Integrand size = 15, antiderivative size = 146 \[ \int x^9 \left (a+b x^4\right )^{5/4} \, dx=-\frac {2 a^3 x^2 \sqrt [4]{a+b x^4}}{231 b^2}+\frac {a^2 x^6 \sqrt [4]{a+b x^4}}{231 b}+\frac {1}{33} a x^{10} \sqrt [4]{a+b x^4}+\frac {1}{15} x^{10} \left (a+b x^4\right )^{5/4}+\frac {4 a^{9/2} \left (1+\frac {b x^4}{a}\right )^{3/4} \operatorname {EllipticF}\left (\frac {1}{2} \arctan \left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right ),2\right )}{231 b^{5/2} \left (a+b x^4\right )^{3/4}} \]
-2/231*a^3*x^2*(b*x^4+a)^(1/4)/b^2+1/231*a^2*x^6*(b*x^4+a)^(1/4)/b+1/33*a* x^10*(b*x^4+a)^(1/4)+1/15*x^10*(b*x^4+a)^(5/4)+4/231*a^(9/2)*(1+b*x^4/a)^( 3/4)*(cos(1/2*arctan(x^2*b^(1/2)/a^(1/2)))^2)^(1/2)/cos(1/2*arctan(x^2*b^( 1/2)/a^(1/2)))*EllipticF(sin(1/2*arctan(x^2*b^(1/2)/a^(1/2))),2^(1/2))/b^( 5/2)/(b*x^4+a)^(3/4)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 9.01 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.55 \[ \int x^9 \left (a+b x^4\right )^{5/4} \, dx=\frac {x^2 \sqrt [4]{a+b x^4} \left (-\left (\left (6 a-11 b x^4\right ) \left (a+b x^4\right )^2\right )+\frac {6 a^3 \operatorname {Hypergeometric2F1}\left (-\frac {5}{4},\frac {1}{2},\frac {3}{2},-\frac {b x^4}{a}\right )}{\sqrt [4]{1+\frac {b x^4}{a}}}\right )}{165 b^2} \]
(x^2*(a + b*x^4)^(1/4)*(-((6*a - 11*b*x^4)*(a + b*x^4)^2) + (6*a^3*Hyperge ometric2F1[-5/4, 1/2, 3/2, -((b*x^4)/a)])/(1 + (b*x^4)/a)^(1/4)))/(165*b^2 )
Time = 0.25 (sec) , antiderivative size = 164, normalized size of antiderivative = 1.12, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.467, Rules used = {807, 248, 248, 262, 262, 231, 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.
\(\displaystyle \int x^9 \left (a+b x^4\right )^{5/4} \, dx\) |
\(\Big \downarrow \) 807 |
\(\displaystyle \frac {1}{2} \int x^8 \left (b x^4+a\right )^{5/4}dx^2\) |
\(\Big \downarrow \) 248 |
\(\displaystyle \frac {1}{2} \left (\frac {1}{3} a \int x^8 \sqrt [4]{b x^4+a}dx^2+\frac {2}{15} x^{10} \left (a+b x^4\right )^{5/4}\right )\) |
\(\Big \downarrow \) 248 |
\(\displaystyle \frac {1}{2} \left (\frac {1}{3} a \left (\frac {1}{11} a \int \frac {x^8}{\left (b x^4+a\right )^{3/4}}dx^2+\frac {2}{11} x^{10} \sqrt [4]{a+b x^4}\right )+\frac {2}{15} x^{10} \left (a+b x^4\right )^{5/4}\right )\) |
\(\Big \downarrow \) 262 |
\(\displaystyle \frac {1}{2} \left (\frac {1}{3} a \left (\frac {1}{11} a \left (\frac {2 x^6 \sqrt [4]{a+b x^4}}{7 b}-\frac {6 a \int \frac {x^4}{\left (b x^4+a\right )^{3/4}}dx^2}{7 b}\right )+\frac {2}{11} x^{10} \sqrt [4]{a+b x^4}\right )+\frac {2}{15} x^{10} \left (a+b x^4\right )^{5/4}\right )\) |
\(\Big \downarrow \) 262 |
\(\displaystyle \frac {1}{2} \left (\frac {1}{3} a \left (\frac {1}{11} a \left (\frac {2 x^6 \sqrt [4]{a+b x^4}}{7 b}-\frac {6 a \left (\frac {2 x^2 \sqrt [4]{a+b x^4}}{3 b}-\frac {2 a \int \frac {1}{\left (b x^4+a\right )^{3/4}}dx^2}{3 b}\right )}{7 b}\right )+\frac {2}{11} x^{10} \sqrt [4]{a+b x^4}\right )+\frac {2}{15} x^{10} \left (a+b x^4\right )^{5/4}\right )\) |
\(\Big \downarrow \) 231 |
