Optimal. Leaf size=344 \[ \frac {\left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} \left (\sqrt {a} e+3 \sqrt {b} c\right ) F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{4 a^{7/4} \sqrt [4]{b} \sqrt {a+b x^4}}-\frac {3 \sqrt [4]{b} c \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{2 a^{7/4} \sqrt {a+b x^4}}-\frac {d \tanh ^{-1}\left (\frac {\sqrt {a+b x^4}}{\sqrt {a}}\right )}{2 a^{3/2}}+\frac {x \left (a e+a f x-b c x^2-b d x^3\right )}{2 a^2 \sqrt {a+b x^4}}-\frac {c \sqrt {a+b x^4}}{a^2 x}+\frac {3 \sqrt {b} c x \sqrt {a+b x^4}}{2 a^2 \left (\sqrt {a}+\sqrt {b} x^2\right )}+\frac {d \sqrt {a+b x^4}}{2 a^2} \]
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Rubi [A] time = 0.38, 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 = {1829, 1833, 1835, 1584, 1198, 220, 1196, 21, 266, 50, 63, 208} \[ \frac {x \left (a e+a f x-b c x^2-b d x^3\right )}{2 a^2 \sqrt {a+b x^4}}+\frac {\left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} \left (\sqrt {a} e+3 \sqrt {b} c\right ) F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{4 a^{7/4} \sqrt [4]{b} \sqrt {a+b x^4}}-\frac {c \sqrt {a+b x^4}}{a^2 x}+\frac {3 \sqrt {b} c x \sqrt {a+b x^4}}{2 a^2 \left (\sqrt {a}+\sqrt {b} x^2\right )}-\frac {3 \sqrt [4]{b} c \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{2 a^{7/4} \sqrt {a+b x^4}}+\frac {d \sqrt {a+b x^4}}{2 a^2}-\frac {d \tanh ^{-1}\left (\frac {\sqrt {a+b x^4}}{\sqrt {a}}\right )}{2 a^{3/2}} \]
Antiderivative was successfully verified.
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Rule 21
Rule 50
Rule 63
Rule 208
Rule 220
Rule 266
Rule 1196
Rule 1198
Rule 1584
Rule 1829
Rule 1833
Rule 1835
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 \operatorname {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 \operatorname {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 \operatorname {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] time = 0.13, size = 123, normalized size = 0.36 \[ \frac {-2 c \sqrt {\frac {b x^4}{a}+1} \, _2F_1\left (-\frac {1}{4},\frac {3}{2};\frac {3}{4};-\frac {b x^4}{a}\right )+d x \, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};\frac {b x^4}{a}+1\right )+x^2 \left (e \sqrt {\frac {b x^4}{a}+1} \, _2F_1\left (\frac {1}{4},\frac {1}{2};\frac {5}{4};-\frac {b x^4}{a}\right )+e+f x\right )}{2 a x \sqrt {a+b x^4}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.49, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {b x^{4} + a} {\left (f x^{3} + e x^{2} + d x + c\right )}}{b^{2} x^{10} + 2 \, a b x^{6} + a^{2} x^{2}}, x\right ) \]
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 \[ \int \frac {f x^{3} + e x^{2} + d x + c}{{\left (b x^{4} + a\right )}^{\frac {3}{2}} x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.19, size = 355, normalized size = 1.03 \[ -\frac {b c \,x^{3}}{2 \sqrt {\left (x^{4}+\frac {a}{b}\right ) b}\, a^{2}}+\frac {f \,x^{2}}{2 \sqrt {b \,x^{4}+a}\, a}+\frac {e x}{2 \sqrt {\left (x^{4}+\frac {a}{b}\right ) b}\, a}+\frac {\sqrt {-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}+1}\, \sqrt {\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}+1}\, e \EllipticF \left (\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, x , i\right )}{2 \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}\, a}-\frac {3 i \sqrt {-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}+1}\, \sqrt {\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}+1}\, \sqrt {b}\, c \EllipticE \left (\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, x , i\right )}{2 \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}\, a^{\frac {3}{2}}}+\frac {3 i \sqrt {-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}+1}\, \sqrt {\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}+1}\, \sqrt {b}\, c \EllipticF \left (\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, x , i\right )}{2 \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}\, a^{\frac {3}{2}}}-\frac {d \ln \left (\frac {2 a +2 \sqrt {b \,x^{4}+a}\, \sqrt {a}}{x^{2}}\right )}{2 a^{\frac {3}{2}}}+\frac {d}{2 \sqrt {b \,x^{4}+a}\, a}-\frac {\sqrt {b \,x^{4}+a}\, c}{a^{2} x} \]
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 \[ \int \frac {f x^{3} + e x^{2} + d x + c}{{\left (b x^{4} + a\right )}^{\frac {3}{2}} x^{2}}\,{d x} \]
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 \[ \frac {d}{2\,a\,\sqrt {b\,x^4+a}}-\frac {d\,\mathrm {atanh}\left (\frac {\sqrt {b\,x^4+a}}{\sqrt {a}}\right )}{2\,a^{3/2}}+\frac {f\,x^2}{2\,a\,\sqrt {b\,x^4+a}}-\frac {c\,{\left (\frac {a}{b\,x^4}+1\right )}^{3/2}\,{{}}_2{\mathrm {F}}_1\left (\frac {3}{2},\frac {7}{4};\ \frac {11}{4};\ -\frac {a}{b\,x^4}\right )}{7\,x\,{\left (b\,x^4+a\right )}^{3/2}}+\frac {e\,x\,{\left (\frac {b\,x^4}{a}+1\right )}^{3/2}\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{4},\frac {3}{2};\ \frac {5}{4};\ -\frac {b\,x^4}{a}\right )}{{\left (b\,x^4+a\right )}^{3/2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 28.56, size = 291, normalized size = 0.85 \[ d \left (\frac {2 a^{3} \sqrt {1 + \frac {b x^{4}}{a}}}{4 a^{\frac {9}{2}} + 4 a^{\frac {7}{2}} b x^{4}} + \frac {a^{3} \log {\left (\frac {b x^{4}}{a} \right )}}{4 a^{\frac {9}{2}} + 4 a^{\frac {7}{2}} b x^{4}} - \frac {2 a^{3} \log {\left (\sqrt {1 + \frac {b x^{4}}{a}} + 1 \right )}}{4 a^{\frac {9}{2}} + 4 a^{\frac {7}{2}} b x^{4}} + \frac {a^{2} b x^{4} \log {\left (\frac {b x^{4}}{a} \right )}}{4 a^{\frac {9}{2}} + 4 a^{\frac {7}{2}} b x^{4}} - \frac {2 a^{2} b x^{4} \log {\left (\sqrt {1 + \frac {b x^{4}}{a}} + 1 \right )}}{4 a^{\frac {9}{2}} + 4 a^{\frac {7}{2}} b x^{4}}\right ) + \frac {c \Gamma \left (- \frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{4}, \frac {3}{2} \\ \frac {3}{4} \end {matrix}\middle | {\frac {b x^{4} e^{i \pi }}{a}} \right )}}{4 a^{\frac {3}{2}} x \Gamma \left (\frac {3}{4}\right )} + \frac {e x \Gamma \left (\frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{4}, \frac {3}{2} \\ \frac {5}{4} \end {matrix}\middle | {\frac {b x^{4} e^{i \pi }}{a}} \right )}}{4 a^{\frac {3}{2}} \Gamma \left (\frac {5}{4}\right )} + \frac {f x^{2}}{2 a^{\frac {3}{2}} \sqrt {1 + \frac {b x^{4}}{a}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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