Optimal. Leaf size=425 \[ -\frac {b^{7/4} \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} \left (5 \sqrt {a} e+7 \sqrt {b} c\right ) F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{105 a^{7/4} \sqrt {a+b x^4}}+\frac {2 b^{9/4} c \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{15 a^{7/4} \sqrt {a+b x^4}}+\frac {b^2 d \tanh ^{-1}\left (\frac {\sqrt {a+b x^4}}{\sqrt {a}}\right )}{16 a^{3/2}}-\frac {2 b^{5/2} c x \sqrt {a+b x^4}}{15 a^2 \left (\sqrt {a}+\sqrt {b} x^2\right )}+\frac {2 b^2 c \sqrt {a+b x^4}}{15 a^2 x}-\frac {1}{504} \sqrt {a+b x^4} \left (\frac {56 c}{x^9}+\frac {63 d}{x^8}+\frac {72 e}{x^7}+\frac {84 f}{x^6}\right )-\frac {2 b c \sqrt {a+b x^4}}{45 a x^5}-\frac {b d \sqrt {a+b x^4}}{16 a x^4}-\frac {2 b e \sqrt {a+b x^4}}{21 a x^3}-\frac {b f \sqrt {a+b x^4}}{6 a x^2} \]
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Rubi [A] time = 0.43, antiderivative size = 425, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 13, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.433, Rules used = {14, 1825, 1833, 1282, 1198, 220, 1196, 1252, 835, 807, 266, 63, 208} \[ -\frac {b^{7/4} \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} \left (5 \sqrt {a} e+7 \sqrt {b} c\right ) F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{105 a^{7/4} \sqrt {a+b x^4}}-\frac {2 b^{5/2} c x \sqrt {a+b x^4}}{15 a^2 \left (\sqrt {a}+\sqrt {b} x^2\right )}+\frac {2 b^2 c \sqrt {a+b x^4}}{15 a^2 x}+\frac {2 b^{9/4} c \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{15 a^{7/4} \sqrt {a+b x^4}}+\frac {b^2 d \tanh ^{-1}\left (\frac {\sqrt {a+b x^4}}{\sqrt {a}}\right )}{16 a^{3/2}}-\frac {1}{504} \sqrt {a+b x^4} \left (\frac {56 c}{x^9}+\frac {63 d}{x^8}+\frac {72 e}{x^7}+\frac {84 f}{x^6}\right )-\frac {2 b c \sqrt {a+b x^4}}{45 a x^5}-\frac {b d \sqrt {a+b x^4}}{16 a x^4}-\frac {2 b e \sqrt {a+b x^4}}{21 a x^3}-\frac {b f \sqrt {a+b x^4}}{6 a x^2} \]
Antiderivative was successfully verified.
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Rule 14
Rule 63
Rule 208
Rule 220
Rule 266
Rule 807
Rule 835
Rule 1196
Rule 1198
Rule 1252
Rule 1282
Rule 1825
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^{10}} \, dx &=-\frac {1}{504} \left (\frac {56 c}{x^9}+\frac {63 d}{x^8}+\frac {72 e}{x^7}+\frac {84 f}{x^6}\right ) \sqrt {a+b x^4}-(2 b) \int \frac {-\frac {c}{9}-\frac {d x}{8}-\frac {e x^2}{7}-\frac {f x^3}{6}}{x^6 \sqrt {a+b x^4}} \, dx\\ &=-\frac {1}{504} \left (\frac {56 c}{x^9}+\frac {63 d}{x^8}+\frac {72 e}{x^7}+\frac {84 f}{x^6}\right ) \sqrt {a+b x^4}-(2 b) \int \left (\frac {-\frac {c}{9}-\frac {e x^2}{7}}{x^6 \sqrt {a+b x^4}}+\frac {-\frac {d}{8}-\frac {f x^2}{6}}{x^5 \sqrt {a+b x^4}}\right ) \, dx\\ &=-\frac {1}{504} \left (\frac {56 c}{x^9}+\frac {63 d}{x^8}+\frac {72 e}{x^7}+\frac {84 f}{x^6}\right ) \sqrt {a+b x^4}-(2 b) \int \frac {-\frac {c}{9}-\frac {e x^2}{7}}{x^6 \sqrt {a+b x^4}} \, dx-(2 b) \int \frac {-\frac {d}{8}-\frac {f x^2}{6}}{x^5 \sqrt {a+b x^4}} \, dx\\ &=-\frac {1}{504} \left (\frac {56 c}{x^9}+\frac {63 d}{x^8}+\frac {72 e}{x^7}+\frac {84 f}{x^6}\right ) \sqrt {a+b x^4}-\frac {2 b c \sqrt {a+b x^4}}{45 a x^5}-b \operatorname {Subst}\left (\int \frac {-\frac {d}{8}-\frac {f x}{6}}{x^3 \sqrt {a+b x^2}} \, dx,x,x^2\right )+\frac {(2 b) \int \frac {\frac {5 a e}{7}-\frac {1}{3} b c x^2}{x^4 \sqrt {a+b x^4}} \, dx}{5 a}\\ &=-\frac {1}{504} \left (\frac {56 c}{x^9}+\frac {63 d}{x^8}+\frac {72 e}{x^7}+\frac {84 f}{x^6}\right ) \sqrt {a+b x^4}-\frac {2 b c \sqrt {a+b x^4}}{45 a x^5}-\frac {b d \sqrt {a+b x^4}}{16 a x^4}-\frac {2 b e \sqrt {a+b x^4}}{21 a x^3}-\frac {(2 b) \int \frac {a b c+\frac {5}{7} a b e x^2}{x^2 \sqrt {a+b x^4}} \, dx}{15 a^2}+\frac {b \operatorname {Subst}\left (\int \frac {\frac {a f}{3}-\frac {b d x}{8}}{x^2 \sqrt {a+b x^2}} \, dx,x,x^2\right )}{2 a}\\ &=-\frac {1}{504} \left (\frac {56 c}{x^9}+\frac {63 d}{x^8}+\frac {72 e}{x^7}+\frac {84 f}{x^6}\right ) \sqrt {a+b x^4}-\frac {2 b c \sqrt {a+b x^4}}{45 a x^5}-\frac {b d \sqrt {a+b x^4}}{16 a x^4}-\frac {2 b e \sqrt {a+b x^4}}{21 a x^3}-\frac {b f \sqrt {a+b x^4}}{6 a x^2}+\frac {2 b^2 c \sqrt {a+b x^4}}{15 a^2 x}+\frac {(2 b) \int \frac {-\frac {5}{7} a^2 b e-a b^2 c x^2}{\sqrt {a+b x^4}} \, dx}{15 a^3}-\frac {\left (b^2 d\right ) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {a+b x^2}} \, dx,x,x^2\right )}{16 a}\\ &=-\frac {1}{504} \left (\frac {56 c}{x^9}+\frac {63 d}{x^8}+\frac {72 e}{x^7}+\frac {84 f}{x^6}\right ) \sqrt {a+b x^4}-\frac {2 b c \sqrt {a+b x^4}}{45 a x^5}-\frac {b d \sqrt {a+b x^4}}{16 a x^4}-\frac {2 b e \sqrt {a+b x^4}}{21 a x^3}-\frac {b f \sqrt {a+b x^4}}{6 a x^2}+\frac {2 b^2 c \sqrt {a+b x^4}}{15 a^2 x}+\frac {\left (2 b^{5/2} c\right ) \int \frac {1-\frac {\sqrt {b} x^2}{\sqrt {a}}}{\sqrt {a+b x^4}} \, dx}{15 a^{3/2}}-\frac {\left (b^2 d\right ) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,x^4\right )}{32 a}-\frac {\left (2 b^2 \left (7 \sqrt {b} c+5 \sqrt {a} e\right )\right ) \int \frac {1}{\sqrt {a+b x^4}} \, dx}{105 a^{3/2}}\\ &=-\frac {1}{504} \left (\frac {56 c}{x^9}+\frac {63 d}{x^8}+\frac {72 e}{x^7}+\frac {84 f}{x^6}\right ) \sqrt {a+b x^4}-\frac {2 b c \sqrt {a+b x^4}}{45 a x^5}-\frac {b d \sqrt {a+b x^4}}{16 a x^4}-\frac {2 b e \sqrt {a+b x^4}}{21 a x^3}-\frac {b f \sqrt {a+b x^4}}{6 a x^2}+\frac {2 b^2 c \sqrt {a+b x^4}}{15 a^2 x}-\frac {2 b^{5/2} c x \sqrt {a+b x^4}}{15 a^2 \left (\sqrt {a}+\sqrt {b} x^2\right )}+\frac {2 b^{9/4} 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 )}{15 a^{7/4} \sqrt {a+b x^4}}-\frac {b^{7/4} \left (7 \sqrt {b} c+5 \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 )}{105 a^{7/4} \sqrt {a+b x^4}}-\frac {(b d) \operatorname {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b x^4}\right )}{16 a}\\ &=-\frac {1}{504} \left (\frac {56 c}{x^9}+\frac {63 d}{x^8}+\frac {72 e}{x^7}+\frac {84 f}{x^6}\right ) \sqrt {a+b x^4}-\frac {2 b c \sqrt {a+b x^4}}{45 a x^5}-\frac {b d \sqrt {a+b x^4}}{16 a x^4}-\frac {2 b e \sqrt {a+b x^4}}{21 a x^3}-\frac {b f \sqrt {a+b x^4}}{6 a x^2}+\frac {2 b^2 c \sqrt {a+b x^4}}{15 a^2 x}-\frac {2 b^{5/2} c x \sqrt {a+b x^4}}{15 a^2 \left (\sqrt {a}+\sqrt {b} x^2\right )}+\frac {b^2 d \tanh ^{-1}\left (\frac {\sqrt {a+b x^4}}{\sqrt {a}}\right )}{16 a^{3/2}}+\frac {2 b^{9/4} 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 )}{15 a^{7/4} \sqrt {a+b x^4}}-\frac {b^{7/4} \left (7 \sqrt {b} c+5 \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 )}{105 a^{7/4} \sqrt {a+b x^4}}\\ \end {align*}
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Mathematica [C] time = 0.18, size = 148, normalized size = 0.35 \[ -\frac {\sqrt {a+b x^4} \left (14 a^3 c \, _2F_1\left (-\frac {9}{4},-\frac {1}{2};-\frac {5}{4};-\frac {b x^4}{a}\right )+3 x^2 \left (6 a^3 e \, _2F_1\left (-\frac {7}{4},-\frac {1}{2};-\frac {3}{4};-\frac {b x^4}{a}\right )+7 x \left (a+b x^4\right ) \sqrt {\frac {b x^4}{a}+1} \left (a^2 f+b^2 d x^6 \, _2F_1\left (\frac {3}{2},3;\frac {5}{2};\frac {b x^4}{a}+1\right )\right )\right )\right )}{126 a^3 x^9 \sqrt {\frac {b x^4}{a}+1}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.48, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {b x^{4} + a} {\left (f x^{3} + e x^{2} + d x + c\right )}}{x^{10}}, x\right ) \]
