Optimal. Leaf size=134 \[ \frac {4389 b^5 \tan ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4+b x^3}}\right )}{4096 a^{23/4}}-\frac {4389 b^5 \tanh ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4+b x^3}}\right )}{4096 a^{23/4}}+\frac {\sqrt [4]{a x^4+b x^3} \left (2048 a^4 x^4-2432 a^3 b x^3+3040 a^2 b^2 x^2-4180 a b^3 x+7315 b^4\right )}{10240 a^5} \]
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Rubi [A] time = 0.32, antiderivative size = 255, normalized size of antiderivative = 1.90, number of steps used = 11, number of rules used = 7, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.269, Rules used = {2056, 50, 63, 331, 298, 203, 206} \begin {gather*} \frac {4389 b^5 \sqrt [4]{a x^4+b x^3} \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{a x+b}}\right )}{4096 a^{23/4} x^{3/4} \sqrt [4]{a x+b}}-\frac {4389 b^5 \sqrt [4]{a x^4+b x^3} \tanh ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{a x+b}}\right )}{4096 a^{23/4} x^{3/4} \sqrt [4]{a x+b}}+\frac {1463 b^4 \sqrt [4]{a x^4+b x^3}}{2048 a^5}-\frac {209 b^3 x \sqrt [4]{a x^4+b x^3}}{512 a^4}+\frac {19 b^2 x^2 \sqrt [4]{a x^4+b x^3}}{64 a^3}-\frac {19 b x^3 \sqrt [4]{a x^4+b x^3}}{80 a^2}+\frac {x^4 \sqrt [4]{a x^4+b x^3}}{5 a} \end {gather*}
Antiderivative was successfully verified.
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Rule 50
Rule 63
Rule 203
Rule 206
Rule 298
Rule 331
Rule 2056
Rubi steps
\begin {align*} \int \frac {x^4 \sqrt [4]{b x^3+a x^4}}{b+a x} \, dx &=\frac {\sqrt [4]{b x^3+a x^4} \int \frac {x^{19/4}}{(b+a x)^{3/4}} \, dx}{x^{3/4} \sqrt [4]{b+a x}}\\ &=\frac {x^4 \sqrt [4]{b x^3+a x^4}}{5 a}-\frac {\left (19 b \sqrt [4]{b x^3+a x^4}\right ) \int \frac {x^{15/4}}{(b+a x)^{3/4}} \, dx}{20 a x^{3/4} \sqrt [4]{b+a x}}\\ &=-\frac {19 b x^3 \sqrt [4]{b x^3+a x^4}}{80 a^2}+\frac {x^4 \sqrt [4]{b x^3+a x^4}}{5 a}+\frac {\left (57 b^2 \sqrt [4]{b x^3+a x^4}\right ) \int \frac {x^{11/4}}{(b+a x)^{3/4}} \, dx}{64 a^2 x^{3/4} \sqrt [4]{b+a x}}\\ &=\frac {19 b^2 x^2 \sqrt [4]{b x^3+a x^4}}{64 a^3}-\frac {19 b x^3 \sqrt [4]{b x^3+a x^4}}{80 a^2}+\frac {x^4 \sqrt [4]{b x^3+a x^4}}{5 a}-\frac {\left (209 b^3 \sqrt [4]{b x^3+a x^4}\right ) \int \frac {x^{7/4}}{(b+a x)^{3/4}} \, dx}{256 a^3 x^{3/4} \sqrt [4]{b+a x}}\\ &=-\frac {209 b^3 x \sqrt [4]{b x^3+a x^4}}{512 a^4}+\frac {19 b^2 x^2 \sqrt [4]{b x^3+a x^4}}{64 a^3}-\frac {19 b x^3 \sqrt [4]{b x^3+a x^4}}{80 a^2}+\frac {x^4 \sqrt [4]{b x^3+a x^4}}{5 a}+\frac {\left (1463 b^4 \sqrt [4]{b x^3+a x^4}\right ) \int \frac {x^{3/4}}{(b+a x)^{3/4}} \, dx}{2048 a^4 x^{3/4} \sqrt [4]{b+a x}}\\ &=\frac {1463 b^4 \sqrt [4]{b x^3+a x^4}}{2048 a^5}-\frac {209 b^3 x \sqrt [4]{b x^3+a x^4}}{512 a^4}+\frac {19 b^2 x^2 \sqrt [4]{b x^3+a