Integrand size = 16, antiderivative size = 108 \[ \int \frac {x^9}{\left (a-b x^4\right )^{3/4}} \, dx=-\frac {2 a x^2 \sqrt [4]{a-b x^4}}{7 b^2}-\frac {x^6 \sqrt [4]{a-b x^4}}{7 b}+\frac {4 a^{5/2} \left (1-\frac {b x^4}{a}\right )^{3/4} \operatorname {EllipticF}\left (\frac {1}{2} \arcsin \left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right ),2\right )}{7 b^{5/2} \left (a-b x^4\right )^{3/4}} \] Output:
-2/7*a*x^2*(-b*x^4+a)^(1/4)/b^2-1/7*x^6*(-b*x^4+a)^(1/4)/b+4/7*a^(5/2)*(1- b*x^4/a)^(3/4)*InverseJacobiAM(1/2*arcsin(b^(1/2)*x^2/a^(1/2)),2^(1/2))/b^ (5/2)/(-b*x^4+a)^(3/4)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 7.06 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.73 \[ \int \frac {x^9}{\left (a-b x^4\right )^{3/4}} \, dx=\frac {x^2 \left (-2 a^2+a b x^4+b^2 x^8+2 a^2 \left (1-\frac {b x^4}{a}\right )^{3/4} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3}{4},\frac {3}{2},\frac {b x^4}{a}\right )\right )}{7 b^2 \left (a-b x^4\right )^{3/4}} \] Input:
Integrate[x^9/(a - b*x^4)^(3/4),x]
Output:
(x^2*(-2*a^2 + a*b*x^4 + b^2*x^8 + 2*a^2*(1 - (b*x^4)/a)^(3/4)*Hypergeomet ric2F1[1/2, 3/4, 3/2, (b*x^4)/a]))/(7*b^2*(a - b*x^4)^(3/4))
Time = 0.22 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.11, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.312, Rules used = {807, 262, 262, 231, 230}
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.
\(\displaystyle \int \frac {x^9}{\left (a-b x^4\right )^{3/4}} \, dx\) |
\(\Big \downarrow \) 807 |
\(\displaystyle \frac {1}{2} \int \frac {x^8}{\left (a-b x^4\right )^{3/4}}dx^2\) |
\(\Big \downarrow \) 262 |
\(\displaystyle \frac {1}{2} \left (\frac {6 a \int \frac {x^4}{\left (a-b x^4\right )^{3/4}}dx^2}{7 b}-\frac {2 x^6 \sqrt [4]{a-b x^4}}{7 b}\right )\) |
\(\Big \downarrow \) 262 |
\(\displaystyle \frac {1}{2} \left (\frac {6 a \left (\frac {2 a \int \frac {1}{\left (a-b x^4\right )^{3/4}}dx^2}{3 b}-\frac {2 x^2 \sqrt [4]{a-b x^4}}{3 b}\right )}{7 b}-\frac {2 x^6 \sqrt [4]{a-b x^4}}{7 b}\right )\) |
\(\Big \downarrow \) 231 |
\(\displaystyle \frac {1}{2} \left (\frac {6 a \left (\frac {2 a \left (1-\frac {b x^4}{a}\right )^{3/4} \int \frac {1}{\left (1-\frac {b x^4}{a}\right )^{3/4}}dx^2}{3 b \left (a-b x^4\right )^{3/4}}-\frac {2 x^2 \sqrt [4]{a-b x^4}}{3 b}\right )}{7 b}-\frac {2 x^6 \sqrt [4]{a-b x^4}}{7 b}\right )\) |
\(\Big \downarrow \) 230 |
\(\displaystyle \frac {1}{2} \left (\frac {6 a \left (\frac {4 a^{3/2} \left (1-\frac {b x^4}{a}\right )^{3/4} \operatorname {EllipticF}\left (\frac {1}{2} \arcsin \left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right ),2\right )}{3 b^{3/2} \left (a-b x^4\right )^{3/4}}-\frac {2 x^2 \sqrt [4]{a-b x^4}}{3 b}\right )}{7 b}-\frac {2 x^6 \sqrt [4]{a-b x^4}}{7 b}\right )\) |
