\(\int \frac {x^4}{(a+b x)^{10}} \, dx\) [230]

Optimal result

Integrand size = 11, antiderivative size = 81 \[ \int \frac {x^4}{(a+b x)^{10}} \, dx=-\frac {a^4}{9 b^5 (a+b x)^9}+\frac {a^3}{2 b^5 (a+b x)^8}-\frac {6 a^2}{7 b^5 (a+b x)^7}+\frac {2 a}{3 b^5 (a+b x)^6}-\frac {1}{5 b^5 (a+b x)^5} \]

[Out]

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

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {45} \[ \int \frac {x^4}{(a+b x)^{10}} \, dx=-\frac {a^4}{9 b^5 (a+b x)^9}+\frac {a^3}{2 b^5 (a+b x)^8}-\frac {6 a^2}{7 b^5 (a+b x)^7}+\frac {2 a}{3 b^5 (a+b x)^6}-\frac {1}{5 b^5 (a+b x)^5} \]

[In]

Int[x^4/(a + b*x)^10,x]

[Out]

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

Rule 45

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {a^4}{b^4 (a+b x)^{10}}-\frac {4 a^3}{b^4 (a+b x)^9}+\frac {6 a^2}{b^4 (a+b x)^8}-\frac {4 a}{b^4 (a+b x)^7}+\frac {1}{b^4 (a+b x)^6}\right ) \, dx \\ & = -\frac {a^4}{9 b^5 (a+b x)^9}+\frac {a^3}{2 b^5 (a+b x)^8}-\frac {6 a^2}{7 b^5 (a+b x)^7}+\frac {2 a}{3 b^5 (a+b x)^6}-\frac {1}{5 b^5 (a+b x)^5} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.65 \[ \int \frac {x^4}{(a+b x)^{10}} \, dx=-\frac {a^4+9 a^3 b x+36 a^2 b^2 x^2+84 a b^3 x^3+126 b^4 x^4}{630 b^5 (a+b x)^9} \]

[In]

Integrate[x^4/(a + b*x)^10,x]

[Out]

-1/630*(a^4 + 9*a^3*b*x + 36*a^2*b^2*x^2 + 84*a*b^3*x^3 + 126*b^4*x^4)/(b^5*(a + b*x)^9)

Maple [A] (verified)

Time = 0.04 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.64

method	result	size
gosper	\(-\frac {126 b^{4} x^{4}+84 a \,b^{3} x^{3}+36 a^{2} b^{2} x^{2}+9 a^{3} b x +a^{4}}{630 \left (b x +a \right )^{9} b^{5}}\)	\(52\)
norman	\(\frac {-\frac {x^{4}}{5 b}-\frac {2 a \,x^{3}}{15 b^{2}}-\frac {2 a^{2} x^{2}}{35 b^{3}}-\frac {a^{3} x}{70 b^{4}}-\frac {a^{4}}{630 b^{5}}}{\left (b x +a \right )^{9}}\)	\(55\)
risch	\(\frac {-\frac {x^{4}}{5 b}-\frac {2 a \,x^{3}}{15 b^{2}}-\frac {2 a^{2} x^{2}}{35 b^{3}}-\frac {a^{3} x}{70 b^{4}}-\frac {a^{4}}{630 b^{5}}}{\left (b x +a \right )^{9}}\)	\(55\)
parallelrisch	\(\frac {-126 b^{8} x^{4}-84 a \,b^{7} x^{3}-36 a^{2} b^{6} x^{2}-9 a^{3} b^{5} x -a^{4} b^{4}}{630 b^{9} \left (b x +a \right )^{9}}\)	\(59\)
default	\(-\frac {a^{4}}{9 b^{5} \left (b x +a \right )^{9}}+\frac {a^{3}}{2 b^{5} \left (b x +a \right )^{8}}-\frac {6 a^{2}}{7 b^{5} \left (b x +a \right )^{7}}+\frac {2 a}{3 b^{5} \left (b x +a \right )^{6}}-\frac {1}{5 b^{5} \left (b x +a \right )^{5}}\)	\(72\)

[In]

int(x^4/(b*x+a)^10,x,method=_RETURNVERBOSE)

[Out]

-1/630*(126*b^4*x^4+84*a*b^3*x^3+36*a^2*b^2*x^2+9*a^3*b*x+a^4)/(b*x+a)^9/b^5

Fricas [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 142, normalized size of antiderivative = 1.75 \[ \int \frac {x^4}{(a+b x)^{10}} \, dx=-\frac {126 \, b^{4} x^{4} + 84 \, a b^{3} x^{3} + 36 \, a^{2} b^{2} x^{2} + 9 \, a^{3} b x + a^{4}}{630 \, {\left (b^{14} x^{9} + 9 \, a b^{13} x^{8} + 36 \, a^{2} b^{12} x^{7} + 84 \, a^{3} b^{11} x^{6} + 126 \, a^{4} b^{10} x^{5} + 126 \, a^{5} b^{9} x^{4} + 84 \, a^{6} b^{8} x^{3} + 36 \, a^{7} b^{7} x^{2} + 9 \, a^{8} b^{6} x + a^{9} b^{5}\right )}} \]

