Integrand size = 22, antiderivative size = 213 \[ \int \frac {c+d x^4}{x^6 \left (a+b x^4\right )^{17/4}} \, dx=-\frac {c}{5 a x^5 \left (a+b x^4\right )^{13/4}}-\frac {18 b c-5 a d}{65 a^2 x \left (a+b x^4\right )^{13/4}}-\frac {14 (18 b c-5 a d)}{585 a^3 x \left (a+b x^4\right )^{9/4}}-\frac {28 (18 b c-5 a d)}{585 a^4 x \left (a+b x^4\right )^{5/4}}+\frac {56 (18 b c-5 a d)}{195 a^5 x \sqrt [4]{a+b x^4}}-\frac {112 \sqrt {b} (18 b c-5 a d) \sqrt [4]{1+\frac {a}{b x^4}} x E\left (\left .\frac {1}{2} \cot ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )\right |2\right )}{195 a^{11/2} \sqrt [4]{a+b x^4}} \] Output:
-1/5*c/a/x^5/(b*x^4+a)^(13/4)-1/65*(-5*a*d+18*b*c)/a^2/x/(b*x^4+a)^(13/4)- 14/585*(-5*a*d+18*b*c)/a^3/x/(b*x^4+a)^(9/4)-28/585*(-5*a*d+18*b*c)/a^4/x/ (b*x^4+a)^(5/4)+56/195*(-5*a*d+18*b*c)/a^5/x/(b*x^4+a)^(1/4)-112/195*b^(1/ 2)*(-5*a*d+18*b*c)*(1+a/b/x^4)^(1/4)*x*EllipticE(sin(1/2*arccot(b^(1/2)*x^ 2/a^(1/2))),2^(1/2))/a^(11/2)/(b*x^4+a)^(1/4)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 10.04 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.39 \[ \int \frac {c+d x^4}{x^6 \left (a+b x^4\right )^{17/4}} \, dx=\frac {-a^4 c+(18 b c-5 a d) x^4 \left (a+b x^4\right )^3 \sqrt [4]{1+\frac {b x^4}{a}} \operatorname {Hypergeometric2F1}\left (-\frac {1}{4},\frac {17}{4},\frac {3}{4},-\frac {b x^4}{a}\right )}{5 a^5 x^5 \left (a+b x^4\right )^{13/4}} \] Input:
Integrate[(c + d*x^4)/(x^6*(a + b*x^4)^(17/4)),x]
Output:
(-(a^4*c) + (18*b*c - 5*a*d)*x^4*(a + b*x^4)^3*(1 + (b*x^4)/a)^(1/4)*Hyper geometric2F1[-1/4, 17/4, 3/4, -((b*x^4)/a)])/(5*a^5*x^5*(a + b*x^4)^(13/4) )
Time = 0.65 (sec) , antiderivative size = 205, normalized size of antiderivative = 0.96, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.409, Rules used = {955, 819, 819, 819, 816, 813, 858, 807, 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.