Input:
Int[x^9/(a - b*x^4)^(3/4),x]
Output:
((-2*x^6*(a - b*x^4)^(1/4))/(7*b) + (6*a*((-2*x^2*(a - b*x^4)^(1/4))/(3*b) + (4*a^(3/2)*(1 - (b*x^4)/a)^(3/4)*EllipticF[ArcSin[(Sqrt[b]*x^2)/Sqrt[a] ]/2, 2])/(3*b^(3/2)*(a - b*x^4)^(3/4))))/(7*b))/2
Int[((a_) + (b_.)*(x_)^2)^(-3/4), x_Symbol] :> Simp[(2/(a^(3/4)*Rt[-b/a, 2] ))*EllipticF[(1/2)*ArcSin[Rt[-b/a, 2]*x], 2], x] /; FreeQ[{a, b}, x] && GtQ [a, 0] && NegQ[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*(c*x) ^(m - 1)*((a + b*x^2)^(p + 1)/(b*(m + 2*p + 1))), x] - Simp[a*c^2*((m - 1)/ (b*(m + 2*p + 1))) Int[(c*x)^(m - 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b , c, p}, x] && GtQ[m, 2 - 1] && NeQ[m + 2*p + 1, 0] && 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]
\[\int \frac {x^{9}}{\left (-b \,x^{4}+a \right )^{\frac {3}{4}}}d x\]
Input:
int(x^9/(-b*x^4+a)^(3/4),x)
Output:
int(x^9/(-b*x^4+a)^(3/4),x)
\[ \int \frac {x^9}{\left (a-b x^4\right )^{3/4}} \, dx=\int { \frac {x^{9}}{{\left (-b x^{4} + a\right )}^{\frac {3}{4}}} \,d x } \] Input:
integrate(x^9/(-b*x^4+a)^(3/4),x, algorithm="fricas")
Output:
integral(-(-b*x^4 + a)^(1/4)*x^9/(b*x^4 - a), x)
Result contains complex when optimal does not.
Time = 0.57 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.27 \[ \int \frac {x^9}{\left (a-b x^4\right )^{3/4}} \, dx=\frac {x^{10} {{}_{2}F_{1}\left (\begin {matrix} \frac {3}{4}, \frac {5}{2} \\ \frac {7}{2} \end {matrix}\middle | {\frac {b x^{4} e^{2 i \pi }}{a}} \right )}}{10 a^{\frac {3}{4}}} \] Input:
integrate(x**9/(-b*x**4+a)**(3/4),x)
Output:
x**10*hyper((3/4, 5/2), (7/2,), b*x**4*exp_polar(2*I*pi)/a)/(10*a**(3/4))
\[ \int \frac {x^9}{\left (a-b x^4\right )^{3/4}} \, dx=\int { \frac {x^{9}}{{\left (-b x^{4} + a\right )}^{\frac {3}{4}}} \,d x } \] Input:
integrate(x^9/(-b*x^4+a)^(3/4),x, algorithm="maxima")
Output:
integrate(x^9/(-b*x^4 + a)^(3/4), x)
\[ \int \frac {x^9}{\left (a-b x^4\right )^{3/4}} \, dx=\int { \frac {x^{9}}{{\left (-b x^{4} + a\right )}^{\frac {3}{4}}} \,d x } \] Input:
integrate(x^9/(-b*x^4+a)^(3/4),x, algorithm="giac")
Output:
integrate(x^9/(-b*x^4 + a)^(3/4), x)
Timed out. \[ \int \frac {x^9}{\left (a-b x^4\right )^{3/4}} \, dx=\int \frac {x^9}{{\left (a-b\,x^4\right )}^{3/4}} \,d x \] Input:
int(x^9/(a - b*x^4)^(3/4),x)
Output:
int(x^9/(a - b*x^4)^(3/4), x)
\[ \int \frac {x^9}{\left (a-b x^4\right )^{3/4}} \, dx=\int \frac {x^{9}}{\left (-b \,x^{4}+a \right )^{\frac {3}{4}}}d x \] Input:
int(x^9/(-b*x^4+a)^(3/4),x)
Output:
int(x**9/(a - b*x**4)**(3/4),x)