x^4}}{64 a^3}-\frac {19 b x^3 \sqrt [4]{b x^3+a x^4}}{80 a^2}+\frac {x^4 \sqrt [4]{b x^3+a x^4}}{5 a}-\frac {\left (4389 b^5 \sqrt [4]{b x^3+a x^4}\right ) \int \frac {1}{\sqrt [4]{x} (b+a x)^{3/4}} \, dx}{8192 a^5 x^{3/4} \sqrt [4]{b+a x}}\\ &=\frac {1463 b^4 \sqrt [4]{b x^3+a x^4}}{2048 a^5}-\frac {209 b^3 x \sqrt [4]{b x^3+a x^4}}{512 a^4}+\frac {19 b^2 x^2 \sqrt [4]{b x^3+a x^4}}{64 a^3}-\frac {19 b x^3 \sqrt [4]{b x^3+a x^4}}{80 a^2}+\frac {x^4 \sqrt [4]{b x^3+a x^4}}{5 a}-\frac {\left (4389 b^5 \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 )}{2048 a^5 x^{3/4} \sqrt [4]{b+a x}}\\ &=\frac {1463 b^4 \sqrt [4]{b x^3+a x^4}}{2048 a^5}-\frac {209 b^3 x \sqrt [4]{b x^3+a x^4}}{512 a^4}+\frac {19 b^2 x^2 \sqrt [4]{b x^3+a x^4}}{64 a^3}-\frac {19 b x^3 \sqrt [4]{b x^3+a x^4}}{80 a^2}+\frac {x^4 \sqrt [4]{b x^3+a x^4}}{5 a}-\frac {\left (4389 b^5 \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 )}{2048 a^5 x^{3/4} \sqrt [4]{b+a x}}\\ &=\frac {1463 b^4 \sqrt [4]{b x^3+a x^4}}{2048 a^5}-\frac {209 b^3 x \sqrt [4]{b x^3+a x^4}}{512 a^4}+\frac {19 b^2 x^2 \sqrt [4]{b x^3+a x^4}}{64 a^3}-\frac {19 b x^3 \sqrt [4]{b x^3+a x^4}}{80 a^2}+\frac {x^4 \sqrt [4]{b x^3+a x^4}}{5 a}-\frac {\left (4389 b^5 \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 )}{4096 a^{11/2} x^{3/4} \sqrt [4]{b+a x}}+\frac {\left (4389 b^5 \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 )}{4096 a^{11/2} x^{3/4} \sqrt [4]{b+a x}}\\ &=\frac {1463 b^4 \sqrt [4]{b x^3+a x^4}}{2048 a^5}-\frac {209 b^3 x \sqrt [4]{b x^3+a x^4}}{512 a^4}+\frac {19 b^2 x^2 \sqrt [4]{b x^3+a x^4}}{64 a^3}-\frac {19 b x^3 \sqrt [4]{b x^3+a x^4}}{80 a^2}+\frac {x^4 \sqrt [4]{b x^3+a x^4}}{5 a}+\frac {4389 b^5 \sqrt [4]{b x^3+a x^4} \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{4096 a^{23/4} x^{3/4} \sqrt [4]{b+a x}}-\frac {4389 b^5 \sqrt [4]{b x^3+a x^4} \tanh ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{4096 a^{23/4} x^{3/4} \sqrt [4]{b+a x}}\\ \end {align*}
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Mathematica [C] time = 0.02, size = 49, normalized size = 0.37 \begin {gather*} \frac {4 x^8 \left (\frac {a x}{b}+1\right )^{3/4} \, _2F_1\left (\frac {3}{4},\frac {23}{4};\frac {27}{4};-\frac {a x}{b}\right )}{23 \left (x^3 (a x+b)\right )^{3/4}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.55, size = 134, normalized size = 1.00 \begin {gather*} \frac {4389 b^5 \tan ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4+b x^3}}\right )}{4096 a^{23/4}}-\frac {4389 b^5 \tanh ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4+b x^3}}\right )}{4096 a^{23/4}}+\frac {\sqrt [4]{a x^4+b x^3} \left (2048 a^4 x^4-2432 a^3 b x^3+3040 a^2 b^2 x^2-4180 a b^3 x+7315 b^4\right )}{10240 