Integrand size = 15, antiderivative size = 71 \[ \int \frac {\left (a+\frac {b}{x^4}\right )^{3/2}}{x^3} \, dx=-\frac {3 a \sqrt {a+\frac {b}{x^4}}}{16 x^2}-\frac {\left (a+\frac {b}{x^4}\right )^{3/2}}{8 x^2}-\frac {3 a^2 \text {arctanh}\left (\frac {\sqrt {b}}{\sqrt {a+\frac {b}{x^4}} x^2}\right )}{16 \sqrt {b}} \] Output:
-3/16*a*(a+b/x^4)^(1/2)/x^2-1/8*(a+b/x^4)^(3/2)/x^2-3/16*a^2*arctanh(b^(1/ 2)/(a+b/x^4)^(1/2)/x^2)/b^(1/2)
Time = 0.13 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.99 \[ \int \frac {\left (a+\frac {b}{x^4}\right )^{3/2}}{x^3} \, dx=\frac {\sqrt {a+\frac {b}{x^4}} \left (-2 b-5 a x^4-\frac {3 a^2 x^8 \text {arctanh}\left (\frac {\sqrt {b+a x^4}}{\sqrt {b}}\right )}{\sqrt {b} \sqrt {b+a x^4}}\right )}{16 x^6} \] Input:
Integrate[(a + b/x^4)^(3/2)/x^3,x]
Output:
(Sqrt[a + b/x^4]*(-2*b - 5*a*x^4 - (3*a^2*x^8*ArcTanh[Sqrt[b + a*x^4]/Sqrt [b]])/(Sqrt[b]*Sqrt[b + a*x^4])))/(16*x^6)
Time = 0.34 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.10, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {858, 807, 211, 211, 224, 219}
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.
\(\displaystyle \int \frac {\left (a+\frac {b}{x^4}\right )^{3/2}}{x^3} \, dx\) |
\(\Big \downarrow \) 858 |
\(\displaystyle -\int \frac {\left (a+\frac {b}{x^4}\right )^{3/2}}{x}d\frac {1}{x}\) |
\(\Big \downarrow \) 807 |
\(\displaystyle -\frac {1}{2} \int \left (a+\frac {b}{x^2}\right )^{3/2}d\frac {1}{x^2}\) |
\(\Big \downarrow \) 211 |
\(\displaystyle \frac {1}{2} \left (-\frac {3}{4} a \int \sqrt {a+\frac {b}{x^2}}d\frac {1}{x^2}-\frac {\left (a+\frac {b}{x^2}\right )^{3/2}}{4 x^2}\right )\) |
\(\Big \downarrow \) 211 |
\(\displaystyle \frac {1}{2} \left (-\frac {3}{4} a \left (\frac {1}{2} a \int \frac {1}{\sqrt {a+\frac {b}{x^2}}}d\frac {1}{x^2}+\frac {\sqrt {a+\frac {b}{x^2}}}{2 x^2}\right )-\frac {\left (a+\frac {b}{x^2}\right )^{3/2}}{4 x^2}\right )\) |
\(\Big \downarrow \) 224 |
\(\displaystyle \frac {1}{2} \left (-\frac {3}{4} a \left (\frac {1}{2} a \int \frac {1}{1-\frac {b}{\sqrt {a+\frac {b}{x^2}} x^2}}d\frac {1}{\sqrt {a+\frac {b}{x^2}} x^2}+\frac {\sqrt {a+\frac {b}{x^2}}}{2 x^2}\right )-\frac {\left (a+\frac {b}{x^2}\right )^{3/2}}{4 x^2}\right )\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {1}{2} \left (-\frac {3}{4} a \left (\frac {a \text {arctanh}\left (\frac {\sqrt {b}}{x^2 \sqrt {a+\frac {b}{x^2}}}\right )}{2 \sqrt {b}}+\frac {\sqrt {a+\frac {b}{x^2}}}{2 x^2}\right )-\frac {\left (a+\frac {b}{x^2}\right )^{3/2}}{4 x^2}\right )\) |
Input:
Int[(a + b/x^4)^(3/2)/x^3,x]
Output:
(-1/4*(a + b/x^2)^(3/2)/x^2 - (3*a*(Sqrt[a + b/x^2]/(2*x^2) + (a*ArcTanh[S qrt[b]/(Sqrt[a + b/x^2]*x^2)])/(2*Sqrt[b])))/4)/2
Int[((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[x*((a + b*x^2)^p/(2*p + 1 )), x] + Simp[2*a*(p/(2*p + 1)) Int[(a + b*x^2)^(p - 1), x], x] /; FreeQ[ {a, b}, x] && GtQ[p, 0] && (IntegerQ[4*p] || IntegerQ[6*p])
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a, b}, x] && !GtQ[a, 0]
