Optimal. Leaf size=474 \[ -\frac{2 b^{11/4} \left (\sqrt{a}+\sqrt{b} x^2\right ) \sqrt{\frac{a+b x^4}{\left (\sqrt{a}+\sqrt{b} x^2\right )^2}} \left (65 \sqrt{a} e+77 \sqrt{b} c\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{5005 a^{7/4} \sqrt{a+b x^4}}+\frac{4 b^{13/4} c \left (\sqrt{a}+\sqrt{b} x^2\right ) \sqrt{\frac{a+b x^4}{\left (\sqrt{a}+\sqrt{b} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{65 a^{7/4} \sqrt{a+b x^4}}+\frac{b^3 d \tanh ^{-1}\left (\frac{\sqrt{a+b x^4}}{\sqrt{a}}\right )}{32 a^{3/2}}-\frac{4 b^{7/2} c x \sqrt{a+b x^4}}{65 a^2 \left (\sqrt{a}+\sqrt{b} x^2\right )}+\frac{4 b^3 c \sqrt{a+b x^4}}{65 a^2 x}-\frac{4 b^2 c \sqrt{a+b x^4}}{195 a x^5}-\frac{b^2 d \sqrt{a+b x^4}}{32 a x^4}-\frac{4 b^2 e \sqrt{a+b x^4}}{77 a x^3}-\frac{b^2 f \sqrt{a+b x^4}}{10 a x^2}-\frac{\left (a+b x^4\right )^{3/2} \left (\frac{660 c}{x^{13}}+\frac{715 d}{x^{12}}+\frac{780 e}{x^{11}}+\frac{858 f}{x^{10}}\right )}{8580}-\frac{b \sqrt{a+b x^4} \left (\frac{12320 c}{x^9}+\frac{15015 d}{x^8}+\frac{18720 e}{x^7}+\frac{24024 f}{x^6}\right )}{240240} \]
[Out]
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Rubi [A] time = 1.20802, antiderivative size = 474, normalized size of antiderivative = 1., number of steps used = 16, number of rules used = 13, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.433 \[ -\frac{2 b^{11/4} \left (\sqrt{a}+\sqrt{b} x^2\right ) \sqrt{\frac{a+b x^4}{\left (\sqrt{a}+\sqrt{b} x^2\right )^2}} \left (65 \sqrt{a} e+77 \sqrt{b} c\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{5005 a^{7/4} \sqrt{a+b x^4}}+\frac{4 b^{13/4} c \left (\sqrt{a}+\sqrt{b} x^2\right ) \sqrt{\frac{a+b x^4}{\left (\sqrt{a}+\sqrt{b} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{65 a^{7/4} \sqrt{a+b x^4}}+\frac{b^3 d \tanh ^{-1}\left (\frac{\sqrt{a+b x^4}}{\sqrt{a}}\right )}{32 a^{3/2}}-\frac{4 b^{7/2} c x \sqrt{a+b x^4}}{65 a^2 \left (\sqrt{a}+\sqrt{b} x^2\right )}+\frac{4 b^3 c \sqrt{a+b x^4}}{65 a^2 x}-\frac{4 b^2 c \sqrt{a+b x^4}}{195 a x^5}-\frac{b^2 d \sqrt{a+b x^4}}{32 a x^4}-\frac{4 b^2 e \sqrt{a+b x^4}}{77 a x^3}-\frac{b^2 f \sqrt{a+b x^4}}{10 a x^2}-\frac{\left (a+b x^4\right )^{3/2} \left (\frac{660 c}{x^{13}}+\frac{715 d}{x^{12}}+\frac{780 e}{x^{11}}+\frac{858 f}{x^{10}}\right )}{8580}-\frac{b \sqrt{a+b x^4} \left (\frac{12320 c}{x^9}+\frac{15015 d}{x^8}+\frac{18720 e}{x^7}+\frac{24024 f}{x^6}\right )}{240240} \]
Antiderivative was successfully verified.
