Optimal. Leaf size=190 \[ \frac {3 b \sqrt {x}}{2 c}+\frac {\sqrt {3} b \text {ArcTan}\left (\frac {1-2 \sqrt [3]{c} \sqrt {x}}{\sqrt {3}}\right )}{4 c^{4/3}}-\frac {\sqrt {3} b \text {ArcTan}\left (\frac {1+2 \sqrt [3]{c} \sqrt {x}}{\sqrt {3}}\right )}{4 c^{4/3}}-\frac {b \tanh ^{-1}\left (\sqrt [3]{c} \sqrt {x}\right )}{2 c^{4/3}}+\frac {1}{2} x^2 \left (a+b \tanh ^{-1}\left (c x^{3/2}\right )\right )+\frac {b \log \left (1-\sqrt [3]{c} \sqrt {x}+c^{2/3} x\right )}{8 c^{4/3}}-\frac {b \log \left (1+\sqrt [3]{c} \sqrt {x}+c^{2/3} x\right )}{8 c^{4/3}} \]
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Rubi [A]
time = 0.25, antiderivative size = 190, normalized size of antiderivative = 1.00, number of steps
used = 13, number of rules used = 9, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.643, Rules used = {6037, 327,
335, 216, 648, 632, 210, 642, 212} \begin {gather*} \frac {1}{2} x^2 \left (a+b \tanh ^{-1}\left (c x^{3/2}\right )\right )+\frac {\sqrt {3} b \text {ArcTan}\left (\frac {1-2 \sqrt [3]{c} \sqrt {x}}{\sqrt {3}}\right )}{4 c^{4/3}}-\frac {\sqrt {3} b \text {ArcTan}\left (\frac {2 \sqrt [3]{c} \sqrt {x}+1}{\sqrt {3}}\right )}{4 c^{4/3}}+\frac {b \log \left (c^{2/3} x-\sqrt [3]{c} \sqrt {x}+1\right )}{8 c^{4/3}}-\frac {b \log \left (c^{2/3} x+\sqrt [3]{c} \sqrt {x}+1\right )}{8 c^{4/3}}-\frac {b \tanh ^{-1}\left (\sqrt [3]{c} \sqrt {x}\right )}{2 c^{4/3}}+\frac {3 b \sqrt {x}}{2 c} \end {gather*}
Antiderivative was successfully verified.
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Rule 210
Rule 212
Rule 216
Rule 327
Rule 335
Rule 632
Rule 642
Rule 648
Rule 6037
Rubi steps
\begin {align*} \int x \left (a+b \tanh ^{-1}\left (c x^{3/2}\right )\right ) \, dx &=\frac {1}{2} x^2 \left (a+b \tanh ^{-1}\left (c x^{3/2}\right )\right )-\frac {1}{4} (3 b c) \int \frac {x^{5/2}}{1-c^2 x^3} \, dx\\ &=\frac {3 b \sqrt {x}}{2 c}+\frac {1}{2} x^2 \left (a+b \tanh ^{-1}\left (c x^{3/2}\right )\right )-\frac {(3 b) \int \frac {1}{\sqrt {x} \left (1-c^2 x^3\right )} \, dx}{4 c}\\ &=\frac {3 b \sqrt {x}}{2 c}+\frac {1}{2} x^2 \left (a+b \tanh ^{-1}\left (c x^{3/2}\right )\right )-\frac {(3 b) \text {Subst}\left (\int \frac {1}{1-c^2 x^6} \, dx,x,\sqrt {x}\right )}{2 c}\\ &=\frac {3 b \sqrt {x}}{2 c}+\frac {1}{2} x^2 \left (a+b \tanh ^{-1}\left (c x^{3/2}\right )\right )-\frac {b \text {Subst}\left (\int \frac {1}{1-c^{2/3} x^2} \, dx,x,\sqrt {x}\right )}{2 c}-\frac {b \text {Subst}\left (\int \frac {1-\frac {\sqrt [3]{c} x}{2}}{1-\sqrt [3]{c} x+c^{2/3} x^2} \, dx,x,\sqrt {x}\right )}{2 c}-\frac {b \text {Subst}\left (\int \frac {1+\frac {\sqrt [3]{c} x}{2}}{1+\sqrt [3]{c} x+c^{2/3} x^2} \, dx,x,\sqrt {x}\right )}{2 c}\\ &=\frac {3 b \sqrt {x}}{2 c}-\frac {b \tanh ^{-1}\left (\sqrt [3]{c} \sqrt {x}\right )}{2 c^{4/3}}+\frac {1}{2} x^2 \left (a+b \tanh ^{-1}\left (c x^{3/2}\right )\right )+\frac {b \text {Subst}\left (\int \frac {-\sqrt [3]{c}+2 c^{2/3} x}{1-\sqrt [3]{c} x+c^{2/3} x^2} \, dx,x,\sqrt {x}\right )}{8 c^{4/3}}-\frac {b \text {Subst}\left (\int \frac {\sqrt [3]{c}+2 c^{2/3} x}{1+\sqrt [3]{c} x+c^{2/3} x^2} \, dx,x,\sqrt {x}\right )}{8 c^{4/3}}-\frac {(3 b) \text {Subst}\left (\int \frac {1}{1-\sqrt [3]{c} x+c^{2/3} x^2} \, dx,x,\sqrt {x}\right )}{8 c}-\frac {(3 b) \text {Subst}\left (\int \frac {1}{1+\sqrt [3]{c} x+c^{2/3} x^2} \, dx,x,\sqrt {x}\right )}{8 c}\\ &=\frac {3 b \sqrt {x}}{2 c}-\frac {b \tanh ^{-1}\left (\sqrt [3]{c} \sqrt {x}\right )}{2 c^{4/3}}+\frac {1}{2} x^2 \left (a+b \tanh ^{-1}\left (c x^{3/2}\right )\right )+\frac {b \log \left (1-\sqrt [3]{c} \sqrt {x}+c^{2/3} x\right )}{8 c^{4/3}}-\frac {b \log \left (1+\sqrt [3]{c} \sqrt {x}+c^{2/3} x\right )}{8 c^{4/3}}-\frac {(3 b) \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-2 \sqrt [3]{c} \sqrt {x}\right )}{4 c^{4/3}}+\frac {(3 b) \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+2 \sqrt [3]{c} \sqrt {x}\right )}{4 c^{4/3}}\\ &=\frac {3 b \sqrt {x}}{2 c}+\frac {\sqrt {3} b \tan ^{-1}\left (\frac {1-2 \sqrt [3]{c} \sqrt {x}}{\sqrt {3}}\right )}{4 c^{4/3}}-\frac {\sqrt {3} b \tan ^{-1}\left (\frac {1+2 \sqrt [3]{c} \sqrt {x}}{\sqrt {3}}\right )}{4 c^{4/3}}-\frac {b \tanh ^{-1}\left (\sqrt [3]{c} \sqrt {x}\right )}{2 c^{4/3}}+\frac {1}{2} x^2 \left (a+b \tanh ^{-1}\left (c x^{3/2}\right )\right )+\frac {b \log \left (1-\sqrt [3]{c} \sqrt {x}+c^{2/3} x\right )}{8 c^{4/3}}-\frac {b \log \left (1+\sqrt [3]{c} \sqrt {x}+c^{2/3} x\right )}{8 c^{4/3}}\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 222, normalized size = 1.17 \begin {gather*} \frac {3 b \sqrt {x}}{2 c}+\frac {a x^2}{2}-\frac {\sqrt {3} b \text {ArcTan}\left (\frac {-1+2 \sqrt [3]{c} \sqrt {x}}{\sqrt {3}}\right )}{4 c^{4/3}}-\frac {\sqrt {3} b \text {ArcTan}\left (\frac {1+2 \sqrt [3]{c} \sqrt {x}}{\sqrt {3}}\right )}{4 c^{4/3}}+\frac {1}{2} b x^2 \tanh ^{-1}\left (c x^{3/2}\right )+\frac {b \log \left (1-\sqrt [3]{c} \sqrt {x}\right )}{4 c^{4/3}}-\frac {b \log \left (1+\sqrt [3]{c} \sqrt {x}\right )}{4 c^{4/3}}+\frac {b \log \left (1-\sqrt [3]{c} \sqrt {x}+c^{2/3} x\right )}{8 c^{4/3}}-\frac {b \log \left (1+\sqrt [3]{c} \sqrt {x}+c^{2/3} x\right )}{8 c^{4/3}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.06, size = 194, normalized size = 1.02
