, size = 79, normalized size = 2.19 \begin {gather*} -\frac {36 \, b^{7} x^{7} + 168 \, a b^{6} x^{6} + 378 \, a^{2} b^{5} x^{5} + 504 \, a^{3} b^{4} x^{4} + 420 \, a^{4} b^{3} x^{3} + 216 \, a^{5} b^{2} x^{2} + 63 \, a^{6} b x + 8 \, a^{7}}{72 \, x^{9}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^7/x^10,x, algorithm="fricas")

[Out]

-1/72*(36*b^7*x^7 + 168*a*b^6*x^6 + 378*a^2*b^5*x^5 + 504*a^3*b^4*x^4 + 420*a^4*b^3*x^3 + 216*a^5*b^2*x^2 + 63
*a^6*b*x + 8*a^7)/x^9

________________________________________________________________________________________

giac [B]  time = 1.01, size = 79, normalized size = 2.19 \begin {gather*} -\frac {36 \, b^{7} x^{7} + 168 \, a b^{6} x^{6} + 378 \, a^{2} b^{5} x^{5} + 504 \, a^{3} b^{4} x^{4} + 420 \, a^{4} b^{3} x^{3} + 216 \, a^{5} b^{2} x^{2} + 63 \, a^{6} b x + 8 \, a^{7}}{72 \, x^{9}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^7/x^10,x, algorithm="giac")

[Out]

-1/72*(36*b^7*x^7 + 168*a*b^6*x^6 + 378*a^2*b^5*x^5 + 504*a^3*b^4*x^4 + 420*a^4*b^3*x^3 + 216*a^5*b^2*x^2 + 63
*a^6*b*x + 8*a^7)/x^9

________________________________________________________________________________________

maple [B]  time = 0.01, size = 80, normalized size = 2.22 \begin {gather*} -\frac {b^{7}}{2 x^{2}}-\frac {7 a \,b^{6}}{3 x^{3}}-\frac {21 a^{2} b^{5}}{4 x^{4}}-\frac {7 a^{3} b^{4}}{x^{5}}-\frac {35 a^{4} b^{3}}{6 x^{6}}-\frac {3 a^{5} b^{2}}{x^{7}}-\frac {7 a^{6} b}{8 x^{8}}-\frac {a^{7}}{9 x^{9}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^7/x^10,x)

[Out]

-7*a^3*b^4/x^5-35/6*a^4*b^3/x^6-1/2*b^7/x^2-7/8*a^6*b/x^8-3*a^5*b^2/x^7-21/4*a^2*b^5/x^4-7/3*a*b^6/x^3-1/9*a^7
/x^9

________________________________________________________________________________________

maxima [B]  time = 1.31, size = 79, normalized size = 2.19 \begin {gather*} -\frac {36 \, b^{7} x^{7} + 168 \, a b^{6} x^{6} + 378 \, a^{2} b^{5} x^{5} + 504 \, a^{3} b^{4} x^{4} + 420 \, a^{4} b^{3} x^{3} + 216 \, a^{5} b^{2} x^{2} + 63 \, a^{6} b x + 8 \, a^{7}}{72 \, x^{9}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^7/x^10,x, algorithm="maxima")

[Out]

-1/72*(36*b^7*x^7 + 168*a*b^6*x^6 + 378*a^2*b^5*x^5 + 504*a^3*b^4*x^4 + 420*a^4*b^3*x^3 + 216*a^5*b^2*x^2 + 63
*a^6*b*x + 8*a^7)/x^9

________________________________________________________________________________________

mupad [B]  time = 0.09, size = 23, normalized size = 0.64 \begin {gather*} -\frac {\left (8\,a-b\,x\right )\,{\left (a+b\,x\right )}^8}{72\,a^2\,x^9} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x)^7/x^10,x)

[Out]

-((8*a - b*x)*(a + b*x)^8)/(72*a^2*x^9)

________________________________________________________________________________________

sympy [B]  time = 0.73, size = 85, normalized size = 2.36 \begin {gather*} \frac {- 8 a^{7} - 63 a^{6} b x - 216 a^{5} b^{2} x^{2} - 420 a^{4} b^{3} x^{3} - 504 a^{3} b^{4} x^{4} - 378 a^{2} b^{5} x^{5} - 168 a b^{6} x^{6} - 36 b^{7} x^{7}}{72 x^{9}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**7/x**10,x)

[Out]

(-8*a**7 - 63*a**6*b*x - 216*a**5*b**2*x**2 - 420*a**4*b**3*x**3 - 504*a**3*b**4*x**4 - 378*a**2*b**5*x**5 - 1
68*a*b**6*x**6 - 36*b**7*x**7)/(72*x**9)

________________________________________________________________________________________