\(\int \frac {1}{x^2 (a-b x^2)^5} \, dx\) [262]

Optimal result

Integrand size = 14, antiderivative size = 112 \[ \int \frac {1}{x^2 \left (a-b x^2\right )^5} \, dx=-\frac {1}{a^5 x}+\frac {b x}{8 a^2 \left (a-b x^2\right )^4}+\frac {5 b x}{16 a^3 \left (a-b x^2\right )^3}+\frac {41 b x}{64 a^4 \left (a-b x^2\right )^2}+\frac {187 b x}{128 a^5 \left (a-b x^2\right )}+\frac {315 \sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{128 a^{11/2}} \] Output:

-1/a^5/x+1/8*b*x/a^2/(-b*x^2+a)^4+5/16*b*x/a^3/(-b*x^2+a)^3+41/64*b*x/a^4/ 
(-b*x^2+a)^2+187/128*b*x/a^5/(-b*x^2+a)+315/128*b^(1/2)*arctanh(b^(1/2)*x/ 
a^(1/2))/a^(11/2)
 

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.82 \[ \int \frac {1}{x^2 \left (a-b x^2\right )^5} \, dx=\frac {\frac {\sqrt {a} \left (-128 a^4+837 a^3 b x^2-1533 a^2 b^2 x^4+1155 a b^3 x^6-315 b^4 x^8\right )}{x \left (a-b x^2\right )^4}+315 \sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{128 a^{11/2}} \] Input:

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

Output:

((Sqrt[a]*(-128*a^4 + 837*a^3*b*x^2 - 1533*a^2*b^2*x^4 + 1155*a*b^3*x^6 - 
315*b^4*x^8))/(x*(a - b*x^2)^4) + 315*Sqrt[b]*ArcTanh[(Sqrt[b]*x)/Sqrt[a]] 
)/(128*a^(11/2))
 

Rubi [A] (verified)

Time = 0.22 (sec) , antiderivative size = 145, normalized size of antiderivative = 1.29, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {253, 253, 253, 253, 264, 221}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {1}{x^2 \left (a-b x^2\right )^5} \, dx\)

\(\Big \downarrow \) 253

\(\displaystyle \frac {9 \int \frac {1}{x^2 \left (a-b x^2\right )^4}dx}{8 a}+\frac {1}{8 a x \left (a-b x^2\right )^4}\)

\(\Big \downarrow \) 253

\(\displaystyle \frac {9 \left (\frac {7 \int \frac {1}{x^2 \left (a-b x^2\right )^3}dx}{6 a}+\frac {1}{6 a x \left (a-b x^2\right )^3}\right )}{8 a}+\frac {1}{8 a x \left (a-b x^2\right )^4}\)

\(\Big \downarrow \) 253

\(\displaystyle \frac {9 \left (\frac {7 \left (\frac {5 \int \frac {1}{x^2 \left (a-b x^2\right )^2}dx}{4 a}+\frac {1}{4 a x \left (a-b x^2\right )^2}\right )}{6 a}+\frac {1}{6 a x \left (a-b x^2\right )^3}\right )}{8 a}+\frac {1}{8 a x \left (a-b x^2\right )^4}\)

\(\Big \downarrow \) 253

\(\displaystyle \frac {9 \left (\frac {7 \left (\frac {5 \left (\frac {3 \int \frac {1}{x^2 \left (a-b x^2\right )}dx}{2 a}+\frac {1}{2 a x \left (a-b x^2\right )}\right )}{4 a}+\frac {1}{4 a x \left (a-b x^2\right )^2}\right )}{6 a}+\frac {1}{6 a x \left (a-b x^2\right )^3}\right )}{8 a}+\frac {1}{8 a x \left (a-b x^2\right )^4}\)

