3.1267 \(\int \frac {x^{24}}{a+b x^5} \, dx\)

Optimal. Leaf size=66 \[ \frac {a^4 \log \left (a+b x^5\right )}{5 b^5}-\frac {a^3 x^5}{5 b^4}+\frac {a^2 x^{10}}{10 b^3}-\frac {a x^{15}}{15 b^2}+\frac {x^{20}}{20 b} \]

[Out]

-1/5*a^3*x^5/b^4+1/10*a^2*x^10/b^3-1/15*a*x^15/b^2+1/20*x^20/b+1/5*a^4*ln(b*x^5+a)/b^5

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Rubi [A]  time = 0.05, antiderivative size = 66, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {266, 43} \[ \frac {a^2 x^{10}}{10 b^3}-\frac {a^3 x^5}{5 b^4}+\frac {a^4 \log \left (a+b x^5\right )}{5 b^5}-\frac {a x^{15}}{15 b^2}+\frac {x^{20}}{20 b} \]

Antiderivative was successfully verified.

[In]

Int[x^24/(a + b*x^5),x]

[Out]

-(a^3*x^5)/(5*b^4) + (a^2*x^10)/(10*b^3) - (a*x^15)/(15*b^2) + x^20/(20*b) + (a^4*Log[a + b*x^5])/(5*b^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])

Rule 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rubi steps

\begin {align*} \int \frac {x^{24}}{a+b x^5} \, dx &=\frac {1}{5} \operatorname {Subst}\left (\int \frac {x^4}{a+b x} \, dx,x,x^5\right )\\ &=\frac {1}{5} \operatorname {Subst}\left (\int \left (-\frac {a^3}{b^4}+\frac {a^2 x}{b^3}-\frac {a x^2}{b^2}+\frac {x^3}{b}+\frac {a^4}{b^4 (a+b x)}\right ) \, dx,x,x^5\right )\\ &=-\frac {a^3 x^5}{5 b^4}+\frac {a^2 x^{10}}{10 b^3}-\frac {a x^{15}}{15 b^2}+\frac {x^{20}}{20 b}+\frac {a^4 \log \left (a+b x^5\right )}{5 b^5}\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 66, normalized size = 1.00 \[ \frac {a^4 \log \left (a+b x^5\right )}{5 b^5}-\frac {a^3 x^5}{5 b^4}+\frac {a^2 x^{10}}{10 b^3}-\frac {a x^{15}}{15 b^2}+\frac {x^{20}}{20 b} \]

Antiderivative was successfully verified.

[In]

Integrate[x^24/(a + b*x^5),x]

[Out]

-1/5*(a^3*x^5)/b^4 + (a^2*x^10)/(10*b^3) - (a*x^15)/(15*b^2) + x^20/(20*b) + (a^4*Log[a + b*x^5])/(5*b^5)

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fricas [A]  time = 0.59, size = 56, normalized size = 0.85 \[ \frac {3 \, b^{4} x^{20} - 4 \, a b^{3} x^{15} + 6 \, a^{2} b^{2} x^{10} - 12 \, a^{3} b x^{5} + 12 \, a^{4} \log \left (b x^{5} + a\right )}{60 \, b^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^24/(b*x^5+a),x, algorithm="fricas")

[Out]

1/60*(3*b^4*x^20 - 4*a*b^3*x^15 + 6*a^2*b^2*x^10 - 12*a^3*b*x^5 + 12*a^4*log(b*x^5 + a))/b^5

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giac [A]  time = 0.17, size = 58, normalized size = 0.88 \[ \frac {a^{4} \log \left ({\left | b x^{5} + a \right |}\right )}{5 \, b^{5}} + \frac {3 \, b^{3} x^{20} - 4 \, a b^{2} x^{15} + 6 \, a^{2} b x^{10} - 12 \, a^{3} x^{5}}{60 \, b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^24/(b*x^5+a),x, algorithm="giac")

[Out]

1/5*a^4*log(abs(b*x^5 + a))/b^5 + 1/60*(3*b^3*x^20 - 4*a*b^2*x^15 + 6*a^2*b*x^10 - 12*a^3*x^5)/b^4

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maple [A]  time = 0.00, size = 57, normalized size = 0.86 \[ \frac {x^{20}}{20 b}-\frac {a \,x^{15}}{15 b^{2}}+\frac {a^{2} x^{10}}{10 b^{3}}-\frac {a^{3} x^{5}}{5 b^{4}}+\frac {a^{4} \ln \left (b \,x^{5}+a \right )}{5 b^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^24/(b*x^5+a),x)

[Out]

-1/5*a^3*x^5/b^4+1/10*a^2*x^10/b^3-1/15*a*x^15/b^2+1/20*x^20/b+1/5*a^4*ln(b*x^5+a)/b^5

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maxima [A]  time = 1.02, size = 57, normalized size = 0.86 \[ \frac {a^{4} \log \left (b x^{5} + a\right )}{5 \, b^{5}} + \frac {3 \, b^{3} x^{20} - 4 \, a b^{2} x^{15} + 6 \, a^{2} b x^{10} - 12 \, a^{3} x^{5}}{60 \, b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^24/(b*x^5+a),x, algorithm="maxima")

[Out]

1/5*a^4*log(b*x^5 + a)/b^5 + 1/60*(3*b^3*x^20 - 4*a*b^2*x^15 + 6*a^2*b*x^10 - 12*a^3*x^5)/b^4

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mupad [B]  time = 0.08, size = 56, normalized size = 0.85 \[ \frac {x^{20}}{20\,b}-\frac {a\,x^{15}}{15\,b^2}+\frac {a^4\,\ln \left (b\,x^5+a\right )}{5\,b^5}-\frac {a^3\,x^5}{5\,b^4}+\frac {a^2\,x^{10}}{10\,b^3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^24/(a + b*x^5),x)

[Out]

x^20/(20*b) - (a*x^15)/(15*b^2) + (a^4*log(a + b*x^5))/(5*b^5) - (a^3*x^5)/(5*b^4) + (a^2*x^10)/(10*b^3)

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sympy [A]  time = 0.32, size = 56, normalized size = 0.85 \[ \frac {a^{4} \log {\left (a + b x^{5} \right )}}{5 b^{5}} - \frac {a^{3} x^{5}}{5 b^{4}} + \frac {a^{2} x^{10}}{10 b^{3}} - \frac {a x^{15}}{15 b^{2}} + \frac {x^{20}}{20 b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**24/(b*x**5+a),x)

[Out]

a**4*log(a + b*x**5)/(5*b**5) - a**3*x**5/(5*b**4) + a**2*x**10/(10*b**3) - a*x**15/(15*b**2) + x**20/(20*b)

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