Integrand size = 26, antiderivative size = 181 \[ \int \frac {c+d x^2}{(e x)^{7/2} \left (a+b x^2\right )^{9/4}} \, dx=-\frac {2 c}{5 a e (e x)^{5/2} \left (a+b x^2\right )^{5/4}}-\frac {2 (2 b c-a d)}{5 a^2 e^3 \sqrt {e x} \left (a+b x^2\right )^{5/4}}+\frac {12 (2 b c-a d)}{5 a^3 e^3 \sqrt {e x} \sqrt [4]{a+b x^2}}-\frac {24 \sqrt {b} (2 b c-a d) \sqrt [4]{1+\frac {a}{b x^2}} \sqrt {e x} E\left (\left .\frac {1}{2} \cot ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right |2\right )}{5 a^{7/2} e^4 \sqrt [4]{a+b x^2}} \]
-2/5*c/a/e/(e*x)^(5/2)/(b*x^2+a)^(5/4)-2/5*(-a*d+2*b*c)/a^2/e^3/(b*x^2+a)^ (5/4)/(e*x)^(1/2)+12/5*(-a*d+2*b*c)/a^3/e^3/(b*x^2+a)^(1/4)/(e*x)^(1/2)-24 /5*(-a*d+2*b*c)*(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))*b^(1/2)*(e*x)^(1/2)/a^(7/2)/e^4/(b*x^2+a)^(1/4)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 10.03 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.48 \[ \int \frac {c+d x^2}{(e x)^{7/2} \left (a+b x^2\right )^{9/4}} \, dx=\frac {2 x \left (-a^2 c-5 (-2 b c+a d) x^2 \left (a+b x^2\right ) \sqrt [4]{1+\frac {b x^2}{a}} \operatorname {Hypergeometric2F1}\left (-\frac {1}{4},\frac {9}{4},\frac {3}{4},-\frac {b x^2}{a}\right )\right )}{5 a^3 (e x)^{7/2} \left (a+b x^2\right )^{5/4}} \]
(2*x*(-(a^2*c) - 5*(-2*b*c + a*d)*x^2*(a + b*x^2)*(1 + (b*x^2)/a)^(1/4)*Hy pergeometric2F1[-1/4, 9/4, 3/4, -((b*x^2)/a)]))/(5*a^3*(e*x)^(7/2)*(a + b* x^2)^(5/4))
Time = 0.28 (sec) , antiderivative size = 178, normalized size of antiderivative = 0.98, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {359, 253, 251, 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.
\(\displaystyle \int \frac {c+d x^2}{(e x)^{7/2} \left (a+b x^2\right )^{9/4}} \, dx\) |
\(\Big \downarrow \) 359 |
\(\displaystyle -\frac {(2 b c-a d) \int \frac {1}{(e x)^{3/2} \left (b x^2+a\right )^{9/4}}dx}{a e^2}-\frac {2 c}{5 a e (e x)^{5/2} \left (a+b x^2\right )^{5/4}}\) |
\(\Big \downarrow \) 253 |
\(\displaystyle -\frac {(2 b c-a d) \left (\frac {6 \int \frac {1}{(e x)^{3/2} \left (b x^2+a\right )^{5/4}}dx}{5 a}+\frac {2}{5 a e \sqrt {e x} \left (a+b x^2\right )^{5/4}}\right )}{a e^2}-\frac {2 c}{5 a e (e x)^{5/2} \left (a+b x^2\right )^{5/4}}\) |
\(\Big \downarrow \) 251 |
\(\displaystyle -\frac {(2 b c-a d) \left (\frac {6 \left (-\frac {2 b \int \frac {\sqrt {e x}}{\left (b x^2+a\right )^{5/4}}dx}{a e^2}-\frac {2}{a e \sqrt {e x} \sqrt [4]{a+b x^2}}\right )}{5 a}+\frac {2}{5 a e \sqrt {e x} \left (a+b x^2\right )^{5/4}}\right )}{a e^2}-\frac {2 c}{5 a e (e x)^{5/2} \left (a+b x^2\right )^{5/4}}\) |
\(\Big \downarrow \) 249 |
\(\displaystyle -\frac {(2 b c-a d) \left (\frac {6 \left (-\frac {2 \sqrt {e x} \sqrt [4]{\frac {a}{b x^2}+1} \int \frac {1}{\left (\frac {a}{b x^2}+1\right )^{5/4} x^2}dx}{a e^2 \sqrt [4]{a+b x^2}}-\frac {2}{a e \sqrt {e x} \sqrt [4]{a+b x^2}}\right )}{5 a}+\frac {2}{5 a e \sqrt {e x} \left (a+b x^2\right )^{5/4}}\right )}{a e^2}-\frac {2 c}{5 a e (e x)^{5/2} \left (a+b x^2\right )^{5/4}}\) |
