Integrand size = 19, antiderivative size = 157 \[ \int \frac {1}{(c x)^{11/2} \sqrt [4]{a+b x^2}} \, dx=-\frac {8 b^2}{15 a^2 c^5 \sqrt {c x} \sqrt [4]{a+b x^2}}-\frac {2 \left (a+b x^2\right )^{3/4}}{9 a c (c x)^{9/2}}+\frac {4 b \left (a+b x^2\right )^{3/4}}{15 a^2 c^3 (c x)^{5/2}}+\frac {8 b^{5/2} \sqrt [4]{1+\frac {a}{b x^2}} \sqrt {c x} E\left (\left .\frac {1}{2} \cot ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right |2\right )}{15 a^{5/2} c^6 \sqrt [4]{a+b x^2}} \]
-2/9*(b*x^2+a)^(3/4)/a/c/(c*x)^(9/2)+4/15*b*(b*x^2+a)^(3/4)/a^2/c^3/(c*x)^ (5/2)-8/15*b^2/a^2/c^5/(b*x^2+a)^(1/4)/(c*x)^(1/2)+8/15*b^(5/2)*(1+a/b/x^2 )^(1/4)*(cos(1/2*arccot(x*b^(1/2)/a^(1/2)))^2)^(1/2)/cos(1/2*arccot(x*b^(1 /2)/a^(1/2)))*EllipticE(sin(1/2*arccot(x*b^(1/2)/a^(1/2))),2^(1/2))*(c*x)^ (1/2)/a^(5/2)/c^6/(b*x^2+a)^(1/4)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 10.01 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.36 \[ \int \frac {1}{(c x)^{11/2} \sqrt [4]{a+b x^2}} \, dx=-\frac {2 x \sqrt [4]{1+\frac {b x^2}{a}} \operatorname {Hypergeometric2F1}\left (-\frac {9}{4},\frac {1}{4},-\frac {5}{4},-\frac {b x^2}{a}\right )}{9 (c x)^{11/2} \sqrt [4]{a+b x^2}} \]
(-2*x*(1 + (b*x^2)/a)^(1/4)*Hypergeometric2F1[-9/4, 1/4, -5/4, -((b*x^2)/a )])/(9*(c*x)^(11/2)*(a + b*x^2)^(1/4))
Time = 0.27 (sec) , antiderivative size = 172, normalized size of antiderivative = 1.10, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.316, Rules used = {264, 264, 257, 249, 858, 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.
\(\displaystyle \int \frac {1}{(c x)^{11/2} \sqrt [4]{a+b x^2}} \, dx\) |
\(\Big \downarrow \) 264 |
\(\displaystyle -\frac {2 b \int \frac {1}{(c x)^{7/2} \sqrt [4]{b x^2+a}}dx}{3 a c^2}-\frac {2 \left (a+b x^2\right )^{3/4}}{9 a c (c x)^{9/2}}\) |
\(\Big \downarrow \) 264 |
\(\displaystyle -\frac {2 b \left (-\frac {2 b \int \frac {1}{(c x)^{3/2} \sqrt [4]{b x^2+a}}dx}{5 a c^2}-\frac {2 \left (a+b x^2\right )^{3/4}}{5 a c (c x)^{5/2}}\right )}{3 a c^2}-\frac {2 \left (a+b x^2\right )^{3/4}}{9 a c (c x)^{9/2}}\) |
\(\Big \downarrow \) 257 |
\(\displaystyle -\frac {2 b \left (-\frac {2 b \left (-\frac {b \int \frac {\sqrt {c x}}{\left (b x^2+a\right )^{5/4}}dx}{c^2}-\frac {2}{c \sqrt {c x} \sqrt [4]{a+b x^2}}\right )}{5 a c^2}-\frac {2 \left (a+b x^2\right )^{3/4}}{5 a c (c x)^{5/2}}\right )}{3 a c^2}-\frac {2 \left (a+b x^2\right )^{3/4}}{9 a c (c x)^{9/2}}\) |
\(\Big \downarrow \) 249 |
\(\displaystyle -\frac {2 b \left (-\frac {2 b \left (-\frac {\sqrt {c x} \sqrt [4]{\frac {a}{b x^2}+1} \int \frac {1}{\left (\frac {a}{b x^2}+1\right )^{5/4} x^2}dx}{c^2 \sqrt [4]{a+b x^2}}-\frac {2}{c \sqrt {c x} \sqrt [4]{a+b x^2}}\right )}{5 a c^2}-\frac {2 \left (a+b x^2\right )^{3/4}}{5 a c (c x)^{5/2}}\right )}{3 a c^2}-\frac {2 \left (a+b x^2\right )^{3/4}}{9 a c (c x)^{9/2}}\) |
\(\Big \downarrow \) 858 |
