3.13.68 \(\int \frac {x^{19}}{a+b x^5} \, dx\) [1268]

Optimal. Leaf size=53 \[ \frac {a^2 x^5}{5 b^3}-\frac {a x^{10}}{10 b^2}+\frac {x^{15}}{15 b}-\frac {a^3 \log \left (a+b x^5\right )}{5 b^4} \]

[Out]

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

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Rubi [A]
time = 0.02, antiderivative size = 53, 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^3 \log \left (a+b x^5\right )}{5 b^4}+\frac {a^2 x^5}{5 b^3}-\frac {a x^{10}}{10 b^2}+\frac {x^{15}}{15 b} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^19/(a + b*x^5),x]

[Out]

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

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^{19}}{a+b x^5} \, dx &=\frac {1}{5} \text {Subst}\left (\int \frac {x^3}{a+b x} \, dx,x,x^5\right )\\ &=\frac {1}{5} \text {Subst}\left (\int \left (\frac {a^2}{b^3}-\frac {a x}{b^2}+\frac {x^2}{b}-\frac {a^3}{b^3 (a+b x)}\right ) \, dx,x,x^5\right )\\ &=\frac {a^2 x^5}{5 b^3}-\frac {a x^{10}}{10 b^2}+\frac {x^{15}}{15 b}-\frac {a^3 \log \left (a+b x^5\right )}{5 b^4}\\ \end {align*}

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Mathematica [A]
time = 0.01, size = 53, normalized size = 1.00 \begin {gather*} \frac {a^2 x^5}{5 b^3}-\frac {a x^{10}}{10 b^2}+\frac {x^{15}}{15 b}-\frac {a^3 \log \left (a+b x^5\right )}{5 b^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^19/(a + b*x^5),x]

[Out]

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

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Maple [A]
time = 0.17, size = 46, normalized size = 0.87

method	result	size
default	\(\frac {\frac {1}{3} b^{2} x^{15}-\frac {1}{2} a b \,x^{10}+a^{2} x^{5}}{5 b^{3}}-\frac {a^{3} \ln \left (b \,x^{5}+a \right )}{5 b^{4}}\)	\(46\)
norman	\(\frac {a^{2} x^{5}}{5 b^{3}}-\frac {a \,x^{10}}{10 b^{2}}+\frac {x^{15}}{15 b}-\frac {a^{3} \ln \left (b \,x^{5}+a \right )}{5 b^{4}}\)	\(46\)
risch	\(\frac {a^{2} x^{5}}{5 b^{3}}-\frac {a \,x^{10}}{10 b^{2}}+\frac {x^{15}}{15 b}-\frac {a^{3} \ln \left (b \,x^{5}+a \right )}{5 b^{4}}\)	\(46\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^19/(b*x^5+a),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.32, size = 46, normalized size = 0.87 \begin {gather*} -\frac {a^{3} \log \left (b x^{5} + a\right )}{5 \, b^{4}} + \frac {2 \, b^{2} x^{15} - 3 \, a b x^{10} + 6 \, a^{2} x^{5}}{30 \, b^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^19/(b*x^5+a),x, algorithm="maxima")

[Out]

-1/5*a^3*log(b*x^5 + a)/b^4 + 1/30*(2*b^2*x^15 - 3*a*b*x^10 + 6*a^2*x^5)/b^3

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Fricas [A]
time = 0.36, size = 45, normalized size = 0.85 \begin {gather*} \frac {2 \, b^{3} x^{15} - 3 \, a b^{2} x^{10} + 6 \, a^{2} b x^{5} - 6 \, a^{3} \log \left (b x^{5} + a\right )}{30 \, b^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^19/(b*x^5+a),x, algorithm="fricas")

[Out]

1/30*(2*b^3*x^15 - 3*a*b^2*x^10 + 6*a^2*b*x^5 - 6*a^3*log(b*x^5 + a))/b^4

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Sympy [A]
time = 0.10, size = 44, normalized size = 0.83 \begin {gather*} - \frac {a^{3} \log {\left (a + b x^{5} \right )}}{5 b^{4}} + \frac {a^{2} x^{5}}{5 b^{3}} - \frac {a x^{10}}{10 b^{2}} + \frac {x^{15}}{15 b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**19/(b*x**5+a),x)

[Out]

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

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Giac [A]
time = 1.17, size = 47, normalized size = 0.89 \begin {gather*} -\frac {a^{3} \log \left ({\left | b x^{5} + a \right |}\right )}{5 \, b^{4}} + \frac {2 \, b^{2} x^{15} - 3 \, a b x^{10} + 6 \, a^{2} x^{5}}{30 \, b^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^19/(b*x^5+a),x, algorithm="giac")

[Out]

-1/5*a^3*log(abs(b*x^5 + a))/b^4 + 1/30*(2*b^2*x^15 - 3*a*b*x^10 + 6*a^2*x^5)/b^3

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Mupad [B]
time = 0.05, size = 45, normalized size = 0.85 \begin {gather*} \frac {x^{15}}{15\,b}-\frac {a\,x^{10}}{10\,b^2}-\frac {a^3\,\ln \left (b\,x^5+a\right )}{5\,b^4}+\frac {a^2\,x^5}{5\,b^3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^19/(a + b*x^5),x)

[Out]

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

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