Optimal. Leaf size=71 \[ \frac {2 (e x)^{3/2} \sqrt {c+d x^4} F_1\left (\frac {3}{8};1,-\frac {1}{2};\frac {11}{8};-\frac {b x^4}{a},-\frac {d x^4}{c}\right )}{3 a e \sqrt {\frac {d x^4}{c}+1}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.11, antiderivative size = 71, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.107, Rules used = {466, 511, 510} \[ \frac {2 (e x)^{3/2} \sqrt {c+d x^4} F_1\left (\frac {3}{8};1,-\frac {1}{2};\frac {11}{8};-\frac {b x^4}{a},-\frac {d x^4}{c}\right )}{3 a e \sqrt {\frac {d x^4}{c}+1}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 466
Rule 510
Rule 511
Rubi steps
\begin {align*} \int \frac {\sqrt {e x} \sqrt {c+d x^4}}{a+b x^4} \, dx &=\frac {2 \operatorname {Subst}\left (\int \frac {x^2 \sqrt {c+\frac {d x^8}{e^4}}}{a+\frac {b x^8}{e^4}} \, dx,x,\sqrt {e x}\right )}{e}\\ &=\frac {\left (2 \sqrt {c+d x^4}\right ) \operatorname {Subst}\left (\int \frac {x^2 \sqrt {1+\frac {d x^8}{c e^4}}}{a+\frac {b x^8}{e^4}} \, dx,x,\sqrt {e x}\right )}{e \sqrt {1+\frac {d x^4}{c}}}\\ &=\frac {2 (e x)^{3/2} \sqrt {c+d x^4} F_1\left (\frac {3}{8};1,-\frac {1}{2};\frac {11}{8};-\frac {b x^4}{a},-\frac {d x^4}{c}\right )}{3 a e \sqrt {1+\frac {d x^4}{c}}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.04, size = 70, normalized size = 0.99 \[ \frac {2 x \sqrt {e x} \sqrt {c+d x^4} F_1\left (\frac {3}{8};-\frac {1}{2},1;\frac {11}{8};-\frac {d x^4}{c},-\frac {b x^4}{a}\right )}{3 a \sqrt {\frac {c+d x^4}{c}}} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
fricas [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {d x^{4} + c} \sqrt {e x}}{b x^{4} + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.65, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {e x}\, \sqrt {d \,x^{4}+c}}{b \,x^{4}+a}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {d x^{4} + c} \sqrt {e x}}{b x^{4} + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\sqrt {e\,x}\,\sqrt {d\,x^4+c}}{b\,x^4+a} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {e x} \sqrt {c + d x^{4}}}{a + b x^{4}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________