Optimal. Leaf size=123 \[ -\frac {4 \left (a x^4+b x\right )^{3/4} \left (4 a x^3+3 b x^3+b\right )}{9 b x^3 \left (a x^3+b\right )}+\frac {2 \tan ^{-1}\left (\frac {\sqrt [4]{a} \left (a x^4+b x\right )^{3/4}}{a x^3+b}\right )}{3 \sqrt [4]{a}}+\frac {2 \tanh ^{-1}\left (\frac {\sqrt [4]{a} \left (a x^4+b x\right )^{3/4}}{a x^3+b}\right )}{3 \sqrt [4]{a}} \]
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Rubi [A] time = 0.37, antiderivative size = 183, normalized size of antiderivative = 1.49, number of steps used = 12, number of rules used = 11, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {2056, 1489, 271, 264, 288, 329, 275, 240, 212, 206, 203} \begin {gather*} -\frac {16 a x}{9 b \sqrt [4]{a x^4+b x}}-\frac {4 x}{3 \sqrt [4]{a x^4+b x}}-\frac {4}{9 x^2 \sqrt [4]{a x^4+b x}}+\frac {2 \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} \sqrt [4]{a x^4+b x}}+\frac {2 \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} \sqrt [4]{a x^4+b x}} \end {gather*}
Antiderivative was successfully verified.
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Rule 203
Rule 206
Rule 212
Rule 240
Rule 264
Rule 271
Rule 275
Rule 288
Rule 329
Rule 1489
Rule 2056
Rubi steps
\begin {align*} \int \frac {b+a x^6}{x^3 \left (b+a x^3\right ) \sqrt [4]{b x+a x^4}} \, dx &=\frac {\left (\sqrt [4]{x} \sqrt [4]{b+a x^3}\right ) \int \frac {b+a x^6}{x^{13/4} \left (b+a x^3\right )^{5/4}} \, dx}{\sqrt [4]{b x+a x^4}}\\ &=\frac {\left (\sqrt [4]{x} \sqrt [4]{b+a x^3}\right ) \int \left (\frac {b}{x^{13/4} \left (b+a x^3\right )^{5/4}}+\frac {a x^{11/4}}{\left (b+a x^3\right )^{5/4}}\right ) \, dx}{\sqrt [4]{b x+a x^4}}\\ &=\frac {\left (a \sqrt [4]{x} \sqrt [4]{b+a x^3}\right ) \int \frac {x^{11/4}}{\left (b+a x^3\right )^{5/4}} \, dx}{\sqrt [4]{b x+a x^4}}+\frac {\left (b \sqrt [4]{x} \sqrt [4]{b+a x^3}\right ) \int \frac {1}{x^{13/4} \left (b+a x^3\right )^{5/4}} \, dx}{\sqrt [4]{b x+a x^4}}\\ &=-\frac {4}{9 x^2 \sqrt [4]{b x+a x^4}}-\frac {4 x}{3 \sqrt [4]{b x+a x^4}}+\frac {\left (\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 {\left (4 a \sqrt [4]{x} \sqrt [4]{b+a x^3}\right ) \int \frac {1}{\sqrt [4]{x} \left (b+a x^3\right )^{5/4}} \, dx}{3 \sqrt [4]{b x+a x^4}}\\ &=-\frac {4}{9 x^2 \sqrt [4]{b x+a x^4}}-\frac {4 x}{3 \sqrt [4]{b x+a x^4}}-\frac {16 a x}{9 b \sqrt [4]{b x+a x^4}}+\frac {\left (4 \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}{9 x^2 \sqrt [4]{b x+a x^4}}-\frac {4 x}{3 \sqrt [4]{b x+a x^4}}-\frac {16 a x}{9 b \sqrt [4]{b x+a x^4}}+\frac {\left (4 \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}{9 x^2 \sqrt [4]{b x+a x^4}}-\frac {4 x}{3 \sqrt [4]{b x+a x^4}}-\frac {16 a x}{9 b \sqrt [4]{b x+a x^4}}+\frac {\left (4 \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}{9 x^2 \sqrt [4]{b x+a x^4}}-\frac {4 x}{3 \sqrt [4]{b x+a x^4}}-\frac {16 a x}{9 b \sqrt [4]{b x+a x^4}}+\frac {\left (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 (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}{9 x^2 \sqrt [4]{b x+a x^4}}-\frac {4 x}{3 \sqrt [4]{b x+a x^4}}-\frac {16 a x}{9 b \sqrt [4]{b x+a x^4}}+\frac {2 \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]{a} \sqrt [4]{b x+a x^4}}+\frac {2 \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]{a} \sqrt [4]{b x+a x^4}}\\ \end {align*}
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Mathematica [C] time = 0.12, size = 73, normalized size = 0.59 \begin {gather*} \frac {4 \left (3 a x^6 \sqrt [4]{\frac {a x^3}{b}+1} \, _2F_1\left (\frac {5}{4},\frac {5}{4};\frac {9}{4};-\frac {a x^3}{b}\right )-5 \left (4 a x^3+b\right )\right )}{45 b x^2 \sqrt [4]{x \left (a x^3+b\right )}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.47, size = 123, normalized size = 1.00 \begin {gather*} -\frac {4 \left (a x^4+b x\right )^{3/4} \left (4 a x^3+3 b x^3+b\right )}{9 b x^3 \left (a x^3+b\right )}+\frac {2 \tan ^{-1}\left (\frac {\sqrt [4]{a} \left (a x^4+b x\right )^{3/4}}{a x^3+b}\right )}{3 \sqrt [4]{a}}+\frac {2 \tanh ^{-1}\left (\frac {\sqrt [4]{a} \left (a x^4+b x\right )^{3/4}}{a x^3+b}\right )}{3 \sqrt [4]{a}} \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 = 0.39, size = 217, normalized size = 1.76 \begin {gather*} \frac {\sqrt {2} \left (-a\right )^{\frac {3}{4}} \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 )}{3 \, a} + \frac {\sqrt {2} \left (-a\right )^{\frac {3}{4}} \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 )}{3 \, a} - \frac {\sqrt {2} \left (-a\right )^{\frac {3}{4}} \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 )}{6 \, a} + \frac {\sqrt {2} \left (-a\right )^{\frac {3}{4}} \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 )}{6 \, a} - \frac {4 \, {\left (a + b\right )}}{3 \, {\left (a + \frac {b}{x^{3}}\right )}^{\frac {1}{4}} b} - \frac {4 \, {\left (a + \frac {b}{x^{3}}\right )}^{\frac {3}{4}}}{9 \, b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.35, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {a \,x^{6}+b}{x^{3} \left (a \,x^{3}+b \right ) \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 {a x^{6} + b}{{\left (a x^{4} + b x\right )}^{\frac {1}{4}} {\left (a x^{3} + b\right )} x^{3}}\,{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 {a\,x^6+b}{x^3\,{\left (a\,x^4+b\,x\right )}^{1/4}\,\left (a\,x^3+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 {a x^{6} + b}{x^{3} \sqrt [4]{x \left (a x^{3} + b\right )} \left (a x^{3} + b\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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