Optimal. Leaf size=346 \[ \frac {b c \tanh ^{-1}\left (\frac {\sqrt {a+b x^4}}{\sqrt {a}}\right )}{4 a^{3/2}}-\frac {\sqrt [4]{b} \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} \left (\sqrt {b} d-3 \sqrt {a} f\right ) F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{6 a^{5/4} \sqrt {a+b x^4}}-\frac {\sqrt [4]{b} f \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{a^{3/4} \sqrt {a+b x^4}}-\frac {c \sqrt {a+b x^4}}{4 a x^4}-\frac {d \sqrt {a+b x^4}}{3 a x^3}-\frac {e \sqrt {a+b x^4}}{2 a x^2}-\frac {f \sqrt {a+b x^4}}{a x}+\frac {\sqrt {b} f x \sqrt {a+b x^4}}{a \left (\sqrt {a}+\sqrt {b} x^2\right )} \]
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Rubi [A] time = 0.28, antiderivative size = 346, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 11, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.367, Rules used = {1833, 1252, 835, 807, 266, 63, 208, 1282, 1198, 220, 1196} \[ \frac {b c \tanh ^{-1}\left (\frac {\sqrt {a+b x^4}}{\sqrt {a}}\right )}{4 a^{3/2}}-\frac {\sqrt [4]{b} \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} \left (\sqrt {b} d-3 \sqrt {a} f\right ) F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{6 a^{5/4} \sqrt {a+b x^4}}-\frac {\sqrt [4]{b} f \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{a^{3/4} \sqrt {a+b x^4}}-\frac {c \sqrt {a+b x^4}}{4 a x^4}-\frac {d \sqrt {a+b x^4}}{3 a x^3}-\frac {e \sqrt {a+b x^4}}{2 a x^2}-\frac {f \sqrt {a+b x^4}}{a x}+\frac {\sqrt {b} f x \sqrt {a+b x^4}}{a \left (\sqrt {a}+\sqrt {b} x^2\right )} \]
Antiderivative was successfully verified.
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Rule 63
Rule 208
Rule 220
Rule 266
Rule 807
Rule 835
Rule 1196
Rule 1198
Rule 1252
Rule 1282
Rule 1833
Rubi steps
\begin {align*} \int \frac {c+d x+e x^2+f x^3}{x^5 \sqrt {a+b x^4}} \, dx &=\int \left (\frac {c+e x^2}{x^5 \sqrt {a+b x^4}}+\frac {d+f x^2}{x^4 \sqrt {a+b x^4}}\right ) \, dx\\ &=\int \frac {c+e x^2}{x^5 \sqrt {a+b x^4}} \, dx+\int \frac {d+f x^2}{x^4 \sqrt {a+b x^4}} \, dx\\ &=-\frac {d \sqrt {a+b x^4}}{3 a x^3}+\frac {1}{2} \operatorname {Subst}\left (\int \frac {c+e x}{x^3 \sqrt {a+b x^2}} \, dx,x,x^2\right )-\frac {\int \frac {-3 a f+b d x^2}{x^2 \sqrt {a+b x^4}} \, dx}{3 a}\\ &=-\frac {c \sqrt {a+b x^4}}{4 a x^4}-\frac {d \sqrt {a+b x^4}}{3 a x^3}-\frac {f \sqrt {a+b x^4}}{a x}+\frac {\int \frac {-a b d+3 a b f x^2}{\sqrt {a+b x^4}} \, dx}{3 a^2}-\frac {\operatorname {Subst}\left (\int \frac {-2 a e+b c x}{x^2 \sqrt {a+b x^2}} \, dx,x,x^2\right )}{4 a}\\ &=-\frac {c \sqrt {a+b x^4}}{4 a x^4}-\frac {d \sqrt {a+b x^4}}{3 a x^3}-\frac {e \sqrt {a+b x^4}}{2 a x^2}-\frac {f \sqrt {a+b x^4}}{a x}-\frac {(b c) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {a+b x^2}} \, dx,x,x^2\right )}{4 a}-\frac {\left (\sqrt {b} f\right ) \int \frac {1-\frac {\sqrt {b} x^2}{\sqrt {a}}}{\sqrt {a+b x^4}} \, dx}{\sqrt {a}}-\frac {\left (\sqrt {b} \left (\sqrt {b} d-3 \sqrt {a} f\right )\right ) \int \frac {1}{\sqrt {a+b x^4}} \, dx}{3 a}\\ &=-\frac {c \sqrt {a+b x^4}}{4 a x^4}-\frac {d \sqrt {a+b x^4}}{3 a x^3}-\frac {e \sqrt {a+b x^4}}{2 a x^2}-\frac {f \sqrt {a+b x^4}}{a x}+\frac {\sqrt {b} f x \sqrt {a+b x^4}}{a \left (\sqrt {a}+\sqrt {b} x^2\right )}-\frac {\sqrt [4]{b} f \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 )}{a^{3/4} \sqrt {a+b x^4}}-\frac {\sqrt [4]{b} \left (\sqrt {b} d-3 \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 )}{6 a^{5/4} \sqrt {a+b x^4}}-\frac {(b c) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,x^4\right )}{8 a}\\ &=-\frac {c \sqrt {a+b x^4}}{4 a x^4}-\frac {d \sqrt {a+b x^4}}{3 a x^3}-\frac {e \sqrt {a+b x^4}}{2 a x^2}-\frac {f \sqrt {a+b x^4}}{a x}+\frac {\sqrt {b} f x \sqrt {a+b x^4}}{a \left (\sqrt {a}+\sqrt {b} x^2\right )}-\frac {\sqrt [4]{b} f \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 )}{a^{3/4} \sqrt {a+b x^4}}-\frac {\sqrt [4]{b} \left (\sqrt {b} d-3 \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 )}{6 a^{5/4} \sqrt {a+b x^4}}-\frac {c \operatorname {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b x^4}\right )}{4 a}\\ &=-\frac {c \sqrt {a+b x^4}}{4 a x^4}-\frac {d \sqrt {a+b x^4}}{3 a x^3}-\frac {e \sqrt {a+b x^4}}{2 a x^2}-\frac {f \sqrt {a+b x^4}}{a x}+\frac {\sqrt {b} f x \sqrt {a+b x^4}}{a \left (\sqrt {a}+\sqrt {b} x^2\right )}+\frac {b c \tanh ^{-1}\left (\frac {\sqrt {a+b x^4}}{\sqrt {a}}\right )}{4 a^{3/2}}-\frac {\sqrt [4]{b} f \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 )}{a^{3/4} \sqrt {a+b x^4}}-\frac {\sqrt [4]{b} \left (\sqrt {b} d-3 \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 )}{6 a^{5/4} \sqrt {a+b x^4}}\\ \end {align*}
