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

Optimal. Leaf size=81 \[ -\frac {a^4}{9 b^5 (a+b x)^9}+\frac {a^3}{2 b^5 (a+b x)^8}-\frac {6 a^2}{7 b^5 (a+b x)^7}+\frac {2 a}{3 b^5 (a+b x)^6}-\frac {1}{5 b^5 (a+b x)^5} \]

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Rubi [A]  time = 0.04, antiderivative size = 81, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {43} \begin {gather*} -\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 {gather*}

Antiderivative was successfully verified.

[In]

Int[x^4/(a + b*x)^10,x]

[Out]

-a^4/(9*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 43

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*} \int \frac {x^4}{(a+b x)^{10}} \, dx &=\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*}

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Mathematica [A]  time = 0.01, size = 53, normalized size = 0.65 \begin {gather*} -\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} \end {gather*}

Antiderivative was successfully verified.

[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)

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^4}{(a+b x)^{10}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[x^4/(a + b*x)^10,x]

[Out]

IntegrateAlgebraic[x^4/(a + b*x)^10, x]

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fricas [A]  time = 1.13, size = 142, normalized size = 1.75 \begin {gather*} -\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 )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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giac [A]  time = 1.22, size = 51, normalized size = 0.63 \begin {gather*} -\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}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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maple [A]  time = 0.01, size = 72, normalized size = 0.89 \begin {gather*} -\frac {a^{4}}{9 \left (b x +a \right )^{9} b^{5}}+\frac {a^{3}}{2 \left (b x +a \right )^{8} b^{5}}-\frac {6 a^{2}}{7 \left (b x +a \right )^{7} b^{5}}+\frac {2 a}{3 \left (b x +a \right )^{6} b^{5}}-\frac {1}{5 \left (b x +a \right )^{5} b^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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

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maxima [A]  time = 1.46, size = 142, normalized size = 1.75 \begin {gather*} -\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 )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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mupad [B]  time = 0.08, size = 61, normalized size = 0.75 \begin {gather*} -\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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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sympy [B]  time = 0.79, size = 151, normalized size = 1.86 \begin {gather*} \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}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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