Integrand size = 22, antiderivative size = 148 \[ \int \frac {x^4 \left (c+d x^4\right )}{\sqrt {a+b x^4}} \, dx=\frac {(7 b c-5 a d) x \sqrt {a+b x^4}}{21 b^2}+\frac {d x^5 \sqrt {a+b x^4}}{7 b}-\frac {a^{3/4} (7 b c-5 a d) \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{42 b^{9/4} \sqrt {a+b x^4}} \] Output:
1/21*(-5*a*d+7*b*c)*x*(b*x^4+a)^(1/2)/b^2+1/7*d*x^5*(b*x^4+a)^(1/2)/b-1/42 *a^(3/4)*(-5*a*d+7*b*c)*(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))/b^( 9/4)/(b*x^4+a)^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 10.12 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.60 \[ \int \frac {x^4 \left (c+d x^4\right )}{\sqrt {a+b x^4}} \, dx=\frac {x \left (-\left (\left (a+b x^4\right ) \left (-7 b c+5 a d-3 b d x^4\right )\right )+a (-7 b c+5 a d) \sqrt {1+\frac {b x^4}{a}} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1}{2},\frac {5}{4},-\frac {b x^4}{a}\right )\right )}{21 b^2 \sqrt {a+b x^4}} \] Input:
Integrate[(x^4*(c + d*x^4))/Sqrt[a + b*x^4],x]
Output:
(x*(-((a + b*x^4)*(-7*b*c + 5*a*d - 3*b*d*x^4)) + a*(-7*b*c + 5*a*d)*Sqrt[ 1 + (b*x^4)/a]*Hypergeometric2F1[1/4, 1/2, 5/4, -((b*x^4)/a)]))/(21*b^2*Sq rt[a + b*x^4])
Time = 0.39 (sec) , antiderivative size = 147, normalized size of antiderivative = 0.99, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {959, 843, 761}
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.
\(\displaystyle \int \frac {x^4 \left (c+d x^4\right )}{\sqrt {a+b x^4}} \, dx\) |
\(\Big \downarrow \) 959 |
\(\displaystyle \frac {(7 b c-5 a d) \int \frac {x^4}{\sqrt {b x^4+a}}dx}{7 b}+\frac {d x^5 \sqrt {a+b x^4}}{7 b}\) |
\(\Big \downarrow \) 843 |
\(\displaystyle \frac {(7 b c-5 a d) \left (\frac {x \sqrt {a+b x^4}}{3 b}-\frac {a \int \frac {1}{\sqrt {b x^4+a}}dx}{3 b}\right )}{7 b}+\frac {d x^5 \sqrt {a+b x^4}}{7 b}\) |
\(\Big \downarrow \) 761 |
\(\displaystyle \frac {(7 b c-5 a d) \left (\frac {x \sqrt {a+b x^4}}{3 b}-\frac {a^{3/4} \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{6 b^{5/4} \sqrt {a+b x^4}}\right )}{7 b}+\frac {d x^5 \sqrt {a+b x^4}}{7 b}\) |
Input:
Int[(x^4*(c + d*x^4))/Sqrt[a + b*x^4],x]
Output:
(d*x^5*Sqrt[a + b*x^4])/(7*b) + ((7*b*c - 5*a*d)*((x*Sqrt[a + b*x^4])/(3*b ) - (a^(3/4)*(Sqrt[a] + Sqrt[b]*x^2)*Sqrt[(a + b*x^4)/(Sqrt[a] + Sqrt[b]*x ^2)^2]*EllipticF[2*ArcTan[(b^(1/4)*x)/a^(1/4)], 1/2])/(6*b^(5/4)*Sqrt[a + b*x^4])))/(7*b)
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^(n - 1)*(c*x)^(m - n + 1)*((a + b*x^n)^(p + 1)/(b*(m + n*p + 1))), x] - Simp[ a*c^n*((m - n + 1)/(b*(m + n*p + 1))) Int[(c*x)^(m - n)*(a + b*x^n)^p, x] , x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n* p + 1, 0] && IntBinomialQ[a, b, c, n, m, p, x]
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n _)), x_Symbol] :> Simp[d*(e*x)^(m + 1)*((a + b*x^n)^(p + 1)/(b*e*(m + n*(p + 1) + 1))), x] - Simp[(a*d*(m + 1) - b*c*(m + n*(p + 1) + 1))/(b*(m + n*(p + 1) + 1)) Int[(e*x)^m*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, c, d, e, m, n, p}, x] && NeQ[b*c - a*d, 0] && NeQ[m + n*(p + 1) + 1, 0]
Result contains complex when optimal does not.
