Optimal. Leaf size=347 \[ -\frac {a^{3/4} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \left (\frac {\sqrt {c} d}{\sqrt {a}}+e\right )^2 \Pi \left (-\frac {\left (\sqrt {c} d-\sqrt {a} e\right )^2}{4 \sqrt {a} \sqrt {c} d e};2 \tan ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{4 \sqrt [4]{c} d \sqrt {-a-c x^4} \left (c d^2-a e^2\right )}+\frac {\sqrt {e} \tan ^{-1}\left (\frac {x \sqrt {-a e^2-c d^2}}{\sqrt {d} \sqrt {e} \sqrt {-a-c x^4}}\right )}{2 \sqrt {d} \sqrt {-a e^2-c d^2}}+\frac {\sqrt [4]{c} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{2 \sqrt [4]{a} \sqrt {-a-c x^4} \left (\sqrt {c} d-\sqrt {a} e\right )} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.30, antiderivative size = 347, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {1217, 220, 1707} \[ -\frac {a^{3/4} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \left (\frac {\sqrt {c} d}{\sqrt {a}}+e\right )^2 \Pi \left (-\frac {\left (\sqrt {c} d-\sqrt {a} e\right )^2}{4 \sqrt {a} \sqrt {c} d e};2 \tan ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{4 \sqrt [4]{c} d \sqrt {-a-c x^4} \left (c d^2-a e^2\right )}+\frac {\sqrt {e} \tan ^{-1}\left (\frac {x \sqrt {-a e^2-c d^2}}{\sqrt {d} \sqrt {e} \sqrt {-a-c x^4}}\right )}{2 \sqrt {d} \sqrt {-a e^2-c d^2}}+\frac {\sqrt [4]{c} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{2 \sqrt [4]{a} \sqrt {-a-c x^4} \left (\sqrt {c} d-\sqrt {a} e\right )} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 220
Rule 1217
Rule 1707
Rubi steps
\begin {align*} \int \frac {1}{\left (d+e x^2\right ) \sqrt {-a-c x^4}} \, dx &=\frac {\sqrt {c} \int \frac {1}{\sqrt {-a-c x^4}} \, dx}{\sqrt {c} d-\sqrt {a} e}-\frac {\left (\sqrt {a} e\right ) \int \frac {1+\frac {\sqrt {c} x^2}{\sqrt {a}}}{\left (d+e x^2\right ) \sqrt {-a-c x^4}} \, dx}{\sqrt {c} d-\sqrt {a} e}\\ &=\frac {\sqrt {e} \tan ^{-1}\left (\frac {\sqrt {-c d^2-a e^2} x}{\sqrt {d} \sqrt {e} \sqrt {-a-c x^4}}\right )}{2 \sqrt {d} \sqrt {-c d^2-a e^2}}+\frac {\sqrt [4]{c} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{2 \sqrt [4]{a} \left (\sqrt {c} d-\sqrt {a} e\right ) \sqrt {-a-c x^4}}-\frac {\sqrt [4]{a} \left (\frac {\sqrt {c} d}{\sqrt {a}}+e\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \Pi \left (-\frac {\left (\sqrt {c} d-\sqrt {a} e\right )^2}{4 \sqrt {a} \sqrt {c} d e};2 \tan ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{4 \sqrt [4]{c} d \left (\sqrt {c} d-\sqrt {a} e\right ) \sqrt {-a-c x^4}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] time = 0.15, size = 98, normalized size = 0.28 \[ -\frac {i \sqrt {\frac {c x^4}{a}+1} \Pi \left (-\frac {i \sqrt {a} e}{\sqrt {c} d};\left .i \sinh ^{-1}\left (\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}} x\right )\right |-1\right )}{d \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}} \sqrt {-a-c x^4}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 10.93, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {-c x^{4} - a}}{c e x^{6} + c d x^{4} + a e x^{2} + a d}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {-c x^{4} - a} {\left (e x^{2} + d\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [C] time = 0.02, size = 110, normalized size = 0.32 \[ \frac {\sqrt {\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}+1}\, \sqrt {-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}+1}\, \EllipticPi \left (\sqrt {-\frac {i \sqrt {c}}{\sqrt {a}}}\, x , -\frac {i \sqrt {a}\, e}{\sqrt {c}\, d}, \frac {\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}}{\sqrt {-\frac {i \sqrt {c}}{\sqrt {a}}}}\right )}{\sqrt {-\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}-a}\, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {-c x^{4} - a} {\left (e x^{2} + d\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {1}{\sqrt {-c\,x^4-a}\,\left (e\,x^2+d\right )} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {- a - c x^{4}} \left (d + e x^{2}\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________