\(\Big \downarrow \) 858 |
\(\displaystyle -\frac {(2 b c-a d) \left (\frac {6 \left (\frac {2 \sqrt {e x} \sqrt [4]{\frac {a}{b x^2}+1} \int \frac {1}{\left (\frac {a}{b x^2}+1\right )^{5/4}}d\frac {1}{x}}{a e^2 \sqrt [4]{a+b x^2}}-\frac {2}{a e \sqrt {e x} \sqrt [4]{a+b x^2}}\right )}{5 a}+\frac {2}{5 a e \sqrt {e x} \left (a+b x^2\right )^{5/4}}\right )}{a e^2}-\frac {2 c}{5 a e (e x)^{5/2} \left (a+b x^2\right )^{5/4}}\) |
\(\Big \downarrow \) 212 |
\(\displaystyle -\frac {(2 b c-a d) \left (\frac {6 \left (\frac {4 \sqrt {b} \sqrt {e x} \sqrt [4]{\frac {a}{b x^2}+1} E\left (\left .\frac {1}{2} \arctan \left (\frac {\sqrt {a}}{\sqrt {b} x}\right )\right |2\right )}{a^{3/2} e^2 \sqrt [4]{a+b x^2}}-\frac {2}{a e \sqrt {e x} \sqrt [4]{a+b x^2}}\right )}{5 a}+\frac {2}{5 a e \sqrt {e x} \left (a+b x^2\right )^{5/4}}\right )}{a e^2}-\frac {2 c}{5 a e (e x)^{5/2} \left (a+b x^2\right )^{5/4}}\) |
(-2*c)/(5*a*e*(e*x)^(5/2)*(a + b*x^2)^(5/4)) - ((2*b*c - a*d)*(2/(5*a*e*Sq rt[e*x]*(a + b*x^2)^(5/4)) + (6*(-2/(a*e*Sqrt[e*x]*(a + b*x^2)^(1/4)) + (4 *Sqrt[b]*(1 + a/(b*x^2))^(1/4)*Sqrt[e*x]*EllipticE[ArcTan[Sqrt[a]/(Sqrt[b] *x)]/2, 2])/(a^(3/2)*e^2*(a + b*x^2)^(1/4))))/(5*a)))/(a*e^2)
3.12.37.3.1 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[((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]
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]
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2), x _Symbol] :> Simp[c*(e*x)^(m + 1)*((a + b*x^2)^(p + 1)/(a*e*(m + 1))), x] + Simp[(a*d*(m + 1) - b*c*(m + 2*p + 3))/(a*e^2*(m + 1)) Int[(e*x)^(m + 2)* (a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[b*c - a*d, 0] && LtQ[m, -1] && !ILtQ[p, -1]
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]
\[\int \frac {d \,x^{2}+c}{\left (e x \right )^{\frac {7}{2}} \left (b \,x^{2}+a \right )^{\frac {9}{4}}}d x\]
\[ \int \frac {c+d x^2}{(e x)^{7/2} \left (a+b x^2\right )^{9/4}} \, dx=\int { \frac {d x^{2} + c}{{\left (b x^{2} + a\right )}^{\frac {9}{4}} \left (e x\right )^{\frac {7}{2}}} \,d x } \]
integral((b*x^2 + a)^(3/4)*(d*x^2 + c)*sqrt(e*x)/(b^3*e^4*x^10 + 3*a*b^2*e ^4*x^8 + 3*a^2*b*e^4*x^6 + a^3*e^4*x^4), x)
Timed out. \[ \int \frac {c+d x^2}{(e x)^{7/2} \left (a+b x^2\right )^{9/4}} \, dx=\text {Timed out} \]
\[ \int \frac {c+d x^2}{(e x)^{7/2} \left (a+b x^2\right )^{9/4}} \, dx=\int { \frac {d x^{2} + c}{{\left (b x^{2} + a\right )}^{\frac {9}{4}} \left (e x\right )^{\frac {7}{2}}} \,d x } \]
\[ \int \frac {c+d x^2}{(e x)^{7/2} \left (a+b x^2\right )^{9/4}} \, dx=\int { \frac {d x^{2} + c}{{\left (b x^{2} + a\right )}^{\frac {9}{4}} \left (e x\right )^{\frac {7}{2}}} \,d x } \]
Timed out. \[ \int \frac {c+d x^2}{(e x)^{7/2} \left (a+b x^2\right )^{9/4}} \, dx=\int \frac {d\,x^2+c}{{\left (e\,x\right )}^{7/2}\,{\left (b\,x^2+a\right )}^{9/4}} \,d x \]