Optimal. Leaf size=382 \[ \frac{2 a^{7/4} \left (\sqrt{a}+\sqrt{b} x^2\right ) \sqrt{\frac{a+b x^4}{\left (\sqrt{a}+\sqrt{b} x^2\right )^2}} \left (7 \sqrt{a} e+15 \sqrt{b} c\right ) \text{EllipticF}\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),\frac{1}{2}\right )}{105 b^{3/4} \sqrt{a+b x^4}}-\frac{4 a^{9/4} e \left (\sqrt{a}+\sqrt{b} x^2\right ) \sqrt{\frac{a+b x^4}{\left (\sqrt{a}+\sqrt{b} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{15 b^{3/4} \sqrt{a+b x^4}}+\frac{3 a^2 d \tanh ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a+b x^4}}\right )}{16 \sqrt{b}}+\frac{4 a^2 e x \sqrt{a+b x^4}}{15 \sqrt{b} \left (\sqrt{a}+\sqrt{b} x^2\right )}+\frac{1}{63} x \left (a+b x^4\right )^{3/2} \left (9 c+7 e x^2\right )+\frac{2}{105} a x \sqrt{a+b x^4} \left (15 c+7 e x^2\right )+\frac{1}{8} d x^2 \left (a+b x^4\right )^{3/2}+\frac{3}{16} a d x^2 \sqrt{a+b x^4}+\frac{f \left (a+b x^4\right )^{5/2}}{10 b} \]
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Rubi [A] time = 0.254981, antiderivative size = 382, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 10, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.37, Rules used = {1885, 1177, 1198, 220, 1196, 1248, 641, 195, 217, 206} \[ \frac{2 a^{7/4} \left (\sqrt{a}+\sqrt{b} x^2\right ) \sqrt{\frac{a+b x^4}{\left (\sqrt{a}+\sqrt{b} x^2\right )^2}} \left (7 \sqrt{a} e+15 \sqrt{b} c\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{105 b^{3/4} \sqrt{a+b x^4}}-\frac{4 a^{9/4} e \left (\sqrt{a}+\sqrt{b} x^2\right ) \sqrt{\frac{a+b x^4}{\left (\sqrt{a}+\sqrt{b} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{15 b^{3/4} \sqrt{a+b x^4}}+\frac{3 a^2 d \tanh ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a+b x^4}}\right )}{16 \sqrt{b}}+\frac{4 a^2 e x \sqrt{a+b x^4}}{15 \sqrt{b} \left (\sqrt{a}+\sqrt{b} x^2\right )}+\frac{1}{63} x \left (a+b x^4\right )^{3/2} \left (9 c+7 e x^2\right )+\frac{2}{105} a x \sqrt{a+b x^4} \left (15 c+7 e x^2\right )+\frac{1}{8} d x^2 \left (a+b x^4\right )^{3/2}+\frac{3}{16} a d x^2 \sqrt{a+b x^4}+\frac{f \left (a+b x^4\right )^{5/2}}{10 b} \]
Antiderivative was successfully verified.
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Rule 1885
Rule 1177
Rule 1198
Rule 220
Rule 1196
Rule 1248
Rule 641
Rule 195
Rule 217
Rule 206
Rubi steps
\begin{align*} \int \left (c+d x+e x^2+f x^3\right ) \left (a+b x^4\right )^{3/2} \, dx &=\int \left (\left (c+e x^2\right ) \left (a+b x^4\right )^{3/2}+x \left (d+f x^2\right ) \left (a+b x^4\right )^{3/2}\right ) \, dx\\ &=\int \left (c+e x^2\right ) \left (a+b x^4\right )^{3/2} \, dx+\int x \left (d+f x^2\right ) \left (a+b x^4\right )^{3/2} \, dx\\ &=\frac{1}{63} x \left (9 c+7 e x^2\right ) \left (a+b x^4\right )^{3/2}+\frac{1}{21} \int \left (18 a c+14 a e x^2\right ) \sqrt{a+b x^4} \, dx+\frac{1}{2} \operatorname{Subst}\left (\int (d+f x) \left (a+b x^2\right )^{3/2} \, dx,x,x^2\right )\\ &=\frac{2}{105} a x \left (15 c+7 e x^2\right ) \sqrt{a+b x^4}+\frac{1}{63} x \left (9 c+7 e x^2\right ) \left (a+b x^4\right )^{3/2}+\frac{f \left (a+b x^4\right )^{5/2}}{10 b}+\frac{1}{315} \int \frac{180 a^2 c+84 a^2 e x^2}{\sqrt{a+b x^4}} \, dx+\frac{1}{2} d \operatorname{Subst}\left (\int \left (a+b x^2\right )^{3/2} \, dx,x,x^2\right )\\ &=\frac{2}{105} a x \left (15 c+7 e x^2\right ) \sqrt{a+b x^4}+\frac{1}{8} d x^2 \left (a+b x^4\right )^{3/2}+\frac{1}{63} x \left (9 c+7 e x^2\right ) \left (a+b x^4\right )^{3/2}+\frac{f \left (a+b x^4\right )^{5/2}}{10 b}+\frac{1}{8} (3 a d) \operatorname{Subst}\left (\int \sqrt{a+b x^2} \, dx,x,x^2\right )-\frac{\left (4 a^{5/2} e\right ) \int \frac{1-\frac{\sqrt{b} x^2}{\sqrt{a}}}{\sqrt{a+b x^4}} \, dx}{15 \sqrt{b}}+\frac{1}{105} \left (4 a^2 \left (15 c+\frac{7 \sqrt{a} e}{\sqrt{b}}\right )\right ) \int \frac{1}{\sqrt{a+b x^4}} \, dx\\ &=\frac{3}{16} a d x^2 \sqrt{a+b x^4}+\frac{4 a^2 e x \sqrt{a+b x^4}}{15 \sqrt{b} \left (\sqrt{a}+\sqrt{b} x^2\right )}+\frac{2}{105} a x \left (15 c+7 e x^2\right ) \sqrt{a+b x^4}+\frac{1}{8} d x^2 \left (a+b x^4\right )^{3/2}+\frac{1}{63} x \left (9 c+7 e x^2\right ) \left (a+b x^4\right )^{3/2}+\frac{f \left (a+b x^4\right )^{5/2}}{10 b}-\frac{4 a^{9/4} 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 )}{15 b^{3/4} \sqrt{a+b x^4}}+\frac{2 a^{7/4} \left (15 \sqrt{b} c+7 \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 b^{3/4} \sqrt{a+b x^4}}+\frac{1}{16} \left (3 a^2 d\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+b x^2}} \, dx,x,x^2\right )\\ &=\frac{3}{16} a d x^2 \sqrt{a+b x^4}+\frac{4 a^2 e x \sqrt{a+b x^4}}{15 \sqrt{b} \left (\sqrt{a}+\sqrt{b} x^2\right )}+\frac{2}{105} a x \left (15 c+7 e x^2\right ) \sqrt{a+b x^4}+\frac{1}{8} d x^2 \left (a+b x^4\right )^{3/2}+\frac{1}{63} x \left (9 c+7 e x^2\right ) \left (a+b x^4\right )^{3/2}+\frac{f \left (a+b x^4\right )^{5/2}}{10 b}-\frac{4 a^{9/4} 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 )}{15 b^{3/4} \sqrt{a+b x^4}}+\frac{2 a^{7/4} \left (15 \sqrt{b} c+7 \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 b^{3/4} \sqrt{a+b x^4}}+\frac{1}{16} \left (3 a^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{3}{16} a d x^2 \sqrt{a+b x^4}+\frac{4 a^2 e x \sqrt{a+b x^4}}{15 \sqrt{b} \left (\sqrt{a}+\sqrt{b} x^2\right )}+\frac{2}{105} a x \left (15 c+7 e x^2\right ) \sqrt{a+b x^4}+\frac{1}{8} d x^2 \left (a+b x^4\right )^{3/2}+\frac{1}{63} x \left (9 c+7 e x^2\right ) \left (a+b x^4\right )^{3/2}+\frac{f \left (a+b x^4\right )^{5/2}}{10 b}+\frac{3 a^2 d \tanh ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a+b x^4}}\right )}{16 \sqrt{b}}-\frac{4 a^{9/4} 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 )}{15 b^{3/4} \sqrt{a+b x^4}}+\frac{2 a^{7/4} \left (15 \sqrt{b} c+7 \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 b^{3/4} \sqrt{a+b x^4}}\\ \end{align*}
Mathematica [C] time = 0.552603, size = 175, normalized size = 0.46 \[ \frac{1}{240} \sqrt{a+b x^4} \left (15 d \left (\frac{3 a^{5/2} \sqrt{\frac{b x^4}{a}+1} \sinh ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )}{\sqrt{b} \left (a+b x^4\right )}+5 a x^2+2 b x^6\right )+\frac{240 a c x \, _2F_1\left (-\frac{3}{2},\frac{1}{4};\frac{5}{4};-\frac{b x^4}{a}\right )}{\sqrt{\frac{b x^4}{a}+1}}+\frac{80 a e x^3 \, _2F_1\left (-\frac{3}{2},\frac{3}{4};\frac{7}{4};-\frac{b x^4}{a}\right )}{\sqrt{\frac{b x^4}{a}+1}}+\frac{24 f \left (a+b x^4\right )^2}{b}\right ) \]
Antiderivative was successfully verified.
