Optimal. Leaf size=399 \[ \frac{2 b^{7/4} \left (\sqrt{a}+\sqrt{b} x^2\right ) \sqrt{\frac{a+b x^4}{\left (\sqrt{a}+\sqrt{b} x^2\right )^2}} \left (15 \sqrt{a} f+7 \sqrt{b} d\right ) \text{EllipticF}\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),\frac{1}{2}\right )}{105 a^{3/4} \sqrt{a+b x^4}}-\frac{4 b^{9/4} d \left (\sqrt{a}+\sqrt{b} x^2\right ) \sqrt{\frac{a+b x^4}{\left (\sqrt{a}+\sqrt{b} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{15 a^{3/4} \sqrt{a+b x^4}}-\frac{b^2 c \sqrt{a+b x^4}}{10 a x^2}+\frac{4 b^{5/2} d x \sqrt{a+b x^4}}{15 a \left (\sqrt{a}+\sqrt{b} x^2\right )}-\frac{4 b^2 d \sqrt{a+b x^4}}{15 a x}-\frac{3 b^2 e \tanh ^{-1}\left (\frac{\sqrt{a+b x^4}}{\sqrt{a}}\right )}{16 \sqrt{a}}-\frac{b \sqrt{a+b x^4} \left (\frac{168 c}{x^6}+\frac{224 d}{x^5}+\frac{315 e}{x^4}+\frac{480 f}{x^3}\right )}{1680}-\frac{\left (a+b x^4\right )^{3/2} \left (\frac{252 c}{x^{10}}+\frac{280 d}{x^9}+\frac{315 e}{x^8}+\frac{360 f}{x^7}\right )}{2520} \]
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Rubi [A] time = 0.420172, antiderivative size = 399, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 12, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4, Rules used = {14, 1825, 1833, 1252, 807, 266, 63, 208, 1282, 1198, 220, 1196} \[ \frac{2 b^{7/4} \left (\sqrt{a}+\sqrt{b} x^2\right ) \sqrt{\frac{a+b x^4}{\left (\sqrt{a}+\sqrt{b} x^2\right )^2}} \left (15 \sqrt{a} f+7 \sqrt{b} d\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{105 a^{3/4} \sqrt{a+b x^4}}-\frac{4 b^{9/4} d \left (\sqrt{a}+\sqrt{b} x^2\right ) \sqrt{\frac{a+b x^4}{\left (\sqrt{a}+\sqrt{b} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{15 a^{3/4} \sqrt{a+b x^4}}-\frac{b^2 c \sqrt{a+b x^4}}{10 a x^2}+\frac{4 b^{5/2} d x \sqrt{a+b x^4}}{15 a \left (\sqrt{a}+\sqrt{b} x^2\right )}-\frac{4 b^2 d \sqrt{a+b x^4}}{15 a x}-\frac{3 b^2 e \tanh ^{-1}\left (\frac{\sqrt{a+b x^4}}{\sqrt{a}}\right )}{16 \sqrt{a}}-\frac{b \sqrt{a+b x^4} \left (\frac{168 c}{x^6}+\frac{224 d}{x^5}+\frac{315 e}{x^4}+\frac{480 f}{x^3}\right )}{1680}-\frac{\left (a+b x^4\right )^{3/2} \left (\frac{252 c}{x^{10}}+\frac{280 d}{x^9}+\frac{315 e}{x^8}+\frac{360 f}{x^7}\right )}{2520} \]
Antiderivative was successfully verified.
