Integrand size = 22, antiderivative size = 109 \[ \int \frac {c+d x^2}{x^2 \sqrt [4]{a+b x^2}} \, dx=\frac {(b c+2 a d) x}{a \sqrt [4]{a+b x^2}}-\frac {c \left (a+b x^2\right )^{3/4}}{a x}-\frac {(b c+2 a d) \sqrt [4]{1+\frac {b x^2}{a}} E\left (\left .\frac {1}{2} \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right |2\right )}{\sqrt {a} \sqrt {b} \sqrt [4]{a+b x^2}} \] Output:
(2*a*d+b*c)*x/a/(b*x^2+a)^(1/4)-c*(b*x^2+a)^(3/4)/a/x-(2*a*d+b*c)*(1+b*x^2 /a)^(1/4)*EllipticE(sin(1/2*arctan(b^(1/2)*x/a^(1/2))),2^(1/2))/a^(1/2)/b^ (1/2)/(b*x^2+a)^(1/4)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 10.02 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.71 \[ \int \frac {c+d x^2}{x^2 \sqrt [4]{a+b x^2}} \, dx=\frac {-2 c \left (a+b x^2\right )+(b c+2 a d) x^2 \sqrt [4]{1+\frac {b x^2}{a}} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1}{2},\frac {3}{2},-\frac {b x^2}{a}\right )}{2 a x \sqrt [4]{a+b x^2}} \] Input:
Integrate[(c + d*x^2)/(x^2*(a + b*x^2)^(1/4)),x]
Output:
(-2*c*(a + b*x^2) + (b*c + 2*a*d)*x^2*(1 + (b*x^2)/a)^(1/4)*Hypergeometric 2F1[1/4, 1/2, 3/2, -((b*x^2)/a)])/(2*a*x*(a + b*x^2)^(1/4))
Time = 0.21 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.01, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {359, 227, 225, 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.
\(\displaystyle \int \frac {c+d x^2}{x^2 \sqrt [4]{a+b x^2}} \, dx\) |
\(\Big \downarrow \) 359 |
\(\displaystyle \frac {(2 a d+b c) \int \frac {1}{\sqrt [4]{b x^2+a}}dx}{2 a}-\frac {c \left (a+b x^2\right )^{3/4}}{a x}\) |
\(\Big \downarrow \) 227 |
\(\displaystyle \frac {\sqrt [4]{\frac {b x^2}{a}+1} (2 a d+b c) \int \frac {1}{\sqrt [4]{\frac {b x^2}{a}+1}}dx}{2 a \sqrt [4]{a+b x^2}}-\frac {c \left (a+b x^2\right )^{3/4}}{a x}\) |
\(\Big \downarrow \) 225 |
\(\displaystyle \frac {\sqrt [4]{\frac {b x^2}{a}+1} (2 a d+b c) \left (\frac {2 x}{\sqrt [4]{\frac {b x^2}{a}+1}}-\int \frac {1}{\left (\frac {b x^2}{a}+1\right )^{5/4}}dx\right )}{2 a \sqrt [4]{a+b x^2}}-\frac {c \left (a+b x^2\right )^{3/4}}{a x}\) |
\(\Big \downarrow \) 212 |
\(\displaystyle \frac {\sqrt [4]{\frac {b x^2}{a}+1} (2 a d+b c) \left (\frac {2 x}{\sqrt [4]{\frac {b x^2}{a}+1}}-\frac {2 \sqrt {a} E\left (\left .\frac {1}{2} \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right |2\right )}{\sqrt {b}}\right )}{2 a \sqrt [4]{a+b x^2}}-\frac {c \left (a+b x^2\right )^{3/4}}{a x}\) |
Input:
Int[(c + d*x^2)/(x^2*(a + b*x^2)^(1/4)),x]
Output:
-((c*(a + b*x^2)^(3/4))/(a*x)) + ((b*c + 2*a*d)*(1 + (b*x^2)/a)^(1/4)*((2* x)/(1 + (b*x^2)/a)^(1/4) - (2*Sqrt[a]*EllipticE[ArcTan[(Sqrt[b]*x)/Sqrt[a] ]/2, 2])/Sqrt[b]))/(2*a*(a + b*x^2)^(1/4))
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[((a_) + (b_.)*(x_)^2)^(-1/4), x_Symbol] :> Simp[2*(x/(a + b*x^2)^(1/4)) , x] - Simp[a Int[1/(a + b*x^2)^(5/4), x], x] /; FreeQ[{a, b}, x] && GtQ[ a, 0] && PosQ[b/a]
Int[((a_) + (b_.)*(x_)^2)^(-1/4), x_Symbol] :> Simp[(1 + b*(x^2/a))^(1/4)/( a + b*x^2)^(1/4) Int[1/(1 + b*(x^2/a))^(1/4), x], x] /; FreeQ[{a, b}, x] && PosQ[a]
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]
\[\int \frac {x^{2} d +c}{x^{2} \left (b \,x^{2}+a \right )^{\frac {1}{4}}}d x\]
Input:
int((d*x^2+c)/x^2/(b*x^2+a)^(1/4),x)
Output:
int((d*x^2+c)/x^2/(b*x^2+a)^(1/4),x)
\[ \int \frac {c+d x^2}{x^2 \sqrt [4]{a+b x^2}} \, dx=\int { \frac {d x^{2} + c}{{\left (b x^{2} + a\right )}^{\frac {1}{4}} x^{2}} \,d x } \] Input:
integrate((d*x^2+c)/x^2/(b*x^2+a)^(1/4),x, algorithm="fricas")
Output:
integral((b*x^2 + a)^(3/4)*(d*x^2 + c)/(b*x^4 + a*x^2), x)
Result contains complex when optimal does not.
