Optimal. Leaf size=320 \[ -\frac{9}{4 a^2 \sqrt{x}}+\frac{1}{4 a \sqrt{x} \left (a+c x^4\right )}-\frac{9 \sqrt [8]{c} \log \left (-\sqrt{2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt{x}+\sqrt [4]{-a}+\sqrt [4]{c} x\right )}{32 \sqrt{2} (-a)^{17/8}}+\frac{9 \sqrt [8]{c} \log \left (\sqrt{2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt{x}+\sqrt [4]{-a}+\sqrt [4]{c} x\right )}{32 \sqrt{2} (-a)^{17/8}}+\frac{9 \sqrt [8]{c} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}\right )}{16 \sqrt{2} (-a)^{17/8}}-\frac{9 \sqrt [8]{c} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}+1\right )}{16 \sqrt{2} (-a)^{17/8}}-\frac{9 \sqrt [8]{c} \tan ^{-1}\left (\frac{\sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}\right )}{16 (-a)^{17/8}}+\frac{9 \sqrt [8]{c} \tanh ^{-1}\left (\frac{\sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}\right )}{16 (-a)^{17/8}} \]
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Rubi [A] time = 0.29857, antiderivative size = 320, normalized size of antiderivative = 1., number of steps used = 16, number of rules used = 13, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.867, Rules used = {290, 325, 329, 301, 297, 1162, 617, 204, 1165, 628, 298, 205, 208} \[ -\frac{9}{4 a^2 \sqrt{x}}+\frac{1}{4 a \sqrt{x} \left (a+c x^4\right )}-\frac{9 \sqrt [8]{c} \log \left (-\sqrt{2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt{x}+\sqrt [4]{-a}+\sqrt [4]{c} x\right )}{32 \sqrt{2} (-a)^{17/8}}+\frac{9 \sqrt [8]{c} \log \left (\sqrt{2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt{x}+\sqrt [4]{-a}+\sqrt [4]{c} x\right )}{32 \sqrt{2} (-a)^{17/8}}+\frac{9 \sqrt [8]{c} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}\right )}{16 \sqrt{2} (-a)^{17/8}}-\frac{9 \sqrt [8]{c} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}+1\right )}{16 \sqrt{2} (-a)^{17/8}}-\frac{9 \sqrt [8]{c} \tan ^{-1}\left (\frac{\sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}\right )}{16 (-a)^{17/8}}+\frac{9 \sqrt [8]{c} \tanh ^{-1}\left (\frac{\sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}\right )}{16 (-a)^{17/8}} \]
Antiderivative was successfully verified.
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Rule 290
Rule 325
Rule 329
Rule 301
Rule 297
Rule 1162
Rule 617
Rule 204
Rule 1165
Rule 628
Rule 298
Rule 205
Rule 208
Rubi steps
\begin{align*} \int \frac{1}{x^{3/2} \left (a+c x^4\right )^2} \, dx &=\frac{1}{4 a \sqrt{x} \left (a+c x^4\right )}+\frac{9 \int \frac{1}{x^{3/2} \left (a+c x^4\right )} \, dx}{8 a}\\ &=-\frac{9}{4 a^2 \sqrt{x}}+\frac{1}{4 a \sqrt{x} \left (a+c x^4\right )}-\frac{(9 c) \int \frac{x^{5/2}}{a+c x^4} \, dx}{8 a^2}\\ &=-\frac{9}{4 a^2 \sqrt{x}}+\frac{1}{4 a \sqrt{x} \left (a+c x^4\right )}-\frac{(9 c) \operatorname{Subst}\left (\int \frac{x^6}{a+c x^8} \, dx,x,\sqrt{x}\right )}{4 a^2}\\ &=-\frac{9}{4 a^2 \sqrt{x}}+\frac{1}{4 a \sqrt{x} \left (a+c x^4\right )}+\frac{\left (9 \sqrt{c}\right ) \operatorname{Subst}\left (\int \frac{x^2}{\sqrt{-a}-\sqrt{c} x^4} \, dx,x,\sqrt{x}\right )}{8 a^2}-\frac{\left (9 \sqrt{c}\right ) \operatorname{Subst}\left (\int \frac{x^2}{\sqrt{-a}+\sqrt{c} x^4} \, dx,x,\sqrt{x}\right )}{8 a^2}\\ &=-\frac{9}{4 a^2 \sqrt{x}}+\frac{1}{4 a \sqrt{x} \left (a+c x^4\right )}+\frac{\left (9 \sqrt [4]{c}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt [4]{-a}-\sqrt [4]{c} x^2} \, dx,x,\sqrt{x}\right )}{16 a^2}-\frac{\left (9 \sqrt [4]{c}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt [4]{-a}+\sqrt [4]{c} x^2} \, dx,x,\sqrt{x}\right )}{16 a^2}+\frac{\left (9 \sqrt [4]{c}\right ) \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^2}-\frac{\left (9 \sqrt [4]{c}\right ) \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^2}\\ &=-\frac{9}{4 a^2 \sqrt{x}}+\frac{1}{4 a \sqrt{x} \left (a+c x^4\right )}-\frac{9 \sqrt [8]{c} \tan ^{-1}\left (\frac{\sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}\right )}{16 (-a)^{17/8}}+\frac{9 \sqrt [8]{c} \tanh ^{-1}\left (\frac{\sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}\right )}{16 (-a)^{17/8}}-\frac{9 \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^2}-\frac{9 \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^2}-\frac{\left (9 \sqrt [8]{c}\right ) \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)^{17/8}}-\frac{\left (9 \sqrt [8]{c}\right ) \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)^{17/8}}\\ &=-\frac{9}{4 a^2 \sqrt{x}}+\frac{1}{4 a \sqrt{x} \left (a+c x^4\right )}-\frac{9 \sqrt [8]{c} \tan ^{-1}\left (\frac{\sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}\right )}{16 (-a)^{17/8}}+\frac{9 \sqrt [8]{c} \tanh ^{-1}\left (\frac{\sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}\right )}{16 (-a)^{17/8}}-\frac{9 \sqrt [8]{c} \log \left (\sqrt [4]{-a}-\sqrt{2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt{x}+\sqrt [4]{c} x\right )}{32 \sqrt{2} (-a)^{17/8}}+\frac{9 \sqrt [8]{c} \log \left (\sqrt [4]{-a}+\sqrt{2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt{x}+\sqrt [4]{c} x\right )}{32 \sqrt{2} (-a)^{17/8}}-\frac{\left (9 \sqrt [8]{c}\right ) \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)^{17/8}}+\frac{\left (9 \sqrt [8]{c}\right ) \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)^{17/8}}\\ &=-\frac{9}{4 a^2 \sqrt{x}}+\frac{1}{4 a \sqrt{x} \left (a+c x^4\right )}+\frac{9 \sqrt [8]{c} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}\right )}{16 \sqrt{2} (-a)^{17/8}}-\frac{9 \sqrt [8]{c} \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}\right )}{16 \sqrt{2} (-a)^{17/8}}-\frac{9 \sqrt [8]{c} \tan ^{-1}\left (\frac{\sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}\right )}{16 (-a)^{17/8}}+\frac{9 \sqrt [8]{c} \tanh ^{-1}\left (\frac{\sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}\right )}{16 (-a)^{17/8}}-\frac{9 \sqrt [8]{c} \log \left (\sqrt [4]{-a}-\sqrt{2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt{x}+\sqrt [4]{c} x\right )}{32 \sqrt{2} (-a)^{17/8}}+\frac{9 \sqrt [8]{c} \log \left (\sqrt [4]{-a}+\sqrt{2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt{x}+\sqrt [4]{c} x\right )}{32 \sqrt{2} (-a)^{17/8}}\\ \end{align*}
Mathematica [C] time = 0.0057917, size = 27, normalized size = 0.08 \[ -\frac{2 \, _2F_1\left (-\frac{1}{8},2;\frac{7}{8};-\frac{c x^4}{a}\right )}{a^2 \sqrt{x}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.015, size = 56, normalized size = 0.2 \begin{align*} -2\,{\frac{1}{{a}^{2}\sqrt{x}}}-{\frac{c}{4\,{a}^{2} \left ( c{x}^{4}+a \right ) }{x}^{{\frac{7}{2}}}}-{\frac{9}{32\,{a}^{2}}\sum _{{\it \_R}={\it RootOf} \left ({{\it \_Z}}^{8}c+a \right ) }{\frac{1}{{\it \_R}}\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*} -9 \, c \int \frac{x^{\frac{5}{2}}}{8 \,{\left (a^{2} c x^{4} + a^{3}\right )}}\,{d x} - \frac{9 \, c x^{\frac{7}{2}} + \frac{8 \, a}{\sqrt{x}}}{4 \,{\left (a^{2} c x^{4} + a^{3}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.79596, size = 1721, normalized size = 5.38 \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 = 1.42837, size = 639, normalized size = 2. \begin{align*} -\frac{9 \, c \sqrt{\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{7}{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^{3}} - \frac{9 \, c \sqrt{\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{7}{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^{3}} - \frac{9 \, c \sqrt{-\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{7}{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^{3}} - \frac{9 \, c \sqrt{-\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{7}{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^{3}} + \frac{9 \, c \sqrt{\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{7}{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^{3}} - \frac{9 \, c \sqrt{\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{7}{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^{3}} + \frac{9 \, c \sqrt{-\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{7}{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^{3}} - \frac{9 \, c \sqrt{-\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{7}{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^{3}} - \frac{9 \, c x^{4} + 8 \, a}{4 \,{\left (c x^{\frac{9}{2}} + a \sqrt{x}\right )} a^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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