Optimal. Leaf size=242 \[ -\frac{3 \log \left (-\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{64 \sqrt{2} b^{7/4} c^{5/4}}+\frac{3 \log \left (\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{64 \sqrt{2} b^{7/4} c^{5/4}}-\frac{3 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )}{32 \sqrt{2} b^{7/4} c^{5/4}}+\frac{3 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}+1\right )}{32 \sqrt{2} b^{7/4} c^{5/4}}+\frac{\sqrt{x}}{16 b c \left (b+c x^2\right )}-\frac{\sqrt{x}}{4 c \left (b+c x^2\right )^2} \]
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Rubi [A] time = 0.183075, antiderivative size = 242, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 10, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.526, Rules used = {1584, 288, 290, 329, 211, 1165, 628, 1162, 617, 204} \[ -\frac{3 \log \left (-\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{64 \sqrt{2} b^{7/4} c^{5/4}}+\frac{3 \log \left (\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{64 \sqrt{2} b^{7/4} c^{5/4}}-\frac{3 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )}{32 \sqrt{2} b^{7/4} c^{5/4}}+\frac{3 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}+1\right )}{32 \sqrt{2} b^{7/4} c^{5/4}}+\frac{\sqrt{x}}{16 b c \left (b+c x^2\right )}-\frac{\sqrt{x}}{4 c \left (b+c x^2\right )^2} \]
Antiderivative was successfully verified.
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Rule 1584
Rule 288
Rule 290
Rule 329
Rule 211
Rule 1165
Rule 628
Rule 1162
Rule 617
Rule 204
Rubi steps
\begin{align*} \int \frac{x^{15/2}}{\left (b x^2+c x^4\right )^3} \, dx &=\int \frac{x^{3/2}}{\left (b+c x^2\right )^3} \, dx\\ &=-\frac{\sqrt{x}}{4 c \left (b+c x^2\right )^2}+\frac{\int \frac{1}{\sqrt{x} \left (b+c x^2\right )^2} \, dx}{8 c}\\ &=-\frac{\sqrt{x}}{4 c \left (b+c x^2\right )^2}+\frac{\sqrt{x}}{16 b c \left (b+c x^2\right )}+\frac{3 \int \frac{1}{\sqrt{x} \left (b+c x^2\right )} \, dx}{32 b c}\\ &=-\frac{\sqrt{x}}{4 c \left (b+c x^2\right )^2}+\frac{\sqrt{x}}{16 b c \left (b+c x^2\right )}+\frac{3 \operatorname{Subst}\left (\int \frac{1}{b+c x^4} \, dx,x,\sqrt{x}\right )}{16 b c}\\ &=-\frac{\sqrt{x}}{4 c \left (b+c x^2\right )^2}+\frac{\sqrt{x}}{16 b c \left (b+c x^2\right )}+\frac{3 \operatorname{Subst}\left (\int \frac{\sqrt{b}-\sqrt{c} x^2}{b+c x^4} \, dx,x,\sqrt{x}\right )}{32 b^{3/2} c}+\frac{3 \operatorname{Subst}\left (\int \frac{\sqrt{b}+\sqrt{c} x^2}{b+c x^4} \, dx,x,\sqrt{x}\right )}{32 b^{3/2} c}\\ &=-\frac{\sqrt{x}}{4 c \left (b+c x^2\right )^2}+\frac{\sqrt{x}}{16 b c \left (b+c x^2\right )}+\frac{3 \operatorname{Subst}\left (\int \frac{1}{\frac{\sqrt{b}}{\sqrt{c}}-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{c}}+x^2} \, dx,x,\sqrt{x}\right )}{64 b^{3/2} c^{3/2}}+\frac{3 \operatorname{Subst}\left (\int \frac{1}{\frac{\sqrt{b}}{\sqrt{c}}+\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{c}}+x^2} \, dx,x,\sqrt{x}\right )}{64 b^{3/2} c^{3/2}}-\frac{3 \operatorname{Subst}\left (\int \frac{\frac{\sqrt{2} \sqrt [4]{b}}{\sqrt [4]{c}}+2 x}{-\frac{\sqrt{b}}{\sqrt{c}}-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{c}}-x^2} \, dx,x,\sqrt{x}\right )}{64 \sqrt{2} b^{7/4} c^{5/4}}-\frac{3 \operatorname{Subst}\left (\int \frac{\frac{\sqrt{2} \sqrt [4]{b}}{\sqrt [4]{c}}-2 x}{-\frac{\sqrt{b}}{\sqrt{c}}+\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{c}}-x^2} \, dx,x,\sqrt{x}\right )}{64 \sqrt{2} b^{7/4} c^{5/4}}\\ &=-\frac{\sqrt{x}}{4 c \left (b+c x^2\right )^2}+\frac{\sqrt{x}}{16 b c \left (b+c x^2\right )}-\frac{3 \log \left (\sqrt{b}-\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{c} x\right )}{64 \sqrt{2} b^{7/4} c^{5/4}}+\frac{3 \log \left (\sqrt{b}+\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{c} x\right )}{64 \sqrt{2} b^{7/4} c^{5/4}}+\frac{3 \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )}{32 \sqrt{2} b^{7/4} c^{5/4}}-\frac{3 \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1+\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )}{32 \sqrt{2} b^{7/4} c^{5/4}}\\ &=-\frac{\sqrt{x}}{4 c \left (b+c x^2\right )^2}+\frac{\sqrt{x}}{16 b c \left (b+c x^2\right )}-\frac{3 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )}{32 \sqrt{2} b^{7/4} c^{5/4}}+\frac{3 \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )}{32 \sqrt{2} b^{7/4} c^{5/4}}-\frac{3 \log \left (\sqrt{b}-\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{c} x\right )}{64 \sqrt{2} b^{7/4} c^{5/4}}+\frac{3 \log \left (\sqrt{b}+\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{c} x\right )}{64 \sqrt{2} b^{7/4} c^{5/4}}\\ \end{align*}
