\(\int \frac {c+d x^4}{x^5 (a+b x^4)^{7/4}} \, dx\) [129]

Optimal result

Integrand size = 22, antiderivative size = 121 \[ \int \frac {c+d x^4}{x^5 \left (a+b x^4\right )^{7/4}} \, dx=-\frac {b c-a d}{3 a^2 \left (a+b x^4\right )^{3/4}}-\frac {c \sqrt [4]{a+b x^4}}{4 a^2 x^4}+\frac {(7 b c-4 a d) \arctan \left (\frac {\sqrt [4]{a+b x^4}}{\sqrt [4]{a}}\right )}{8 a^{11/4}}+\frac {(7 b c-4 a d) \text {arctanh}\left (\frac {\sqrt [4]{a+b x^4}}{\sqrt [4]{a}}\right )}{8 a^{11/4}} \] Output:

-1/3*(-a*d+b*c)/a^2/(b*x^4+a)^(3/4)-1/4*c*(b*x^4+a)^(1/4)/a^2/x^4+1/8*(-4* 
a*d+7*b*c)*arctan((b*x^4+a)^(1/4)/a^(1/4))/a^(11/4)+1/8*(-4*a*d+7*b*c)*arc 
tanh((b*x^4+a)^(1/4)/a^(1/4))/a^(11/4)
 

Mathematica [A] (verified)

Time = 0.26 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.89 \[ \int \frac {c+d x^4}{x^5 \left (a+b x^4\right )^{7/4}} \, dx=\frac {\frac {2 a^{3/4} \left (-3 a c-7 b c x^4+4 a d x^4\right )}{x^4 \left (a+b x^4\right )^{3/4}}+3 (7 b c-4 a d) \arctan \left (\frac {\sqrt [4]{a+b x^4}}{\sqrt [4]{a}}\right )+3 (7 b c-4 a d) \text {arctanh}\left (\frac {\sqrt [4]{a+b x^4}}{\sqrt [4]{a}}\right )}{24 a^{11/4}} \] Input:

Integrate[(c + d*x^4)/(x^5*(a + b*x^4)^(7/4)),x]
 

Output:

((2*a^(3/4)*(-3*a*c - 7*b*c*x^4 + 4*a*d*x^4))/(x^4*(a + b*x^4)^(3/4)) + 3* 
(7*b*c - 4*a*d)*ArcTan[(a + b*x^4)^(1/4)/a^(1/4)] + 3*(7*b*c - 4*a*d)*ArcT 
anh[(a + b*x^4)^(1/4)/a^(1/4)])/(24*a^(11/4))
 

Rubi [A] (verified)

Time = 0.40 (sec) , antiderivative size = 125, normalized size of antiderivative = 1.03, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.318, Rules used = {948, 87, 61, 73, 756, 216, 219}

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.

\(\displaystyle \int \frac {c+d x^4}{x^5 \left (a+b x^4\right )^{7/4}} \, dx\)

\(\Big \downarrow \) 948

\(\displaystyle \frac {1}{4} \int \frac {d x^4+c}{x^8 \left (b x^4+a\right )^{7/4}}dx^4\)

\(\Big \downarrow \) 87

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

\(\Big \downarrow \) 61

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

\(\Big \downarrow \) 73

\(\displaystyle \frac {1}{4} \left (-\frac {(7 b c-4 a d) \left (\frac {4 \int \frac {1}{\frac {x^{16}}{b}-\frac {a}{b}}d\sqrt [4]{b x^4+a}}{a b}+\frac {4}{3 a \left (a+b x^4\right )^{3/4}}\right )}{4 a}-\frac {c}{a x^4 \left (a+b x^4\right )^{3/4}}\right )\)

\(\Big \downarrow \) 756

\(\displaystyle \frac {1}{4} \left (-\frac {(7 b c-4 a d) \left (\frac {4 \left (-\frac {b \int \frac {1}{\sqrt {a}-x^8}d\sqrt [4]{b x^4+a}}{2 \sqrt {a}}-\frac {b \int \frac {1}{x^8+\sqrt {a}}d\sqrt [4]{b x^4+a}}{2 \sqrt {a}}\right )}{a b}+\frac {4}{3 a \left (a+b x^4\right )^{3/4}}\right )}{4 a}-\frac {c}{a x^4 \left (a+b x^4\right )^{3/4}}\right )\)

