3.658 \(\int \frac {1}{x^4 (a^2+2 a b x^2+b^2 x^4)^{5/2}} \, dx\)

Optimal. Leaf size=291 \[ \frac {11}{48 a^2 x^3 \left (a+b x^2\right )^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {1}{8 a x^3 \left (a+b x^2\right )^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {1155 b^{3/2} \left (a+b x^2\right ) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{128 a^{13/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {1155 b \left (a+b x^2\right )}{128 a^6 x \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {385 \left (a+b x^2\right )}{128 a^5 x^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {231}{128 a^4 x^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {33}{64 a^3 x^3 \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}} \]

[Out]

231/128/a^4/x^3/((b*x^2+a)^2)^(1/2)+1/8/a/x^3/(b*x^2+a)^3/((b*x^2+a)^2)^(1/2)+11/48/a^2/x^3/(b*x^2+a)^2/((b*x^
2+a)^2)^(1/2)+33/64/a^3/x^3/(b*x^2+a)/((b*x^2+a)^2)^(1/2)-385/128*(b*x^2+a)/a^5/x^3/((b*x^2+a)^2)^(1/2)+1155/1
28*b*(b*x^2+a)/a^6/x/((b*x^2+a)^2)^(1/2)+1155/128*b^(3/2)*(b*x^2+a)*arctan(x*b^(1/2)/a^(1/2))/a^(13/2)/((b*x^2
+a)^2)^(1/2)

________________________________________________________________________________________

Rubi [A]  time = 0.12, antiderivative size = 291, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 4, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {1112, 290, 325, 205} \[ \frac {1155 b \left (a+b x^2\right )}{128 a^6 x \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {385 \left (a+b x^2\right )}{128 a^5 x^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {33}{64 a^3 x^3 \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {11}{48 a^2 x^3 \left (a+b x^2\right )^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {231}{128 a^4 x^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {1}{8 a x^3 \left (a+b x^2\right )^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {1155 b^{3/2} \left (a+b x^2\right ) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{128 a^{13/2} \sqrt {a^2+2 a b x^2+b^2 x^4}} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^4*(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2)),x]

[Out]

231/(128*a^4*x^3*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) + 1/(8*a*x^3*(a + b*x^2)^3*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])
+ 11/(48*a^2*x^3*(a + b*x^2)^2*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) + 33/(64*a^3*x^3*(a + b*x^2)*Sqrt[a^2 + 2*a*b*
x^2 + b^2*x^4]) - (385*(a + b*x^2))/(128*a^5*x^3*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) + (1155*b*(a + b*x^2))/(128*
a^6*x*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) + (1155*b^(3/2)*(a + b*x^2)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(128*a^(13/2)*
Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])

Rule 205

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]*ArcTan[x/Rt[a/b, 2]])/a, x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rule 290

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> -Simp[((c*x)^(m + 1)*(a + b*x^n)^(p + 1))/(
a*c*n*(p + 1)), x] + Dist[(m + n*(p + 1) + 1)/(a*n*(p + 1)), Int[(c*x)^m*(a + b*x^n)^(p + 1), x], x] /; FreeQ[
{a, b, c, m}, x] && IGtQ[n, 0] && LtQ[p, -1] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 325

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[((c*x)^(m + 1)*(a + b*x^n)^(p + 1))/(a*
c*(m + 1)), x] - Dist[(b*(m + n*(p + 1) + 1))/(a*c^n*(m + 1)), Int[(c*x)^(m + n)*(a + b*x^n)^p, x], x] /; Free
Q[{a, b, c, p}, x] && IGtQ[n, 0] && LtQ[m, -1] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 1112

Int[((d_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Dist[(a + b*x^2 + c*x^4)^FracPa
rt[p]/(c^IntPart[p]*(b/2 + c*x^2)^(2*FracPart[p])), Int[(d*x)^m*(b/2 + c*x^2)^(2*p), x], x] /; FreeQ[{a, b, c,
d, m, p}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p - 1/2]

