Integrand size = 16, antiderivative size = 81 \[ \int \frac {1}{x^5 \left (a-b x^4\right )^{3/4}} \, dx=-\frac {\sqrt [4]{a-b x^4}}{4 a x^4}-\frac {3 b \arctan \left (\frac {\sqrt [4]{a-b x^4}}{\sqrt [4]{a}}\right )}{8 a^{7/4}}-\frac {3 b \text {arctanh}\left (\frac {\sqrt [4]{a-b x^4}}{\sqrt [4]{a}}\right )}{8 a^{7/4}} \] Output:
-1/4*(-b*x^4+a)^(1/4)/a/x^4-3/8*b*arctan((-b*x^4+a)^(1/4)/a^(1/4))/a^(7/4) -3/8*b*arctanh((-b*x^4+a)^(1/4)/a^(1/4))/a^(7/4)
Time = 0.08 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x^5 \left (a-b x^4\right )^{3/4}} \, dx=-\frac {\sqrt [4]{a-b x^4}}{4 a x^4}-\frac {3 b \arctan \left (\frac {\sqrt [4]{a-b x^4}}{\sqrt [4]{a}}\right )}{8 a^{7/4}}-\frac {3 b \text {arctanh}\left (\frac {\sqrt [4]{a-b x^4}}{\sqrt [4]{a}}\right )}{8 a^{7/4}} \] Input:
Integrate[1/(x^5*(a - b*x^4)^(3/4)),x]
Output:
-1/4*(a - b*x^4)^(1/4)/(a*x^4) - (3*b*ArcTan[(a - b*x^4)^(1/4)/a^(1/4)])/( 8*a^(7/4)) - (3*b*ArcTanh[(a - b*x^4)^(1/4)/a^(1/4)])/(8*a^(7/4))
Time = 0.20 (sec) , antiderivative size = 89, normalized size of antiderivative = 1.10, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {798, 52, 73, 756, 216, 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.
\(\displaystyle \int \frac {1}{x^5 \left (a-b x^4\right )^{3/4}} \, dx\) |
\(\Big \downarrow \) 798 |
\(\displaystyle \frac {1}{4} \int \frac {1}{x^8 \left (a-b x^4\right )^{3/4}}dx^4\) |
\(\Big \downarrow \) 52 |
\(\displaystyle \frac {1}{4} \left (\frac {3 b \int \frac {1}{x^4 \left (a-b x^4\right )^{3/4}}dx^4}{4 a}-\frac {\sqrt [4]{a-b x^4}}{a x^4}\right )\) |
\(\Big \downarrow \) 73 |
\(\displaystyle \frac {1}{4} \left (-\frac {3 \int \frac {1}{\frac {a}{b}-\frac {x^{16}}{b}}d\sqrt [4]{a-b x^4}}{a}-\frac {\sqrt [4]{a-b x^4}}{a x^4}\right )\) |
\(\Big \downarrow \) 756 |
\(\displaystyle \frac {1}{4} \left (-\frac {3 \left (\frac {b \int \frac {1}{\sqrt {a}-x^8}d\sqrt [4]{a-b x^4}}{2 \sqrt {a}}+\frac {b \int \frac {1}{x^8+\sqrt {a}}d\sqrt [4]{a-b x^4}}{2 \sqrt {a}}\right )}{a}-\frac {\sqrt [4]{a-b x^4}}{a x^4}\right )\) |
\(\Big \downarrow \) 216 |