\(\Big \downarrow \) 216

\(\displaystyle \frac {1}{4} \left (-\frac {(7 b c-4 a d) \left (\frac {4 \left (-\frac {b \int \frac {1}{\sqrt {a}-x^8}d\sqrt [4]{b x^4+a}}{2 \sqrt {a}}-\frac {b \arctan \left (\frac {\sqrt [4]{a+b x^4}}{\sqrt [4]{a}}\right )}{2 a^{3/4}}\right )}{a b}+\frac {4}{3 a \left (a+b x^4\right )^{3/4}}\right )}{4 a}-\frac {c}{a x^4 \left (a+b x^4\right )^{3/4}}\right )\)

\(\Big \downarrow \) 219

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

Input:

Int[(c + d*x^4)/(x^5*(a + b*x^4)^(7/4)),x]
 

Output:

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

Defintions of rubi rules used

rule 61

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ 
(a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((b*c - a*d)*(m + 1))), x] - Simp[d*(( 
m + n + 2)/((b*c - a*d)*(m + 1)))   Int[(a + b*x)^(m + 1)*(c + d*x)^n, x], 
x] /; FreeQ[{a, b, c, d, n}, x] && LtQ[m, -1] &&  !(LtQ[n, -1] && (EqQ[a, 0 
] || (NeQ[c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && IntLinearQ[a, b, c, d 
, m, n, x]
 

rule 73

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ 
{p = Denominator[m]}, Simp[p/b   Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + 
 d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt 
Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL 
inearQ[a, b, c, d, m, n, x]
 

rule 87

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p 
_.), x_] :> Simp[(-(b*e - a*f))*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(f*(p 
+ 1)*(c*f - d*e))), x] - Simp[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p 
+ 1)))/(f*(p + 1)*(c*f - d*e))   Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] 
/; FreeQ[{a, b, c, d, e, f, n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || Intege 
rQ[p] ||  !(IntegerQ[n] ||  !(EqQ[e, 0] ||  !(EqQ[c, 0] || LtQ[p, n]))))
 

rule 216

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*A 
rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a 
, 0] || GtQ[b, 0])
 

rule 219

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* 
ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt 
Q[a, 0] || LtQ[b, 0])
 

rule 756

Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[-a/b, 2 
]], s = Denominator[Rt[-a/b, 2]]}, Simp[r/(2*a)   Int[1/(r - s*x^2), x], x] 
 + Simp[r/(2*a)   Int[1/(r + s*x^2), x], x]] /; FreeQ[{a, b}, x] &&  !GtQ[a 
/b, 0]
 

rule 948

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_. 
), x_Symbol] :> Simp[1/n   Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a + b*x)^ 
p*(c + d*x)^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] && NeQ 
[b*c - a*d, 0] && IntegerQ[Simplify[(m + 1)/n]]
 

Maple [A] (verified)

Time = 0.28 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.12

method	result	size
pseudoelliptic	\(-\frac {\left (a d -\frac {7 c b}{4}\right ) \left (b \,x^{4}+a \right )^{\frac {3}{4}} x^{4} a^{2} \ln \left (\frac {\left (b \,x^{4}+a \right )^{\frac {1}{4}}+a^{\frac {1}{4}}}{\left (b \,x^{4}+a \right )^{\frac {1}{4}}-a^{\frac {1}{4}}}\right )+2 \left (a d -\frac {7 c b}{4}\right ) \left (b \,x^{4}+a \right )^{\frac {3}{4}} x^{4} a^{2} \arctan \left (\frac {\left (b \,x^{4}+a \right )^{\frac {1}{4}}}{a^{\frac {1}{4}}}\right )+a^{\frac {11}{4}} \left (\left (-\frac {4 d \,x^{4}}{3}+c \right ) a +\frac {7 b c \,x^{4}}{3}\right )}{4 \left (b \,x^{4}+a \right )^{\frac {3}{4}} x^{4} a^{\frac {19}{4}}}\)	\(136\)

Input:

int((d*x^4+c)/x^5/(b*x^4+a)^(7/4),x,method=_RETURNVERBOSE)
 

Output:

-1/4/(b*x^4+a)^(3/4)*((a*d-7/4*c*b)*(b*x^4+a)^(3/4)*x^4*a^2*ln(((b*x^4+a)^ 
(1/4)+a^(1/4))/((b*x^4+a)^(1/4)-a^(1/4)))+2*(a*d-7/4*c*b)*(b*x^4+a)^(3/4)* 
x^4*a^2*arctan((b*x^4+a)^(1/4)/a^(1/4))+a^(11/4)*((-4/3*d*x^4+c)*a+7/3*b*c 
*x^4))/x^4/a^(19/4)
 

