Optimal. Leaf size=126 \[ -\frac {2\ 2^{3/4} a^{3/4} \tan ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{a} \left (a x^4+b x\right )^{3/4}}{a x^3+b}\right )}{3 b}-\frac {2\ 2^{3/4} a^{3/4} \tanh ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{a} \left (a x^4+b x\right )^{3/4}}{a x^3+b}\right )}{3 b}+\frac {4 \left (a x^4+b x\right )^{3/4}}{9 b x^3} \]
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Rubi [C] time = 0.38, antiderivative size = 53, normalized size of antiderivative = 0.42, number of steps used = 5, number of rules used = 5, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {2056, 466, 465, 511, 510} \begin {gather*} \frac {4 \left (a x^3+b\right ) \, _2F_1\left (-\frac {3}{4},1;\frac {1}{4};\frac {2 a x^3}{a x^3+b}\right )}{9 b x^2 \sqrt [4]{a x^4+b x}} \end {gather*}
Antiderivative was successfully verified.
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Rule 465
Rule 466
Rule 510
Rule 511
Rule 2056
Rubi steps
\begin {align*} \int \frac {b+a x^3}{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 {\left (b+a x^3\right )^{3/4}}{x^{13/4} \left (-b+a x^3\right )} \, dx}{\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 {\left (b+a x^{12}\right )^{3/4}}{x^{10} \left (-b+a x^{12}\right )} \, dx,x,\sqrt [4]{x}\right )}{\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 {\left (b+a x^4\right )^{3/4}}{x^4 \left (-b+a x^4\right )} \, dx,x,x^{3/4}\right )}{3 \sqrt [4]{b x+a x^4}}\\ &=\frac {\left (4 \sqrt [4]{x} \left (b+a x^3\right )\right ) \operatorname {Subst}\left (\int \frac {\left (1+\frac {a x^4}{b}\right )^{3/4}}{x^4 \left (-b+a x^4\right )} \, dx,x,x^{3/4}\right )}{3 \left (1+\frac {a x^3}{b}\right )^{3/4} \sqrt [4]{b x+a x^4}}\\ &=\frac {4 \left (b+a x^3\right ) \, _2F_1\left (-\frac {3}{4},1;\frac {1}{4};\frac {2 a x^3}{b+a x^3}\right )}{9 b x^2 \sqrt [4]{b x+a x^4}}\\ \end {align*}
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Mathematica [C] time = 0.03, size = 46, normalized size = 0.37 \begin {gather*} \frac {4 \left (x \left (a x^3+b\right )\right )^{3/4} \, _2F_1\left (-\frac {3}{4},1;\frac {1}{4};\frac {2 a x^3}{a x^3+b}\right )}{9 b x^3} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.44, size = 126, normalized size = 1.00 \begin {gather*} \frac {4 \left (b x+a x^4\right )^{3/4}}{9 b x^3}-\frac {2\ 2^{3/4} a^{3/4} \tan ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{a} \left (b x+a x^4\right )^{3/4}}{b+a x^3}\right )}{3 b}-\frac {2\ 2^{3/4} a^{3/4} \tanh ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{a} \left (b x+a x^4\right )^{3/4}}{b+a x^3}\right )}{3 b} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 89.33, size = 479, normalized size = 3.80 \begin {gather*} \frac {12 \cdot 8^{\frac {1}{4}} b x^{3} \left (\frac {a^{3}}{b^{4}}\right )^{\frac {1}{4}} \arctan \left (\frac {16 \cdot 8^{\frac {1}{4}} {\left (a x^{4} + b x\right )}^{\frac {1}{4}} a^{4} b x^{2} \left (\frac {a^{3}}{b^{4}}\right )^{\frac {1}{4}} + 4 \cdot 8^{\frac {3}{4}} {\left (a x^{4} + b x\right )}^{\frac {3}{4}} a^{2} b^{3} \left (\frac {a^{3}}{b^{4}}\right )^{\frac {3}{4}} + \sqrt {2} {\left (8 \cdot 8^{\frac {1}{4}} \sqrt {a x^{4} + b x} a^{2} b x \left (\frac {a^{3}}{b^{4}}\right )^{\frac {1}{4}} + 8^{\frac {3}{4}} {\left (3 \, a b^{3} x^{3} + b^{4}\right )} \left (\frac {a^{3}}{b^{4}}\right )^{\frac {3}{4}}\right )} \sqrt {\sqrt {2} a^{2} b^{2} \sqrt {\frac {a^{3}}{b^{4}}}}}{8 \, {\left (a^{5} x^{3} - a^{4} b\right )}}\right ) - 3 \cdot 8^{\frac {1}{4}} b x^{3} \left (\frac {a^{3}}{b^{4}}\right )^{\frac {1}{4}} \log \left (\frac {4 \, \sqrt {2} {\left (a x^{4} + b x\right )}^{\frac {1}{4}} a b^{2} x^{2} \sqrt {\frac {a^{3}}{b^{4}}} + 8^{\frac {3}{4}} \sqrt {a x^{4} + b x} b^{3} x \left (\frac {a^{3}}{b^{4}}\right )^{\frac {3}{4}} + 4 \, {\left (a x^{4} + b x\right )}^{\frac {3}{4}} a^{2} + 8^{\frac {1}{4}} {\left (3 \, a^{2} b x^{3} + a b^{2}\right )} \left (\frac {a^{3}}{b^{4}}\right )^{\frac {1}{4}}}{a x^{3} - b}\right ) + 3 \cdot 8^{\frac {1}{4}} b x^{3} \left (\frac {a^{3}}{b^{4}}\right )^{\frac {1}{4}} \log \left (\frac {4 \, \sqrt {2} {\left (a x^{4} + b x\right )}^{\frac {1}{4}} a b^{2} x^{2} \sqrt {\frac {a^{3}}{b^{4}}} - 8^{\frac {3}{4}} \sqrt {a x^{4} + b x} b^{3} x \left (\frac {a^{3}}{b^{4}}\right )^{\frac {3}{4}} + 4 \, {\left (a x^{4} + b x\right )}^{\frac {3}{4}} a^{2} - 8^{\frac {1}{4}} {\left (3 \, a^{2} b x^{3} + a b^{2}\right )} \left (\frac {a^{3}}{b^{4}}\right )^{\frac {1}{4}}}{a x^{3} - b}\right ) + 8 \, {\left (a x^{4} + b x\right )}^{\frac {3}{4}}}{18 \, b x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.40, size = 208, normalized size = 1.65 \begin {gather*} -\frac {2 \cdot 2^{\frac {1}{4}} \left (-a\right )^{\frac {3}{4}} \arctan \left (\frac {2^{\frac {1}{4}} {\left (2^{\frac {3}{4}} \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 \, b} - \frac {2 \cdot 2^{\frac {1}{4}} \left (-a\right )^{\frac {3}{4}} \arctan \left (-\frac {2^{\frac {1}{4}} {\left (2^{\frac {3}{4}} \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 \, b} + \frac {2^{\frac {1}{4}} \left (-a\right )^{\frac {3}{4}} \log \left (2^{\frac {3}{4}} \left (-a\right )^{\frac {1}{4}} {\left (a + \frac {b}{x^{3}}\right )}^{\frac {1}{4}} + \sqrt {2} \sqrt {-a} + \sqrt {a + \frac {b}{x^{3}}}\right )}{3 \, b} - \frac {2^{\frac {1}{4}} \left (-a\right )^{\frac {3}{4}} \log \left (-2^{\frac {3}{4}} \left (-a\right )^{\frac {1}{4}} {\left (a + \frac {b}{x^{3}}\right )}^{\frac {1}{4}} + \sqrt {2} \sqrt {-a} + \sqrt {a + \frac {b}{x^{3}}}\right )}{3 \, 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 = 0.01, size = 0, normalized size = 0.00 \[\int \frac {a \,x^{3}+b}{x^{3} \left (a \,x^{3}-b \right ) \left (a \,x^{4}+b x \right )^{\frac {1}{4}}}\, dx\]
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 {a x^{3} + 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^3+b}{x^3\,{\left (a\,x^4+b\,x\right )}^{1/4}\,\left (b-a\,x^3\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^{3} + 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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