Optimal. Leaf size=308 \[ \frac{3 \log \left (-\sqrt{2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt{x}+\sqrt [4]{-a}+\sqrt [4]{c} x\right )}{32 \sqrt{2} (-a)^{11/8} c^{5/8}}-\frac{3 \log \left (\sqrt{2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt{x}+\sqrt [4]{-a}+\sqrt [4]{c} x\right )}{32 \sqrt{2} (-a)^{11/8} c^{5/8}}+\frac{3 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}\right )}{16 \sqrt{2} (-a)^{11/8} c^{5/8}}-\frac{3 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}+1\right )}{16 \sqrt{2} (-a)^{11/8} c^{5/8}}+\frac{3 \tan ^{-1}\left (\frac{\sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}\right )}{16 (-a)^{11/8} c^{5/8}}+\frac{3 \tanh ^{-1}\left (\frac{\sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}\right )}{16 (-a)^{11/8} c^{5/8}}+\frac{x^{5/2}}{4 a \left (a+c x^4\right )} \]
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Rubi [A] time = 0.260625, antiderivative size = 308, normalized size of antiderivative = 1., number of steps used = 15, number of rules used = 12, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.8, Rules used = {290, 329, 301, 211, 1165, 628, 1162, 617, 204, 212, 208, 205} \[ \frac{3 \log \left (-\sqrt{2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt{x}+\sqrt [4]{-a}+\sqrt [4]{c} x\right )}{32 \sqrt{2} (-a)^{11/8} c^{5/8}}-\frac{3 \log \left (\sqrt{2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt{x}+\sqrt [4]{-a}+\sqrt [4]{c} x\right )}{32 \sqrt{2} (-a)^{11/8} c^{5/8}}+\frac{3 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}\right )}{16 \sqrt{2} (-a)^{11/8} c^{5/8}}-\frac{3 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}+1\right )}{16 \sqrt{2} (-a)^{11/8} c^{5/8}}+\frac{3 \tan ^{-1}\left (\frac{\sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}\right )}{16 (-a)^{11/8} c^{5/8}}+\frac{3 \tanh ^{-1}\left (\frac{\sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}\right )}{16 (-a)^{11/8} c^{5/8}}+\frac{x^{5/2}}{4 a \left (a+c x^4\right )} \]
Antiderivative was successfully verified.
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Rule 290
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}}{\left (a+c x^4\right )^2} \, dx &=\frac{x^{5/2}}{4 a \left (a+c x^4\right )}+\frac{3 \int \frac{x^{3/2}}{a+c x^4} \, dx}{8 a}\\ &=\frac{x^{5/2}}{4 a \left (a+c x^4\right )}+\frac{3 \operatorname{Subst}\left (\int \frac{x^4}{a+c x^8} \, dx,x,\sqrt{x}\right )}{4 a}\\ &=\frac{x^{5/2}}{4 a \left (a+c x^4\right )}-\frac{3 \operatorname{Subst}\left (\int \frac{1}{\sqrt{-a}-\sqrt{c} x^4} \, dx,x,\sqrt{x}\right )}{8 a \sqrt{c}}+\frac{3 \operatorname{Subst}\left (\int \frac{1}{\sqrt{-a}+\sqrt{c} x^4} \, dx,x,\sqrt{x}\right )}{8 a \sqrt{c}}\\ &=\frac{x^{5/2}}{4 a \left (a+c x^4\right )}+\frac{3 \operatorname{Subst}\left (\int \frac{1}{\sqrt [4]{-a}-\sqrt [4]{c} x^2} \, dx,x,\sqrt{x}\right )}{16 (-a)^{5/4} \sqrt{c}}+\frac{3 \operatorname{Subst}\left (\int \frac{1}{\sqrt [4]{-a}+\sqrt [4]{c} x^2} \, dx,x,\sqrt{x}\right )}{16 (-a)^{5/4} \sqrt{c}}-\frac{3 \operatorname{Subst}\left (\int \frac{\sqrt [4]{-a}-\sqrt [4]{c} x^2}{\sqrt{-a}+\sqrt{c} x^4} \, dx,x,\sqrt{x}\right )}{16 (-a)^{5/4} \sqrt{c}}-\frac{3 \operatorname{Subst}\left (\int \frac{\sqrt [4]{-a}+\sqrt [4]{c} x^2}{\sqrt{-a}+\sqrt{c} x^4} \, dx,x,\sqrt{x}\right )}{16 (-a)^{5/4} \sqrt{c}}\\ &=\frac{x^{5/2}}{4 a \left (a+c x^4\right )}+\frac{3 \tan ^{-1}\left (\frac{\sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}\right )}{16 (-a)^{11/8} c^{5/8}}+\frac{3 \tanh ^{-1}\left (\frac{\sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}\right )}{16 (-a)^{11/8} c^{5/8}}-\frac{3 \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 )}{32 (-a)^{5/4} c^{3/4}}-\frac{3 \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 )}{32 (-a)^{5/4} c^{3/4}}+\frac{3 \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 )}{32 \sqrt{2} (-a)^{11/8} c^{5/8}}+\frac{3 \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 )}{32 \sqrt{2} (-a)^{11/8} c^{5/8}}\\ &=\frac{x^{5/2}}{4 a \left (a+c x^4\right )}+\frac{3 \tan ^{-1}\left (\frac{\sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}\right )}{16 (-a)^{11/8} c^{5/8}}+\frac{3 \tanh ^{-1}\left (\frac{\sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}\right )}{16 (-a)^{11/8} c^{5/8}}+\frac{3 \log \left (\sqrt [4]{-a}-\sqrt{2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt{x}+\sqrt [4]{c} x\right )}{32 \sqrt{2} (-a)^{11/8} c^{5/8}}-\frac{3 \log \left (\sqrt [4]{-a}+\sqrt{2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt{x}+\sqrt [4]{c} x\right )}{32 \sqrt{2} (-a)^{11/8} c^{5/8}}-\frac{3 \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1-\frac{\sqrt{2} \sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}\right )}{16 \sqrt{2} (-a)^{11/8} c^{5/8}}+\frac{3 \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1+\frac{\sqrt{2} \sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}\right )}{16 \sqrt{2} (-a)^{11/8} c^{5/8}}\\ &=\frac{x^{5/2}}{4 a \left (a+c x^4\right )}+\frac{3 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}\right )}{16 \sqrt{2} (-a)^{11/8} c^{5/8}}-\frac{3 \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}\right )}{16 \sqrt{2} (-a)^{11/8} c^{5/8}}+\frac{3 \tan ^{-1}\left (\frac{\sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}\right )}{16 (-a)^{11/8} c^{5/8}}+\frac{3 \tanh ^{-1}\left (\frac{\sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}\right )}{16 (-a)^{11/8} c^{5/8}}+\frac{3 \log \left (\sqrt [4]{-a}-\sqrt{2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt{x}+\sqrt [4]{c} x\right )}{32 \sqrt{2} (-a)^{11/8} c^{5/8}}-\frac{3 \log \left (\sqrt [4]{-a}+\sqrt{2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt{x}+\sqrt [4]{c} x\right )}{32 \sqrt{2} (-a)^{11/8} c^{5/8}}\\ \end{align*}
Mathematica [C] time = 0.0061858, size = 29, normalized size = 0.09 \[ \frac{2 x^{5/2} \, _2F_1\left (\frac{5}{8},2;\frac{13}{8};-\frac{c x^4}{a}\right )}{5 a^2} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.011, size = 50, normalized size = 0.2 \begin{align*}{\frac{1}{4\,a \left ( c{x}^{4}+a \right ) }{x}^{{\frac{5}{2}}}}+{\frac{3}{32\,ac}\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*} \frac{x^{\frac{5}{2}}}{4 \,{\left (a c x^{4} + a^{2}\right )}} + 3 \, \int \frac{x^{\frac{3}{2}}}{8 \,{\left (a c x^{4} + a^{2}\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.73104, size = 1501, normalized size = 4.87 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.54997, size = 613, normalized size = 1.99 \begin{align*} \frac{x^{\frac{5}{2}}}{4 \,{\left (c x^{4} + a\right )} a} - \frac{3 \, \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 )}{32 \, a^{2}} - \frac{3 \, \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 )}{32 \, a^{2}} + \frac{3 \, \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 )}{32 \, a^{2}} + \frac{3 \, \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 )}{32 \, a^{2}} - \frac{3 \, \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 )}{64 \, a^{2}} + \frac{3 \, \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 )}{64 \, a^{2}} + \frac{3 \, \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 )}{64 \, a^{2}} - \frac{3 \, \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 )}{64 \, a^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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