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.11 (sec) , antiderivative size = 689, normalized size of antiderivative = 5.69 \[ \int \frac {c+d x^4}{x^5 \left (a+b x^4\right )^{7/4}} \, dx =\text {Too large to display} \] Input:

integrate((d*x^4+c)/x^5/(b*x^4+a)^(7/4),x, algorithm="fricas")
 

Output:

-1/48*(3*(a^2*b*x^8 + a^3*x^4)*((2401*b^4*c^4 - 5488*a*b^3*c^3*d + 4704*a^ 
2*b^2*c^2*d^2 - 1792*a^3*b*c*d^3 + 256*a^4*d^4)/a^11)^(1/4)*log(a^3*((2401 
*b^4*c^4 - 5488*a*b^3*c^3*d + 4704*a^2*b^2*c^2*d^2 - 1792*a^3*b*c*d^3 + 25 
6*a^4*d^4)/a^11)^(1/4) - (b*x^4 + a)^(1/4)*(7*b*c - 4*a*d)) + 3*(I*a^2*b*x 
^8 + I*a^3*x^4)*((2401*b^4*c^4 - 5488*a*b^3*c^3*d + 4704*a^2*b^2*c^2*d^2 - 
 1792*a^3*b*c*d^3 + 256*a^4*d^4)/a^11)^(1/4)*log(I*a^3*((2401*b^4*c^4 - 54 
88*a*b^3*c^3*d + 4704*a^2*b^2*c^2*d^2 - 1792*a^3*b*c*d^3 + 256*a^4*d^4)/a^ 
11)^(1/4) - (b*x^4 + a)^(1/4)*(7*b*c - 4*a*d)) + 3*(-I*a^2*b*x^8 - I*a^3*x 
^4)*((2401*b^4*c^4 - 5488*a*b^3*c^3*d + 4704*a^2*b^2*c^2*d^2 - 1792*a^3*b* 
c*d^3 + 256*a^4*d^4)/a^11)^(1/4)*log(-I*a^3*((2401*b^4*c^4 - 5488*a*b^3*c^ 
3*d + 4704*a^2*b^2*c^2*d^2 - 1792*a^3*b*c*d^3 + 256*a^4*d^4)/a^11)^(1/4) - 
 (b*x^4 + a)^(1/4)*(7*b*c - 4*a*d)) - 3*(a^2*b*x^8 + a^3*x^4)*((2401*b^4*c 
^4 - 5488*a*b^3*c^3*d + 4704*a^2*b^2*c^2*d^2 - 1792*a^3*b*c*d^3 + 256*a^4* 
d^4)/a^11)^(1/4)*log(-a^3*((2401*b^4*c^4 - 5488*a*b^3*c^3*d + 4704*a^2*b^2 
*c^2*d^2 - 1792*a^3*b*c*d^3 + 256*a^4*d^4)/a^11)^(1/4) - (b*x^4 + a)^(1/4) 
*(7*b*c - 4*a*d)) + 4*((7*b*c - 4*a*d)*x^4 + 3*a*c)*(b*x^4 + a)^(1/4))/(a^ 
2*b*x^8 + a^3*x^4)
 

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 42.27 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.68 \[ \int \frac {c+d x^4}{x^5 \left (a+b x^4\right )^{7/4}} \, dx=- \frac {c \Gamma \left (\frac {11}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {7}{4}, \frac {11}{4} \\ \frac {15}{4} \end {matrix}\middle | {\frac {a e^{i \pi }}{b x^{4}}} \right )}}{4 b^{\frac {7}{4}} x^{11} \Gamma \left (\frac {15}{4}\right )} - \frac {d \Gamma \left (\frac {7}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {7}{4}, \frac {7}{4} \\ \frac {11}{4} \end {matrix}\middle | {\frac {a e^{i \pi }}{b x^{4}}} \right )}}{4 b^{\frac {7}{4}} x^{7} \Gamma \left (\frac {11}{4}\right )} \] Input:

integrate((d*x**4+c)/x**5/(b*x**4+a)**(7/4),x)
 

Output:

-c*gamma(11/4)*hyper((7/4, 11/4), (15/4,), a*exp_polar(I*pi)/(b*x**4))/(4* 
b**(7/4)*x**11*gamma(15/4)) - d*gamma(7/4)*hyper((7/4, 7/4), (11/4,), a*ex 
p_polar(I*pi)/(b*x**4))/(4*b**(7/4)*x**7*gamma(11/4))
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 196 vs. \(2 (97) = 194\).

