3.13.67 \(\int \frac {x^{24}}{a+b x^5} \, dx\) [1267]

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

[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 = 0.03, 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 = {272, 45} \begin {gather*} \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} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[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)

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])

Rule 272

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} \text {Subst}\left (\int \frac {x^4}{a+b x} \, dx,x,x^5\right )\\ &=\frac {1}{5} \text {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 \begin {gather*} -\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 {gather*}

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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Maple [A]
time = 0.20, size = 57, normalized size = 0.86

method	result	size
default	\(-\frac {-\frac {1}{4} b^{3} x^{20}+\frac {1}{3} a \,b^{2} x^{15}-\frac {1}{2} a^{2} b \,x^{10}+a^{3} x^{5}}{5 b^{4}}+\frac {a^{4} \ln \left (b \,x^{5}+a \right )}{5 b^{5}}\)	\(57\)
norman	\(-\frac {a^{3} x^{5}}{5 b^{4}}+\frac {a^{2} x^{10}}{10 b^{3}}-\frac {a \,x^{15}}{15 b^{2}}+\frac {x^{20}}{20 b}+\frac {a^{4} \ln \left (b \,x^{5}+a \right )}{5 b^{5}}\)	\(57\)
risch	\(\frac {a^{2} x^{10}}{10 b^{3}}-\frac {a^{3} x^{5}}{5 b^{4}}+\frac {7 a^{4}}{60 b^{5}}-\frac {a \,x^{15}}{15 b^{2}}+\frac {x^{20}}{20 b}+\frac {a^{4} \ln \left (b \,x^{5}+a \right )}{5 b^{5}}\)	\(65\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.30, size = 57, normalized size = 0.86 \begin {gather*} \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}} \end {gather*}

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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Fricas [A]
time = 0.36, size = 56, normalized size = 0.85 \begin {gather*} \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}} \end {gather*}

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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Sympy [A]
time = 0.11, size = 56, normalized size = 0.85 \begin {gather*} \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} \end {gather*}

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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Giac [A]
time = 1.46, size = 58, normalized size = 0.88 \begin {gather*} \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}} \end {gather*}

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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Mupad [B]
time = 0.08, size = 56, normalized size = 0.85 \begin {gather*} \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} \end {gather*}

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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