a^5} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.44, size = 275, normalized size = 2.05 \begin {gather*} \frac {87780 \, a^{5} \left (\frac {b^{20}}{a^{23}}\right )^{\frac {1}{4}} \arctan \left (-\frac {{\left (a x^{4} + b x^{3}\right )}^{\frac {1}{4}} a^{17} b^{5} \left (\frac {b^{20}}{a^{23}}\right )^{\frac {3}{4}} - a^{17} x \sqrt {\frac {a^{12} x^{2} \sqrt {\frac {b^{20}}{a^{23}}} + \sqrt {a x^{4} + b x^{3}} b^{10}}{x^{2}}} \left (\frac {b^{20}}{a^{23}}\right )^{\frac {3}{4}}}{b^{20} x}\right ) - 21945 \, a^{5} \left (\frac {b^{20}}{a^{23}}\right )^{\frac {1}{4}} \log \left (\frac {4389 \, {\left (a^{6} x \left (\frac {b^{20}}{a^{23}}\right )^{\frac {1}{4}} + {\left (a x^{4} + b x^{3}\right )}^{\frac {1}{4}} b^{5}\right )}}{x}\right ) + 21945 \, a^{5} \left (\frac {b^{20}}{a^{23}}\right )^{\frac {1}{4}} \log \left (-\frac {4389 \, {\left (a^{6} x \left (\frac {b^{20}}{a^{23}}\right )^{\frac {1}{4}} - {\left (a x^{4} + b x^{3}\right )}^{\frac {1}{4}} b^{5}\right )}}{x}\right ) + 4 \, {\left (2048 \, a^{4} x^{4} - 2432 \, a^{3} b x^{3} + 3040 \, a^{2} b^{2} x^{2} - 4180 \, a b^{3} x + 7315 \, b^{4}\right )} {\left (a x^{4} + b x^{3}\right )}^{\frac {1}{4}}}{40960 \, a^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.74, size = 295, normalized size = 2.20 \begin {gather*} -\frac {\frac {43890 \, \sqrt {2} \left (-a\right )^{\frac {1}{4}} b^{6} \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^{6}} + \frac {43890 \, \sqrt {2} \left (-a\right )^{\frac {1}{4}} b^{6} \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^{6}} + \frac {21945 \, \sqrt {2} \left (-a\right )^{\frac {1}{4}} b^{6} \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^{6}} + \frac {21945 \, \sqrt {2} b^{6} \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}} a^{5}} - \frac {8 \, {\left (7315 \, {\left (a + \frac {b}{x}\right )}^{\frac {17}{4}} b^{6} - 33440 \, {\left (a + \frac {b}{x}\right )}^{\frac {13}{4}} a b^{6} + 59470 \, {\left (a + \frac {b}{x}\right )}^{\frac {9}{4}} a^{2} b^{6} - 50312 \, {\left (a + \frac {b}{x}\right )}^{\frac {5}{4}} a^{3} b^{6} + 19015 \, {\left (a + \frac {b}{x}\right )}^{\frac {1}{4}} a^{4} b^{6}\right )} x^{5}}{a^{5} b^{5}}}{81920 \, 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 {x^{4} \left (a \,x^{4}+b \,x^{3}\right )^{\frac {1}{4}}}{a x +b}\, 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^{4}}{a x + b}\,{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 {x^4\,{\left (a\,x^4+b\,x^3\right )}^{1/4}}{b+a\,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 {x^{4} \sqrt [4]{x^{3} \left (a x + b\right )}}{a x + b}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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