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Rule 14
Rule 1825
Rule 1833
Rule 1252
Rule 807
Rule 266
Rule 63
Rule 208
Rule 1282
Rule 1198
Rule 220
Rule 1196
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^{11}} \, dx &=-\frac{\left (\frac{252 c}{x^{10}}+\frac{280 d}{x^9}+\frac{315 e}{x^8}+\frac{360 f}{x^7}\right ) \left (a+b x^4\right )^{3/2}}{2520}-(6 b) \int \frac{\left (-\frac{c}{10}-\frac{d x}{9}-\frac{e x^2}{8}-\frac{f x^3}{7}\right ) \sqrt{a+b x^4}}{x^7} \, dx\\ &=-\frac{b \left (\frac{168 c}{x^6}+\frac{224 d}{x^5}+\frac{315 e}{x^4}+\frac{480 f}{x^3}\right ) \sqrt{a+b x^4}}{1680}-\frac{\left (\frac{252 c}{x^{10}}+\frac{280 d}{x^9}+\frac{315 e}{x^8}+\frac{360 f}{x^7}\right ) \left (a+b x^4\right )^{3/2}}{2520}+\left (12 b^2\right ) \int \frac{\frac{c}{60}+\frac{d x}{45}+\frac{e x^2}{32}+\frac{f x^3}{21}}{x^3 \sqrt{a+b x^4}} \, dx\\ &=-\frac{b \left (\frac{168 c}{x^6}+\frac{224 d}{x^5}+\frac{315 e}{x^4}+\frac{480 f}{x^3}\right ) \sqrt{a+b x^4}}{1680}-\frac{\left (\frac{252 c}{x^{10}}+\frac{280 d}{x^9}+\frac{315 e}{x^8}+\frac{360 f}{x^7}\right ) \left (a+b x^4\right )^{3/2}}{2520}+\left (12 b^2\right ) \int \left (\frac{\frac{c}{60}+\frac{e x^2}{32}}{x^3 \sqrt{a+b x^4}}+\frac{\frac{d}{45}+\frac{f x^2}{21}}{x^2 \sqrt{a+b x^4}}\right ) \, dx\\ &=-\frac{b \left (\frac{168 c}{x^6}+\frac{224 d}{x^5}+\frac{315 e}{x^4}+\frac{480 f}{x^3}\right ) \sqrt{a+b x^4}}{1680}-\frac{\left (\frac{252 c}{x^{10}}+\frac{280 d}{x^9}+\frac{315 e}{x^8}+\frac{360 f}{x^7}\right ) \left (a+b x^4\right )^{3/2}}{2520}+\left (12 b^2\right ) \int \frac{\frac{c}{60}+\frac{e x^2}{32}}{x^3 \sqrt{a+b x^4}} \, dx+\left (12 b^2\right ) \int \frac{\frac{d}{45}+\frac{f x^2}{21}}{x^2 \sqrt{a+b x^4}} \, dx\\ &=-\frac{b \left (\frac{168 c}{x^6}+\frac{224 d}{x^5}+\frac{315 e}{x^4}+\frac{480 f}{x^3}\right ) \sqrt{a+b x^4}}{1680}-\frac{4 b^2 d \sqrt{a+b x^4}}{15 a x}-\frac{\left (\frac{252 c}{x^{10}}+\frac{280 d}{x^9}+\frac{315 e}{x^8}+\frac{360 f}{x^7}\right ) \left (a+b x^4\right )^{3/2}}{2520}+\left (6 b^2\right ) \operatorname{Subst}\left (\int \frac{\frac{c}{60}+\frac{e x}{32}}{x^2 \sqrt{a+b x^2}} \, dx,x,x^2\right )-\frac{\left (12 b^2\right ) \int \frac{-\frac{a f}{21}-\frac{1}{45} b d x^2}{\sqrt{a+b x^4}} \, dx}{a}\\ &=-\frac{b \left (\frac{168 c}{x^6}+\frac{224 d}{x^5}+\frac{315 e}{x^4}+\frac{480 f}{x^3}\right ) \sqrt{a+b x^4}}{1680}-\frac{b^2 c \sqrt{a+b x^4}}{10 a x^2}-\frac{4 b^2 d \sqrt{a+b x^4}}{15 a x}-\frac{\left (\frac{252 c}{x^{10}}+\frac{280 d}{x^9}+\frac{315 e}{x^8}+\frac{360 f}{x^7}\right ) \left (a+b x^4\right )^{3/2}}{2520}-\frac{\left (4 b^{5/2} d\right ) \int \frac{1-\frac{\sqrt{b} x^2}{\sqrt{a}}}{\sqrt{a+b x^4}} \, dx}{15 \sqrt{a}}+\frac{1}{16} \left (3 b^2 e\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+b