[In] Int[((c + d*x + e*x^2 + f*x^3)*(a + b*x^4)^(3/2))/x^14,x]
[Out]
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Rubi in Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((f*x**3+e*x**2+d*x+c)*(b*x**4+a)**(3/2)/x**14,x)
[Out]
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Mathematica [C] time = 1.09918, size = 339, normalized size = 0.72 \[ \frac{\sqrt{\frac{i \sqrt{b}}{\sqrt{a}}} \left (15015 \sqrt{a} b^3 d x^{13} \sqrt{a+b x^4} \tanh ^{-1}\left (\frac{\sqrt{a+b x^4}}{\sqrt{a}}\right )-\left (a+b x^4\right ) \left (56 a^3 \left (660 c+13 x \left (55 d+60 e x+66 f x^2\right )\right )+2 a^2 b x^4 (30800 c+13 x (2695 d+48 x (65 e+77 f x)))+a b^2 x^8 (9856 c+39 x (385 d+16 x (40 e+77 f x)))-29568 b^3 c x^{12}\right )\right )-29568 \sqrt{a} b^{7/2} c x^{13} \sqrt{\frac{b x^4}{a}+1} E\left (\left .i \sinh ^{-1}\left (\sqrt{\frac{i \sqrt{b}}{\sqrt{a}}} x\right )\right |-1\right )+384 \sqrt{a} b^3 x^{13} \sqrt{\frac{b x^4}{a}+1} \left (77 \sqrt{b} c+65 i \sqrt{a} e\right ) F\left (\left .i \sinh ^{-1}\left (\sqrt{\frac{i \sqrt{b}}{\sqrt{a}}} x\right )\right |-1\right )}{480480 a^2 x^{13} \sqrt{\frac{i \sqrt{b}}{\sqrt{a}}} \sqrt{a+b x^4}} \]
Antiderivative was successfully verified.
[In] Integrate[((c + d*x + e*x^2 + f*x^3)*(a + b*x^4)^(3/2))/x^14,x]
[Out]
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Maple [C] time = 0.034, size = 483, normalized size = 1. \[ -{\frac{ac}{13\,{x}^{13}}\sqrt{b{x}^{4}+a}}-{\frac{5\,bc}{39\,{x}^{9}}\sqrt{b{x}^{4}+a}}-{\frac{4\,{b}^{2}c}{195\,a{x}^{5}}\sqrt{b{x}^{4}+a}}+{\frac{4\,{b}^{3}c}{65\,{a}^{2}x}\sqrt{b{x}^{4}+a}}-{{\frac{4\,i}{65}}c{b}^{{\frac{7}{2}}}\sqrt{1-{i{x}^{2}\sqrt{b}{\frac{1}{\sqrt{a}}}}}\sqrt{1+{i{x}^{2}\sqrt{b}{\frac{1}{\sqrt{a}}}}}{\it EllipticF} \left ( x\sqrt{{i\sqrt{b}{\frac{1}{\sqrt{a}}}}},i \right ){a}^{-{\frac{3}{2}}}{\frac{1}{\sqrt{{i\sqrt{b}{\frac{1}{\sqrt{a}}}}}}}{\frac{1}{\sqrt{b{x}^{4}+a}}}}+{{\frac{4\,i}{65}}c{b}^{{\frac{7}{2}}}\sqrt{1-{i{x}^{2}\sqrt{b}{\frac{1}{\sqrt{a}}}}}\sqrt{1+{i{x}^{2}\sqrt{b}{\frac{1}{\sqrt{a}}}}}{\it EllipticE} \left ( x\sqrt{{i\sqrt{b}{\frac{1}{\sqrt{a}}}}},i \right ){a}^{-{\frac{3}{2}}}{\frac{1}{\sqrt{{i\sqrt{b}{\frac{1}{\sqrt{a}}}}}}}{\frac{1}{\sqrt{b{x}^{4}+a}}}}-{\frac{ad}{12\,{x}^{12}}\sqrt{b{x}^{4}+a}}-{\frac{7\,bd}{48\,{x}^{8}}\sqrt{b{x}^{4}+a}}-{\frac{{b}^{2}d}{32\,a{x}^{4}}\sqrt{b{x}^{4}+a}}+{\frac{{b}^{3}d}{32}\ln \left ({\frac{1}{{x}^{2}} \left ( 2\,a+2\,\sqrt{a}\sqrt{b{x}^{4}+a} \right ) } \right ){a}^{-{\frac{3}{2}}}}-{\frac{ae}{11\,{x}^{11}}\sqrt{b{x}^{4}+a}}-{\frac{13\,be}{77\,{x}^{7}}\sqrt{b{x}^{4}+a}}-{\frac{4\,{b}^{2}e}{77\,a{x}^{3}}\sqrt{b{x}^{4}+a}}-{\frac{4\,e{b}^{3}}{77\,a}\sqrt{1-{i{x}^{2}\sqrt{b}{\frac{1}{\sqrt{a}}}}}\sqrt{1+{i{x}^{2}\sqrt{b}{\frac{1}{\sqrt{a}}}}}{\it EllipticF} \left ( x\sqrt{{i\sqrt{b}{\frac{1}{\sqrt{a}}}}},i \right ){\frac{1}{\sqrt{{i\sqrt{b}{\frac{1}{\sqrt{a}}}}}}}{\frac{1}{\sqrt{b{x}^{4}+a}}}}-{\frac{f \left ({b}^{2}{x}^{8}+2\,ab{x}^{4}+{a}^{2} \right ) }{10\,{x}^{10}a}\sqrt{b{x}^{4}+a}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((f*x^3+e*x^2+d*x+c)*(b*x^4+a)^(3/2)/x^14,x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x^{4} + a\right )}^{\frac{3}{2}}{\left (f x^{3} + e x^{2} + d x + c\right )}}{x^{14}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^4 + a)^(3/2)*(f*x^3 + e*x^2 + d*x + c)/x^14,x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (b f x^{7} + b e x^{6} + b d x^{5} + b c x^{4} + a f x^{3} + a e x^{2} + a d x + a c\right )} \sqrt{b x^{4} + a}}{x^{14}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^4 + a)^(3/2)*(f*x^3 + e*x^2 + d*x + c)/x^14,x, algorithm="fricas")
[Out]
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Sympy [A] time = 42.4552, size = 403, normalized size = 0.85 \[ \frac{a^{\frac{3}{2}} c \Gamma \left (- \frac{13}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{13}{4}, - \frac{1}{2} \\ - \frac{9}{4} \end{matrix}\middle |{\frac{b x^{4} e^{i \pi }}{a}} \right )}}{4 x^{13} \Gamma \left (- \frac{9}{4}\right )} + \frac{a^{\frac{3}{2}} e \Gamma \left (- \frac{11}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{11}{4}, - \frac{1}{2} \\ - \frac{7}{4} \end{matrix}\middle |{\frac{b x^{4} e^{i \pi }}{a}} \right )}}{4 x^{11} \Gamma \left (- \frac{7}{4}\right )} + \frac{\sqrt{a} b c \Gamma \left (- \frac{9}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{9}{4}, - \frac{1}{2} \\ - \frac{5}{4} \end{matrix}\middle |{\frac{b x^{4} e^{i \pi }}{a}} \right )}}{4 x^{9} \Gamma \left (- \frac{5}{4}\right )} + \frac{\sqrt{a} b e \Gamma \left (- \frac{7}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{7}{4}, - \frac{1}{2} \\ - \frac{3}{4} \end{matrix}\middle |{\frac{b x^{4} e^{i \pi }}{a}} \right )}}{4 x^{7} \Gamma \left (- \frac{3}{4}\right )} - \frac{a^{2} d}{12 \sqrt{b} x^{14} \sqrt{\frac{a}{b x^{4}} + 1}} - \frac{11 a \sqrt{b} d}{48 x^{10} \sqrt{\frac{a}{b x^{4}} + 1}} - \frac{a \sqrt{b} f \sqrt{\frac{a}{b x^{4}} + 1}}{10 x^{8}} - \frac{17 b^{\frac{3}{2}} d}{96 x^{6} \sqrt{\frac{a}{b x^{4}} + 1}} - \frac{b^{\frac{3}{2}} f \sqrt{\frac{a}{b x^{4}} + 1}}{5 x^{4}} - \frac{b^{\frac{5}{2}} d}{32 a x^{2} \sqrt{\frac{a}{b x^{4}} + 1}} - \frac{b^{\frac{5}{2}} f \sqrt{\frac{a}{b x^{4}} + 1}}{10 a} + \frac{b^{3} d \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{b} x^{2}} \right )}}{32 a^{\frac{3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((f*x**3+e*x**2+d*x+c)*(b*x**4+a)**(3/2)/x**14,x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x^{4} + a\right )}^{\frac{3}{2}}{\left (f x^{3} + e x^{2} + d x + c\right )}}{x^{14}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^4 + a)^(3/2)*(f*x^3 + e*x^2 + d*x + c)/x^14,x, algorithm="giac")
[Out]