\(\displaystyle \int \frac {c+d x^4}{x^6 \left (a+b x^4\right )^{17/4}} \, dx\) |
\(\Big \downarrow \) 955 |
\(\displaystyle -\frac {(18 b c-5 a d) \int \frac {1}{x^2 \left (b x^4+a\right )^{17/4}}dx}{5 a}-\frac {c}{5 a x^5 \left (a+b x^4\right )^{13/4}}\) |
\(\Big \downarrow \) 819 |
\(\displaystyle -\frac {(18 b c-5 a d) \left (\frac {14 \int \frac {1}{x^2 \left (b x^4+a\right )^{13/4}}dx}{13 a}+\frac {1}{13 a x \left (a+b x^4\right )^{13/4}}\right )}{5 a}-\frac {c}{5 a x^5 \left (a+b x^4\right )^{13/4}}\) |
\(\Big \downarrow \) 819 |
\(\displaystyle -\frac {(18 b c-5 a d) \left (\frac {14 \left (\frac {10 \int \frac {1}{x^2 \left (b x^4+a\right )^{9/4}}dx}{9 a}+\frac {1}{9 a x \left (a+b x^4\right )^{9/4}}\right )}{13 a}+\frac {1}{13 a x \left (a+b x^4\right )^{13/4}}\right )}{5 a}-\frac {c}{5 a x^5 \left (a+b x^4\right )^{13/4}}\) |
\(\Big \downarrow \) 819 |
\(\displaystyle -\frac {(18 b c-5 a d) \left (\frac {14 \left (\frac {10 \left (\frac {6 \int \frac {1}{x^2 \left (b x^4+a\right )^{5/4}}dx}{5 a}+\frac {1}{5 a x \left (a+b x^4\right )^{5/4}}\right )}{9 a}+\frac {1}{9 a x \left (a+b x^4\right )^{9/4}}\right )}{13 a}+\frac {1}{13 a x \left (a+b x^4\right )^{13/4}}\right )}{5 a}-\frac {c}{5 a x^5 \left (a+b x^4\right )^{13/4}}\) |
\(\Big \downarrow \) 816 |
\(\displaystyle -\frac {(18 b c-5 a d) \left (\frac {14 \left (\frac {10 \left (\frac {6 \left (-\frac {2 b \int \frac {x^2}{\left (b x^4+a\right )^{5/4}}dx}{a}-\frac {1}{a x \sqrt [4]{a+b x^4}}\right )}{5 a}+\frac {1}{5 a x \left (a+b x^4\right )^{5/4}}\right )}{9 a}+\frac {1}{9 a x \left (a+b x^4\right )^{9/4}}\right )}{13 a}+\frac {1}{13 a x \left (a+b x^4\right )^{13/4}}\right )}{5 a}-\frac {c}{5 a x^5 \left (a+b x^4\right )^{13/4}}\) |
\(\Big \downarrow \) 813 |
\(\displaystyle -\frac {(18 b c-5 a d) \left (\frac {14 \left (\frac {10 \left (\frac {6 \left (-\frac {2 x \sqrt [4]{\frac {a}{b x^4}+1} \int \frac {1}{\left (\frac {a}{b x^4}+1\right )^{5/4} x^3}dx}{a \sqrt [4]{a+b x^4}}-\frac {1}{a x \sqrt [4]{a+b x^4}}\right )}{5 a}+\frac {1}{5 a x \left (a+b x^4\right )^{5/4}}\right )}{9 a}+\frac {1}{9 a x \left (a+b x^4\right )^{9/4}}\right )}{13 a}+\frac {1}{13 a x \left (a+b x^4\right )^{13/4}}\right )}{5 a}-\frac {c}{5 a x^5 \left (a+b x^4\right )^{13/4}}\) |
\(\Big \downarrow \) 858 |
\(\displaystyle -\frac {(18 b c-5 a d) \left (\frac {14 \left (\frac {10 \left (\frac {6 \left (\frac {2 x \sqrt [4]{\frac {a}{b x^4}+1} \int \frac {1}{\left (\frac {a}{b x^4}+1\right )^{5/4} x}d\frac {1}{x}}{a \sqrt [4]{a+b x^4}}-\frac {1}{a x \sqrt [4]{a+b x^4}}\right )}{5 a}+\frac {1}{5 a x \left (a+b x^4\right )^{5/4}}\right )}{9 a}+\frac {1}{9 a x \left (a+b x^4\right )^{9/4}}\right )}{13 a}+\frac {1}{13 a x \left (a+b x^4\right )^{13/4}}\right )}{5 a}-\frac {c}{5 a x^5 \left (a+b x^4\right )^{13/4}}\) |