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Mathematica [C] time = 0.16, size = 147, normalized size = 0.42 \[ -\frac {\sqrt {a+b x^4} \left (3 a c \sqrt {\frac {b x^4}{a}+1}-3 b c x^4 \tanh ^{-1}\left (\sqrt {\frac {b x^4}{a}+1}\right )+4 a d x \, _2F_1\left (-\frac {3}{4},\frac {1}{2};\frac {1}{4};-\frac {b x^4}{a}\right )+6 a e x^2 \sqrt {\frac {b x^4}{a}+1}+12 a f x^3 \, _2F_1\left (-\frac {1}{4},\frac {1}{2};\frac {3}{4};-\frac {b x^4}{a}\right )\right )}{12 a^2 x^4 \sqrt {\frac {b x^4}{a}+1}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.52, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {b x^{4} + a} {\left (f x^{3} + e x^{2} + d x + c\right )}}{b x^{9} + a x^{5}}, x\right ) \]
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 \[ \int \frac {f x^{3} + e x^{2} + d x + c}{\sqrt {b x^{4} + a} x^{5}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.21, size = 335, normalized size = 0.97 \[ -\frac {i \sqrt {-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}+1}\, \sqrt {\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}+1}\, \sqrt {b}\, f \EllipticE \left (\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, x , i\right )}{\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}\, \sqrt {a}}+\frac {i \sqrt {-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}+1}\, \sqrt {\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}+1}\, \sqrt {b}\, f \EllipticF \left (\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, x , i\right )}{\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}\, \sqrt {a}}-\frac {\sqrt {-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}+1}\, \sqrt {\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}+1}\, b d \EllipticF \left (\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, x , i\right )}{3 \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}\, a}+\frac {b c \ln \left (\frac {2 a +2 \sqrt {b \,x^{4}+a}\, \sqrt {a}}{x^{2}}\right )}{4 a^{\frac {3}{2}}}-\frac {\sqrt {b \,x^{4}+a}\, f}{a x}-\frac {\sqrt {b \,x^{4}+a}\, e}{2 a \,x^{2}}-\frac {\sqrt {b \,x^{4}+a}\, d}{3 a \,x^{3}}-\frac {\sqrt {b \,x^{4}+a}\, c}{4 a \,x^{4}} \]
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 \[ -\frac {1}{8} \, c {\left (\frac {2 \, \sqrt {b x^{4} + a} b}{{\left (b x^{4} + a\right )} a - a^{2}} + \frac {b \log \left (\frac {\sqrt {b x^{4} + a} - \sqrt {a}}{\sqrt {b x^{4} + a} + \sqrt {a}}\right )}{a^{\frac {3}{2}}}\right )} + \int \frac {f x^{2} + e x + d}{\sqrt {b x^{4} + a} x^{4}}\,{d x} \]
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 \[ \int \frac {f\,x^3+e\,x^2+d\,x+c}{x^5\,\sqrt {b\,x^4+a}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 8.55, size = 158, normalized size = 0.46 \[ - \frac {\sqrt {b} c \sqrt {\frac {a}{b x^{4}} + 1}}{4 a x^{2}} - \frac {\sqrt {b} e \sqrt {\frac {a}{b x^{4}} + 1}}{2 a} + \frac {d \Gamma \left (- \frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {3}{4}, \frac {1}{2} \\ \frac {1}{4} \end {matrix}\middle | {\frac {b x^{4} e^{i \pi }}{a}} \right )}}{4 \sqrt {a} x^{3} \Gamma \left (\frac {1}{4}\right )} + \frac {f \Gamma \left (- \frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{4}, \frac {1}{2} \\ \frac {3}{4} \end {matrix}\middle | {\frac {b x^{4} e^{i \pi }}{a}} \right )}}{4 \sqrt {a} x \Gamma \left (\frac {3}{4}\right )} + \frac {b c \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {b} x^{2}} \right )}}{4 a^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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