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

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

[Out]

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

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Rubi [A]  time = 0.0048768, 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} \[ \frac{b (a+b x)^8}{72 a^2 x^8}-\frac{(a+b x)^8}{9 a x^9} \]

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 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)^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 = 0.0039336, size = 91, normalized size = 2.53 \[ -\frac{3 a^5 b^2}{x^7}-\frac{35 a^4 b^3}{6 x^6}-\frac{7 a^3 b^4}{x^5}-\frac{21 a^2 b^5}{4 x^4}-\frac{7 a^6 b}{8 x^8}-\frac{a^7}{9 x^9}-\frac{7 a b^6}{3 x^3}-\frac{b^7}{2 x^2} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^7/x^10,x]

[Out]

-a^7/(9*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)

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Maple [B]  time = 0.005, size = 80, normalized size = 2.2 \begin{align*} -{\frac{7\,a{b}^{6}}{3\,{x}^{3}}}-7\,{\frac{{a}^{3}{b}^{4}}{{x}^{5}}}-{\frac{21\,{a}^{2}{b}^{5}}{4\,{x}^{4}}}-{\frac{35\,{a}^{4}{b}^{3}}{6\,{x}^{6}}}-{\frac{7\,{a}^{6}b}{8\,{x}^{8}}}-{\frac{{b}^{7}}{2\,{x}^{2}}}-3\,{\frac{{a}^{5}{b}^{2}}{{x}^{7}}}-{\frac{{a}^{7}}{9\,{x}^{9}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B]  time = 1.03955, size = 107, normalized size = 2.97 \begin{align*} -\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{align*}

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

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Fricas [B]  time = 1.72973, size = 180, normalized size = 5. \begin{align*} -\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{align*}

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

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Sympy [B]  time = 0.826221, size = 85, normalized size = 2.36 \begin{align*} - \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{align*}

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)

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Giac [B]  time = 1.19416, size = 107, normalized size = 2.97 \begin{align*} -\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{align*}

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