\(\Big \downarrow \) 807 |
\(\displaystyle -\frac {(18 b c-5 a d) \left (\frac {14 \left (\frac {10 \left (\frac {6 \left (\frac {x \sqrt [4]{\frac {a}{b x^4}+1} \int \frac {1}{\left (\frac {a}{b x^2}+1\right )^{5/4}}d\frac {1}{x^2}}{a \sqrt [4]{a+b x^4}}-\frac {1}{a x \sqrt [4]{a+b x^4}}\right )}{5 a}+\frac {1}{5 a x \left (a+b x^4\right )^{5/4}}\right )}{9 a}+\frac {1}{9 a x \left (a+b x^4\right )^{9/4}}\right )}{13 a}+\frac {1}{13 a x \left (a+b x^4\right )^{13/4}}\right )}{5 a}-\frac {c}{5 a x^5 \left (a+b x^4\right )^{13/4}}\) |
\(\Big \downarrow \) 212 |
\(\displaystyle -\frac {(18 b c-5 a d) \left (\frac {14 \left (\frac {10 \left (\frac {6 \left (\frac {2 \sqrt {b} x \sqrt [4]{\frac {a}{b x^4}+1} E\left (\left .\frac {1}{2} \arctan \left (\frac {\sqrt {a}}{\sqrt {b} x^2}\right )\right |2\right )}{a^{3/2} \sqrt [4]{a+b x^4}}-\frac {1}{a x \sqrt [4]{a+b x^4}}\right )}{5 a}+\frac {1}{5 a x \left (a+b x^4\right )^{5/4}}\right )}{9 a}+\frac {1}{9 a x \left (a+b x^4\right )^{9/4}}\right )}{13 a}+\frac {1}{13 a x \left (a+b x^4\right )^{13/4}}\right )}{5 a}-\frac {c}{5 a x^5 \left (a+b x^4\right )^{13/4}}\) |
Input:
Int[(c + d*x^4)/(x^6*(a + b*x^4)^(17/4)),x]
Output:
-1/5*c/(a*x^5*(a + b*x^4)^(13/4)) - ((18*b*c - 5*a*d)*(1/(13*a*x*(a + b*x^ 4)^(13/4)) + (14*(1/(9*a*x*(a + b*x^4)^(9/4)) + (10*(1/(5*a*x*(a + b*x^4)^ (5/4)) + (6*(-(1/(a*x*(a + b*x^4)^(1/4))) + (2*Sqrt[b]*(1 + a/(b*x^4))^(1/ 4)*x*EllipticE[ArcTan[Sqrt[a]/(Sqrt[b]*x^2)]/2, 2])/(a^(3/2)*(a + b*x^4)^( 1/4))))/(5*a)))/(9*a)))/(13*a)))/(5*a)
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]
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]
Int[(x_)^2/((a_) + (b_.)*(x_)^4)^(5/4), x_Symbol] :> Simp[x*((1 + a/(b*x^4) )^(1/4)/(b*(a + b*x^4)^(1/4))) Int[1/(x^3*(1 + a/(b*x^4))^(5/4)), x], x] /; FreeQ[{a, b}, x] && PosQ[b/a]
Int[(x_)^(m_)/((a_) + (b_.)*(x_)^4)^(5/4), x_Symbol] :> Simp[x^(m + 1)/(a*( m + 1)*(a + b*x^4)^(1/4)), x] - Simp[b*(m/(a*(m + 1))) Int[x^(m + 4)/(a + b*x^4)^(5/4), x], x] /; FreeQ[{a, b}, x] && PosQ[b/a] && ILtQ[(m - 2)/4, 0 ]
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(-( c*x)^(m + 1))*((a + b*x^n)^(p + 1)/(a*c*n*(p + 1))), x] + Simp[(m + n*(p + 1) + 1)/(a*n*(p + 1)) Int[(c*x)^m*(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a , b, c, m}, x] && IGtQ[n, 0] && LtQ[p, -1] && IntBinomialQ[a, b, c, n, m, p , x]
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]
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]
\[\int \frac {d \,x^{4}+c}{x^{6} \left (b \,x^{4}+a \right )^{\frac {17}{4}}}d x\]