Mathematica [A] time = 0.116256, size = 223, normalized size = 0.92 \[ \frac{\frac{8 \sqrt [4]{c} \sqrt{x}}{b^2+b c x^2}-\frac{3 \sqrt{2} \log \left (-\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{b^{7/4}}+\frac{3 \sqrt{2} \log \left (\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{b^{7/4}}-\frac{6 \sqrt{2} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )}{b^{7/4}}+\frac{6 \sqrt{2} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}+1\right )}{b^{7/4}}-\frac{32 \sqrt [4]{c} \sqrt{x}}{\left (b+c x^2\right )^2}}{128 c^{5/4}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.06, size = 169, normalized size = 0.7 \begin{align*} 2\,{\frac{1}{ \left ( c{x}^{2}+b \right ) ^{2}} \left ( 1/32\,{\frac{{x}^{5/2}}{b}}-{\frac{3\,\sqrt{x}}{32\,c}} \right ) }+{\frac{3\,\sqrt{2}}{128\,{b}^{2}c}\sqrt [4]{{\frac{b}{c}}}\ln \left ({ \left ( x+\sqrt [4]{{\frac{b}{c}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{b}{c}}} \right ) \left ( x-\sqrt [4]{{\frac{b}{c}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{b}{c}}} \right ) ^{-1}} \right ) }+{\frac{3\,\sqrt{2}}{64\,{b}^{2}c}\sqrt [4]{{\frac{b}{c}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}}+1 \right ) }+{\frac{3\,\sqrt{2}}{64\,{b}^{2}c}\sqrt [4]{{\frac{b}{c}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}}-1 \right ) } \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 [A] time = 1.38107, size = 608, normalized size = 2.51 \begin{align*} \frac{12 \,{\left (b c^{3} x^{4} + 2 \, b^{2} c^{2} x^{2} + b^{3} c\right )} \left (-\frac{1}{b^{7} c^{5}}\right )^{\frac{1}{4}} \arctan \left (\sqrt{b^{4} c^{2} \sqrt{-\frac{1}{b^{7} c^{5}}} + x} b^{5} c^{4} \left (-\frac{1}{b^{7} c^{5}}\right )^{\frac{3}{4}} - b^{5} c^{4} \sqrt{x} \left (-\frac{1}{b^{7} c^{5}}\right )^{\frac{3}{4}}\right ) + 3 \,{\left (b c^{3} x^{4} + 2 \, b^{2} c^{2} x^{2} + b^{3} c\right )} \left (-\frac{1}{b^{7} c^{5}}\right )^{\frac{1}{4}} \log \left (b^{2} c \left (-\frac{1}{b^{7} c^{5}}\right )^{\frac{1}{4}} + \sqrt{x}\right ) - 3 \,{\left (b c^{3} x^{4} + 2 \, b^{2} c^{2} x^{2} + b^{3} c\right )} \left (-\frac{1}{b^{7} c^{5}}\right )^{\frac{1}{4}} \log \left (-b^{2} c \left (-\frac{1}{b^{7} c^{5}}\right )^{\frac{1}{4}} + \sqrt{x}\right ) + 4 \,{\left (c x^{2} - 3 \, b\right )} \sqrt{x}}{64 \,{\left (b c^{3} x^{4} + 2 \, b^{2} c^{2} x^{2} + b^{3} c\right )}} \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 [A] time = 1.13433, size = 285, normalized size = 1.18 \begin{align*} \frac{3 \, \sqrt{2} \left (b c^{3}\right )^{\frac{1}{4}} \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{b}{c}\right )^{\frac{1}{4}} + 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{b}{c}\right )^{\frac{1}{4}}}\right )}{64 \, b^{2} c^{2}} + \frac{3 \, \sqrt{2} \left (b c^{3}\right )^{\frac{1}{4}} \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{b}{c}\right )^{\frac{1}{4}} - 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{b}{c}\right )^{\frac{1}{4}}}\right )}{64 \, b^{2} c^{2}} + \frac{3 \, \sqrt{2} \left (b c^{3}\right )^{\frac{1}{4}} \log \left (\sqrt{2} \sqrt{x} \left (\frac{b}{c}\right )^{\frac{1}{4}} + x + \sqrt{\frac{b}{c}}\right )}{128 \, b^{2} c^{2}} - \frac{3 \, \sqrt{2} \left (b c^{3}\right )^{\frac{1}{4}} \log \left (-\sqrt{2} \sqrt{x} \left (\frac{b}{c}\right )^{\frac{1}{4}} + x + \sqrt{\frac{b}{c}}\right )}{128 \, b^{2} c^{2}} + \frac{c x^{\frac{5}{2}} - 3 \, b \sqrt{x}}{16 \,{\left (c x^{2} + b\right )}^{2} b c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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