\(\displaystyle \frac {1}{4} \left (-\frac {3 \left (\frac {b \int \frac {1}{\sqrt {a}-x^8}d\sqrt [4]{a-b x^4}}{2 \sqrt {a}}+\frac {b \arctan \left (\frac {\sqrt [4]{a-b x^4}}{\sqrt [4]{a}}\right )}{2 a^{3/4}}\right )}{a}-\frac {\sqrt [4]{a-b x^4}}{a x^4}\right )\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {1}{4} \left (-\frac {3 \left (\frac {b \arctan \left (\frac {\sqrt [4]{a-b x^4}}{\sqrt [4]{a}}\right )}{2 a^{3/4}}+\frac {b \text {arctanh}\left (\frac {\sqrt [4]{a-b x^4}}{\sqrt [4]{a}}\right )}{2 a^{3/4}}\right )}{a}-\frac {\sqrt [4]{a-b x^4}}{a x^4}\right )\) |
Input:
Int[1/(x^5*(a - b*x^4)^(3/4)),x]
Output:
(-((a - b*x^4)^(1/4)/(a*x^4)) - (3*((b*ArcTan[(a - b*x^4)^(1/4)/a^(1/4)])/ (2*a^(3/4)) + (b*ArcTanh[(a - b*x^4)^(1/4)/a^(1/4)])/(2*a^(3/4))))/a)/4
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((b*c - a*d)*(m + 1))), x] - Simp[d*(( m + n + 2)/((b*c - a*d)*(m + 1))) Int[(a + b*x)^(m + 1)*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && ILtQ[m, -1] && FractionQ[n] && LtQ[n, 0]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ {p = Denominator[m]}, Simp[p/b Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL inearQ[a, b, c, d, m, n, x]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*A rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a , 0] || GtQ[b, 0])
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[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[-a/b, 2 ]], s = Denominator[Rt[-a/b, 2]]}, Simp[r/(2*a) Int[1/(r - s*x^2), x], x] + Simp[r/(2*a) Int[1/(r + s*x^2), x], x]] /; FreeQ[{a, b}, x] && !GtQ[a /b, 0]
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[1/n Subst [Int[x^(Simplify[(m + 1)/n] - 1)*(a + b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]
Time = 0.69 (sec) , antiderivative size = 89, normalized size of antiderivative = 1.10
method | result | size |
pseudoelliptic | \(-\frac {3 \left (\ln \left (\frac {-\left (-b \,x^{4}+a \right )^{\frac {1}{4}}-a^{\frac {1}{4}}}{-\left (-b \,x^{4}+a \right )^{\frac {1}{4}}+a^{\frac {1}{4}}}\right ) b \,x^{4}+2 \arctan \left (\frac {\left (-b \,x^{4}+a \right )^{\frac {1}{4}}}{a^{\frac {1}{4}}}\right ) b \,x^{4}+\frac {4 \left (-b \,x^{4}+a \right )^{\frac {1}{4}} a^{\frac {3}{4}}}{3}\right )}{16 a^{\frac {7}{4}} x^{4}}\) | \(89\) |
Input:
int(1/x^5/(-b*x^4+a)^(3/4),x,method=_RETURNVERBOSE)
Output:
-3/16*(ln((-(-b*x^4+a)^(1/4)-a^(1/4))/(-(-b*x^4+a)^(1/4)+a^(1/4)))*b*x^4+2 *arctan((-b*x^4+a)^(1/4)/a^(1/4))*b*x^4+4/3*(-b*x^4+a)^(1/4)*a^(3/4))/a^(7 /4)/x^4
Result contains complex when optimal does not.
