3.168 \(\int \frac {x^{15}}{(a+b x^2)^3} \, dx\)

Optimal. Leaf size=114 \[ \frac {a^7}{4 b^8 \left (a+b x^2\right )^2}-\frac {7 a^6}{2 b^8 \left (a+b x^2\right )}-\frac {21 a^5 \log \left (a+b x^2\right )}{2 b^8}+\frac {15 a^4 x^2}{2 b^7}-\frac {5 a^3 x^4}{2 b^6}+\frac {a^2 x^6}{b^5}-\frac {3 a x^8}{8 b^4}+\frac {x^{10}}{10 b^3} \]

[Out]

15/2*a^4*x^2/b^7-5/2*a^3*x^4/b^6+a^2*x^6/b^5-3/8*a*x^8/b^4+1/10*x^10/b^3+1/4*a^7/b^8/(b*x^2+a)^2-7/2*a^6/b^8/(
b*x^2+a)-21/2*a^5*ln(b*x^2+a)/b^8

________________________________________________________________________________________

Rubi [A]  time = 0.10, antiderivative size = 114, 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} \[ \frac {a^2 x^6}{b^5}-\frac {5 a^3 x^4}{2 b^6}+\frac {15 a^4 x^2}{2 b^7}-\frac {7 a^6}{2 b^8 \left (a+b x^2\right )}+\frac {a^7}{4 b^8 \left (a+b x^2\right )^2}-\frac {21 a^5 \log \left (a+b x^2\right )}{2 b^8}-\frac {3 a x^8}{8 b^4}+\frac {x^{10}}{10 b^3} \]

Antiderivative was successfully verified.

[In]

Int[x^15/(a + b*x^2)^3,x]

[Out]

(15*a^4*x^2)/(2*b^7) - (5*a^3*x^4)/(2*b^6) + (a^2*x^6)/b^5 - (3*a*x^8)/(8*b^4) + x^10/(10*b^3) + a^7/(4*b^8*(a
+ b*x^2)^2) - (7*a^6)/(2*b^8*(a + b*x^2)) - (21*a^5*Log[a + b*x^2])/(2*b^8)

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^{15}}{\left (a+b x^2\right )^3} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {x^7}{(a+b x)^3} \, dx,x,x^2\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \left (\frac {15 a^4}{b^7}-\frac {10 a^3 x}{b^6}+\frac {6 a^2 x^2}{b^5}-\frac {3 a x^3}{b^4}+\frac {x^4}{b^3}-\frac {a^7}{b^7 (a+b x)^3}+\frac {7 a^6}{b^7 (a+b x)^2}-\frac {21 a^5}{b^7 (a+b x)}\right ) \, dx,x,x^2\right )\\ &=\frac {15 a^4 x^2}{2 b^7}-\frac {5 a^3 x^4}{2 b^6}+\frac {a^2 x^6}{b^5}-\frac {3 a x^8}{8 b^4}+\frac {x^{10}}{10 b^3}+\frac {a^7}{4 b^8 \left (a+b x^2\right )^2}-\frac {7 a^6}{2 b^8 \left (a+b x^2\right )}-\frac {21 a^5 \log \left (a+b x^2\right )}{2 b^8}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.03, size = 97, normalized size = 0.85 \[ \frac {\frac {10 a^7}{\left (a+b x^2\right )^2}-\frac {140 a^6}{a+b x^2}-420 a^5 \log \left (a+b x^2\right )+300 a^4 b x^2-100 a^3 b^2 x^4+40 a^2 b^3 x^6-15 a b^4 x^8+4 b^5 x^{10}}{40 b^8} \]

Antiderivative was successfully verified.

[In]

Integrate[x^15/(a + b*x^2)^3,x]

[Out]

(300*a^4*b*x^2 - 100*a^3*b^2*x^4 + 40*a^2*b^3*x^6 - 15*a*b^4*x^8 + 4*b^5*x^10 + (10*a^7)/(a + b*x^2)^2 - (140*
a^6)/(a + b*x^2) - 420*a^5*Log[a + b*x^2])/(40*b^8)

________________________________________________________________________________________

fricas [A]  time = 0.80, size = 137, normalized size = 1.20 \[ \frac {4 \, b^{7} x^{14} - 7 \, a b^{6} x^{12} + 14 \, a^{2} b^{5} x^{10} - 35 \, a^{3} b^{4} x^{8} + 140 \, a^{4} b^{3} x^{6} + 500 \, a^{5} b^{2} x^{4} + 160 \, a^{6} b x^{2} - 130 \, a^{7} - 420 \, {\left (a^{5} b^{2} x^{4} + 2 \, a^{6} b x^{2} + a^{7}\right )} \log \left (b x^{2} + a\right )}{40 \, {\left (b^{10} x^{4} + 2 \, a b^{9} x^{2} + a^{2} b^{8}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^15/(b*x^2+a)^3,x, algorithm="fricas")

[Out]

1/40*(4*b^7*x^14 - 7*a*b^6*x^12 + 14*a^2*b^5*x^10 - 35*a^3*b^4*x^8 + 140*a^4*b^3*x^6 + 500*a^5*b^2*x^4 + 160*a
^6*b*x^2 - 130*a^7 - 420*(a^5*b^2*x^4 + 2*a^6*b*x^2 + a^7)*log(b*x^2 + a))/(b^10*x^4 + 2*a*b^9*x^2 + a^2*b^8)

