Optimal. Leaf size=116 \[ -\sqrt {2} \text {ArcTan}\left (\frac {\sqrt {2} x \sqrt [4]{-b x^3+a x^5}}{-x^2+\sqrt {-b x^3+a x^5}}\right )-\sqrt {2} \tanh ^{-1}\left (\frac {\frac {x^2}{\sqrt {2}}+\frac {\sqrt {-b x^3+a x^5}}{\sqrt {2}}}{x \sqrt [4]{-b x^3+a x^5}}\right ) \]
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Rubi [C] Result contains higher order function than in optimal. Order 6 vs. order 3 in
optimal.
time = 1.38, antiderivative size = 397, normalized size of antiderivative = 3.42, number of steps
used = 21, number of rules used = 10, integrand size = 36, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.278, Rules used = {2081, 6860,
372, 371, 973, 477, 441, 440, 525, 524} \begin {gather*} \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 {4 a b+1}\right )^2},\frac {a x^2}{b}\right )}{5 \left (1-\sqrt {4 a b+1}\right ) \sqrt [4]{a x^5-b x^3}}+\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 (\sqrt {4 a b+1}+1\right )^2},\frac {a x^2}{b}\right )}{5 \left (\sqrt {4 a b+1}+1\right ) \sqrt [4]{a x^5-b x^3}}-\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 {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]{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 (\sqrt {4 a b+1}+1\right )^2},\frac {a x^2}{b}\right )}{\sqrt [4]{a x^5-b x^3}}+\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]{a x^5-b x^3}} \end {gather*}
Antiderivative was successfully verified.
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Rule 371
Rule 372
Rule 440
Rule 441
Rule 477
Rule 524
Rule 525
Rule 973
Rule 2081
Rule 6860
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 (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 ) \left (-1+\sqrt {1+4 a b}\right ) 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 (\left (-1-\sqrt {1+4 a b}\right ) \left (1+\sqrt {1+4 a b}\right ) 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 ) \text {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 (8 a \left (-1+\sqrt {1+4 a b}\right ) x^{3/4} \sqrt [4]{-b+a x^2}\right ) \text {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 ) \left (-1+\sqrt {1+4 a b}\right ) x^{3/4} \sqrt [4]{-b+a x^2}\right ) \text {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 (4 \left (-1-\sqrt {1+4 a b}\right ) \left (1+\sqrt {1+4 a b}\right ) x^{3/4} \sqrt [4]{-b+a x^2}\right ) \text {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 ) \text {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 (8 a \left (-1+\sqrt {1+4 a b}\right ) x^{3/4} \sqrt [4]{1-\frac {a x^2}{b}}\right ) \text {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 ) \left (-1+\sqrt {1+4 a b}\right ) x^{3/4} \sqrt [4]{1-\frac {a x^2}{b}}\right ) \text {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 (4 \left (-1-\sqrt {1+4 a b}\right ) \left (1+\sqrt {1+4 a b}\right ) x^{3/4} \sqrt [4]{1-\frac {a x^2}{b}}\right ) \text {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 = 10.83, 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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Maple [F]
time = 0.10, size = 0, normalized size = 0.00 \[\int \frac {a \,x^{2}+b}{\left (a \,x^{2}-b +x \right ) \left (a \,x^{5}-b \,x^{3}\right )^{\frac {1}{4}}}\, dx\]
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*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-1)] Timed out
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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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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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \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^2+b}{{\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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