\(\int \frac {c+d x^4}{x^7 (a+b x^4)^{9/4}} \, dx\) [164]

Optimal result

Integrand size = 22, antiderivative size = 152 \[ \int \frac {c+d x^4}{x^7 \left (a+b x^4\right )^{9/4}} \, dx=-\frac {c}{6 a x^6 \left (a+b x^4\right )^{5/4}}-\frac {11 b c-6 a d}{30 a^2 x^2 \left (a+b x^4\right )^{5/4}}+\frac {7 (11 b c-6 a d)}{60 a^3 x^2 \sqrt [4]{a+b x^4}}+\frac {7 \sqrt {b} (11 b c-6 a d) \sqrt [4]{1+\frac {b x^4}{a}} E\left (\left .\frac {1}{2} \arctan \left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )\right |2\right )}{20 a^{7/2} \sqrt [4]{a+b x^4}} \] Output:

-1/6*c/a/x^6/(b*x^4+a)^(5/4)-1/30*(-6*a*d+11*b*c)/a^2/x^2/(b*x^4+a)^(5/4)+ 
7/60*(-6*a*d+11*b*c)/a^3/x^2/(b*x^4+a)^(1/4)+7/20*b^(1/2)*(-6*a*d+11*b*c)* 
(1+b*x^4/a)^(1/4)*EllipticE(sin(1/2*arctan(b^(1/2)*x^2/a^(1/2))),2^(1/2))/ 
a^(7/2)/(b*x^4+a)^(1/4)
 

Mathematica [C] (verified)

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

Time = 10.04 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.53 \[ \int \frac {c+d x^4}{x^7 \left (a+b x^4\right )^{9/4}} \, dx=\frac {-2 a^2 c+(11 b c-6 a d) x^4 \left (a+b x^4\right ) \sqrt [4]{1+\frac {b x^4}{a}} \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},\frac {9}{4},\frac {1}{2},-\frac {b x^4}{a}\right )}{12 a^3 x^6 \left (a+b x^4\right )^{5/4}} \] Input:

Integrate[(c + d*x^4)/(x^7*(a + b*x^4)^(9/4)),x]
 

Output:

(-2*a^2*c + (11*b*c - 6*a*d)*x^4*(a + b*x^4)*(1 + (b*x^4)/a)^(1/4)*Hyperge 
ometric2F1[-1/2, 9/4, 1/2, -((b*x^4)/a)])/(12*a^3*x^6*(a + b*x^4)^(5/4))
 

Rubi [A] (verified)

Time = 0.42 (sec) , antiderivative size = 146, normalized size of antiderivative = 0.96, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {955, 807, 253, 251, 213, 212}

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.

\(\displaystyle \int \frac {c+d x^4}{x^7 \left (a+b x^4\right )^{9/4}} \, dx\)

\(\Big \downarrow \) 955

\(\displaystyle -\frac {(11 b c-6 a d) \int \frac {1}{x^3 \left (b x^4+a\right )^{9/4}}dx}{6 a}-\frac {c}{6 a x^6 \left (a+b x^4\right )^{5/4}}\)

\(\Big \downarrow \) 807

\(\displaystyle -\frac {(11 b c-6 a d) \int \frac {1}{x^4 \left (b x^4+a\right )^{9/4}}dx^2}{12 a}-\frac {c}{6 a x^6 \left (a+b x^4\right )^{5/4}}\)

\(\Big \downarrow \) 253

\(\displaystyle -\frac {(11 b c-6 a d) \left (\frac {7 \int \frac {1}{x^4 \left (b x^4+a\right )^{5/4}}dx^2}{5 a}+\frac {2}{5 a x^2 \left (a+b x^4\right )^{5/4}}\right )}{12 a}-\frac {c}{6 a x^6 \left (a+b x^4\right )^{5/4}}\)

