3.2.17 \(\int \frac {(a+b x)^7}{x^{11}} \, dx\) [117]

Optimal. Leaf size=56 \[ -\frac {(a+b x)^8}{10 a x^{10}}+\frac {b (a+b x)^8}{45 a^2 x^9}-\frac {b^2 (a+b x)^8}{360 a^3 x^8} \]

[Out]

-1/10*(b*x+a)^8/a/x^10+1/45*b*(b*x+a)^8/a^2/x^9-1/360*b^2*(b*x+a)^8/a^3/x^8

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Rubi [A]
time = 0.01, antiderivative size = 56, normalized size of antiderivative = 1.00, number of steps used = 3, 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^2 (a+b x)^8}{360 a^3 x^8}+\frac {b (a+b x)^8}{45 a^2 x^9}-\frac {(a+b x)^8}{10 a x^{10}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^7/x^11,x]

[Out]

-1/10*(a + b*x)^8/(a*x^10) + (b*(a + b*x)^8)/(45*a^2*x^9) - (b^2*(a + b*x)^8)/(360*a^3*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^{11}} \, dx &=-\frac {(a+b x)^8}{10 a x^{10}}-\frac {b \int \frac {(a+b x)^7}{x^{10}} \, dx}{5 a}\\ &=-\frac {(a+b x)^8}{10 a x^{10}}+\frac {b (a+b x)^8}{45 a^2 x^9}+\frac {b^2 \int \frac {(a+b x)^7}{x^9} \, dx}{45 a^2}\\ &=-\frac {(a+b x)^8}{10 a x^{10}}+\frac {b (a+b x)^8}{45 a^2 x^9}-\frac {b^2 (a+b x)^8}{360 a^3 x^8}\\ \end {align*}

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Mathematica [A]
time = 0.00, size = 93, normalized size = 1.66 \begin {gather*} -\frac {a^7}{10 x^{10}}-\frac {7 a^6 b}{9 x^9}-\frac {21 a^5 b^2}{8 x^8}-\frac {5 a^4 b^3}{x^7}-\frac {35 a^3 b^4}{6 x^6}-\frac {21 a^2 b^5}{5 x^5}-\frac {7 a b^6}{4 x^4}-\frac {b^7}{3 x^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^7/x^11,x]

[Out]

-1/10*a^7/x^10 - (7*a^6*b)/(9*x^9) - (21*a^5*b^2)/(8*x^8) - (5*a^4*b^3)/x^7 - (35*a^3*b^4)/(6*x^6) - (21*a^2*b
^5)/(5*x^5) - (7*a*b^6)/(4*x^4) - b^7/(3*x^3)

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Mathics [A]
time = 2.28, size = 79, normalized size = 1.41 \begin {gather*} \frac {-36 a^7-280 a^6 b x-945 a^5 b^2 x^2-1800 a^4 b^3 x^3-2100 a^3 b^4 x^4-1512 a^2 b^5 x^5-630 a b^6 x^6-120 b^7 x^7}{360 x^{10}} \end {gather*}

Antiderivative was successfully verified.

[In]

mathics('Integrate[(a + b*x)^7/x^11,x]')

[Out]

(-36 a ^ 7 - 280 a ^ 6 b x - 945 a ^ 5 b ^ 2 x ^ 2 - 1800 a ^ 4 b ^ 3 x ^ 3 - 2100 a ^ 3 b ^ 4 x ^ 4 - 1512 a
^ 2 b ^ 5 x ^ 5 - 630 a b ^ 6 x ^ 6 - 120 b ^ 7 x ^ 7) / (360 x ^ 10)

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Maple [A]
time = 0.07, size = 80, normalized size = 1.43

method	result	size
norman	\(\frac {-\frac {1}{3} b^{7} x^{7}-\frac {7}{4} a \,b^{6} x^{6}-\frac {21}{5} a^{2} b^{5} x^{5}-\frac {35}{6} a^{3} b^{4} x^{4}-5 a^{4} b^{3} x^{3}-\frac {21}{8} a^{5} b^{2} x^{2}-\frac {7}{9} a^{6} b x -\frac {1}{10} a^{7}}{x^{10}}\)	\(79\)
risch	\(\frac {-\frac {1}{3} b^{7} x^{7}-\frac {7}{4} a \,b^{6} x^{6}-\frac {21}{5} a^{2} b^{5} x^{5}-\frac {35}{6} a^{3} b^{4} x^{4}-5 a^{4} b^{3} x^{3}-\frac {21}{8} a^{5} b^{2} x^{2}-\frac {7}{9} a^{6} b x -\frac {1}{10} a^{7}}{x^{10}}\)	\(79\)
gosper	\(-\frac {120 b^{7} x^{7}+630 a \,b^{6} x^{6}+1512 a^{2} b^{5} x^{5}+2100 a^{3} b^{4} x^{4}+1800 a^{4} b^{3} x^{3}+945 a^{5} b^{2} x^{2}+280 a^{6} b x +36 a^{7}}{360 x^{10}}\)	\(80\)
default	\(-\frac {a^{7}}{10 x^{10}}-\frac {b^{7}}{3 x^{3}}-\frac {7 a \,b^{6}}{4 x^{4}}-\frac {7 a^{6} b}{9 x^{9}}-\frac {21 a^{5} b^{2}}{8 x^{8}}-\frac {21 a^{2} b^{5}}{5 x^{5}}-\frac {35 a^{3} b^{4}}{6 x^{6}}-\frac {5 a^{4} b^{3}}{x^{7}}\)	\(80\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^7/x^11,x,method=_RETURNVERBOSE)