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = GCD[m + 1, n]}, Simp[1/k Subst[Int[x^((m + 1)/k - 1)*(a + b*x^(n/k))^p, x], x, x^k], x] /; k != 1] /; FreeQ[{a, b, p}, x] && IGtQ[n, 0] && IntegerQ[m]
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> -Subst[Int[(a + b/x^n)^p/x^(m + 2), x], x, 1/x] /; FreeQ[{a, b, p}, x] && ILtQ[n, 0] && Int egerQ[m]
Time = 0.15 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.21
method | result | size |
risch | \(-\frac {\left (5 a \,x^{4}+2 b \right ) \sqrt {\frac {a \,x^{4}+b}{x^{4}}}}{16 x^{6}}-\frac {3 a^{2} \ln \left (\frac {2 b +2 \sqrt {b}\, \sqrt {a \,x^{4}+b}}{x^{2}}\right ) \sqrt {\frac {a \,x^{4}+b}{x^{4}}}\, x^{2}}{16 \sqrt {b}\, \sqrt {a \,x^{4}+b}}\) | \(86\) |
default | \(-\frac {\left (\frac {a \,x^{4}+b}{x^{4}}\right )^{\frac {3}{2}} \left (3 a^{2} \ln \left (\frac {2 b +2 \sqrt {b}\, \sqrt {a \,x^{4}+b}}{x^{2}}\right ) x^{8}+5 a \sqrt {a \,x^{4}+b}\, x^{4} \sqrt {b}+2 b^{\frac {3}{2}} \sqrt {a \,x^{4}+b}\right )}{16 x^{2} \left (a \,x^{4}+b \right )^{\frac {3}{2}} \sqrt {b}}\) | \(93\) |
Input:
int((a+b/x^4)^(3/2)/x^3,x,method=_RETURNVERBOSE)
Output:
-1/16*(5*a*x^4+2*b)/x^6*((a*x^4+b)/x^4)^(1/2)-3/16*a^2/b^(1/2)*ln((2*b+2*b ^(1/2)*(a*x^4+b)^(1/2))/x^2)*((a*x^4+b)/x^4)^(1/2)*x^2/(a*x^4+b)^(1/2)
Time = 0.08 (sec) , antiderivative size = 166, normalized size of antiderivative = 2.34 \[ \int \frac {\left (a+\frac {b}{x^4}\right )^{3/2}}{x^3} \, dx=\left [\frac {3 \, a^{2} \sqrt {b} x^{6} \log \left (\frac {a x^{4} - 2 \, \sqrt {b} x^{2} \sqrt {\frac {a x^{4} + b}{x^{4}}} + 2 \, b}{x^{4}}\right ) - 2 \, {\left (5 \, a b x^{4} + 2 \, b^{2}\right )} \sqrt {\frac {a x^{4} + b}{x^{4}}}}{32 \, b x^{6}}, \frac {3 \, a^{2} \sqrt {-b} x^{6} \arctan \left (\frac {\sqrt {-b} x^{2} \sqrt {\frac {a x^{4} + b}{x^{4}}}}{a x^{4} + b}\right ) - {\left (5 \, a b x^{4} + 2 \, b^{2}\right )} \sqrt {\frac {a x^{4} + b}{x^{4}}}}{16 \, b x^{6}}\right ] \] Input:
integrate((a+b/x^4)^(3/2)/x^3,x, algorithm="fricas")
Output:
[1/32*(3*a^2*sqrt(b)*x^6*log((a*x^4 - 2*sqrt(b)*x^2*sqrt((a*x^4 + b)/x^4) + 2*b)/x^4) - 2*(5*a*b*x^4 + 2*b^2)*sqrt((a*x^4 + b)/x^4))/(b*x^6), 1/16*( 3*a^2*sqrt(-b)*x^6*arctan(sqrt(-b)*x^2*sqrt((a*x^4 + b)/x^4)/(a*x^4 + b)) - (5*a*b*x^4 + 2*b^2)*sqrt((a*x^4 + b)/x^4))/(b*x^6)]
Time = 1.82 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.06 \[ \int \frac {\left (a+\frac {b}{x^4}\right )^{3/2}}{x^3} \, dx=- \frac {5 a^{\frac {3}{2}} \sqrt {1 + \frac {b}{a x^{4}}}}{16 x^{2}} - \frac {\sqrt {a} b \sqrt {1 + \frac {b}{a x^{4}}}}{8 x^{6}} - \frac {3 a^{2} \operatorname {asinh}{\left (\frac {\sqrt {b}}{\sqrt {a} x^{2}} \right )}}{16 \sqrt {b}} \] Input:
integrate((a+b/x**4)**(3/2)/x**3,x)
Output:
-5*a**(3/2)*sqrt(1 + b/(a*x**4))/(16*x**2) - sqrt(a)*b*sqrt(1 + b/(a*x**4) )/(8*x**6) - 3*a**2*asinh(sqrt(b)/(sqrt(a)*x**2))/(16*sqrt(b))
Leaf count of result is larger than twice the leaf count of optimal. 119 vs. \(2 (55) = 110\).