\(\Big \downarrow \) 251

\(\displaystyle -\frac {(11 b c-6 a d) \left (\frac {7 \left (-\frac {3 b \int \frac {1}{\left (b x^4+a\right )^{5/4}}dx^2}{2 a}-\frac {1}{a x^2 \sqrt [4]{a+b x^4}}\right )}{5 a}+\frac {2}{5 a x^2 \left (a+b x^4\right )^{5/4}}\right )}{12 a}-\frac {c}{6 a x^6 \left (a+b x^4\right )^{5/4}}\)

\(\Big \downarrow \) 213

\(\displaystyle -\frac {(11 b c-6 a d) \left (\frac {7 \left (-\frac {3 b \sqrt [4]{\frac {b x^4}{a}+1} \int \frac {1}{\left (\frac {b x^4}{a}+1\right )^{5/4}}dx^2}{2 a^2 \sqrt [4]{a+b x^4}}-\frac {1}{a x^2 \sqrt [4]{a+b x^4}}\right )}{5 a}+\frac {2}{5 a x^2 \left (a+b x^4\right )^{5/4}}\right )}{12 a}-\frac {c}{6 a x^6 \left (a+b x^4\right )^{5/4}}\)

\(\Big \downarrow \) 212

\(\displaystyle -\frac {(11 b c-6 a d) \left (\frac {7 \left (-\frac {3 \sqrt {b} \sqrt [4]{\frac {b x^4}{a}+1} E\left (\left .\frac {1}{2} \arctan \left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )\right |2\right )}{a^{3/2} \sqrt [4]{a+b x^4}}-\frac {1}{a x^2 \sqrt [4]{a+b x^4}}\right )}{5 a}+\frac {2}{5 a x^2 \left (a+b x^4\right )^{5/4}}\right )}{12 a}-\frac {c}{6 a x^6 \left (a+b x^4\right )^{5/4}}\)

Input:

Int[(c + d*x^4)/(x^7*(a + b*x^4)^(9/4)),x]
 

Output:

-1/6*c/(a*x^6*(a + b*x^4)^(5/4)) - ((11*b*c - 6*a*d)*(2/(5*a*x^2*(a + b*x^ 
4)^(5/4)) + (7*(-(1/(a*x^2*(a + b*x^4)^(1/4))) - (3*Sqrt[b]*(1 + (b*x^4)/a 
)^(1/4)*EllipticE[ArcTan[(Sqrt[b]*x^2)/Sqrt[a]]/2, 2])/(a^(3/2)*(a + b*x^4 
)^(1/4))))/(5*a)))/(12*a)
 

Defintions of rubi rules used

rule 212

Int[((a_) + (b_.)*(x_)^2)^(-5/4), x_Symbol] :> Simp[(2/(a^(5/4)*Rt[b/a, 2]) 
)*EllipticE[(1/2)*ArcTan[Rt[b/a, 2]*x], 2], x] /; FreeQ[{a, b}, x] && GtQ[a 
, 0] && PosQ[b/a]
 

rule 213

Int[((a_) + (b_.)*(x_)^2)^(-5/4), x_Symbol] :> Simp[(1 + b*(x^2/a))^(1/4)/( 
a*(a + b*x^2)^(1/4))   Int[1/(1 + b*(x^2/a))^(5/4), x], x] /; FreeQ[{a, b}, 
 x] && PosQ[a] && PosQ[b/a]
 

rule 251

Int[((c_.)*(x_))^(m_)/((a_) + (b_.)*(x_)^2)^(5/4), x_Symbol] :> Simp[(c*x)^ 
(m + 1)/(a*c*(m + 1)*(a + b*x^2)^(1/4)), x] - Simp[b*((2*m + 1)/(2*a*c^2*(m 
 + 1)))   Int[(c*x)^(m + 2)/(a + b*x^2)^(5/4), x], x] /; FreeQ[{a, b, c}, x 
] && PosQ[b/a] && IntegerQ[2*m] && LtQ[m, -1]
 

rule 253

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-(c*x 
)^(m + 1))*((a + b*x^2)^(p + 1)/(2*a*c*(p + 1))), x] + Simp[(m + 2*p + 3)/( 
2*a*(p + 1))   Int[(c*x)^m*(a + b*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, m 
}, x] && LtQ[p, -1] && IntBinomialQ[a, b, c, 2, m, p, 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 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]

\[\int \frac {d \,x^{4}+c}{x^{7} \left (b \,x^{4}+a \right )^{\frac {9}{4}}}d x\]

