3.5.73 \(\int \frac {x^{11}}{(a^2+2 a b x^2+b^2 x^4)^{5/2}} \, dx\)
Optimal. Leaf size=238 \[ -\frac {5 a^2}{b^6 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {5 a \left (a+b x^2\right ) \log \left (a+b x^2\right )}{2 b^6 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {x^2 \left (a+b x^2\right )}{2 b^5 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {a^5}{8 b^6 \left (a+b x^2\right )^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {5 a^4}{6 b^6 \left (a+b x^2\right )^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {5 a^3}{2 b^6 \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}} \]
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Rubi [A] time = 0.19, antiderivative size = 238, normalized size of antiderivative = 1.00,
number of steps used = 4, number of rules used = 3, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used =
{1111, 646, 43} \begin {gather*} \frac {a^5}{8 b^6 \left (a+b x^2\right )^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {5 a^4}{6 b^6 \left (a+b x^2\right )^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {5 a^3}{2 b^6 \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {5 a^2}{b^6 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {x^2 \left (a+b x^2\right )}{2 b^5 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {5 a \left (a+b x^2\right ) \log \left (a+b x^2\right )}{2 b^6 \sqrt {a^2+2 a b x^2+b^2 x^4}} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[x^11/(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2),x]
[Out]
(-5*a^2)/(b^6*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) + a^5/(8*b^6*(a + b*x^2)^3*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) - (
5*a^4)/(6*b^6*(a + b*x^2)^2*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) + (5*a^3)/(2*b^6*(a + b*x^2)*Sqrt[a^2 + 2*a*b*x^2
+ b^2*x^4]) + (x^2*(a + b*x^2))/(2*b^5*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) - (5*a*(a + b*x^2)*Log[a + b*x^2])/(2
*b^6*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])
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 646
Int[((d_.) + (e_.)*(x_))^(m_)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Dist[(a + b*x + c*x^2)^Fra
cPart[p]/(c^IntPart[p]*(b/2 + c*x)^(2*FracPart[p])), Int[(d + e*x)^m*(b/2 + c*x)^(2*p), x], x] /; FreeQ[{a, b,
c, d, e, m, p}, x] && EqQ[b^2 - 4*a*c, 0] && !IntegerQ[p] && NeQ[2*c*d - b*e, 0]
Rule 1111
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Dist[1/2, Subst[Int[x^((m - 1)/2)*(a +
b*x + c*x^2)^p, x], x, x^2], x] /; FreeQ[{a, b, c, p}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p - 1/2] && Integ
erQ[(m - 1)/2] && (GtQ[m, 0] || LtQ[0, 4*p, -m - 1])
Rubi steps
\begin {align*} \int \frac {x^{11}}{\left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {x^5}{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx,x,x^2\right )\\ &=\frac {\left (b^4 \left (a b+b^2 x^2\right )\right ) \operatorname {Subst}\left (\int \frac {x^5}{\left (a b+b^2 x\right )^5} \, dx,x,x^2\right )}{2 \sqrt {a^2+2 a b x^2+b^2 x^4}}\\ &=\frac {\left (b^4 \left (a b+b^2 x^2\right )\right ) \operatorname {Subst}\left (\int \left (\frac {1}{b^{10}}-\frac {a^5}{b^{10} (a+b x)^5}+\frac {5 a^4}{b^{10} (a+b x)^4}-\frac {10 a^3}{b^{10} (a+b x)^3}+\frac {10 a^2}{b^{10} (a+b x)^2}-\frac {5 a}{b^{10} (a+b x)}\right ) \, dx,x,x^2\right )}{2 \sqrt {a^2+2 a b x^2+b^2 x^4}}\\ &=-\frac {5 a^2}{b^6 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {a^5}{8 b^6 \left (a+b x^2\right )^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {5 a^4}{6 b^6 \left (a+b x^2\right )^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {5 a^3}{2 b^6 \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {x^2 \left (a+b x^2\right )}{2 b^5 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {5 a \left (a+b x^2\right ) \log \left (a+b x^2\right )}{2 b^6 \sqrt {a^2+2 a b x^2+b^2 x^4}}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 103, normalized size = 0.43 \begin {gather*} \frac {-77 a^5-248 a^4 b x^2-252 a^3 b^2 x^4-48 a^2 b^3 x^6+48 a b^4 x^8-60 a \left (a+b x^2\right )^4 \log \left (a+b x^2\right )+12 b^5 x^{10}}{24 b^6 \left (a+b x^2\right )^3 \sqrt {\left (a+b x^2\right )^2}} \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[x^11/(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2),x]
[Out]
(-77*a^5 - 248*a^4*b*x^2 - 252*a^3*b^2*x^4 - 48*a^2*b^3*x^6 + 48*a*b^4*x^8 + 12*b^5*x^10 - 60*a*(a + b*x^2)^4*
Log[a + b*x^2])/(24*b^6*(a + b*x^2)^3*Sqrt[(a + b*x^2)^2])
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IntegrateAlgebraic [B] time = 2.50, size = 2541, normalized size = 10.68 \begin {gather*} \text {Result too large to show} \end {gather*}
Antiderivative was successfully verified.
