Optimal. Leaf size=82 \[ -\frac {b \tan ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4+b x^3}}\right )}{2 a^{3/4}}+\frac {b \tanh ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4+b x^3}}\right )}{2 a^{3/4}}+\sqrt [4]{a x^4+b x^3} \]
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Rubi [A] time = 0.15, antiderivative size = 136, normalized size of antiderivative = 1.66, number of steps used = 7, number of rules used = 7, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.368, Rules used = {2021, 2032, 63, 331, 298, 203, 206} \begin {gather*} -\frac {b x^{9/4} (a x+b)^{3/4} \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{a x+b}}\right )}{2 a^{3/4} \left (a x^4+b x^3\right )^{3/4}}+\frac {b x^{9/4} (a x+b)^{3/4} \tanh ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{a x+b}}\right )}{2 a^{3/4} \left (a x^4+b x^3\right )^{3/4}}+\sqrt [4]{a x^4+b x^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 63
Rule 203
Rule 206
Rule 298
Rule 331
Rule 2021
Rule 2032
Rubi steps
\begin {align*} \int \frac {\sqrt [4]{b x^3+a x^4}}{x} \, dx &=\sqrt [4]{b x^3+a x^4}+\frac {1}{4} b \int \frac {x^2}{\left (b x^3+a x^4\right )^{3/4}} \, dx\\ &=\sqrt [4]{b x^3+a x^4}+\frac {\left (b x^{9/4} (b+a x)^{3/4}\right ) \int \frac {1}{\sqrt [4]{x} (b+a x)^{3/4}} \, dx}{4 \left (b x^3+a x^4\right )^{3/4}}\\ &=\sqrt [4]{b x^3+a x^4}+\frac {\left (b x^{9/4} (b+a x)^{3/4}\right ) \operatorname {Subst}\left (\int \frac {x^2}{\left (b+a x^4\right )^{3/4}} \, dx,x,\sqrt [4]{x}\right )}{\left (b x^3+a x^4\right )^{3/4}}\\ &=\sqrt [4]{b x^3+a x^4}+\frac {\left (b x^{9/4} (b+a x)^{3/4}\right ) \operatorname {Subst}\left (\int \frac {x^2}{1-a x^4} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{\left (b x^3+a x^4\right )^{3/4}}\\ &=\sqrt [4]{b x^3+a x^4}+\frac {\left (b x^{9/4} (b+a x)^{3/4}\right ) \operatorname {Subst}\left (\int \frac {1}{1-\sqrt {a} x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{2 \sqrt {a} \left (b x^3+a x^4\right )^{3/4}}-\frac {\left (b x^{9/4} (b+a x)^{3/4}\right ) \operatorname {Subst}\left (\int \frac {1}{1+\sqrt {a} x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{2 \sqrt {a} \left (b x^3+a x^4\right )^{3/4}}\\ &=\sqrt [4]{b x^3+a x^4}-\frac {b x^{9/4} (b+a x)^{3/4} \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{2 a^{3/4} \left (b x^3+a x^4\right )^{3/4}}+\frac {b x^{9/4} (b+a x)^{3/4} \tanh ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{2 a^{3/4} \left (b x^3+a x^4\right )^{3/4}}\\ \end {align*}
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Mathematica [C] time = 0.02, size = 46, normalized size = 0.56 \begin {gather*} \frac {4 \sqrt [4]{x^3 (a x+b)} \, _2F_1\left (-\frac {1}{4},\frac {3}{4};\frac {7}{4};-\frac {a x}{b}\right )}{3 \sqrt [4]{\frac {a x}{b}+1}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.36, size = 82, normalized size = 1.00 \begin {gather*} -\frac {b \tan ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4+b x^3}}\right )}{2 a^{3/4}}+\frac {b \tanh ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4+b x^3}}\right )}{2 a^{3/4}}+\sqrt [4]{a x^4+b x^3} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.43, size = 206, normalized size = 2.51 \begin {gather*} -\left (\frac {b^{4}}{a^{3}}\right )^{\frac {1}{4}} \arctan \left (\frac {a^{2} \left (\frac {b^{4}}{a^{3}}\right )^{\frac {3}{4}} x \sqrt {\frac {a^{2} \sqrt {\frac {b^{4}}{a^{3}}} x^{2} + \sqrt {a x^{4} + b x^{3}} b^{2}}{x^{2}}} - {\left (a x^{4} + b x^{3}\right )}^{\frac {1}{4}} a^{2} b \left (\frac {b^{4}}{a^{3}}\right )^{\frac {3}{4}}}{b^{4} x}\right ) + \frac {1}{4} \, \left (\frac {b^{4}}{a^{3}}\right )^{\frac {1}{4}} \log \left (\frac {a \left (\frac {b^{4}}{a^{3}}\right )^{\frac {1}{4}} x + {\left (a x^{4} + b x^{3}\right )}^{\frac {1}{4}} b}{x}\right ) - \frac {1}{4} \, \left (\frac {b^{4}}{a^{3}}\right )^{\frac {1}{4}} \log \left (-\frac {a \left (\frac {b^{4}}{a^{3}}\right )^{\frac {1}{4}} x - {\left (a x^{4} + b x^{3}\right )}^{\frac {1}{4}} b}{x}\right ) + {\left (a x^{4} + b x^{3}\right )}^{\frac {1}{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.21, size = 211, normalized size = 2.57 \begin {gather*} \frac {\frac {2 \, \sqrt {2} \left (-a\right )^{\frac {1}{4}} b^{2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (-a\right )^{\frac {1}{4}} + 2 \, {\left (a + \frac {b}{x}\right )}^{\frac {1}{4}}\right )}}{2 \, \left (-a\right )^{\frac {1}{4}}}\right )}{a} + \frac {2 \, \sqrt {2} \left (-a\right )^{\frac {1}{4}} b^{2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (-a\right )^{\frac {1}{4}} - 2 \, {\left (a + \frac {b}{x}\right )}^{\frac {1}{4}}\right )}}{2 \, \left (-a\right )^{\frac {1}{4}}}\right )}{a} + \frac {\sqrt {2} \left (-a\right )^{\frac {1}{4}} b^{2} \log \left (\sqrt {2} \left (-a\right )^{\frac {1}{4}} {\left (a + \frac {b}{x}\right )}^{\frac {1}{4}} + \sqrt {-a} + \sqrt {a + \frac {b}{x}}\right )}{a} + \frac {\sqrt {2} b^{2} \log \left (-\sqrt {2} \left (-a\right )^{\frac {1}{4}} {\left (a + \frac {b}{x}\right )}^{\frac {1}{4}} + \sqrt {-a} + \sqrt {a + \frac {b}{x}}\right )}{\left (-a\right )^{\frac {3}{4}}} + 8 \, {\left (a + \frac {b}{x}\right )}^{\frac {1}{4}} b x}{8 \, b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.06, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a \,x^{4}+b \,x^{3}\right )^{\frac {1}{4}}}{x}\, 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} + b x^{3}\right )}^{\frac {1}{4}}}{x}\,{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\,x^3\right )}^{1/4}}{x} \,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 {\sqrt [4]{x^{3} \left (a x + b\right )}}{x}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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