Optimal. Leaf size=109 \[ -\frac {5 b^3 \tan ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4-b}}\right )}{256 a^{9/4}}-\frac {5 b^3 \tanh ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4-b}}\right )}{256 a^{9/4}}+\frac {\left (a x^4-b\right )^{3/4} \left (32 a^2 x^9-12 a b x^5-15 b^2 x\right )}{384 a^2} \]
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Rubi [A] time = 0.06, antiderivative size = 135, normalized size of antiderivative = 1.24, number of steps used = 7, number of rules used = 6, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.353, Rules used = {279, 321, 240, 212, 206, 203} \begin {gather*} -\frac {5 b^3 \tan ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4-b}}\right )}{256 a^{9/4}}-\frac {5 b^3 \tanh ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4-b}}\right )}{256 a^{9/4}}-\frac {5 b^2 x \left (a x^4-b\right )^{3/4}}{128 a^2}+\frac {1}{12} x^9 \left (a x^4-b\right )^{3/4}-\frac {b x^5 \left (a x^4-b\right )^{3/4}}{32 a} \end {gather*}
Antiderivative was successfully verified.
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Rule 203
Rule 206
Rule 212
Rule 240
Rule 279
Rule 321
Rubi steps
\begin {align*} \int x^8 \left (-b+a x^4\right )^{3/4} \, dx &=\frac {1}{12} x^9 \left (-b+a x^4\right )^{3/4}-\frac {1}{4} b \int \frac {x^8}{\sqrt [4]{-b+a x^4}} \, dx\\ &=-\frac {b x^5 \left (-b+a x^4\right )^{3/4}}{32 a}+\frac {1}{12} x^9 \left (-b+a x^4\right )^{3/4}-\frac {\left (5 b^2\right ) \int \frac {x^4}{\sqrt [4]{-b+a x^4}} \, dx}{32 a}\\ &=-\frac {5 b^2 x \left (-b+a x^4\right )^{3/4}}{128 a^2}-\frac {b x^5 \left (-b+a x^4\right )^{3/4}}{32 a}+\frac {1}{12} x^9 \left (-b+a x^4\right )^{3/4}-\frac {\left (5 b^3\right ) \int \frac {1}{\sqrt [4]{-b+a x^4}} \, dx}{128 a^2}\\ &=-\frac {5 b^2 x \left (-b+a x^4\right )^{3/4}}{128 a^2}-\frac {b x^5 \left (-b+a x^4\right )^{3/4}}{32 a}+\frac {1}{12} x^9 \left (-b+a x^4\right )^{3/4}-\frac {\left (5 b^3\right ) \operatorname {Subst}\left (\int \frac {1}{1-a x^4} \, dx,x,\frac {x}{\sqrt [4]{-b+a x^4}}\right )}{128 a^2}\\ &=-\frac {5 b^2 x \left (-b+a x^4\right )^{3/4}}{128 a^2}-\frac {b x^5 \left (-b+a x^4\right )^{3/4}}{32 a}+\frac {1}{12} x^9 \left (-b+a x^4\right )^{3/4}-\frac {\left (5 b^3\right ) \operatorname {Subst}\left (\int \frac {1}{1-\sqrt {a} x^2} \, dx,x,\frac {x}{\sqrt [4]{-b+a x^4}}\right )}{256 a^2}-\frac {\left (5 b^3\right ) \operatorname {Subst}\left (\int \frac {1}{1+\sqrt {a} x^2} \, dx,x,\frac {x}{\sqrt [4]{-b+a x^4}}\right )}{256 a^2}\\ &=-\frac {5 b^2 x \left (-b+a x^4\right )^{3/4}}{128 a^2}-\frac {b x^5 \left (-b+a x^4\right )^{3/4}}{32 a}+\frac {1}{12} x^9 \left (-b+a x^4\right )^{3/4}-\frac {5 b^3 \tan ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{-b+a x^4}}\right )}{256 a^{9/4}}-\frac {5 b^3 \tanh ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{-b+a x^4}}\right )}{256 a^{9/4}}\\ \end {align*}
