Integrand size = 16, antiderivative size = 126 \[ \int x^4 \left (a-b x^2\right )^{3/4} \, dx=-\frac {4 a^2 x \left (a-b x^2\right )^{3/4}}{65 b^2}-\frac {2 a x^3 \left (a-b x^2\right )^{3/4}}{39 b}+\frac {2}{13} x^5 \left (a-b x^2\right )^{3/4}+\frac {8 a^{7/2} \sqrt [4]{1-\frac {b x^2}{a}} E\left (\left .\frac {1}{2} \arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right |2\right )}{65 b^{5/2} \sqrt [4]{a-b x^2}} \] Output:
-4/65*a^2*x*(-b*x^2+a)^(3/4)/b^2-2/39*a*x^3*(-b*x^2+a)^(3/4)/b+2/13*x^5*(- b*x^2+a)^(3/4)+8/65*a^(7/2)*(1-b*x^2/a)^(1/4)*EllipticE(sin(1/2*arcsin(b^( 1/2)*x/a^(1/2))),2^(1/2))/b^(5/2)/(-b*x^2+a)^(1/4)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 7.25 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.75 \[ \int x^4 \left (a-b x^2\right )^{3/4} \, dx=-\frac {2 x \left (a-b x^2\right )^{3/4} \left (\left (1-\frac {b x^2}{a}\right )^{3/4} \left (2 a^2+a b x^2-3 b^2 x^4\right )-2 a^2 \operatorname {Hypergeometric2F1}\left (-\frac {3}{4},\frac {1}{2},\frac {3}{2},\frac {b x^2}{a}\right )\right )}{39 b^2 \left (1-\frac {b x^2}{a}\right )^{3/4}} \] Input:
Integrate[x^4*(a - b*x^2)^(3/4),x]
Output:
(-2*x*(a - b*x^2)^(3/4)*((1 - (b*x^2)/a)^(3/4)*(2*a^2 + a*b*x^2 - 3*b^2*x^ 4) - 2*a^2*Hypergeometric2F1[-3/4, 1/2, 3/2, (b*x^2)/a]))/(39*b^2*(1 - (b* x^2)/a)^(3/4))
Time = 0.21 (sec) , antiderivative size = 137, normalized size of antiderivative = 1.09, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.312, Rules used = {248, 262, 262, 227, 226}
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.
\(\displaystyle \int x^4 \left (a-b x^2\right )^{3/4} \, dx\) |
\(\Big \downarrow \) 248 |
\(\displaystyle \frac {3}{13} a \int \frac {x^4}{\sqrt [4]{a-b x^2}}dx+\frac {2}{13} x^5 \left (a-b x^2\right )^{3/4}\) |
\(\Big \downarrow \) 262 |
\(\displaystyle \frac {3}{13} a \left (\frac {2 a \int \frac {x^2}{\sqrt [4]{a-b x^2}}dx}{3 b}-\frac {2 x^3 \left (a-b x^2\right )^{3/4}}{9 b}\right )+\frac {2}{13} x^5 \left (a-b x^2\right )^{3/4}\) |
\(\Big \downarrow \) 262 |
\(\displaystyle \frac {3}{13} a \left (\frac {2 a \left (\frac {2 a \int \frac {1}{\sqrt [4]{a-b x^2}}dx}{5 b}-\frac {2 x \left (a-b x^2\right )^{3/4}}{5 b}\right )}{3 b}-\frac {2 x^3 \left (a-b x^2\right )^{3/4}}{9 b}\right )+\frac {2}{13} x^5 \left (a-b x^2\right )^{3/4}\) |
\(\Big \downarrow \) 227 |
\(\displaystyle \frac {3}{13} a \left (\frac {2 a \left (\frac {2 a \sqrt [4]{1-\frac {b x^2}{a}} \int \frac {1}{\sqrt [4]{1-\frac {b x^2}{a}}}dx}{5 b \sqrt [4]{a-b x^2}}-\frac {2 x \left (a-b x^2\right )^{3/4}}{5 b}\right )}{3 b}-\frac {2 x^3 \left (a-b x^2\right )^{3/4}}{9 b}\right )+\frac {2}{13} x^5 \left (a-b x^2\right )^{3/4}\) |
\(\Big \downarrow \) 226 |
\(\displaystyle \frac {3}{13} a \left (\frac {2 a \left (\frac {4 a^{3/2} \sqrt [4]{1-\frac {b x^2}{a}} E\left (\left .\frac {1}{2} \arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right |2\right )}{5 b^{3/2} \sqrt [4]{a-b x^2}}-\frac {2 x \left (a-b x^2\right )^{3/4}}{5 b}\right )}{3 b}-\frac {2 x^3 \left (a-b x^2\right )^{3/4}}{9 b}\right )+\frac {2}{13} x^5 \left (a-b x^2\right )^{3/4}\) |