\(\Big \downarrow \) 264

\(\displaystyle \frac {9 \left (\frac {7 \left (\frac {5 \left (\frac {3 \left (\frac {b \int \frac {1}{a-b x^2}dx}{a}-\frac {1}{a x}\right )}{2 a}+\frac {1}{2 a x \left (a-b x^2\right )}\right )}{4 a}+\frac {1}{4 a x \left (a-b x^2\right )^2}\right )}{6 a}+\frac {1}{6 a x \left (a-b x^2\right )^3}\right )}{8 a}+\frac {1}{8 a x \left (a-b x^2\right )^4}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {9 \left (\frac {7 \left (\frac {5 \left (\frac {3 \left (\frac {\sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{a^{3/2}}-\frac {1}{a x}\right )}{2 a}+\frac {1}{2 a x \left (a-b x^2\right )}\right )}{4 a}+\frac {1}{4 a x \left (a-b x^2\right )^2}\right )}{6 a}+\frac {1}{6 a x \left (a-b x^2\right )^3}\right )}{8 a}+\frac {1}{8 a x \left (a-b x^2\right )^4}\)

Input:

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

Output:

1/(8*a*x*(a - b*x^2)^4) + (9*(1/(6*a*x*(a - b*x^2)^3) + (7*(1/(4*a*x*(a - 
b*x^2)^2) + (5*(1/(2*a*x*(a - b*x^2)) + (3*(-(1/(a*x)) + (Sqrt[b]*ArcTanh[ 
(Sqrt[b]*x)/Sqrt[a]])/a^(3/2)))/(2*a)))/(4*a)))/(6*a)))/(8*a)
 

Defintions of rubi rules used

rule 221

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

rule 253

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-(c*x 
)^(m + 1))*((a + b*x^2)^(p + 1)/(2*a*c*(p + 1))), x] + Simp[(m + 2*p + 3)/( 
2*a*(p + 1))   Int[(c*x)^m*(a + b*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, m 
}, x] && LtQ[p, -1] && IntBinomialQ[a, b, c, 2, m, p, x]
 

rule 264

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(c*x)^( 
m + 1)*((a + b*x^2)^(p + 1)/(a*c*(m + 1))), x] - Simp[b*((m + 2*p + 3)/(a*c 
^2*(m + 1)))   Int[(c*x)^(m + 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, p 
}, x] && LtQ[m, -1] && IntBinomialQ[a, b, c, 2, m, p, x]
 

Maple [A] (verified)

Time = 0.34 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.68

method	result	size
default	\(\frac {b \left (\frac {-\frac {187}{128} b^{3} x^{7}+\frac {643}{128} a \,b^{2} x^{5}-\frac {765}{128} a^{2} b \,x^{3}+\frac {325}{128} a^{3} x}{\left (-b \,x^{2}+a \right )^{4}}+\frac {315 \,\operatorname {arctanh}\left (\frac {b x}{\sqrt {a b}}\right )}{128 \sqrt {a b}}\right )}{a^{5}}-\frac {1}{a^{5} x}\)	\(76\)
risch	\(\frac {-\frac {315 b^{4} x^{8}}{128 a^{5}}+\frac {1155 b^{3} x^{6}}{128 a^{4}}-\frac {1533 b^{2} x^{4}}{128 a^{3}}+\frac {837 b \,x^{2}}{128 a^{2}}-\frac {1}{a}}{x \left (-b \,x^{2}+a \right )^{4}}+\frac {315 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (a^{11} \textit {\_Z}^{2}-b \right )}{\sum }\textit {\_R} \ln \left (\left (3 \textit {\_R}^{2} a^{11}-2 b \right ) x +a^{6} \textit {\_R} \right )\right )}{256}\)	\(104\)

Input:

int(1/x^2/(-b*x^2+a)^5,x,method=_RETURNVERBOSE)
 

Output:

1/a^5*b*((-187/128*b^3*x^7+643/128*a*b^2*x^5-765/128*a^2*b*x^3+325/128*a^3 
*x)/(-b*x^2+a)^4+315/128/(a*b)^(1/2)*arctanh(b*x/(a*b)^(1/2)))-1/a^5/x
 

