Integrand size = 15, antiderivative size = 104 \[ \int \frac {1}{x^7 \left (a+b x^4\right )^{3/4}} \, dx=-\frac {\sqrt [4]{a+b x^4}}{6 a x^6}+\frac {5 b \sqrt [4]{a+b x^4}}{12 a^2 x^2}+\frac {5 b^{3/2} \left (1+\frac {b x^4}{a}\right )^{3/4} \operatorname {EllipticF}\left (\frac {1}{2} \arctan \left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right ),2\right )}{12 a^{3/2} \left (a+b x^4\right )^{3/4}} \] Output:
-1/6*(b*x^4+a)^(1/4)/a/x^6+5/12*b*(b*x^4+a)^(1/4)/a^2/x^2+5/12*b^(3/2)*(1+ b*x^4/a)^(3/4)*InverseJacobiAM(1/2*arctan(b^(1/2)*x^2/a^(1/2)),2^(1/2))/a^ (3/2)/(b*x^4+a)^(3/4)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 10.02 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.49 \[ \int \frac {1}{x^7 \left (a+b x^4\right )^{3/4}} \, dx=-\frac {\left (1+\frac {b x^4}{a}\right )^{3/4} \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},\frac {3}{4},-\frac {1}{2},-\frac {b x^4}{a}\right )}{6 x^6 \left (a+b x^4\right )^{3/4}} \] Input:
Integrate[1/(x^7*(a + b*x^4)^(3/4)),x]
Output:
-1/6*((1 + (b*x^4)/a)^(3/4)*Hypergeometric2F1[-3/2, 3/4, -1/2, -((b*x^4)/a )])/(x^6*(a + b*x^4)^(3/4))
Time = 0.34 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.08, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {807, 264, 264, 231, 229}
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.
\(\displaystyle \int \frac {1}{x^7 \left (a+b x^4\right )^{3/4}} \, dx\) |
\(\Big \downarrow \) 807 |
\(\displaystyle \frac {1}{2} \int \frac {1}{x^8 \left (b x^4+a\right )^{3/4}}dx^2\) |
\(\Big \downarrow \) 264 |
\(\displaystyle \frac {1}{2} \left (-\frac {5 b \int \frac {1}{x^4 \left (b x^4+a\right )^{3/4}}dx^2}{6 a}-\frac {\sqrt [4]{a+b x^4}}{3 a x^6}\right )\) |
\(\Big \downarrow \) 264 |
\(\displaystyle \frac {1}{2} \left (-\frac {5 b \left (-\frac {b \int \frac {1}{\left (b x^4+a\right )^{3/4}}dx^2}{2 a}-\frac {\sqrt [4]{a+b x^4}}{a x^2}\right )}{6 a}-\frac {\sqrt [4]{a+b x^4}}{3 a x^6}\right )\) |
\(\Big \downarrow \) 231 |
\(\displaystyle \frac {1}{2} \left (-\frac {5 b \left (-\frac {b \left (\frac {b x^4}{a}+1\right )^{3/4} \int \frac {1}{\left (\frac {b x^4}{a}+1\right )^{3/4}}dx^2}{2 a \left (a+b x^4\right )^{3/4}}-\frac {\sqrt [4]{a+b x^4}}{a x^2}\right )}{6 a}-\frac {\sqrt [4]{a+b x^4}}{3 a x^6}\right )\) |
\(\Big \downarrow \) 229 |
\(\displaystyle \frac {1}{2} \left (-\frac {5 b \left (-\frac {\sqrt {b} \left (\frac {b x^4}{a}+1\right )^{3/4} \operatorname {EllipticF}\left (\frac {1}{2} \arctan \left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right ),2\right )}{\sqrt {a} \left (a+b x^4\right )^{3/4}}-\frac {\sqrt [4]{a+b x^4}}{a x^2}\right )}{6 a}-\frac {\sqrt [4]{a+b x^4}}{3 a x^6}\right )\) |
