Optimal. Leaf size=112 \[ -\sqrt {2} \tan ^{-1}\left (\frac {\sqrt {2} x \sqrt [4]{a x^5+b x^3}}{\sqrt {a x^5+b x^3}-x^2}\right )-\sqrt {2} \tanh ^{-1}\left (\frac {\frac {\sqrt {a x^5+b x^3}}{\sqrt {2}}+\frac {x^2}{\sqrt {2}}}{x \sqrt [4]{a x^5+b x^3}}\right ) \]
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Rubi [C] time = 1.96, antiderivative size = 392, normalized size of antiderivative = 3.50, number of steps used = 21, number of rules used = 10, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {2056, 6728, 365, 364, 959, 466, 430, 429, 511, 510} \begin {gather*} \frac {8 a x^2 \sqrt [4]{\frac {a x^2}{b}+1} F_1\left (\frac {5}{8};1,\frac {1}{4};\frac {13}{8};\frac {4 a^2 x^2}{\left (1-\sqrt {1-4 a b}\right )^2},-\frac {a x^2}{b}\right )}{5 \left (1-\sqrt {1-4 a b}\right ) \sqrt [4]{a x^5+b x^3}}+\frac {8 a x^2 \sqrt [4]{\frac {a x^2}{b}+1} F_1\left (\frac {5}{8};1,\frac {1}{4};\frac {13}{8};\frac {4 a^2 x^2}{\left (\sqrt {1-4 a b}+1\right )^2},-\frac {a x^2}{b}\right )}{5 \left (\sqrt {1-4 a b}+1\right ) \sqrt [4]{a x^5+b x^3}}-\frac {4 x \sqrt [4]{\frac {a x^2}{b}+1} F_1\left (\frac {1}{8};1,\frac {1}{4};\frac {9}{8};\frac {4 a^2 x^2}{\left (1-\sqrt {1-4 a b}\right )^2},-\frac {a x^2}{b}\right )}{\sqrt [4]{a x^5+b x^3}}-\frac {4 x \sqrt [4]{\frac {a x^2}{b}+1} F_1\left (\frac {1}{8};1,\frac {1}{4};\frac {9}{8};\frac {4 a^2 x^2}{\left (\sqrt {1-4 a b}+1\right )^2},-\frac {a x^2}{b}\right )}{\sqrt [4]{a x^5+b x^3}}+\frac {4 x \sqrt [4]{\frac {a x^2}{b}+1} \, _2F_1\left (\frac {1}{8},\frac {1}{4};\frac {9}{8};-\frac {a x^2}{b}\right )}{\sqrt [4]{a x^5+b x^3}} \end {gather*}
Warning: Unable to verify antiderivative.
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Rule 364
Rule 365
Rule 429
Rule 430
Rule 466
Rule 510
Rule 511
Rule 959
Rule 2056
Rule 6728
Rubi steps
\begin {align*} \int \frac {-b+a x^2}{\left (b+x+a x^2\right ) \sqrt [4]{b x^3+a x^5}} \, dx &=\frac {\left (x^{3/4} \sqrt [4]{b+a x^2}\right ) \int \frac {-b+a x^2}{x^{3/4} \sqrt [4]{b+a x^2} \left (b+x+a x^2\right )} \, dx}{\sqrt [4]{b x^3+a x^5}}\\ &=\frac {\left (x^{3/4} \sqrt [4]{b+a x^2}\right ) \int \left (\frac {1}{x^{3/4} \sqrt [4]{b+a x^2}}-\frac {2 b+x}{x^{3/4} \sqrt [4]{b+a x^2} \left (b+x+a x^2\right )}\right ) \, dx}{\sqrt [4]{b x^3+a x^5}}\\ &=\frac {\left (x^{3/4} \sqrt [4]{b+a x^2}\right ) \int \frac {1}{x^{3/4} \sqrt [4]{b+a x^2}} \, dx}{\sqrt [4]{b x^3+a x^5}}-\frac {\left (x^{3/4} \sqrt [4]{b+a x^2}\right ) \int \frac {2 b+x}{x^{3/4} \sqrt [4]{b+a x^2} \left (b+x+a x^2\right )} \, dx}{\sqrt [4]{b x^3+a x^5}}\\ &=-\frac {\left (x^{3/4} \sqrt [4]{b+a x^2}\right ) \int \left (\frac {1-\sqrt {1-4 a b}}{x^{3/4} \left (1-\sqrt {1-4 a b}+2 a x\right ) \sqrt [4]{b+a x^2}}+\frac {1+\sqrt {1-4 a b}}{x^{3/4} \left (1+\sqrt {1-4 a b}+2 a x\right ) \sqrt [4]{b+a x^2}}\right ) \, dx}{\sqrt [4]{b x^3+a x^5}}+\frac {\left (x^{3/4} \sqrt [4]{1+\frac {a x^2}{b}}\right ) \int \frac {1}{x^{3/4} \sqrt [4]{1+\frac {a x^2}{b}}} \, dx}{\sqrt [4]{b x^3+a x^5}}\\ &=\frac {4 x \sqrt [4]{1+\frac {a x^2}{b}} \, _2F_1\left (\frac {1}{8},\frac {1}{4};\frac {9}{8};-\frac {a x^2}{b}\right )}{\sqrt [4]{b x^3+a x^5}}-\frac {\left (\left (1-\sqrt {1-4 a b}\right ) x^{3/4} \sqrt [4]{b+a x^2}\right ) \int \frac {1}{x^{3/4} \left (1-\sqrt {1-4 a b}+2 a x\right ) \sqrt [4]{b+a x^2}} \, dx}{\sqrt [4]{b x^3+a x^5}}-\frac {\left (\left (1+\sqrt {1-4 a b}\right ) x^{3/4} \sqrt [4]{b+a x^2}\right ) \int \frac {1}{x^{3/4} \left (1+\sqrt {1-4 a b}+2 a x\right ) \sqrt [4]{b+a x^2}} \, dx}{\sqrt [4]{b x^3+a x^5}}\\ &=\frac {4 x \sqrt [4]{1+\frac {a x^2}{b}} \, _2F_1\left (\frac {1}{8},\frac {1}{4};\frac {9}{8};-\frac {a x^2}{b}\right )}{\sqrt [4]{b x^3+a x^5}}+\frac {\left (2 a \left (1-\sqrt {1-4 a b}\right ) x^{3/4} \sqrt [4]{b+a x^2}\right ) \int \frac {\sqrt [4]{x}}{\sqrt [4]{b+a x^2} \left (\left (1-\sqrt {1-4 a b}\right )^2-4 a^2 x^2\right )} \, dx}{\sqrt [4]{b x^3+a x^5}}-\frac {\left (\left (1-\sqrt {1-4 a b}\right )^2 x^{3/4} \sqrt [4]{b+a x^2}\right ) \int \frac {1}{x^{3/4} \sqrt [4]{b+a x^2} \left (\left (1-\sqrt {1-4 a b}\right )^2-4 a^2 x^2\right )} \, dx}{\sqrt [4]{b x^3+a x^5}}+\frac {\left (2 a \left (1+\sqrt {1-4 a b}\right ) x^{3/4} \sqrt [4]{b+a x^2}\right ) \int \frac {\sqrt [4]{x}}{\sqrt [4]{b+a x^2} \left (\left (1+\sqrt {1-4 a b}\right )^2-4 a^2 x^2\right )} \, dx}{\sqrt [4]{b x^3+a x^5}}-\frac {\left (\left (1+\sqrt {1-4 a b}\right )^2 x^{3/4} \sqrt [4]{b+a x^2}\right ) \int \frac {1}{x^{3/4} \sqrt [4]{b+a x^2} \left (\left (1+\sqrt {1-4 a b}\right )^2-4 a^2 x^2\right )} \, dx}{\sqrt [4]{b x^3+a x^5}}\\ &=\frac {4 x \sqrt [4]{1+\frac {a x^2}{b}} \, _2F_1\left (\frac {1}{8},\frac {1}{4};\frac {9}{8};-\frac {a x^2}{b}\right )}{\sqrt [4]{b x^3+a x^5}}+\frac {\left (8 a \left (1-\sqrt {1-4 a b}\right ) x^{3/4} \sqrt [4]{b+a x^2}\right ) \operatorname {Subst}\left (\int \frac {x^4}{\sqrt [4]{b+a x^8} \left (\left (1-\sqrt {1-4 a b}\right )^2-4 a^2 x^8\right )} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{b x^3+a x^5}}-\frac {\left (4 \left (1-\sqrt {1-4 a b}\right )^2 x^{3/4} \sqrt [4]{b+a x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [4]{b+a x^8} \left (\left (1-\sqrt {1-4 a b}\right )^2-4 a^2 x^8\right )} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{b x^3+a x^5}}+\frac {\left (8 a \left (1+\sqrt {1-4 a b}\right ) x^{3/4} \sqrt [4]{b+a x^2}\right ) \operatorname {Subst}\left (\int \frac {x^4}{\sqrt [4]{b+a x^8} \left (\left (1+\sqrt {1-4 a b}\right )^2-4 a^2 x^8\right )} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{b x^3+a x^5}}-\frac {\left (4 \left (1+\sqrt {1-4 a b}\right )^2 x^{3/4} \sqrt [4]{b+a x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [4]{b+a x^8} \left (\left (1+\sqrt {1-4 a b}\right )^2-4 a^2 x^8\right )} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{b x^3+a x^5}}\\ &=\frac {4 x \sqrt [4]{1+\frac {a x^2}{b}} \, _2F_1\left (\frac {1}{8},\frac {1}{4};\frac {9}{8};-\frac {a x^2}{b}\right )}{\sqrt [4]{b x^3+a x^5}}+\frac {\left (8 a \left (1-\sqrt {1-4 a b}\right ) x^{3/4} \sqrt [4]{1+\frac {a x^2}{b}}\right ) \operatorname {Subst}\left (\int \frac {x^4}{\left (\left (1-\sqrt {1-4 a b}\right )^2-4 a^2 x^8\right ) \sqrt [4]{1+\frac {a x^8}{b}}} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{b x^3+a x^5}}-\frac {\left (4 \left (1-\sqrt {1-4 a b}\right )^2 x^{3/4} \sqrt [4]{1+\frac {a x^2}{b}}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (\left (1-\sqrt {1-4 a b}\right )^2-4 a^2 x^8\right ) \sqrt [4]{1+\frac {a