Integrand size = 22, antiderivative size = 182 \[ \int \frac {x^{14} \left (c+d x^4\right )}{\left (a+b x^4\right )^{17/4}} \, dx=-\frac {(b c-a d) x^{11}}{13 b^2 \left (a+b x^4\right )^{13/4}}-\frac {(11 b c-24 a d) x^7}{117 b^3 \left (a+b x^4\right )^{9/4}}-\frac {(77 b c-285 a d) x^3}{585 b^4 \left (a+b x^4\right )^{5/4}}+\frac {d x^3}{2 b^4 \sqrt [4]{a+b x^4}}-\frac {77 (2 b c-15 a d) \sqrt [4]{1+\frac {a}{b x^4}} x E\left (\left .\frac {1}{2} \cot ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )\right |2\right )}{390 \sqrt {a} b^{9/2} \sqrt [4]{a+b x^4}} \] Output:
-1/13*(-a*d+b*c)*x^11/b^2/(b*x^4+a)^(13/4)-1/117*(-24*a*d+11*b*c)*x^7/b^3/ (b*x^4+a)^(9/4)-1/585*(-285*a*d+77*b*c)*x^3/b^4/(b*x^4+a)^(5/4)+1/2*d*x^3/ b^4/(b*x^4+a)^(1/4)-77/390*(-15*a*d+2*b*c)*(1+a/b/x^4)^(1/4)*x*EllipticE(s in(1/2*arccot(b^(1/2)*x^2/a^(1/2))),2^(1/2))/a^(1/2)/b^(9/2)/(b*x^4+a)^(1/ 4)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 10.12 (sec) , antiderivative size = 139, normalized size of antiderivative = 0.76 \[ \int \frac {x^{14} \left (c+d x^4\right )}{\left (a+b x^4\right )^{17/4}} \, dx=\frac {x^3 \left (a \left (1155 a^3 d-22 a^2 b \left (7 c-75 d x^4\right )+120 b^3 x^8 \left (-c+d x^4\right )+20 a b^2 x^4 \left (-11 c+45 d x^4\right )\right )-77 (-2 b c+15 a d) \left (a+b x^4\right )^3 \sqrt [4]{1+\frac {b x^4}{a}} \operatorname {Hypergeometric2F1}\left (\frac {3}{4},\frac {17}{4},\frac {7}{4},-\frac {b x^4}{a}\right )\right )}{240 a b^4 \left (a+b x^4\right )^{13/4}} \] Input:
Integrate[(x^14*(c + d*x^4))/(a + b*x^4)^(17/4),x]
Output:
(x^3*(a*(1155*a^3*d - 22*a^2*b*(7*c - 75*d*x^4) + 120*b^3*x^8*(-c + d*x^4) + 20*a*b^2*x^4*(-11*c + 45*d*x^4)) - 77*(-2*b*c + 15*a*d)*(a + b*x^4)^3*( 1 + (b*x^4)/a)^(1/4)*Hypergeometric2F1[3/4, 17/4, 7/4, -((b*x^4)/a)]))/(24 0*a*b^4*(a + b*x^4)^(13/4))
Time = 0.61 (sec) , antiderivative size = 193, normalized size of antiderivative = 1.06, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {957, 817, 817, 815, 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.