Input:

int((d*x^4+c)/x^7/(b*x^4+a)^(9/4),x)
 

Output:

int((d*x^4+c)/x^7/(b*x^4+a)^(9/4),x)
 

Fricas [F]

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

integrate((d*x^4+c)/x^7/(b*x^4+a)^(9/4),x, algorithm="fricas")
 

Output:

integral((b*x^4 + a)^(3/4)*(d*x^4 + c)/(b^3*x^19 + 3*a*b^2*x^15 + 3*a^2*b* 
x^11 + a^3*x^7), x)
 

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 78.09 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.43 \[ \int \frac {c+d x^4}{x^7 \left (a+b x^4\right )^{9/4}} \, dx=- \frac {c {{}_{2}F_{1}\left (\begin {matrix} - \frac {3}{2}, \frac {9}{4} \\ - \frac {1}{2} \end {matrix}\middle | {\frac {b x^{4} e^{i \pi }}{a}} \right )}}{6 a^{\frac {9}{4}} x^{6}} - \frac {d {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {9}{4} \\ \frac {1}{2} \end {matrix}\middle | {\frac {b x^{4} e^{i \pi }}{a}} \right )}}{2 a^{\frac {9}{4}} x^{2}} \] Input:

integrate((d*x**4+c)/x**7/(b*x**4+a)**(9/4),x)
 

Output:

-c*hyper((-3/2, 9/4), (-1/2,), b*x**4*exp_polar(I*pi)/a)/(6*a**(9/4)*x**6) 
 - d*hyper((-1/2, 9/4), (1/2,), b*x**4*exp_polar(I*pi)/a)/(2*a**(9/4)*x**2 
)
 

Maxima [F]

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

integrate((d*x^4+c)/x^7/(b*x^4+a)^(9/4),x, algorithm="maxima")
 

Output:

integrate((d*x^4 + c)/((b*x^4 + a)^(9/4)*x^7), x)
 

Giac [F]

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

integrate((d*x^4+c)/x^7/(b*x^4+a)^(9/4),x, algorithm="giac")
 

Output:

integrate((d*x^4 + c)/((b*x^4 + a)^(9/4)*x^7), x)
 

Mupad [F(-1)]

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

int((c + d*x^4)/(x^7*(a + b*x^4)^(9/4)),x)
 

Output:

int((c + d*x^4)/(x^7*(a + b*x^4)^(9/4)), x)
 

Reduce [F]

\[ \int \frac {c+d x^4}{x^7 \left (a+b x^4\right )^{9/4}} \, dx=\left (\int \frac {1}{\left (b \,x^{4}+a \right )^{\frac {1}{4}} a^{2} x^{7}+2 \left (b \,x^{4}+a \right )^{\frac {1}{4}} a b \,x^{11}+\left (b \,x^{4}+a \right )^{\frac {1}{4}} b^{2} x^{15}}d x \right ) c +\left (\int \frac {1}{\left (b \,x^{4}+a \right )^{\frac {1}{4}} a^{2} x^{3}+2 \left (b \,x^{4}+a \right )^{\frac {1}{4}} a b \,x^{7}+\left (b \,x^{4}+a \right )^{\frac {1}{4}} b^{2} x^{11}}d x \right ) d \] Input:

int((d*x^4+c)/x^7/(b*x^4+a)^(9/4),x)
 

Output:

int(1/((a + b*x**4)**(1/4)*a**2*x**7 + 2*(a + b*x**4)**(1/4)*a*b*x**11 + ( 
a + b*x**4)**(1/4)*b**2*x**15),x)*c + int(1/((a + b*x**4)**(1/4)*a**2*x**3 
 + 2*(a + b*x**4)**(1/4)*a*b*x**7 + (a + b*x**4)**(1/4)*b**2*x**11),x)*d