Time = 0.12 (sec) , antiderivative size = 196, normalized size of antiderivative = 1.62 \[ \int \frac {c+d x^4}{x^5 \left (a+b x^4\right )^{7/4}} \, dx=-\frac {1}{48} \, c {\left (\frac {4 \, {\left (7 \, {\left (b x^{4} + a\right )} b - 4 \, a b\right )}}{{\left (b x^{4} + a\right )}^{\frac {7}{4}} a^{2} - {\left (b x^{4} + a\right )}^{\frac {3}{4}} a^{3}} - \frac {21 \, {\left (\frac {2 \, b \arctan \left (\frac {{\left (b x^{4} + a\right )}^{\frac {1}{4}}}{a^{\frac {1}{4}}}\right )}{a^{\frac {3}{4}}} - \frac {b \log \left (\frac {{\left (b x^{4} + a\right )}^{\frac {1}{4}} - a^{\frac {1}{4}}}{{\left (b x^{4} + a\right )}^{\frac {1}{4}} + a^{\frac {1}{4}}}\right )}{a^{\frac {3}{4}}}\right )}}{a^{2}}\right )} - \frac {1}{12} \, d {\left (\frac {3 \, {\left (\frac {2 \, \arctan \left (\frac {{\left (b x^{4} + a\right )}^{\frac {1}{4}}}{a^{\frac {1}{4}}}\right )}{a^{\frac {3}{4}}} - \frac {\log \left (\frac {{\left (b x^{4} + a\right )}^{\frac {1}{4}} - a^{\frac {1}{4}}}{{\left (b x^{4} + a\right )}^{\frac {1}{4}} + a^{\frac {1}{4}}}\right )}{a^{\frac {3}{4}}}\right )}}{a} - \frac {4}{{\left (b x^{4} + a\right )}^{\frac {3}{4}} a}\right )} \] Input:

integrate((d*x^4+c)/x^5/(b*x^4+a)^(7/4),x, algorithm="maxima")
 

Output:

-1/48*c*(4*(7*(b*x^4 + a)*b - 4*a*b)/((b*x^4 + a)^(7/4)*a^2 - (b*x^4 + a)^ 
(3/4)*a^3) - 21*(2*b*arctan((b*x^4 + a)^(1/4)/a^(1/4))/a^(3/4) - b*log(((b 
*x^4 + a)^(1/4) - a^(1/4))/((b*x^4 + a)^(1/4) + a^(1/4)))/a^(3/4))/a^2) - 
1/12*d*(3*(2*arctan((b*x^4 + a)^(1/4)/a^(1/4))/a^(3/4) - log(((b*x^4 + a)^ 
(1/4) - a^(1/4))/((b*x^4 + a)^(1/4) + a^(1/4)))/a^(3/4))/a - 4/((b*x^4 + a 
)^(3/4)*a))
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 262 vs. \(2 (97) = 194\).

Time = 0.12 (sec) , antiderivative size = 262, normalized size of antiderivative = 2.17 \[ \int \frac {c+d x^4}{x^5 \left (a+b x^4\right )^{7/4}} \, dx=-\frac {\sqrt {2} {\left (7 \, b c - 4 \, a d\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (-a\right )^{\frac {1}{4}} + 2 \, {\left (b x^{4} + a\right )}^{\frac {1}{4}}\right )}}{2 \, \left (-a\right )^{\frac {1}{4}}}\right )}{16 \, \left (-a\right )^{\frac {3}{4}} a^{2}} - \frac {\sqrt {2} {\left (7 \, b c - 4 \, a d\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (-a\right )^{\frac {1}{4}} - 2 \, {\left (b x^{4} + a\right )}^{\frac {1}{4}}\right )}}{2 \, \left (-a\right )^{\frac {1}{4}}}\right )}{16 \, \left (-a\right )^{\frac {3}{4}} a^{2}} - \frac {\sqrt {2} {\left (7 \, b c - 4 \, a d\right )} \log \left (\sqrt {2} {\left (b x^{4} + a\right )}^{\frac {1}{4}} \left (-a\right )^{\frac {1}{4}} + \sqrt {b x^{4} + a} + \sqrt {-a}\right )}{32 \, \left (-a\right )^{\frac {3}{4}} a^{2}} + \frac {\sqrt {2} {\left (7 \, b c - 4 \, a d\right )} \log \left (-\sqrt {2} {\left (b x^{4} + a\right )}^{\frac {1}{4}} \left (-a\right )^{\frac {1}{4}} + \sqrt {b x^{4} + a} + \sqrt {-a}\right )}{32 \, \left (-a\right )^{\frac {3}{4}} a^{2}} - \frac {b c - a d}{3 \, {\left (b x^{4} + a\right )}^{\frac {3}{4}} a^{2}} - \frac {{\left (b x^{4} + a\right )}^{\frac {1}{4}} c}{4 \, a^{2} x^{4}} \] Input:

integrate((d*x^4+c)/x^5/(b*x^4+a)^(7/4),x, algorithm="giac")
 