\(\displaystyle \int \frac {x^{14} \left (c+d x^4\right )}{\left (a+b x^4\right )^{17/4}} \, dx\) |
\(\Big \downarrow \) 957 |
\(\displaystyle \frac {x^{15} (b c-a d)}{13 a b \left (a+b x^4\right )^{13/4}}-\frac {(2 b c-15 a d) \int \frac {x^{14}}{\left (b x^4+a\right )^{13/4}}dx}{13 a b}\) |
\(\Big \downarrow \) 817 |
\(\displaystyle \frac {x^{15} (b c-a d)}{13 a b \left (a+b x^4\right )^{13/4}}-\frac {(2 b c-15 a d) \left (\frac {11 \int \frac {x^{10}}{\left (b x^4+a\right )^{9/4}}dx}{9 b}-\frac {x^{11}}{9 b \left (a+b x^4\right )^{9/4}}\right )}{13 a b}\) |
\(\Big \downarrow \) 817 |
\(\displaystyle \frac {x^{15} (b c-a d)}{13 a b \left (a+b x^4\right )^{13/4}}-\frac {(2 b c-15 a d) \left (\frac {11 \left (\frac {7 \int \frac {x^6}{\left (b x^4+a\right )^{5/4}}dx}{5 b}-\frac {x^7}{5 b \left (a+b x^4\right )^{5/4}}\right )}{9 b}-\frac {x^{11}}{9 b \left (a+b x^4\right )^{9/4}}\right )}{13 a b}\) |
\(\Big \downarrow \) 815 |
\(\displaystyle \frac {x^{15} (b c-a d)}{13 a b \left (a+b x^4\right )^{13/4}}-\frac {(2 b c-15 a d) \left (\frac {11 \left (\frac {7 \left (\frac {x^3}{2 b \sqrt [4]{a+b x^4}}-\frac {3 a \int \frac {x^2}{\left (b x^4+a\right )^{5/4}}dx}{2 b}\right )}{5 b}-\frac {x^7}{5 b \left (a+b x^4\right )^{5/4}}\right )}{9 b}-\frac {x^{11}}{9 b \left (a+b x^4\right )^{9/4}}\right )}{13 a b}\) |
\(\Big \downarrow \) 813 |
\(\displaystyle \frac {x^{15} (b c-a d)}{13 a b \left (a+b x^4\right )^{13/4}}-\frac {(2 b c-15 a d) \left (\frac {11 \left (\frac {7 \left (\frac {x^3}{2 b \sqrt [4]{a+b x^4}}-\frac {3 a x \sqrt [4]{\frac {a}{b x^4}+1} \int \frac {1}{\left (\frac {a}{b x^4}+1\right )^{5/4} x^3}dx}{2 b^2 \sqrt [4]{a+b x^4}}\right )}{5 b}-\frac {x^7}{5 b \left (a+b x^4\right )^{5/4}}\right )}{9 b}-\frac {x^{11}}{9 b \left (a+b x^4\right )^{9/4}}\right )}{13 a b}\) |
\(\Big \downarrow \) 858 |
\(\displaystyle \frac {x^{15} (b c-a d)}{13 a b \left (a+b x^4\right )^{13/4}}-\frac {(2 b c-15 a d) \left (\frac {11 \left (\frac {7 \left (\frac {3 a x \sqrt [4]{\frac {a}{b x^4}+1} \int \frac {1}{\left (\frac {a}{b x^4}+1\right )^{5/4} x}d\frac {1}{x}}{2 b^2 \sqrt [4]{a+b x^4}}+\frac {x^3}{2 b \sqrt [4]{a+b x^4}}\right )}{5 b}-\frac {x^7}{5 b \left (a+b x^4\right )^{5/4}}\right )}{9 b}-\frac {x^{11}}{9 b \left (a+b x^4\right )^{9/4}}\right )}{13 a b}\) |
\(\Big \downarrow \) 807 |
