Optimal. Leaf size=343 \[ \frac{\sqrt [4]{a} \left (\sqrt{a}+\sqrt{b} x^2\right ) \sqrt{\frac{a+b x^4}{\left (\sqrt{a}+\sqrt{b} x^2\right )^2}} \left (9 \sqrt{b} d-5 \sqrt{a} f\right ) \text{EllipticF}\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),\frac{1}{2}\right )}{12 b^{9/4} \sqrt{a+b x^4}}+\frac{x \left (a f-b c x-b d x^2-b e x^3\right )}{2 b^2 \sqrt{a+b x^4}}+\frac{c \tanh ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a+b x^4}}\right )}{2 b^{3/2}}+\frac{3 d x \sqrt{a+b x^4}}{2 b^{3/2} \left (\sqrt{a}+\sqrt{b} x^2\right )}-\frac{3 \sqrt [4]{a} d \left (\sqrt{a}+\sqrt{b} x^2\right ) \sqrt{\frac{a+b x^4}{\left (\sqrt{a}+\sqrt{b} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{2 b^{7/4} \sqrt{a+b x^4}}+\frac{e \sqrt{a+b x^4}}{b^2}+\frac{f x \sqrt{a+b x^4}}{3 b^2} \]
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Rubi [A] time = 0.367229, antiderivative size = 343, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 10, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {1828, 1885, 1248, 641, 217, 206, 1888, 1198, 220, 1196} \[ \frac{x \left (a f-b c x-b d x^2-b e x^3\right )}{2 b^2 \sqrt{a+b x^4}}+\frac{c \tanh ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a+b x^4}}\right )}{2 b^{3/2}}+\frac{\sqrt [4]{a} \left (\sqrt{a}+\sqrt{b} x^2\right ) \sqrt{\frac{a+b x^4}{\left (\sqrt{a}+\sqrt{b} x^2\right )^2}} \left (9 \sqrt{b} d-5 \sqrt{a} f\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{12 b^{9/4} \sqrt{a+b x^4}}+\frac{3 d x \sqrt{a+b x^4}}{2 b^{3/2} \left (\sqrt{a}+\sqrt{b} x^2\right )}-\frac{3 \sqrt [4]{a} d \left (\sqrt{a}+\sqrt{b} x^2\right ) \sqrt{\frac{a+b x^4}{\left (\sqrt{a}+\sqrt{b} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{2 b^{7/4} \sqrt{a+b x^4}}+\frac{e \sqrt{a+b x^4}}{b^2}+\frac{f x \sqrt{a+b x^4}}{3 b^2} \]
Antiderivative was successfully verified.
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Rule 1828
Rule 1885
Rule 1248
Rule 641
Rule 217
Rule 206
Rule 1888
Rule 1198
Rule 220
Rule 1196
Rubi steps
\begin{align*} \int \frac{x^5 \left (c+d x+e x^2+f x^3\right )}{\left (a+b x^4\right )^{3/2}} \, dx &=\frac{x \left (a f-b c x-b d x^2-b e x^3\right )}{2 b^2 \sqrt{a+b x^4}}-\frac{\int \frac{a^2 f-2 a b c x-3 a b d x^2-4 a b e x^3-2 a b f x^4}{\sqrt{a+b x^4}} \, dx}{2 a b^2}\\ &=\frac{x \left (a f-b c x-b d x^2-b e x^3\right )}{2 b^2 \sqrt{a+b x^4}}-\frac{\int \left (\frac{x \left (-2 a b c-4 a b e x^2\right )}{\sqrt{a+b x^4}}+\frac{a^2 f-3 a b d x^2-2 a b f x^4}{\sqrt{a+b x^4}}\right ) \, dx}{2 a b^2}\\ &=\frac{x \left (a f-b c x-b d x^2-b e x^3\right )}{2 b^2 \sqrt{a+b x^4}}-\frac{\int \frac{x \left (-2 a b c-4 a b e x^2\right )}{\sqrt{a+b x^4}} \, dx}{2 a b^2}-\frac{\int \frac{a^2 f-3 a b d x^2-2 a b f x^4}{\sqrt{a+b x^4}} \, dx}{2 a b^2}\\ &=\frac{x \left (a f-b c x-b d x^2-b e x^3\right )}{2 b^2 \sqrt{a+b x^4}}+\frac{f x \sqrt{a+b x^4}}{3 b^2}-\frac{\int \frac{5 a^2 b f-9 a b^2 d x^2}{\sqrt{a+b x^4}} \, dx}{6 a b^3}-\frac{\operatorname{Subst}\left (\int \frac{-2 a b c-4 a b e x}{\sqrt{a+b x^2}} \, dx,x,x^2\right )}{4 a b^2}\\ &=\frac{x \left (a f-b c x-b d x^2-b e x^3\right )}{2 b^2 \sqrt{a+b x^4}}+\frac{e \sqrt{a+b x^4}}{b^2}+\frac{f x \sqrt{a+b x^4}}{3 b^2}+\frac{c \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+b x^2}} \, dx,x,x^2\right )}{2 b}-\frac{\left (3 \sqrt{a} d\right ) \int \frac{1-\frac{\sqrt{b} x^2}{\sqrt{a}}}{\sqrt{a+b x^4}} \, dx}{2 b^{3/2}}+\frac{\left (\sqrt{a} \left (9 \sqrt{b} d-5 \sqrt{a} f\right )\right ) \int \frac{1}{\sqrt{a+b x^4}} \, dx}{6 b^2}\\ &=\frac{x \left (a f-b c x-b d x^2-b e x^3\right )}{2 b^2 \sqrt{a+b x^4}}+\frac{e \sqrt{a+b x^4}}{b^2}+\frac{f x \sqrt{a+b x^4}}{3 b^2}+\frac{3 d x \sqrt{a+b x^4}}{2 b^{3/2} \left (\sqrt{a}+\sqrt{b} x^2\right )}-\frac{3 \sqrt [4]{a} 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 )}{2 b^{7/4} \sqrt{a+b x^4}}+\frac{\sqrt [4]{a} \left (9 \sqrt{b} d-5 \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 )}{12 b^{9/4} \sqrt{a+b x^4}}+\frac{c \operatorname{Subst}\left (\int \frac{1}{1-b x^2} \, dx,x,\frac{x^2}{\sqrt{a+b x^4}}\right )}{2 b}\\ &=\frac{x \left (a f-b c x-b d x^2-b e x^3\right )}{2 b^2 \sqrt{a+b x^4}}+\frac{e \sqrt{a+b x^4}}{b^2}+\frac{f x \sqrt{a+b x^4}}{3 b^2}+\frac{3 d x \sqrt{a+b x^4}}{2 b^{3/2} \left (\sqrt{a}+\sqrt{b} x^2\right )}+\frac{c \tanh ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a+b x^4}}\right )}{2 b^{3/2}}-\frac{3 \sqrt [4]{a} 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 )}{2 b^{7/4} \sqrt{a+b x^4}}+\frac{\sqrt [4]{a} \left (9 \sqrt{b} d-5 \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 )}{12 b^{9/4} \sqrt{a+b x^4}}\\ \end{align*}
Mathematica [C] time = 0.151856, size = 176, normalized size = 0.51 \[ \frac{3 \sqrt{a} \sqrt{b} c \sqrt{\frac{b x^4}{a}+1} \sinh ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )-6 b d x^3 \sqrt{\frac{b x^4}{a}+1} \, _2F_1\left (\frac{3}{4},\frac{3}{2};\frac{7}{4};-\frac{b x^4}{a}\right )-5 a f x \sqrt{\frac{b x^4}{a}+1} \, _2F_1\left (\frac{1}{4},\frac{1}{2};\frac{5}{4};-\frac{b x^4}{a}\right )+6 a e+5 a f x-3 b c x^2+6 b d x^3+3 b e x^4+2 b f x^5}{6 b^2 \sqrt{a+b x^4}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.016, size = 358, normalized size = 1. \begin{align*}{\frac{afx}{2\,{b}^{2}}{\frac{1}{\sqrt{ \left ({x}^{4}+{\frac{a}{b}} \right ) b}}}}+{\frac{fx}{3\,{b}^{2}}\sqrt{b{x}^{4}+a}}-{\frac{5\,af}{6\,{b}^{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{e \left ( b{x}^{4}+2\,a \right ) }{2\,{b}^{2}}{\frac{1}{\sqrt{b{x}^{4}+a}}}}-{\frac{d{x}^{3}}{2\,b}{\frac{1}{\sqrt{ \left ({x}^{4}+{\frac{a}{b}} \right ) b}}}}+{{\frac{3\,i}{2}}d\sqrt{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 ){b}^{-{\frac{3}{2}}}{\frac{1}{\sqrt{{i\sqrt{b}{\frac{1}{\sqrt{a}}}}}}}{\frac{1}{\sqrt{b{x}^{4}+a}}}}-{{\frac{3\,i}{2}}d\sqrt{a}\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 ){b}^{-{\frac{3}{2}}}{\frac{1}{\sqrt{{i\sqrt{b}{\frac{1}{\sqrt{a}}}}}}}{\frac{1}{\sqrt{b{x}^{4}+a}}}}-{\frac{c{x}^{2}}{2\,b}{\frac{1}{\sqrt{b{x}^{4}+a}}}}+{\frac{c}{2}\ln \left ({x}^{2}\sqrt{b}+\sqrt{b{x}^{4}+a} \right ){b}^{-{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \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 (f x^{8} + e x^{7} + d x^{6} + c x^{5}\right )} \sqrt{b x^{4} + a}}{b^{2} x^{8} + 2 \, a b x^{4} + a^{2}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 25.7538, size = 172, normalized size = 0.5 \begin{align*} c \left (\frac{\operatorname{asinh}{\left (\frac{\sqrt{b} x^{2}}{\sqrt{a}} \right )}}{2 b^{\frac{3}{2}}} - \frac{x^{2}}{2 \sqrt{a} b \sqrt{1 + \frac{b x^{4}}{a}}}\right ) + e \left (\begin{cases} \frac{a}{b^{2} \sqrt{a + b x^{4}}} + \frac{x^{4}}{2 b \sqrt{a + b x^{4}}} & \text{for}\: b \neq 0 \\\frac{x^{8}}{8 a^{\frac{3}{2}}} & \text{otherwise} \end{cases}\right ) + \frac{d x^{7} \Gamma \left (\frac{7}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{3}{2}, \frac{7}{4} \\ \frac{11}{4} \end{matrix}\middle |{\frac{b x^{4} e^{i \pi }}{a}} \right )}}{4 a^{\frac{3}{2}} \Gamma \left (\frac{11}{4}\right )} + \frac{f x^{9} \Gamma \left (\frac{9}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{3}{2}, \frac{9}{4} \\ \frac{13}{4} \end{matrix}\middle |{\frac{b x^{4} e^{i \pi }}{a}} \right )}}{4 a^{\frac{3}{2}} \Gamma \left (\frac{13}{4}\right )} \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 (f x^{3} + e x^{2} + d x + c\right )} x^{5}}{{\left (b x^{4} + a\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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