Output:

-1/16*sqrt(2)*(7*b*c - 4*a*d)*arctan(1/2*sqrt(2)*(sqrt(2)*(-a)^(1/4) + 2*( 
b*x^4 + a)^(1/4))/(-a)^(1/4))/((-a)^(3/4)*a^2) - 1/16*sqrt(2)*(7*b*c - 4*a 
*d)*arctan(-1/2*sqrt(2)*(sqrt(2)*(-a)^(1/4) - 2*(b*x^4 + a)^(1/4))/(-a)^(1 
/4))/((-a)^(3/4)*a^2) - 1/32*sqrt(2)*(7*b*c - 4*a*d)*log(sqrt(2)*(b*x^4 + 
a)^(1/4)*(-a)^(1/4) + sqrt(b*x^4 + a) + sqrt(-a))/((-a)^(3/4)*a^2) + 1/32* 
sqrt(2)*(7*b*c - 4*a*d)*log(-sqrt(2)*(b*x^4 + a)^(1/4)*(-a)^(1/4) + sqrt(b 
*x^4 + a) + sqrt(-a))/((-a)^(3/4)*a^2) - 1/3*(b*c - a*d)/((b*x^4 + a)^(3/4 
)*a^2) - 1/4*(b*x^4 + a)^(1/4)*c/(a^2*x^4)
 

Mupad [B] (verification not implemented)

Time = 4.33 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.21 \[ \int \frac {c+d x^4}{x^5 \left (a+b x^4\right )^{7/4}} \, dx=\frac {d}{3\,a\,{\left (b\,x^4+a\right )}^{3/4}}-\frac {\frac {b\,c}{3\,a}-\frac {7\,b\,c\,\left (b\,x^4+a\right )}{12\,a^2}}{a\,{\left (b\,x^4+a\right )}^{3/4}-{\left (b\,x^4+a\right )}^{7/4}}-\frac {d\,\mathrm {atan}\left (\frac {{\left (b\,x^4+a\right )}^{1/4}}{a^{1/4}}\right )}{2\,a^{7/4}}-\frac {d\,\mathrm {atanh}\left (\frac {{\left (b\,x^4+a\right )}^{1/4}}{a^{1/4}}\right )}{2\,a^{7/4}}+\frac {7\,b\,c\,\mathrm {atan}\left (\frac {{\left (b\,x^4+a\right )}^{1/4}}{a^{1/4}}\right )}{8\,a^{11/4}}+\frac {7\,b\,c\,\mathrm {atanh}\left (\frac {{\left (b\,x^4+a\right )}^{1/4}}{a^{1/4}}\right )}{8\,a^{11/4}} \] Input:

int((c + d*x^4)/(x^5*(a + b*x^4)^(7/4)),x)
 

Output:

d/(3*a*(a + b*x^4)^(3/4)) - ((b*c)/(3*a) - (7*b*c*(a + b*x^4))/(12*a^2))/( 
a*(a + b*x^4)^(3/4) - (a + b*x^4)^(7/4)) - (d*atan((a + b*x^4)^(1/4)/a^(1/ 
4)))/(2*a^(7/4)) - (d*atanh((a + b*x^4)^(1/4)/a^(1/4)))/(2*a^(7/4)) + (7*b 
*c*atan((a + b*x^4)^(1/4)/a^(1/4)))/(8*a^(11/4)) + (7*b*c*atanh((a + b*x^4 
)^(1/4)/a^(1/4)))/(8*a^(11/4))
 

Reduce [F]

\[ \int \frac {c+d x^4}{x^5 \left (a+b x^4\right )^{7/4}} \, dx=\left (\int \frac {1}{\left (b \,x^{4}+a \right )^{\frac {3}{4}} a \,x^{5}+\left (b \,x^{4}+a \right )^{\frac {3}{4}} b \,x^{9}}d x \right ) c +\left (\int \frac {1}{\left (b \,x^{4}+a \right )^{\frac {3}{4}} a x +\left (b \,x^{4}+a \right )^{\frac {3}{4}} b \,x^{5}}d x \right ) d \] Input:

int((d*x^4+c)/x^5/(b*x^4+a)^(7/4),x)
 

Output:

int(1/((a + b*x**4)**(3/4)*a*x**5 + (a + b*x**4)**(3/4)*b*x**9),x)*c + int 
(1/((a + b*x**4)**(3/4)*a*x + (a + b*x**4)**(3/4)*b*x**5),x)*d