Time = 0.87 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.50 \[ \int \frac {c+d x^2}{x^2 \sqrt [4]{a+b x^2}} \, dx=- \frac {c {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {1}{4} \\ \frac {1}{2} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{\sqrt [4]{a} x} + \frac {d x {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{4}, \frac {1}{2} \\ \frac {3}{2} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{\sqrt [4]{a}} \] Input:
integrate((d*x**2+c)/x**2/(b*x**2+a)**(1/4),x)
Output:
-c*hyper((-1/2, 1/4), (1/2,), b*x**2*exp_polar(I*pi)/a)/(a**(1/4)*x) + d*x *hyper((1/4, 1/2), (3/2,), b*x**2*exp_polar(I*pi)/a)/a**(1/4)
\[ \int \frac {c+d x^2}{x^2 \sqrt [4]{a+b x^2}} \, dx=\int { \frac {d x^{2} + c}{{\left (b x^{2} + a\right )}^{\frac {1}{4}} x^{2}} \,d x } \] Input:
integrate((d*x^2+c)/x^2/(b*x^2+a)^(1/4),x, algorithm="maxima")
Output:
integrate((d*x^2 + c)/((b*x^2 + a)^(1/4)*x^2), x)
\[ \int \frac {c+d x^2}{x^2 \sqrt [4]{a+b x^2}} \, dx=\int { \frac {d x^{2} + c}{{\left (b x^{2} + a\right )}^{\frac {1}{4}} x^{2}} \,d x } \] Input:
integrate((d*x^2+c)/x^2/(b*x^2+a)^(1/4),x, algorithm="giac")
Output:
integrate((d*x^2 + c)/((b*x^2 + a)^(1/4)*x^2), x)
Time = 0.87 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.73 \[ \int \frac {c+d x^2}{x^2 \sqrt [4]{a+b x^2}} \, dx=\frac {d\,x\,{\left (\frac {b\,x^2}{a}+1\right )}^{1/4}\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{4},\frac {1}{2};\ \frac {3}{2};\ -\frac {b\,x^2}{a}\right )}{{\left (b\,x^2+a\right )}^{1/4}}-\frac {2\,c\,{\left (\frac {a}{b\,x^2}+1\right )}^{1/4}\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{4},\frac {3}{4};\ \frac {7}{4};\ -\frac {a}{b\,x^2}\right )}{3\,x\,{\left (b\,x^2+a\right )}^{1/4}} \] Input:
int((c + d*x^2)/(x^2*(a + b*x^2)^(1/4)),x)
Output:
(d*x*((b*x^2)/a + 1)^(1/4)*hypergeom([1/4, 1/2], 3/2, -(b*x^2)/a))/(a + b* x^2)^(1/4) - (2*c*(a/(b*x^2) + 1)^(1/4)*hypergeom([1/4, 3/4], 7/4, -a/(b*x ^2)))/(3*x*(a + b*x^2)^(1/4))
\[ \int \frac {c+d x^2}{x^2 \sqrt [4]{a+b x^2}} \, dx=\left (\int \frac {1}{\left (b \,x^{2}+a \right )^{\frac {1}{4}}}d x \right ) d +\left (\int \frac {1}{\left (b \,x^{2}+a \right )^{\frac {1}{4}} x^{2}}d x \right ) c \] Input:
int((d*x^2+c)/x^2/(b*x^2+a)^(1/4),x)
Output:
int(1/(a + b*x**2)**(1/4),x)*d + int(1/((a + b*x**2)**(1/4)*x**2),x)*c