Optimal. Leaf size=287 \[ -\frac{\log \left (-\sqrt{2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt{x}+\sqrt [4]{-a}+\sqrt [4]{c} x\right )}{4 \sqrt{2} (-a)^{3/8} c^{5/8}}+\frac{\log \left (\sqrt{2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt{x}+\sqrt [4]{-a}+\sqrt [4]{c} x\right )}{4 \sqrt{2} (-a)^{3/8} c^{5/8}}-\frac{\tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}\right )}{2 \sqrt{2} (-a)^{3/8} c^{5/8}}+\frac{\tan ^{-1}\left (\frac{\sqrt{2} \sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}+1\right )}{2 \sqrt{2} (-a)^{3/8} c^{5/8}}-\frac{\tan ^{-1}\left (\frac{\sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}\right )}{2 (-a)^{3/8} c^{5/8}}-\frac{\tanh ^{-1}\left (\frac{\sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}\right )}{2 (-a)^{3/8} c^{5/8}} \]
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Rubi [A] time = 0.233087, antiderivative size = 287, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 11, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.733, Rules used = {329, 301, 211, 1165, 628, 1162, 617, 204, 212, 208, 205} \[ -\frac{\log \left (-\sqrt{2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt{x}+\sqrt [4]{-a}+\sqrt [4]{c} x\right )}{4 \sqrt{2} (-a)^{3/8} c^{5/8}}+\frac{\log \left (\sqrt{2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt{x}+\sqrt [4]{-a}+\sqrt [4]{c} x\right )}{4 \sqrt{2} (-a)^{3/8} c^{5/8}}-\frac{\tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}\right )}{2 \sqrt{2} (-a)^{3/8} c^{5/8}}+\frac{\tan ^{-1}\left (\frac{\sqrt{2} \sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}+1\right )}{2 \sqrt{2} (-a)^{3/8} c^{5/8}}-\frac{\tan ^{-1}\left (\frac{\sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}\right )}{2 (-a)^{3/8} c^{5/8}}-\frac{\tanh ^{-1}\left (\frac{\sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}\right )}{2 (-a)^{3/8} c^{5/8}} \]
Antiderivative was successfully verified.
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Rule 329
Rule 301
Rule 211
Rule 1165
Rule 628
Rule 1162
Rule 617
Rule 204
Rule 212
Rule 208
Rule 205
Rubi steps
\begin{align*} \int \frac{x^{3/2}}{a+c x^4} \, dx &=2 \operatorname{Subst}\left (\int \frac{x^4}{a+c x^8} \, dx,x,\sqrt{x}\right )\\ &=-\frac{\operatorname{Subst}\left (\int \frac{1}{\sqrt{-a}-\sqrt{c} x^4} \, dx,x,\sqrt{x}\right )}{\sqrt{c}}+\frac{\operatorname{Subst}\left (\int \frac{1}{\sqrt{-a}+\sqrt{c} x^4} \, dx,x,\sqrt{x}\right )}{\sqrt{c}}\\ &=-\frac{\operatorname{Subst}\left (\int \frac{1}{\sqrt [4]{-a}-\sqrt [4]{c} x^2} \, dx,x,\sqrt{x}\right )}{2 \sqrt [4]{-a} \sqrt{c}}-\frac{\operatorname{Subst}\left (\int \frac{1}{\sqrt [4]{-a}+\sqrt [4]{c} x^2} \, dx,x,\sqrt{x}\right )}{2 \sqrt [4]{-a} \sqrt{c}}+\frac{\operatorname{Subst}\left (\int \frac{\sqrt [4]{-a}-\sqrt [4]{c} x^2}{\sqrt{-a}+\sqrt{c} x^4} \, dx,x,\sqrt{x}\right )}{2 \sqrt [4]{-a} \sqrt{c}}+\frac{\operatorname{Subst}\left (\int \frac{\sqrt [4]{-a}+\sqrt [4]{c} x^2}{\sqrt{-a}+\sqrt{c} x^4} \, dx,x,\sqrt{x}\right )}{2 \sqrt [4]{-a} \sqrt{c}}\\ &=-\frac{\tan ^{-1}\left (\frac{\sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}\right )}{2 (-a)^{3/8} c^{5/8}}-\frac{\tanh ^{-1}\left (\frac{\sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}\right )}{2 (-a)^{3/8} c^{5/8}}+\frac{\operatorname{Subst}\left (\int \frac{1}{\frac{\sqrt [4]{-a}}{\sqrt [4]{c}}-\frac{\sqrt{2} \sqrt [8]{-a} x}{\sqrt [8]{c}}+x^2} \, dx,x,\sqrt{x}\right )}{4 \sqrt [4]{-a} c^{3/4}}+\frac{\operatorname{Subst}\left (\int \frac{1}{\frac{\sqrt [4]{-a}}{\sqrt [4]{c}}+\frac{\sqrt{2} \sqrt [8]{-a} x}{\sqrt [8]{c}}+x^2} \, dx,x,\sqrt{x}\right )}{4 \sqrt [4]{-a} c^{3/4}}-\frac{\operatorname{Subst}\left (\int \frac{\frac{\sqrt{2} \sqrt [8]{-a}}{\sqrt [8]{c}}+2 x}{-\frac{\sqrt [4]{-a}}{\sqrt [4]{c}}-\frac{\sqrt{2} \sqrt [8]{-a} x}{\sqrt [8]{c}}-x^2} \, dx,x,\sqrt{x}\right )}{4 \sqrt{2} (-a)^{3/8} c^{5/8}}-\frac{\operatorname{Subst}\left (\int \frac{\frac{\sqrt{2} \sqrt [8]{-a}}{\sqrt [8]{c}}-2 x}{-\frac{\sqrt [4]{-a}}{\sqrt [4]{c}}+\frac{\sqrt{2} \sqrt [8]{-a} x}{\sqrt [8]{c}}-x^2} \, dx,x,\sqrt{x}\right )}{4 \sqrt{2} (-a)^{3/8} c^{5/8}}\\ &=-\frac{\tan ^{-1}\left (\frac{\sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}\right )}{2 (-a)^{3/8} c^{5/8}}-\frac{\tanh ^{-1}\left (\frac{\sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}\right )}{2 (-a)^{3/8} c^{5/8}}-\frac{\log \left (\sqrt [4]{-a}-\sqrt{2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt{x}+\sqrt [4]{c} x\right )}{4 \sqrt{2} (-a)^{3/8} c^{5/8}}+\frac{\log \left (\sqrt [4]{-a}+\sqrt{2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt{x}+\sqrt [4]{c} x\right )}{4 \sqrt{2} (-a)^{3/8} c^{5/8}}+\frac{\operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1-\frac{\sqrt{2} \sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}\right )}{2 \sqrt{2} (-a)^{3/8} c^{5/8}}-\frac{\operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1+\frac{\sqrt{2} \sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}\right )}{2 \sqrt{2} (-a)^{3/8} c^{5/8}}\\ &=-\frac{\tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}\right )}{2 \sqrt{2} (-a)^{3/8} c^{5/8}}+\frac{\tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}\right )}{2 \sqrt{2} (-a)^{3/8} c^{5/8}}-\frac{\tan ^{-1}\left (\frac{\sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}\right )}{2 (-a)^{3/8} c^{5/8}}-\frac{\tanh ^{-1}\left (\frac{\sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}\right )}{2 (-a)^{3/8} c^{5/8}}-\frac{\log \left (\sqrt [4]{-a}-\sqrt{2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt{x}+\sqrt [4]{c} x\right )}{4 \sqrt{2} (-a)^{3/8} c^{5/8}}+\frac{\log \left (\sqrt [4]{-a}+\sqrt{2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt{x}+\sqrt [4]{c} x\right )}{4 \sqrt{2} (-a)^{3/8} c^{5/8}}\\ \end{align*}