Time = 0.09 (sec) , antiderivative size = 197, normalized size of antiderivative = 2.43 \[ \int \frac {1}{x^5 \left (a-b x^4\right )^{3/4}} \, dx=-\frac {3 \, a x^{4} \left (\frac {b^{4}}{a^{7}}\right )^{\frac {1}{4}} \log \left (3 \, a^{2} \left (\frac {b^{4}}{a^{7}}\right )^{\frac {1}{4}} + 3 \, {\left (-b x^{4} + a\right )}^{\frac {1}{4}} b\right ) + 3 i \, a x^{4} \left (\frac {b^{4}}{a^{7}}\right )^{\frac {1}{4}} \log \left (3 i \, a^{2} \left (\frac {b^{4}}{a^{7}}\right )^{\frac {1}{4}} + 3 \, {\left (-b x^{4} + a\right )}^{\frac {1}{4}} b\right ) - 3 i \, a x^{4} \left (\frac {b^{4}}{a^{7}}\right )^{\frac {1}{4}} \log \left (-3 i \, a^{2} \left (\frac {b^{4}}{a^{7}}\right )^{\frac {1}{4}} + 3 \, {\left (-b x^{4} + a\right )}^{\frac {1}{4}} b\right ) - 3 \, a x^{4} \left (\frac {b^{4}}{a^{7}}\right )^{\frac {1}{4}} \log \left (-3 \, a^{2} \left (\frac {b^{4}}{a^{7}}\right )^{\frac {1}{4}} + 3 \, {\left (-b x^{4} + a\right )}^{\frac {1}{4}} b\right ) + 4 \, {\left (-b x^{4} + a\right )}^{\frac {1}{4}}}{16 \, a x^{4}} \] Input:
integrate(1/x^5/(-b*x^4+a)^(3/4),x, algorithm="fricas")
Output:
-1/16*(3*a*x^4*(b^4/a^7)^(1/4)*log(3*a^2*(b^4/a^7)^(1/4) + 3*(-b*x^4 + a)^ (1/4)*b) + 3*I*a*x^4*(b^4/a^7)^(1/4)*log(3*I*a^2*(b^4/a^7)^(1/4) + 3*(-b*x ^4 + a)^(1/4)*b) - 3*I*a*x^4*(b^4/a^7)^(1/4)*log(-3*I*a^2*(b^4/a^7)^(1/4) + 3*(-b*x^4 + a)^(1/4)*b) - 3*a*x^4*(b^4/a^7)^(1/4)*log(-3*a^2*(b^4/a^7)^( 1/4) + 3*(-b*x^4 + a)^(1/4)*b) + 4*(-b*x^4 + a)^(1/4))/(a*x^4)
Result contains complex when optimal does not.
Time = 0.80 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.48 \[ \int \frac {1}{x^5 \left (a-b x^4\right )^{3/4}} \, dx=\frac {e^{\frac {i \pi }{4}} \Gamma \left (\frac {7}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {3}{4}, \frac {7}{4} \\ \frac {11}{4} \end {matrix}\middle | {\frac {a}{b x^{4}}} \right )}}{4 b^{\frac {3}{4}} x^{7} \Gamma \left (\frac {11}{4}\right )} \] Input:
integrate(1/x**5/(-b*x**4+a)**(3/4),x)
Output:
exp(I*pi/4)*gamma(7/4)*hyper((3/4, 7/4), (11/4,), a/(b*x**4))/(4*b**(3/4)* x**7*gamma(11/4))
Time = 0.11 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.21 \[ \int \frac {1}{x^5 \left (a-b x^4\right )^{3/4}} \, dx=-\frac {{\left (-b x^{4} + a\right )}^{\frac {1}{4}} b}{4 \, {\left ({\left (b x^{4} - a\right )} a + a^{2}\right )}} - \frac {3 \, {\left (\frac {2 \, b \arctan \left (\frac {{\left (-b x^{4} + a\right )}^{\frac {1}{4}}}{a^{\frac {1}{4}}}\right )}{a^{\frac {3}{4}}} - \frac {b \log \left (\frac {{\left (-b x^{4} + a\right )}^{\frac {1}{4}} - a^{\frac {1}{4}}}{{\left (-b x^{4} + a\right )}^{\frac {1}{4}} + a^{\frac {1}{4}}}\right )}{a^{\frac {3}{4}}}\right )}}{16 \, a} \] Input:
integrate(1/x^5/(-b*x^4+a)^(3/4),x, algorithm="maxima")
Output:
-1/4*(-b*x^4 + a)^(1/4)*b/((b*x^4 - a)*a + a^2) - 3/16*(2*b*arctan((-b*x^4 + a)^(1/4)/a^(1/4))/a^(3/4) - b*log(((-b*x^4 + a)^(1/4) - a^(1/4))/((-b*x ^4 + a)^(1/4) + a^(1/4)))/a^(3/4))/a
Leaf count of result is larger than twice the leaf count of optimal. 228 vs. \(2 (61) = 122\).