x^8}{b}}} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{b x^3+a x^5}}+\frac {\left (8 a \left (1+\sqrt {1-4 a b}\right ) x^{3/4} \sqrt [4]{1+\frac {a x^2}{b}}\right ) \operatorname {Subst}\left (\int \frac {x^4}{\left (\left (1+\sqrt {1-4 a b}\right )^2-4 a^2 x^8\right ) \sqrt [4]{1+\frac {a x^8}{b}}} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{b x^3+a x^5}}-\frac {\left (4 \left (1+\sqrt {1-4 a b}\right )^2 x^{3/4} \sqrt [4]{1+\frac {a x^2}{b}}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (\left (1+\sqrt {1-4 a b}\right )^2-4 a^2 x^8\right ) \sqrt [4]{1+\frac {a x^8}{b}}} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{b x^3+a x^5}}\\ &=-\frac {4 x \sqrt [4]{1+\frac {a x^2}{b}} F_1\left (\frac {1}{8};1,\frac {1}{4};\frac {9}{8};\frac {4 a^2 x^2}{\left (1-\sqrt {1-4 a b}\right )^2},-\frac {a x^2}{b}\right )}{\sqrt [4]{b x^3+a x^5}}-\frac {4 x \sqrt [4]{1+\frac {a x^2}{b}} F_1\left (\frac {1}{8};1,\frac {1}{4};\frac {9}{8};\frac {4 a^2 x^2}{\left (1+\sqrt {1-4 a b}\right )^2},-\frac {a x^2}{b}\right )}{\sqrt [4]{b x^3+a x^5}}+\frac {8 a x^2 \sqrt [4]{1+\frac {a x^2}{b}} F_1\left (\frac {5}{8};1,\frac {1}{4};\frac {13}{8};\frac {4 a^2 x^2}{\left (1-\sqrt {1-4 a b}\right )^2},-\frac {a x^2}{b}\right )}{5 \left (1-\sqrt {1-4 a b}\right ) \sqrt [4]{b x^3+a x^5}}+\frac {8 a x^2 \sqrt [4]{1+\frac {a x^2}{b}} F_1\left (\frac {5}{8};1,\frac {1}{4};\frac {13}{8};\frac {4 a^2 x^2}{\left (1+\sqrt {1-4 a b}\right )^2},-\frac {a x^2}{b}\right )}{5 \left (1+\sqrt {1-4 a b}\right ) \sqrt [4]{b x^3+a x^5}}+\frac {4 x \sqrt [4]{1+\frac {a x^2}{b}} \, _2F_1\left (\frac {1}{8},\frac {1}{4};\frac {9}{8};-\frac {a x^2}{b}\right )}{\sqrt [4]{b x^3+a x^5}}\\ \end {align*}
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Mathematica [F] time = 1.13, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {-b+a x^2}{\left (b+x+a x^2\right ) \sqrt [4]{b x^3+a x^5}} \, dx \end {gather*}
Verification is not applicable to the result.
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IntegrateAlgebraic [A] time = 2.87, size = 112, normalized size = 1.00 \begin {gather*} -\sqrt {2} \tan ^{-1}\left (\frac {\sqrt {2} x \sqrt [4]{a x^5+b x^3}}{\sqrt {a x^5+b x^3}-x^2}\right )-\sqrt {2} \tanh ^{-1}\left (\frac {\frac {\sqrt {a x^5+b x^3}}{\sqrt {2}}+\frac {x^2}{\sqrt {2}}}{x \sqrt [4]{a x^5+b x^3}}\right ) \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 [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {a x^{2} - b}{{\left (a x^{5} + b x^{3}\right )}^{\frac {1}{4}} {\left (a x^{2} + b + x\right )}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.45, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {a \,x^{2}-b}{\left (a \,x^{2}+b +x \right ) \left (a \,x^{5}+b \,x^{3}\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^{2} - b}{{\left (a x^{5} + b x^{3}\right )}^{\frac {1}{4}} {\left (a x^{2} + b + x\right )}}\,{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 {b-a\,x^2}{{\left (a\,x^5+b\,x^3\right )}^{1/4}\,\left (a\,x^2+x+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^{2} - b}{\sqrt [4]{x^{3} \left (a x^{2} + b\right )} \left (a x^{2} + b + x\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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