[In]

integrate(x^4/(b*x+a)^10,x, algorithm="fricas")

[Out]

-1/630*(126*b^4*x^4 + 84*a*b^3*x^3 + 36*a^2*b^2*x^2 + 9*a^3*b*x + a^4)/(b^14*x^9 + 9*a*b^13*x^8 + 36*a^2*b^12*
x^7 + 84*a^3*b^11*x^6 + 126*a^4*b^10*x^5 + 126*a^5*b^9*x^4 + 84*a^6*b^8*x^3 + 36*a^7*b^7*x^2 + 9*a^8*b^6*x + a
^9*b^5)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 151 vs. \(2 (75) = 150\).

Time = 0.41 (sec) , antiderivative size = 151, normalized size of antiderivative = 1.86 \[ \int \frac {x^4}{(a+b x)^{10}} \, dx=\frac {- a^{4} - 9 a^{3} b x - 36 a^{2} b^{2} x^{2} - 84 a b^{3} x^{3} - 126 b^{4} x^{4}}{630 a^{9} b^{5} + 5670 a^{8} b^{6} x + 22680 a^{7} b^{7} x^{2} + 52920 a^{6} b^{8} x^{3} + 79380 a^{5} b^{9} x^{4} + 79380 a^{4} b^{10} x^{5} + 52920 a^{3} b^{11} x^{6} + 22680 a^{2} b^{12} x^{7} + 5670 a b^{13} x^{8} + 630 b^{14} x^{9}} \]

[In]

integrate(x**4/(b*x+a)**10,x)

[Out]

(-a**4 - 9*a**3*b*x - 36*a**2*b**2*x**2 - 84*a*b**3*x**3 - 126*b**4*x**4)/(630*a**9*b**5 + 5670*a**8*b**6*x +
22680*a**7*b**7*x**2 + 52920*a**6*b**8*x**3 + 79380*a**5*b**9*x**4 + 79380*a**4*b**10*x**5 + 52920*a**3*b**11*
x**6 + 22680*a**2*b**12*x**7 + 5670*a*b**13*x**8 + 630*b**14*x**9)

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 142, normalized size of antiderivative = 1.75 \[ \int \frac {x^4}{(a+b x)^{10}} \, dx=-\frac {126 \, b^{4} x^{4} + 84 \, a b^{3} x^{3} + 36 \, a^{2} b^{2} x^{2} + 9 \, a^{3} b x + a^{4}}{630 \, {\left (b^{14} x^{9} + 9 \, a b^{13} x^{8} + 36 \, a^{2} b^{12} x^{7} + 84 \, a^{3} b^{11} x^{6} + 126 \, a^{4} b^{10} x^{5} + 126 \, a^{5} b^{9} x^{4} + 84 \, a^{6} b^{8} x^{3} + 36 \, a^{7} b^{7} x^{2} + 9 \, a^{8} b^{6} x + a^{9} b^{5}\right )}} \]

[In]

integrate(x^4/(b*x+a)^10,x, algorithm="maxima")

[Out]

-1/630*(126*b^4*x^4 + 84*a*b^3*x^3 + 36*a^2*b^2*x^2 + 9*a^3*b*x + a^4)/(b^14*x^9 + 9*a*b^13*x^8 + 36*a^2*b^12*
x^7 + 84*a^3*b^11*x^6 + 126*a^4*b^10*x^5 + 126*a^5*b^9*x^4 + 84*a^6*b^8*x^3 + 36*a^7*b^7*x^2 + 9*a^8*b^6*x + a
^9*b^5)

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.63 \[ \int \frac {x^4}{(a+b x)^{10}} \, dx=-\frac {126 \, b^{4} x^{4} + 84 \, a b^{3} x^{3} + 36 \, a^{2} b^{2} x^{2} + 9 \, a^{3} b x + a^{4}}{630 \, {\left (b x + a\right )}^{9} b^{5}} \]

[In]

integrate(x^4/(b*x+a)^10,x, algorithm="giac")

[Out]

-1/630*(126*b^4*x^4 + 84*a*b^3*x^3 + 36*a^2*b^2*x^2 + 9*a^3*b*x + a^4)/((b*x + a)^9*b^5)

Mupad [B] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.75 \[ \int \frac {x^4}{(a+b x)^{10}} \, dx=-\frac {\frac {1}{5\,{\left (a+b\,x\right )}^5}-\frac {2\,a}{3\,{\left (a+b\,x\right )}^6}+\frac {6\,a^2}{7\,{\left (a+b\,x\right )}^7}-\frac {a^3}{2\,{\left (a+b\,x\right )}^8}+\frac {a^4}{9\,{\left (a+b\,x\right )}^9}}{b^5} \]

[In]

int(x^4/(a + b*x)^10,x)

[Out]

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