Integrand size = 26, antiderivative size = 113 \[ \int \frac {c+d x^2}{(e x)^{3/2} \left (a+b x^2\right )^{3/4}} \, dx=-\frac {2 c \sqrt [4]{a+b x^2}}{a e \sqrt {e x}}-\frac {d \arctan \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt {e} \sqrt [4]{a+b x^2}}\right )}{b^{3/4} e^{3/2}}+\frac {d \text {arctanh}\left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt {e} \sqrt [4]{a+b x^2}}\right )}{b^{3/4} e^{3/2}} \]
-d*arctan(b^(1/4)*(e*x)^(1/2)/(b*x^2+a)^(1/4)/e^(1/2))/b^(3/4)/e^(3/2)+d*a rctanh(b^(1/4)*(e*x)^(1/2)/(b*x^2+a)^(1/4)/e^(1/2))/b^(3/4)/e^(3/2)-2*c*(b *x^2+a)^(1/4)/a/e/(e*x)^(1/2)
Time = 0.37 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.88 \[ \int \frac {c+d x^2}{(e x)^{3/2} \left (a+b x^2\right )^{3/4}} \, dx=\frac {x \left (-2 b^{3/4} c \sqrt [4]{a+b x^2}-a d \sqrt {x} \arctan \left (\frac {\sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a+b x^2}}\right )+a d \sqrt {x} \text {arctanh}\left (\frac {\sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a+b x^2}}\right )\right )}{a b^{3/4} (e x)^{3/2}} \]
(x*(-2*b^(3/4)*c*(a + b*x^2)^(1/4) - a*d*Sqrt[x]*ArcTan[(b^(1/4)*Sqrt[x])/ (a + b*x^2)^(1/4)] + a*d*Sqrt[x]*ArcTanh[(b^(1/4)*Sqrt[x])/(a + b*x^2)^(1/ 4)]))/(a*b^(3/4)*(e*x)^(3/2))
Time = 0.25 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.09, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.269, Rules used = {358, 266, 854, 27, 827, 218, 221}
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.
\(\displaystyle \int \frac {c+d x^2}{(e x)^{3/2} \left (a+b x^2\right )^{3/4}} \, dx\) |
\(\Big \downarrow \) 358 |
\(\displaystyle \frac {d \int \frac {\sqrt {e x}}{\left (b x^2+a\right )^{3/4}}dx}{e^2}-\frac {2 c \sqrt [4]{a+b x^2}}{a e \sqrt {e x}}\) |
\(\Big \downarrow \) 266 |
\(\displaystyle \frac {2 d \int \frac {e x}{\left (b x^2+a\right )^{3/4}}d\sqrt {e x}}{e^3}-\frac {2 c \sqrt [4]{a+b x^2}}{a e \sqrt {e x}}\) |
\(\Big \downarrow \) 854 |
\(\displaystyle \frac {2 d \int \frac {e^3 x}{e^2-b e^2 x^2}d\frac {\sqrt {e x}}{\sqrt [4]{b x^2+a}}}{e^3}-\frac {2 c \sqrt [4]{a+b x^2}}{a e \sqrt {e x}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {2 d \int \frac {e x}{e^2-b e^2 x^2}d\frac {\sqrt {e x}}{\sqrt [4]{b x^2+a}}}{e}-\frac {2 c \sqrt [4]{a+b x^2}}{a e \sqrt {e x}}\) |
\(\Big \downarrow \) 827 |
\(\displaystyle \frac {2 d \left (\frac {\int \frac {1}{e-\sqrt {b} e x}d\frac {\sqrt {e x}}{\sqrt [4]{b x^2+a}}}{2 \sqrt {b}}-\frac {\int \frac {1}{\sqrt {b} x e+e}d\frac {\sqrt {e x}}{\sqrt [4]{b x^2+a}}}{2 \sqrt {b}}\right )}{e}-\frac {2 c \sqrt [4]{a+b x^2}}{a e \sqrt {e x}}\) |
\(\Big \downarrow \) 218 |
\(\displaystyle \frac {2 d \left (\frac {\int \frac {1}{e-\sqrt {b} e x}d\frac {\sqrt {e x}}{\sqrt [4]{b x^2+a}}}{2 \sqrt {b}}-\frac {\arctan \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt {e} \sqrt [4]{a+b x^2}}\right )}{2 b^{3/4} \sqrt {e}}\right )}{e}-\frac {2 c \sqrt [4]{a+b x^2}}{a e \sqrt {e x}}\) |
\(\Big \downarrow \) 221 |
