Optimal. Leaf size=71 \[ \frac {2 (e x)^{5/2} \sqrt {c+d x^4} F_1\left (\frac {5}{8};1,-\frac {1}{2};\frac {13}{8};-\frac {b x^4}{a},-\frac {d x^4}{c}\right )}{5 a e \sqrt {1+\frac {d x^4}{c}}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.08, 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 = {477, 525, 524}
\begin {gather*} \frac {2 (e x)^{5/2} \sqrt {c+d x^4} F_1\left (\frac {5}{8};1,-\frac {1}{2};\frac {13}{8};-\frac {b x^4}{a},-\frac {d x^4}{c}\right )}{5 a e \sqrt {\frac {d x^4}{c}+1}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 477
Rule 524
Rule 525
Rubi steps
\begin {align*} \int \frac {(e x)^{3/2} \sqrt {c+d x^4}}{a+b x^4} \, dx &=\frac {2 \text {Subst}\left (\int \frac {x^4 \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 ) \text {Subst}\left (\int \frac {x^4 \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)^{5/2} \sqrt {c+d x^4} F_1\left (\frac {5}{8};1,-\frac {1}{2};\frac {13}{8};-\frac {b x^4}{a},-\frac {d x^4}{c}\right )}{5 a e \sqrt {1+\frac {d x^4}{c}}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 10.05, size = 70, normalized size = 0.99 \begin {gather*} \frac {2 x (e x)^{3/2} \sqrt {c+d x^4} F_1\left (\frac {5}{8};-\frac {1}{2},1;\frac {13}{8};-\frac {d x^4}{c},-\frac {b x^4}{a}\right )}{5 a \sqrt {\frac {c+d x^4}{c}}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F]
time = 0.07, size = 0, normalized size = 0.00 \[\int \frac {\left (e x \right )^{\frac {3}{2}} \sqrt {d \,x^{4}+c}}{b \,x^{4}+a}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (e x\right )^{\frac {3}{2}} \sqrt {c + d x^{4}}}{a + b x^{4}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (e\,x\right )}^{3/2}\,\sqrt {d\,x^4+c}}{b\,x^4+a} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________