3.2.16 \(\int \frac {(a+b x)^7}{x^{10}} \, dx\) [116]

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

[Out]

-1/9*(b*x+a)^8/a/x^9+1/72*b*(b*x+a)^8/a^2/x^8

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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 = {47, 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]

-1/9*(a + b*x)^8/(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 47

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*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(91\) vs. \(2(36)=72\).
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)

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Mathics [B] Leaf count is larger than twice the leaf count of optimal. \(81\) vs. \(2(36)=72\).
time = 2.29, size = 79, normalized size = 2.19 \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*}

Antiderivative was successfully verified.

[In]

mathics('Integrate[(a + b*x)^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 - 168 a b ^ 6 x ^ 6 - 36 b ^ 7 x ^ 7) / (72 x ^ 9)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(79\) vs. \(2(32)=64\).
time = 0.07, size = 80, normalized size = 2.22

method	result	size
norman	\(\frac {-\frac {1}{2} b^{7} x^{7}-\frac {7}{3} a \,b^{6} x^{6}-\frac {21}{4} a^{2} b^{5} x^{5}-7 a^{3} b^{4} x^{4}-\frac {35}{6} a^{4} b^{3} x^{3}-3 a^{5} b^{2} x^{2}-\frac {7}{8} a^{6} b x -\frac {1}{9} a^{7}}{x^{9}}\)	\(79\)
risch	\(\frac {-\frac {1}{2} b^{7} x^{7}-\frac {7}{3} a \,b^{6} x^{6}-\frac {21}{4} a^{2} b^{5} x^{5}-7 a^{3} b^{4} x^{4}-\frac {35}{6} a^{4} b^{3} x^{3}-3 a^{5} b^{2} x^{2}-\frac {7}{8} a^{6} b x -\frac {1}{9} a^{7}}{x^{9}}\)	\(79\)
gosper	\(-\frac {36 b^{7} x^{7}+168 a \,b^{6} x^{6}+378 a^{2} b^{5} x^{5}+504 a^{3} b^{4} x^{4}+420 a^{4} b^{3} x^{3}+216 a^{5} b^{2} x^{2}+63 a^{6} b x +8 a^{7}}{72 x^{9}}\)	\(80\)
default	\(-\frac {7 a \,b^{6}}{3 x^{3}}-\frac {21 a^{2} b^{5}}{4 x^{4}}-\frac {a^{7}}{9 x^{9}}-\frac {b^{7}}{2 x^{2}}-\frac {7 a^{6} b}{8 x^{8}}-\frac {7 a^{3} b^{4}}{x^{5}}-\frac {35 a^{4} b^{3}}{6 x^{6}}-\frac {3 a^{5} b^{2}}{x^{7}}\)	\(80\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 79 vs. \(2 (32) = 64\).
time = 0.25, 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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 79 vs. \(2 (32) = 64\).
time = 0.29, 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

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 85 vs. \(2 (29) = 58\).
time = 0.29, 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)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 79 vs. \(2 (32) = 64\).
time = 0.00, size = 89, normalized size = 2.47 \begin {gather*} \frac {-36 x^{7} b^{7}-168 x^{6} b^{6} a-378 x^{5} b^{5} a^{2}-504 x^{4} b^{4} a^{3}-420 x^{3} b^{3} a^{4}-216 x^{2} b^{2} a^{5}-63 x b a^{6}-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)

[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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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)

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