3.169 \(\int \frac {x^{13}}{(a+b x^2)^3} \, dx\)

Optimal. Leaf size=100 \[ -\frac {a^6}{4 b^7 \left (a+b x^2\right )^2}+\frac {3 a^5}{b^7 \left (a+b x^2\right )}+\frac {15 a^4 \log \left (a+b x^2\right )}{2 b^7}-\frac {5 a^3 x^2}{b^6}+\frac {3 a^2 x^4}{2 b^5}-\frac {a x^6}{2 b^4}+\frac {x^8}{8 b^3} \]

[Out]

-5*a^3*x^2/b^6+3/2*a^2*x^4/b^5-1/2*a*x^6/b^4+1/8*x^8/b^3-1/4*a^6/b^7/(b*x^2+a)^2+3*a^5/b^7/(b*x^2+a)+15/2*a^4*
ln(b*x^2+a)/b^7

________________________________________________________________________________________

Rubi [A]  time = 0.08, antiderivative size = 100, 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} \[ \frac {3 a^2 x^4}{2 b^5}-\frac {5 a^3 x^2}{b^6}+\frac {3 a^5}{b^7 \left (a+b x^2\right )}-\frac {a^6}{4 b^7 \left (a+b x^2\right )^2}+\frac {15 a^4 \log \left (a+b x^2\right )}{2 b^7}-\frac {a x^6}{2 b^4}+\frac {x^8}{8 b^3} \]

Antiderivative was successfully verified.

[In]

Int[x^13/(a + b*x^2)^3,x]

[Out]

(-5*a^3*x^2)/b^6 + (3*a^2*x^4)/(2*b^5) - (a*x^6)/(2*b^4) + x^8/(8*b^3) - a^6/(4*b^7*(a + b*x^2)^2) + (3*a^5)/(
b^7*(a + b*x^2)) + (15*a^4*Log[a + b*x^2])/(2*b^7)

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^{13}}{\left (a+b x^2\right )^3} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {x^6}{(a+b x)^3} \, dx,x,x^2\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \left (-\frac {10 a^3}{b^6}+\frac {6 a^2 x}{b^5}-\frac {3 a x^2}{b^4}+\frac {x^3}{b^3}+\frac {a^6}{b^6 (a+b x)^3}-\frac {6 a^5}{b^6 (a+b x)^2}+\frac {15 a^4}{b^6 (a+b x)}\right ) \, dx,x,x^2\right )\\ &=-\frac {5 a^3 x^2}{b^6}+\frac {3 a^2 x^4}{2 b^5}-\frac {a x^6}{2 b^4}+\frac {x^8}{8 b^3}-\frac {a^6}{4 b^7 \left (a+b x^2\right )^2}+\frac {3 a^5}{b^7 \left (a+b x^2\right )}+\frac {15 a^4 \log \left (a+b x^2\right )}{2 b^7}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.03, size = 85, normalized size = 0.85 \[ \frac {-\frac {2 a^6}{\left (a+b x^2\right )^2}+\frac {24 a^5}{a+b x^2}+60 a^4 \log \left (a+b x^2\right )-40 a^3 b x^2+12 a^2 b^2 x^4-4 a b^3 x^6+b^4 x^8}{8 b^7} \]

Antiderivative was successfully verified.

[In]

Integrate[x^13/(a + b*x^2)^3,x]

[Out]

(-40*a^3*b*x^2 + 12*a^2*b^2*x^4 - 4*a*b^3*x^6 + b^4*x^8 - (2*a^6)/(a + b*x^2)^2 + (24*a^5)/(a + b*x^2) + 60*a^
4*Log[a + b*x^2])/(8*b^7)

________________________________________________________________________________________

fricas [A]  time = 0.82, size = 125, normalized size = 1.25 \[ \frac {b^{6} x^{12} - 2 \, a b^{5} x^{10} + 5 \, a^{2} b^{4} x^{8} - 20 \, a^{3} b^{3} x^{6} - 68 \, a^{4} b^{2} x^{4} - 16 \, a^{5} b x^{2} + 22 \, a^{6} + 60 \, {\left (a^{4} b^{2} x^{4} + 2 \, a^{5} b x^{2} + a^{6}\right )} \log \left (b x^{2} + a\right )}{8 \, {\left (b^{9} x^{4} + 2 \, a b^{8} x^{2} + a^{2} b^{7}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^13/(b*x^2+a)^3,x, algorithm="fricas")

[Out]

1/8*(b^6*x^12 - 2*a*b^5*x^10 + 5*a^2*b^4*x^8 - 20*a^3*b^3*x^6 - 68*a^4*b^2*x^4 - 16*a^5*b*x^2 + 22*a^6 + 60*(a
^4*b^2*x^4 + 2*a^5*b*x^2 + a^6)*log(b*x^2 + a))/(b^9*x^4 + 2*a*b^8*x^2 + a^2*b^7)