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 \[ \int \frac {\sqrt {b x^{4} + a} {\left (f x^{3} + e x^{2} + d x + c\right )}}{x^{10}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.21, size = 429, normalized size = 1.01 \[ -\frac {2 \sqrt {-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}+1}\, \sqrt {\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}+1}\, b^{2} e \EllipticF \left (\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, x , i\right )}{21 \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}\, a}+\frac {2 i \sqrt {-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}+1}\, \sqrt {\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}+1}\, b^{\frac {5}{2}} c \EllipticE \left (\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, x , i\right )}{15 \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}\, a^{\frac {3}{2}}}-\frac {2 i \sqrt {-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}+1}\, \sqrt {\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}+1}\, b^{\frac {5}{2}} c \EllipticF \left (\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, x , i\right )}{15 \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}\, a^{\frac {3}{2}}}+\frac {b^{2} d \ln \left (\frac {2 a +2 \sqrt {b \,x^{4}+a}\, \sqrt {a}}{x^{2}}\right )}{16 a^{\frac {3}{2}}}-\frac {\sqrt {b \,x^{4}+a}\, b^{2} d}{16 a^{2}}+\frac {2 \sqrt {b \,x^{4}+a}\, b^{2} c}{15 a^{2} x}-\frac {2 \sqrt {b \,x^{4}+a}\, b e}{21 a \,x^{3}}-\frac {2 \sqrt {b \,x^{4}+a}\, b c}{45 a \,x^{5}}+\frac {\left (b \,x^{4}+a \right )^{\frac {3}{2}} b d}{16 a^{2} x^{4}}-\frac {\left (b \,x^{4}+a \right )^{\frac {3}{2}} f}{6 a \,x^{6}}-\frac {\sqrt {b \,x^{4}+a}\, e}{7 x^{7}}-\frac {\left (b \,x^{4}+a \right )^{\frac {3}{2}} d}{8 a \,x^{8}}-\frac {\sqrt {b \,x^{4}+a}\, c}{9 x^{9}} \]
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 \[ \int \frac {\sqrt {b x^{4} + a} {\left (f x^{3} + e x^{2} + d x + c\right )}}{x^{10}}\,{d x} \]
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 \[ \int \frac {\sqrt {b\,x^4+a}\,\left (f\,x^3+e\,x^2+d\,x+c\right )}{x^{10}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 10.71, size = 246, normalized size = 0.58 \[ \frac {\sqrt {a} c \Gamma \left (- \frac {9}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {9}{4}, - \frac {1}{2} \\ - \frac {5}{4} \end {matrix}\middle | {\frac {b x^{4} e^{i \pi }}{a}} \right )}}{4 x^{9} \Gamma \left (- \frac {5}{4}\right )} + \frac {\sqrt {a} e \Gamma \left (- \frac {7}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {7}{4}, - \frac {1}{2} \\ - \frac {3}{4} \end {matrix}\middle | {\frac {b x^{4} e^{i \pi }}{a}} \right )}}{4 x^{7} \Gamma \left (- \frac {3}{4}\right )} - \frac {a d}{8 \sqrt {b} x^{10} \sqrt {\frac {a}{b x^{4}} + 1}} - \frac {3 \sqrt {b} d}{16 x^{6} \sqrt {\frac {a}{b x^{4}} + 1}} - \frac {\sqrt {b} f \sqrt {\frac {a}{b x^{4}} + 1}}{6 x^{4}} - \frac {b^{\frac {3}{2}} d}{16 a x^{2} \sqrt {\frac {a}{b x^{4}} + 1}} - \frac {b^{\frac {3}{2}} f \sqrt {\frac {a}{b x^{4}} + 1}}{6 a} + \frac {b^{2} d \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {b} x^{2}} \right )}}{16 a^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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