Optimal. Leaf size=91 \[ \frac {b \tan ^{-1}\left (\frac {\sqrt [4]{a x^8+b x^5}}{\sqrt [4]{a} x^2}\right )}{6 a^{3/4}}+\frac {b \tanh ^{-1}\left (\frac {\sqrt [4]{a x^8+b x^5}}{\sqrt [4]{a} x^2}\right )}{6 a^{3/4}}+\frac {1}{3} x \sqrt [4]{a x^8+b x^5} \]
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Rubi [A] time = 0.14, antiderivative size = 149, normalized size of antiderivative = 1.64, number of steps used = 8, number of rules used = 8, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.533, Rules used = {2004, 2032, 329, 275, 331, 298, 203, 206} \begin {gather*} -\frac {b x^{15/4} \left (a x^3+b\right )^{3/4} \tan ^{-1}\left (\frac {\sqrt [4]{a} x^{3/4}}{\sqrt [4]{a x^3+b}}\right )}{6 a^{3/4} \left (a x^8+b x^5\right )^{3/4}}+\frac {b x^{15/4} \left (a x^3+b\right )^{3/4} \tanh ^{-1}\left (\frac {\sqrt [4]{a} x^{3/4}}{\sqrt [4]{a x^3+b}}\right )}{6 a^{3/4} \left (a x^8+b x^5\right )^{3/4}}+\frac {1}{3} x \sqrt [4]{a x^8+b x^5} \end {gather*}
Antiderivative was successfully verified.
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Rule 203
Rule 206
Rule 275
Rule 298
Rule 329
Rule 331
Rule 2004
Rule 2032
Rubi steps
\begin {align*} \int \sqrt [4]{b x^5+a x^8} \, dx &=\frac {1}{3} x \sqrt [4]{b x^5+a x^8}+\frac {1}{4} b \int \frac {x^5}{\left (b x^5+a x^8\right )^{3/4}} \, dx\\ &=\frac {1}{3} x \sqrt [4]{b x^5+a x^8}+\frac {\left (b x^{15/4} \left (b+a x^3\right )^{3/4}\right ) \int \frac {x^{5/4}}{\left (b+a x^3\right )^{3/4}} \, dx}{4 \left (b x^5+a x^8\right )^{3/4}}\\ &=\frac {1}{3} x \sqrt [4]{b x^5+a x^8}+\frac {\left (b x^{15/4} \left (b+a x^3\right )^{3/4}\right ) \operatorname {Subst}\left (\int \frac {x^8}{\left (b+a x^{12}\right )^{3/4}} \, dx,x,\sqrt [4]{x}\right )}{\left (b x^5+a x^8\right )^{3/4}}\\ &=\frac {1}{3} x \sqrt [4]{b x^5+a x^8}+\frac {\left (b x^{15/4} \left (b+a x^3\right )^{3/4}\right ) \operatorname {Subst}\left (\int \frac {x^2}{\left (b+a x^4\right )^{3/4}} \, dx,x,x^{3/4}\right )}{3 \left (b x^5+a x^8\right )^{3/4}}\\ &=\frac {1}{3} x \sqrt [4]{b x^5+a x^8}+\frac {\left (b x^{15/4} \left (b+a x^3\right )^{3/4}\right ) \operatorname {Subst}\left (\int \frac {x^2}{1-a x^4} \, dx,x,\frac {x^{3/4}}{\sqrt [4]{b+a x^3}}\right )}{3 \left (b x^5+a x^8\right )^{3/4}}\\ &=\frac {1}{3} x \sqrt [4]{b x^5+a x^8}+\frac {\left (b x^{15/4} \left (b+a x^3\right )^{3/4}\right ) \operatorname {Subst}\left (\int \frac {1}{1-\sqrt {a} x^2} \, dx,x,\frac {x^{3/4}}{\sqrt [4]{b+a x^3}}\right )}{6 \sqrt {a} \left (b x^5+a x^8\right )^{3/4}}-\frac {\left (b x^{15/4} \left (b+a x^3\right )^{3/4}\right ) \operatorname {Subst}\left (\int \frac {1}{1+\sqrt {a} x^2} \, dx,x,\frac {x^{3/4}}{\sqrt [4]{b+a x^3}}\right )}{6 \sqrt {a} \left (b x^5+a x^8\right )^{3/4}}\\ &=\frac {1}{3} x \sqrt [4]{b x^5+a x^8}-\frac {b x^{15/4} \left (b+a x^3\right )^{3/4} \tan ^{-1}\left (\frac {\sqrt [4]{a} x^{3/4}}{\sqrt [4]{b+a x^3}}\right )}{6 a^{3/4} \left (b x^5+a x^8\right )^{3/4}}+\frac {b x^{15/4} \left (b+a x^3\right )^{3/4} \tanh ^{-1}\left (\frac {\sqrt [4]{a} x^{3/4}}{\sqrt [4]{b+a x^3}}\right )}{6 a^{3/4} \left (b x^5+a x^8\right )^{3/4}}\\ \end {align*}