[In]
IntegrateAlgebraic[x^11/(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2),x]
[Out]
((256*a^8*x^2)/(3*b^4*Sqrt[b^2]) + (1264*a^7*x^4)/(3*b^3*Sqrt[b^2]) + (1312*a^6*x^6)/(b^2)^(3/2) + (7904*a^5*x
^8)/(3*b*Sqrt[b^2]) + (9344*a^4*x^10)/(3*Sqrt[b^2]) + (1792*a^3*b*x^12)/Sqrt[b^2] + 128*a^2*Sqrt[b^2]*x^14 - (
288*a*b^3*x^16)/Sqrt[b^2] - (64*b^4*x^18)/Sqrt[b^2] - (16*a^8*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/b^6 - (208*a^7*
x^2*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(3*b^5) - (352*a^6*x^4*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/b^4 - (960*a^5*x^
6*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/b^3 - (5024*a^4*x^8*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(3*b^2) - (1440*a^3*x^
10*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/b - 352*a^2*x^12*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4] + 224*a*b*x^14*Sqrt[a^2 +
2*a*b*x^2 + b^2*x^4] + 64*b^2*x^16*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4] - (320*a^5*x^8*ArcTanh[(-(Sqrt[b^2]*x^2) +
Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/a])/b^2 - (1280*a^4*x^10*ArcTanh[(-(Sqrt[b^2]*x^2) + Sqrt[a^2 + 2*a*b*x^2 +
b^2*x^4])/a])/b - 1920*a^3*x^12*ArcTanh[(-(Sqrt[b^2]*x^2) + Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/a] - 1280*a^2*b*x
^14*ArcTanh[(-(Sqrt[b^2]*x^2) + Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/a] - 320*a*b^2*x^16*ArcTanh[(-(Sqrt[b^2]*x^2)
+ Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/a] + (320*a^4*x^8*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]*ArcTanh[(-(Sqrt[b^2]*x^2
) + Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/a])/(b*Sqrt[b^2]) + (960*a^3*x^10*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]*ArcTanh
[(-(Sqrt[b^2]*x^2) + Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/a])/Sqrt[b^2] + (960*a^2*b*x^12*Sqrt[a^2 + 2*a*b*x^2 + b
^2*x^4]*ArcTanh[(-(Sqrt[b^2]*x^2) + Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/a])/Sqrt[b^2] + 320*a*Sqrt[b^2]*x^14*Sqrt
[a^2 + 2*a*b*x^2 + b^2*x^4]*ArcTanh[(-(Sqrt[b^2]*x^2) + Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/a])/((-a - Sqrt[b^2]*
x^2 + Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])^4*(a - Sqrt[b^2]*x^2 + Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])^4) + ((-16*a^9)
/(b^5*Sqrt[b^2]) - (256*a^8*x^2)/(3*b^4*Sqrt[b^2]) - (1264*a^7*x^4)/(3*b^3*Sqrt[b^2]) - (1312*a^6*x^6)/(b^2)^(
3/2) - (2256*a^5*x^8)/(b*Sqrt[b^2]) - (1920*a^4*x^10)/Sqrt[b^2] - (640*a^3*b*x^12)/Sqrt[b^2] + (256*a^7*x^2*Sq
rt[a^2 + 2*a*b*x^2 + b^2*x^4])/(3*b^5) + (336*a^6*x^4*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/b^4 + (976*a^5*x^6*Sqrt
[a^2 + 2*a*b*x^2 + b^2*x^4])/b^3 + (1280*a^4*x^8*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/b^2 + (640*a^3*x^10*Sqrt[a^2
+ 2*a*b*x^2 + b^2*x^4])/b + (160*a^5*x^8*Log[-a - Sqrt[b^2]*x^2 + Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]])/(b*Sqrt[b
^2]) + (640*a^4*x^10*Log[-a - Sqrt[b^2]*x^2 + Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]])/Sqrt[b^2] + (960*a^3*b*x^12*Lo
