Optimal. Leaf size=80 \[ \frac {3 b \tan ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4+b}}\right )}{8 a^{7/4}}-\frac {3 b \tanh ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4+b}}\right )}{8 a^{7/4}}+\frac {x^3 \sqrt [4]{a x^4+b}}{4 a} \]
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Rubi [A] time = 0.04, antiderivative size = 80, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {321, 331, 298, 203, 206} \begin {gather*} \frac {3 b \tan ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4+b}}\right )}{8 a^{7/4}}-\frac {3 b \tanh ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4+b}}\right )}{8 a^{7/4}}+\frac {x^3 \sqrt [4]{a x^4+b}}{4 a} \end {gather*}
Antiderivative was successfully verified.
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Rule 203
Rule 206
Rule 298
Rule 321
Rule 331
Rubi steps
\begin {align*} \int \frac {x^6}{\left (b+a x^4\right )^{3/4}} \, dx &=\frac {x^3 \sqrt [4]{b+a x^4}}{4 a}-\frac {(3 b) \int \frac {x^2}{\left (b+a x^4\right )^{3/4}} \, dx}{4 a}\\ &=\frac {x^3 \sqrt [4]{b+a x^4}}{4 a}-\frac {(3 b) \operatorname {Subst}\left (\int \frac {x^2}{1-a x^4} \, dx,x,\frac {x}{\sqrt [4]{b+a x^4}}\right )}{4 a}\\ &=\frac {x^3 \sqrt [4]{b+a x^4}}{4 a}-\frac {(3 b) \operatorname {Subst}\left (\int \frac {1}{1-\sqrt {a} x^2} \, dx,x,\frac {x}{\sqrt [4]{b+a x^4}}\right )}{8 a^{3/2}}+\frac {(3 b) \operatorname {Subst}\left (\int \frac {1}{1+\sqrt {a} x^2} \, dx,x,\frac {x}{\sqrt [4]{b+a x^4}}\right )}{8 a^{3/2}}\\ &=\frac {x^3 \sqrt [4]{b+a x^4}}{4 a}+\frac {3 b \tan ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b+a x^4}}\right )}{8 a^{7/4}}-\frac {3 b \tanh ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b+a x^4}}\right )}{8 a^{7/4}}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 75, normalized size = 0.94 \begin {gather*} \frac {2 a^{3/4} x^3 \sqrt [4]{a x^4+b}+3 b \tan ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4+b}}\right )-3 b \tanh ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4+b}}\right )}{8 a^{7/4}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.41, size = 80, normalized size = 1.00 \begin {gather*} \frac {x^3 \sqrt [4]{b+a x^4}}{4 a}+\frac {3 b \tan ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b+a x^4}}\right )}{8 a^{7/4}}-\frac {3 b \tanh ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b+a x^4}}\right )}{8 a^{7/4}} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.50, size = 204, normalized size = 2.55 \begin {gather*} \frac {4 \, {\left (a x^{4} + b\right )}^{\frac {1}{4}} x^{3} + 12 \, a \left (\frac {b^{4}}{a^{7}}\right )^{\frac {1}{4}} \arctan \left (\frac {a^{5} x \sqrt {\frac {a^{4} x^{2} \sqrt {\frac {b^{4}}{a^{7}}} + \sqrt {a x^{4} + b} b^{2}}{x^{2}}} \left (\frac {b^{4}}{a^{7}}\right )^{\frac {3}{4}} - {\left (a x^{4} + b\right )}^{\frac {1}{4}} a^{5} b \left (\frac {b^{4}}{a^{7}}\right )^{\frac {3}{4}}}{b^{4} x}\right ) - 3 \, a \left (\frac {b^{4}}{a^{7}}\right )^{\frac {1}{4}} \log \left (\frac {3 \, {\left (a^{2} x \left (\frac {b^{4}}{a^{7}}\right )^{\frac {1}{4}} + {\left (a x^{4} + b\right )}^{\frac {1}{4}} b\right )}}{x}\right ) + 3 \, a \left (\frac {b^{4}}{a^{7}}\right )^{\frac {1}{4}} \log \left (-\frac {3 \, {\left (a^{2} x \left (\frac {b^{4}}{a^{7}}\right )^{\frac {1}{4}} - {\left (a x^{4} + b\right )}^{\frac {1}{4}} b\right )}}{x}\right )}{16 \, a} \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 \frac {x^{6}}{{\left (a x^{4} + b\right )}^{\frac {3}{4}}}\,{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 \[\int \frac {x^{6}}{\left (a \,x^{4}+b \right )^{\frac {3}{4}}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.42, size = 110, normalized size = 1.38 \begin {gather*} -\frac {3 \, {\left (\frac {2 \, b \arctan \left (\frac {{\left (a x^{4} + b\right )}^{\frac {1}{4}}}{a^{\frac {1}{4}} x}\right )}{a^{\frac {3}{4}}} - \frac {b \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 {3}{4}}}\right )}}{16 \, a} - \frac {{\left (a x^{4} + b\right )}^{\frac {1}{4}} b}{4 \, {\left (a^{2} - \frac {{\left (a x^{4} + b\right )} a}{x^{4}}\right )} x} \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 \frac {x^6}{{\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 = 1.08, size = 37, normalized size = 0.46 \begin {gather*} \frac {x^{7} \Gamma \left (\frac {7}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {3}{4}, \frac {7}{4} \\ \frac {11}{4} \end {matrix}\middle | {\frac {a x^{4} e^{i \pi }}{b}} \right )}}{4 b^{\frac {3}{4}} \Gamma \left (\frac {11}{4}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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