Optimal. Leaf size=107 \[ \frac {4}{3} a^{7/4} \tan ^{-1}\left (\frac {\sqrt [4]{a} \left (a x^4+b x\right )^{3/4}}{a x^3+b}\right )+\frac {4}{3} a^{7/4} \tanh ^{-1}\left (\frac {\sqrt [4]{a} \left (a x^4+b x\right )^{3/4}}{a x^3+b}\right )-\frac {4 \left (17 a x^3+3 b\right ) \left (a x^4+b x\right )^{3/4}}{63 x^6} \]
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Rubi [A] time = 0.35, antiderivative size = 165, normalized size of antiderivative = 1.54, number of steps used = 12, number of rules used = 10, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.312, Rules used = {2052, 2011, 329, 275, 240, 212, 206, 203, 2016, 2014} \begin {gather*} \frac {4 a^{7/4} \sqrt [4]{x} \sqrt [4]{a x^3+b} \tan ^{-1}\left (\frac {\sqrt [4]{a} x^{3/4}}{\sqrt [4]{a x^3+b}}\right )}{3 \sqrt [4]{a x^4+b x}}+\frac {4 a^{7/4} \sqrt [4]{x} \sqrt [4]{a x^3+b} \tanh ^{-1}\left (\frac {\sqrt [4]{a} x^{3/4}}{\sqrt [4]{a x^3+b}}\right )}{3 \sqrt [4]{a x^4+b x}}-\frac {4 b \left (a x^4+b x\right )^{3/4}}{21 x^6}-\frac {68 a \left (a x^4+b x\right )^{3/4}}{63 x^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 203
Rule 206
Rule 212
Rule 240
Rule 275
Rule 329
Rule 2011
Rule 2014
Rule 2016
Rule 2052
Rubi steps
\begin {align*} \int \frac {\left (b+a x^3\right ) \left (b+2 a x^3\right )}{x^6 \sqrt [4]{b x+a x^4}} \, dx &=\int \left (\frac {2 a^2}{\sqrt [4]{b x+a x^4}}+\frac {b^2}{x^6 \sqrt [4]{b x+a x^4}}+\frac {3 a b}{x^3 \sqrt [4]{b x+a x^4}}\right ) \, dx\\ &=\left (2 a^2\right ) \int \frac {1}{\sqrt [4]{b x+a x^4}} \, dx+(3 a b) \int \frac {1}{x^3 \sqrt [4]{b x+a x^4}} \, dx+b^2 \int \frac {1}{x^6 \sqrt [4]{b x+a x^4}} \, dx\\ &=-\frac {4 b \left (b x+a x^4\right )^{3/4}}{21 x^6}-\frac {4 a \left (b x+a x^4\right )^{3/4}}{3 x^3}-\frac {1}{7} (4 a b) \int \frac {1}{x^3 \sqrt [4]{b x+a x^4}} \, dx+\frac {\left (2 a^2 \sqrt [4]{x} \sqrt [4]{b+a x^3}\right ) \int \frac {1}{\sqrt [4]{x} \sqrt [4]{b+a x^3}} \, dx}{\sqrt [4]{b x+a x^4}}\\ &=-\frac {4 b \left (b x+a x^4\right )^{3/4}}{21 x^6}-\frac {68 a \left (b x+a x^4\right )^{3/4}}{63 x^3}+\frac {\left (8 a^2 \sqrt [4]{x} \sqrt [4]{b+a x^3}\right ) \operatorname {Subst}\left (\int \frac {x^2}{\sqrt [4]{b+a x^{12}}} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{b x+a x^4}}\\ &=-\frac {4 b \left (b x+a x^4\right )^{3/4}}{21 x^6}-\frac {68 a \left (b x+a x^4\right )^{3/4}}{63 x^3}+\frac {\left (8 a^2 \sqrt [4]{x} \sqrt [4]{b+a x^3}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [4]{b+a x^4}} \, dx,x,x^{3/4}\right )}{3 \sqrt [4]{b x+a x^4}}\\ &=-\frac {4 b \left (b x+a x^4\right )^{3/4}}{21 x^6}-\frac {68 a \left (b x+a x^4\right )^{3/4}}{63 x^3}+\frac {\left (8 a^2 \sqrt [4]{x} \sqrt [4]{b+a x^3}\right ) \operatorname {Subst}\left (\int \frac {1}{1-a x^4} \, dx,x,\frac {x^{3/4}}{\sqrt [4]{b+a x^3}}\right )}{3 \sqrt [4]{b x+a x^4}}\\ &=-\frac {4 b \left (b x+a x^4\right )^{3/4}}{21 x^6}-\frac {68 a \left (b x+a x^4\right )^{3/4}}{63 x^3}+\frac {\left (4 a^2 \sqrt [4]{x} \sqrt [4]{b+a x^3}\right ) \operatorname {Subst}\left (\int \frac {1}{1-\sqrt {a} x^2} \, dx,x,\frac {x^{3/4}}{\sqrt [4]{b+a x^3}}\right )}{3 \sqrt [4]{b x+a x^4}}+\frac {\left (4 a^2 \sqrt [4]{x} \sqrt [4]{b+a x^3}\right ) \operatorname {Subst}\left (\int \frac {1}{1+\sqrt {a} x^2} \, dx,x,\frac {x^{3/4}}{\sqrt [4]{b+a x^3}}\right )}{3 \sqrt [4]{b x+a x^4}}\\ &=-\frac {4 b \left (b x+a x^4\right )^{3/4}}{21 x^6}-\frac {68 a \left (b x+a x^4\right )^{3/4}}{63 x^3}+\frac {4 a^{7/4} \sqrt [4]{x} \sqrt [4]{b+a x^3} \tan ^{-1}\left (\frac {\sqrt [4]{a} x^{3/4}}{\sqrt [4]{b+a x^3}}\right )}{3 \sqrt [4]{b x+a x^4}}+\frac {4 a^{7/4} \sqrt [4]{x} \sqrt [4]{b+a x^3} \tanh ^{-1}\left (\frac {\sqrt [4]{a} x^{3/4}}{\sqrt [4]{b+a x^3}}\right )}{3 \sqrt [4]{b x+a x^4}}\\ \end {align*}
