Integrand size = 15, antiderivative size = 282 \[ \int \frac {1}{x^6 \left (a+b x^4\right )^{3/2}} \, dx=\frac {1}{2 a x^5 \sqrt {a+b x^4}}-\frac {7 \sqrt {a+b x^4}}{10 a^2 x^5}+\frac {21 b \sqrt {a+b x^4}}{10 a^3 x}-\frac {21 b^{3/2} x \sqrt {a+b x^4}}{10 a^3 \left (\sqrt {a}+\sqrt {b} x^2\right )}+\frac {21 b^{5/4} \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{10 a^{11/4} \sqrt {a+b x^4}}-\frac {21 b^{5/4} \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{20 a^{11/4} \sqrt {a+b x^4}} \] Output:
1/2/a/x^5/(b*x^4+a)^(1/2)-7/10*(b*x^4+a)^(1/2)/a^2/x^5+21/10*b*(b*x^4+a)^( 1/2)/a^3/x-21/10*b^(3/2)*x*(b*x^4+a)^(1/2)/a^3/(a^(1/2)+b^(1/2)*x^2)+21/10 *b^(5/4)*(a^(1/2)+b^(1/2)*x^2)*((b*x^4+a)/(a^(1/2)+b^(1/2)*x^2)^2)^(1/2)*E llipticE(sin(2*arctan(b^(1/4)*x/a^(1/4))),1/2*2^(1/2))/a^(11/4)/(b*x^4+a)^ (1/2)-21/20*b^(5/4)*(a^(1/2)+b^(1/2)*x^2)*((b*x^4+a)/(a^(1/2)+b^(1/2)*x^2) ^2)^(1/2)*InverseJacobiAM(2*arctan(b^(1/4)*x/a^(1/4)),1/2*2^(1/2))/a^(11/4 )/(b*x^4+a)^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 10.01 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.19 \[ \int \frac {1}{x^6 \left (a+b x^4\right )^{3/2}} \, dx=-\frac {\sqrt {1+\frac {b x^4}{a}} \operatorname {Hypergeometric2F1}\left (-\frac {5}{4},\frac {3}{2},-\frac {1}{4},-\frac {b x^4}{a}\right )}{5 a x^5 \sqrt {a+b x^4}} \] Input:
Integrate[1/(x^6*(a + b*x^4)^(3/2)),x]
Output:
-1/5*(Sqrt[1 + (b*x^4)/a]*Hypergeometric2F1[-5/4, 3/2, -1/4, -((b*x^4)/a)] )/(a*x^5*Sqrt[a + b*x^4])
Time = 0.63 (sec) , antiderivative size = 297, normalized size of antiderivative = 1.05, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.467, Rules used = {819, 847, 847, 834, 27, 761, 1510}
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.
\(\displaystyle \int \frac {1}{x^6 \left (a+b x^4\right )^{3/2}} \, dx\) |
\(\Big \downarrow \) 819 |
\(\displaystyle \frac {7 \int \frac {1}{x^6 \sqrt {b x^4+a}}dx}{2 a}+\frac {1}{2 a x^5 \sqrt {a+b x^4}}\) |
\(\Big \downarrow \) 847 |
\(\displaystyle \frac {7 \left (-\frac {3 b \int \frac {1}{x^2 \sqrt {b x^4+a}}dx}{5 a}-\frac {\sqrt {a+b x^4}}{5 a x^5}\right )}{2 a}+\frac {1}{2 a x^5 \sqrt {a+b x^4}}\) |
\(\Big \downarrow \) 847 |
\(\displaystyle \frac {7 \left (-\frac {3 b \left (\frac {b \int \frac {x^2}{\sqrt {b x^4+a}}dx}{a}-\frac {\sqrt {a+b x^4}}{a x}\right )}{5 a}-\frac {\sqrt {a+b x^4}}{5 a x^5}\right )}{2 a}+\frac {1}{2 a x^5 \sqrt {a+b x^4}}\) |
\(\Big \downarrow \) 834 |
