Optimal. Leaf size=109 \[ -\frac {5 a x \sqrt [4]{a-b x^4}}{12 b^2}-\frac {x^5 \sqrt [4]{a-b x^4}}{6 b}-\frac {5 a^{3/2} \left (1-\frac {a}{b x^4}\right )^{3/4} x^3 F\left (\left .\frac {1}{2} \csc ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )\right |2\right )}{12 b^{3/2} \left (a-b x^4\right )^{3/4}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.04, antiderivative size = 109, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 5, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.312, Rules used = {327, 243, 342,
281, 238} \begin {gather*} -\frac {5 a^{3/2} x^3 \left (1-\frac {a}{b x^4}\right )^{3/4} F\left (\left .\frac {1}{2} \csc ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )\right |2\right )}{12 b^{3/2} \left (a-b x^4\right )^{3/4}}-\frac {5 a x \sqrt [4]{a-b x^4}}{12 b^2}-\frac {x^5 \sqrt [4]{a-b x^4}}{6 b} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 238
Rule 243
Rule 281
Rule 327
Rule 342
Rubi steps
\begin {align*} \int \frac {x^8}{\left (a-b x^4\right )^{3/4}} \, dx &=-\frac {x^5 \sqrt [4]{a-b x^4}}{6 b}+\frac {(5 a) \int \frac {x^4}{\left (a-b x^4\right )^{3/4}} \, dx}{6 b}\\ &=-\frac {5 a x \sqrt [4]{a-b x^4}}{12 b^2}-\frac {x^5 \sqrt [4]{a-b x^4}}{6 b}+\frac {\left (5 a^2\right ) \int \frac {1}{\left (a-b x^4\right )^{3/4}} \, dx}{12 b^2}\\ &=-\frac {5 a x \sqrt [4]{a-b x^4}}{12 b^2}-\frac {x^5 \sqrt [4]{a-b x^4}}{6 b}+\frac {\left (5 a^2 \left (1-\frac {a}{b x^4}\right )^{3/4} x^3\right ) \int \frac {1}{\left (1-\frac {a}{b x^4}\right )^{3/4} x^3} \, dx}{12 b^2 \left (a-b x^4\right )^{3/4}}\\ &=-\frac {5 a x \sqrt [4]{a-b x^4}}{12 b^2}-\frac {x^5 \sqrt [4]{a-b x^4}}{6 b}-\frac {\left (5 a^2 \left (1-\frac {a}{b x^4}\right )^{3/4} x^3\right ) \text {Subst}\left (\int \frac {x}{\left (1-\frac {a x^4}{b}\right )^{3/4}} \, dx,x,\frac {1}{x}\right )}{12 b^2 \left (a-b x^4\right )^{3/4}}\\ &=-\frac {5 a x \sqrt [4]{a-b x^4}}{12 b^2}-\frac {x^5 \sqrt [4]{a-b x^4}}{6 b}-\frac {\left (5 a^2 \left (1-\frac {a}{b x^4}\right )^{3/4} x^3\right ) \text {Subst}\left (\int \frac {1}{\left (1-\frac {a x^2}{b}\right )^{3/4}} \, dx,x,\frac {1}{x^2}\right )}{24 b^2 \left (a-b x^4\right )^{3/4}}\\ &=-\frac {5 a x \sqrt [4]{a-b x^4}}{12 b^2}-\frac {x^5 \sqrt [4]{a-b x^4}}{6 b}-\frac {5 a^{3/2} \left (1-\frac {a}{b x^4}\right )^{3/4} x^3 F\left (\left .\frac {1}{2} \csc ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )\right |2\right )}{12 b^{3/2} \left (a-b x^4\right )^{3/4}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 4 in
optimal.
time = 7.18, size = 80, normalized size = 0.73 \begin {gather*} \frac {-5 a^2 x+3 a b x^5+2 b^2 x^9+5 a^2 x \left (1-\frac {b x^4}{a}\right )^{3/4} \, _2F_1\left (\frac {1}{4},\frac {3}{4};\frac {5}{4};\frac {b x^4}{a}\right )}{12 b^2 \left (a-b x^4\right )^{3/4}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int \frac {x^{8}}{\left (-b \,x^{4}+a \right )^{\frac {3}{4}}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F]
time = 0.08, size = 28, normalized size = 0.26 \begin {gather*} {\rm integral}\left (-\frac {{\left (-b x^{4} + a\right )}^{\frac {1}{4}} x^{8}}{b x^{4} - a}, x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [C] Result contains complex when optimal does not.
time = 0.50, size = 39, normalized size = 0.36 \begin {gather*} \frac {x^{9} \Gamma \left (\frac {9}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {3}{4}, \frac {9}{4} \\ \frac {13}{4} \end {matrix}\middle | {\frac {b x^{4} e^{2 i \pi }}{a}} \right )}}{4 a^{\frac {3}{4}} \Gamma \left (\frac {13}{4}\right )} \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 {x^8}{{\left (a-b\,x^4\right )}^{3/4}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________