\(\displaystyle \frac {x^{15} (b c-a d)}{13 a b \left (a+b x^4\right )^{13/4}}-\frac {(2 b c-15 a d) \left (\frac {11 \left (\frac {7 \left (\frac {3 a x \sqrt [4]{\frac {a}{b x^4}+1} \int \frac {1}{\left (\frac {a}{b x^2}+1\right )^{5/4}}d\frac {1}{x^2}}{4 b^2 \sqrt [4]{a+b x^4}}+\frac {x^3}{2 b \sqrt [4]{a+b x^4}}\right )}{5 b}-\frac {x^7}{5 b \left (a+b x^4\right )^{5/4}}\right )}{9 b}-\frac {x^{11}}{9 b \left (a+b x^4\right )^{9/4}}\right )}{13 a b}\) |
\(\Big \downarrow \) 212 |
\(\displaystyle \frac {x^{15} (b c-a d)}{13 a b \left (a+b x^4\right )^{13/4}}-\frac {(2 b c-15 a d) \left (\frac {11 \left (\frac {7 \left (\frac {3 \sqrt {a} x \sqrt [4]{\frac {a}{b x^4}+1} E\left (\left .\frac {1}{2} \arctan \left (\frac {\sqrt {a}}{\sqrt {b} x^2}\right )\right |2\right )}{2 b^{3/2} \sqrt [4]{a+b x^4}}+\frac {x^3}{2 b \sqrt [4]{a+b x^4}}\right )}{5 b}-\frac {x^7}{5 b \left (a+b x^4\right )^{5/4}}\right )}{9 b}-\frac {x^{11}}{9 b \left (a+b x^4\right )^{9/4}}\right )}{13 a b}\) |
Input:
Int[(x^14*(c + d*x^4))/(a + b*x^4)^(17/4),x]
Output:
((b*c - a*d)*x^15)/(13*a*b*(a + b*x^4)^(13/4)) - ((2*b*c - 15*a*d)*(-1/9*x ^11/(b*(a + b*x^4)^(9/4)) + (11*(-1/5*x^7/(b*(a + b*x^4)^(5/4)) + (7*(x^3/ (2*b*(a + b*x^4)^(1/4)) + (3*Sqrt[a]*(1 + a/(b*x^4))^(1/4)*x*EllipticE[Arc Tan[Sqrt[a]/(Sqrt[b]*x^2)]/2, 2])/(2*b^(3/2)*(a + b*x^4)^(1/4))))/(5*b)))/ (9*b)))/(13*a*b)
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 - 3)/(b*( m - 4)*(a + b*x^4)^(1/4)), x] - Simp[a*((m - 3)/(b*(m - 4))) Int[x^(m - 4 )/(a + b*x^4)^(5/4), x], x] /; FreeQ[{a, b}, x] && PosQ[b/a] && IGtQ[(m - 2 )/4, 0]
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[c^( n - 1)*(c*x)^(m - n + 1)*((a + b*x^n)^(p + 1)/(b*n*(p + 1))), x] - Simp[c^n *((m - n + 1)/(b*n*(p + 1))) Int[(c*x)^(m - n)*(a + b*x^n)^(p + 1), x], x ] /; FreeQ[{a, b, c}, x] && IGtQ[n, 0] && LtQ[p, -1] && GtQ[m + 1, n] && ! ILtQ[(m + n*(p + 1) + 1)/n, 0] && 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[(-(b*c - a*d))*(e*x)^(m + 1)*((a + b*x^n)^(p + 1)/(a *b*e*n*(p + 1))), x] - Simp[(a*d*(m + 1) - b*c*(m + n*(p + 1) + 1))/(a*b*n* (p + 1)) Int[(e*x)^m*(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] && (( !IntegerQ[p + 1/2] && N eQ[p, -5/4]) || !RationalQ[m] || (IGtQ[n, 0] && ILtQ[p + 1/2, 0] && LeQ[-1 , m, (-n)*(p + 1)]))
\[\int \frac {x^{14} \left (d \,x^{4}+c \right )}{\left (b \,x^{4}+a \right )^{\frac {17}{4}}}d x\]
Input:
int(x^14*(d*x^4+c)/(b*x^4+a)^(17/4),x)
Output:
int(x^14*(d*x^4+c)/(b*x^4+a)^(17/4),x)
\[ \int \frac {x^{14} \left (c+d x^4\right )}{\left (a+b x^4\right )^{17/4}} \, dx=\int { \frac {{\left (d x^{4} + c\right )} x^{14}}{{\left (b x^{4} + a\right )}^{\frac {17}{4}}} \,d x } \] Input:
integrate(x^14*(d*x^4+c)/(b*x^4+a)^(17/4),x, algorithm="fricas")
Output:
integral((d*x^18 + c*x^14)*(b*x^4 + a)^(3/4)/(b^5*x^20 + 5*a*b^4*x^16 + 10 *a^2*b^3*x^12 + 10*a^3*b^2*x^8 + 5*a^4*b*x^4 + a^5), x)
Timed out. \[ \int \frac {x^{14} \left (c+d x^4\right )}{\left (a+b x^4\right )^{17/4}} \, dx=\text {Timed out} \] Input:
integrate(x**14*(d*x**4+c)/(b*x**4+a)**(17/4),x)
Output:
Timed out
\[ \int \frac {x^{14} \left (c+d x^4\right )}{\left (a+b x^4\right )^{17/4}} \, dx=\int { \frac {{\left (d x^{4} + c\right )} x^{14}}{{\left (b x^{4} + a\right )}^{\frac {17}{4}}} \,d x } \] Input:
integrate(x^14*(d*x^4+c)/(b*x^4+a)^(17/4),x, algorithm="maxima")
Output:
integrate((d*x^4 + c)*x^14/(b*x^4 + a)^(17/4), x)
\[ \int \frac {x^{14} \left (c+d x^4\right )}{\left (a+b x^4\right )^{17/4}} \, dx=\int { \frac {{\left (d x^{4} + c\right )} x^{14}}{{\left (b x^{4} + a\right )}^{\frac {17}{4}}} \,d x } \] Input:
integrate(x^14*(d*x^4+c)/(b*x^4+a)^(17/4),x, algorithm="giac")
Output:
integrate((d*x^4 + c)*x^14/(b*x^4 + a)^(17/4), x)
Timed out. \[ \int \frac {x^{14} \left (c+d x^4\right )}{\left (a+b x^4\right )^{17/4}} \, dx=\int \frac {x^{14}\,\left (d\,x^4+c\right )}{{\left (b\,x^4+a\right )}^{17/4}} \,d x \] Input:
int((x^14*(c + d*x^4))/(a + b*x^4)^(17/4),x)
Output:
int((x^14*(c + d*x^4))/(a + b*x^4)^(17/4), x)
\[ \int \frac {x^{14} \left (c+d x^4\right )}{\left (a+b x^4\right )^{17/4}} \, dx=\left (\int \frac {x^{18}}{\left (b \,x^{4}+a \right )^{\frac {1}{4}} a^{4}+4 \left (b \,x^{4}+a \right )^{\frac {1}{4}} a^{3} b \,x^{4}+6 \left (b \,x^{4}+a \right )^{\frac {1}{4}} a^{2} b^{2} x^{8}+4 \left (b \,x^{4}+a \right )^{\frac {1}{4}} a \,b^{3} x^{12}+\left (b \,x^{4}+a \right )^{\frac {1}{4}} b^{4} x^{16}}d x \right ) d +\left (\int \frac {x^{14}}{\left (b \,x^{4}+a \right )^{\frac {1}{4}} a^{4}+4 \left (b \,x^{4}+a \right )^{\frac {1}{4}} a^{3} b \,x^{4}+6 \left (b \,x^{4}+a \right )^{\frac {1}{4}} a^{2} b^{2} x^{8}+4 \left (b \,x^{4}+a \right )^{\frac {1}{4}} a \,b^{3} x^{12}+\left (b \,x^{4}+a \right )^{\frac {1}{4}} b^{4} x^{16}}d x \right ) c \] Input:
int(x^14*(d*x^4+c)/(b*x^4+a)^(17/4),x)
Output:
int(x**18/((a + b*x**4)**(1/4)*a**4 + 4*(a + b*x**4)**(1/4)*a**3*b*x**4 + 6*(a + b*x**4)**(1/4)*a**2*b**2*x**8 + 4*(a + b*x**4)**(1/4)*a*b**3*x**12 + (a + b*x**4)**(1/4)*b**4*x**16),x)*d + int(x**14/((a + b*x**4)**(1/4)*a* *4 + 4*(a + b*x**4)**(1/4)*a**3*b*x**4 + 6*(a + b*x**4)**(1/4)*a**2*b**2*x **8 + 4*(a + b*x**4)**(1/4)*a*b**3*x**12 + (a + b*x**4)**(1/4)*b**4*x**16) ,x)*c