Integrand size = 26, antiderivative size = 116 \[ \int \frac {c+d x^2}{\sqrt {e x} \left (a+b x^2\right )^{7/4}} \, dx=\frac {2 (b c-a d) \sqrt {e x}}{3 a b e \left (a+b x^2\right )^{3/4}}-\frac {2 (2 b c+a d) \left (1+\frac {a}{b x^2}\right )^{3/4} (e x)^{3/2} \operatorname {EllipticF}\left (\frac {1}{2} \cot ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),2\right )}{3 a^{3/2} \sqrt {b} e^2 \left (a+b x^2\right )^{3/4}} \]
-2/3*(a*d+2*b*c)*(1+a/b/x^2)^(3/4)*(e*x)^(3/2)*(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)))*EllipticF(sin(1/2*arc cot(x*b^(1/2)/a^(1/2))),2^(1/2))/a^(3/2)/e^2/(b*x^2+a)^(3/4)/b^(1/2)+2/3*( -a*d+b*c)*(e*x)^(1/2)/a/b/e/(b*x^2+a)^(3/4)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 10.05 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.68 \[ \int \frac {c+d x^2}{\sqrt {e x} \left (a+b x^2\right )^{7/4}} \, dx=\frac {2 x \left (b c-a d+(2 b c+a d) \left (1+\frac {b x^2}{a}\right )^{3/4} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {3}{4},\frac {5}{4},-\frac {b x^2}{a}\right )\right )}{3 a b \sqrt {e x} \left (a+b x^2\right )^{3/4}} \]
(2*x*(b*c - a*d + (2*b*c + a*d)*(1 + (b*x^2)/a)^(3/4)*Hypergeometric2F1[1/ 4, 3/4, 5/4, -((b*x^2)/a)]))/(3*a*b*Sqrt[e*x]*(a + b*x^2)^(3/4))
Time = 0.29 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.03, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {362, 266, 768, 858, 807, 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.
\(\displaystyle \int \frac {c+d x^2}{\sqrt {e x} \left (a+b x^2\right )^{7/4}} \, dx\) |
\(\Big \downarrow \) 362 |
\(\displaystyle \frac {(a d+2 b c) \int \frac {1}{\sqrt {e x} \left (b x^2+a\right )^{3/4}}dx}{3 a b}+\frac {2 \sqrt {e x} (b c-a d)}{3 a b e \left (a+b x^2\right )^{3/4}}\) |
\(\Big \downarrow \) 266 |
\(\displaystyle \frac {2 (a d+2 b c) \int \frac {1}{\left (b x^2+a\right )^{3/4}}d\sqrt {e x}}{3 a b e}+\frac {2 \sqrt {e x} (b c-a d)}{3 a b e \left (a+b x^2\right )^{3/4}}\) |
\(\Big \downarrow \) 768 |
\(\displaystyle \frac {2 (e x)^{3/2} \left (\frac {a}{b x^2}+1\right )^{3/4} (a d+2 b c) \int \frac {1}{\left (\frac {a}{b x^2}+1\right )^{3/4} (e x)^{3/2}}d\sqrt {e x}}{3 a b e \left (a+b x^2\right )^{3/4}}+\frac {2 \sqrt {e x} (b c-a d)}{3 a b e \left (a+b x^2\right )^{3/4}}\) |
\(\Big \downarrow \) 858 |
\(\displaystyle \frac {2 \sqrt {e x} (b c-a d)}{3 a b e \left (a+b x^2\right )^{3/4}}-\frac {2 (e x)^{3/2} \left (\frac {a}{b x^2}+1\right )^{3/4} (a d+2 b c) \int \frac {1}{\sqrt {e x} \left (\frac {a x^2 e^4}{b}+1\right )^{3/4}}d\frac {1}{\sqrt {e x}}}{3 a b e \left (a+b x^2\right )^{3/4}}\) |
\(\Big \downarrow \) 807 |
\(\displaystyle \frac {2 \sqrt {e x} (b c-a d)}{3 a b e \left (a+b x^2\right )^{3/4}}-\frac {(e x)^{3/2} \left (\frac {a}{b x^2}+1\right )^{3/4} (a d+2 b c) \int \frac {1}{\left (\frac {a x e^3}{b}+1\right )^{3/4}}d(e x)}{3 a b e \left (a+b x^2\right )^{3/4}}\) |
\(\Big \downarrow \) 229 |
