Optimal. Leaf size=342 \[ -\frac {\sqrt {a+b x^4} \left (c-e x^2\right )}{2 x^2}+\frac {1}{2} \sqrt {b} c \tanh ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a+b x^4}}\right )-\frac {\sqrt {a+b x^4} \left (3 d-f x^2\right )}{3 x}+\frac {\sqrt [4]{a} \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} \left (\sqrt {a} f+3 \sqrt {b} d\right ) F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{3 \sqrt [4]{b} \sqrt {a+b x^4}}+\frac {2 \sqrt {b} d x \sqrt {a+b x^4}}{\sqrt {a}+\sqrt {b} x^2}-\frac {2 \sqrt [4]{a} \sqrt [4]{b} d \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{\sqrt {a+b x^4}}-\frac {1}{2} \sqrt {a} e \tanh ^{-1}\left (\frac {\sqrt {a+b x^4}}{\sqrt {a}}\right ) \]
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Rubi [A] time = 0.26, antiderivative size = 342, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 13, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.433, Rules used = {1833, 1252, 813, 844, 217, 206, 266, 63, 208, 1272, 1198, 220, 1196} \[ -\frac {\sqrt {a+b x^4} \left (c-e x^2\right )}{2 x^2}+\frac {1}{2} \sqrt {b} c \tanh ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a+b x^4}}\right )-\frac {\sqrt {a+b x^4} \left (3 d-f x^2\right )}{3 x}+\frac {\sqrt [4]{a} \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} \left (\sqrt {a} f+3 \sqrt {b} d\right ) F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{3 \sqrt [4]{b} \sqrt {a+b x^4}}+\frac {2 \sqrt {b} d x \sqrt {a+b x^4}}{\sqrt {a}+\sqrt {b} x^2}-\frac {2 \sqrt [4]{a} \sqrt [4]{b} d \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{\sqrt {a+b x^4}}-\frac {1}{2} \sqrt {a} e \tanh ^{-1}\left (\frac {\sqrt {a+b x^4}}{\sqrt {a}}\right ) \]
Antiderivative was successfully verified.
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Rule 63
Rule 206
Rule 208
Rule 217
Rule 220
Rule 266
Rule 813
Rule 844
Rule 1196
Rule 1198
Rule 1252
Rule 1272
Rule 1833
Rubi steps
\begin {align*} \int \frac {\left (c+d x+e x^2+f x^3\right ) \sqrt {a+b x^4}}{x^3} \, dx &=\int \left (\frac {\left (c+e x^2\right ) \sqrt {a+b x^4}}{x^3}+\frac {\left (d+f x^2\right ) \sqrt {a+b x^4}}{x^2}\right ) \, dx\\ &=\int \frac {\left (c+e x^2\right ) \sqrt {a+b x^4}}{x^3} \, dx+\int \frac {\left (d+f x^2\right ) \sqrt {a+b x^4}}{x^2} \, dx\\ &=-\frac {\left (3 d-f x^2\right ) \sqrt {a+b x^4}}{3 x}+\frac {1}{2} \operatorname {Subst}\left (\int \frac {(c+e x) \sqrt {a+b x^2}}{x^2} \, dx,x,x^2\right )-\frac {2}{3} \int \frac {-a f-3 b d x^2}{\sqrt {a+b x^4}} \, dx\\ &=-\frac {\left (c-e x^2\right ) \sqrt {a+b x^4}}{2 x^2}-\frac {\left (3 d-f x^2\right ) \sqrt {a+b x^4}}{3 x}-\frac {1}{4} \operatorname {Subst}\left (\int \frac {-2 a e-2 b c x}{x \sqrt {a+b x^2}} \, dx,x,x^2\right )-\left (2 \sqrt {a} \sqrt {b} d\right ) \int \frac {1-\frac {\sqrt {b} x^2}{\sqrt {a}}}{\sqrt {a+b x^4}} \, dx+\frac {1}{3} \left (2 \sqrt {a} \left (3 \sqrt {b} d+\sqrt {a} f\right )\right ) \int \frac {1}{\sqrt {a+b x^4}} \, dx\\ &=\frac {2 \sqrt {b} d x \sqrt {a+b x^4}}{\sqrt {a}+\sqrt {b} x^2}-\frac {\left (c-e x^2\right ) \sqrt {a+b x^4}}{2 x^2}-\frac {\left (3 d-f x^2\right ) \sqrt {a+b x^4}}{3 x}-\frac {2 \sqrt [4]{a} \sqrt [4]{b} d \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 )}{\sqrt {a+b x^4}}+\frac {\sqrt [4]{a} \left (3 \sqrt {b} d+\sqrt {a} f\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 )}{3 \sqrt [4]{b} \sqrt {a+b x^4}}+\frac {1}{2} (b c) \operatorname {Subst}\left (\int \frac {1}{\sqrt {a+b x^2}} \, dx,x,x^2\right )+\frac {1}{2} (a e) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {a+b x^2}} \, dx,x,x^2\right )\\ &=\frac {2 \sqrt {b} d x \sqrt {a+b x^4}}{\sqrt {a}+\sqrt {b} x^2}-\frac {\left (c-e x^2\right ) \sqrt {a+b x^4}}{2 x^2}-\frac {\left (3 d-f x^2\right ) \sqrt {a+b x^4}}{3 x}-\frac {2 \sqrt [4]{a} \sqrt [4]{b} d \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 )}{\sqrt {a+b x^4}}+\frac {\sqrt [4]{a} \left (3 \sqrt {b} d+\sqrt {a} f\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 )}{3 \sqrt [4]{b} \sqrt {a+b x^4}}+\frac {1}{2} (b c) \operatorname {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {x^2}{\sqrt {a+b x^4}}\right )+\frac {1}{4} (a e) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,x^4\right )\\ &=\frac {2 \sqrt {b} d x \sqrt {a+b x^4}}{\sqrt {a}+\sqrt {b} x^2}-\frac {\left (c-e x^2\right ) \sqrt {a+b x^4}}{2 x^2}-\frac {\left (3 d-f x^2\right ) \sqrt {a+b x^4}}{3 x}+\frac {1}{2} \sqrt {b} c \tanh ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a+b x^4}}\right )-\frac {2 \sqrt [4]{a} \sqrt [4]{b} d \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 )}{\sqrt {a+b x^4}}+\frac {\sqrt [4]{a} \left (3 \sqrt {b} d+\sqrt {a} f\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 )}{3 \sqrt [4]{b} \sqrt {a+b x^4}}+\frac {(a e) \operatorname {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b x^4}\right )}{2 b}\\ &=\frac {2 \sqrt {b} d x \sqrt {a+b x^4}}{\sqrt {a}+\sqrt {b} x^2}-\frac {\left (c-e x^2\right ) \sqrt {a+b x^4}}{2 x^2}-\frac {\left (3 d-f x^2\right ) \sqrt {a+b x^4}}{3 x}+\frac {1}{2} \sqrt {b} c \tanh ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a+b x^4}}\right )-\frac {1}{2} \sqrt {a} e \tanh ^{-1}\left (\frac {\sqrt {a+b x^4}}{\sqrt {a}}\right )-\frac {2 \sqrt [4]{a} \sqrt [4]{b} d \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 )}{\sqrt {a+b x^4}}+\frac {\sqrt [4]{a} \left (3 \sqrt {b} d+\sqrt {a} f\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 )}{3 \sqrt [4]{b} \sqrt {a+b x^4}}\\ \end {align*}
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Mathematica [C] time = 0.26, size = 204, normalized size = 0.60 \[ \frac {\sqrt {a} \sqrt {b} c x^2 \sqrt {\frac {b x^4}{a}+1} \sinh ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )-2 a d x \sqrt {\frac {b x^4}{a}+1} \, _2F_1\left (-\frac {1}{2},-\frac {1}{4};\frac {3}{4};-\frac {b x^4}{a}\right )-\sqrt {a} e x^2 \sqrt {a+b x^4} \tanh ^{-1}\left (\frac {\sqrt {a+b x^4}}{\sqrt {a}}\right )+2 a f x^3 \sqrt {\frac {b x^4}{a}+1} \, _2F_1\left (-\frac {1}{2},\frac {1}{4};\frac {5}{4};-\frac {b x^4}{a}\right )-a c+a e x^2-b c x^4+b e x^6}{2 x^2 \sqrt {a+b x^4}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.75, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {b x^{4} + a} {\left (f x^{3} + e x^{2} + d x + c\right )}}{x^{3}}, x\right ) \]
