Optimal. Leaf size=64 \[ -\frac {\sqrt {\frac {d x^8}{c}+1} F_1\left (-\frac {3}{8};2,\frac {1}{2};\frac {5}{8};-\frac {b x^8}{a},-\frac {d x^8}{c}\right )}{3 a^2 x^3 \sqrt {c+d x^8}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.06, 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 = {511, 510} \[ -\frac {\sqrt {\frac {d x^8}{c}+1} F_1\left (-\frac {3}{8};2,\frac {1}{2};\frac {5}{8};-\frac {b x^8}{a},-\frac {d x^8}{c}\right )}{3 a^2 x^3 \sqrt {c+d x^8}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 510
Rule 511
Rubi steps
\begin {align*} \int \frac {1}{x^4 \left (a+b x^8\right )^2 \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 )^2 \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};2,\frac {1}{2};\frac {5}{8};-\frac {b x^8}{a},-\frac {d x^8}{c}\right )}{3 a^2 x^3 \sqrt {c+d x^8}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [B] time = 0.32, size = 226, normalized size = 3.53 \[ \frac {-13 x^8 \left (a+b x^8\right ) \sqrt {\frac {d x^8}{c}+1} \left (8 a^2 d^2-56 a b c d+33 b^2 c^2\right ) F_1\left (\frac {5}{8};\frac {1}{2},1;\frac {13}{8};-\frac {d x^8}{c},-\frac {b x^8}{a}\right )+65 a \left (c+d x^8\right ) \left (8 a^2 d-8 a b \left (c-d x^8\right )-11 b^2 c x^8\right )+5 b d x^{16} \left (a+b x^8\right ) \sqrt {\frac {d x^8}{c}+1} (11 b c-8 a d) F_1\left (\frac {13}{8};\frac {1}{2},1;\frac {21}{8};-\frac {d x^8}{c},-\frac {b x^8}{a}\right )}{1560 a^3 c x^3 \left (a+b x^8\right ) \sqrt {c+d x^8} (b c-a d)} \]
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 \[ \int \frac {1}{{\left (b x^{8} + a\right )}^{2} \sqrt {d x^{8} + c} x^{4}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.46, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (b \,x^{8}+a \right )^{2} \sqrt {d \,x^{8}+c}\, x^{4}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (b x^{8} + a\right )}^{2} \sqrt {d x^{8} + c} x^{4}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {1}{x^4\,{\left (b\,x^8+a\right )}^2\,\sqrt {d\,x^8+c}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [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]
________________________________________________________________________________________