method | result | size |
derivativedivides | \(\frac {a \,x^{2}}{2}+\frac {b \,x^{2} \arctanh \left (c \,x^{\frac {3}{2}}\right )}{2}+\frac {3 b \sqrt {x}}{2 c}+\frac {b \ln \left (\sqrt {x}-\left (\frac {1}{c}\right )^{\frac {1}{3}}\right )}{4 c^{2} \left (\frac {1}{c}\right )^{\frac {2}{3}}}-\frac {b \ln \left (x +\left (\frac {1}{c}\right )^{\frac {1}{3}} \sqrt {x}+\left (\frac {1}{c}\right )^{\frac {2}{3}}\right )}{8 c^{2} \left (\frac {1}{c}\right )^{\frac {2}{3}}}-\frac {b \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \sqrt {x}}{\left (\frac {1}{c}\right )^{\frac {1}{3}}}+1\right )}{3}\right )}{4 c^{2} \left (\frac {1}{c}\right )^{\frac {2}{3}}}-\frac {b \ln \left (\sqrt {x}+\left (\frac {1}{c}\right )^{\frac {1}{3}}\right )}{4 c^{2} \left (\frac {1}{c}\right )^{\frac {2}{3}}}+\frac {b \ln \left (x -\left (\frac {1}{c}\right )^{\frac {1}{3}} \sqrt {x}+\left (\frac {1}{c}\right )^{\frac {2}{3}}\right )}{8 c^{2} \left (\frac {1}{c}\right )^{\frac {2}{3}}}-\frac {b \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \sqrt {x}}{\left (\frac {1}{c}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{4 c^{2} \left (\frac {1}{c}\right )^{\frac {2}{3}}}\) | \(194\) |
default | \(\frac {a \,x^{2}}{2}+\frac {b \,x^{2} \arctanh \left (c \,x^{\frac {3}{2}}\right )}{2}+\frac {3 b \sqrt {x}}{2 c}+\frac {b \ln \left (\sqrt {x}-\left (\frac {1}{c}\right )^{\frac {1}{3}}\right )}{4 c^{2} \left (\frac {1}{c}\right )^{\frac {2}{3}}}-\frac {b \ln \left (x +\left (\frac {1}{c}\right )^{\frac {1}{3}} \sqrt {x}+\left (\frac {1}{c}\right )^{\frac {2}{3}}\right )}{8 c^{2} \left (\frac {1}{c}\right )^{\frac {2}{3}}}-\frac {b \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \sqrt {x}}{\left (\frac {1}{c}\right )^{\frac {1}{3}}}+1\right )}{3}\right )}{4 c^{2} \left (\frac {1}{c}\right )^{\frac {2}{3}}}-\frac {b \ln \left (\sqrt {x}+\left (\frac {1}{c}\right )^{\frac {1}{3}}\right )}{4 c^{2} \left (\frac {1}{c}\right )^{\frac {2}{3}}}+\frac {b \ln \left (x -\left (\frac {1}{c}\right )^{\frac {1}{3}} \sqrt {x}+\left (\frac {1}{c}\right )^{\frac {2}{3}}\right )}{8 c^{2} \left (\frac {1}{c}\right )^{\frac {2}{3}}}-\frac {b \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \sqrt {x}}{\left (\frac {1}{c}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{4 c^{2} \left (\frac {1}{c}\right )^{\frac {2}{3}}}\) | \(194\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.47, size = 172, normalized size = 0.91 \begin {gather*} \frac {1}{2} \, a x^{2} + \frac {1}{8} \, {\left (4 \, x^{2} \operatorname {artanh}\left (c x^{\frac {3}{2}}\right ) - c {\left (\frac {2 \, \sqrt {3} \arctan \left (\frac {\sqrt {3} {\left (2 \, c^{\frac {2}{3}} \sqrt {x} + c^{\frac {1}{3}}\right )}}{3 \, c^{\frac {1}{3}}}\right )}{c^{\frac {7}{3}}} + \frac {2 \, \sqrt {3} \arctan \left (\frac {\sqrt {3} {\left (2 \, c^{\frac {2}{3}} \sqrt {x} - c^{\frac {1}{3}}\right )}}{3 \, c^{\frac {1}{3}}}\right )}{c^{\frac {7}{3}}} + \frac {\log \left (c^{\frac {2}{3}} x + c^{\frac {1}{3}} \sqrt {x} + 1\right )}{c^{\frac {7}{3}}} - \frac {\log \left (c^{\frac {2}{3}} x - c^{\frac {1}{3}} \sqrt {x} + 1\right )}{c^{\frac {7}{3}}} + \frac {2 \, \log \left (\frac {c^{\frac {1}{3}} \sqrt {x} + 1}{c^{\frac {1}{3}}}\right )}{c^{\frac {7}{3}}} - \frac {2 \, \log \left (\frac {c^{\frac {1}{3}} \sqrt {x} - 1}{c^{\frac {1}{3}}}\right )}{c^{\frac {7}{3}}} - \frac {12 \, \sqrt {x}}{c^{2}}\right )}\right )} b \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [C] Result contains complex when optimal does not.
time = 1.22, size = 1848, normalized size = 9.73 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 11.97, size = 247, normalized size = 1.30 \begin {gather*} \frac {a\,x^2}{2}+\frac {3\,b\,\sqrt {x}}{2\,c}+\frac {b\,\ln \left (\frac {c^{1/3}\,\sqrt {x}-1}{c^{1/3}\,\sqrt {x}+1}\right )}{4\,c^{4/3}}+\frac {\ln \left (1-c\,x^{3/2}\right )\,\left (\frac {b\,x^2}{2}-\frac {b\,c^2\,x^5}{2}\right )}{2\,c^2\,x^3-2}+\frac {b\,x^2\,\ln \left (c\,x^{3/2}+1\right )}{4}+\frac {b\,\ln \left (\frac {\sqrt {3}\,c^{2/3}\,x+c^{2/3}\,x\,1{}\mathrm {i}-c^{1/3}\,\sqrt {x}\,4{}\mathrm {i}-\sqrt {3}+1{}\mathrm {i}}{2\,c^{2/3}\,x+1-\sqrt {3}\,1{}\mathrm {i}}\right )\,\sqrt {-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}{4\,c^{4/3}}+\frac {\sqrt {2}\,b\,\ln \left (\frac {2\,\sqrt {2}-c^{1/3}\,\sqrt {x}\,{\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}^{5/2}\,1{}\mathrm {i}-\sqrt {2}\,c^{2/3}\,x+\sqrt {2}\,\sqrt {3}\,c^{2/3}\,x\,1{}\mathrm {i}}{2\,c^{2/3}\,x+1+\sqrt {3}\,1{}\mathrm {i}}\right )\,\sqrt {1+\sqrt {3}\,1{}\mathrm {i}}\,1{}\mathrm {i}}{8\,c^{4/3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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