Optimal. Leaf size=147 \[ -\frac {5 a^{7/2} x^3 \left (\frac {a}{b x^4}+1\right )^{3/4} F\left (\left .\frac {1}{2} \cot ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )\right |2\right )}{336 b^{3/2} \left (a+b x^4\right )^{3/4}}-\frac {5 a^3 x \sqrt [4]{a+b x^4}}{336 b^2}+\frac {a^2 x^5 \sqrt [4]{a+b x^4}}{168 b}+\frac {1}{14} x^9 \left (a+b x^4\right )^{5/4}+\frac {1}{28} a x^9 \sqrt [4]{a+b x^4} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.07, antiderivative size = 147, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {279, 321, 237, 335, 275, 231} \[ -\frac {5 a^3 x \sqrt [4]{a+b x^4}}{336 b^2}-\frac {5 a^{7/2} x^3 \left (\frac {a}{b x^4}+1\right )^{3/4} F\left (\left .\frac {1}{2} \cot ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )\right |2\right )}{336 b^{3/2} \left (a+b x^4\right )^{3/4}}+\frac {a^2 x^5 \sqrt [4]{a+b x^4}}{168 b}+\frac {1}{14} x^9 \left (a+b x^4\right )^{5/4}+\frac {1}{28} a x^9 \sqrt [4]{a+b x^4} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 231
Rule 237
Rule 275
Rule 279
Rule 321
Rule 335
Rubi steps
\begin {align*} \int x^8 \left (a+b x^4\right )^{5/4} \, dx &=\frac {1}{14} x^9 \left (a+b x^4\right )^{5/4}+\frac {1}{14} (5 a) \int x^8 \sqrt [4]{a+b x^4} \, dx\\ &=\frac {1}{28} a x^9 \sqrt [4]{a+b x^4}+\frac {1}{14} x^9 \left (a+b x^4\right )^{5/4}+\frac {1}{28} a^2 \int \frac {x^8}{\left (a+b x^4\right )^{3/4}} \, dx\\ &=\frac {a^2 x^5 \sqrt [4]{a+b x^4}}{168 b}+\frac {1}{28} a x^9 \sqrt [4]{a+b x^4}+\frac {1}{14} x^9 \left (a+b x^4\right )^{5/4}-\frac {\left (5 a^3\right ) \int \frac {x^4}{\left (a+b x^4\right )^{3/4}} \, dx}{168 b}\\ &=-\frac {5 a^3 x \sqrt [4]{a+b x^4}}{336 b^2}+\frac {a^2 x^5 \sqrt [4]{a+b x^4}}{168 b}+\frac {1}{28} a x^9 \sqrt [4]{a+b x^4}+\frac {1}{14} x^9 \left (a+b x^4\right )^{5/4}+\frac {\left (5 a^4\right ) \int \frac {1}{\left (a+b x^4\right )^{3/4}} \, dx}{336 b^2}\\ &=-\frac {5 a^3 x \sqrt [4]{a+b x^4}}{336 b^2}+\frac {a^2 x^5 \sqrt [4]{a+b x^4}}{168 b}+\frac {1}{28} a x^9 \sqrt [4]{a+b x^4}+\frac {1}{14} x^9 \left (a+b x^4\right )^{5/4}+\frac {\left (5 a^4 \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}{336 b^2 \left (a+b x^4\right )^{3/4}}\\ &=-\frac {5 a^3 x \sqrt [4]{a+b x^4}}{336 b^2}+\frac {a^2 x^5 \sqrt [4]{a+b x^4}}{168 b}+\frac {1}{28} a x^9 \sqrt [4]{a+b x^4}+\frac {1}{14} x^9 \left (a+b x^4\right )^{5/4}-\frac {\left (5 a^4 \left (1+\frac {a}{b x^4}\right )^{3/4} x^3\right ) \operatorname {Subst}\left (\int \frac {x}{\left (1+\frac {a x^4}{b}\right )^{3/4}} \, dx,x,\frac {1}{x}\right )}{336 b^2 \left (a+b x^4\right )^{3/4}}\\ &=-\frac {5 a^3 x \sqrt [4]{a+b x^4}}{336 b^2}+\frac {a^2 x^5 \sqrt [4]{a+b x^4}}{168 b}+\frac {1}{28} a x^9 \sqrt [4]{a+b x^4}+\frac {1}{14} x^9 \left (a+b x^4\right )^{5/4}-\frac {\left (5 a^4 \left (1+\frac {a}{b x^4}\right )^{3/4} x^3\right ) \operatorname {Subst}\left (\int \frac {1}{\left (1+\frac {a x^2}{b}\right )^{3/4}} \, dx,x,\frac {1}{x^2}\right )}{672 b^2 \left (a+b x^4\right )^{3/4}}\\ &=-\frac {5 a^3 x \sqrt [4]{a+b x^4}}{336 b^2}+\frac {a^2 x^5 \sqrt [4]{a+b x^4}}{168 b}+\frac {1}{28} a x^9 \sqrt [4]{a+b x^4}+\frac {1}{14} x^9 \left (a+b x^4\right )^{5/4}-\frac {5 a^{7/2} \left (1+\frac {a}{b x^4}\right )^{3/4} x^3 F\left (\left .\frac {1}{2} \cot ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )\right |2\right )}{336 b^{3/2} \left (a+b x^4\right )^{3/4}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] time = 0.12, size = 76, normalized size = 0.52 \[ \frac {x \sqrt [4]{a+b x^4} \left (\frac {a^3 \, _2F_1\left (-\frac {5}{4},\frac {1}{4};\frac {5}{4};-\frac {b x^4}{a}\right )}{\sqrt [4]{\frac {b x^4}{a}+1}}-\left (a-2 b x^4\right ) \left (a+b x^4\right )^2\right )}{28 b^2} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.84, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (b x^{12} + a x^{8}\right )} {\left (b x^{4} + a\right )}^{\frac {1}{4}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (b x^{4} + a\right )}^{\frac {5}{4}} x^{8}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.14, size = 0, normalized size = 0.00 \[ \int \left (b \,x^{4}+a \right )^{\frac {5}{4}} x^{8}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (b x^{4} + a\right )}^{\frac {5}{4}} x^{8}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int x^8\,{\left (b\,x^4+a\right )}^{5/4} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [C] time = 3.37, size = 39, normalized size = 0.27 \[ \frac {a^{\frac {5}{4}} x^{9} \Gamma \left (\frac {9}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {5}{4}, \frac {9}{4} \\ \frac {13}{4} \end {matrix}\middle | {\frac {b x^{4} e^{i \pi }}{a}} \right )}}{4 \Gamma \left (\frac {13}{4}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________