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Maple [C] time = 0.006, size = 368, normalized size = 1. \begin{align*}{\frac{f}{10\,b} \left ( b{x}^{4}+a \right ) ^{{\frac{5}{2}}}}+{\frac{be{x}^{7}}{9}\sqrt{b{x}^{4}+a}}+{\frac{11\,ae{x}^{3}}{45}\sqrt{b{x}^{4}+a}}+{{\frac{4\,i}{15}}e{a}^{{\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{{i\sqrt{b}{\frac{1}{\sqrt{a}}}}}}}{\frac{1}{\sqrt{b{x}^{4}+a}}}{\frac{1}{\sqrt{b}}}}-{{\frac{4\,i}{15}}e{a}^{{\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{{i\sqrt{b}{\frac{1}{\sqrt{a}}}}}}}{\frac{1}{\sqrt{b{x}^{4}+a}}}{\frac{1}{\sqrt{b}}}}+{\frac{bd{x}^{6}}{8}\sqrt{b{x}^{4}+a}}+{\frac{5\,ad{x}^{2}}{16}\sqrt{b{x}^{4}+a}}+{\frac{3\,{a}^{2}d}{16}\ln \left ({x}^{2}\sqrt{b}+\sqrt{b{x}^{4}+a} \right ){\frac{1}{\sqrt{b}}}}+{\frac{bc{x}^{5}}{7}\sqrt{b{x}^{4}+a}}+{\frac{3\,acx}{7}\sqrt{b{x}^{4}+a}}+{\frac{4\,{a}^{2}c}{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}}}} \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*} \int{\left (b x^{4} + a\right )}^{\frac{3}{2}}{\left (f x^{3} + e x^{2} + d x + c\right )}\,{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 ({\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\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 9.64242, size = 394, normalized size = 1.03 \begin{align*} \frac{a^{\frac{3}{2}} c x \Gamma \left (\frac{1}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, \frac{1}{4} \\ \frac{5}{4} \end{matrix}\middle |{\frac{b x^{4} e^{i \pi }}{a}} \right )}}{4 \Gamma \left (\frac{5}{4}\right )} + \frac{a^{\frac{3}{2}} d x^{2} \sqrt{1 + \frac{b x^{4}}{a}}}{4} + \frac{a^{\frac{3}{2}} d x^{2}}{16 \sqrt{1 + \frac{b x^{4}}{a}}} + \frac{a^{\frac{3}{2}} e x^{3} \Gamma \left (\frac{3}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, \frac{3}{4} \\ \frac{7}{4} \end{matrix}\middle |{\frac{b x^{4} e^{i \pi }}{a}} \right )}}{4 \Gamma \left (\frac{7}{4}\right )} + \frac{\sqrt{a} b c x^{5} \Gamma \left (\frac{5}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, \frac{5}{4} \\ \frac{9}{4} \end{matrix}\middle |{\frac{b x^{4} e^{i \pi }}{a}} \right )}}{4 \Gamma \left (\frac{9}{4}\right )} + \frac{3 \sqrt{a} b d x^{6}}{16 \sqrt{1 + \frac{b x^{4}}{a}}} + \frac{\sqrt{a} b e x^{7} \Gamma \left (\frac{7}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, \frac{7}{4} \\ \frac{11}{4} \end{matrix}\middle |{\frac{b x^{4} e^{i \pi }}{a}} \right )}}{4 \Gamma \left (\frac{11}{4}\right )} + \frac{3 a^{2} d \operatorname{asinh}{\left (\frac{\sqrt{b} x^{2}}{\sqrt{a}} \right )}}{16 \sqrt{b}} + a f \left (\begin{cases} \frac{\sqrt{a} x^{4}}{4} & \text{for}\: b = 0 \\\frac{\left (a + b x^{4}\right )^{\frac{3}{2}}}{6 b} & \text{otherwise} \end{cases}\right ) + b f \left (\begin{cases} - \frac{a^{2} \sqrt{a + b x^{4}}}{15 b^{2}} + \frac{a x^{4} \sqrt{a + b x^{4}}}{30 b} + \frac{x^{8} \sqrt{a + b x^{4}}}{10} & \text{for}\: b \neq 0 \\\frac{\sqrt{a} x^{8}}{8} & \text{otherwise} \end{cases}\right ) + \frac{b^{2} d x^{10}}{8 \sqrt{a} \sqrt{1 + \frac{b x^{4}}{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{\left (b x^{4} + a\right )}^{\frac{3}{2}}{\left (f x^{3} + e x^{2} + d x + c\right )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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