x^2}} \, dx,x,x^2\right )+\frac{1}{105} \left (4 b^2 \left (\frac{7 \sqrt{b} d}{\sqrt{a}}+15 f\right )\right ) \int \frac{1}{\sqrt{a+b x^4}} \, dx\\ &=-\frac{b \left (\frac{168 c}{x^6}+\frac{224 d}{x^5}+\frac{315 e}{x^4}+\frac{480 f}{x^3}\right ) \sqrt{a+b x^4}}{1680}-\frac{b^2 c \sqrt{a+b x^4}}{10 a x^2}-\frac{4 b^2 d \sqrt{a+b x^4}}{15 a x}+\frac{4 b^{5/2} d x \sqrt{a+b x^4}}{15 a \left (\sqrt{a}+\sqrt{b} x^2\right )}-\frac{\left (\frac{252 c}{x^{10}}+\frac{280 d}{x^9}+\frac{315 e}{x^8}+\frac{360 f}{x^7}\right ) \left (a+b x^4\right )^{3/2}}{2520}-\frac{4 b^{9/4} 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 )}{15 a^{3/4} \sqrt{a+b x^4}}+\frac{2 b^{7/4} \left (7 \sqrt{b} d+15 \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 )}{105 a^{3/4} \sqrt{a+b x^4}}+\frac{1}{32} \left (3 b^2 e\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+b x}} \, dx,x,x^4\right )\\ &=-\frac{b \left (\frac{168 c}{x^6}+\frac{224 d}{x^5}+\frac{315 e}{x^4}+\frac{480 f}{x^3}\right ) \sqrt{a+b x^4}}{1680}-\frac{b^2 c \sqrt{a+b x^4}}{10 a x^2}-\frac{4 b^2 d \sqrt{a+b x^4}}{15 a x}+\frac{4 b^{5/2} d x \sqrt{a+b x^4}}{15 a \left (\sqrt{a}+\sqrt{b} x^2\right )}-\frac{\left (\frac{252 c}{x^{10}}+\frac{280 d}{x^9}+\frac{315 e}{x^8}+\frac{360 f}{x^7}\right ) \left (a+b x^4\right )^{3/2}}{2520}-\frac{4 b^{9/4} 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 )}{15 a^{3/4} \sqrt{a+b x^4}}+\frac{2 b^{7/4} \left (7 \sqrt{b} d+15 \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 )}{105 a^{3/4} \sqrt{a+b x^4}}+\frac{1}{16} (3 b e) \operatorname{Subst}\left (\int \frac{1}{-\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{a+b x^4}\right )\\ &=-\frac{b \left (\frac{168 c}{x^6}+\frac{224 d}{x^5}+\frac{315 e}{x^4}+\frac{480 f}{x^3}\right ) \sqrt{a+b x^4}}{1680}-\frac{b^2 c \sqrt{a+b x^4}}{10 a x^2}-\frac{4 b^2 d \sqrt{a+b x^4}}{15 a x}+\frac{4 b^{5/2} d x \sqrt{a+b x^4}}{15 a \left (\sqrt{a}+\sqrt{b} x^2\right )}-\frac{\left (\frac{252 c}{x^{10}}+\frac{280 d}{x^9}+\frac{315 e}{x^8}+\frac{360 f}{x^7}\right ) \left (a+b x^4\right )^{3/2}}{2520}-\frac{3 b^2 e \tanh ^{-1}\left (\frac{\sqrt{a+b x^4}}{\sqrt{a}}\right )}{16 \sqrt{a}}-\frac{4 b^{9/4} 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 )}{15 a^{3/4} \sqrt{a+b x^4}}+\frac{2 b^{7/4} \left (7 \sqrt{b} d+15 \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 )}{105 a^{3/4} \sqrt{a+b x^4}}\\ \end{align*}
