Optimal. Leaf size=412 \[ \frac{2 b^{5/4} \left (\sqrt{a}+\sqrt{b} x^2\right ) \sqrt{\frac{a+b x^4}{\left (\sqrt{a}+\sqrt{b} x^2\right )^2}} \left (21 \sqrt{a} e+5 \sqrt{b} c\right ) \text{EllipticF}\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),\frac{1}{2}\right )}{35 \sqrt [4]{a} \sqrt{a+b x^4}}+\frac{1}{2} b^{3/2} d \tanh ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a+b x^4}}\right )+\frac{12 b^{3/2} e x \sqrt{a+b x^4}}{5 \left (\sqrt{a}+\sqrt{b} x^2\right )}-\frac{12 \sqrt [4]{a} b^{5/4} e \left (\sqrt{a}+\sqrt{b} x^2\right ) \sqrt{\frac{a+b x^4}{\left (\sqrt{a}+\sqrt{b} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{5 \sqrt{a+b x^4}}-\frac{1}{420} \left (a+b x^4\right )^{3/2} \left (\frac{60 c}{x^7}+\frac{70 d}{x^6}+\frac{84 e}{x^5}+\frac{105 f}{x^4}\right )-\frac{2 b \sqrt{a+b x^4} \left (5 c-21 e x^2\right )}{35 x^3}-\frac{b \sqrt{a+b x^4} \left (2 d-3 f x^2\right )}{4 x^2}-\frac{12 b e \sqrt{a+b x^4}}{5 x}-\frac{3}{4} \sqrt{a} b f \tanh ^{-1}\left (\frac{\sqrt{a+b x^4}}{\sqrt{a}}\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.393938, antiderivative size = 412, normalized size of antiderivative = 1., 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.
[In]
[Out]
Rule 14
Rule 1825
Rule 1833
Rule 1272
Rule 1282
Rule 1198
Rule 220
Rule 1196
Rule 1252
Rule 813
Rule 844
Rule 217
Rule 206
Rule 266
Rule 63
Rule 208
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*}
Mathematica [C] time = 0.221303, size = 164, normalized size = 0.4 \[ \frac{\sqrt{a+b x^4} \left (7 x \left (-5 a^3 d \, _2F_1\left (-\frac{3}{2},-\frac{3}{2};-\frac{1}{2};-\frac{b x^4}{a}\right )-6 a^3 e x \, _2F_1\left (-\frac{3}{2},-\frac{5}{4};-\frac{1}{4};-\frac{b x^4}{a}\right )+3 b f x^6 \left (a+b x^4\right )^2 \sqrt{\frac{b x^4}{a}+1} \, _2F_1\left (2,\frac{5}{2};\frac{7}{2};\frac{b x^4}{a}+1\right )\right )-30 a^3 c \, _2F_1\left (-\frac{7}{4},-\frac{3}{2};-\frac{3}{4};-\frac{b x^4}{a}\right )\right )}{210 a^2 x^7 \sqrt{\frac{b x^4}{a}+1}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [C] time = 0.015, size = 411, normalized size = 1. \begin{align*} -{\frac{ae}{5\,{x}^{5}}\sqrt{b{x}^{4}+a}}-{\frac{7\,be}{5\,x}\sqrt{b{x}^{4}+a}}+{{\frac{12\,i}{5}}e{b}^{{\frac{3}{2}}}\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 ){\frac{1}{\sqrt{{i\sqrt{b}{\frac{1}{\sqrt{a}}}}}}}{\frac{1}{\sqrt{b{x}^{4}+a}}}}-{{\frac{12\,i}{5}}e{b}^{{\frac{3}{2}}}\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 ){\frac{1}{\sqrt{{i\sqrt{b}{\frac{1}{\sqrt{a}}}}}}}{\frac{1}{\sqrt{b{x}^{4}+a}}}}+{\frac{fb}{2}\sqrt{b{x}^{4}+a}}-{\frac{3\,fb}{4}\sqrt{a}\ln \left ({\frac{1}{{x}^{2}} \left ( 2\,a+2\,\sqrt{a}\sqrt{b{x}^{4}+a} \right ) } \right ) }-{\frac{af}{4\,{x}^{4}}\sqrt{b{x}^{4}+a}}+{\frac{d}{2}{b}^{{\frac{3}{2}}}\ln \left ({x}^{2}\sqrt{b}+\sqrt{b{x}^{4}+a} \right ) }-{\frac{ad}{6\,{x}^{6}}\sqrt{b{x}^{4}+a}}-{\frac{2\,bd}{3\,{x}^{2}}\sqrt{b{x}^{4}+a}}-{\frac{ac}{7\,{x}^{7}}\sqrt{b{x}^{4}+a}}-{\frac{3\,bc}{7\,{x}^{3}}\sqrt{b{x}^{4}+a}}+{\frac{4\,{b}^{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.
[In]
[Out]
________________________________________________________________________________________
Maxima [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^{8}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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^{8}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [C] time = 11.4562, size = 415, normalized size = 1.01 \begin{align*} \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}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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^{8}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]