Optimal. Leaf size=344 \[ \frac {x \left (a e+a f x-b c x^2-b d x^3\right )}{2 a^2 \sqrt {a+b x^4}}+\frac {d \sqrt {a+b x^4}}{2 a^2}-\frac {c \sqrt {a+b x^4}}{a^2 x}+\frac {3 \sqrt {b} c x \sqrt {a+b x^4}}{2 a^2 \left (\sqrt {a}+\sqrt {b} x^2\right )}-\frac {d \tanh ^{-1}\left (\frac {\sqrt {a+b x^4}}{\sqrt {a}}\right )}{2 a^{3/2}}-\frac {3 \sqrt [4]{b} c \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{2 a^{7/4} \sqrt {a+b x^4}}+\frac {\left (3 \sqrt {b} c+\sqrt {a} e\right ) \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{4 a^{7/4} \sqrt [4]{b} \sqrt {a+b x^4}} \]
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Rubi [A]
time = 0.25, antiderivative size = 344, normalized size of antiderivative = 1.00, number of steps
used = 13, number of rules used = 12, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {1843, 1847,
1849, 1598, 1212, 226, 1210, 21, 272, 52, 65, 214} \begin {gather*} \frac {\left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} \left (\sqrt {a} e+3 \sqrt {b} c\right ) F\left (2 \text {ArcTan}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{4 a^{7/4} \sqrt [4]{b} \sqrt {a+b x^4}}-\frac {3 \sqrt [4]{b} c \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} E\left (2 \text {ArcTan}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{2 a^{7/4} \sqrt {a+b x^4}}-\frac {d \tanh ^{-1}\left (\frac {\sqrt {a+b x^4}}{\sqrt {a}}\right )}{2 a^{3/2}}+\frac {x \left (a e+a f x-b c x^2-b d x^3\right )}{2 a^2 \sqrt {a+b x^4}}-\frac {c \sqrt {a+b x^4}}{a^2 x}+\frac {3 \sqrt {b} c x \sqrt {a+b x^4}}{2 a^2 \left (\sqrt {a}+\sqrt {b} x^2\right )}+\frac {d \sqrt {a+b x^4}}{2 a^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 21
Rule 52
Rule 65
Rule 214
Rule 226
Rule 272
Rule 1210
Rule 1212
Rule 1598
Rule 1843
Rule 1847
Rule 1849
Rubi steps
\begin {align*} \int \frac {c+d x+e x^2+f x^3}{x^2 \left (a+b x^4\right )^{3/2}} \, dx &=\frac {x \left (a e+a f x-b c x^2-b d x^3\right )}{2 a^2 \sqrt {a+b x^4}}-\frac {\int \frac {-2 b c-2 b d x-b e x^2-\frac {b^2 c x^4}{a}-\frac {2 b^2 d x^5}{a}}{x^2 \sqrt {a+b x^4}} \, dx}{2 a b}\\ &=\frac {x \left (a e+a f x-b c x^2-b d x^3\right )}{2 a^2 \sqrt {a+b x^4}}-\frac {\int \left (\frac {-2 b c-b e x^2-\frac {b^2 c x^4}{a}}{x^2 \sqrt {a+b x^4}}+\frac {-2 b d-\frac {2 b^2 d x^4}{a}}{x \sqrt {a+b x^4}}\right ) \, dx}{2 a b}\\ &=\frac {x \left (a e+a f x-b c x^2-b d x^3\right )}{2 a^2 \sqrt {a+b x^4}}-\frac {\int \frac {-2 b c-b e x^2-\frac {b^2 c x^4}{a}}{x^2 \sqrt {a+b x^4}} \, dx}{2 a b}-\frac {\int \frac {-2 b d-\frac {2 b^2 d x^4}{a}}{x \sqrt {a+b x^4}} \, dx}{2 a b}\\ &=\frac {x \left (a e+a f x-b c x^2-b d x^3\right )}{2 a^2 \sqrt {a+b x^4}}-\frac {c \sqrt {a+b x^4}}{a^2 x}+\frac {\int \frac {2 a b e x+6 b^2 c x^3}{x \sqrt {a+b x^4}} \, dx}{4 a^2 b}+\frac {d \int \frac {\sqrt {a+b x^4}}{x} \, dx}{a^2}\\ &=\frac {x \left (a e+a f x-b c x^2-b d x^3\right )}{2 a^2 \sqrt {a+b x^4}}-\frac {c \sqrt {a+b x^4}}{a^2 x}+\frac {\int \frac {2 a b e+6 b^2 c x^2}{\sqrt {a+b x^4}} \, dx}{4 a^2 b}+\frac {d \text {Subst}\left (\int \frac {\sqrt {a+b x}}{x} \, dx,x,x^4\right )}{4 a^2}\\ &=\frac {x \left (a e+a f x-b c x^2-b d x^3\right )}{2 a^2 \sqrt {a+b x^4}}+\frac {d \sqrt {a+b x^4}}{2 a^2}-\frac {c \sqrt {a+b x^4}}{a^2 x}-\frac {\left (3 \sqrt {b} c\right ) \int \frac {1-\frac {\sqrt {b} x^2}{\sqrt {a}}}{\sqrt {a+b x^4}} \, dx}{2 a^{3/2}}+\frac {d \text {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,x^4\right )}{4 a}+\frac {\left (3 \sqrt {b} c+\sqrt {a} e\right ) \int \frac {1}{\sqrt {a+b x^4}} \, dx}{2 a^{3/2}}\\ &=\frac {x \left (a e+a f x-b c x^2-b d x^3\right )}{2 a^2 \sqrt {a+b x^4}}+\frac {d \sqrt {a+b x^4}}{2 a^2}-\frac {c \sqrt {a+b x^4}}{a^2 x}+\frac {3 \sqrt {b} c x \sqrt {a+b x^4}}{2 a^2 \left (\sqrt {a}+\sqrt {b} x^2\right )}-\frac {3 \sqrt [4]{b} c \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{2 a^{7/4} \sqrt {a+b x^4}}+\frac {\left (3 \sqrt {b} c+\sqrt {a} e\right ) \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{4 a^{7/4} \sqrt [4]{b} \sqrt {a+b x^4}}+\frac {d \text {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b x^4}\right )}{2 a b}\\ &=\frac {x \left (a e+a f x-b c x^2-b d x^3\right )}{2 a^2 \sqrt {a+b x^4}}+\frac {d \sqrt {a+b x^4}}{2 a^2}-\frac {c \sqrt {a+b x^4}}{a^2 x}+\frac {3 \sqrt {b} c x \sqrt {a+b x^4}}{2 a^2 \left (\sqrt {a}+\sqrt {b} x^2\right )}-\frac {d \tanh ^{-1}\left (\frac {\sqrt {a+b x^4}}{\sqrt {a}}\right )}{2 a^{3/2}}-\frac {3 \sqrt [4]{b} c \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{2 a^{7/4} \sqrt {a+b x^4}}+\frac {\left (3 \sqrt {b} c+\sqrt {a} e\right ) \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{4 a^{7/4} \sqrt [4]{b} \sqrt {a+b x^4}}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 10.30, size = 245, normalized size = 0.71 \begin {gather*} \frac {\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}} \left (-2 a c-3 b c x^4+a x (d+x (e+f x))-\sqrt {a} d x \sqrt {a+b x^4} \tanh ^{-1}\left (\frac {\sqrt {a+b x^4}}{\sqrt {a}}\right )\right )+3 \sqrt {a} \sqrt {b} c x \sqrt {1+\frac {b x^4}{a}} E\left (\left .i \sinh ^{-1}\left (\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}} x\right )\right |-1\right )-i \sqrt {a} \left (-3 i \sqrt {b} c+\sqrt {a} e\right ) x \sqrt {1+\frac {b x^4}{a}} F\left (\left .i \sinh ^{-1}\left (\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}} x\right )\right |-1\right )}{2 a^2 \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}} x \sqrt {a+b x^4}} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains complex when optimal does not.
time = 0.42, size = 298, normalized size = 0.87
method | result | size |
elliptic | \(-\frac {c \sqrt {b \,x^{4}+a}}{a^{2} x}-\frac {2 b \left (\frac {c \,x^{3}}{4 a^{2}}-\frac {x^{2} f}{4 a b}-\frac {x e}{4 a b}-\frac {d}{4 a b}\right )}{\sqrt {\left (x^{4}+\frac {a}{b}\right ) b}}+\frac {e \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \EllipticF \left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )}{2 a \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}+\frac {3 i \sqrt {b}\, c \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \left (\EllipticF \left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )-\EllipticE \left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )\right )}{2 a^{\frac {3}{2}} \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}-\frac {d \arctanh \left (\frac {\sqrt {a}}{\sqrt {b \,x^{4}+a}}\right )}{2 a^{\frac {3}{2}}}\) | \(268\) |
default | \(\frac {f \,x^{2}}{2 a \sqrt {b \,x^{4}+a}}+e \left (\frac {x}{2 a \sqrt {\left (x^{4}+\frac {a}{b}\right ) b}}+\frac {\sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \EllipticF \left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )}{2 a \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}\right )+d \left (\frac {1}{2 a \sqrt {b \,x^{4}+a}}-\frac {\ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{4}+a}}{x^{2}}\right )}{2 a^{\frac {3}{2}}}\right )+c \left (-\frac {b \,x^{3}}{2 a^{2} \sqrt {\left (x^{4}+\frac {a}{b}\right ) b}}-\frac {\sqrt {b \,x^{4}+a}}{a^{2} x}+\frac {3 i \sqrt {b}\, \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \left (\EllipticF \left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )-\EllipticE \left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )\right )}{2 a^{\frac {3}{2}} \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}\right )\) | \(298\) |