Input:
Int[x^4*(a - b*x^2)^(3/4),x]
Output:
(2*x^5*(a - b*x^2)^(3/4))/13 + (3*a*((-2*x^3*(a - b*x^2)^(3/4))/(9*b) + (2 *a*((-2*x*(a - b*x^2)^(3/4))/(5*b) + (4*a^(3/2)*(1 - (b*x^2)/a)^(1/4)*Elli pticE[ArcSin[(Sqrt[b]*x)/Sqrt[a]]/2, 2])/(5*b^(3/2)*(a - b*x^2)^(1/4))))/( 3*b)))/13
Int[((a_) + (b_.)*(x_)^2)^(-1/4), x_Symbol] :> Simp[(2/(a^(1/4)*Rt[-b/a, 2] ))*EllipticE[(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)^(-1/4), x_Symbol] :> Simp[(1 + b*(x^2/a))^(1/4)/( a + b*x^2)^(1/4) Int[1/(1 + b*(x^2/a))^(1/4), 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/(c*(m + 2*p + 1))), x] + Simp[2*a*(p/(m + 2*p + 1)) Int[(c*x)^m*(a + b*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c, m}, x] && GtQ[ p, 0] && NeQ[m + 2*p + 1, 0] && IntBinomialQ[a, b, c, 2, m, p, x]
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]
\[\int x^{4} \left (-b \,x^{2}+a \right )^{\frac {3}{4}}d x\]
Input:
int(x^4*(-b*x^2+a)^(3/4),x)
Output:
int(x^4*(-b*x^2+a)^(3/4),x)
\[ \int x^4 \left (a-b x^2\right )^{3/4} \, dx=\int { {\left (-b x^{2} + a\right )}^{\frac {3}{4}} x^{4} \,d x } \] Input:
integrate(x^4*(-b*x^2+a)^(3/4),x, algorithm="fricas")
Output:
integral((-b*x^2 + a)^(3/4)*x^4, x)
Result contains complex when optimal does not.
Time = 0.59 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.25 \[ \int x^4 \left (a-b x^2\right )^{3/4} \, dx=\frac {a^{\frac {3}{4}} x^{5} {{}_{2}F_{1}\left (\begin {matrix} - \frac {3}{4}, \frac {5}{2} \\ \frac {7}{2} \end {matrix}\middle | {\frac {b x^{2} e^{2 i \pi }}{a}} \right )}}{5} \] Input:
integrate(x**4*(-b*x**2+a)**(3/4),x)
Output:
a**(3/4)*x**5*hyper((-3/4, 5/2), (7/2,), b*x**2*exp_polar(2*I*pi)/a)/5
\[ \int x^4 \left (a-b x^2\right )^{3/4} \, dx=\int { {\left (-b x^{2} + a\right )}^{\frac {3}{4}} x^{4} \,d x } \] Input:
integrate(x^4*(-b*x^2+a)^(3/4),x, algorithm="maxima")
Output:
integrate((-b*x^2 + a)^(3/4)*x^4, x)
\[ \int x^4 \left (a-b x^2\right )^{3/4} \, dx=\int { {\left (-b x^{2} + a\right )}^{\frac {3}{4}} x^{4} \,d x } \] Input:
integrate(x^4*(-b*x^2+a)^(3/4),x, algorithm="giac")
Output:
integrate((-b*x^2 + a)^(3/4)*x^4, x)
Timed out. \[ \int x^4 \left (a-b x^2\right )^{3/4} \, dx=\int x^4\,{\left (a-b\,x^2\right )}^{3/4} \,d x \] Input:
int(x^4*(a - b*x^2)^(3/4),x)
Output:
int(x^4*(a - b*x^2)^(3/4), x)
\[ \int x^4 \left (a-b x^2\right )^{3/4} \, dx=\frac {-\frac {4 \left (-b \,x^{2}+a \right )^{\frac {3}{4}} a^{2} x}{65}-\frac {2 \left (-b \,x^{2}+a \right )^{\frac {3}{4}} a b \,x^{3}}{39}+\frac {2 \left (-b \,x^{2}+a \right )^{\frac {3}{4}} b^{2} x^{5}}{13}+\frac {4 \left (\int \frac {1}{\left (-b \,x^{2}+a \right )^{\frac {1}{4}}}d x \right ) a^{3}}{65}}{b^{2}} \] Input:
int(x^4*(-b*x^2+a)^(3/4),x)
Output:
(2*( - 6*(a - b*x**2)**(3/4)*a**2*x - 5*(a - b*x**2)**(3/4)*a*b*x**3 + 15* (a - b*x**2)**(3/4)*b**2*x**5 + 6*int((a - b*x**2)**(3/4)/(a - b*x**2),x)* a**3))/(195*b**2)