Integrand size = 22, antiderivative size = 150 \[ \int \frac {c+d x^4}{x^8 \sqrt {a+b x^4}} \, dx=-\frac {c \sqrt {a+b x^4}}{7 a x^7}+\frac {(5 b c-7 a d) \sqrt {a+b x^4}}{21 a^2 x^3}+\frac {b^{3/4} (5 b c-7 a d) \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{42 a^{9/4} \sqrt {a+b x^4}} \] Output:
-1/7*c*(b*x^4+a)^(1/2)/a/x^7+1/21*(-7*a*d+5*b*c)*(b*x^4+a)^(1/2)/a^2/x^3+1 /42*b^(3/4)*(-7*a*d+5*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))/ a^(9/4)/(b*x^4+a)^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 10.04 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.52 \[ \int \frac {c+d x^4}{x^8 \sqrt {a+b x^4}} \, dx=\frac {-3 c \left (a+b x^4\right )+(5 b c-7 a d) x^4 \sqrt {1+\frac {b x^4}{a}} \operatorname {Hypergeometric2F1}\left (-\frac {3}{4},\frac {1}{2},\frac {1}{4},-\frac {b x^4}{a}\right )}{21 a x^7 \sqrt {a+b x^4}} \] Input:
Integrate[(c + d*x^4)/(x^8*Sqrt[a + b*x^4]),x]
Output:
(-3*c*(a + b*x^4) + (5*b*c - 7*a*d)*x^4*Sqrt[1 + (b*x^4)/a]*Hypergeometric 2F1[-3/4, 1/2, 1/4, -((b*x^4)/a)])/(21*a*x^7*Sqrt[a + b*x^4])
Time = 0.38 (sec) , antiderivative size = 149, 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 = {955, 847, 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.
\(\displaystyle \int \frac {c+d x^4}{x^8 \sqrt {a+b x^4}} \, dx\) |
\(\Big \downarrow \) 955 |
\(\displaystyle -\frac {(5 b c-7 a d) \int \frac {1}{x^4 \sqrt {b x^4+a}}dx}{7 a}-\frac {c \sqrt {a+b x^4}}{7 a x^7}\) |
\(\Big \downarrow \) 847 |
\(\displaystyle -\frac {(5 b c-7 a d) \left (-\frac {b \int \frac {1}{\sqrt {b x^4+a}}dx}{3 a}-\frac {\sqrt {a+b x^4}}{3 a x^3}\right )}{7 a}-\frac {c \sqrt {a+b x^4}}{7 a x^7}\) |
\(\Big \downarrow \) 761 |
\(\displaystyle -\frac {(5 b c-7 a d) \left (-\frac {b^{3/4} \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{6 a^{5/4} \sqrt {a+b x^4}}-\frac {\sqrt {a+b x^4}}{3 a x^3}\right )}{7 a}-\frac {c \sqrt {a+b x^4}}{7 a x^7}\) |
Input:
Int[(c + d*x^4)/(x^8*Sqrt[a + b*x^4]),x]
Output:
-1/7*(c*Sqrt[a + b*x^4])/(a*x^7) - ((5*b*c - 7*a*d)*(-1/3*Sqrt[a + b*x^4]/ (a*x^3) - (b^(3/4)*(Sqrt[a] + Sqrt[b]*x^2)*Sqrt[(a + b*x^4)/(Sqrt[a] + Sqr t[b]*x^2)^2]*EllipticF[2*ArcTan[(b^(1/4)*x)/a^(1/4)], 1/2])/(6*a^(5/4)*Sqr t[a + b*x^4])))/(7*a)
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*(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[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n _)), x_Symbol] :> Simp[c*(e*x)^(m + 1)*((a + b*x^n)^(p + 1)/(a*e*(m + 1))), x] + Simp[(a*d*(m + 1) - b*c*(m + n*(p + 1) + 1))/(a*e^n*(m + 1)) Int[(e *x)^(m + n)*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[b* c - a*d, 0] && (IntegerQ[n] || GtQ[e, 0]) && ((GtQ[n, 0] && LtQ[m, -1]) || (LtQ[n, 0] && GtQ[m + n, -1])) && !ILtQ[p, -1]
Result contains complex when optimal does not.
