Optimal. Leaf size=64 \[ -\frac {\sqrt {1+\frac {d x^8}{c}} F_1\left (-\frac {3}{8};1,\frac {1}{2};\frac {5}{8};-\frac {b x^8}{a},-\frac {d x^8}{c}\right )}{3 a x^3 \sqrt {c+d x^8}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.03, antiderivative size = 64, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {525, 524}
\begin {gather*} -\frac {\sqrt {\frac {d x^8}{c}+1} F_1\left (-\frac {3}{8};1,\frac {1}{2};\frac {5}{8};-\frac {b x^8}{a},-\frac {d x^8}{c}\right )}{3 a x^3 \sqrt {c+d x^8}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 524
Rule 525
Rubi steps
\begin {align*} \int \frac {1}{x^4 \left (a+b x^8\right ) \sqrt {c+d x^8}} \, dx &=\frac {\sqrt {1+\frac {d x^8}{c}} \int \frac {1}{x^4 \left (a+b x^8\right ) \sqrt {1+\frac {d x^8}{c}}} \, dx}{\sqrt {c+d x^8}}\\ &=-\frac {\sqrt {1+\frac {d x^8}{c}} F_1\left (-\frac {3}{8};1,\frac {1}{2};\frac {5}{8};-\frac {b x^8}{a},-\frac {d x^8}{c}\right )}{3 a x^3 \sqrt {c+d x^8}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(141\) vs. \(2(64)=128\).
time = 10.12, size = 141, normalized size = 2.20 \begin {gather*} \frac {-65 a \left (c+d x^8\right )+13 (-3 b c+a d) x^8 \sqrt {1+\frac {d x^8}{c}} F_1\left (\frac {5}{8};\frac {1}{2},1;\frac {13}{8};-\frac {d x^8}{c},-\frac {b x^8}{a}\right )+5 b d x^{16} \sqrt {1+\frac {d x^8}{c}} F_1\left (\frac {13}{8};\frac {1}{2},1;\frac {21}{8};-\frac {d x^8}{c},-\frac {b x^8}{a}\right )}{195 a^2 c x^3 \sqrt {c+d x^8}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int \frac {1}{x^{4} \left (b \,x^{8}+a \right ) \sqrt {d \,x^{8}+c}}\, dx\]
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(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \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 {1}{x^{4} \left (a + b x^{8}\right ) \sqrt {c + d x^{8}}}\, 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 = -0.02 \begin {gather*} \int \frac {1}{x^4\,\left (b\,x^8+a\right )\,\sqrt {d\,x^8+c}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________