Optimal. Leaf size=412 \[ \frac {2 b^{5/4} \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} \left (21 \sqrt {a} e+5 \sqrt {b} c\right ) F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{35 \sqrt [4]{a} \sqrt {a+b x^4}}+\frac {1}{2} b^{3/2} d \tanh ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a+b x^4}}\right )+\frac {12 b^{3/2} e x \sqrt {a+b x^4}}{5 \left (\sqrt {a}+\sqrt {b} x^2\right )}-\frac {12 \sqrt [4]{a} b^{5/4} e \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{5 \sqrt {a+b x^4}}-\frac {1}{420} \left (a+b x^4\right )^{3/2} \left (\frac {60 c}{x^7}+\frac {70 d}{x^6}+\frac {84 e}{x^5}+\frac {105 f}{x^4}\right )-\frac {2 b \sqrt {a+b x^4} \left (5 c-21 e x^2\right )}{35 x^3}-\frac {b \sqrt {a+b x^4} \left (2 d-3 f x^2\right )}{4 x^2}-\frac {12 b e \sqrt {a+b x^4}}{5 x}-\frac {3}{4} \sqrt {a} b f \tanh ^{-1}\left (\frac {\sqrt {a+b x^4}}{\sqrt {a}}\right ) \]
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Rubi [A] time = 0.39, antiderivative size = 412, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 16, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.533, Rules used = {14, 1825, 1833, 1272, 1282, 1198, 220, 1196, 1252, 813, 844, 217, 206, 266, 63, 208} \[ \frac {2 b^{5/4} \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} \left (21 \sqrt {a} e+5 \sqrt {b} c\right ) F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{35 \sqrt [4]{a} \sqrt {a+b x^4}}+\frac {1}{2} b^{3/2} d \tanh ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a+b x^4}}\right )+\frac {12 b^{3/2} e x \sqrt {a+b x^4}}{5 \left (\sqrt {a}+\sqrt {b} x^2\right )}-\frac {12 \sqrt [4]{a} b^{5/4} e \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{5 \sqrt {a+b x^4}}-\frac {1}{420} \left (a+b x^4\right )^{3/2} \left (\frac {60 c}{x^7}+\frac {70 d}{x^6}+\frac {84 e}{x^5}+\frac {105 f}{x^4}\right )-\frac {2 b \sqrt {a+b x^4} \left (5 c-21 e x^2\right )}{35 x^3}-\frac {b \sqrt {a+b x^4} \left (2 d-3 f x^2\right )}{4 x^2}-\frac {12 b e \sqrt {a+b x^4}}{5 x}-\frac {3}{4} \sqrt {a} b f \tanh ^{-1}\left (\frac {\sqrt {a+b x^4}}{\sqrt {a}}\right ) \]
Antiderivative was successfully verified.
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Rule 14
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 1282
Rule 1825
Rule 1833
Rubi steps
\begin {align*} \int \frac {\left (c+d x+e x^2+f x^3\right ) \left (a+b x^4\right )^{3/2}}{x^8} \, dx &=-\frac {1}{420} \left (\frac {60 c}{x^7}+\frac {70 d}{x^6}+\frac {84 e}{x^5}+\frac {105 f}{x^4}\right ) \left (a+b x^4\right )^{3/2}-(6 b) \int \frac {\left (-\frac {c}{7}-\frac {d x}{6}-\frac {e x^2}{5}-\frac {f x^3}{4}\right ) \sqrt {a+b x^4}}{x^4} \, dx\\ &=-\frac {1}{420} \left (\frac {60 c}{x^7}+\frac {70 d}{x^6}+\frac {84 e}{x^5}+\frac {105 f}{x^4}\right ) \left (a+b x^4\right )^{3/2}-(6 b) \int \left (\frac {\left (-\frac {c}{7}-\frac {e x^2}{5}\right ) \sqrt {a+b x^4}}{x^4}+\frac {\left (-\frac {d}{6}-\frac {f x^2}{4}\right ) \sqrt {a+b x^4}}{x^3}\right ) \, dx\\ &=-\frac {1}{420} \left (\frac {60 c}{x^7}+\frac {70 d}{x^6}+\frac {84 e}{x^5}+\frac {105 