Mathematica [C] time = 0.326002, size = 171, normalized size = 0.43 \[ -\frac{\sqrt{a+b x^4} \left (63 \sqrt{\frac{b x^4}{a}+1} \left (2 a^2 \left (4 c+5 e x^2\right )+a b x^4 \left (16 c+25 e x^2\right )+8 b^2 c x^8\right )+560 a^2 d x \, _2F_1\left (-\frac{9}{4},-\frac{3}{2};-\frac{5}{4};-\frac{b x^4}{a}\right )+720 a^2 f x^3 \, _2F_1\left (-\frac{7}{4},-\frac{3}{2};-\frac{3}{4};-\frac{b x^4}{a}\right )+945 b^2 e x^{10} \tanh ^{-1}\left (\sqrt{\frac{b x^4}{a}+1}\right )\right )}{5040 a x^{10} \sqrt{\frac{b x^4}{a}+1}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.019, size = 417, normalized size = 1.1 \begin{align*} -{\frac{c \left ({b}^{2}{x}^{8}+2\,ab{x}^{4}+{a}^{2} \right ) }{10\,{x}^{10}a}\sqrt{b{x}^{4}+a}}-{\frac{3\,{b}^{2}e}{16}\ln \left ({\frac{1}{{x}^{2}} \left ( 2\,a+2\,\sqrt{a}\sqrt{b{x}^{4}+a} \right ) } \right ){\frac{1}{\sqrt{a}}}}-{\frac{ae}{8\,{x}^{8}}\sqrt{b{x}^{4}+a}}-{\frac{5\,be}{16\,{x}^{4}}\sqrt{b{x}^{4}+a}}-{\frac{af}{7\,{x}^{7}}\sqrt{b{x}^{4}+a}}-{\frac{3\,fb}{7\,{x}^{3}}\sqrt{b{x}^{4}+a}}+{\frac{4\,{b}^{2}f}{7}\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{ad}{9\,{x}^{9}}\sqrt{b{x}^{4}+a}}-{\frac{11\,bd}{45\,{x}^{5}}\sqrt{b{x}^{4}+a}}-{\frac{4\,{b}^{2}d}{15\,ax}\sqrt{b{x}^{4}+a}}+{{\frac{4\,i}{15}}d{b}^{{\frac{5}{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 ){\frac{1}{\sqrt{a}}}{\frac{1}{\sqrt{{i\sqrt{b}{\frac{1}{\sqrt{a}}}}}}}{\frac{1}{\sqrt{b{x}^{4}+a}}}}-{{\frac{4\,i}{15}}d{b}^{{\frac{5}{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 ){\frac{1}{\sqrt{a}}}{\frac{1}{\sqrt{{i\sqrt{b}{\frac{1}{\sqrt{a}}}}}}}{\frac{1}{\sqrt{b{x}^{4}+a}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{{\left (b x^{4} + a\right )}^{\frac{5}{2}} c}{10 \, a x^{10}} + \int \frac{{\left (b f x^{6} + b e x^{5} + b d x^{4} + a f x^{2} + a e x + a d\right )} \sqrt{b x^{4} + a}}{x^{10}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\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^{11}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 15.602, size = 398, normalized size = 1. \begin{align*} \frac{a^{\frac{3}{2}} d \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{a^{\frac{3}{2}} f \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{\sqrt{a} b d \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 f \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{a^{2} e}{8 \sqrt{b} x^{10} \sqrt{\frac{a}{b x^{4}} + 1}} - \frac{a \sqrt{b} c \sqrt{\frac{a}{b x^{4}} + 1}}{10 x^{8}} - \frac{3 a \sqrt{b} e}{16 x^{6} \sqrt{\frac{a}{b x^{4}} + 1}} - \frac{b^{\frac{3}{2}} c \sqrt{\frac{a}{b x^{4}} + 1}}{5 x^{4}} - \frac{b^{\frac{3}{2}} e \sqrt{\frac{a}{b x^{4}} + 1}}{4 x^{2}} - \frac{b^{\frac{3}{2}} e}{16 x^{2} \sqrt{\frac{a}{b x^{4}} + 1}} - \frac{b^{\frac{5}{2}} c \sqrt{\frac{a}{b x^{4}} + 1}}{10 a} - \frac{3 b^{2} e \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{b} x^{2}} \right )}}{16 \sqrt{a}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b x^{4} + a\right )}^{\frac{3}{2}}{\left (f x^{3} + e x^{2} + d x + c\right )}}{x^{11}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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