Optimal. Leaf size=82 \[ \frac {7 b \tan ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4+b x^3}}\right )}{2 a^{3/4}}-\frac {7 b \tanh ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4+b x^3}}\right )}{2 a^{3/4}}+\sqrt [4]{a x^4+b x^3} \]
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Rubi [A] time = 0.36, antiderivative size = 136, normalized size of antiderivative = 1.66, number of steps used = 7, number of rules used = 7, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.212, Rules used = {2056, 80, 63, 331, 298, 203, 206} \begin {gather*} \frac {7 b \sqrt [4]{a x^4+b x^3} \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{a x+b}}\right )}{2 a^{3/4} x^{3/4} \sqrt [4]{a x+b}}-\frac {7 b \sqrt [4]{a x^4+b x^3} \tanh ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{a x+b}}\right )}{2 a^{3/4} x^{3/4} \sqrt [4]{a x+b}}+\sqrt [4]{a x^4+b x^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 63
Rule 80
Rule 203
Rule 206
Rule 298
Rule 331
Rule 2056
Rubi steps
\begin {align*} \int \frac {(-b+a x) \sqrt [4]{b x^3+a x^4}}{x (b+a x)} \, dx &=\frac {\sqrt [4]{b x^3+a x^4} \int \frac {-b+a x}{\sqrt [4]{x} (b+a x)^{3/4}} \, dx}{x^{3/4} \sqrt [4]{b+a x}}\\ &=\sqrt [4]{b x^3+a x^4}-\frac {\left (7 b \sqrt [4]{b x^3+a x^4}\right ) \int \frac {1}{\sqrt [4]{x} (b+a x)^{3/4}} \, dx}{4 x^{3/4} \sqrt [4]{b+a x}}\\ &=\sqrt [4]{b x^3+a x^4}-\frac {\left (7 b \sqrt [4]{b x^3+a x^4}\right ) \operatorname {Subst}\left (\int \frac {x^2}{\left (b+a x^4\right )^{3/4}} \, dx,x,\sqrt [4]{x}\right )}{x^{3/4} \sqrt [4]{b+a x}}\\ &=\sqrt [4]{b x^3+a x^4}-\frac {\left (7 b \sqrt [4]{b x^3+a x^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 )}{x^{3/4} \sqrt [4]{b+a x}}\\ &=\sqrt [4]{b x^3+a x^4}-\frac {\left (7 b \sqrt [4]{b x^3+a x^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} x^{3/4} \sqrt [4]{b+a x}}+\frac {\left (7 b \sqrt [4]{b x^3+a x^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} x^{3/4} \sqrt [4]{b+a x}}\\ &=\sqrt [4]{b x^3+a x^4}+\frac {7 b \sqrt [4]{b x^3+a x^4} \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{2 a^{3/4} x^{3/4} \sqrt [4]{b+a x}}-\frac {7 b \sqrt [4]{b x^3+a x^4} \tanh ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{2 a^{3/4} x^{3/4} \sqrt [4]{b+a x}}\\ \end {align*}
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Mathematica [C] time = 0.03, size = 60, normalized size = 0.73 \begin {gather*} \frac {x^3 \left (3 (a x+b)-7 b \left (\frac {a x}{b}+1\right )^{3/4} \, _2F_1\left (\frac {3}{4},\frac {3}{4};\frac {7}{4};-\frac {a x}{b}\right )\right )}{3 \left (x^3 (a x+b)\right )^{3/4}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.41, size = 82, normalized size = 1.00 \begin {gather*} \sqrt [4]{b x^3+a x^4}+\frac {7 b \tan ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b x^3+a x^4}}\right )}{2 a^{3/4}}-\frac {7 b \tanh ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b x^3+a x^4}}\right )}{2 a^{3/4}} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.49, size = 207, normalized size = 2.52 \begin {gather*} 7 \, \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 {7}{4} \, \left (\frac {b^{4}}{a^{3}}\right )^{\frac {1}{4}} \log \left (\frac {7 \, {\left (a \left (\frac {b^{4}}{a^{3}}\right )^{\frac {1}{4}} x + {\left (a x^{4} + b x^{3}\right )}^{\frac {1}{4}} b\right )}}{x}\right ) + \frac {7}{4} \, \left (\frac {b^{4}}{a^{3}}\right )^{\frac {1}{4}} \log \left (-\frac {7 \, {\left (a \left (\frac {b^{4}}{a^{3}}\right )^{\frac {1}{4}} x - {\left (a x^{4} + b x^{3}\right )}^{\frac {1}{4}} b\right )}}{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 = 0.38, size = 207, normalized size = 2.52 \begin {gather*} \frac {\frac {14 \, \sqrt {2} 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 )}{\left (-a\right )^{\frac {3}{4}}} + \frac {14 \, \sqrt {2} 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 )}{\left (-a\right )^{\frac {3}{4}}} + \frac {7 \, \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}}} + \frac {7 \, \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} + 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 = 0.01, size = 0, normalized size = 0.00 \[\int \frac {\left (a x -b \right ) \left (a \,x^{4}+b \,x^{3}\right )^{\frac {1}{4}}}{x \left (a x +b \right )}\, dx\]
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}} {\left (a x - b\right )}}{{\left (a x + b\right )} 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}\,\left (b-a\,x\right )}{x\,\left (b+a\,x\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 {\sqrt [4]{x^{3} \left (a x + b\right )} \left (a x - b\right )}{x \left (a x + b\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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