Input:
int((d*x^4+c)/x^6/(b*x^4+a)^(17/4),x)
Output:
int((d*x^4+c)/x^6/(b*x^4+a)^(17/4),x)
\[ \int \frac {c+d x^4}{x^6 \left (a+b x^4\right )^{17/4}} \, dx=\int { \frac {d x^{4} + c}{{\left (b x^{4} + a\right )}^{\frac {17}{4}} x^{6}} \,d x } \] Input:
integrate((d*x^4+c)/x^6/(b*x^4+a)^(17/4),x, algorithm="fricas")
Output:
integral((b*x^4 + a)^(3/4)*(d*x^4 + c)/(b^5*x^26 + 5*a*b^4*x^22 + 10*a^2*b ^3*x^18 + 10*a^3*b^2*x^14 + 5*a^4*b*x^10 + a^5*x^6), x)
Timed out. \[ \int \frac {c+d x^4}{x^6 \left (a+b x^4\right )^{17/4}} \, dx=\text {Timed out} \] Input:
integrate((d*x**4+c)/x**6/(b*x**4+a)**(17/4),x)
Output:
Timed out
\[ \int \frac {c+d x^4}{x^6 \left (a+b x^4\right )^{17/4}} \, dx=\int { \frac {d x^{4} + c}{{\left (b x^{4} + a\right )}^{\frac {17}{4}} x^{6}} \,d x } \] Input:
integrate((d*x^4+c)/x^6/(b*x^4+a)^(17/4),x, algorithm="maxima")
Output:
integrate((d*x^4 + c)/((b*x^4 + a)^(17/4)*x^6), x)
\[ \int \frac {c+d x^4}{x^6 \left (a+b x^4\right )^{17/4}} \, dx=\int { \frac {d x^{4} + c}{{\left (b x^{4} + a\right )}^{\frac {17}{4}} x^{6}} \,d x } \] Input:
integrate((d*x^4+c)/x^6/(b*x^4+a)^(17/4),x, algorithm="giac")
Output:
integrate((d*x^4 + c)/((b*x^4 + a)^(17/4)*x^6), x)
Timed out. \[ \int \frac {c+d x^4}{x^6 \left (a+b x^4\right )^{17/4}} \, dx=\int \frac {d\,x^4+c}{x^6\,{\left (b\,x^4+a\right )}^{17/4}} \,d x \] Input:
int((c + d*x^4)/(x^6*(a + b*x^4)^(17/4)),x)
Output:
int((c + d*x^4)/(x^6*(a + b*x^4)^(17/4)), x)
\[ \int \frac {c+d x^4}{x^6 \left (a+b x^4\right )^{17/4}} \, dx=\left (\int \frac {1}{\left (b \,x^{4}+a \right )^{\frac {1}{4}} a^{4} x^{6}+4 \left (b \,x^{4}+a \right )^{\frac {1}{4}} a^{3} b \,x^{10}+6 \left (b \,x^{4}+a \right )^{\frac {1}{4}} a^{2} b^{2} x^{14}+4 \left (b \,x^{4}+a \right )^{\frac {1}{4}} a \,b^{3} x^{18}+\left (b \,x^{4}+a \right )^{\frac {1}{4}} b^{4} x^{22}}d x \right ) c +\left (\int \frac {1}{\left (b \,x^{4}+a \right )^{\frac {1}{4}} a^{4} x^{2}+4 \left (b \,x^{4}+a \right )^{\frac {1}{4}} a^{3} b \,x^{6}+6 \left (b \,x^{4}+a \right )^{\frac {1}{4}} a^{2} b^{2} x^{10}+4 \left (b \,x^{4}+a \right )^{\frac {1}{4}} a \,b^{3} x^{14}+\left (b \,x^{4}+a \right )^{\frac {1}{4}} b^{4} x^{18}}d x \right ) d \] Input:
int((d*x^4+c)/x^6/(b*x^4+a)^(17/4),x)
Output:
int(1/((a + b*x**4)**(1/4)*a**4*x**6 + 4*(a + b*x**4)**(1/4)*a**3*b*x**10 + 6*(a + b*x**4)**(1/4)*a**2*b**2*x**14 + 4*(a + b*x**4)**(1/4)*a*b**3*x** 18 + (a + b*x**4)**(1/4)*b**4*x**22),x)*c + int(1/((a + b*x**4)**(1/4)*a** 4*x**2 + 4*(a + b*x**4)**(1/4)*a**3*b*x**6 + 6*(a + b*x**4)**(1/4)*a**2*b* *2*x**10 + 4*(a + b*x**4)**(1/4)*a*b**3*x**14 + (a + b*x**4)**(1/4)*b**4*x **18),x)*d