________________________________________________________________________________________

giac [A]  time = 0.63, size = 114, normalized size = 1.00 \[ -\frac {21 \, a^{5} \log \left ({\left | b x^{2} + a \right |}\right )}{2 \, b^{8}} + \frac {63 \, a^{5} b^{2} x^{4} + 112 \, a^{6} b x^{2} + 50 \, a^{7}}{4 \, {\left (b x^{2} + a\right )}^{2} b^{8}} + \frac {4 \, b^{12} x^{10} - 15 \, a b^{11} x^{8} + 40 \, a^{2} b^{10} x^{6} - 100 \, a^{3} b^{9} x^{4} + 300 \, a^{4} b^{8} x^{2}}{40 \, b^{15}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^15/(b*x^2+a)^3,x, algorithm="giac")

[Out]

-21/2*a^5*log(abs(b*x^2 + a))/b^8 + 1/4*(63*a^5*b^2*x^4 + 112*a^6*b*x^2 + 50*a^7)/((b*x^2 + a)^2*b^8) + 1/40*(
4*b^12*x^10 - 15*a*b^11*x^8 + 40*a^2*b^10*x^6 - 100*a^3*b^9*x^4 + 300*a^4*b^8*x^2)/b^15

________________________________________________________________________________________

maple [A]  time = 0.01, size = 101, normalized size = 0.89 \[ \frac {x^{10}}{10 b^{3}}-\frac {3 a \,x^{8}}{8 b^{4}}+\frac {a^{2} x^{6}}{b^{5}}-\frac {5 a^{3} x^{4}}{2 b^{6}}+\frac {a^{7}}{4 \left (b \,x^{2}+a \right )^{2} b^{8}}+\frac {15 a^{4} x^{2}}{2 b^{7}}-\frac {7 a^{6}}{2 \left (b \,x^{2}+a \right ) b^{8}}-\frac {21 a^{5} \ln \left (b \,x^{2}+a \right )}{2 b^{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^15/(b*x^2+a)^3,x)

[Out]

15/2*a^4*x^2/b^7-5/2*a^3*x^4/b^6+a^2*x^6/b^5-3/8*a*x^8/b^4+1/10*x^10/b^3+1/4*a^7/b^8/(b*x^2+a)^2-7/2*a^6/b^8/(
b*x^2+a)-21/2*a^5*ln(b*x^2+a)/b^8

________________________________________________________________________________________

maxima [A]  time = 1.36, size = 111, normalized size = 0.97 \[ -\frac {14 \, a^{6} b x^{2} + 13 \, a^{7}}{4 \, {\left (b^{10} x^{4} + 2 \, a b^{9} x^{2} + a^{2} b^{8}\right )}} - \frac {21 \, a^{5} \log \left (b x^{2} + a\right )}{2 \, b^{8}} + \frac {4 \, b^{4} x^{10} - 15 \, a b^{3} x^{8} + 40 \, a^{2} b^{2} x^{6} - 100 \, a^{3} b x^{4} + 300 \, a^{4} x^{2}}{40 \, b^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^15/(b*x^2+a)^3,x, algorithm="maxima")

[Out]

-1/4*(14*a^6*b*x^2 + 13*a^7)/(b^10*x^4 + 2*a*b^9*x^2 + a^2*b^8) - 21/2*a^5*log(b*x^2 + a)/b^8 + 1/40*(4*b^4*x^
10 - 15*a*b^3*x^8 + 40*a^2*b^2*x^6 - 100*a^3*b*x^4 + 300*a^4*x^2)/b^7

________________________________________________________________________________________

mupad [B]  time = 4.72, size = 111, normalized size = 0.97 \[ \frac {x^{10}}{10\,b^3}-\frac {\frac {13\,a^7}{4\,b}+\frac {7\,a^6\,x^2}{2}}{a^2\,b^7+2\,a\,b^8\,x^2+b^9\,x^4}-\frac {3\,a\,x^8}{8\,b^4}-\frac {21\,a^5\,\ln \left (b\,x^2+a\right )}{2\,b^8}+\frac {a^2\,x^6}{b^5}-\frac {5\,a^3\,x^4}{2\,b^6}+\frac {15\,a^4\,x^2}{2\,b^7} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^15/(a + b*x^2)^3,x)

[Out]

x^10/(10*b^3) - ((13*a^7)/(4*b) + (7*a^6*x^2)/2)/(a^2*b^7 + b^9*x^4 + 2*a*b^8*x^2) - (3*a*x^8)/(8*b^4) - (21*a
^5*log(a + b*x^2))/(2*b^8) + (a^2*x^6)/b^5 - (5*a^3*x^4)/(2*b^6) + (15*a^4*x^2)/(2*b^7)

________________________________________________________________________________________

sympy [A]  time = 0.48, size = 119, normalized size = 1.04 \[ - \frac {21 a^{5} \log {\left (a + b x^{2} \right )}}{2 b^{8}} + \frac {15 a^{4} x^{2}}{2 b^{7}} - \frac {5 a^{3} x^{4}}{2 b^{6}} + \frac {a^{2} x^{6}}{b^{5}} - \frac {3 a x^{8}}{8 b^{4}} + \frac {- 13 a^{7} - 14 a^{6} b x^{2}}{4 a^{2} b^{8} + 8 a b^{9} x^{2} + 4 b^{10} x^{4}} + \frac {x^{10}}{10 b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**15/(b*x**2+a)**3,x)

[Out]

-21*a**5*log(a + b*x**2)/(2*b**8) + 15*a**4*x**2/(2*b**7) - 5*a**3*x**4/(2*b**6) + a**2*x**6/b**5 - 3*a*x**8/(
8*b**4) + (-13*a**7 - 14*a**6*b*x**2)/(4*a**2*b**8 + 8*a*b**9*x**2 + 4*b**10*x**4) + x**10/(10*b**3)

________________________________________________________________________________________