Time = 1.45 (sec) , antiderivative size = 116, normalized size of antiderivative = 0.78
method | result | size |
risch | \(-\frac {x \left (-3 d b \,x^{4}+5 a d -7 c b \right ) \sqrt {b \,x^{4}+a}}{21 b^{2}}+\frac {a \left (5 a d -7 c b \right ) \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )}{21 b^{2} \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}\) | \(116\) |
elliptic | \(\frac {d \,x^{5} \sqrt {b \,x^{4}+a}}{7 b}+\frac {\left (c -\frac {5 a d}{7 b}\right ) x \sqrt {b \,x^{4}+a}}{3 b}-\frac {a \left (c -\frac {5 a d}{7 b}\right ) \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )}{3 b \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}\) | \(127\) |
default | \(c \left (\frac {x \sqrt {b \,x^{4}+a}}{3 b}-\frac {a \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )}{3 b \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}\right )+d \left (\frac {x^{5} \sqrt {b \,x^{4}+a}}{7 b}-\frac {5 a x \sqrt {b \,x^{4}+a}}{21 b^{2}}+\frac {5 a^{2} \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )}{21 b^{2} \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}\right )\) | \(206\) |
Input:
int(x^4*(d*x^4+c)/(b*x^4+a)^(1/2),x,method=_RETURNVERBOSE)
Output:
-1/21*x*(-3*b*d*x^4+5*a*d-7*b*c)/b^2*(b*x^4+a)^(1/2)+1/21*a*(5*a*d-7*b*c)/ b^2/(I/a^(1/2)*b^(1/2))^(1/2)*(1-I/a^(1/2)*b^(1/2)*x^2)^(1/2)*(1+I/a^(1/2) *b^(1/2)*x^2)^(1/2)/(b*x^4+a)^(1/2)*EllipticF(x*(I/a^(1/2)*b^(1/2))^(1/2), I)
Time = 0.09 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.49 \[ \int \frac {x^4 \left (c+d x^4\right )}{\sqrt {a+b x^4}} \, dx=-\frac {{\left (7 \, b c - 5 \, a d\right )} \sqrt {b} \left (-\frac {a}{b}\right )^{\frac {3}{4}} F(\arcsin \left (\frac {\left (-\frac {a}{b}\right )^{\frac {1}{4}}}{x}\right )\,|\,-1) - {\left (3 \, b d x^{5} + {\left (7 \, b c - 5 \, a d\right )} x\right )} \sqrt {b x^{4} + a}}{21 \, b^{2}} \] Input:
integrate(x^4*(d*x^4+c)/(b*x^4+a)^(1/2),x, algorithm="fricas")
Output:
-1/21*((7*b*c - 5*a*d)*sqrt(b)*(-a/b)^(3/4)*elliptic_f(arcsin((-a/b)^(1/4) /x), -1) - (3*b*d*x^5 + (7*b*c - 5*a*d)*x)*sqrt(b*x^4 + a))/b^2
Result contains complex when optimal does not.
Time = 1.45 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.54 \[ \int \frac {x^4 \left (c+d x^4\right )}{\sqrt {a+b x^4}} \, dx=\frac {c x^{5} \Gamma \left (\frac {5}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {5}{4} \\ \frac {9}{4} \end {matrix}\middle | {\frac {b x^{4} e^{i \pi }}{a}} \right )}}{4 \sqrt {a} \Gamma \left (\frac {9}{4}\right )} + \frac {d x^{9} \Gamma \left (\frac {9}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {9}{4} \\ \frac {13}{4} \end {matrix}\middle | {\frac {b x^{4} e^{i \pi }}{a}} \right )}}{4 \sqrt {a} \Gamma \left (\frac {13}{4}\right )} \] Input:
integrate(x**4*(d*x**4+c)/(b*x**4+a)**(1/2),x)
Output:
c*x**5*gamma(5/4)*hyper((1/2, 5/4), (9/4,), b*x**4*exp_polar(I*pi)/a)/(4*s qrt(a)*gamma(9/4)) + d*x**9*gamma(9/4)*hyper((1/2, 9/4), (13/4,), b*x**4*e xp_polar(I*pi)/a)/(4*sqrt(a)*gamma(13/4))
\[ \int \frac {x^4 \left (c+d x^4\right )}{\sqrt {a+b x^4}} \, dx=\int { \frac {{\left (d x^{4} + c\right )} x^{4}}{\sqrt {b x^{4} + a}} \,d x } \] Input:
integrate(x^4*(d*x^4+c)/(b*x^4+a)^(1/2),x, algorithm="maxima")
Output:
integrate((d*x^4 + c)*x^4/sqrt(b*x^4 + a), x)
\[ \int \frac {x^4 \left (c+d x^4\right )}{\sqrt {a+b x^4}} \, dx=\int { \frac {{\left (d x^{4} + c\right )} x^{4}}{\sqrt {b x^{4} + a}} \,d x } \] Input:
integrate(x^4*(d*x^4+c)/(b*x^4+a)^(1/2),x, algorithm="giac")
Output:
integrate((d*x^4 + c)*x^4/sqrt(b*x^4 + a), x)
Timed out. \[ \int \frac {x^4 \left (c+d x^4\right )}{\sqrt {a+b x^4}} \, dx=\int \frac {x^4\,\left (d\,x^4+c\right )}{\sqrt {b\,x^4+a}} \,d x \] Input:
int((x^4*(c + d*x^4))/(a + b*x^4)^(1/2),x)
Output:
int((x^4*(c + d*x^4))/(a + b*x^4)^(1/2), x)
\[ \int \frac {x^4 \left (c+d x^4\right )}{\sqrt {a+b x^4}} \, dx=\frac {-5 \sqrt {b \,x^{4}+a}\, a d x +7 \sqrt {b \,x^{4}+a}\, b c x +3 \sqrt {b \,x^{4}+a}\, b d \,x^{5}+5 \left (\int \frac {\sqrt {b \,x^{4}+a}}{b \,x^{4}+a}d x \right ) a^{2} d -7 \left (\int \frac {\sqrt {b \,x^{4}+a}}{b \,x^{4}+a}d x \right ) a b c}{21 b^{2}} \] Input:
int(x^4*(d*x^4+c)/(b*x^4+a)^(1/2),x)
Output:
( - 5*sqrt(a + b*x**4)*a*d*x + 7*sqrt(a + b*x**4)*b*c*x + 3*sqrt(a + b*x** 4)*b*d*x**5 + 5*int(sqrt(a + b*x**4)/(a + b*x**4),x)*a**2*d - 7*int(sqrt(a + b*x**4)/(a + b*x**4),x)*a*b*c)/(21*b**2)