Optimal. Leaf size=134 \[ \frac {3^{3/4} a^{7/4} \tan ^{-1}\left (\frac {\sqrt [4]{3} \sqrt [4]{a} x}{\sqrt {2} \sqrt [4]{a x^4+b}}\right )}{32 \sqrt {2} b}+\frac {3^{3/4} a^{7/4} \tanh ^{-1}\left (\frac {\sqrt [4]{3} \sqrt [4]{a} x}{\sqrt {2} \sqrt [4]{a x^4+b}}\right )}{32 \sqrt {2} b}+\frac {\left (-19 a x^4-12 b\right ) \left (a x^4+b\right )^{3/4}}{168 b x^7} \]
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Rubi [A] time = 0.17, antiderivative size = 143, normalized size of antiderivative = 1.07, number of steps used = 7, number of rules used = 7, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {580, 583, 12, 377, 212, 206, 203} \begin {gather*} \frac {3^{3/4} a^{7/4} \tan ^{-1}\left (\frac {\sqrt [4]{3} \sqrt [4]{a} x}{\sqrt {2} \sqrt [4]{a x^4+b}}\right )}{32 \sqrt {2} b}+\frac {3^{3/4} a^{7/4} \tanh ^{-1}\left (\frac {\sqrt [4]{3} \sqrt [4]{a} x}{\sqrt {2} \sqrt [4]{a x^4+b}}\right )}{32 \sqrt {2} b}-\frac {\left (a x^4+b\right )^{3/4}}{14 x^7}-\frac {19 a \left (a x^4+b\right )^{3/4}}{168 b x^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 203
Rule 206
Rule 212
Rule 377
Rule 580
Rule 583
Rubi steps
\begin {align*} \int \frac {\left (b+a x^4\right )^{3/4} \left (2 b+a x^4\right )}{x^8 \left (4 b+a x^4\right )} \, dx &=-\frac {\left (b+a x^4\right )^{3/4}}{14 x^7}+\frac {\int \frac {38 a b^2+20 a^2 b x^4}{x^4 \sqrt [4]{b+a x^4} \left (4 b+a x^4\right )} \, dx}{28 b}\\ &=-\frac {\left (b+a x^4\right )^{3/4}}{14 x^7}-\frac {19 a \left (b+a x^4\right )^{3/4}}{168 b x^3}-\frac {\int -\frac {126 a^2 b^3}{\sqrt [4]{b+a x^4} \left (4 b+a x^4\right )} \, dx}{336 b^3}\\ &=-\frac {\left (b+a x^4\right )^{3/4}}{14 x^7}-\frac {19 a \left (b+a x^4\right )^{3/4}}{168 b x^3}+\frac {1}{8} \left (3 a^2\right ) \int \frac {1}{\sqrt [4]{b+a x^4} \left (4 b+a x^4\right )} \, dx\\ &=-\frac {\left (b+a x^4\right )^{3/4}}{14 x^7}-\frac {19 a \left (b+a x^4\right )^{3/4}}{168 b x^3}+\frac {1}{8} \left (3 a^2\right ) \operatorname {Subst}\left (\int \frac {1}{4 b-3 a b x^4} \, dx,x,\frac {x}{\sqrt [4]{b+a x^4}}\right )\\ &=-\frac {\left (b+a x^4\right )^{3/4}}{14 x^7}-\frac {19 a \left (b+a x^4\right )^{3/4}}{168 b x^3}+\frac {\left (3 a^2\right ) \operatorname {Subst}\left (\int \frac {1}{2-\sqrt {3} \sqrt {a} x^2} \, dx,x,\frac {x}{\sqrt [4]{b+a x^4}}\right )}{32 b}+\frac {\left (3 a^2\right ) \operatorname {Subst}\left (\int \frac {1}{2+\sqrt {3} \sqrt {a} x^2} \, dx,x,\frac {x}{\sqrt [4]{b+a x^4}}\right )}{32 b}\\ &=-\frac {\left (b+a x^4\right )^{3/4}}{14 x^7}-\frac {19 a \left (b+a x^4\right )^{3/4}}{168 b x^3}+\frac {3^{3/4} a^{7/4} \tan ^{-1}\left (\frac {\sqrt [4]{3} \sqrt [4]{a} x}{\sqrt {2} \sqrt [4]{b+a x^4}}\right )}{32 \sqrt {2} b}+\frac {3^{3/4} a^{7/4} \tanh ^{-1}\left (\frac {\sqrt [4]{3} \sqrt [4]{a} x}{\sqrt {2} \sqrt [4]{b+a x^4}}\right )}{32 \sqrt {2} b}\\ \end {align*}
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Mathematica [A] time = 0.16, size = 151, normalized size = 1.13 \begin {gather*} \frac {3^{3/4} a^{7/4} \left (-\log \left (2-\frac {\sqrt {2} \sqrt [4]{3} \sqrt [4]{a} x}{\sqrt [4]{a+b x^4}}\right )+\log \left (\frac {\sqrt {2} \sqrt [4]{3} \sqrt [4]{a} x}{\sqrt [4]{a+b x^4}}+2\right )+2 \tan ^{-1}\left (\frac {\sqrt [4]{3} \sqrt [4]{a} x}{\sqrt {2} \sqrt [4]{a+b x^4}}\right )\right )}{64 \sqrt {2} b}+\left (-\frac {19 a}{168 b x^3}-\frac {1}{14 x^7}\right ) \left (a x^4+b\right )^{3/4} \end {gather*}
Warning: Unable to verify antiderivative.