\(\displaystyle \frac {2 \sqrt {e x} (b c-a d)}{3 a b e \left (a+b x^2\right )^{3/4}}-\frac {2 (e x)^{3/2} \left (\frac {a}{b x^2}+1\right )^{3/4} (a d+2 b c) \operatorname {EllipticF}\left (\frac {1}{2} \arctan \left (\frac {\sqrt {a} e^2 x}{\sqrt {b}}\right ),2\right )}{3 a^{3/2} \sqrt {b} e^2 \left (a+b x^2\right )^{3/4}}\) |
(2*(b*c - a*d)*Sqrt[e*x])/(3*a*b*e*(a + b*x^2)^(3/4)) - (2*(2*b*c + a*d)*( 1 + a/(b*x^2))^(3/4)*(e*x)^(3/2)*EllipticF[ArcTan[(Sqrt[a]*e^2*x)/Sqrt[b]] /2, 2])/(3*a^(3/2)*Sqrt[b]*e^2*(a + b*x^2)^(3/4))
3.12.23.3.1 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[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{k = De nominator[m]}, Simp[k/c Subst[Int[x^(k*(m + 1) - 1)*(a + b*(x^(2*k)/c^2)) ^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && FractionQ[m] && I ntBinomialQ[a, b, c, 2, m, p, x]
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2), x _Symbol] :> Simp[(-(b*c - a*d))*(e*x)^(m + 1)*((a + b*x^2)^(p + 1)/(2*a*b*e *(p + 1))), x] - Simp[(a*d*(m + 1) - b*c*(m + 2*p + 3))/(2*a*b*(p + 1)) I nt[(e*x)^m*(a + b*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, m}, x] && N eQ[b*c - a*d, 0] && LtQ[p, -1] && (( !IntegerQ[p + 1/2] && NeQ[p, -5/4]) || !RationalQ[m] || (ILtQ[p + 1/2, 0] && LeQ[-1, m, -2*(p + 1)]))
Int[((a_) + (b_.)*(x_)^4)^(-3/4), x_Symbol] :> Simp[x^3*((1 + a/(b*x^4))^(3 /4)/(a + b*x^4)^(3/4)) Int[1/(x^3*(1 + a/(b*x^4))^(3/4)), x], x] /; FreeQ [{a, b}, 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_)^(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]
\[\int \frac {d \,x^{2}+c}{\sqrt {e x}\, \left (b \,x^{2}+a \right )^{\frac {7}{4}}}d x\]
\[ \int \frac {c+d x^2}{\sqrt {e x} \left (a+b x^2\right )^{7/4}} \, dx=\int { \frac {d x^{2} + c}{{\left (b x^{2} + a\right )}^{\frac {7}{4}} \sqrt {e x}} \,d x } \]
Result contains complex when optimal does not.
Time = 18.12 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.67 \[ \int \frac {c+d x^2}{\sqrt {e x} \left (a+b x^2\right )^{7/4}} \, dx=- \frac {d {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {7}{4} \\ \frac {3}{2} \end {matrix}\middle | {\frac {a e^{i \pi }}{b x^{2}}} \right )}}{b^{\frac {7}{4}} \sqrt {e} x} + \frac {c \sqrt {x} \Gamma \left (\frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{4}, \frac {7}{4} \\ \frac {5}{4} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{2 a^{\frac {7}{4}} \sqrt {e} \Gamma \left (\frac {5}{4}\right )} \]
-d*hyper((1/2, 7/4), (3/2,), a*exp_polar(I*pi)/(b*x**2))/(b**(7/4)*sqrt(e) *x) + c*sqrt(x)*gamma(1/4)*hyper((1/4, 7/4), (5/4,), b*x**2*exp_polar(I*pi )/a)/(2*a**(7/4)*sqrt(e)*gamma(5/4))
\[ \int \frac {c+d x^2}{\sqrt {e x} \left (a+b x^2\right )^{7/4}} \, dx=\int { \frac {d x^{2} + c}{{\left (b x^{2} + a\right )}^{\frac {7}{4}} \sqrt {e x}} \,d x } \]
\[ \int \frac {c+d x^2}{\sqrt {e x} \left (a+b x^2\right )^{7/4}} \, dx=\int { \frac {d x^{2} + c}{{\left (b x^{2} + a\right )}^{\frac {7}{4}} \sqrt {e x}} \,d x } \]
Timed out. \[ \int \frac {c+d x^2}{\sqrt {e x} \left (a+b x^2\right )^{7/4}} \, dx=\int \frac {d\,x^2+c}{\sqrt {e\,x}\,{\left (b\,x^2+a\right )}^{7/4}} \,d x \]