Fricas [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 334, normalized size of antiderivative = 2.98 \[ \int \frac {1}{x^2 \left (a-b x^2\right )^5} \, dx=\left [-\frac {630 \, b^{4} x^{8} - 2310 \, a b^{3} x^{6} + 3066 \, a^{2} b^{2} x^{4} - 1674 \, a^{3} b x^{2} + 256 \, a^{4} - 315 \, {\left (b^{4} x^{9} - 4 \, a b^{3} x^{7} + 6 \, a^{2} b^{2} x^{5} - 4 \, a^{3} b x^{3} + a^{4} x\right )} \sqrt {\frac {b}{a}} \log \left (\frac {b x^{2} + 2 \, a x \sqrt {\frac {b}{a}} + a}{b x^{2} - a}\right )}{256 \, {\left (a^{5} b^{4} x^{9} - 4 \, a^{6} b^{3} x^{7} + 6 \, a^{7} b^{2} x^{5} - 4 \, a^{8} b x^{3} + a^{9} x\right )}}, -\frac {315 \, b^{4} x^{8} - 1155 \, a b^{3} x^{6} + 1533 \, a^{2} b^{2} x^{4} - 837 \, a^{3} b x^{2} + 128 \, a^{4} + 315 \, {\left (b^{4} x^{9} - 4 \, a b^{3} x^{7} + 6 \, a^{2} b^{2} x^{5} - 4 \, a^{3} b x^{3} + a^{4} x\right )} \sqrt {-\frac {b}{a}} \arctan \left (x \sqrt {-\frac {b}{a}}\right )}{128 \, {\left (a^{5} b^{4} x^{9} - 4 \, a^{6} b^{3} x^{7} + 6 \, a^{7} b^{2} x^{5} - 4 \, a^{8} b x^{3} + a^{9} x\right )}}\right ] \] Input:

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

Output:

[-1/256*(630*b^4*x^8 - 2310*a*b^3*x^6 + 3066*a^2*b^2*x^4 - 1674*a^3*b*x^2 
+ 256*a^4 - 315*(b^4*x^9 - 4*a*b^3*x^7 + 6*a^2*b^2*x^5 - 4*a^3*b*x^3 + a^4 
*x)*sqrt(b/a)*log((b*x^2 + 2*a*x*sqrt(b/a) + a)/(b*x^2 - a)))/(a^5*b^4*x^9 
 - 4*a^6*b^3*x^7 + 6*a^7*b^2*x^5 - 4*a^8*b*x^3 + a^9*x), -1/128*(315*b^4*x 
^8 - 1155*a*b^3*x^6 + 1533*a^2*b^2*x^4 - 837*a^3*b*x^2 + 128*a^4 + 315*(b^ 
4*x^9 - 4*a*b^3*x^7 + 6*a^2*b^2*x^5 - 4*a^3*b*x^3 + a^4*x)*sqrt(-b/a)*arct 
an(x*sqrt(-b/a)))/(a^5*b^4*x^9 - 4*a^6*b^3*x^7 + 6*a^7*b^2*x^5 - 4*a^8*b*x 
^3 + a^9*x)]
 

Sympy [A] (verification not implemented)

Time = 0.42 (sec) , antiderivative size = 155, normalized size of antiderivative = 1.38 \[ \int \frac {1}{x^2 \left (a-b x^2\right )^5} \, dx=- \frac {315 \sqrt {\frac {b}{a^{11}}} \log {\left (- \frac {a^{6} \sqrt {\frac {b}{a^{11}}}}{b} + x \right )}}{256} + \frac {315 \sqrt {\frac {b}{a^{11}}} \log {\left (\frac {a^{6} \sqrt {\frac {b}{a^{11}}}}{b} + x \right )}}{256} - \frac {128 a^{4} - 837 a^{3} b x^{2} + 1533 a^{2} b^{2} x^{4} - 1155 a b^{3} x^{6} + 315 b^{4} x^{8}}{128 a^{9} x - 512 a^{8} b x^{3} + 768 a^{7} b^{2} x^{5} - 512 a^{6} b^{3} x^{7} + 128 a^{5} b^{4} x^{9}} \] Input:

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

Output:

-315*sqrt(b/a**11)*log(-a**6*sqrt(b/a**11)/b + x)/256 + 315*sqrt(b/a**11)* 
log(a**6*sqrt(b/a**11)/b + x)/256 - (128*a**4 - 837*a**3*b*x**2 + 1533*a** 
2*b**2*x**4 - 1155*a*b**3*x**6 + 315*b**4*x**8)/(128*a**9*x - 512*a**8*b*x 
**3 + 768*a**7*b**2*x**5 - 512*a**6*b**3*x**7 + 128*a**5*b**4*x**9)
 

Maxima [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.16 \[ \int \frac {1}{x^2 \left (a-b x^2\right )^5} \, dx=-\frac {315 \, b^{4} x^{8} - 1155 \, a b^{3} x^{6} + 1533 \, a^{2} b^{2} x^{4} - 837 \, a^{3} b x^{2} + 128 \, a^{4}}{128 \, {\left (a^{5} b^{4} x^{9} - 4 \, a^{6} b^{3} x^{7} + 6 \, a^{7} b^{2} x^{5} - 4 \, a^{8} b x^{3} + a^{9} x\right )}} - \frac {315 \, b \log \left (\frac {b x - \sqrt {a b}}{b x + \sqrt {a b}}\right )}{256 \, \sqrt {a b} a^{5}} \] Input:

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

Output:

-1/128*(315*b^4*x^8 - 1155*a*b^3*x^6 + 1533*a^2*b^2*x^4 - 837*a^3*b*x^2 + 
128*a^4)/(a^5*b^4*x^9 - 4*a^6*b^3*x^7 + 6*a^7*b^2*x^5 - 4*a^8*b*x^3 + a^9* 
x) - 315/256*b*log((b*x - sqrt(a*b))/(b*x + sqrt(a*b)))/(sqrt(a*b)*a^5)
 

Giac [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.74 \[ \int \frac {1}{x^2 \left (a-b x^2\right )^5} \, dx=-\frac {315 \, b \arctan \left (\frac {b x}{\sqrt {-a b}}\right )}{128 \, \sqrt {-a b} a^{5}} - \frac {1}{a^{5} x} - \frac {187 \, b^{4} x^{7} - 643 \, a b^{3} x^{5} + 765 \, a^{2} b^{2} x^{3} - 325 \, a^{3} b x}{128 \, {\left (b x^{2} - a\right )}^{4} a^{5}} \] Input:

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

Output:

-315/128*b*arctan(b*x/sqrt(-a*b))/(sqrt(-a*b)*a^5) - 1/(a^5*x) - 1/128*(18 
7*b^4*x^7 - 643*a*b^3*x^5 + 765*a^2*b^2*x^3 - 325*a^3*b*x)/((b*x^2 - a)^4* 
a^5)
 

Mupad [B] (verification not implemented)

Time = 0.25 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.98 \[ \int \frac {1}{x^2 \left (a-b x^2\right )^5} \, dx=\frac {315\,\sqrt {b}\,\mathrm {atanh}\left (\frac {\sqrt {b}\,x}{\sqrt {a}}\right )}{128\,a^{11/2}}-\frac {\frac {1}{a}-\frac {837\,b\,x^2}{128\,a^2}+\frac {1533\,b^2\,x^4}{128\,a^3}-\frac {1155\,b^3\,x^6}{128\,a^4}+\frac {315\,b^4\,x^8}{128\,a^5}}{a^4\,x-4\,a^3\,b\,x^3+6\,a^2\,b^2\,x^5-4\,a\,b^3\,x^7+b^4\,x^9} \] Input:

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

Output:

(315*b^(1/2)*atanh((b^(1/2)*x)/a^(1/2)))/(128*a^(11/2)) - (1/a - (837*b*x^ 
2)/(128*a^2) + (1533*b^2*x^4)/(128*a^3) - (1155*b^3*x^6)/(128*a^4) + (315* 
b^4*x^8)/(128*a^5))/(a^4*x + b^4*x^9 - 4*a^3*b*x^3 - 4*a*b^3*x^7 + 6*a^2*b 
^2*x^5)
 