Rubi steps

\begin {align*} \int \frac {1}{x^4 \left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}} \, dx &=\frac {\left (b^4 \left (a b+b^2 x^2\right )\right ) \int \frac {1}{x^4 \left (a b+b^2 x^2\right )^5} \, dx}{\sqrt {a^2+2 a b x^2+b^2 x^4}}\\ &=\frac {1}{8 a x^3 \left (a+b x^2\right )^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (11 b^3 \left (a b+b^2 x^2\right )\right ) \int \frac {1}{x^4 \left (a b+b^2 x^2\right )^4} \, dx}{8 a \sqrt {a^2+2 a b x^2+b^2 x^4}}\\ &=\frac {1}{8 a x^3 \left (a+b x^2\right )^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {11}{48 a^2 x^3 \left (a+b x^2\right )^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (33 b^2 \left (a b+b^2 x^2\right )\right ) \int \frac {1}{x^4 \left (a b+b^2 x^2\right )^3} \, dx}{16 a^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}\\ &=\frac {1}{8 a x^3 \left (a+b x^2\right )^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {11}{48 a^2 x^3 \left (a+b x^2\right )^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {33}{64 a^3 x^3 \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (231 b \left (a b+b^2 x^2\right )\right ) \int \frac {1}{x^4 \left (a b+b^2 x^2\right )^2} \, dx}{64 a^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}\\ &=\frac {231}{128 a^4 x^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {1}{8 a x^3 \left (a+b x^2\right )^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {11}{48 a^2 x^3 \left (a+b x^2\right )^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {33}{64 a^3 x^3 \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (1155 \left (a b+b^2 x^2\right )\right ) \int \frac {1}{x^4 \left (a b+b^2 x^2\right )} \, dx}{128 a^4 \sqrt {a^2+2 a b x^2+b^2 x^4}}\\ &=\frac {231}{128 a^4 x^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {1}{8 a x^3 \left (a+b x^2\right )^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {11}{48 a^2 x^3 \left (a+b x^2\right )^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {33}{64 a^3 x^3 \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {385 \left (a+b x^2\right )}{128 a^5 x^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {\left (1155 b \left (a b+b^2 x^2\right )\right ) \int \frac {1}{x^2 \left (a b+b^2 x^2\right )} \, dx}{128 a^5 \sqrt {a^2+2 a b x^2+b^2 x^4}}\\ &=\frac {231}{128 a^4 x^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {1}{8 a x^3 \left (a+b x^2\right )^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {11}{48 a^2 x^3 \left (a+b x^2\right )^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {33}{64 a^3 x^3 \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {385 \left (a+b x^2\right )}{128 a^5 x^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {1155 b \left (a+b x^2\right )}{128 a^6 x \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (1155 b^2 \left (a b+b^2 x^2\right )\right ) \int \frac {1}{a b+b^2 x^2} \, dx}{128 a^6 \sqrt {a^2+2 a b x^2+b^2 x^4}}\\ &=\frac {231}{128 a^4 x^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {1}{8 a x^3 \left (a+b x^2\right )^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {11}{48 a^2 x^3 \left (a+b x^2\right )^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {33}{64 a^3 x^3 \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {385 \left (a+b x^2\right )}{128 a^5 x^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {1155 b \left (a+b x^2\right )}{128 a^6 x \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {1155 b^{3/2} \left (a+b x^2\right ) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{128 a^{13/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.05, size = 127, normalized size = 0.44 \[ \frac {\sqrt {a} \left (-128 a^5+1408 a^4 b x^2+9207 a^3 b^2 x^4+16863 a^2 b^3 x^6+12705 a b^4 x^8+3465 b^5 x^{10}\right )+3465 b^{3/2} x^3 \left (a+b x^2\right )^4 \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{384 a^{13/2} x^3 \left (a+b x^2\right )^3 \sqrt {\left (a+b x^2\right )^2}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^4*(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2)),x]

[Out]

(Sqrt[a]*(-128*a^5 + 1408*a^4*b*x^2 + 9207*a^3*b^2*x^4 + 16863*a^2*b^3*x^6 + 12705*a*b^4*x^8 + 3465*b^5*x^10)
+ 3465*b^(3/2)*x^3*(a + b*x^2)^4*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(384*a^(13/2)*x^3*(a + b*x^2)^3*Sqrt[(a + b*x^2)
^2])