risch | \(-\frac {c \sqrt {b \,x^{4}+a}}{a^{2} x}-\frac {b c \,x^{3}}{2 a^{2} \sqrt {\left (x^{4}+\frac {a}{b}\right ) b}}+\frac {3 i \sqrt {b}\, c \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \EllipticF \left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )}{2 a^{\frac {3}{2}} \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}-\frac {3 i \sqrt {b}\, c \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \EllipticE \left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )}{2 a^{\frac {3}{2}} \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}+\frac {f \,x^{2}}{2 a \sqrt {b \,x^{4}+a}}+\frac {e x}{2 a \sqrt {\left (x^{4}+\frac {a}{b}\right ) b}}+\frac {e \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \EllipticF \left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )}{2 a \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}+\frac {d}{2 a \sqrt {b \,x^{4}+a}}-\frac {d \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{4}+a}}{x^{2}}\right )}{2 a^{\frac {3}{2}}}\) | \(355\) |
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 [A]
time = 0.12, size = 215, normalized size = 0.62 \begin {gather*} -\frac {6 \, {\left (b^{2} c x^{5} + a b c x\right )} \sqrt {a} \left (-\frac {b}{a}\right )^{\frac {3}{4}} E(\arcsin \left (x \left (-\frac {b}{a}\right )^{\frac {1}{4}}\right )\,|\,-1) - 2 \, {\left ({\left (3 \, b^{2} c - a b e\right )} x^{5} + {\left (3 \, a b c - a^{2} e\right )} x\right )} \sqrt {a} \left (-\frac {b}{a}\right )^{\frac {3}{4}} F(\arcsin \left (x \left (-\frac {b}{a}\right )^{\frac {1}{4}}\right )\,|\,-1) - {\left (b^{2} d x^{5} + a b d x\right )} \sqrt {a} \log \left (-\frac {b x^{4} - 2 \, \sqrt {b x^{4} + a} \sqrt {a} + 2 \, a}{x^{4}}\right ) + 2 \, {\left (3 \, b^{2} c x^{4} - a b f x^{3} - a b e x^{2} - a b d x + 2 \, a b c\right )} \sqrt {b x^{4} + a}}{4 \, {\left (a^{2} b^{2} x^{5} + a^{3} b x\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] Result contains complex when optimal does not.
time = 8.67, size = 291, normalized size = 0.85 \begin {gather*} d \left (\frac {2 a^{3} \sqrt {1 + \frac {b x^{4}}{a}}}{4 a^{\frac {9}{2}} + 4 a^{\frac {7}{2}} b x^{4}} + \frac {a^{3} \log {\left (\frac {b x^{4}}{a} \right )}}{4 a^{\frac {9}{2}} + 4 a^{\frac {7}{2}} b x^{4}} - \frac {2 a^{3} \log {\left (\sqrt {1 + \frac {b x^{4}}{a}} + 1 \right )}}{4 a^{\frac {9}{2}} + 4 a^{\frac {7}{2}} b x^{4}} + \frac {a^{2} b x^{4} \log {\left (\frac {b x^{4}}{a} \right )}}{4 a^{\frac {9}{2}} + 4 a^{\frac {7}{2}} b x^{4}} - \frac {2 a^{2} b x^{4} \log {\left (\sqrt {1 + \frac {b x^{4}}{a}} + 1 \right )}}{4 a^{\frac {9}{2}} + 4 a^{\frac {7}{2}} b x^{4}}\right ) + \frac {c \Gamma \left (- \frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{4}, \frac {3}{2} \\ \frac {3}{4} \end {matrix}\middle | {\frac {b x^{4} e^{i \pi }}{a}} \right )}}{4 a^{\frac {3}{2}} x \Gamma \left (\frac {3}{4}\right )} + \frac {e x \Gamma \left (\frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{4}, \frac {3}{2} \\ \frac {5}{4} \end {matrix}\middle | {\frac {b x^{4} e^{i \pi }}{a}} \right )}}{4 a^{\frac {3}{2}} \Gamma \left (\frac {5}{4}\right )} + \frac {f x^{2}}{2 a^{\frac {3}{2}} \sqrt {1 + \frac {b x^{4}}{a}}} \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 [B]
time = 5.94, size = 133, normalized size = 0.39 \begin {gather*} \frac {d}{2\,a\,\sqrt {b\,x^4+a}}-\frac {d\,\mathrm {atanh}\left (\frac {\sqrt {b\,x^4+a}}{\sqrt {a}}\right )}{2\,a^{3/2}}+\frac {f\,x^2}{2\,a\,\sqrt {b\,x^4+a}}-\frac {c\,{\left (\frac {a}{b\,x^4}+1\right )}^{3/2}\,{{}}_2{\mathrm {F}}_1\left (\frac {3}{2},\frac {7}{4};\ \frac {11}{4};\ -\frac {a}{b\,x^4}\right )}{7\,x\,{\left (b\,x^4+a\right )}^{3/2}}+\frac {e\,x\,{\left (\frac {b\,x^4}{a}+1\right )}^{3/2}\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{4},\frac {3}{2};\ \frac {5}{4};\ -\frac {b\,x^4}{a}\right )}{{\left (b\,x^4+a\right )}^{3/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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