Optimal. Leaf size=80 \[ \frac {x^3 \sqrt [4]{a+b x^4}}{4 b}+\frac {3 a \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{8 b^{7/4}}-\frac {3 a \tanh ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{8 b^{7/4}} \]
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Rubi [A]
time = 0.02, 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 = {327, 338, 304,
209, 212} \begin {gather*} \frac {3 a \text {ArcTan}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{8 b^{7/4}}-\frac {3 a \tanh ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{8 b^{7/4}}+\frac {x^3 \sqrt [4]{a+b x^4}}{4 b} \end {gather*}
Antiderivative was successfully verified.
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Rule 209
Rule 212
Rule 304
Rule 327
Rule 338
Rubi steps
\begin {align*} \int \frac {x^6}{\left (a+b x^4\right )^{3/4}} \, dx &=\frac {x^3 \sqrt [4]{a+b x^4}}{4 b}-\frac {(3 a) \int \frac {x^2}{\left (a+b x^4\right )^{3/4}} \, dx}{4 b}\\ &=\frac {x^3 \sqrt [4]{a+b x^4}}{4 b}-\frac {(3 a) \text {Subst}\left (\int \frac {x^2}{1-b x^4} \, dx,x,\frac {x}{\sqrt [4]{a+b x^4}}\right )}{4 b}\\ &=\frac {x^3 \sqrt [4]{a+b x^4}}{4 b}-\frac {(3 a) \text {Subst}\left (\int \frac {1}{1-\sqrt {b} x^2} \, dx,x,\frac {x}{\sqrt [4]{a+b x^4}}\right )}{8 b^{3/2}}+\frac {(3 a) \text {Subst}\left (\int \frac {1}{1+\sqrt {b} x^2} \, dx,x,\frac {x}{\sqrt [4]{a+b x^4}}\right )}{8 b^{3/2}}\\ &=\frac {x^3 \sqrt [4]{a+b x^4}}{4 b}+\frac {3 a \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{8 b^{7/4}}-\frac {3 a \tanh ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{8 b^{7/4}}\\ \end {align*}
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Mathematica [A]
time = 0.28, size = 75, normalized size = 0.94 \begin {gather*} \frac {2 b^{3/4} x^3 \sqrt [4]{a+b x^4}+3 a \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )-3 a \tanh ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{8 b^{7/4}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int \frac {x^{6}}{\left (b \,x^{4}+a \right )^{\frac {3}{4}}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.50, size = 110, normalized size = 1.38 \begin {gather*} -\frac {3 \, {\left (\frac {2 \, a \arctan \left (\frac {{\left (b x^{4} + a\right )}^{\frac {1}{4}}}{b^{\frac {1}{4}} x}\right )}{b^{\frac {3}{4}}} - \frac {a \log \left (-\frac {b^{\frac {1}{4}} - \frac {{\left (b x^{4} + a\right )}^{\frac {1}{4}}}{x}}{b^{\frac {1}{4}} + \frac {{\left (b x^{4} + a\right )}^{\frac {1}{4}}}{x}}\right )}{b^{\frac {3}{4}}}\right )}}{16 \, b} - \frac {{\left (b x^{4} + a\right )}^{\frac {1}{4}} a}{4 \, {\left (b^{2} - \frac {{\left (b x^{4} + a\right )} b}{x^{4}}\right )} x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 204 vs.
\(2 (60) = 120\).
time = 0.38, size = 204, normalized size = 2.55 \begin {gather*} \frac {4 \, {\left (b x^{4} + a\right )}^{\frac {1}{4}} x^{3} + 12 \, b \left (\frac {a^{4}}{b^{7}}\right )^{\frac {1}{4}} \arctan \left (\frac {b^{5} x \sqrt {\frac {b^{4} x^{2} \sqrt {\frac {a^{4}}{b^{7}}} + \sqrt {b x^{4} + a} a^{2}}{x^{2}}} \left (\frac {a^{4}}{b^{7}}\right )^{\frac {3}{4}} - {\left (b x^{4} + a\right )}^{\frac {1}{4}} a b^{5} \left (\frac {a^{4}}{b^{7}}\right )^{\frac {3}{4}}}{a^{4} x}\right ) - 3 \, b \left (\frac {a^{4}}{b^{7}}\right )^{\frac {1}{4}} \log \left (\frac {3 \, {\left (b^{2} x \left (\frac {a^{4}}{b^{7}}\right )^{\frac {1}{4}} + {\left (b x^{4} + a\right )}^{\frac {1}{4}} a\right )}}{x}\right ) + 3 \, b \left (\frac {a^{4}}{b^{7}}\right )^{\frac {1}{4}} \log \left (-\frac {3 \, {\left (b^{2} x \left (\frac {a^{4}}{b^{7}}\right )^{\frac {1}{4}} - {\left (b x^{4} + a\right )}^{\frac {1}{4}} a\right )}}{x}\right )}{16 \, b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] Result contains complex when optimal does not.
time = 0.66, 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 {b x^{4} e^{i \pi }}{a}} \right )}}{4 a^{\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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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.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {x^6}{{\left (b\,x^4+a\right )}^{3/4}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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