________________________________________________________________________________________

fricas [A]  time = 1.04, size = 370, normalized size = 1.27 \[ \left [\frac {6930 \, b^{5} x^{10} + 25410 \, a b^{4} x^{8} + 33726 \, a^{2} b^{3} x^{6} + 18414 \, a^{3} b^{2} x^{4} + 2816 \, a^{4} b x^{2} - 256 \, a^{5} + 3465 \, {\left (b^{5} x^{11} + 4 \, a b^{4} x^{9} + 6 \, a^{2} b^{3} x^{7} + 4 \, a^{3} b^{2} x^{5} + a^{4} b x^{3}\right )} \sqrt {-\frac {b}{a}} \log \left (\frac {b x^{2} + 2 \, a x \sqrt {-\frac {b}{a}} - a}{b x^{2} + a}\right )}{768 \, {\left (a^{6} b^{4} x^{11} + 4 \, a^{7} b^{3} x^{9} + 6 \, a^{8} b^{2} x^{7} + 4 \, a^{9} b x^{5} + a^{10} x^{3}\right )}}, \frac {3465 \, b^{5} x^{10} + 12705 \, a b^{4} x^{8} + 16863 \, a^{2} b^{3} x^{6} + 9207 \, a^{3} b^{2} x^{4} + 1408 \, a^{4} b x^{2} - 128 \, a^{5} + 3465 \, {\left (b^{5} x^{11} + 4 \, a b^{4} x^{9} + 6 \, a^{2} b^{3} x^{7} + 4 \, a^{3} b^{2} x^{5} + a^{4} b x^{3}\right )} \sqrt {\frac {b}{a}} \arctan \left (x \sqrt {\frac {b}{a}}\right )}{384 \, {\left (a^{6} b^{4} x^{11} + 4 \, a^{7} b^{3} x^{9} + 6 \, a^{8} b^{2} x^{7} + 4 \, a^{9} b x^{5} + a^{10} x^{3}\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^4/(b^2*x^4+2*a*b*x^2+a^2)^(5/2),x, algorithm="fricas")

[Out]

[1/768*(6930*b^5*x^10 + 25410*a*b^4*x^8 + 33726*a^2*b^3*x^6 + 18414*a^3*b^2*x^4 + 2816*a^4*b*x^2 - 256*a^5 + 3
465*(b^5*x^11 + 4*a*b^4*x^9 + 6*a^2*b^3*x^7 + 4*a^3*b^2*x^5 + a^4*b*x^3)*sqrt(-b/a)*log((b*x^2 + 2*a*x*sqrt(-b
/a) - a)/(b*x^2 + a)))/(a^6*b^4*x^11 + 4*a^7*b^3*x^9 + 6*a^8*b^2*x^7 + 4*a^9*b*x^5 + a^10*x^3), 1/384*(3465*b^
5*x^10 + 12705*a*b^4*x^8 + 16863*a^2*b^3*x^6 + 9207*a^3*b^2*x^4 + 1408*a^4*b*x^2 - 128*a^5 + 3465*(b^5*x^11 +
4*a*b^4*x^9 + 6*a^2*b^3*x^7 + 4*a^3*b^2*x^5 + a^4*b*x^3)*sqrt(b/a)*arctan(x*sqrt(b/a)))/(a^6*b^4*x^11 + 4*a^7*
b^3*x^9 + 6*a^8*b^2*x^7 + 4*a^9*b*x^5 + a^10*x^3)]

________________________________________________________________________________________

giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \mathit {sage}_{0} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^4/(b^2*x^4+2*a*b*x^2+a^2)^(5/2),x, algorithm="giac")