Time = 0.11 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.68 \[ \int \frac {\left (a+\frac {b}{x^4}\right )^{3/2}}{x^3} \, dx=\frac {3 \, a^{2} \log \left (\frac {\sqrt {a + \frac {b}{x^{4}}} x^{2} - \sqrt {b}}{\sqrt {a + \frac {b}{x^{4}}} x^{2} + \sqrt {b}}\right )}{32 \, \sqrt {b}} - \frac {5 \, {\left (a + \frac {b}{x^{4}}\right )}^{\frac {3}{2}} a^{2} x^{6} - 3 \, \sqrt {a + \frac {b}{x^{4}}} a^{2} b x^{2}}{16 \, {\left ({\left (a + \frac {b}{x^{4}}\right )}^{2} x^{8} - 2 \, {\left (a + \frac {b}{x^{4}}\right )} b x^{4} + b^{2}\right )}} \] Input:
integrate((a+b/x^4)^(3/2)/x^3,x, algorithm="maxima")
Output:
3/32*a^2*log((sqrt(a + b/x^4)*x^2 - sqrt(b))/(sqrt(a + b/x^4)*x^2 + sqrt(b )))/sqrt(b) - 1/16*(5*(a + b/x^4)^(3/2)*a^2*x^6 - 3*sqrt(a + b/x^4)*a^2*b* x^2)/((a + b/x^4)^2*x^8 - 2*(a + b/x^4)*b*x^4 + b^2)
Time = 0.13 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.99 \[ \int \frac {\left (a+\frac {b}{x^4}\right )^{3/2}}{x^3} \, dx=\frac {\frac {3 \, a^{3} \arctan \left (\frac {\sqrt {a x^{4} + b}}{\sqrt {-b}}\right )}{\sqrt {-b}} - \frac {5 \, {\left (a x^{4} + b\right )}^{\frac {3}{2}} a^{3} - 3 \, \sqrt {a x^{4} + b} a^{3} b}{a^{2} x^{8}}}{16 \, a} \] Input:
integrate((a+b/x^4)^(3/2)/x^3,x, algorithm="giac")
Output:
1/16*(3*a^3*arctan(sqrt(a*x^4 + b)/sqrt(-b))/sqrt(-b) - (5*(a*x^4 + b)^(3/ 2)*a^3 - 3*sqrt(a*x^4 + b)*a^3*b)/(a^2*x^8))/a
Timed out. \[ \int \frac {\left (a+\frac {b}{x^4}\right )^{3/2}}{x^3} \, dx=\int \frac {{\left (a+\frac {b}{x^4}\right )}^{3/2}}{x^3} \,d x \] Input:
int((a + b/x^4)^(3/2)/x^3,x)
Output:
int((a + b/x^4)^(3/2)/x^3, x)
Time = 0.26 (sec) , antiderivative size = 554, normalized size of antiderivative = 7.80 \[ \int \frac {\left (a+\frac {b}{x^4}\right )^{3/2}}{x^3} \, dx=\frac {24 \sqrt {a}\, \sqrt {a \,x^{4}+b}\, \mathrm {log}\left (\frac {\sqrt {a \,x^{4}+b}+\sqrt {a}\, x^{2}-\sqrt {b}}{\sqrt {b}}\right ) a^{3} x^{14}+12 \sqrt {a}\, \sqrt {a \,x^{4}+b}\, \mathrm {log}\left (\frac {\sqrt {a \,x^{4}+b}+\sqrt {a}\, x^{2}-\sqrt {b}}{\sqrt {b}}\right ) a^{2} b \,x^{10}-24 \sqrt {a}\, \sqrt {a \,x^{4}+b}\, \mathrm {log}\left (\frac {\sqrt {a \,x^{4}+b}+\sqrt {a}\, x^{2}+\sqrt {b}}{\sqrt {b}}\right ) a^{3} x^{14}-12 \sqrt {a}\, \sqrt {a \,x^{4}+b}\, \mathrm {log}\left (\frac {\sqrt {a \,x^{4}+b}+\sqrt {a}\, x^{2}+\sqrt {b}}{\sqrt {b}}\right ) a^{2} b \,x^{10}-40 \sqrt {b}\, \sqrt {a \,x^{4}+b}\, a^{3} x^{12}-56 \sqrt {b}\, \sqrt {a \,x^{4}+b}\, a^{2} b \,x^{8}-21 \sqrt {b}\, \sqrt {a \,x^{4}+b}\, a \,b^{2} x^{4}-2 \sqrt {b}\, \sqrt {a \,x^{4}+b}\, b^{3}-40 \sqrt {b}\, \sqrt {a}\, a^{3} x^{14}-76 \sqrt {b}\, \sqrt {a}\, a^{2} b \,x^{10}-44 \sqrt {b}\, \sqrt {a}\, a \,b^{2} x^{6}-8 \sqrt {b}\, \sqrt {a}\, b^{3} x^{2}+24 \,\mathrm {log}\left (\frac {\sqrt {a \,x^{4}+b}+\sqrt {a}\, x^{2}-\sqrt {b}}{\sqrt {b}}\right ) a^{4} x^{16}+24 \,\mathrm {log}\left (\frac {\sqrt {a \,x^{4}+b}+\sqrt {a}\, x^{2}-\sqrt {b}}{\sqrt {b}}\right ) a^{3} b \,x^{12}+3 \,\mathrm {log}\left (\frac {\sqrt {a \,x^{4}+b}+\sqrt {a}\, x^{2}-\sqrt {b}}{\sqrt {b}}\right ) a^{2} b^{2} x^{8}-24 \,\mathrm {log}\left (\frac {\sqrt {a \,x^{4}+b}+\sqrt {a}\, x^{2}+\sqrt {b}}{\sqrt {b}}\right ) a^{4} x^{16}-24 \,\mathrm {log}\left (\frac {\sqrt {a \,x^{4}+b}+\sqrt {a}\, x^{2}+\sqrt {b}}{\sqrt {b}}\right ) a^{3} b \,x^{12}-3 \,\mathrm {log}\left (\frac {\sqrt {a \,x^{4}+b}+\sqrt {a}\, x^{2}+\sqrt {b}}{\sqrt {b}}\right ) a^{2} b^{2} x^{8}}{16 \sqrt {b}\, x^{8} \left (8 \sqrt {a}\, \sqrt {a \,x^{4}+b}\, a \,x^{6}+4 \sqrt {a}\, \sqrt {a \,x^{4}+b}\, b \,x^{2}+8 a^{2} x^{8}+8 a b \,x^{4}+b^{2}\right )} \] Input:
int((a+b/x^4)^(3/2)/x^3,x)
Output:
(24*sqrt(a)*sqrt(a*x**4 + b)*log((sqrt(a*x**4 + b) + sqrt(a)*x**2 - sqrt(b ))/sqrt(b))*a**3*x**14 + 12*sqrt(a)*sqrt(a*x**4 + b)*log((sqrt(a*x**4 + b) + sqrt(a)*x**2 - sqrt(b))/sqrt(b))*a**2*b*x**10 - 24*sqrt(a)*sqrt(a*x**4 + b)*log((sqrt(a*x**4 + b) + sqrt(a)*x**2 + sqrt(b))/sqrt(b))*a**3*x**14 - 12*sqrt(a)*sqrt(a*x**4 + b)*log((sqrt(a*x**4 + b) + sqrt(a)*x**2 + sqrt(b ))/sqrt(b))*a**2*b*x**10 - 40*sqrt(b)*sqrt(a*x**4 + b)*a**3*x**12 - 56*sqr t(b)*sqrt(a*x**4 + b)*a**2*b*x**8 - 21*sqrt(b)*sqrt(a*x**4 + b)*a*b**2*x** 4 - 2*sqrt(b)*sqrt(a*x**4 + b)*b**3 - 40*sqrt(b)*sqrt(a)*a**3*x**14 - 76*s qrt(b)*sqrt(a)*a**2*b*x**10 - 44*sqrt(b)*sqrt(a)*a*b**2*x**6 - 8*sqrt(b)*s qrt(a)*b**3*x**2 + 24*log((sqrt(a*x**4 + b) + sqrt(a)*x**2 - sqrt(b))/sqrt (b))*a**4*x**16 + 24*log((sqrt(a*x**4 + b) + sqrt(a)*x**2 - sqrt(b))/sqrt( b))*a**3*b*x**12 + 3*log((sqrt(a*x**4 + b) + sqrt(a)*x**2 - sqrt(b))/sqrt( b))*a**2*b**2*x**8 - 24*log((sqrt(a*x**4 + b) + sqrt(a)*x**2 + sqrt(b))/sq rt(b))*a**4*x**16 - 24*log((sqrt(a*x**4 + b) + sqrt(a)*x**2 + sqrt(b))/sqr t(b))*a**3*b*x**12 - 3*log((sqrt(a*x**4 + b) + sqrt(a)*x**2 + sqrt(b))/sqr t(b))*a**2*b**2*x**8)/(16*sqrt(b)*x**8*(8*sqrt(a)*sqrt(a*x**4 + b)*a*x**6 + 4*sqrt(a)*sqrt(a*x**4 + b)*b*x**2 + 8*a**2*x**8 + 8*a*b*x**4 + b**2))