[Out]

-1/10*a^7/x^10-1/3*b^7/x^3-7/4*a*b^6/x^4-7/9*a^6*b/x^9-21/8*a^5*b^2/x^8-21/5*a^2*b^5/x^5-35/6*a^3*b^4/x^6-5*a^
4*b^3/x^7

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Maxima [A]
time = 0.25, size = 79, normalized size = 1.41 \begin {gather*} -\frac {120 \, b^{7} x^{7} + 630 \, a b^{6} x^{6} + 1512 \, a^{2} b^{5} x^{5} + 2100 \, a^{3} b^{4} x^{4} + 1800 \, a^{4} b^{3} x^{3} + 945 \, a^{5} b^{2} x^{2} + 280 \, a^{6} b x + 36 \, a^{7}}{360 \, x^{10}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^7/x^11,x, algorithm="maxima")

[Out]

-1/360*(120*b^7*x^7 + 630*a*b^6*x^6 + 1512*a^2*b^5*x^5 + 2100*a^3*b^4*x^4 + 1800*a^4*b^3*x^3 + 945*a^5*b^2*x^2
+ 280*a^6*b*x + 36*a^7)/x^10

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Fricas [A]
time = 0.34, size = 79, normalized size = 1.41 \begin {gather*} -\frac {120 \, b^{7} x^{7} + 630 \, a b^{6} x^{6} + 1512 \, a^{2} b^{5} x^{5} + 2100 \, a^{3} b^{4} x^{4} + 1800 \, a^{4} b^{3} x^{3} + 945 \, a^{5} b^{2} x^{2} + 280 \, a^{6} b x + 36 \, a^{7}}{360 \, x^{10}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^7/x^11,x, algorithm="fricas")

[Out]

-1/360*(120*b^7*x^7 + 630*a*b^6*x^6 + 1512*a^2*b^5*x^5 + 2100*a^3*b^4*x^4 + 1800*a^4*b^3*x^3 + 945*a^5*b^2*x^2
+ 280*a^6*b*x + 36*a^7)/x^10

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Sympy [A]
time = 0.31, size = 85, normalized size = 1.52 \begin {gather*} \frac {- 36 a^{7} - 280 a^{6} b x - 945 a^{5} b^{2} x^{2} - 1800 a^{4} b^{3} x^{3} - 2100 a^{3} b^{4} x^{4} - 1512 a^{2} b^{5} x^{5} - 630 a b^{6} x^{6} - 120 b^{7} x^{7}}{360 x^{10}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**7/x**11,x)

[Out]

(-36*a**7 - 280*a**6*b*x - 945*a**5*b**2*x**2 - 1800*a**4*b**3*x**3 - 2100*a**3*b**4*x**4 - 1512*a**2*b**5*x**
5 - 630*a*b**6*x**6 - 120*b**7*x**7)/(360*x**10)

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Giac [A]
time = 0.00, size = 89, normalized size = 1.59 \begin {gather*} \frac {-120 x^{7} b^{7}-630 x^{6} b^{6} a-1512 x^{5} b^{5} a^{2}-2100 x^{4} b^{4} a^{3}-1800 x^{3} b^{3} a^{4}-945 x^{2} b^{2} a^{5}-280 x b a^{6}-36 a^{7}}{360 x^{10}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^7/x^11,x)

[Out]

-1/360*(120*b^7*x^7 + 630*a*b^6*x^6 + 1512*a^2*b^5*x^5 + 2100*a^3*b^4*x^4 + 1800*a^4*b^3*x^3 + 945*a^5*b^2*x^2
+ 280*a^6*b*x + 36*a^7)/x^10

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Mupad [B]
time = 0.11, size = 79, normalized size = 1.41 \begin {gather*} -\frac {\frac {a^7}{10}+\frac {7\,a^6\,b\,x}{9}+\frac {21\,a^5\,b^2\,x^2}{8}+5\,a^4\,b^3\,x^3+\frac {35\,a^3\,b^4\,x^4}{6}+\frac {21\,a^2\,b^5\,x^5}{5}+\frac {7\,a\,b^6\,x^6}{4}+\frac {b^7\,x^7}{3}}{x^{10}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x)^7/x^11,x)

[Out]

-(a^7/10 + (b^7*x^7)/3 + (7*a*b^6*x^6)/4 + (21*a^5*b^2*x^2)/8 + 5*a^4*b^3*x^3 + (35*a^3*b^4*x^4)/6 + (21*a^2*b
^5*x^5)/5 + (7*a^6*b*x)/9)/x^10

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