Time = 1.59 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.81
method | result | size |
risch | \(-\frac {\sqrt {b \,x^{4}+a}\, \left (7 a d \,x^{4}-5 b c \,x^{4}+3 a c \right )}{21 a^{2} x^{7}}-\frac {b \left (7 a d -5 c b \right ) \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )}{21 a^{2} \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}\) | \(121\) |
elliptic | \(-\frac {c \sqrt {b \,x^{4}+a}}{7 a \,x^{7}}-\frac {\left (7 a d -5 c b \right ) \sqrt {b \,x^{4}+a}}{21 a^{2} x^{3}}-\frac {b \left (7 a d -5 c b \right ) \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )}{21 a^{2} \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}\) | \(129\) |
default | \(c \left (-\frac {\sqrt {b \,x^{4}+a}}{7 a \,x^{7}}+\frac {5 b \sqrt {b \,x^{4}+a}}{21 a^{2} x^{3}}+\frac {5 b^{2} \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )}{21 a^{2} \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}\right )+d \left (-\frac {\sqrt {b \,x^{4}+a}}{3 a \,x^{3}}-\frac {b \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )}{3 a \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}\right )\) | \(210\) |
Input:
int((d*x^4+c)/x^8/(b*x^4+a)^(1/2),x,method=_RETURNVERBOSE)
Output:
-1/21*(b*x^4+a)^(1/2)*(7*a*d*x^4-5*b*c*x^4+3*a*c)/a^2/x^7-1/21*b*(7*a*d-5* b*c)/a^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 = 0.08 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.50 \[ \int \frac {c+d x^4}{x^8 \sqrt {a+b x^4}} \, dx=-\frac {{\left (5 \, b c - 7 \, a d\right )} \sqrt {a} x^{7} \left (-\frac {b}{a}\right )^{\frac {3}{4}} F(\arcsin \left (x \left (-\frac {b}{a}\right )^{\frac {1}{4}}\right )\,|\,-1) - {\left ({\left (5 \, b c - 7 \, a d\right )} x^{4} - 3 \, a c\right )} \sqrt {b x^{4} + a}}{21 \, a^{2} x^{7}} \] Input:
integrate((d*x^4+c)/x^8/(b*x^4+a)^(1/2),x, algorithm="fricas")
Output:
-1/21*((5*b*c - 7*a*d)*sqrt(a)*x^7*(-b/a)^(3/4)*elliptic_f(arcsin(x*(-b/a) ^(1/4)), -1) - ((5*b*c - 7*a*d)*x^4 - 3*a*c)*sqrt(b*x^4 + a))/(a^2*x^7)
Result contains complex when optimal does not.
Time = 1.43 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.60 \[ \int \frac {c+d x^4}{x^8 \sqrt {a+b x^4}} \, dx=\frac {c \Gamma \left (- \frac {7}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {7}{4}, \frac {1}{2} \\ - \frac {3}{4} \end {matrix}\middle | {\frac {b x^{4} e^{i \pi }}{a}} \right )}}{4 \sqrt {a} x^{7} \Gamma \left (- \frac {3}{4}\right )} + \frac {d \Gamma \left (- \frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {3}{4}, \frac {1}{2} \\ \frac {1}{4} \end {matrix}\middle | {\frac {b x^{4} e^{i \pi }}{a}} \right )}}{4 \sqrt {a} x^{3} \Gamma \left (\frac {1}{4}\right )} \] Input:
integrate((d*x**4+c)/x**8/(b*x**4+a)**(1/2),x)
Output:
c*gamma(-7/4)*hyper((-7/4, 1/2), (-3/4,), b*x**4*exp_polar(I*pi)/a)/(4*sqr t(a)*x**7*gamma(-3/4)) + d*gamma(-3/4)*hyper((-3/4, 1/2), (1/4,), b*x**4*e xp_polar(I*pi)/a)/(4*sqrt(a)*x**3*gamma(1/4))
\[ \int \frac {c+d x^4}{x^8 \sqrt {a+b x^4}} \, dx=\int { \frac {d x^{4} + c}{\sqrt {b x^{4} + a} x^{8}} \,d x } \] Input:
integrate((d*x^4+c)/x^8/(b*x^4+a)^(1/2),x, algorithm="maxima")
Output:
integrate((d*x^4 + c)/(sqrt(b*x^4 + a)*x^8), x)
\[ \int \frac {c+d x^4}{x^8 \sqrt {a+b x^4}} \, dx=\int { \frac {d x^{4} + c}{\sqrt {b x^{4} + a} x^{8}} \,d x } \] Input:
integrate((d*x^4+c)/x^8/(b*x^4+a)^(1/2),x, algorithm="giac")
Output:
integrate((d*x^4 + c)/(sqrt(b*x^4 + a)*x^8), x)
Timed out. \[ \int \frac {c+d x^4}{x^8 \sqrt {a+b x^4}} \, dx=\int \frac {d\,x^4+c}{x^8\,\sqrt {b\,x^4+a}} \,d x \] Input:
int((c + d*x^4)/(x^8*(a + b*x^4)^(1/2)),x)
Output:
int((c + d*x^4)/(x^8*(a + b*x^4)^(1/2)), x)
\[ \int \frac {c+d x^4}{x^8 \sqrt {a+b x^4}} \, dx=\frac {-\sqrt {b \,x^{4}+a}\, d -7 \left (\int \frac {\sqrt {b \,x^{4}+a}}{b \,x^{12}+a \,x^{8}}d x \right ) a d \,x^{7}+5 \left (\int \frac {\sqrt {b \,x^{4}+a}}{b \,x^{12}+a \,x^{8}}d x \right ) b c \,x^{7}}{5 b \,x^{7}} \] Input:
int((d*x^4+c)/x^8/(b*x^4+a)^(1/2),x)
Output:
( - sqrt(a + b*x**4)*d - 7*int(sqrt(a + b*x**4)/(a*x**8 + b*x**12),x)*a*d* x**7 + 5*int(sqrt(a + b*x**4)/(a*x**8 + b*x**12),x)*b*c*x**7)/(5*b*x**7)