\(\displaystyle \frac {7 \left (-\frac {3 b \left (\frac {b \left (\frac {\sqrt {a} \int \frac {1}{\sqrt {b x^4+a}}dx}{\sqrt {b}}-\frac {\sqrt {a} \int \frac {\sqrt {a}-\sqrt {b} x^2}{\sqrt {a} \sqrt {b x^4+a}}dx}{\sqrt {b}}\right )}{a}-\frac {\sqrt {a+b x^4}}{a x}\right )}{5 a}-\frac {\sqrt {a+b x^4}}{5 a x^5}\right )}{2 a}+\frac {1}{2 a x^5 \sqrt {a+b x^4}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {7 \left (-\frac {3 b \left (\frac {b \left (\frac {\sqrt {a} \int \frac {1}{\sqrt {b x^4+a}}dx}{\sqrt {b}}-\frac {\int \frac {\sqrt {a}-\sqrt {b} x^2}{\sqrt {b x^4+a}}dx}{\sqrt {b}}\right )}{a}-\frac {\sqrt {a+b x^4}}{a x}\right )}{5 a}-\frac {\sqrt {a+b x^4}}{5 a x^5}\right )}{2 a}+\frac {1}{2 a x^5 \sqrt {a+b x^4}}\) |
\(\Big \downarrow \) 761 |
\(\displaystyle \frac {7 \left (-\frac {3 b \left (\frac {b \left (\frac {\sqrt [4]{a} \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{2 b^{3/4} \sqrt {a+b x^4}}-\frac {\int \frac {\sqrt {a}-\sqrt {b} x^2}{\sqrt {b x^4+a}}dx}{\sqrt {b}}\right )}{a}-\frac {\sqrt {a+b x^4}}{a x}\right )}{5 a}-\frac {\sqrt {a+b x^4}}{5 a x^5}\right )}{2 a}+\frac {1}{2 a x^5 \sqrt {a+b x^4}}\) |
\(\Big \downarrow \) 1510 |
\(\displaystyle \frac {7 \left (-\frac {3 b \left (\frac {b \left (\frac {\sqrt [4]{a} \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{2 b^{3/4} \sqrt {a+b x^4}}-\frac {\frac {\sqrt [4]{a} \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{\sqrt [4]{b} \sqrt {a+b x^4}}-\frac {x \sqrt {a+b x^4}}{\sqrt {a}+\sqrt {b} x^2}}{\sqrt {b}}\right )}{a}-\frac {\sqrt {a+b x^4}}{a x}\right )}{5 a}-\frac {\sqrt {a+b x^4}}{5 a x^5}\right )}{2 a}+\frac {1}{2 a x^5 \sqrt {a+b x^4}}\) |
Input:
Int[1/(x^6*(a + b*x^4)^(3/2)),x]
Output:
1/(2*a*x^5*Sqrt[a + b*x^4]) + (7*(-1/5*Sqrt[a + b*x^4]/(a*x^5) - (3*b*(-(S qrt[a + b*x^4]/(a*x)) + (b*(-((-((x*Sqrt[a + b*x^4])/(Sqrt[a] + Sqrt[b]*x^ 2)) + (a^(1/4)*(Sqrt[a] + Sqrt[b]*x^2)*Sqrt[(a + b*x^4)/(Sqrt[a] + Sqrt[b] *x^2)^2]*EllipticE[2*ArcTan[(b^(1/4)*x)/a^(1/4)], 1/2])/(b^(1/4)*Sqrt[a + b*x^4]))/Sqrt[b]) + (a^(1/4)*(Sqrt[a] + Sqrt[b]*x^2)*Sqrt[(a + b*x^4)/(Sqr t[a] + Sqrt[b]*x^2)^2]*EllipticF[2*ArcTan[(b^(1/4)*x)/a^(1/4)], 1/2])/(2*b ^(3/4)*Sqrt[a + b*x^4])))/a))/(5*a)))/(2*a)
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> With[{q = Rt[b/a, 4]}, Simp[( 1 + q^2*x^2)*(Sqrt[(a + b*x^4)/(a*(1 + q^2*x^2)^2)]/(2*q*Sqrt[a + b*x^4]))* EllipticF[2*ArcTan[q*x], 1/2], x]] /; FreeQ[{a, b}, x] && PosQ[b/a]
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] + Simp[(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]