g[-a - Sqrt[b^2]*x^2 + Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]])/Sqrt[b^2] + 640*a^2*Sqrt[b^2]*x^14*Log[-a - Sqrt[b^2]
*x^2 + Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]] + (160*a*b^3*x^16*Log[-a - Sqrt[b^2]*x^2 + Sqrt[a^2 + 2*a*b*x^2 + b^2*
x^4]])/Sqrt[b^2] - (160*a^4*x^8*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]*Log[-a - Sqrt[b^2]*x^2 + Sqrt[a^2 + 2*a*b*x^2
+ b^2*x^4]])/b^2 - (480*a^3*x^10*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]*Log[-a - Sqrt[b^2]*x^2 + Sqrt[a^2 + 2*a*b*x^2
+ b^2*x^4]])/b - 480*a^2*x^12*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]*Log[-a - Sqrt[b^2]*x^2 + Sqrt[a^2 + 2*a*b*x^2 +
b^2*x^4]] - 160*a*b*x^14*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]*Log[-a - Sqrt[b^2]*x^2 + Sqrt[a^2 + 2*a*b*x^2 + b^2*
x^4]] + (160*a^5*x^8*Log[a - Sqrt[b^2]*x^2 + Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]])/(b*Sqrt[b^2]) + (640*a^4*x^10*L
og[a - Sqrt[b^2]*x^2 + Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]])/Sqrt[b^2] + (960*a^3*b*x^12*Log[a - Sqrt[b^2]*x^2 + S
qrt[a^2 + 2*a*b*x^2 + b^2*x^4]])/Sqrt[b^2] + 640*a^2*Sqrt[b^2]*x^14*Log[a - Sqrt[b^2]*x^2 + Sqrt[a^2 + 2*a*b*x
^2 + b^2*x^4]] + (160*a*b^3*x^16*Log[a - Sqrt[b^2]*x^2 + Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]])/Sqrt[b^2] - (160*a^
4*x^8*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]*Log[a - Sqrt[b^2]*x^2 + Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]])/b^2 - (480*a^3
*x^10*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]*Log[a - Sqrt[b^2]*x^2 + Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]])/b - 480*a^2*x^
12*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]*Log[a - Sqrt[b^2]*x^2 + Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]] - 160*a*b*x^14*Sqr
t[a^2 + 2*a*b*x^2 + b^2*x^4]*Log[a - Sqrt[b^2]*x^2 + Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]])/((-a - Sqrt[b^2]*x^2 +
Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])^4*(a - Sqrt[b^2]*x^2 + Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])^4)
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fricas [A] time = 0.63, size = 157, normalized size = 0.66 \begin {gather*} \frac {12 \, b^{5} x^{10} + 48 \, a b^{4} x^{8} - 48 \, a^{2} b^{3} x^{6} - 252 \, a^{3} b^{2} x^{4} - 248 \, a^{4} b x^{2} - 77 \, a^{5} - 60 \, {\left (a b^{4} x^{8} + 4 \, a^{2} b^{3} x^{6} + 6 \, a^{3} b^{2} x^{4} + 4 \, a^{4} b x^{2} + a^{5}\right )} \log \left (b x^{2} + a\right )}{24 \, {\left (b^{10} x^{8} + 4 \, a b^{9} x^{6} + 6 \, a^{2} b^{8} x^{4} + 4 \, a^{3} b^{7} x^{2} + a^{4} b^{6}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x^11/(b^2*x^4+2*a*b*x^2+a^2)^(5/2),x, algorithm="fricas")
[Out]
1/24*(12*b^5*x^10 + 48*a*b^4*x^8 - 48*a^2*b^3*x^6 - 252*a^3*b^2*x^4 - 248*a^4*b*x^2 - 77*a^5 - 60*(a*b^4*x^8 +
4*a^2*b^3*x^6 + 6*a^3*b^2*x^4 + 4*a^4*b*x^2 + a^5)*log(b*x^2 + a))/(b^10*x^8 + 4*a*b^9*x^6 + 6*a^2*b^8*x^4 +
4*a^3*b^7*x^2 + a^4*b^6)