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IntegrateAlgebraic [A] time = 0.61, size = 134, normalized size = 1.00 \begin {gather*} \frac {3^{3/4} a^{7/4} \tan ^{-1}\left (\frac {\sqrt [4]{3} \sqrt [4]{a} x}{\sqrt {2} \sqrt [4]{a x^4+b}}\right )}{32 \sqrt {2} b}+\frac {3^{3/4} a^{7/4} \tanh ^{-1}\left (\frac {\sqrt [4]{3} \sqrt [4]{a} x}{\sqrt {2} \sqrt [4]{a x^4+b}}\right )}{32 \sqrt {2} b}+\frac {\left (-19 a x^4-12 b\right ) \left (a x^4+b\right )^{3/4}}{168 b x^7} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 91.96, size = 495, normalized size = 3.69 \begin {gather*} -\frac {84 \, \left (\frac {27}{4}\right )^{\frac {1}{4}} \left (\frac {a^{7}}{b^{4}}\right )^{\frac {1}{4}} b x^{7} \arctan \left (-\frac {4 \, {\left (27 \, \left (\frac {27}{4}\right )^{\frac {1}{4}} {\left (a x^{4} + b\right )}^{\frac {1}{4}} \left (\frac {a^{7}}{b^{4}}\right )^{\frac {1}{4}} a^{9} b x^{3} + 12 \, \left (\frac {27}{4}\right )^{\frac {3}{4}} {\left (a x^{4} + b\right )}^{\frac {3}{4}} \left (\frac {a^{7}}{b^{4}}\right )^{\frac {3}{4}} a^{5} b^{3} x - \sqrt {\frac {3}{2}} \sqrt {\sqrt {3} \sqrt {\frac {a^{7}}{b^{4}}} a^{6} b^{2}} {\left (18 \, \left (\frac {27}{4}\right )^{\frac {1}{4}} \sqrt {a x^{4} + b} \left (\frac {a^{7}}{b^{4}}\right )^{\frac {1}{4}} a^{4} b x^{2} + \left (\frac {27}{4}\right )^{\frac {3}{4}} {\left (7 \, a b^{3} x^{4} + 4 \, b^{4}\right )} \left (\frac {a^{7}}{b^{4}}\right )^{\frac {3}{4}}\right )}\right )}}{27 \, {\left (a^{11} x^{4} + 4 \, a^{10} b\right )}}\right ) - 21 \, \left (\frac {27}{4}\right )^{\frac {1}{4}} \left (\frac {a^{7}}{b^{4}}\right )^{\frac {1}{4}} b x^{7} \log \left (\frac {18 \, \sqrt {3} {\left (a x^{4} + b\right )}^{\frac {1}{4}} \sqrt {\frac {a^{7}}{b^{4}}} a^{2} b^{2} x^{3} + 8 \, \left (\frac {27}{4}\right )^{\frac {3}{4}} \sqrt {a x^{4} + b} \left (\frac {a^{7}}{b^{4}}\right )^{\frac {3}{4}} b^{3} x^{2} + 36 \, {\left (a x^{4} + b\right )}^{\frac {3}{4}} a^{5} x + 3 \, \left (\frac {27}{4}\right )^{\frac {1}{4}} {\left (7 \, a^{4} b x^{4} + 4 \, a^{3} b^{2}\right )} \left (\frac {a^{7}}{b^{4}}\right )^{\frac {1}{4}}}{4 \, {\left (a x^{4} + 4 \, b\right )}}\right ) + 21 \, \left (\frac {27}{4}\right )^{\frac {1}{4}} \left (\frac {a^{7}}{b^{4}}\right )^{\frac {1}{4}} b x^{7} \log \left (\frac {18 \, \sqrt {3} {\left (a x^{4} + b\right )}^{\frac {1}{4}} \sqrt {\frac {a^{7}}{b^{4}}} a^{2} b^{2} x^{3} - 8 \, \left (\frac {27}{4}\right )^{\frac {3}{4}} \sqrt {a x^{4} + b} \left (\frac {a^{7}}{b^{4}}\right )^{\frac {3}{4}} b^{3} x^{2} + 36 \, {\left (a x^{4} + b\right )}^{\frac {3}{4}} a^{5} x - 3 \, \left (\frac {27}{4}\right )^{\frac {1}{4}} {\left (7 \, a^{4} b x^{4} + 4 \, a^{3} b^{2}\right )} \left (\frac {a^{7}}{b^{4}}\right )^{\frac {1}{4}}}{4 \, {\left (a x^{4} + 4 \, b\right )}}\right ) + 16 \, {\left (19 \, a x^{4} + 12 \, b\right )} {\left (a x^{4} + b\right )}^{\frac {3}{4}}}{2688 \, b x^{7}} \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 {{\left (a x^{4} + 2 \, b\right )} {\left (a x^{4} + b\right )}^{\frac {3}{4}}}{{\left (a x^{4} + 4 \, b\right )} x^{8}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.34, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a \,x^{4}+b \right )^{\frac {3}{4}} \left (a \,x^{4}+2 b \right )}{x^{8} \left (a \,x^{4}+4 b \right )}\, dx \end {gather*}
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 \frac {{\left (a x^{4} + 2 \, b\right )} {\left (a x^{4} + b\right )}^{\frac {3}{4}}}{{\left (a x^{4} + 4 \, b\right )} x^{8}}\,{d 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 {{\left (a\,x^4+b\right )}^{3/4}\,\left (a\,x^4+2\,b\right )}{x^8\,\left (a\,x^4+4\,b\right )} \,d x \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 \frac {\left (a x^{4} + b\right )^{\frac {3}{4}} \left (a x^{4} + 2 b\right )}{x^{8} \left (a x^{4} + 4 b\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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