Input:
Int[1/(x^7*(a + b*x^4)^(3/4)),x]
Output:
(-1/3*(a + b*x^4)^(1/4)/(a*x^6) - (5*b*(-((a + b*x^4)^(1/4)/(a*x^2)) - (Sq rt[b]*(1 + (b*x^4)/a)^(3/4)*EllipticF[ArcTan[(Sqrt[b]*x^2)/Sqrt[a]]/2, 2]) /(Sqrt[a]*(a + b*x^4)^(3/4))))/(6*a))/2
Int[((a_) + (b_.)*(x_)^2)^(-3/4), x_Symbol] :> Simp[(2/(a^(3/4)*Rt[b/a, 2]) )*EllipticF[(1/2)*ArcTan[Rt[b/a, 2]*x], 2], x] /; FreeQ[{a, b}, x] && GtQ[a , 0] && PosQ[b/a]
Int[((a_) + (b_.)*(x_)^2)^(-3/4), x_Symbol] :> Simp[(1 + b*(x^2/a))^(3/4)/( a + b*x^2)^(3/4) Int[1/(1 + b*(x^2/a))^(3/4), x], x] /; FreeQ[{a, b}, x] && PosQ[a]
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]
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]
\[\int \frac {1}{x^{7} \left (b \,x^{4}+a \right )^{\frac {3}{4}}}d x\]
Input:
int(1/x^7/(b*x^4+a)^(3/4),x)
Output:
int(1/x^7/(b*x^4+a)^(3/4),x)
\[ \int \frac {1}{x^7 \left (a+b x^4\right )^{3/4}} \, dx=\int { \frac {1}{{\left (b x^{4} + a\right )}^{\frac {3}{4}} x^{7}} \,d x } \] Input:
integrate(1/x^7/(b*x^4+a)^(3/4),x, algorithm="fricas")
Output:
integral((b*x^4 + a)^(1/4)/(b*x^11 + a*x^7), x)
Result contains complex when optimal does not.
Time = 0.69 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.31 \[ \int \frac {1}{x^7 \left (a+b x^4\right )^{3/4}} \, dx=- \frac {{{}_{2}F_{1}\left (\begin {matrix} - \frac {3}{2}, \frac {3}{4} \\ - \frac {1}{2} \end {matrix}\middle | {\frac {b x^{4} e^{i \pi }}{a}} \right )}}{6 a^{\frac {3}{4}} x^{6}} \] Input:
integrate(1/x**7/(b*x**4+a)**(3/4),x)
Output:
-hyper((-3/2, 3/4), (-1/2,), b*x**4*exp_polar(I*pi)/a)/(6*a**(3/4)*x**6)
\[ \int \frac {1}{x^7 \left (a+b x^4\right )^{3/4}} \, dx=\int { \frac {1}{{\left (b x^{4} + a\right )}^{\frac {3}{4}} x^{7}} \,d x } \] Input:
integrate(1/x^7/(b*x^4+a)^(3/4),x, algorithm="maxima")
Output:
integrate(1/((b*x^4 + a)^(3/4)*x^7), x)
\[ \int \frac {1}{x^7 \left (a+b x^4\right )^{3/4}} \, dx=\int { \frac {1}{{\left (b x^{4} + a\right )}^{\frac {3}{4}} x^{7}} \,d x } \] Input:
integrate(1/x^7/(b*x^4+a)^(3/4),x, algorithm="giac")
Output:
integrate(1/((b*x^4 + a)^(3/4)*x^7), x)
Timed out. \[ \int \frac {1}{x^7 \left (a+b x^4\right )^{3/4}} \, dx=\int \frac {1}{x^7\,{\left (b\,x^4+a\right )}^{3/4}} \,d x \] Input:
int(1/(x^7*(a + b*x^4)^(3/4)),x)
Output:
int(1/(x^7*(a + b*x^4)^(3/4)), x)
\[ \int \frac {1}{x^7 \left (a+b x^4\right )^{3/4}} \, dx=\int \frac {1}{\left (b \,x^{4}+a \right )^{\frac {3}{4}} x^{7}}d x \] Input:
int(1/x^7/(b*x^4+a)^(3/4),x)
Output:
int(1/((a + b*x**4)**(3/4)*x**7),x)