Optimal. Leaf size=76 \[ \frac{8 c^{3/2} \tan ^{-1}\left (\frac{\sqrt{c+d x^3}}{\sqrt{3} \sqrt{c}}\right )}{\sqrt{3} d^2}-\frac{8 c \sqrt{c+d x^3}}{3 d^2}+\frac{2 \left (c+d x^3\right )^{3/2}}{9 d^2} \]
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Rubi [A] time = 0.0623822, antiderivative size = 76, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.192, Rules used = {446, 80, 50, 63, 203} \[ \frac{8 c^{3/2} \tan ^{-1}\left (\frac{\sqrt{c+d x^3}}{\sqrt{3} \sqrt{c}}\right )}{\sqrt{3} d^2}-\frac{8 c \sqrt{c+d x^3}}{3 d^2}+\frac{2 \left (c+d x^3\right )^{3/2}}{9 d^2} \]
Antiderivative was successfully verified.
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Rule 446
Rule 80
Rule 50
Rule 63
Rule 203
Rubi steps
\begin{align*} \int \frac{x^5 \sqrt{c+d x^3}}{4 c+d x^3} \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int \frac{x \sqrt{c+d x}}{4 c+d x} \, dx,x,x^3\right )\\ &=\frac{2 \left (c+d x^3\right )^{3/2}}{9 d^2}-\frac{(4 c) \operatorname{Subst}\left (\int \frac{\sqrt{c+d x}}{4 c+d x} \, dx,x,x^3\right )}{3 d}\\ &=-\frac{8 c \sqrt{c+d x^3}}{3 d^2}+\frac{2 \left (c+d x^3\right )^{3/2}}{9 d^2}+\frac{\left (4 c^2\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{c+d x} (4 c+d x)} \, dx,x,x^3\right )}{d}\\ &=-\frac{8 c \sqrt{c+d x^3}}{3 d^2}+\frac{2 \left (c+d x^3\right )^{3/2}}{9 d^2}+\frac{\left (8 c^2\right ) \operatorname{Subst}\left (\int \frac{1}{3 c+x^2} \, dx,x,\sqrt{c+d x^3}\right )}{d^2}\\ &=-\frac{8 c \sqrt{c+d x^3}}{3 d^2}+\frac{2 \left (c+d x^3\right )^{3/2}}{9 d^2}+\frac{8 c^{3/2} \tan ^{-1}\left (\frac{\sqrt{c+d x^3}}{\sqrt{3} \sqrt{c}}\right )}{\sqrt{3} d^2}\\ \end{align*}
Mathematica [A] time = 0.0283623, size = 65, normalized size = 0.86 \[ \frac{24 \sqrt{3} c^{3/2} \tan ^{-1}\left (\frac{\sqrt{c+d x^3}}{\sqrt{3} \sqrt{c}}\right )+2 \left (d x^3-11 c\right ) \sqrt{c+d x^3}}{9 d^2} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.01, size = 446, normalized size = 5.9 \begin{align*}{\frac{2}{9\,{d}^{2}} \left ( d{x}^{3}+c \right ) ^{{\frac{3}{2}}}}-4\,{\frac{c}{d} \left ( 2/3\,{\frac{\sqrt{d{x}^{3}+c}}{d}}+{\frac{i/3\sqrt{2}}{{d}^{3}}\sum _{{\it \_alpha}={\it RootOf} \left ({{\it \_Z}}^{3}d+4\,c \right ) }{\frac{\sqrt [3]{-{d}^{2}c} \left ( i\sqrt [3]{-{d}^{2}c}{\it \_alpha}\,\sqrt{3}d-i \left ( -{d}^{2}c \right ) ^{2/3}\sqrt{3}+2\,{{\it \_alpha}}^{2}{d}^{2}-\sqrt [3]{-{d}^{2}c}{\it \_alpha}\,d- \left ( -{d}^{2}c \right ) ^{2/3} \right ) }{\sqrt{d{x}^{3}+c}}\sqrt{{\frac{i/2d}{\sqrt [3]{-{d}^{2}c}} \left ( 2\,x+{\frac{-i\sqrt{3}\sqrt [3]{-{d}^{2}c}+\sqrt [3]{-{d}^{2}c}}{d}} \right ) }}\sqrt{{\frac{d}{-3\,\sqrt [3]{-{d}^{2}c}+i\sqrt{3}\sqrt [3]{-{d}^{2}c}} \left ( x-{\frac{\sqrt [3]{-{d}^{2}c}}{d}} \right ) }}\sqrt{{\frac{-i/2d}{\sqrt [3]{-{d}^{2}c}} \left ( 2\,x+{\frac{i\sqrt{3}\sqrt [3]{-{d}^{2}c}+\sqrt [3]{-{d}^{2}c}}{d}} \right ) }}{\it EllipticPi} \left ( 1/3\,\sqrt{3}\sqrt{{\frac{i\sqrt{3}d}{\sqrt [3]{-{d}^{2}c}} \left ( x+1/2\,{\frac{\sqrt [3]{-{d}^{2}c}}{d}}-{\frac{i/2\sqrt{3}\sqrt [3]{-{d}^{2}c}}{d}} \right ) }},1/6\,{\frac{2\,i\sqrt [3]{-{d}^{2}c}\sqrt{3}{{\it \_alpha}}^{2}d-i \left ( -{d}^{2}c \right ) ^{2/3}\sqrt{3}{\it \_alpha}+i\sqrt{3}cd-3\, \left ( -{d}^{2}c \right ) ^{2/3}{\it \_alpha}-3\,cd}{cd}},\sqrt{{\frac{i\sqrt{3}\sqrt [3]{-{d}^{2}c}}{d} \left ( -3/2\,{\frac{\sqrt [3]{-{d}^{2}c}}{d}}+{\frac{i/2\sqrt{3}\sqrt [3]{-{d}^{2}c}}{d}} \right ) ^{-1}}} \right ) }} \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 = 2.09525, size = 332, normalized size = 4.37 \begin{align*} \left [\frac{2 \,{\left (6 \, \sqrt{3} \sqrt{-c} c \log \left (\frac{d x^{3} + 2 \, \sqrt{3} \sqrt{d x^{3} + c} \sqrt{-c} - 2 \, c}{d x^{3} + 4 \, c}\right ) + \sqrt{d x^{3} + c}{\left (d x^{3} - 11 \, c\right )}\right )}}{9 \, d^{2}}, \frac{2 \,{\left (12 \, \sqrt{3} c^{\frac{3}{2}} \arctan \left (\frac{\sqrt{3} \sqrt{d x^{3} + c}}{3 \, \sqrt{c}}\right ) + \sqrt{d x^{3} + c}{\left (d x^{3} - 11 \, c\right )}\right )}}{9 \, d^{2}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 19.2291, size = 68, normalized size = 0.89 \begin{align*} \frac{2 \left (\frac{4 \sqrt{3} c^{\frac{3}{2}} \operatorname{atan}{\left (\frac{\sqrt{3} \sqrt{c + d x^{3}}}{3 \sqrt{c}} \right )}}{3} - \frac{4 c \sqrt{c + d x^{3}}}{3} + \frac{\left (c + d x^{3}\right )^{\frac{3}{2}}}{9}\right )}{d^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.13672, size = 92, normalized size = 1.21 \begin{align*} \frac{2 \,{\left (\frac{12 \, \sqrt{3} c^{\frac{3}{2}} \arctan \left (\frac{\sqrt{3} \sqrt{d x^{3} + c}}{3 \, \sqrt{c}}\right )}{d} + \frac{{\left (d x^{3} + c\right )}^{\frac{3}{2}} d^{2} - 12 \, \sqrt{d x^{3} + c} c d^{2}}{d^{3}}\right )}}{9 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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