Integrand size = 23, antiderivative size = 150 \[ \int \frac {\sqrt {c+d \sqrt {a+b x^2}}}{x^2} \, dx=-\frac {\sqrt {c+d \sqrt {a+b x^2}}}{x}-\frac {\sqrt {a} d \sqrt {-\frac {b x^2}{a}} \sqrt {\frac {c+d \sqrt {a+b x^2}}{c+\sqrt {a} d}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {a+b x^2}}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} d}{c+\sqrt {a} d}\right )}{x \sqrt {c+d \sqrt {a+b x^2}}} \] Output:
-(c+d*(b*x^2+a)^(1/2))^(1/2)/x-a^(1/2)*d*(-b*x^2/a)^(1/2)*((c+d*(b*x^2+a)^ (1/2))/(c+a^(1/2)*d))^(1/2)*EllipticF(1/2*(1-(b*x^2+a)^(1/2)/a^(1/2))^(1/2 )*2^(1/2),2^(1/2)*(a^(1/2)*d/(c+a^(1/2)*d))^(1/2))/x/(c+d*(b*x^2+a)^(1/2)) ^(1/2)
Result contains complex when optimal does not.
Time = 31.74 (sec) , antiderivative size = 207, normalized size of antiderivative = 1.38 \[ \int \frac {\sqrt {c+d \sqrt {a+b x^2}}}{x^2} \, dx=\frac {-\sqrt {c+d \sqrt {a+b x^2}}+\frac {i \sqrt {\frac {d \left (-\sqrt {a}+\sqrt {a+b x^2}\right )}{c+d \sqrt {a+b x^2}}} \sqrt {\frac {d \left (\sqrt {a}+\sqrt {a+b x^2}\right )}{c+d \sqrt {a+b x^2}}} \left (c+d \sqrt {a+b x^2}\right ) \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {-c-\sqrt {a} d}}{\sqrt {c+d \sqrt {a+b x^2}}}\right ),\frac {c-\sqrt {a} d}{c+\sqrt {a} d}\right )}{\sqrt {-c-\sqrt {a} d}}}{x} \] Input:
Integrate[Sqrt[c + d*Sqrt[a + b*x^2]]/x^2,x]
Output:
(-Sqrt[c + d*Sqrt[a + b*x^2]] + (I*Sqrt[(d*(-Sqrt[a] + Sqrt[a + b*x^2]))/( c + d*Sqrt[a + b*x^2])]*Sqrt[(d*(Sqrt[a] + Sqrt[a + b*x^2]))/(c + d*Sqrt[a + b*x^2])]*(c + d*Sqrt[a + b*x^2])*EllipticF[I*ArcSinh[Sqrt[-c - Sqrt[a]* d]/Sqrt[c + d*Sqrt[a + b*x^2]]], (c - Sqrt[a]*d)/(c + Sqrt[a]*d)])/Sqrt[-c - Sqrt[a]*d])/x
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.
\(\displaystyle \int \frac {\sqrt {d \sqrt {a+b x^2}+c}}{x^2} \, dx\) |
\(\Big \downarrow \) 7299 |
\(\displaystyle \int \frac {\sqrt {d \sqrt {a+b x^2}+c}}{x^2}dx\) |
Input:
Int[Sqrt[c + d*Sqrt[a + b*x^2]]/x^2,x]
Output:
$Aborted
\[\int \frac {\sqrt {c +d \sqrt {b \,x^{2}+a}}}{x^{2}}d x\]
Input:
int((c+d*(b*x^2+a)^(1/2))^(1/2)/x^2,x)
Output:
int((c+d*(b*x^2+a)^(1/2))^(1/2)/x^2,x)
\[ \int \frac {\sqrt {c+d \sqrt {a+b x^2}}}{x^2} \, dx=\int { \frac {\sqrt {\sqrt {b x^{2} + a} d + c}}{x^{2}} \,d x } \] Input:
integrate((c+d*(b*x^2+a)^(1/2))^(1/2)/x^2,x, algorithm="fricas")
Output:
integral(sqrt(sqrt(b*x^2 + a)*d + c)/x^2, x)
\[ \int \frac {\sqrt {c+d \sqrt {a+b x^2}}}{x^2} \, dx=\int \frac {\sqrt {c + d \sqrt {a + b x^{2}}}}{x^{2}}\, dx \] Input:
integrate((c+d*(b*x**2+a)**(1/2))**(1/2)/x**2,x)
Output:
Integral(sqrt(c + d*sqrt(a + b*x**2))/x**2, x)
\[ \int \frac {\sqrt {c+d \sqrt {a+b x^2}}}{x^2} \, dx=\int { \frac {\sqrt {\sqrt {b x^{2} + a} d + c}}{x^{2}} \,d x } \] Input:
integrate((c+d*(b*x^2+a)^(1/2))^(1/2)/x^2,x, algorithm="maxima")
Output:
integrate(sqrt(sqrt(b*x^2 + a)*d + c)/x^2, x)
\[ \int \frac {\sqrt {c+d \sqrt {a+b x^2}}}{x^2} \, dx=\int { \frac {\sqrt {\sqrt {b x^{2} + a} d + c}}{x^{2}} \,d x } \] Input:
integrate((c+d*(b*x^2+a)^(1/2))^(1/2)/x^2,x, algorithm="giac")
Output:
integrate(sqrt(sqrt(b*x^2 + a)*d + c)/x^2, x)
Timed out. \[ \int \frac {\sqrt {c+d \sqrt {a+b x^2}}}{x^2} \, dx=\int \frac {\sqrt {c+d\,\sqrt {b\,x^2+a}}}{x^2} \,d x \] Input:
int((c + d*(a + b*x^2)^(1/2))^(1/2)/x^2,x)
Output:
int((c + d*(a + b*x^2)^(1/2))^(1/2)/x^2, x)
\[ \int \frac {\sqrt {c+d \sqrt {a+b x^2}}}{x^2} \, dx=\int \frac {\sqrt {\sqrt {b \,x^{2}+a}\, d +c}}{x^{2}}d x \] Input:
int((c+d*(b*x^2+a)^(1/2))^(1/2)/x^2,x)
Output:
int(sqrt(sqrt(a + b*x**2)*d + c)/x**2,x)