[Out]

sage0*x

________________________________________________________________________________________

maple [A]  time = 0.02, size = 211, normalized size = 0.73 \[ \frac {\left (3465 b^{6} x^{11} \arctan \left (\frac {b x}{\sqrt {a b}}\right )+13860 a \,b^{5} x^{9} \arctan \left (\frac {b x}{\sqrt {a b}}\right )+3465 \sqrt {a b}\, b^{5} x^{10}+20790 a^{2} b^{4} x^{7} \arctan \left (\frac {b x}{\sqrt {a b}}\right )+12705 \sqrt {a b}\, a \,b^{4} x^{8}+13860 a^{3} b^{3} x^{5} \arctan \left (\frac {b x}{\sqrt {a b}}\right )+16863 \sqrt {a b}\, a^{2} b^{3} x^{6}+3465 a^{4} b^{2} x^{3} \arctan \left (\frac {b x}{\sqrt {a b}}\right )+9207 \sqrt {a b}\, a^{3} b^{2} x^{4}+1408 \sqrt {a b}\, a^{4} b \,x^{2}-128 \sqrt {a b}\, a^{5}\right ) \left (b \,x^{2}+a \right )}{384 \sqrt {a b}\, \left (\left (b \,x^{2}+a \right )^{2}\right )^{\frac {5}{2}} a^{6} x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^4/(b^2*x^4+2*a*b*x^2+a^2)^(5/2),x)

[Out]

1/384*(3465*arctan(1/(a*b)^(1/2)*b*x)*x^11*b^6+3465*(a*b)^(1/2)*x^10*b^5+13860*arctan(1/(a*b)^(1/2)*b*x)*x^9*a
*b^5+12705*(a*b)^(1/2)*x^8*a*b^4+20790*arctan(1/(a*b)^(1/2)*b*x)*x^7*a^2*b^4+16863*(a*b)^(1/2)*x^6*a^2*b^3+138
60*arctan(1/(a*b)^(1/2)*b*x)*x^5*a^3*b^3+9207*(a*b)^(1/2)*x^4*a^3*b^2+3465*arctan(1/(a*b)^(1/2)*b*x)*x^3*a^4*b
^2+1408*(a*b)^(1/2)*x^2*a^4*b-128*(a*b)^(1/2)*a^5)*(b*x^2+a)/(a*b)^(1/2)/x^3/a^6/((b*x^2+a)^2)^(5/2)

________________________________________________________________________________________

maxima [A]  time = 3.13, size = 130, normalized size = 0.45 \[ \frac {3465 \, b^{5} x^{10} + 12705 \, a b^{4} x^{8} + 16863 \, a^{2} b^{3} x^{6} + 9207 \, a^{3} b^{2} x^{4} + 1408 \, a^{4} b x^{2} - 128 \, a^{5}}{384 \, {\left (a^{6} b^{4} x^{11} + 4 \, a^{7} b^{3} x^{9} + 6 \, a^{8} b^{2} x^{7} + 4 \, a^{9} b x^{5} + a^{10} x^{3}\right )}} + \frac {1155 \, b^{2} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{128 \, \sqrt {a b} a^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^4/(b^2*x^4+2*a*b*x^2+a^2)^(5/2),x, algorithm="maxima")

[Out]

1/384*(3465*b^5*x^10 + 12705*a*b^4*x^8 + 16863*a^2*b^3*x^6 + 9207*a^3*b^2*x^4 + 1408*a^4*b*x^2 - 128*a^5)/(a^6
*b^4*x^11 + 4*a^7*b^3*x^9 + 6*a^8*b^2*x^7 + 4*a^9*b*x^5 + a^10*x^3) + 1155/128*b^2*arctan(b*x/sqrt(a*b))/(sqrt
(a*b)*a^6)

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {1}{x^4\,{\left (a^2+2\,a\,b\,x^2+b^2\,x^4\right )}^{5/2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^4*(a^2 + b^2*x^4 + 2*a*b*x^2)^(5/2)),x)

[Out]

int(1/(x^4*(a^2 + b^2*x^4 + 2*a*b*x^2)^(5/2)), x)

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{x^{4} \left (\left (a + b x^{2}\right )^{2}\right )^{\frac {5}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**4/(b**2*x**4+2*a*b*x**2+a**2)**(5/2),x)

[Out]

Integral(1/(x**4*((a + b*x**2)**2)**(5/2)), x)

________________________________________________________________________________________