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Mathematica [C] time = 0.08, size = 69, normalized size = 0.64 \begin {gather*} \frac {4 \left (x \left (a x^3+b\right )\right )^{3/4} \left (-\frac {14 a x^3 \, _2F_1\left (-\frac {3}{4},-\frac {3}{4};\frac {1}{4};-\frac {a x^3}{b}\right )}{\left (\frac {a x^3}{b}+1\right )^{3/4}}-3 \left (a x^3+b\right )\right )}{63 x^6} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.41, size = 107, normalized size = 1.00 \begin {gather*} \frac {4}{3} a^{7/4} \tan ^{-1}\left (\frac {\sqrt [4]{a} \left (a x^4+b x\right )^{3/4}}{a x^3+b}\right )+\frac {4}{3} a^{7/4} \tanh ^{-1}\left (\frac {\sqrt [4]{a} \left (a x^4+b x\right )^{3/4}}{a x^3+b}\right )-\frac {4 \left (17 a x^3+3 b\right ) \left (a x^4+b x\right )^{3/4}}{63 x^6} \end {gather*}
Antiderivative was successfully verified.
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fricas [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.28, size = 201, normalized size = 1.88 \begin {gather*} \frac {2}{3} \, \sqrt {2} \left (-a\right )^{\frac {3}{4}} a \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (-a\right )^{\frac {1}{4}} + 2 \, {\left (a + \frac {b}{x^{3}}\right )}^{\frac {1}{4}}\right )}}{2 \, \left (-a\right )^{\frac {1}{4}}}\right ) + \frac {2}{3} \, \sqrt {2} \left (-a\right )^{\frac {3}{4}} a \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (-a\right )^{\frac {1}{4}} - 2 \, {\left (a + \frac {b}{x^{3}}\right )}^{\frac {1}{4}}\right )}}{2 \, \left (-a\right )^{\frac {1}{4}}}\right ) - \frac {1}{3} \, \sqrt {2} \left (-a\right )^{\frac {3}{4}} a \log \left (\sqrt {2} \left (-a\right )^{\frac {1}{4}} {\left (a + \frac {b}{x^{3}}\right )}^{\frac {1}{4}} + \sqrt {-a} + \sqrt {a + \frac {b}{x^{3}}}\right ) + \frac {1}{3} \, \sqrt {2} \left (-a\right )^{\frac {3}{4}} a \log \left (-\sqrt {2} \left (-a\right )^{\frac {1}{4}} {\left (a + \frac {b}{x^{3}}\right )}^{\frac {1}{4}} + \sqrt {-a} + \sqrt {a + \frac {b}{x^{3}}}\right ) - \frac {4}{21} \, {\left (a + \frac {b}{x^{3}}\right )}^{\frac {7}{4}} - \frac {8}{9} \, {\left (a + \frac {b}{x^{3}}\right )}^{\frac {3}{4}} a \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.31, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a \,x^{3}+b \right ) \left (2 a \,x^{3}+b \right )}{x^{6} \left (a \,x^{4}+b x \right )^{\frac {1}{4}}}\, 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 (2 \, a x^{3} + b\right )} {\left (a x^{3} + b\right )}}{{\left (a x^{4} + b x\right )}^{\frac {1}{4}} x^{6}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.81, size = 104, normalized size = 0.97 \begin {gather*} -\frac {12\,b\,\left (a\,x^4+b\,x\right )+68\,a\,x^3\,\left (a\,x^4+b\,x\right )-168\,a^2\,x^7\,{\left (\frac {a\,x^3}{b}+1\right )}^{1/4}\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{4},\frac {1}{4};\ \frac {5}{4};\ -\frac {a\,x^3}{b}\right )}{{\left (a\,x^4+b\,x\right )}^{1/4}\,\left (\frac {63\,x^2\,\left (a\,x^4+b\,x\right )}{a}-\frac {63\,b\,x^3}{a}\right )} \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^{3} + b\right ) \left (2 a x^{3} + b\right )}{x^{6} \sqrt [4]{x \left (a x^{3} + b\right )}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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