Time = 0.13 (sec) , antiderivative size = 228, normalized size of antiderivative = 2.81 \[ \int \frac {1}{x^5 \left (a-b x^4\right )^{3/4}} \, dx=-\frac {\frac {6 \, \sqrt {2} \left (-a\right )^{\frac {1}{4}} b^{2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (-a\right )^{\frac {1}{4}} + 2 \, {\left (-b x^{4} + a\right )}^{\frac {1}{4}}\right )}}{2 \, \left (-a\right )^{\frac {1}{4}}}\right )}{a^{2}} + \frac {6 \, \sqrt {2} \left (-a\right )^{\frac {1}{4}} b^{2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (-a\right )^{\frac {1}{4}} - 2 \, {\left (-b x^{4} + a\right )}^{\frac {1}{4}}\right )}}{2 \, \left (-a\right )^{\frac {1}{4}}}\right )}{a^{2}} + \frac {3 \, \sqrt {2} \left (-a\right )^{\frac {1}{4}} b^{2} \log \left (\sqrt {2} {\left (-b x^{4} + a\right )}^{\frac {1}{4}} \left (-a\right )^{\frac {1}{4}} + \sqrt {-b x^{4} + a} + \sqrt {-a}\right )}{a^{2}} + \frac {3 \, \sqrt {2} b^{2} \log \left (-\sqrt {2} {\left (-b x^{4} + a\right )}^{\frac {1}{4}} \left (-a\right )^{\frac {1}{4}} + \sqrt {-b x^{4} + a} + \sqrt {-a}\right )}{\left (-a\right )^{\frac {3}{4}} a} + \frac {8 \, {\left (-b x^{4} + a\right )}^{\frac {1}{4}} b}{a x^{4}}}{32 \, b} \] Input:
integrate(1/x^5/(-b*x^4+a)^(3/4),x, algorithm="giac")
Output:
-1/32*(6*sqrt(2)*(-a)^(1/4)*b^2*arctan(1/2*sqrt(2)*(sqrt(2)*(-a)^(1/4) + 2 *(-b*x^4 + a)^(1/4))/(-a)^(1/4))/a^2 + 6*sqrt(2)*(-a)^(1/4)*b^2*arctan(-1/ 2*sqrt(2)*(sqrt(2)*(-a)^(1/4) - 2*(-b*x^4 + a)^(1/4))/(-a)^(1/4))/a^2 + 3* sqrt(2)*(-a)^(1/4)*b^2*log(sqrt(2)*(-b*x^4 + a)^(1/4)*(-a)^(1/4) + sqrt(-b *x^4 + a) + sqrt(-a))/a^2 + 3*sqrt(2)*b^2*log(-sqrt(2)*(-b*x^4 + a)^(1/4)* (-a)^(1/4) + sqrt(-b*x^4 + a) + sqrt(-a))/((-a)^(3/4)*a) + 8*(-b*x^4 + a)^ (1/4)*b/(a*x^4))/b
Time = 0.49 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.75 \[ \int \frac {1}{x^5 \left (a-b x^4\right )^{3/4}} \, dx=-\frac {{\left (a-b\,x^4\right )}^{1/4}}{4\,a\,x^4}-\frac {3\,b\,\mathrm {atan}\left (\frac {{\left (a-b\,x^4\right )}^{1/4}}{a^{1/4}}\right )}{8\,a^{7/4}}-\frac {3\,b\,\mathrm {atanh}\left (\frac {{\left (a-b\,x^4\right )}^{1/4}}{a^{1/4}}\right )}{8\,a^{7/4}} \] Input:
int(1/(x^5*(a - b*x^4)^(3/4)),x)
Output:
- (a - b*x^4)^(1/4)/(4*a*x^4) - (3*b*atan((a - b*x^4)^(1/4)/a^(1/4)))/(8*a ^(7/4)) - (3*b*atanh((a - b*x^4)^(1/4)/a^(1/4)))/(8*a^(7/4))
\[ \int \frac {1}{x^5 \left (a-b x^4\right )^{3/4}} \, dx=\int \frac {1}{\left (-b \,x^{4}+a \right )^{\frac {3}{4}} x^{5}}d x \] Input:
int(1/x^5/(-b*x^4+a)^(3/4),x)
Output:
int(1/((a - b*x**4)**(3/4)*x**5),x)