\(\int \frac {c+d x^4}{x^8 (a+b x^4)^{3/4}} \, dx\) [106]

Optimal result

Integrand size = 22, antiderivative size = 125 \[ \int \frac {c+d x^4}{x^8 \left (a+b x^4\right )^{3/4}} \, dx=-\frac {c \sqrt [4]{a+b x^4}}{7 a x^7}+\frac {(6 b c-7 a d) \sqrt [4]{a+b x^4}}{21 a^2 x^3}-\frac {2 b^{3/2} (6 b c-7 a d) \left (1+\frac {a}{b x^4}\right )^{3/4} x^3 \operatorname {EllipticF}\left (\frac {1}{2} \cot ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right ),2\right )}{21 a^{5/2} \left (a+b x^4\right )^{3/4}} \] Output:

-1/7*c*(b*x^4+a)^(1/4)/a/x^7+1/21*(-7*a*d+6*b*c)*(b*x^4+a)^(1/4)/a^2/x^3-2 
/21*b^(3/2)*(-7*a*d+6*b*c)*(1+a/b/x^4)^(3/4)*x^3*InverseJacobiAM(1/2*arcco 
t(b^(1/2)*x^2/a^(1/2)),2^(1/2))/a^(5/2)/(b*x^4+a)^(3/4)
 

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.

Time = 10.03 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.62 \[ \int \frac {c+d x^4}{x^8 \left (a+b x^4\right )^{3/4}} \, dx=\frac {-3 c \left (a+b x^4\right )+(6 b c-7 a d) x^4 \left (1+\frac {b x^4}{a}\right )^{3/4} \operatorname {Hypergeometric2F1}\left (-\frac {3}{4},\frac {3}{4},\frac {1}{4},-\frac {b x^4}{a}\right )}{21 a x^7 \left (a+b x^4\right )^{3/4}} \] Input:

Integrate[(c + d*x^4)/(x^8*(a + b*x^4)^(3/4)),x]
 

Output:

(-3*c*(a + b*x^4) + (6*b*c - 7*a*d)*x^4*(1 + (b*x^4)/a)^(3/4)*Hypergeometr 
ic2F1[-3/4, 3/4, 1/4, -((b*x^4)/a)])/(21*a*x^7*(a + b*x^4)^(3/4))
 

Rubi [A] (verified)

Time = 0.46 (sec) , antiderivative size = 124, normalized size of antiderivative = 0.99, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {955, 847, 768, 858, 807, 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.

\(\displaystyle \int \frac {c+d x^4}{x^8 \left (a+b x^4\right )^{3/4}} \, dx\)

\(\Big \downarrow \) 955

\(\displaystyle -\frac {(6 b c-7 a d) \int \frac {1}{x^4 \left (b x^4+a\right )^{3/4}}dx}{7 a}-\frac {c \sqrt [4]{a+b x^4}}{7 a x^7}\)

\(\Big \downarrow \) 847

\(\displaystyle -\frac {(6 b c-7 a d) \left (-\frac {2 b \int \frac {1}{\left (b x^4+a\right )^{3/4}}dx}{3 a}-\frac {\sqrt [4]{a+b x^4}}{3 a x^3}\right )}{7 a}-\frac {c \sqrt [4]{a+b x^4}}{7 a x^7}\)

\(\Big \downarrow \) 768

\(\displaystyle -\frac {(6 b c-7 a d) \left (-\frac {2 b x^3 \left (\frac {a}{b x^4}+1\right )^{3/4} \int \frac {1}{\left (\frac {a}{b x^4}+1\right )^{3/4} x^3}dx}{3 a \left (a+b x^4\right )^{3/4}}-\frac {\sqrt [4]{a+b x^4}}{3 a x^3}\right )}{7 a}-\frac {c \sqrt [4]{a+b x^4}}{7 a x^7}\)

\(\Big \downarrow \) 858

\(\displaystyle -\frac {(6 b c-7 a d) \left (\frac {2 b x^3 \left (\frac {a}{b x^4}+1\right )^{3/4} \int \frac {1}{\left (\frac {a}{b x^4}+1\right )^{3/4} x}d\frac {1}{x}}{3 a \left (a+b x^4\right )^{3/4}}-\frac {\sqrt [4]{a+b x^4}}{3 a x^3}\right )}{7 a}-\frac {c \sqrt [4]{a+b x^4}}{7 a x^7}\)

\(\Big \downarrow \) 807

\(\displaystyle -\frac {(6 b c-7 a d) \left (\frac {b x^3 \left (\frac {a}{b x^4}+1\right )^{3/4} \int \frac {1}{\left (\frac {a}{b x^2}+1\right )^{3/4}}d\frac {1}{x^2}}{3 a \left (a+b x^4\right )^{3/4}}-\frac {\sqrt [4]{a+b x^4}}{3 a x^3}\right )}{7 a}-\frac {c \sqrt [4]{a+b x^4}}{7 a x^7}\)

\(\Big \downarrow \) 229

\(\displaystyle -\frac {(6 b c-7 a d) \left (\frac {2 b^{3/2} x^3 \left (\frac {a}{b x^4}+1\right )^{3/4} \operatorname {EllipticF}\left (\frac {1}{2} \arctan \left (\frac {\sqrt {a}}{\sqrt {b} x^2}\right ),2\right )}{3 a^{3/2} \left (a+b x^4\right )^{3/4}}-\frac {\sqrt [4]{a+b x^4}}{3 a x^3}\right )}{7 a}-\frac {c \sqrt [4]{a+b x^4}}{7 a x^7}\)

Input:

Int[(c + d*x^4)/(x^8*(a + b*x^4)^(3/4)),x]
 

Output:

-1/7*(c*(a + b*x^4)^(1/4))/(a*x^7) - ((6*b*c - 7*a*d)*(-1/3*(a + b*x^4)^(1 
/4)/(a*x^3) + (2*b^(3/2)*(1 + a/(b*x^4))^(3/4)*x^3*EllipticF[ArcTan[Sqrt[a 
]/(Sqrt[b]*x^2)]/2, 2])/(3*a^(3/2)*(a + b*x^4)^(3/4))))/(7*a)
 

Defintions of rubi rules used

rule 229

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]
 

rule 768

Int[((a_) + (b_.)*(x_)^4)^(-3/4), x_Symbol] :> Simp[x^3*((1 + a/(b*x^4))^(3 
/4)/(a + b*x^4)^(3/4))   Int[1/(x^3*(1 + a/(b*x^4))^(3/4)), x], x] /; FreeQ 
[{a, b}, x]
 

rule 807

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]
 

rule 847

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]
 

rule 858

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> -Subst[Int[(a + 
b/x^n)^p/x^(m + 2), x], x, 1/x] /; FreeQ[{a, b, p}, x] && ILtQ[n, 0] && Int 
egerQ[m]
 

rule 955

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]
 

Maple [F]

\[\int \frac {d \,x^{4}+c}{x^{8} \left (b \,x^{4}+a \right )^{\frac {3}{4}}}d x\]

Input:

int((d*x^4+c)/x^8/(b*x^4+a)^(3/4),x)
 

Output:

int((d*x^4+c)/x^8/(b*x^4+a)^(3/4),x)
 

Fricas [F]

\[ \int \frac {c+d x^4}{x^8 \left (a+b x^4\right )^{3/4}} \, dx=\int { \frac {d x^{4} + c}{{\left (b x^{4} + a\right )}^{\frac {3}{4}} x^{8}} \,d x } \] Input:

integrate((d*x^4+c)/x^8/(b*x^4+a)^(3/4),x, algorithm="fricas")
 

Output:

integral((b*x^4 + a)^(1/4)*(d*x^4 + c)/(b*x^12 + a*x^8), x)
 

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 2.41 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.72 \[ \int \frac {c+d x^4}{x^8 \left (a+b x^4\right )^{3/4}} \, dx=\frac {c \Gamma \left (- \frac {7}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {7}{4}, \frac {3}{4} \\ - \frac {3}{4} \end {matrix}\middle | {\frac {b x^{4} e^{i \pi }}{a}} \right )}}{4 a^{\frac {3}{4}} x^{7} \Gamma \left (- \frac {3}{4}\right )} + \frac {d \Gamma \left (- \frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {3}{4}, \frac {3}{4} \\ \frac {1}{4} \end {matrix}\middle | {\frac {b x^{4} e^{i \pi }}{a}} \right )}}{4 a^{\frac {3}{4}} x^{3} \Gamma \left (\frac {1}{4}\right )} \] Input:

integrate((d*x**4+c)/x**8/(b*x**4+a)**(3/4),x)
 

Output:

c*gamma(-7/4)*hyper((-7/4, 3/4), (-3/4,), b*x**4*exp_polar(I*pi)/a)/(4*a** 
(3/4)*x**7*gamma(-3/4)) + d*gamma(-3/4)*hyper((-3/4, 3/4), (1/4,), b*x**4* 
exp_polar(I*pi)/a)/(4*a**(3/4)*x**3*gamma(1/4))
 

Maxima [F]

\[ \int \frac {c+d x^4}{x^8 \left (a+b x^4\right )^{3/4}} \, dx=\int { \frac {d x^{4} + c}{{\left (b x^{4} + a\right )}^{\frac {3}{4}} x^{8}} \,d x } \] Input:

integrate((d*x^4+c)/x^8/(b*x^4+a)^(3/4),x, algorithm="maxima")
 

Output:

integrate((d*x^4 + c)/((b*x^4 + a)^(3/4)*x^8), x)
 

Giac [F]

\[ \int \frac {c+d x^4}{x^8 \left (a+b x^4\right )^{3/4}} \, dx=\int { \frac {d x^{4} + c}{{\left (b x^{4} + a\right )}^{\frac {3}{4}} x^{8}} \,d x } \] Input:

integrate((d*x^4+c)/x^8/(b*x^4+a)^(3/4),x, algorithm="giac")
 

Output:

integrate((d*x^4 + c)/((b*x^4 + a)^(3/4)*x^8), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {c+d x^4}{x^8 \left (a+b x^4\right )^{3/4}} \, dx=\int \frac {d\,x^4+c}{x^8\,{\left (b\,x^4+a\right )}^{3/4}} \,d x \] Input:

int((c + d*x^4)/(x^8*(a + b*x^4)^(3/4)),x)
 

Output:

int((c + d*x^4)/(x^8*(a + b*x^4)^(3/4)), x)
 

Reduce [F]

\[ \int \frac {c+d x^4}{x^8 \left (a+b x^4\right )^{3/4}} \, dx=\left (\int \frac {1}{\left (b \,x^{4}+a \right )^{\frac {3}{4}} x^{8}}d x \right ) c +\left (\int \frac {1}{\left (b \,x^{4}+a \right )^{\frac {3}{4}} x^{4}}d x \right ) d \] Input:

int((d*x^4+c)/x^8/(b*x^4+a)^(3/4),x)
 

Output:

int(1/((a + b*x**4)**(3/4)*x**8),x)*c + int(1/((a + b*x**4)**(3/4)*x**4),x 
)*d