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giac [A] time = 0.25, size = 105, normalized size = 0.44 \begin {gather*} \frac {x^{2}}{2 \, b^{5} \mathrm {sgn}\left (b x^{2} + a\right )} - \frac {5 \, a \log \left ({\left | b x^{2} + a \right |}\right )}{2 \, b^{6} \mathrm {sgn}\left (b x^{2} + a\right )} - \frac {120 \, a^{2} b^{3} x^{6} + 300 \, a^{3} b^{2} x^{4} + 260 \, a^{4} b x^{2} + 77 \, a^{5}}{24 \, {\left (b x^{2} + a\right )}^{4} b^{6} \mathrm {sgn}\left (b x^{2} + a\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x^11/(b^2*x^4+2*a*b*x^2+a^2)^(5/2),x, algorithm="giac")
[Out]
1/2*x^2/(b^5*sgn(b*x^2 + a)) - 5/2*a*log(abs(b*x^2 + a))/(b^6*sgn(b*x^2 + a)) - 1/24*(120*a^2*b^3*x^6 + 300*a^
3*b^2*x^4 + 260*a^4*b*x^2 + 77*a^5)/((b*x^2 + a)^4*b^6*sgn(b*x^2 + a))
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maple [A] time = 0.02, size = 163, normalized size = 0.68 \begin {gather*} -\frac {\left (-12 b^{5} x^{10}+60 a \,b^{4} x^{8} \ln \left (b \,x^{2}+a \right )-48 a \,b^{4} x^{8}+240 a^{2} b^{3} x^{6} \ln \left (b \,x^{2}+a \right )+48 a^{2} b^{3} x^{6}+360 a^{3} b^{2} x^{4} \ln \left (b \,x^{2}+a \right )+252 a^{3} b^{2} x^{4}+240 a^{4} b \,x^{2} \ln \left (b \,x^{2}+a \right )+248 a^{4} b \,x^{2}+60 a^{5} \ln \left (b \,x^{2}+a \right )+77 a^{5}\right ) \left (b \,x^{2}+a \right )}{24 \left (\left (b \,x^{2}+a \right )^{2}\right )^{\frac {5}{2}} b^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(x^11/(b^2*x^4+2*a*b*x^2+a^2)^(5/2),x)
[Out]
-1/24*(-12*b^5*x^10+60*ln(b*x^2+a)*x^8*a*b^4-48*a*b^4*x^8+240*ln(b*x^2+a)*x^6*a^2*b^3+48*a^2*b^3*x^6+360*ln(b*
x^2+a)*x^4*a^3*b^2+252*a^3*b^2*x^4+240*ln(b*x^2+a)*x^2*a^4*b+248*a^4*b*x^2+60*ln(b*x^2+a)*a^5+77*a^5)*(b*x^2+a
)/b^6/((b*x^2+a)^2)^(5/2)
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maxima [A] time = 1.38, size = 110, normalized size = 0.46 \begin {gather*} -\frac {120 \, a^{2} b^{3} x^{6} + 300 \, a^{3} b^{2} x^{4} + 260 \, a^{4} b x^{2} + 77 \, a^{5}}{24 \, {\left (b^{10} x^{8} + 4 \, a b^{9} x^{6} + 6 \, a^{2} b^{8} x^{4} + 4 \, a^{3} b^{7} x^{2} + a^{4} b^{6}\right )}} + \frac {x^{2}}{2 \, b^{5}} - \frac {5 \, a \log \left (b x^{2} + a\right )}{2 \, b^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x^11/(b^2*x^4+2*a*b*x^2+a^2)^(5/2),x, algorithm="maxima")
[Out]
-1/24*(120*a^2*b^3*x^6 + 300*a^3*b^2*x^4 + 260*a^4*b*x^2 + 77*a^5)/(b^10*x^8 + 4*a*b^9*x^6 + 6*a^2*b^8*x^4 + 4
*a^3*b^7*x^2 + a^4*b^6) + 1/2*x^2/b^5 - 5/2*a*log(b*x^2 + a)/b^6
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {x^{11}}{{\left (a^2+2\,a\,b\,x^2+b^2\,x^4\right )}^{5/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(x^11/(a^2 + b^2*x^4 + 2*a*b*x^2)^(5/2),x)
[Out]
int(x^11/(a^2 + b^2*x^4 + 2*a*b*x^2)^(5/2), x)
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{11}}{\left (\left (a + b x^{2}\right )^{2}\right )^{\frac {5}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x**11/(b**2*x**4+2*a*b*x**2+a**2)**(5/2),x)
[Out]
Integral(x**11/((a + b*x**2)**2)**(5/2), x)
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