Int[(x_)^2/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> With[{q = Rt[b/a, 2]}, S imp[1/q Int[1/Sqrt[a + b*x^4], x], x] - Simp[1/q Int[(1 - q*x^2)/Sqrt[a + b*x^4], x], x]] /; FreeQ[{a, b}, x] && PosQ[b/a]
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] - Simp[b*((m + n*(p + 1) + 1)/(a*c^n*(m + 1))) Int[(c*x)^(m + n)*(a + b*x^n)^p, x], x] /; FreeQ[{a , b, c, p}, x] && IGtQ[n, 0] && LtQ[m, -1] && IntBinomialQ[a, b, c, n, m, p , x]
Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[c/a, 4]}, Simp[(-d)*x*(Sqrt[a + c*x^4]/(a*(1 + q^2*x^2))), x] + Simp[d* (1 + q^2*x^2)*(Sqrt[(a + c*x^4)/(a*(1 + q^2*x^2)^2)]/(q*Sqrt[a + c*x^4]))*E llipticE[2*ArcTan[q*x], 1/2], x] /; EqQ[e + d*q^2, 0]] /; FreeQ[{a, c, d, e }, x] && PosQ[c/a]
Result contains complex when optimal does not.
Time = 1.85 (sec) , antiderivative size = 157, normalized size of antiderivative = 0.56
method | result | size |
default | \(-\frac {\sqrt {b \,x^{4}+a}}{5 a^{2} x^{5}}+\frac {8 b \sqrt {b \,x^{4}+a}}{5 a^{3} x}+\frac {b^{2} x^{3}}{2 a^{3} \sqrt {\left (x^{4}+\frac {a}{b}\right ) b}}-\frac {21 i b^{\frac {3}{2}} \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )\right )}{10 a^{\frac {5}{2}} \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}\) | \(157\) |
elliptic | \(-\frac {\sqrt {b \,x^{4}+a}}{5 a^{2} x^{5}}+\frac {8 b \sqrt {b \,x^{4}+a}}{5 a^{3} x}+\frac {b^{2} x^{3}}{2 a^{3} \sqrt {\left (x^{4}+\frac {a}{b}\right ) b}}-\frac {21 i b^{\frac {3}{2}} \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )\right )}{10 a^{\frac {5}{2}} \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}\) | \(157\) |
risch | \(-\frac {\sqrt {b \,x^{4}+a}\, \left (-8 b \,x^{4}+a \right )}{5 a^{3} x^{5}}-\frac {b^{2} \left (3 a \left (\frac {x^{3}}{2 a \sqrt {\left (x^{4}+\frac {a}{b}\right ) b}}-\frac {i \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )\right )}{2 \sqrt {a}\, \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}\, \sqrt {b}}\right )+8 b \left (-\frac {x^{3}}{2 b \sqrt {\left (x^{4}+\frac {a}{b}\right ) b}}+\frac {3 i \sqrt {a}\, \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )\right )}{2 b^{\frac {3}{2}} \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}\right )\right )}{5 a^{3}}\) | \(278\) |
Input:
int(1/x^6/(b*x^4+a)^(3/2),x,method=_RETURNVERBOSE)
Output:
-1/5*(b*x^4+a)^(1/2)/a^2/x^5+8/5*b*(b*x^4+a)^(1/2)/a^3/x+1/2*b^2/a^3*x^3/( (x^4+a/b)*b)^(1/2)-21/10*I*b^(3/2)/a^(5/2)/(I/a^(1/2)*b^(1/2))^(1/2)*(1-I* b^(1/2)*x^2/a^(1/2))^(1/2)*(1+I*b^(1/2)*x^2/a^(1/2))^(1/2)/(b*x^4+a)^(1/2) *(EllipticF(x*(I/a^(1/2)*b^(1/2))^(1/2),I)-EllipticE(x*(I/a^(1/2)*b^(1/2)) ^(1/2),I))