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 \[ \int \frac {\sqrt {b x^{4} + a} {\left (f x^{3} + e x^{2} + d x + c\right )}}{x^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.18, size = 360, normalized size = 1.05 \[ \frac {2 \sqrt {-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}+1}\, \sqrt {\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}+1}\, a f \EllipticF \left (\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, x , i\right )}{3 \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}-\frac {2 i \sqrt {-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}+1}\, \sqrt {\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}+1}\, \sqrt {a}\, \sqrt {b}\, d \EllipticE \left (\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, x , i\right )}{\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}+\frac {2 i \sqrt {-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}+1}\, \sqrt {\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}+1}\, \sqrt {a}\, \sqrt {b}\, d \EllipticF \left (\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, x , i\right )}{\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}+\frac {\sqrt {b \,x^{4}+a}\, b c \,x^{2}}{2 a}-\frac {\sqrt {a}\, e \ln \left (\frac {2 a +2 \sqrt {b \,x^{4}+a}\, \sqrt {a}}{x^{2}}\right )}{2}+\frac {\sqrt {b}\, c \ln \left (\sqrt {b}\, x^{2}+\sqrt {b \,x^{4}+a}\right )}{2}+\frac {\sqrt {b \,x^{4}+a}\, f x}{3}+\frac {\sqrt {b \,x^{4}+a}\, e}{2}-\frac {\sqrt {b \,x^{4}+a}\, d}{x}-\frac {\left (b \,x^{4}+a \right )^{\frac {3}{2}} c}{2 a \,x^{2}} \]
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 \[ \int \frac {\sqrt {b x^{4} + a} {\left (f x^{3} + e x^{2} + d x + c\right )}}{x^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {\sqrt {b\,x^4+a}\,\left (f\,x^3+e\,x^2+d\,x+c\right )}{x^3} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 6.41, size = 230, normalized size = 0.67 \[ - \frac {\sqrt {a} c}{2 x^{2} \sqrt {1 + \frac {b x^{4}}{a}}} + \frac {\sqrt {a} d \Gamma \left (- \frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, - \frac {1}{4} \\ \frac {3}{4} \end {matrix}\middle | {\frac {b x^{4} e^{i \pi }}{a}} \right )}}{4 x \Gamma \left (\frac {3}{4}\right )} - \frac {\sqrt {a} e \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {b} x^{2}} \right )}}{2} + \frac {\sqrt {a} f x \Gamma \left (\frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {1}{4} \\ \frac {5}{4} \end {matrix}\middle | {\frac {b x^{4} e^{i \pi }}{a}} \right )}}{4 \Gamma \left (\frac {5}{4}\right )} + \frac {a e}{2 \sqrt {b} x^{2} \sqrt {\frac {a}{b x^{4}} + 1}} + \frac {\sqrt {b} c \operatorname {asinh}{\left (\frac {\sqrt {b} x^{2}}{\sqrt {a}} \right )}}{2} + \frac {\sqrt {b} e x^{2}}{2 \sqrt {\frac {a}{b x^{4}} + 1}} - \frac {b c x^{2}}{2 \sqrt {a} \sqrt {1 + \frac {b x^{4}}{a}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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