Integrand size = 10, antiderivative size = 100 \[ \int \frac {1}{\left (a-b x^2\right )^5} \, dx=\frac {x}{8 a \left (a-b x^2\right )^4}+\frac {7 x}{48 a^2 \left (a-b x^2\right )^3}+\frac {35 x}{192 a^3 \left (a-b x^2\right )^2}+\frac {35 x}{128 a^4 \left (a-b x^2\right )}+\frac {35 \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{128 a^{9/2} \sqrt {b}} \] Output:
1/8*x/a/(-b*x^2+a)^4+7/48*x/a^2/(-b*x^2+a)^3+35/192*x/a^3/(-b*x^2+a)^2+35/ 128*x/a^4/(-b*x^2+a)+35/128*arctanh(b^(1/2)*x/a^(1/2))/a^(9/2)/b^(1/2)
Time = 0.03 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.79 \[ \int \frac {1}{\left (a-b x^2\right )^5} \, dx=\frac {\frac {\sqrt {a} x \left (279 a^3-511 a^2 b x^2+385 a b^2 x^4-105 b^3 x^6\right )}{\left (a-b x^2\right )^4}+\frac {105 \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {b}}}{384 a^{9/2}} \] Input:
Integrate[(a - b*x^2)^(-5),x]
Output:
((Sqrt[a]*x*(279*a^3 - 511*a^2*b*x^2 + 385*a*b^2*x^4 - 105*b^3*x^6))/(a - b*x^2)^4 + (105*ArcTanh[(Sqrt[b]*x)/Sqrt[a]])/Sqrt[b])/(384*a^(9/2))
Time = 0.19 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.24, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {215, 215, 215, 215, 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.
\(\displaystyle \int \frac {1}{\left (a-b x^2\right )^5} \, dx\) |
\(\Big \downarrow \) 215 |
\(\displaystyle \frac {7 \int \frac {1}{\left (a-b x^2\right )^4}dx}{8 a}+\frac {x}{8 a \left (a-b x^2\right )^4}\) |
\(\Big \downarrow \) 215 |
\(\displaystyle \frac {7 \left (\frac {5 \int \frac {1}{\left (a-b x^2\right )^3}dx}{6 a}+\frac {x}{6 a \left (a-b x^2\right )^3}\right )}{8 a}+\frac {x}{8 a \left (a-b x^2\right )^4}\) |
\(\Big \downarrow \) 215 |
\(\displaystyle \frac {7 \left (\frac {5 \left (\frac {3 \int \frac {1}{\left (a-b x^2\right )^2}dx}{4 a}+\frac {x}{4 a \left (a-b x^2\right )^2}\right )}{6 a}+\frac {x}{6 a \left (a-b x^2\right )^3}\right )}{8 a}+\frac {x}{8 a \left (a-b x^2\right )^4}\) |
\(\Big \downarrow \) 215 |
\(\displaystyle \frac {7 \left (\frac {5 \left (\frac {3 \left (\frac {\int \frac {1}{a-b x^2}dx}{2 a}+\frac {x}{2 a \left (a-b x^2\right )}\right )}{4 a}+\frac {x}{4 a \left (a-b x^2\right )^2}\right )}{6 a}+\frac {x}{6 a \left (a-b x^2\right )^3}\right )}{8 a}+\frac {x}{8 a \left (a-b x^2\right )^4}\) |
\(\Big \downarrow \) 221 |
\(\displaystyle \frac {7 \left (\frac {5 \left (\frac {3 \left (\frac {\text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 a^{3/2} \sqrt {b}}+\frac {x}{2 a \left (a-b x^2\right )}\right )}{4 a}+\frac {x}{4 a \left (a-b x^2\right )^2}\right )}{6 a}+\frac {x}{6 a \left (a-b x^2\right )^3}\right )}{8 a}+\frac {x}{8 a \left (a-b x^2\right )^4}\) |
Input:
Int[(a - b*x^2)^(-5),x]
Output:
x/(8*a*(a - b*x^2)^4) + (7*(x/(6*a*(a - b*x^2)^3) + (5*(x/(4*a*(a - b*x^2) ^2) + (3*(x/(2*a*(a - b*x^2)) + ArcTanh[(Sqrt[b]*x)/Sqrt[a]]/(2*a^(3/2)*Sq rt[b])))/(4*a)))/(6*a)))/(8*a)
Int[((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-x)*((a + b*x^2)^(p + 1) /(2*a*(p + 1))), x] + Simp[(2*p + 3)/(2*a*(p + 1)) Int[(a + b*x^2)^(p + 1 ), x], x] /; FreeQ[{a, b}, x] && LtQ[p, -1] && (IntegerQ[4*p] || IntegerQ[6 *p])
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]
Time = 0.33 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.92
method | result | size |
risch | \(\frac {-\frac {35 b^{3} x^{7}}{128 a^{4}}+\frac {385 b^{2} x^{5}}{384 a^{3}}-\frac {511 b \,x^{3}}{384 a^{2}}+\frac {93 x}{128 a}}{\left (-b \,x^{2}+a \right )^{4}}+\frac {35 \ln \left (b x +\sqrt {a b}\right )}{256 \sqrt {a b}\, a^{4}}-\frac {35 \ln \left (-b x +\sqrt {a b}\right )}{256 \sqrt {a b}\, a^{4}}\) | \(92\) |
default | \(\frac {x}{8 a \left (-b \,x^{2}+a \right )^{4}}+\frac {\frac {7 x}{48 a \left (-b \,x^{2}+a \right )^{3}}+\frac {7 \left (\frac {5 x}{24 a \left (-b \,x^{2}+a \right )^{2}}+\frac {5 \left (\frac {3 x}{8 a \left (-b \,x^{2}+a \right )}+\frac {3 \,\operatorname {arctanh}\left (\frac {b x}{\sqrt {a b}}\right )}{8 a \sqrt {a b}}\right )}{6 a}\right )}{8 a}}{a}\) | \(103\) |
Input:
int(1/(-b*x^2+a)^5,x,method=_RETURNVERBOSE)
Output:
(-35/128*b^3/a^4*x^7+385/384*b^2/a^3*x^5-511/384*b/a^2*x^3+93/128*x/a)/(-b *x^2+a)^4+35/256/(a*b)^(1/2)/a^4*ln(b*x+(a*b)^(1/2))-35/256/(a*b)^(1/2)/a^ 4*ln(-b*x+(a*b)^(1/2))
Time = 0.07 (sec) , antiderivative size = 320, normalized size of antiderivative = 3.20 \[ \int \frac {1}{\left (a-b x^2\right )^5} \, dx=\left [-\frac {210 \, a b^{4} x^{7} - 770 \, a^{2} b^{3} x^{5} + 1022 \, a^{3} b^{2} x^{3} - 558 \, a^{4} b x - 105 \, {\left (b^{4} x^{8} - 4 \, a b^{3} x^{6} + 6 \, a^{2} b^{2} x^{4} - 4 \, a^{3} b x^{2} + a^{4}\right )} \sqrt {a b} \log \left (\frac {b x^{2} + 2 \, \sqrt {a b} x + a}{b x^{2} - a}\right )}{768 \, {\left (a^{5} b^{5} x^{8} - 4 \, a^{6} b^{4} x^{6} + 6 \, a^{7} b^{3} x^{4} - 4 \, a^{8} b^{2} x^{2} + a^{9} b\right )}}, -\frac {105 \, a b^{4} x^{7} - 385 \, a^{2} b^{3} x^{5} + 511 \, a^{3} b^{2} x^{3} - 279 \, a^{4} b x + 105 \, {\left (b^{4} x^{8} - 4 \, a b^{3} x^{6} + 6 \, a^{2} b^{2} x^{4} - 4 \, a^{3} b x^{2} + a^{4}\right )} \sqrt {-a b} \arctan \left (\frac {\sqrt {-a b} x}{a}\right )}{384 \, {\left (a^{5} b^{5} x^{8} - 4 \, a^{6} b^{4} x^{6} + 6 \, a^{7} b^{3} x^{4} - 4 \, a^{8} b^{2} x^{2} + a^{9} b\right )}}\right ] \] Input:
integrate(1/(-b*x^2+a)^5,x, algorithm="fricas")
Output:
[-1/768*(210*a*b^4*x^7 - 770*a^2*b^3*x^5 + 1022*a^3*b^2*x^3 - 558*a^4*b*x - 105*(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)*sqrt(a*b )*log((b*x^2 + 2*sqrt(a*b)*x + a)/(b*x^2 - a)))/(a^5*b^5*x^8 - 4*a^6*b^4*x ^6 + 6*a^7*b^3*x^4 - 4*a^8*b^2*x^2 + a^9*b), -1/384*(105*a*b^4*x^7 - 385*a ^2*b^3*x^5 + 511*a^3*b^2*x^3 - 279*a^4*b*x + 105*(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)*sqrt(-a*b)*arctan(sqrt(-a*b)*x/a))/(a^5 *b^5*x^8 - 4*a^6*b^4*x^6 + 6*a^7*b^3*x^4 - 4*a^8*b^2*x^2 + a^9*b)]
Time = 0.28 (sec) , antiderivative size = 146, normalized size of antiderivative = 1.46 \[ \int \frac {1}{\left (a-b x^2\right )^5} \, dx=- \frac {35 \sqrt {\frac {1}{a^{9} b}} \log {\left (- a^{5} \sqrt {\frac {1}{a^{9} b}} + x \right )}}{256} + \frac {35 \sqrt {\frac {1}{a^{9} b}} \log {\left (a^{5} \sqrt {\frac {1}{a^{9} b}} + x \right )}}{256} - \frac {- 279 a^{3} x + 511 a^{2} b x^{3} - 385 a b^{2} x^{5} + 105 b^{3} x^{7}}{384 a^{8} - 1536 a^{7} b x^{2} + 2304 a^{6} b^{2} x^{4} - 1536 a^{5} b^{3} x^{6} + 384 a^{4} b^{4} x^{8}} \] Input:
integrate(1/(-b*x**2+a)**5,x)
Output:
-35*sqrt(1/(a**9*b))*log(-a**5*sqrt(1/(a**9*b)) + x)/256 + 35*sqrt(1/(a**9 *b))*log(a**5*sqrt(1/(a**9*b)) + x)/256 - (-279*a**3*x + 511*a**2*b*x**3 - 385*a*b**2*x**5 + 105*b**3*x**7)/(384*a**8 - 1536*a**7*b*x**2 + 2304*a**6 *b**2*x**4 - 1536*a**5*b**3*x**6 + 384*a**4*b**4*x**8)
Time = 0.10 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.17 \[ \int \frac {1}{\left (a-b x^2\right )^5} \, dx=-\frac {105 \, b^{3} x^{7} - 385 \, a b^{2} x^{5} + 511 \, a^{2} b x^{3} - 279 \, a^{3} x}{384 \, {\left (a^{4} b^{4} x^{8} - 4 \, a^{5} b^{3} x^{6} + 6 \, a^{6} b^{2} x^{4} - 4 \, a^{7} b x^{2} + a^{8}\right )}} - \frac {35 \, \log \left (\frac {b x - \sqrt {a b}}{b x + \sqrt {a b}}\right )}{256 \, \sqrt {a b} a^{4}} \] Input:
integrate(1/(-b*x^2+a)^5,x, algorithm="maxima")
Output:
-1/384*(105*b^3*x^7 - 385*a*b^2*x^5 + 511*a^2*b*x^3 - 279*a^3*x)/(a^4*b^4* x^8 - 4*a^5*b^3*x^6 + 6*a^6*b^2*x^4 - 4*a^7*b*x^2 + a^8) - 35/256*log((b*x - sqrt(a*b))/(b*x + sqrt(a*b)))/(sqrt(a*b)*a^4)
Time = 0.12 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.71 \[ \int \frac {1}{\left (a-b x^2\right )^5} \, dx=-\frac {35 \, \arctan \left (\frac {b x}{\sqrt {-a b}}\right )}{128 \, \sqrt {-a b} a^{4}} - \frac {105 \, b^{3} x^{7} - 385 \, a b^{2} x^{5} + 511 \, a^{2} b x^{3} - 279 \, a^{3} x}{384 \, {\left (b x^{2} - a\right )}^{4} a^{4}} \] Input:
integrate(1/(-b*x^2+a)^5,x, algorithm="giac")
Output:
-35/128*arctan(b*x/sqrt(-a*b))/(sqrt(-a*b)*a^4) - 1/384*(105*b^3*x^7 - 385 *a*b^2*x^5 + 511*a^2*b*x^3 - 279*a^3*x)/((b*x^2 - a)^4*a^4)