Reduce [B] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 337, normalized size of antiderivative = 3.01 \[ \int \frac {1}{x^2 \left (a-b x^2\right )^5} \, dx=\frac {315 \sqrt {b}\, \sqrt {a}\, \mathrm {log}\left (-\sqrt {b}\, \sqrt {a}-b x \right ) a^{4} x -1260 \sqrt {b}\, \sqrt {a}\, \mathrm {log}\left (-\sqrt {b}\, \sqrt {a}-b x \right ) a^{3} b \,x^{3}+1890 \sqrt {b}\, \sqrt {a}\, \mathrm {log}\left (-\sqrt {b}\, \sqrt {a}-b x \right ) a^{2} b^{2} x^{5}-1260 \sqrt {b}\, \sqrt {a}\, \mathrm {log}\left (-\sqrt {b}\, \sqrt {a}-b x \right ) a \,b^{3} x^{7}+315 \sqrt {b}\, \sqrt {a}\, \mathrm {log}\left (-\sqrt {b}\, \sqrt {a}-b x \right ) b^{4} x^{9}-315 \sqrt {b}\, \sqrt {a}\, \mathrm {log}\left (\sqrt {b}\, \sqrt {a}-b x \right ) a^{4} x +1260 \sqrt {b}\, \sqrt {a}\, \mathrm {log}\left (\sqrt {b}\, \sqrt {a}-b x \right ) a^{3} b \,x^{3}-1890 \sqrt {b}\, \sqrt {a}\, \mathrm {log}\left (\sqrt {b}\, \sqrt {a}-b x \right ) a^{2} b^{2} x^{5}+1260 \sqrt {b}\, \sqrt {a}\, \mathrm {log}\left (\sqrt {b}\, \sqrt {a}-b x \right ) a \,b^{3} x^{7}-315 \sqrt {b}\, \sqrt {a}\, \mathrm {log}\left (\sqrt {b}\, \sqrt {a}-b x \right ) b^{4} x^{9}-256 a^{5}+1674 a^{4} b \,x^{2}-3066 a^{3} b^{2} x^{4}+2310 a^{2} b^{3} x^{6}-630 a \,b^{4} x^{8}}{256 a^{6} x \left (b^{4} x^{8}-4 a \,b^{3} x^{6}+6 a^{2} b^{2} x^{4}-4 a^{3} b \,x^{2}+a^{4}\right )} \] Input:

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

Output:

(315*sqrt(b)*sqrt(a)*log( - sqrt(b)*sqrt(a) - b*x)*a**4*x - 1260*sqrt(b)*s 
qrt(a)*log( - sqrt(b)*sqrt(a) - b*x)*a**3*b*x**3 + 1890*sqrt(b)*sqrt(a)*lo 
g( - sqrt(b)*sqrt(a) - b*x)*a**2*b**2*x**5 - 1260*sqrt(b)*sqrt(a)*log( - s 
qrt(b)*sqrt(a) - b*x)*a*b**3*x**7 + 315*sqrt(b)*sqrt(a)*log( - sqrt(b)*sqr 
t(a) - b*x)*b**4*x**9 - 315*sqrt(b)*sqrt(a)*log(sqrt(b)*sqrt(a) - b*x)*a** 
4*x + 1260*sqrt(b)*sqrt(a)*log(sqrt(b)*sqrt(a) - b*x)*a**3*b*x**3 - 1890*s 
qrt(b)*sqrt(a)*log(sqrt(b)*sqrt(a) - b*x)*a**2*b**2*x**5 + 1260*sqrt(b)*sq 
rt(a)*log(sqrt(b)*sqrt(a) - b*x)*a*b**3*x**7 - 315*sqrt(b)*sqrt(a)*log(sqr 
t(b)*sqrt(a) - b*x)*b**4*x**9 - 256*a**5 + 1674*a**4*b*x**2 - 3066*a**3*b* 
*2*x**4 + 2310*a**2*b**3*x**6 - 630*a*b**4*x**8)/(256*a**6*x*(a**4 - 4*a** 
3*b*x**2 + 6*a**2*b**2*x**4 - 4*a*b**3*x**6 + b**4*x**8))