\(\int \frac {c+d x^4}{x^4 (a+b x^4)^{7/4}} \, dx\) [143]

Optimal result

Integrand size = 22, antiderivative size = 123 \[ \int \frac {c+d x^4}{x^4 \left (a+b x^4\right )^{7/4}} \, dx=-\frac {c}{3 a x^3 \left (a+b x^4\right )^{3/4}}-\frac {(2 b c-a d) x}{3 a^2 \left (a+b x^4\right )^{3/4}}+\frac {2 \sqrt {b} (2 b c-a d) \left (1+\frac {a}{b x^4}\right )^{3/4} x^3 \operatorname {EllipticF}\left (\frac {1}{2} \cot ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right ),2\right )}{3 a^{5/2} \left (a+b x^4\right )^{3/4}} \] Output:

-1/3*c/a/x^3/(b*x^4+a)^(3/4)-1/3*(-a*d+2*b*c)*x/a^2/(b*x^4+a)^(3/4)+2/3*b^ 
(1/2)*(-a*d+2*b*c)*(1+a/b/x^4)^(3/4)*x^3*InverseJacobiAM(1/2*arccot(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 = 85, normalized size of antiderivative = 0.69 \[ \int \frac {c+d x^4}{x^4 \left (a+b x^4\right )^{7/4}} \, dx=\frac {-a c-2 b c x^4+a d x^4+2 (-2 b c+a d) x^4 \left (1+\frac {b x^4}{a}\right )^{3/4} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {3}{4},\frac {5}{4},-\frac {b x^4}{a}\right )}{3 a^2 x^3 \left (a+b x^4\right )^{3/4}} \] Input:

Integrate[(c + d*x^4)/(x^4*(a + b*x^4)^(7/4)),x]
 

Output:

(-(a*c) - 2*b*c*x^4 + a*d*x^4 + 2*(-2*b*c + a*d)*x^4*(1 + (b*x^4)/a)^(3/4) 
*Hypergeometric2F1[1/4, 3/4, 5/4, -((b*x^4)/a)])/(3*a^2*x^3*(a + b*x^4)^(3 
/4))
 

Rubi [A] (verified)

Time = 0.46 (sec) , antiderivative size = 120, normalized size of antiderivative = 0.98, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {955, 749, 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.

\(\displaystyle \int \frac {c+d x^4}{x^4 \left (a+b x^4\right )^{7/4}} \, dx\)

\(\Big \downarrow \) 955

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

\(\Big \downarrow \) 749

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

\(\Big \downarrow \) 768

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

\(\Big \downarrow \) 858

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

\(\Big \downarrow \) 807

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

\(\Big \downarrow \) 229

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

Input:

Int[(c + d*x^4)/(x^4*(a + b*x^4)^(7/4)),x]
 

Output:

-1/3*c/(a*x^3*(a + b*x^4)^(3/4)) - ((2*b*c - a*d)*(x/(3*a*(a + b*x^4)^(3/4 
)) - (2*Sqrt[b]*(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))))/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 749

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(-x)*((a + b*x^n)^(p + 
 1)/(a*n*(p + 1))), x] + Simp[(n*(p + 1) + 1)/(a*n*(p + 1))   Int[(a + b*x^ 
n)^(p + 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && LtQ[p, -1] && (Inte 
gerQ[2*p] || Denominator[p + 1/n] < Denominator[p])
 

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 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]

\[\int \frac {d \,x^{4}+c}{x^{4} \left (b \,x^{4}+a \right )^{\frac {7}{4}}}d x\]

Input:

int((d*x^4+c)/x^4/(b*x^4+a)^(7/4),x)
 

Output:

int((d*x^4+c)/x^4/(b*x^4+a)^(7/4),x)
 

Fricas [F]

\[ \int \frac {c+d x^4}{x^4 \left (a+b x^4\right )^{7/4}} \, dx=\int { \frac {d x^{4} + c}{{\left (b x^{4} + a\right )}^{\frac {7}{4}} x^{4}} \,d x } \] Input:

integrate((d*x^4+c)/x^4/(b*x^4+a)^(7/4),x, algorithm="fricas")
 

Output:

integral((b*x^4 + a)^(1/4)*(d*x^4 + c)/(b^2*x^12 + 2*a*b*x^8 + a^2*x^4), x 
)
 

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 29.40 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.67 \[ \int \frac {c+d x^4}{x^4 \left (a+b x^4\right )^{7/4}} \, dx=\frac {c \Gamma \left (- \frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {3}{4}, \frac {7}{4} \\ \frac {1}{4} \end {matrix}\middle | {\frac {b x^{4} e^{i \pi }}{a}} \right )}}{4 a^{\frac {7}{4}} x^{3} \Gamma \left (\frac {1}{4}\right )} + \frac {d x \Gamma \left (\frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{4}, \frac {7}{4} \\ \frac {5}{4} \end {matrix}\middle | {\frac {b x^{4} e^{i \pi }}{a}} \right )}}{4 a^{\frac {7}{4}} \Gamma \left (\frac {5}{4}\right )} \] Input:

integrate((d*x**4+c)/x**4/(b*x**4+a)**(7/4),x)
 

Output:

c*gamma(-3/4)*hyper((-3/4, 7/4), (1/4,), b*x**4*exp_polar(I*pi)/a)/(4*a**( 
7/4)*x**3*gamma(1/4)) + d*x*gamma(1/4)*hyper((1/4, 7/4), (5/4,), b*x**4*ex 
p_polar(I*pi)/a)/(4*a**(7/4)*gamma(5/4))
 

Maxima [F]

\[ \int \frac {c+d x^4}{x^4 \left (a+b x^4\right )^{7/4}} \, dx=\int { \frac {d x^{4} + c}{{\left (b x^{4} + a\right )}^{\frac {7}{4}} x^{4}} \,d x } \] Input:

integrate((d*x^4+c)/x^4/(b*x^4+a)^(7/4),x, algorithm="maxima")
 

Output:

integrate((d*x^4 + c)/((b*x^4 + a)^(7/4)*x^4), x)
 

Giac [F]

\[ \int \frac {c+d x^4}{x^4 \left (a+b x^4\right )^{7/4}} \, dx=\int { \frac {d x^{4} + c}{{\left (b x^{4} + a\right )}^{\frac {7}{4}} x^{4}} \,d x } \] Input:

integrate((d*x^4+c)/x^4/(b*x^4+a)^(7/4),x, algorithm="giac")
 

Output:

integrate((d*x^4 + c)/((b*x^4 + a)^(7/4)*x^4), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {c+d x^4}{x^4 \left (a+b x^4\right )^{7/4}} \, dx=\int \frac {d\,x^4+c}{x^4\,{\left (b\,x^4+a\right )}^{7/4}} \,d x \] Input:

int((c + d*x^4)/(x^4*(a + b*x^4)^(7/4)),x)
 

Output:

int((c + d*x^4)/(x^4*(a + b*x^4)^(7/4)), x)
 

Reduce [F]

\[ \int \frac {c+d x^4}{x^4 \left (a+b x^4\right )^{7/4}} \, dx=\left (\int \frac {1}{\left (b \,x^{4}+a \right )^{\frac {3}{4}} a \,x^{4}+\left (b \,x^{4}+a \right )^{\frac {3}{4}} b \,x^{8}}d x \right ) c +\left (\int \frac {1}{\left (b \,x^{4}+a \right )^{\frac {3}{4}} a +\left (b \,x^{4}+a \right )^{\frac {3}{4}} b \,x^{4}}d x \right ) d \] Input:

int((d*x^4+c)/x^4/(b*x^4+a)^(7/4),x)
 

Output:

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