Time = 0.20 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.99 \[ \int \frac {1}{\left (a-b x^2\right )^5} \, dx=\frac {\frac {93\,x}{128\,a}-\frac {511\,b\,x^3}{384\,a^2}+\frac {385\,b^2\,x^5}{384\,a^3}-\frac {35\,b^3\,x^7}{128\,a^4}}{a^4-4\,a^3\,b\,x^2+6\,a^2\,b^2\,x^4-4\,a\,b^3\,x^6+b^4\,x^8}+\frac {35\,\mathrm {atanh}\left (\frac {\sqrt {b}\,x}{\sqrt {a}}\right )}{128\,a^{9/2}\,\sqrt {b}} \] Input:
int(1/(a - b*x^2)^5,x)
Output:
((93*x)/(128*a) - (511*b*x^3)/(384*a^2) + (385*b^2*x^5)/(384*a^3) - (35*b^ 3*x^7)/(128*a^4))/(a^4 + b^4*x^8 - 4*a^3*b*x^2 - 4*a*b^3*x^6 + 6*a^2*b^2*x ^4) + (35*atanh((b^(1/2)*x)/a^(1/2)))/(128*a^(9/2)*b^(1/2))
Time = 0.21 (sec) , antiderivative size = 328, normalized size of antiderivative = 3.28 \[ \int \frac {1}{\left (a-b x^2\right )^5} \, dx=\frac {105 \sqrt {b}\, \sqrt {a}\, \mathrm {log}\left (-\sqrt {b}\, \sqrt {a}-b x \right ) a^{4}-420 \sqrt {b}\, \sqrt {a}\, \mathrm {log}\left (-\sqrt {b}\, \sqrt {a}-b x \right ) a^{3} b \,x^{2}+630 \sqrt {b}\, \sqrt {a}\, \mathrm {log}\left (-\sqrt {b}\, \sqrt {a}-b x \right ) a^{2} b^{2} x^{4}-420 \sqrt {b}\, \sqrt {a}\, \mathrm {log}\left (-\sqrt {b}\, \sqrt {a}-b x \right ) a \,b^{3} x^{6}+105 \sqrt {b}\, \sqrt {a}\, \mathrm {log}\left (-\sqrt {b}\, \sqrt {a}-b x \right ) b^{4} x^{8}-105 \sqrt {b}\, \sqrt {a}\, \mathrm {log}\left (\sqrt {b}\, \sqrt {a}-b x \right ) a^{4}+420 \sqrt {b}\, \sqrt {a}\, \mathrm {log}\left (\sqrt {b}\, \sqrt {a}-b x \right ) a^{3} b \,x^{2}-630 \sqrt {b}\, \sqrt {a}\, \mathrm {log}\left (\sqrt {b}\, \sqrt {a}-b x \right ) a^{2} b^{2} x^{4}+420 \sqrt {b}\, \sqrt {a}\, \mathrm {log}\left (\sqrt {b}\, \sqrt {a}-b x \right ) a \,b^{3} x^{6}-105 \sqrt {b}\, \sqrt {a}\, \mathrm {log}\left (\sqrt {b}\, \sqrt {a}-b x \right ) b^{4} x^{8}+558 a^{4} b x -1022 a^{3} b^{2} x^{3}+770 a^{2} b^{3} x^{5}-210 a \,b^{4} x^{7}}{768 a^{5} b \left (b^{4} x^{8}-4 a \,b^{3} x^{6}+6 a^{2} b^{2} x^{4}-4 a^{3} b \,x^{2}+a^{4}\right )} \] Input:
int(1/(-b*x^2+a)^5,x)
Output:
(105*sqrt(b)*sqrt(a)*log( - sqrt(b)*sqrt(a) - b*x)*a**4 - 420*sqrt(b)*sqrt (a)*log( - sqrt(b)*sqrt(a) - b*x)*a**3*b*x**2 + 630*sqrt(b)*sqrt(a)*log( - sqrt(b)*sqrt(a) - b*x)*a**2*b**2*x**4 - 420*sqrt(b)*sqrt(a)*log( - sqrt(b )*sqrt(a) - b*x)*a*b**3*x**6 + 105*sqrt(b)*sqrt(a)*log( - sqrt(b)*sqrt(a) - b*x)*b**4*x**8 - 105*sqrt(b)*sqrt(a)*log(sqrt(b)*sqrt(a) - b*x)*a**4 + 4 20*sqrt(b)*sqrt(a)*log(sqrt(b)*sqrt(a) - b*x)*a**3*b*x**2 - 630*sqrt(b)*sq rt(a)*log(sqrt(b)*sqrt(a) - b*x)*a**2*b**2*x**4 + 420*sqrt(b)*sqrt(a)*log( sqrt(b)*sqrt(a) - b*x)*a*b**3*x**6 - 105*sqrt(b)*sqrt(a)*log(sqrt(b)*sqrt( a) - b*x)*b**4*x**8 + 558*a**4*b*x - 1022*a**3*b**2*x**3 + 770*a**2*b**3*x **5 - 210*a*b**4*x**7)/(768*a**5*b*(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))