\(\displaystyle \frac {1}{2} \left (\frac {1}{3} a \left (\frac {1}{11} a \left (\frac {2 x^6 \sqrt [4]{a+b x^4}}{7 b}-\frac {6 a \left (\frac {2 x^2 \sqrt [4]{a+b x^4}}{3 b}-\frac {2 a \left (\frac {b x^4}{a}+1\right )^{3/4} \int \frac {1}{\left (\frac {b x^4}{a}+1\right )^{3/4}}dx^2}{3 b \left (a+b x^4\right )^{3/4}}\right )}{7 b}\right )+\frac {2}{11} x^{10} \sqrt [4]{a+b x^4}\right )+\frac {2}{15} x^{10} \left (a+b x^4\right )^{5/4}\right )\) |
\(\Big \downarrow \) 229 |
\(\displaystyle \frac {1}{2} \left (\frac {1}{3} a \left (\frac {1}{11} a \left (\frac {2 x^6 \sqrt [4]{a+b x^4}}{7 b}-\frac {6 a \left (\frac {2 x^2 \sqrt [4]{a+b x^4}}{3 b}-\frac {4 a^{3/2} \left (\frac {b x^4}{a}+1\right )^{3/4} \operatorname {EllipticF}\left (\frac {1}{2} \arctan \left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right ),2\right )}{3 b^{3/2} \left (a+b x^4\right )^{3/4}}\right )}{7 b}\right )+\frac {2}{11} x^{10} \sqrt [4]{a+b x^4}\right )+\frac {2}{15} x^{10} \left (a+b x^4\right )^{5/4}\right )\) |
((2*x^10*(a + b*x^4)^(5/4))/15 + (a*((2*x^10*(a + b*x^4)^(1/4))/11 + (a*(( 2*x^6*(a + b*x^4)^(1/4))/(7*b) - (6*a*((2*x^2*(a + b*x^4)^(1/4))/(3*b) - ( 4*a^(3/2)*(1 + (b*x^4)/a)^(3/4)*EllipticF[ArcTan[(Sqrt[b]*x^2)/Sqrt[a]]/2, 2])/(3*b^(3/2)*(a + b*x^4)^(3/4))))/(7*b)))/11))/3)/2
3.11.55.3.1 Defintions of rubi rules used
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]
Int[((a_) + (b_.)*(x_)^2)^(-3/4), x_Symbol] :> Simp[(1 + b*(x^2/a))^(3/4)/( a + b*x^2)^(3/4) Int[1/(1 + b*(x^2/a))^(3/4), x], x] /; FreeQ[{a, b}, x] && PosQ[a]
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(c*x)^ (m + 1)*((a + b*x^2)^p/(c*(m + 2*p + 1))), x] + Simp[2*a*(p/(m + 2*p + 1)) Int[(c*x)^m*(a + b*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c, m}, x] && GtQ[ p, 0] && NeQ[m + 2*p + 1, 0] && IntBinomialQ[a, b, c, 2, m, p, x]
Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[c*(c*x) ^(m - 1)*((a + b*x^2)^(p + 1)/(b*(m + 2*p + 1))), x] - Simp[a*c^2*((m - 1)/ (b*(m + 2*p + 1))) Int[(c*x)^(m - 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b , c, p}, x] && GtQ[m, 2 - 1] && NeQ[m + 2*p + 1, 0] && IntBinomialQ[a, b, c , 2, m, p, x]
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]
\[\int x^{9} \left (b \,x^{4}+a \right )^{\frac {5}{4}}d x\]
\[ \int x^9 \left (a+b x^4\right )^{5/4} \, dx=\int { {\left (b x^{4} + a\right )}^{\frac {5}{4}} x^{9} \,d x } \]
Result contains complex when optimal does not.
Time = 1.00 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.20 \[ \int x^9 \left (a+b x^4\right )^{5/4} \, dx=\frac {a^{\frac {5}{4}} x^{10} {{}_{2}F_{1}\left (\begin {matrix} - \frac {5}{4}, \frac {5}{2} \\ \frac {7}{2} \end {matrix}\middle | {\frac {b x^{4} e^{i \pi }}{a}} \right )}}{10} \]
\[ \int x^9 \left (a+b x^4\right )^{5/4} \, dx=\int { {\left (b x^{4} + a\right )}^{\frac {5}{4}} x^{9} \,d x } \]
\[ \int x^9 \left (a+b x^4\right )^{5/4} \, dx=\int { {\left (b x^{4} + a\right )}^{\frac {5}{4}} x^{9} \,d x } \]
Timed out. \[ \int x^9 \left (a+b x^4\right )^{5/4} \, dx=\int x^9\,{\left (b\,x^4+a\right )}^{5/4} \,d x \]