3.13.78 \(\int \frac {x^{19}}{(a+b x^5)^2} \, dx\) [1278]

Optimal. Leaf size=59 \[ -\frac {2 a x^5}{5 b^3}+\frac {x^{10}}{10 b^2}+\frac {a^3}{5 b^4 \left (a+b x^5\right )}+\frac {3 a^2 \log \left (a+b x^5\right )}{5 b^4} \]

[Out]

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

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Rubi [A]
time = 0.03, antiderivative size = 59, 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}{5 b^4 \left (a+b x^5\right )}+\frac {3 a^2 \log \left (a+b x^5\right )}{5 b^4}-\frac {2 a x^5}{5 b^3}+\frac {x^{10}}{10 b^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^19/(a + b*x^5)^2,x]

[Out]

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

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Mathematica [A]
time = 0.01, size = 49, normalized size = 0.83 \begin {gather*} \frac {-4 a b x^5+b^2 x^{10}+\frac {2 a^3}{a+b x^5}+6 a^2 \log \left (a+b x^5\right )}{10 b^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^19/(a + b*x^5)^2,x]

[Out]

(-4*a*b*x^5 + b^2*x^10 + (2*a^3)/(a + b*x^5) + 6*a^2*Log[a + b*x^5])/(10*b^4)

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Maple [A]
time = 0.17, size = 55, normalized size = 0.93

method	result	size
norman	\(\frac {\frac {x^{15}}{10 b}-\frac {3 a \,x^{10}}{10 b^{2}}+\frac {3 a^{3}}{5 b^{4}}}{b \,x^{5}+a}+\frac {3 a^{2} \ln \left (b \,x^{5}+a \right )}{5 b^{4}}\)	\(54\)
default	\(\frac {\left (-b \,x^{5}+2 a \right )^{2}}{10 b^{4}}+\frac {a^{2} \left (\frac {a}{b \left (b \,x^{5}+a \right )}+\frac {3 \ln \left (b \,x^{5}+a \right )}{b}\right )}{5 b^{3}}\)	\(55\)
risch	\(\frac {x^{10}}{10 b^{2}}-\frac {2 a \,x^{5}}{5 b^{3}}+\frac {2 a^{2}}{5 b^{4}}+\frac {a^{3}}{5 b^{4} \left (b \,x^{5}+a \right )}+\frac {3 a^{2} \ln \left (b \,x^{5}+a \right )}{5 b^{4}}\)	\(60\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^19/(b*x^5+a)^2,x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.29, size = 54, normalized size = 0.92 \begin {gather*} \frac {a^{3}}{5 \, {\left (b^{5} x^{5} + a b^{4}\right )}} + \frac {3 \, a^{2} \log \left (b x^{5} + a\right )}{5 \, b^{4}} + \frac {b x^{10} - 4 \, a x^{5}}{10 \, b^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^19/(b*x^5+a)^2,x, algorithm="maxima")

[Out]

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

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Fricas [A]
time = 0.37, size = 70, normalized size = 1.19 \begin {gather*} \frac {b^{3} x^{15} - 3 \, a b^{2} x^{10} - 4 \, a^{2} b x^{5} + 2 \, a^{3} + 6 \, {\left (a^{2} b x^{5} + a^{3}\right )} \log \left (b x^{5} + a\right )}{10 \, {\left (b^{5} x^{5} + a b^{4}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^19/(b*x^5+a)^2,x, algorithm="fricas")

[Out]

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

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Sympy [A]
time = 0.20, size = 56, normalized size = 0.95 \begin {gather*} \frac {a^{3}}{5 a b^{4} + 5 b^{5} x^{5}} + \frac {3 a^{2} \log {\left (a + b x^{5} \right )}}{5 b^{4}} - \frac {2 a x^{5}}{5 b^{3}} + \frac {x^{10}}{10 b^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**19/(b*x**5+a)**2,x)

[Out]

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

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Giac [A]
time = 1.50, size = 67, normalized size = 1.14 \begin {gather*} \frac {3 \, a^{2} \log \left ({\left | b x^{5} + a \right |}\right )}{5 \, b^{4}} + \frac {b^{2} x^{10} - 4 \, a b x^{5}}{10 \, b^{4}} - \frac {3 \, a^{2} b x^{5} + 2 \, a^{3}}{5 \, {\left (b x^{5} + a\right )} b^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^19/(b*x^5+a)^2,x, algorithm="giac")

[Out]

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

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Mupad [B]
time = 0.09, size = 57, normalized size = 0.97 \begin {gather*} \frac {x^{10}}{10\,b^2}+\frac {a^3}{5\,b\,\left (b^4\,x^5+a\,b^3\right )}-\frac {2\,a\,x^5}{5\,b^3}+\frac {3\,a^2\,\ln \left (b\,x^5+a\right )}{5\,b^4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^19/(a + b*x^5)^2,x)

[Out]

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

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