\(\displaystyle \frac {2 d \left (\frac {\text {arctanh}\left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt {e} \sqrt [4]{a+b x^2}}\right )}{2 b^{3/4} \sqrt {e}}-\frac {\arctan \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt {e} \sqrt [4]{a+b x^2}}\right )}{2 b^{3/4} \sqrt {e}}\right )}{e}-\frac {2 c \sqrt [4]{a+b x^2}}{a e \sqrt {e x}}\) |
(-2*c*(a + b*x^2)^(1/4))/(a*e*Sqrt[e*x]) + (2*d*(-1/2*ArcTan[(b^(1/4)*Sqrt [e*x])/(Sqrt[e]*(a + b*x^2)^(1/4))]/(b^(3/4)*Sqrt[e]) + ArcTanh[(b^(1/4)*S qrt[e*x])/(Sqrt[e]*(a + b*x^2)^(1/4))]/(2*b^(3/4)*Sqrt[e])))/e
3.11.95.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/R t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
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[c*(e*x)^(m + 1)*((a + b*x^2)^(p + 1)/(a*e*(m + 1))), x] + S imp[d/e^2 Int[(e*x)^(m + 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e , m, p}, x] && NeQ[b*c - a*d, 0] && EqQ[Simplify[m + 2*p + 3], 0] && NeQ[m, -1]
Int[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[-a/b, 2]], s = Denominator[Rt[-a/b, 2]]}, Simp[s/(2*b) Int[1/(r + s*x^2), x], x] - Simp[s/(2*b) Int[1/(r - s*x^2), x], x]] /; FreeQ[{a, b}, x] && !GtQ [a/b, 0]
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^(p + (m + 1)/n) Subst[Int[x^m/(1 - b*x^n)^(p + (m + 1)/n + 1), x], x, x/(a + b*x^n )^(1/n)], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && LtQ[-1, p, 0] && NeQ[p, - 2^(-1)] && IntegersQ[m, p + (m + 1)/n]
\[\int \frac {d \,x^{2}+c}{\left (e x \right )^{\frac {3}{2}} \left (b \,x^{2}+a \right )^{\frac {3}{4}}}d x\]
Timed out. \[ \int \frac {c+d x^2}{(e x)^{3/2} \left (a+b x^2\right )^{3/4}} \, dx=\text {Timed out} \]
Result contains complex when optimal does not.
Time = 3.92 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.75 \[ \int \frac {c+d x^2}{(e x)^{3/2} \left (a+b x^2\right )^{3/4}} \, dx=\frac {\sqrt [4]{b} c \sqrt [4]{\frac {a}{b x^{2}} + 1} \Gamma \left (- \frac {1}{4}\right )}{2 a e^{\frac {3}{2}} \Gamma \left (\frac {3}{4}\right )} + \frac {d x^{\frac {3}{2}} \Gamma \left (\frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {3}{4}, \frac {3}{4} \\ \frac {7}{4} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{2 a^{\frac {3}{4}} e^{\frac {3}{2}} \Gamma \left (\frac {7}{4}\right )} \]
b**(1/4)*c*(a/(b*x**2) + 1)**(1/4)*gamma(-1/4)/(2*a*e**(3/2)*gamma(3/4)) + d*x**(3/2)*gamma(3/4)*hyper((3/4, 3/4), (7/4,), b*x**2*exp_polar(I*pi)/a) /(2*a**(3/4)*e**(3/2)*gamma(7/4))
\[ \int \frac {c+d x^2}{(e x)^{3/2} \left (a+b x^2\right )^{3/4}} \, dx=\int { \frac {d x^{2} + c}{{\left (b x^{2} + a\right )}^{\frac {3}{4}} \left (e x\right )^{\frac {3}{2}}} \,d x } \]
\[ \int \frac {c+d x^2}{(e x)^{3/2} \left (a+b x^2\right )^{3/4}} \, dx=\int { \frac {d x^{2} + c}{{\left (b x^{2} + a\right )}^{\frac {3}{4}} \left (e x\right )^{\frac {3}{2}}} \,d x } \]
Timed out. \[ \int \frac {c+d x^2}{(e x)^{3/2} \left (a+b x^2\right )^{3/4}} \, dx=\int \frac {d\,x^2+c}{{\left (e\,x\right )}^{3/2}\,{\left (b\,x^2+a\right )}^{3/4}} \,d x \]