Mathematica [C] time = 0.0058144, size = 29, normalized size = 0.1 \[ \frac{2 x^{5/2} \, _2F_1\left (\frac{5}{8},1;\frac{13}{8};-\frac{c x^4}{a}\right )}{5 a} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.022, size = 29, normalized size = 0.1 \begin{align*}{\frac{1}{4\,c}\sum _{{\it \_R}={\it RootOf} \left ({{\it \_Z}}^{8}c+a \right ) }{\frac{1}{{{\it \_R}}^{3}}\ln \left ( \sqrt{x}-{\it \_R} \right ) }} \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 \frac{x^{\frac{3}{2}}}{c x^{4} + a}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.0078, size = 1249, normalized size = 4.35 \begin{align*} \frac{1}{2} \, \sqrt{2} \left (-\frac{1}{a^{3} c^{5}}\right )^{\frac{1}{8}} \arctan \left (\sqrt{2} \sqrt{\sqrt{2} a^{2} c^{3} \sqrt{x} \left (-\frac{1}{a^{3} c^{5}}\right )^{\frac{5}{8}} - a c \left (-\frac{1}{a^{3} c^{5}}\right )^{\frac{1}{4}} + x} a c^{2} \left (-\frac{1}{a^{3} c^{5}}\right )^{\frac{3}{8}} - \sqrt{2} a c^{2} \sqrt{x} \left (-\frac{1}{a^{3} c^{5}}\right )^{\frac{3}{8}} + 1\right ) + \frac{1}{2} \, \sqrt{2} \left (-\frac{1}{a^{3} c^{5}}\right )^{\frac{1}{8}} \arctan \left (\sqrt{2} \sqrt{-\sqrt{2} a^{2} c^{3} \sqrt{x} \left (-\frac{1}{a^{3} c^{5}}\right )^{\frac{5}{8}} - a c \left (-\frac{1}{a^{3} c^{5}}\right )^{\frac{1}{4}} + x} a c^{2} \left (-\frac{1}{a^{3} c^{5}}\right )^{\frac{3}{8}} - \sqrt{2} a c^{2} \sqrt{x} \left (-\frac{1}{a^{3} c^{5}}\right )^{\frac{3}{8}} - 1\right ) + \frac{1}{8} \, \sqrt{2} \left (-\frac{1}{a^{3} c^{5}}\right )^{\frac{1}{8}} \log \left (\sqrt{2} a^{2} c^{3} \sqrt{x} \left (-\frac{1}{a^{3} c^{5}}\right )^{\frac{5}{8}} - a c \left (-\frac{1}{a^{3} c^{5}}\right )^{\frac{1}{4}} + x\right ) - \frac{1}{8} \, \sqrt{2} \left (-\frac{1}{a^{3} c^{5}}\right )^{\frac{1}{8}} \log \left (-\sqrt{2} a^{2} c^{3} \sqrt{x} \left (-\frac{1}{a^{3} c^{5}}\right )^{\frac{5}{8}} - a c \left (-\frac{1}{a^{3} c^{5}}\right )^{\frac{1}{4}} + x\right ) - \left (-\frac{1}{a^{3} c^{5}}\right )^{\frac{1}{8}} \arctan \left (\sqrt{-a c \left (-\frac{1}{a^{3} c^{5}}\right )^{\frac{1}{4}} + x} a c^{2} \left (-\frac{1}{a^{3} c^{5}}\right )^{\frac{3}{8}} - a c^{2} \sqrt{x} \left (-\frac{1}{a^{3} c^{5}}\right )^{\frac{3}{8}}\right ) - \frac{1}{4} \, \left (-\frac{1}{a^{3} c^{5}}\right )^{\frac{1}{8}} \log \left (a^{2} c^{3} \left (-\frac{1}{a^{3} c^{5}}\right )^{\frac{5}{8}} + \sqrt{x}\right ) + \frac{1}{4} \, \left (-\frac{1}{a^{3} c^{5}}\right )^{\frac{1}{8}} \log \left (-a^{2} c^{3} \left (-\frac{1}{a^{3} c^{5}}\right )^{\frac{5}{8}} + \sqrt{x}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 76.0411, size = 466, normalized size = 1.62 \begin{align*} \begin{cases} \frac{\tilde{\infty }}{x^{\frac{3}{2}}} & \text{for}\: a = 0 \wedge c = 0 \\\frac{2 x^{\frac{5}{2}}}{5 a} & \text{for}\: c = 0 \\- \frac{2}{3 c x^{\frac{3}{2}}} & \text{for}\: a = 0 \\- \frac{\left (-1\right )^{\frac{5}{8}} c^{29} \left (\frac{1}{c}\right )^{\frac{237}{8}} \log{\left (- \sqrt [8]{-1} \sqrt [8]{a} \sqrt [8]{\frac{1}{c}} + \sqrt{x} \right )}}{4 a^{\frac{3}{8}}} - \frac{\left (-1\right )^{\frac{5}{8}} \sqrt{2} c^{25} \left (\frac{1}{c}\right )^{\frac{205}{8}} \log{\left (4 \sqrt [8]{-1} \sqrt{2} \sqrt [8]{a} \sqrt{x} \sqrt [8]{\frac{1}{c}} + 4 \sqrt [4]{-1} \sqrt [4]{a} \sqrt [4]{\frac{1}{c}} + 4 x \right )}}{8 a^{\frac{3}{8}}} + \frac{\left (-1\right )^{\frac{5}{8}} \log{\left (\sqrt [8]{-1} \sqrt [8]{a} \sqrt [8]{\frac{1}{c}} + \sqrt{x} \right )}}{4 a^{\frac{3}{8}} c^{10} \left (\frac{1}{c}\right )^{\frac{75}{8}}} - \frac{\left (-1\right )^{\frac{5}{8}} \operatorname{atan}{\left (\frac{\left (-1\right )^{\frac{7}{8}} \sqrt{x}}{\sqrt [8]{a} \sqrt [8]{\frac{1}{c}}} \right )}}{2 a^{\frac{3}{8}} c^{10} \left (\frac{1}{c}\right )^{\frac{75}{8}}} - \frac{\left (-1\right )^{\frac{5}{8}} \sqrt{2} \operatorname{atan}{\left (1 - \frac{\left (-1\right )^{\frac{7}{8}} \sqrt{2} \sqrt{x}}{\sqrt [8]{a} \sqrt [8]{\frac{1}{c}}} \right )}}{4 a^{\frac{3}{8}} c^{10} \left (\frac{1}{c}\right )^{\frac{75}{8}}} + \frac{\left (-1\right )^{\frac{5}{8}} \sqrt{2} \operatorname{atan}{\left (1 + \frac{\left (-1\right )^{\frac{7}{8}} \sqrt{2} \sqrt{x}}{\sqrt [8]{a} \sqrt [8]{\frac{1}{c}}} \right )}}{4 a^{\frac{3}{8}} c^{10} \left (\frac{1}{c}\right )^{\frac{75}{8}}} + \frac{\left (-1\right )^{\frac{5}{8}} \sqrt{2} \log{\left (- 4 \sqrt [8]{-1} \sqrt{2} \sqrt [8]{a} \sqrt{x} \sqrt [8]{\frac{1}{c}} + 4 \sqrt [4]{-1} \sqrt [4]{a} \sqrt [4]{\frac{1}{c}} + 4 x \right )}}{8 a^{\frac{3}{8}} c^{24} \left (\frac{1}{c}\right )^{\frac{187}{8}}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.31592, size = 590, normalized size = 2.06 \begin{align*} -\frac{\sqrt{-\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{5}{8}} \arctan \left (\frac{\sqrt{-\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{1}{8}} + 2 \, \sqrt{x}}{\sqrt{\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{1}{8}}}\right )}{4 \, a} - \frac{\sqrt{-\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{5}{8}} \arctan \left (-\frac{\sqrt{-\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{1}{8}} - 2 \, \sqrt{x}}{\sqrt{\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{1}{8}}}\right )}{4 \, a} + \frac{\sqrt{\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{5}{8}} \arctan \left (\frac{\sqrt{\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{1}{8}} + 2 \, \sqrt{x}}{\sqrt{-\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{1}{8}}}\right )}{4 \, a} + \frac{\sqrt{\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{5}{8}} \arctan \left (-\frac{\sqrt{\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{1}{8}} - 2 \, \sqrt{x}}{\sqrt{-\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{1}{8}}}\right )}{4 \, a} - \frac{\sqrt{-\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{5}{8}} \log \left (\sqrt{x} \sqrt{\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{1}{8}} + x + \left (\frac{a}{c}\right )^{\frac{1}{4}}\right )}{8 \, a} + \frac{\sqrt{-\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{5}{8}} \log \left (-\sqrt{x} \sqrt{\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{1}{8}} + x + \left (\frac{a}{c}\right )^{\frac{1}{4}}\right )}{8 \, a} + \frac{\sqrt{\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{5}{8}} \log \left (\sqrt{x} \sqrt{-\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{1}{8}} + x + \left (\frac{a}{c}\right )^{\frac{1}{4}}\right )}{8 \, a} - \frac{\sqrt{\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{5}{8}} \log \left (-\sqrt{x} \sqrt{-\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{1}{8}} + x + \left (\frac{a}{c}\right )^{\frac{1}{4}}\right )}{8 \, a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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