Time = 0.08 (sec) , antiderivative size = 132, normalized size of antiderivative = 0.47 \[ \int \frac {1}{x^6 \left (a+b x^4\right )^{3/2}} \, dx=\frac {21 \, {\left (b^{2} x^{9} + a b x^{5}\right )} \sqrt {a} \left (-\frac {b}{a}\right )^{\frac {3}{4}} E(\arcsin \left (x \left (-\frac {b}{a}\right )^{\frac {1}{4}}\right )\,|\,-1) - 21 \, {\left (b^{2} x^{9} + a b x^{5}\right )} \sqrt {a} \left (-\frac {b}{a}\right )^{\frac {3}{4}} F(\arcsin \left (x \left (-\frac {b}{a}\right )^{\frac {1}{4}}\right )\,|\,-1) + {\left (21 \, b^{2} x^{8} + 14 \, a b x^{4} - 2 \, a^{2}\right )} \sqrt {b x^{4} + a}}{10 \, {\left (a^{3} b x^{9} + a^{4} x^{5}\right )}} \] Input:
integrate(1/x^6/(b*x^4+a)^(3/2),x, algorithm="fricas")
Output:
1/10*(21*(b^2*x^9 + a*b*x^5)*sqrt(a)*(-b/a)^(3/4)*elliptic_e(arcsin(x*(-b/ a)^(1/4)), -1) - 21*(b^2*x^9 + a*b*x^5)*sqrt(a)*(-b/a)^(3/4)*elliptic_f(ar csin(x*(-b/a)^(1/4)), -1) + (21*b^2*x^8 + 14*a*b*x^4 - 2*a^2)*sqrt(b*x^4 + a))/(a^3*b*x^9 + a^4*x^5)
Result contains complex when optimal does not.
Time = 0.66 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.16 \[ \int \frac {1}{x^6 \left (a+b x^4\right )^{3/2}} \, dx=\frac {\Gamma \left (- \frac {5}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {5}{4}, \frac {3}{2} \\ - \frac {1}{4} \end {matrix}\middle | {\frac {b x^{4} e^{i \pi }}{a}} \right )}}{4 a^{\frac {3}{2}} x^{5} \Gamma \left (- \frac {1}{4}\right )} \] Input:
integrate(1/x**6/(b*x**4+a)**(3/2),x)
Output:
gamma(-5/4)*hyper((-5/4, 3/2), (-1/4,), b*x**4*exp_polar(I*pi)/a)/(4*a**(3 /2)*x**5*gamma(-1/4))
\[ \int \frac {1}{x^6 \left (a+b x^4\right )^{3/2}} \, dx=\int { \frac {1}{{\left (b x^{4} + a\right )}^{\frac {3}{2}} x^{6}} \,d x } \] Input:
integrate(1/x^6/(b*x^4+a)^(3/2),x, algorithm="maxima")
Output:
integrate(1/((b*x^4 + a)^(3/2)*x^6), x)
\[ \int \frac {1}{x^6 \left (a+b x^4\right )^{3/2}} \, dx=\int { \frac {1}{{\left (b x^{4} + a\right )}^{\frac {3}{2}} x^{6}} \,d x } \] Input:
integrate(1/x^6/(b*x^4+a)^(3/2),x, algorithm="giac")
Output:
integrate(1/((b*x^4 + a)^(3/2)*x^6), x)
Timed out. \[ \int \frac {1}{x^6 \left (a+b x^4\right )^{3/2}} \, dx=\int \frac {1}{x^6\,{\left (b\,x^4+a\right )}^{3/2}} \,d x \] Input:
int(1/(x^6*(a + b*x^4)^(3/2)),x)
Output:
int(1/(x^6*(a + b*x^4)^(3/2)), x)
\[ \int \frac {1}{x^6 \left (a+b x^4\right )^{3/2}} \, dx=\int \frac {\sqrt {b \,x^{4}+a}}{b^{2} x^{14}+2 a b \,x^{10}+a^{2} x^{6}}d x \] Input:
int(1/x^6/(b*x^4+a)^(3/2),x)
Output:
int(sqrt(a + b*x**4)/(a**2*x**6 + 2*a*b*x**10 + b**2*x**14),x)