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Mathematica [C] time = 0.02, size = 53, normalized size = 0.58 \begin {gather*} \frac {4 x \sqrt [4]{x^5 \left (a x^3+b\right )} \, _2F_1\left (-\frac {1}{4},\frac {3}{4};\frac {7}{4};-\frac {a x^3}{b}\right )}{9 \sqrt [4]{\frac {a x^3}{b}+1}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.46, size = 91, normalized size = 1.00 \begin {gather*} \frac {1}{3} x \sqrt [4]{b x^5+a x^8}+\frac {b \tan ^{-1}\left (\frac {\sqrt [4]{b x^5+a x^8}}{\sqrt [4]{a} x^2}\right )}{6 a^{3/4}}+\frac {b \tanh ^{-1}\left (\frac {\sqrt [4]{b x^5+a x^8}}{\sqrt [4]{a} x^2}\right )}{6 a^{3/4}} \end {gather*}
Antiderivative was successfully verified.
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fricas [F(-1)] 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.
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giac [B] time = 0.29, size = 213, normalized size = 2.34 \begin {gather*} \frac {8 \, {\left (a + \frac {b}{x^{3}}\right )}^{\frac {1}{4}} b x^{3} + \frac {2 \, \sqrt {2} \left (-a\right )^{\frac {1}{4}} b^{2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (-a\right )^{\frac {1}{4}} + 2 \, {\left (a + \frac {b}{x^{3}}\right )}^{\frac {1}{4}}\right )}}{2 \, \left (-a\right )^{\frac {1}{4}}}\right )}{a} + \frac {2 \, \sqrt {2} \left (-a\right )^{\frac {1}{4}} b^{2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (-a\right )^{\frac {1}{4}} - 2 \, {\left (a + \frac {b}{x^{3}}\right )}^{\frac {1}{4}}\right )}}{2 \, \left (-a\right )^{\frac {1}{4}}}\right )}{a} + \frac {\sqrt {2} \left (-a\right )^{\frac {1}{4}} b^{2} \log \left (\sqrt {2} \left (-a\right )^{\frac {1}{4}} {\left (a + \frac {b}{x^{3}}\right )}^{\frac {1}{4}} + \sqrt {-a} + \sqrt {a + \frac {b}{x^{3}}}\right )}{a} + \frac {\sqrt {2} b^{2} \log \left (-\sqrt {2} \left (-a\right )^{\frac {1}{4}} {\left (a + \frac {b}{x^{3}}\right )}^{\frac {1}{4}} + \sqrt {-a} + \sqrt {a + \frac {b}{x^{3}}}\right )}{\left (-a\right )^{\frac {3}{4}}}}{24 \, b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.02, size = 0, normalized size = 0.00 \[\int \left (a \,x^{8}+b \,x^{5}\right )^{\frac {1}{4}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int {\left (a x^{8} + b x^{5}\right )}^{\frac {1}{4}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.93, size = 42, normalized size = 0.46 \begin {gather*} \frac {4\,x\,{\left (a\,x^8+b\,x^5\right )}^{1/4}\,{{}}_2{\mathrm {F}}_1\left (-\frac {1}{4},\frac {3}{4};\ \frac {7}{4};\ -\frac {a\,x^3}{b}\right )}{9\,{\left (\frac {a\,x^3}{b}+1\right )}^{1/4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \sqrt [4]{a x^{8} + b x^{5}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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