3.3.34 \(\int \frac {1}{(a+b x)^{10}} \, dx\) [234]

Optimal. Leaf size=14 \[ -\frac {1}{9 b (a+b x)^9} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.00, antiderivative size = 14, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {32} \begin {gather*} -\frac {1}{9 b (a+b x)^9} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^(-10),x]

[Out]

-1/9*1/(b*(a + b*x)^9)

Rule 32

Int[((a_.) + (b_.)*(x_))^(m_), x_Symbol] :> Simp[(a + b*x)^(m + 1)/(b*(m + 1)), x] /; FreeQ[{a, b, m}, x] && N
eQ[m, -1]

Rubi steps

\begin {align*} \int \frac {1}{(a+b x)^{10}} \, dx &=-\frac {1}{9 b (a+b x)^9}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 0.00, size = 14, normalized size = 1.00 \begin {gather*} -\frac {1}{9 b (a+b x)^9} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^(-10),x]

[Out]

-1/9*1/(b*(a + b*x)^9)

________________________________________________________________________________________

Mathics [B] Leaf count is larger than twice the leaf count of optimal. \(102\) vs. \(2(14)=28\).
time = 2.57, size = 100, normalized size = 7.14 \begin {gather*} -\frac {1}{9 b \left (a^9+9 a^8 b x+36 a^7 b^2 x^2+84 a^6 b^3 x^3+126 a^5 b^4 x^4+126 a^4 b^5 x^5+84 a^3 b^6 x^6+36 a^2 b^7 x^7+9 a b^8 x^8+b^9 x^9\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

mathics('Integrate[x^0/(a + b*x)^10,x]')

[Out]

-1 / (9 b (a ^ 9 + 9 a ^ 8 b x + 36 a ^ 7 b ^ 2 x ^ 2 + 84 a ^ 6 b ^ 3 x ^ 3 + 126 a ^ 5 b ^ 4 x ^ 4 + 126 a ^
4 b ^ 5 x ^ 5 + 84 a ^ 3 b ^ 6 x ^ 6 + 36 a ^ 2 b ^ 7 x ^ 7 + 9 a b ^ 8 x ^ 8 + b ^ 9 x ^ 9))

________________________________________________________________________________________

Maple [A]
time = 0.09, size = 13, normalized size = 0.93

method	result	size
gosper	\(-\frac {1}{9 b \left (b x +a \right )^{9}}\)	\(13\)
default	\(-\frac {1}{9 b \left (b x +a \right )^{9}}\)	\(13\)
norman	\(-\frac {1}{9 b \left (b x +a \right )^{9}}\)	\(13\)
risch	\(-\frac {1}{9 b \left (b x +a \right )^{9}}\)	\(13\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(b*x+a)^10,x,method=_RETURNVERBOSE)

[Out]

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

________________________________________________________________________________________

Maxima [A]
time = 0.26, size = 12, normalized size = 0.86 \begin {gather*} -\frac {1}{9 \, {\left (b x + a\right )}^{9} b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x+a)^10,x, algorithm="maxima")

[Out]

-1/9/((b*x + a)^9*b)

________________________________________________________________________________________

Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 101 vs. \(2 (12) = 24\).
time = 0.30, size = 101, normalized size = 7.21 \begin {gather*} -\frac {1}{9 \, {\left (b^{10} x^{9} + 9 \, a b^{9} x^{8} + 36 \, a^{2} b^{8} x^{7} + 84 \, a^{3} b^{7} x^{6} + 126 \, a^{4} b^{6} x^{5} + 126 \, a^{5} b^{5} x^{4} + 84 \, a^{6} b^{4} x^{3} + 36 \, a^{7} b^{3} x^{2} + 9 \, a^{8} b^{2} x + a^{9} b\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x+a)^10,x, algorithm="fricas")

[Out]

-1/9/(b^10*x^9 + 9*a*b^9*x^8 + 36*a^2*b^8*x^7 + 84*a^3*b^7*x^6 + 126*a^4*b^6*x^5 + 126*a^5*b^5*x^4 + 84*a^6*b^
4*x^3 + 36*a^7*b^3*x^2 + 9*a^8*b^2*x + a^9*b)

________________________________________________________________________________________

Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 109 vs. \(2 (12) = 24\).
time = 0.31, size = 109, normalized size = 7.79 \begin {gather*} - \frac {1}{9 a^{9} b + 81 a^{8} b^{2} x + 324 a^{7} b^{3} x^{2} + 756 a^{6} b^{4} x^{3} + 1134 a^{5} b^{5} x^{4} + 1134 a^{4} b^{6} x^{5} + 756 a^{3} b^{7} x^{6} + 324 a^{2} b^{8} x^{7} + 81 a b^{9} x^{8} + 9 b^{10} x^{9}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x+a)**10,x)

[Out]

-1/(9*a**9*b + 81*a**8*b**2*x + 324*a**7*b**3*x**2 + 756*a**6*b**4*x**3 + 1134*a**5*b**5*x**4 + 1134*a**4*b**6
*x**5 + 756*a**3*b**7*x**6 + 324*a**2*b**8*x**7 + 81*a*b**9*x**8 + 9*b**10*x**9)

________________________________________________________________________________________

Giac [A]
time = 0.00, size = 12, normalized size = 0.86 \begin {gather*} -\frac 1{9 b \left (x b+a\right )^{9}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x+a)^10,x)

[Out]

-1/9/((b*x + a)^9*b)

________________________________________________________________________________________

Mupad [B]
time = 0.14, size = 103, normalized size = 7.36 \begin {gather*} -\frac {1}{9\,a^9\,b+81\,a^8\,b^2\,x+324\,a^7\,b^3\,x^2+756\,a^6\,b^4\,x^3+1134\,a^5\,b^5\,x^4+1134\,a^4\,b^6\,x^5+756\,a^3\,b^7\,x^6+324\,a^2\,b^8\,x^7+81\,a\,b^9\,x^8+9\,b^{10}\,x^9} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a + b*x)^10,x)

[Out]

-1/(9*a^9*b + 9*b^10*x^9 + 81*a^8*b^2*x + 81*a*b^9*x^8 + 324*a^7*b^3*x^2 + 756*a^6*b^4*x^3 + 1134*a^5*b^5*x^4
+ 1134*a^4*b^6*x^5 + 756*a^3*b^7*x^6 + 324*a^2*b^8*x^7)

________________________________________________________________________________________