\(\displaystyle -\frac {2 b \left (-\frac {2 b \left (\frac {\sqrt {c x} \sqrt [4]{\frac {a}{b x^2}+1} \int \frac {1}{\left (\frac {a}{b x^2}+1\right )^{5/4}}d\frac {1}{x}}{c^2 \sqrt [4]{a+b x^2}}-\frac {2}{c \sqrt {c x} \sqrt [4]{a+b x^2}}\right )}{5 a c^2}-\frac {2 \left (a+b x^2\right )^{3/4}}{5 a c (c x)^{5/2}}\right )}{3 a c^2}-\frac {2 \left (a+b x^2\right )^{3/4}}{9 a c (c x)^{9/2}}\) |
\(\Big \downarrow \) 212 |
\(\displaystyle -\frac {2 b \left (-\frac {2 b \left (\frac {2 \sqrt {b} \sqrt {c x} \sqrt [4]{\frac {a}{b x^2}+1} E\left (\left .\frac {1}{2} \arctan \left (\frac {\sqrt {a}}{\sqrt {b} x}\right )\right |2\right )}{\sqrt {a} c^2 \sqrt [4]{a+b x^2}}-\frac {2}{c \sqrt {c x} \sqrt [4]{a+b x^2}}\right )}{5 a c^2}-\frac {2 \left (a+b x^2\right )^{3/4}}{5 a c (c x)^{5/2}}\right )}{3 a c^2}-\frac {2 \left (a+b x^2\right )^{3/4}}{9 a c (c x)^{9/2}}\) |
(-2*(a + b*x^2)^(3/4))/(9*a*c*(c*x)^(9/2)) - (2*b*((-2*(a + b*x^2)^(3/4))/ (5*a*c*(c*x)^(5/2)) - (2*b*(-2/(c*Sqrt[c*x]*(a + b*x^2)^(1/4)) + (2*Sqrt[b ]*(1 + a/(b*x^2))^(1/4)*Sqrt[c*x]*EllipticE[ArcTan[Sqrt[a]/(Sqrt[b]*x)]/2, 2])/(Sqrt[a]*c^2*(a + b*x^2)^(1/4))))/(5*a*c^2)))/(3*a*c^2)
3.10.54.3.1 Defintions of rubi rules used
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[Sqrt[(c_.)*(x_)]/((a_) + (b_.)*(x_)^2)^(5/4), x_Symbol] :> Simp[Sqrt[c* x]*((1 + a/(b*x^2))^(1/4)/(b*(a + b*x^2)^(1/4))) Int[1/(x^2*(1 + a/(b*x^2 ))^(5/4)), x], x] /; FreeQ[{a, b, c}, x] && PosQ[b/a]
Int[1/(((c_.)*(x_))^(3/2)*((a_) + (b_.)*(x_)^2)^(1/4)), x_Symbol] :> Simp[- 2/(c*Sqrt[c*x]*(a + b*x^2)^(1/4)), x] - Simp[b/c^2 Int[Sqrt[c*x]/(a + b*x ^2)^(5/4), x], x] /; FreeQ[{a, b, c}, x] && PosQ[b/a]
Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(c*x)^( m + 1)*((a + b*x^2)^(p + 1)/(a*c*(m + 1))), x] - Simp[b*((m + 2*p + 3)/(a*c ^2*(m + 1))) Int[(c*x)^(m + 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, p }, x] && LtQ[m, -1] && IntBinomialQ[a, b, c, 2, 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]
\[\int \frac {1}{\left (c x \right )^{\frac {11}{2}} \left (b \,x^{2}+a \right )^{\frac {1}{4}}}d x\]
\[ \int \frac {1}{(c x)^{11/2} \sqrt [4]{a+b x^2}} \, dx=\int { \frac {1}{{\left (b x^{2} + a\right )}^{\frac {1}{4}} \left (c x\right )^{\frac {11}{2}}} \,d x } \]
Result contains complex when optimal does not.
Time = 60.71 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.22 \[ \int \frac {1}{(c x)^{11/2} \sqrt [4]{a+b x^2}} \, dx=- \frac {{{}_{2}F_{1}\left (\begin {matrix} \frac {1}{4}, \frac {5}{2} \\ \frac {7}{2} \end {matrix}\middle | {\frac {a e^{i \pi }}{b x^{2}}} \right )}}{5 \sqrt [4]{b} c^{\frac {11}{2}} x^{5}} \]
\[ \int \frac {1}{(c x)^{11/2} \sqrt [4]{a+b x^2}} \, dx=\int { \frac {1}{{\left (b x^{2} + a\right )}^{\frac {1}{4}} \left (c x\right )^{\frac {11}{2}}} \,d x } \]
\[ \int \frac {1}{(c x)^{11/2} \sqrt [4]{a+b x^2}} \, dx=\int { \frac {1}{{\left (b x^{2} + a\right )}^{\frac {1}{4}} \left (c x\right )^{\frac {11}{2}}} \,d x } \]
Timed out. \[ \int \frac {1}{(c x)^{11/2} \sqrt [4]{a+b x^2}} \, dx=\int \frac {1}{{\left (c\,x\right )}^{11/2}\,{\left (b\,x^2+a\right )}^{1/4}} \,d x \]