f}{x^4}\right ) \left (a+b x^4\right )^{3/2}-(6 b) \int \frac {\left (-\frac {c}{7}-\frac {e x^2}{5}\right ) \sqrt {a+b x^4}}{x^4} \, dx-(6 b) \int \frac {\left (-\frac {d}{6}-\frac {f x^2}{4}\right ) \sqrt {a+b x^4}}{x^3} \, dx\\ &=-\frac {2 b \left (5 c-21 e x^2\right ) \sqrt {a+b x^4}}{35 x^3}-\frac {1}{420} \left (\frac {60 c}{x^7}+\frac {70 d}{x^6}+\frac {84 e}{x^5}+\frac {105 f}{x^4}\right ) \left (a+b x^4\right )^{3/2}-(3 b) \operatorname {Subst}\left (\int \frac {\left (-\frac {d}{6}-\frac {f x}{4}\right ) \sqrt {a+b x^2}}{x^2} \, dx,x,x^2\right )+(4 b) \int \frac {\frac {3 a e}{5}+\frac {1}{7} b c x^2}{x^2 \sqrt {a+b x^4}} \, dx\\ &=-\frac {12 b e \sqrt {a+b x^4}}{5 x}-\frac {2 b \left (5 c-21 e x^2\right ) \sqrt {a+b x^4}}{35 x^3}-\frac {b \left (2 d-3 f x^2\right ) \sqrt {a+b x^4}}{4 x^2}-\frac {1}{420} \left (\frac {60 c}{x^7}+\frac {70 d}{x^6}+\frac {84 e}{x^5}+\frac {105 f}{x^4}\right ) \left (a+b x^4\right )^{3/2}+\frac {1}{2} (3 b) \operatorname {Subst}\left (\int \frac {\frac {a f}{2}+\frac {b d x}{3}}{x \sqrt {a+b x^2}} \, dx,x,x^2\right )-\frac {(4 b) \int \frac {-\frac {1}{7} a b c-\frac {3}{5} a b e x^2}{\sqrt {a+b x^4}} \, dx}{a}\\ &=-\frac {12 b e \sqrt {a+b x^4}}{5 x}-\frac {2 b \left (5 c-21 e x^2\right ) \sqrt {a+b x^4}}{35 x^3}-\frac {b \left (2 d-3 f x^2\right ) \sqrt {a+b x^4}}{4 x^2}-\frac {1}{420} \left (\frac {60 c}{x^7}+\frac {70 d}{x^6}+\frac {84 e}{x^5}+\frac {105 f}{x^4}\right ) \left (a+b x^4\right )^{3/2}+\frac {1}{2} \left (b^2 d\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {a+b x^2}} \, dx,x,x^2\right )-\frac {1}{5} \left (12 \sqrt {a} b^{3/2} e\right ) \int \frac {1-\frac {\sqrt {b} x^2}{\sqrt {a}}}{\sqrt {a+b x^4}} \, dx+\frac {1}{35} \left (4 b^{3/2} \left (5 \sqrt {b} c+21 \sqrt {a} e\right )\right ) \int \frac {1}{\sqrt {a+b x^4}} \, dx+\frac {1}{4} (3 a b f) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {a+b x^2}} \, dx,x,x^2\right )\\ &=-\frac {12 b e \sqrt {a+b x^4}}{5 x}+\frac {12 b^{3/2} e x \sqrt {a+b x^4}}{5 \left (\sqrt {a}+\sqrt {b} x^2\right )}-\frac {2 b \left (5 c-21 e x^2\right ) \sqrt {a+b x^4}}{35 x^3}-\frac {b \left (2 d-3 f x^2\right ) \sqrt {a+b x^4}}{4 x^2}-\frac {1}{420} \left (\frac {60 c}{x^7}+\frac {70 d}{x^6}+\frac {84 e}{x^5}+\frac {105 f}{x^4}\right ) \left (a+b x^4\right )^{3/2}-\frac {12 \sqrt [4]{a} b^{5/4} e \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 )}{5 \sqrt {a+b x^4}}+\frac {2 b^{5/4} \left (5 \sqrt {b} c+21 \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 )}{35 \sqrt [4]{a} \sqrt {a+b x^4}}+\frac {1}{2} \left (b^2 d\right ) \operatorname {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {x^2}{\sqrt {a+b x^4}}\right )+\frac {1}{8} (3 a b f) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,x^4\right )\\ &=-\frac {12 b e \sqrt {a+b x^4}}{5 x}+\frac {12 b^{3/2} e x \sqrt {a+b x^4}}{5 \left (\sqrt {a}+\sqrt {b} x^2\right )}-\frac {2 b \left (5 c-21 e x^2\right ) \sqrt {a+b x^4}}{35 x^3}-\frac {b \left (2 d-3 f x^2\right ) \sqrt {a+b x^4}}{4 x^2}-\frac {1}{420} \left (\frac {60 c}{x^7}+\frac {70 d}{x^6}+\frac {84 e}{x^5}+\frac {105 f}{x^4}\right ) \left (a+b x^4\right )^{3/2}+\frac {1}{2} b^{3/2} d \tanh ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a+b x^4}}\right )-\frac {12 \sqrt [4]{a} b^{5/4} e \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 )}{5 \sqrt {a+b x^4}}+\frac {2 b^{5/4} \left (5 \sqrt {b} c+21 \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 )}{35 \sqrt [4]{a} \sqrt {a+b x^4}}+\frac {1}{4} (3 a f) \operatorname {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b x^4}\right )\\ &=-\frac {12 b e \sqrt {a+b x^4}}{5 x}+\frac {12 b^{3/2} e x \sqrt {a+b x^4}}{5 \left (\sqrt {a}+\sqrt {b} x^2\right )}-\frac {2 b \left (5 c-21 e x^2\right ) \sqrt {a+b x^4}}{35 x^3}-\frac {b \left (2 d-3 f x^2\right ) \sqrt {a+b x^4}}{4 x^2}-\frac {1}{420} \left (\frac {60 c}{x^7}+\frac {70 d}{x^6}+\frac {84 e}{x^5}+\frac {105 f}{x^4}\right ) \left (a+b x^4\right )^{3/2}+\frac {1}{2} b^{3/2} d \tanh ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a+b x^4}}\right )-\frac {3}{4} \sqrt {a} b f \tanh ^{-1}\left (\frac {\sqrt {a+b x^4}}{\sqrt {a}}\right )-\frac {12 \sqrt [4]{a} b^{5/4} e \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 )}{5 \sqrt {a+b x^4}}+\frac {2 b^{5/4} \left (5 \sqrt {b} c+21 \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 )}{35 \sqrt [4]{a} \sqrt {a+b x^4}}\\ \end {align*}
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Mathematica [C] time = 0.25, size = 164, normalized size = 0.40 \[ \frac {\sqrt {a+b x^4} \left (7 x \left (-5 a^3 d \, _2F_1\left (-\frac {3}{2},-\frac {3}{2};-\frac {1}{2};-\frac {b x^4}{a}\right )-6 a^3 e x \, _2F_1\left (-\frac {3}{2},-\frac {5}{4};-\frac {1}{4};-\frac {b x^4}{a}\right )+3 b f x^6 \left (a+b x^4\right )^2 \sqrt {\frac {b x^4}{a}+1} \, _2F_1\left (2,\frac {5}{2};\frac {7}{2};\frac {b x^4}{a}+1\right )\right )-30 a^3 c \, _2F_1\left (-\frac {7}{4},-\frac {3}{2};-\frac {3}{4};-\frac {b x^4}{a}\right )\right )}{210 a^2 x^7 \sqrt {\frac {b x^4}{a}+1}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.75, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (b f x^{7} + b e x^{6} + b d x^{5} + b c x^{4} + a f x^{3} + a e x^{2} + a d x + a c\right )} \sqrt {b x^{4} + a}}{x^{8}}, x\right ) \]
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 \[ \int \frac {{\left (b x^{4} + a\right )}^{\frac {3}{2}} {\left (f x^{3} + e x^{2} + d x + c\right )}}{x^{8}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.22, size = 411, normalized size = 1.00 \[ -\frac {12 i \sqrt {-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}+1}\, \sqrt {\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}+1}\, \sqrt {a}\, b^{\frac {3}{2}} e \EllipticE \left (\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, x , i\right )}{5 \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}+\frac {12 i \sqrt {-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}+1}\, \sqrt {\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}+1}\, \sqrt {a}\, b^{\frac {3}{2}} e \EllipticF \left (\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, x , i\right )}{5 \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}+\frac {4 \sqrt {-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}+1}\, \sqrt {\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}+1}\, b^{2} c \EllipticF \left (\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, x , i\right )}{7 \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}-\frac {3 \sqrt {a}\, b f \ln \left (\frac {2 a +2 \sqrt {b \,x^{4}+a}\, \sqrt {a}}{x^{2}}\right )}{4}+\frac {b^{\frac {3}{2}} d \ln \left (\sqrt {b}\, x^{2}+\sqrt {b \,x^{4}+a}\right )}{2}+\frac {\sqrt {b \,x^{4}+a}\, b f}{2}-\frac {7 \sqrt {b \,x^{4}+a}\, b e}{5 x}-\frac {2 \sqrt {b \,x^{4}+a}\, b d}{3 x^{2}}-\frac {3 \sqrt {b \,x^{4}+a}\, b c}{7 x^{3}}-\frac {\sqrt {b \,x^{4}+a}\, a f}{4 x^{4}}-\frac {\sqrt {b \,x^{4}+a}\, a e}{5 x^{5}}-\frac {\sqrt {b \,x^{4}+a}\, a d}{6 x^{6}}-\frac {\sqrt {b \,x^{4}+a}\, a c}{7 x^{7}} \]
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 \[ \int \frac {{\left (b x^{4} + a\right )}^{\frac {3}{2}} {\left (f x^{3} + e x^{2} + d x + c\right )}}{x^{8}}\,{d x} \]
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 \[ \int \frac {{\left (b\,x^4+a\right )}^{3/2}\,\left (f\,x^3+e\,x^2+d\,x+c\right )}{x^8} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 13.00, size = 415, normalized size = 1.01 \[ \frac {a^{\frac {3}{2}} c \Gamma \left (- \frac {7}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {7}{4}, - \frac {1}{2} \\ - \frac {3}{4} \end {matrix}\middle | {\frac {b x^{4} e^{i \pi }}{a}} \right )}}{4 x^{7} \Gamma \left (- \frac {3}{4}\right )} + \frac {a^{\frac {3}{2}} e \Gamma \left (- \frac {5}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {5}{4}, - \frac {1}{2} \\ - \frac {1}{4} \end {matrix}\middle | {\frac {b x^{4} e^{i \pi }}{a}} \right )}}{4 x^{5} \Gamma \left (- \frac {1}{4}\right )} + \frac {\sqrt {a} b c \Gamma \left (- \frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {3}{4}, - \frac {1}{2} \\ \frac {1}{4} \end {matrix}\middle | {\frac {b x^{4} e^{i \pi }}{a}} \right )}}{4 x^{3} \Gamma \left (\frac {1}{4}\right )} - \frac {\sqrt {a} b d}{2 x^{2} \sqrt {1 + \frac {b x^{4}}{a}}} + \frac {\sqrt {a} b e \Gamma \left (- \frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, - \frac {1}{4} \\ \frac {3}{4} \end {matrix}\middle | {\frac {b x^{4} e^{i \pi }}{a}} \right )}}{4 x \Gamma \left (\frac {3}{4}\right )} - \frac {3 \sqrt {a} b f \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {b} x^{2}} \right )}}{4} - \frac {a \sqrt {b} d \sqrt {\frac {a}{b x^{4}} + 1}}{6 x^{4}} - \frac {a \sqrt {b} f \sqrt {\frac {a}{b x^{4}} + 1}}{4 x^{2}} + \frac {a \sqrt {b} f}{2 x^{2} \sqrt {\frac {a}{b x^{4}} + 1}} - \frac {b^{\frac {3}{2}} d \sqrt {\frac {a}{b x^{4}} + 1}}{6} + \frac {b^{\frac {3}{2}} d \operatorname {asinh}{\left (\frac {\sqrt {b} x^{2}}{\sqrt {a}} \right )}}{2} + \frac {b^{\frac {3}{2}} f x^{2}}{2 \sqrt {\frac {a}{b x^{4}} + 1}} - \frac {b^{2} d x^{2}}{2 \sqrt {a} \sqrt {1 + \frac {b x^{4}}{a}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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