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Mathematica [C] time = 0.06, size = 97, normalized size = 0.89 \begin {gather*} \frac {x \left (a x^4-b\right )^{3/4} \left (\left (1-\frac {a x^4}{b}\right )^{3/4} \left (8 a^2 x^8-3 a b x^4-5 b^2\right )+5 b^2 \, _2F_1\left (-\frac {3}{4},\frac {1}{4};\frac {5}{4};\frac {a x^4}{b}\right )\right )}{96 a^2 \left (1-\frac {a x^4}{b}\right )^{3/4}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.43, size = 109, normalized size = 1.00 \begin {gather*} \frac {\left (-b+a x^4\right )^{3/4} \left (-15 b^2 x-12 a b x^5+32 a^2 x^9\right )}{384 a^2}-\frac {5 b^3 \tan ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{-b+a x^4}}\right )}{256 a^{9/4}}-\frac {5 b^3 \tanh ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{-b+a x^4}}\right )}{256 a^{9/4}} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.49, size = 249, normalized size = 2.28 \begin {gather*} -\frac {60 \, \left (\frac {b^{12}}{a^{9}}\right )^{\frac {1}{4}} a^{2} \arctan \left (-\frac {{\left (a x^{4} - b\right )}^{\frac {1}{4}} \left (\frac {b^{12}}{a^{9}}\right )^{\frac {1}{4}} a^{2} b^{9} - \left (\frac {b^{12}}{a^{9}}\right )^{\frac {1}{4}} a^{2} x \sqrt {\frac {\sqrt {\frac {b^{12}}{a^{9}}} a^{5} b^{12} x^{2} + \sqrt {a x^{4} - b} b^{18}}{x^{2}}}}{b^{12} x}\right ) + 15 \, \left (\frac {b^{12}}{a^{9}}\right )^{\frac {1}{4}} a^{2} \log \left (\frac {125 \, {\left ({\left (a x^{4} - b\right )}^{\frac {1}{4}} b^{9} + \left (\frac {b^{12}}{a^{9}}\right )^{\frac {3}{4}} a^{7} x\right )}}{x}\right ) - 15 \, \left (\frac {b^{12}}{a^{9}}\right )^{\frac {1}{4}} a^{2} \log \left (\frac {125 \, {\left ({\left (a x^{4} - b\right )}^{\frac {1}{4}} b^{9} - \left (\frac {b^{12}}{a^{9}}\right )^{\frac {3}{4}} a^{7} x\right )}}{x}\right ) - 4 \, {\left (32 \, a^{2} x^{9} - 12 \, a b x^{5} - 15 \, b^{2} x\right )} {\left (a x^{4} - b\right )}^{\frac {3}{4}}}{1536 \, a^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int {\left (a x^{4} - b\right )}^{\frac {3}{4}} x^{8}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.01, size = 0, normalized size = 0.00 \[\int x^{8} \left (a \,x^{4}-b \right )^{\frac {3}{4}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.41, size = 207, normalized size = 1.90 \begin {gather*} \frac {5 \, b^{3} {\left (\frac {2 \, \arctan \left (\frac {{\left (a x^{4} - b\right )}^{\frac {1}{4}}}{a^{\frac {1}{4}} x}\right )}{a^{\frac {1}{4}}} + \frac {\log \left (-\frac {a^{\frac {1}{4}} - \frac {{\left (a x^{4} - b\right )}^{\frac {1}{4}}}{x}}{a^{\frac {1}{4}} + \frac {{\left (a x^{4} - b\right )}^{\frac {1}{4}}}{x}}\right )}{a^{\frac {1}{4}}}\right )}}{512 \, a^{2}} + \frac {\frac {5 \, {\left (a x^{4} - b\right )}^{\frac {3}{4}} a^{2} b^{3}}{x^{3}} + \frac {42 \, {\left (a x^{4} - b\right )}^{\frac {7}{4}} a b^{3}}{x^{7}} - \frac {15 \, {\left (a x^{4} - b\right )}^{\frac {11}{4}} b^{3}}{x^{11}}}{384 \, {\left (a^{5} - \frac {3 \, {\left (a x^{4} - b\right )} a^{4}}{x^{4}} + \frac {3 \, {\left (a x^{4} - b\right )}^{2} a^{3}}{x^{8}} - \frac {{\left (a x^{4} - b\right )}^{3} a^{2}}{x^{12}}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int x^8\,{\left (a\,x^4-b\right )}^{3/4} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 1.74, size = 42, normalized size = 0.39 \begin {gather*} - \frac {b^{\frac {3}{4}} x^{9} e^{- \frac {i \pi }{4}} \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 {a x^{4}}{b}} \right )}}{4 \Gamma \left (\frac {13}{4}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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