Optimal. Leaf size=128 \[ -\frac {\sqrt {a-b x^4}}{a x}+\frac {\sqrt [4]{b} \sqrt {1-\frac {b x^4}{a}} F\left (\left .\sin ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{\sqrt [4]{a} \sqrt {a-b x^4}}-\frac {\sqrt [4]{b} \sqrt {1-\frac {b x^4}{a}} E\left (\left .\sin ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{\sqrt [4]{a} \sqrt {a-b x^4}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.08, antiderivative size = 128, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.438, Rules used = {325, 307, 224, 221, 1200, 1199, 424} \[ -\frac {\sqrt {a-b x^4}}{a x}+\frac {\sqrt [4]{b} \sqrt {1-\frac {b x^4}{a}} F\left (\left .\sin ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{\sqrt [4]{a} \sqrt {a-b x^4}}-\frac {\sqrt [4]{b} \sqrt {1-\frac {b x^4}{a}} E\left (\left .\sin ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{\sqrt [4]{a} \sqrt {a-b x^4}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 221
Rule 224
Rule 307
Rule 325
Rule 424
Rule 1199
Rule 1200
Rubi steps
\begin {align*} \int \frac {1}{x^2 \sqrt {a-b x^4}} \, dx &=-\frac {\sqrt {a-b x^4}}{a x}-\frac {b \int \frac {x^2}{\sqrt {a-b x^4}} \, dx}{a}\\ &=-\frac {\sqrt {a-b x^4}}{a x}+\frac {\sqrt {b} \int \frac {1}{\sqrt {a-b x^4}} \, dx}{\sqrt {a}}-\frac {\sqrt {b} \int \frac {1+\frac {\sqrt {b} x^2}{\sqrt {a}}}{\sqrt {a-b x^4}} \, dx}{\sqrt {a}}\\ &=-\frac {\sqrt {a-b x^4}}{a x}+\frac {\left (\sqrt {b} \sqrt {1-\frac {b x^4}{a}}\right ) \int \frac {1}{\sqrt {1-\frac {b x^4}{a}}} \, dx}{\sqrt {a} \sqrt {a-b x^4}}-\frac {\left (\sqrt {b} \sqrt {1-\frac {b x^4}{a}}\right ) \int \frac {1+\frac {\sqrt {b} x^2}{\sqrt {a}}}{\sqrt {1-\frac {b x^4}{a}}} \, dx}{\sqrt {a} \sqrt {a-b x^4}}\\ &=-\frac {\sqrt {a-b x^4}}{a x}+\frac {\sqrt [4]{b} \sqrt {1-\frac {b x^4}{a}} F\left (\left .\sin ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{\sqrt [4]{a} \sqrt {a-b x^4}}-\frac {\left (\sqrt {b} \sqrt {1-\frac {b x^4}{a}}\right ) \int \frac {\sqrt {1+\frac {\sqrt {b} x^2}{\sqrt {a}}}}{\sqrt {1-\frac {\sqrt {b} x^2}{\sqrt {a}}}} \, dx}{\sqrt {a} \sqrt {a-b x^4}}\\ &=-\frac {\sqrt {a-b x^4}}{a x}-\frac {\sqrt [4]{b} \sqrt {1-\frac {b x^4}{a}} E\left (\left .\sin ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{\sqrt [4]{a} \sqrt {a-b x^4}}+\frac {\sqrt [4]{b} \sqrt {1-\frac {b x^4}{a}} F\left (\left .\sin ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{\sqrt [4]{a} \sqrt {a-b x^4}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] time = 0.01, size = 50, normalized size = 0.39 \[ -\frac {\sqrt {1-\frac {b x^4}{a}} \, _2F_1\left (-\frac {1}{4},\frac {1}{2};\frac {3}{4};\frac {b x^4}{a}\right )}{x \sqrt {a-b x^4}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.93, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {-b x^{4} + a}}{b x^{6} - a x^{2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {-b x^{4} + a} x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.01, size = 106, normalized size = 0.83 \[ \frac {\sqrt {-\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}+1}\, \sqrt {\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}+1}\, \left (-\EllipticE \left (\sqrt {\frac {\sqrt {b}}{\sqrt {a}}}\, x , i\right )+\EllipticF \left (\sqrt {\frac {\sqrt {b}}{\sqrt {a}}}\, x , i\right )\right ) \sqrt {b}}{\sqrt {\frac {\sqrt {b}}{\sqrt {a}}}\, \sqrt {-b \,x^{4}+a}\, \sqrt {a}}-\frac {\sqrt {-b \,x^{4}+a}}{a x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {-b x^{4} + a} x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 1.28, size = 41, normalized size = 0.32 \[ -\frac {\sqrt {1-\frac {a}{b\,x^4}}\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},\frac {3}{4};\ \frac {7}{4};\ \frac {a}{b\,x^4}\right )}{3\,x\,\sqrt {a-b\,x^4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 2.04, size = 41, normalized size = 0.32 \[ \frac {\Gamma \left (- \frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{4}, \frac {1}{2} \\ \frac {3}{4} \end {matrix}\middle | {\frac {b x^{4} e^{2 i \pi }}{a}} \right )}}{4 \sqrt {a} x \Gamma \left (\frac {3}{4}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________