________________________________________________________________________________________

giac [A]  time = 0.61, size = 102, normalized size = 1.02 \[ \frac {15 \, a^{4} \log \left ({\left | b x^{2} + a \right |}\right )}{2 \, b^{7}} - \frac {45 \, a^{4} b^{2} x^{4} + 78 \, a^{5} b x^{2} + 34 \, a^{6}}{4 \, {\left (b x^{2} + a\right )}^{2} b^{7}} + \frac {b^{9} x^{8} - 4 \, a b^{8} x^{6} + 12 \, a^{2} b^{7} x^{4} - 40 \, a^{3} b^{6} x^{2}}{8 \, b^{12}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^13/(b*x^2+a)^3,x, algorithm="giac")

[Out]

15/2*a^4*log(abs(b*x^2 + a))/b^7 - 1/4*(45*a^4*b^2*x^4 + 78*a^5*b*x^2 + 34*a^6)/((b*x^2 + a)^2*b^7) + 1/8*(b^9
*x^8 - 4*a*b^8*x^6 + 12*a^2*b^7*x^4 - 40*a^3*b^6*x^2)/b^12

________________________________________________________________________________________

maple [A]  time = 0.01, size = 91, normalized size = 0.91 \[ \frac {x^{8}}{8 b^{3}}-\frac {a \,x^{6}}{2 b^{4}}+\frac {3 a^{2} x^{4}}{2 b^{5}}-\frac {a^{6}}{4 \left (b \,x^{2}+a \right )^{2} b^{7}}-\frac {5 a^{3} x^{2}}{b^{6}}+\frac {3 a^{5}}{\left (b \,x^{2}+a \right ) b^{7}}+\frac {15 a^{4} \ln \left (b \,x^{2}+a \right )}{2 b^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^13/(b*x^2+a)^3,x)

[Out]

-5*a^3*x^2/b^6+3/2*a^2*x^4/b^5-1/2*a*x^6/b^4+1/8*x^8/b^3-1/4*a^6/b^7/(b*x^2+a)^2+3*a^5/b^7/(b*x^2+a)+15/2*a^4*
ln(b*x^2+a)/b^7

________________________________________________________________________________________

maxima [A]  time = 1.29, size = 99, normalized size = 0.99 \[ \frac {12 \, a^{5} b x^{2} + 11 \, a^{6}}{4 \, {\left (b^{9} x^{4} + 2 \, a b^{8} x^{2} + a^{2} b^{7}\right )}} + \frac {15 \, a^{4} \log \left (b x^{2} + a\right )}{2 \, b^{7}} + \frac {b^{3} x^{8} - 4 \, a b^{2} x^{6} + 12 \, a^{2} b x^{4} - 40 \, a^{3} x^{2}}{8 \, b^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^13/(b*x^2+a)^3,x, algorithm="maxima")

[Out]

1/4*(12*a^5*b*x^2 + 11*a^6)/(b^9*x^4 + 2*a*b^8*x^2 + a^2*b^7) + 15/2*a^4*log(b*x^2 + a)/b^7 + 1/8*(b^3*x^8 - 4
*a*b^2*x^6 + 12*a^2*b*x^4 - 40*a^3*x^2)/b^6

________________________________________________________________________________________

mupad [B]  time = 0.08, size = 100, normalized size = 1.00 \[ \frac {\frac {11\,a^6}{4\,b}+3\,a^5\,x^2}{a^2\,b^6+2\,a\,b^7\,x^2+b^8\,x^4}+\frac {x^8}{8\,b^3}-\frac {a\,x^6}{2\,b^4}+\frac {15\,a^4\,\ln \left (b\,x^2+a\right )}{2\,b^7}+\frac {3\,a^2\,x^4}{2\,b^5}-\frac {5\,a^3\,x^2}{b^6} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^13/(a + b*x^2)^3,x)

[Out]

((11*a^6)/(4*b) + 3*a^5*x^2)/(a^2*b^6 + b^8*x^4 + 2*a*b^7*x^2) + x^8/(8*b^3) - (a*x^6)/(2*b^4) + (15*a^4*log(a
+ b*x^2))/(2*b^7) + (3*a^2*x^4)/(2*b^5) - (5*a^3*x^2)/b^6

________________________________________________________________________________________

sympy [A]  time = 0.47, size = 104, normalized size = 1.04 \[ \frac {15 a^{4} \log {\left (a + b x^{2} \right )}}{2 b^{7}} - \frac {5 a^{3} x^{2}}{b^{6}} + \frac {3 a^{2} x^{4}}{2 b^{5}} - \frac {a x^{6}}{2 b^{4}} + \frac {11 a^{6} + 12 a^{5} b x^{2}}{4 a^{2} b^{7} + 8 a b^{8} x^{2} + 4 b^{9} x^{4}} + \frac {x^{8}}{8 b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**13/(b*x**2+a)**3,x)

[Out]

15*a**4*log(a + b*x**2)/(2*b**7) - 5*a**3*x**2/b**6 + 3*a**2*x**4/(2*b**5) - a*x**6/(2*b**4) + (11*a**6 + 12*a
**5*b*x**2)/(4*a**2*b**7 + 8*a*b**8*x**2 + 4*b**9*x**4) + x**8/(8*b**3)

________________________________________________________________________________________