Optimal. Leaf size=106 \[ -\frac {7 a x^3 \sqrt [4]{a+b x^4}}{32 b^2}+\frac {x^7 \sqrt [4]{a+b x^4}}{8 b}-\frac {21 a^2 \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{64 b^{11/4}}+\frac {21 a^2 \tanh ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{64 b^{11/4}} \]
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Rubi [A]
time = 0.03, antiderivative size = 106, normalized size of antiderivative = 1.00, number of steps
used = 6, 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 {21 a^2 \text {ArcTan}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{64 b^{11/4}}+\frac {21 a^2 \tanh ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{64 b^{11/4}}-\frac {7 a x^3 \sqrt [4]{a+b x^4}}{32 b^2}+\frac {x^7 \sqrt [4]{a+b x^4}}{8 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^{10}}{\left (a+b x^4\right )^{3/4}} \, dx &=\frac {x^7 \sqrt [4]{a+b x^4}}{8 b}-\frac {(7 a) \int \frac {x^6}{\left (a+b x^4\right )^{3/4}} \, dx}{8 b}\\ &=-\frac {7 a x^3 \sqrt [4]{a+b x^4}}{32 b^2}+\frac {x^7 \sqrt [4]{a+b x^4}}{8 b}+\frac {\left (21 a^2\right ) \int \frac {x^2}{\left (a+b x^4\right )^{3/4}} \, dx}{32 b^2}\\ &=-\frac {7 a x^3 \sqrt [4]{a+b x^4}}{32 b^2}+\frac {x^7 \sqrt [4]{a+b x^4}}{8 b}+\frac {\left (21 a^2\right ) \text {Subst}\left (\int \frac {x^2}{1-b x^4} \, dx,x,\frac {x}{\sqrt [4]{a+b x^4}}\right )}{32 b^2}\\ &=-\frac {7 a x^3 \sqrt [4]{a+b x^4}}{32 b^2}+\frac {x^7 \sqrt [4]{a+b x^4}}{8 b}+\frac {\left (21 a^2\right ) \text {Subst}\left (\int \frac {1}{1-\sqrt {b} x^2} \, dx,x,\frac {x}{\sqrt [4]{a+b x^4}}\right )}{64 b^{5/2}}-\frac {\left (21 a^2\right ) \text {Subst}\left (\int \frac {1}{1+\sqrt {b} x^2} \, dx,x,\frac {x}{\sqrt [4]{a+b x^4}}\right )}{64 b^{5/2}}\\ &=-\frac {7 a x^3 \sqrt [4]{a+b x^4}}{32 b^2}+\frac {x^7 \sqrt [4]{a+b x^4}}{8 b}-\frac {21 a^2 \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{64 b^{11/4}}+\frac {21 a^2 \tanh ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{64 b^{11/4}}\\ \end {align*}
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Mathematica [A]
time = 0.34, size = 89, normalized size = 0.84 \begin {gather*} \frac {2 b^{3/4} x^3 \sqrt [4]{a+b x^4} \left (-7 a+4 b x^4\right )-21 a^2 \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )+21 a^2 \tanh ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{64 b^{11/4}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int \frac {x^{10}}{\left (b \,x^{4}+a \right )^{\frac {3}{4}}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.51, size = 155, normalized size = 1.46 \begin {gather*} \frac {\frac {11 \, {\left (b x^{4} + a\right )}^{\frac {1}{4}} a^{2} b}{x} - \frac {7 \, {\left (b x^{4} + a\right )}^{\frac {5}{4}} a^{2}}{x^{5}}}{32 \, {\left (b^{4} - \frac {2 \, {\left (b x^{4} + a\right )} b^{3}}{x^{4}} + \frac {{\left (b x^{4} + a\right )}^{2} b^{2}}{x^{8}}\right )}} + \frac {21 \, {\left (\frac {2 \, a^{2} \arctan \left (\frac {{\left (b x^{4} + a\right )}^{\frac {1}{4}}}{b^{\frac {1}{4}} x}\right )}{b^{\frac {3}{4}}} - \frac {a^{2} \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 )}}{128 \, b^{2}} \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. 227 vs.
\(2 (82) = 164\).
time = 0.39, size = 227, normalized size = 2.14 \begin {gather*} -\frac {84 \, b^{2} \left (\frac {a^{8}}{b^{11}}\right )^{\frac {1}{4}} \arctan \left (-\frac {{\left (b x^{4} + a\right )}^{\frac {1}{4}} a^{2} b^{8} \left (\frac {a^{8}}{b^{11}}\right )^{\frac {3}{4}} - b^{8} x \sqrt {\frac {b^{6} x^{2} \sqrt {\frac {a^{8}}{b^{11}}} + \sqrt {b x^{4} + a} a^{4}}{x^{2}}} \left (\frac {a^{8}}{b^{11}}\right )^{\frac {3}{4}}}{a^{8} x}\right ) - 21 \, b^{2} \left (\frac {a^{8}}{b^{11}}\right )^{\frac {1}{4}} \log \left (\frac {21 \, {\left (b^{3} x \left (\frac {a^{8}}{b^{11}}\right )^{\frac {1}{4}} + {\left (b x^{4} + a\right )}^{\frac {1}{4}} a^{2}\right )}}{x}\right ) + 21 \, b^{2} \left (\frac {a^{8}}{b^{11}}\right )^{\frac {1}{4}} \log \left (-\frac {21 \, {\left (b^{3} x \left (\frac {a^{8}}{b^{11}}\right )^{\frac {1}{4}} - {\left (b x^{4} + a\right )}^{\frac {1}{4}} a^{2}\right )}}{x}\right ) - 4 \, {\left (4 \, b x^{7} - 7 \, a x^{3}\right )} {\left (b x^{4} + a\right )}^{\frac {1}{4}}}{128 \, b^{2}} \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 = 1.55, size = 37, normalized size = 0.35 \begin {gather*} \frac {x^{11} \Gamma \left (\frac {11}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {3}{4}, \frac {11}{4} \\ \frac {15}{4} \end {matrix}\middle | {\frac {b x^{4} e^{i \pi }}{a}} \right )}}{4 a^{\frac {3}{4}} \Gamma \left (\frac {15}{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^{10}}{{\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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