Optimal. Leaf size=190 \[ \frac {1}{4} x^4 \left (a+b \tanh ^{-1}\left (c x^{3/2}\right )\right )+\frac {b \log \left (c^{2/3} x-\sqrt [3]{c} \sqrt {x}+1\right )}{16 c^{8/3}}-\frac {b \log \left (c^{2/3} x+\sqrt [3]{c} \sqrt {x}+1\right )}{16 c^{8/3}}-\frac {\sqrt {3} b \tan ^{-1}\left (\frac {1-2 \sqrt [3]{c} \sqrt {x}}{\sqrt {3}}\right )}{8 c^{8/3}}+\frac {\sqrt {3} b \tan ^{-1}\left (\frac {2 \sqrt [3]{c} \sqrt {x}+1}{\sqrt {3}}\right )}{8 c^{8/3}}-\frac {b \tanh ^{-1}\left (\sqrt [3]{c} \sqrt {x}\right )}{4 c^{8/3}}+\frac {3 b x^{5/2}}{20 c} \]
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Rubi [A] time = 0.30, antiderivative size = 190, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 9, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.562, Rules used = {6097, 321, 329, 296, 634, 618, 204, 628, 206} \[ \frac {1}{4} x^4 \left (a+b \tanh ^{-1}\left (c x^{3/2}\right )\right )+\frac {b \log \left (c^{2/3} x-\sqrt [3]{c} \sqrt {x}+1\right )}{16 c^{8/3}}-\frac {b \log \left (c^{2/3} x+\sqrt [3]{c} \sqrt {x}+1\right )}{16 c^{8/3}}-\frac {\sqrt {3} b \tan ^{-1}\left (\frac {1-2 \sqrt [3]{c} \sqrt {x}}{\sqrt {3}}\right )}{8 c^{8/3}}+\frac {\sqrt {3} b \tan ^{-1}\left (\frac {2 \sqrt [3]{c} \sqrt {x}+1}{\sqrt {3}}\right )}{8 c^{8/3}}-\frac {b \tanh ^{-1}\left (\sqrt [3]{c} \sqrt {x}\right )}{4 c^{8/3}}+\frac {3 b x^{5/2}}{20 c} \]
Antiderivative was successfully verified.
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Rule 204
Rule 206
Rule 296
Rule 321
Rule 329
Rule 618
Rule 628
Rule 634
Rule 6097
Rubi steps
\begin {align*} \int x^3 \left (a+b \tanh ^{-1}\left (c x^{3/2}\right )\right ) \, dx &=\frac {1}{4} x^4 \left (a+b \tanh ^{-1}\left (c x^{3/2}\right )\right )-\frac {1}{8} (3 b c) \int \frac {x^{9/2}}{1-c^2 x^3} \, dx\\ &=\frac {3 b x^{5/2}}{20 c}+\frac {1}{4} x^4 \left (a+b \tanh ^{-1}\left (c x^{3/2}\right )\right )-\frac {(3 b) \int \frac {x^{3/2}}{1-c^2 x^3} \, dx}{8 c}\\ &=\frac {3 b x^{5/2}}{20 c}+\frac {1}{4} x^4 \left (a+b \tanh ^{-1}\left (c x^{3/2}\right )\right )-\frac {(3 b) \operatorname {Subst}\left (\int \frac {x^4}{1-c^2 x^6} \, dx,x,\sqrt {x}\right )}{4 c}\\ &=\frac {3 b x^{5/2}}{20 c}+\frac {1}{4} x^4 \left (a+b \tanh ^{-1}\left (c x^{3/2}\right )\right )-\frac {b \operatorname {Subst}\left (\int \frac {1}{1-c^{2/3} x^2} \, dx,x,\sqrt {x}\right )}{4 c^{7/3}}-\frac {b \operatorname {Subst}\left (\int \frac {-\frac {1}{2}-\frac {\sqrt [3]{c} x}{2}}{1-\sqrt [3]{c} x+c^{2/3} x^2} \, dx,x,\sqrt {x}\right )}{4 c^{7/3}}-\frac {b \operatorname {Subst}\left (\int \frac {-\frac {1}{2}+\frac {\sqrt [3]{c} x}{2}}{1+\sqrt [3]{c} x+c^{2/3} x^2} \, dx,x,\sqrt {x}\right )}{4 c^{7/3}}\\ &=\frac {3 b x^{5/2}}{20 c}-\frac {b \tanh ^{-1}\left (\sqrt [3]{c} \sqrt {x}\right )}{4 c^{8/3}}+\frac {1}{4} x^4 \left (a+b \tanh ^{-1}\left (c x^{3/2}\right )\right )+\frac {b \operatorname {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 )}{16 c^{8/3}}-\frac {b \operatorname {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 )}{16 c^{8/3}}+\frac {(3 b) \operatorname {Subst}\left (\int \frac {1}{1-\sqrt [3]{c} x+c^{2/3} x^2} \, dx,x,\sqrt {x}\right )}{16 c^{7/3}}+\frac {(3 b) \operatorname {Subst}\left (\int \frac {1}{1+\sqrt [3]{c} x+c^{2/3} x^2} \, dx,x,\sqrt {x}\right )}{16 c^{7/3}}\\ &=\frac {3 b x^{5/2}}{20 c}-\frac {b \tanh ^{-1}\left (\sqrt [3]{c} \sqrt {x}\right )}{4 c^{8/3}}+\frac {1}{4} x^4 \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 )}{16 c^{8/3}}-\frac {b \log \left (1+\sqrt [3]{c} \sqrt {x}+c^{2/3} x\right )}{16 c^{8/3}}+\frac {(3 b) \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-2 \sqrt [3]{c} \sqrt {x}\right )}{8 c^{8/3}}-\frac {(3 b) \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+2 \sqrt [3]{c} \sqrt {x}\right )}{8 c^{8/3}}\\ &=\frac {3 b x^{5/2}}{20 c}-\frac {\sqrt {3} b \tan ^{-1}\left (\frac {1-2 \sqrt [3]{c} \sqrt {x}}{\sqrt {3}}\right )}{8 c^{8/3}}+\frac {\sqrt {3} b \tan ^{-1}\left (\frac {1+2 \sqrt [3]{c} \sqrt {x}}{\sqrt {3}}\right )}{8 c^{8/3}}-\frac {b \tanh ^{-1}\left (\sqrt [3]{c} \sqrt {x}\right )}{4 c^{8/3}}+\frac {1}{4} x^4 \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 )}{16 c^{8/3}}-\frac {b \log \left (1+\sqrt [3]{c} \sqrt {x}+c^{2/3} x\right )}{16 c^{8/3}}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 222, normalized size = 1.17 \[ \frac {a x^4}{4}+\frac {b \log \left (1-\sqrt [3]{c} \sqrt {x}\right )}{8 c^{8/3}}-\frac {b \log \left (\sqrt [3]{c} \sqrt {x}+1\right )}{8 c^{8/3}}+\frac {b \log \left (c^{2/3} x-\sqrt [3]{c} \sqrt {x}+1\right )}{16 c^{8/3}}-\frac {b \log \left (c^{2/3} x+\sqrt [3]{c} \sqrt {x}+1\right )}{16 c^{8/3}}+\frac {\sqrt {3} b \tan ^{-1}\left (\frac {2 \sqrt [3]{c} \sqrt {x}-1}{\sqrt {3}}\right )}{8 c^{8/3}}+\frac {\sqrt {3} b \tan ^{-1}\left (\frac {2 \sqrt [3]{c} \sqrt {x}+1}{\sqrt {3}}\right )}{8 c^{8/3}}+\frac {3 b x^{5/2}}{20 c}+\frac {1}{4} b x^4 \tanh ^{-1}\left (c x^{3/2}\right ) \]
Antiderivative was successfully verified.
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fricas [C] time = 3.71, size = 1803, normalized size = 9.49 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [C] time = 0.37, size = 227, normalized size = 1.19 \[ \frac {1}{4} \, a x^{4} + \frac {1}{320} \, {\left (40 \, x^{4} \log \left (-\frac {c x^{\frac {3}{2}} + 1}{c x^{\frac {3}{2}} - 1}\right ) + c {\left (\frac {48 \, x^{\frac {5}{2}}}{c^{2}} - \frac {10 \, \sqrt {3} {\left (-i \, \sqrt {3} - 1\right )}^{2} {\left | c \right |}^{\frac {4}{3}} \arctan \left (\frac {\sqrt {3} {\left (2 \, \sqrt {x} + \left (-\frac {1}{c}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {1}{c}\right )^{\frac {1}{3}}}\right )}{c^{5}} + \frac {5 \, {\left (-i \, \sqrt {3} - 1\right )}^{2} {\left | c \right |}^{\frac {4}{3}} \log \left (x + \sqrt {x} \left (-\frac {1}{c}\right )^{\frac {1}{3}} + \left (-\frac {1}{c}\right )^{\frac {2}{3}}\right )}{c^{5}} - \frac {40 \, \left (-\frac {1}{c}\right )^{\frac {2}{3}} \log \left ({\left | \sqrt {x} - \left (-\frac {1}{c}\right )^{\frac {1}{3}} \right |}\right )}{c^{3}} + \frac {40 \, \sqrt {3} {\left | c \right |}^{\frac {4}{3}} \arctan \left (\frac {1}{3} \, \sqrt {3} c^{\frac {1}{3}} {\left (2 \, \sqrt {x} + \frac {1}{c^{\frac {1}{3}}}\right )}\right )}{c^{5}} - \frac {20 \, {\left | c \right |}^{\frac {4}{3}} \log \left (x + \frac {\sqrt {x}}{c^{\frac {1}{3}}} + \frac {1}{c^{\frac {2}{3}}}\right )}{c^{5}} + \frac {40 \, \log \left ({\left | \sqrt {x} - \frac {1}{c^{\frac {1}{3}}} \right |}\right )}{c^{\frac {11}{3}}}\right )}\right )} b \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 194, normalized size = 1.02 \[ \frac {x^{4} a}{4}+\frac {b \,x^{4} \arctanh \left (c \,x^{\frac {3}{2}}\right )}{4}+\frac {3 b \,x^{\frac {5}{2}}}{20 c}+\frac {b \ln \left (\sqrt {x}-\left (\frac {1}{c}\right )^{\frac {1}{3}}\right )}{8 c^{3} \left (\frac {1}{c}\right )^{\frac {1}{3}}}-\frac {b \ln \left (x +\left (\frac {1}{c}\right )^{\frac {1}{3}} \sqrt {x}+\left (\frac {1}{c}\right )^{\frac {2}{3}}\right )}{16 c^{3} \left (\frac {1}{c}\right )^{\frac {1}{3}}}+\frac {b \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \sqrt {x}}{\left (\frac {1}{c}\right )^{\frac {1}{3}}}+1\right )}{3}\right )}{8 c^{3} \left (\frac {1}{c}\right )^{\frac {1}{3}}}-\frac {b \ln \left (\sqrt {x}+\left (\frac {1}{c}\right )^{\frac {1}{3}}\right )}{8 c^{3} \left (\frac {1}{c}\right )^{\frac {1}{3}}}+\frac {b \ln \left (x -\left (\frac {1}{c}\right )^{\frac {1}{3}} \sqrt {x}+\left (\frac {1}{c}\right )^{\frac {2}{3}}\right )}{16 c^{3} \left (\frac {1}{c}\right )^{\frac {1}{3}}}+\frac {b \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \sqrt {x}}{\left (\frac {1}{c}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{8 c^{3} \left (\frac {1}{c}\right )^{\frac {1}{3}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.43, size = 172, normalized size = 0.91 \[ \frac {1}{4} \, a x^{4} + \frac {1}{80} \, {\left (20 \, x^{4} \operatorname {artanh}\left (c x^{\frac {3}{2}}\right ) + c {\left (\frac {12 \, x^{\frac {5}{2}}}{c^{2}} + \frac {10 \, \sqrt {3} \arctan \left (\frac {\sqrt {3} {\left (2 \, c^{\frac {2}{3}} \sqrt {x} + c^{\frac {1}{3}}\right )}}{3 \, c^{\frac {1}{3}}}\right )}{c^{\frac {11}{3}}} + \frac {10 \, \sqrt {3} \arctan \left (\frac {\sqrt {3} {\left (2 \, c^{\frac {2}{3}} \sqrt {x} - c^{\frac {1}{3}}\right )}}{3 \, c^{\frac {1}{3}}}\right )}{c^{\frac {11}{3}}} - \frac {5 \, \log \left (c^{\frac {2}{3}} x + c^{\frac {1}{3}} \sqrt {x} + 1\right )}{c^{\frac {11}{3}}} + \frac {5 \, \log \left (c^{\frac {2}{3}} x - c^{\frac {1}{3}} \sqrt {x} + 1\right )}{c^{\frac {11}{3}}} - \frac {10 \, \log \left (\frac {c^{\frac {1}{3}} \sqrt {x} + 1}{c^{\frac {1}{3}}}\right )}{c^{\frac {11}{3}}} + \frac {10 \, \log \left (\frac {c^{\frac {1}{3}} \sqrt {x} - 1}{c^{\frac {1}{3}}}\right )}{c^{\frac {11}{3}}}\right )}\right )} b \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 13.38, size = 231, normalized size = 1.22 \[ \frac {a\,x^4}{4}+\frac {3\,b\,x^{5/2}}{20\,c}+\frac {b\,\ln \left (\frac {c^{1/3}\,\sqrt {x}-1}{c^{1/3}\,\sqrt {x}+1}\right )}{8\,c^{8/3}}+\frac {\ln \left (1-c\,x^{3/2}\right )\,\left (\frac {b\,x^4}{4}-\frac {b\,c^2\,x^7}{4}\right )}{2\,c^2\,x^3-2}+\frac {b\,x^4\,\ln \left (c\,x^{3/2}+1\right )}{8}+\frac {b\,\ln \left (\frac {\sqrt {3}+c^{2/3}\,x\,1{}\mathrm {i}-c^{1/3}\,\sqrt {x}\,4{}\mathrm {i}-\sqrt {3}\,c^{2/3}\,x+1{}\mathrm {i}}{2\,c^{2/3}\,x+1+\sqrt {3}\,1{}\mathrm {i}}\right )\,\sqrt {-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}{8\,c^{8/3}}+\frac {\sqrt {2}\,b\,\ln \left (\frac {\sqrt {3}\,c^{2/3}\,x+c^{2/3}\,x\,1{}\mathrm {i}+c^{1/3}\,\sqrt {x}\,4{}\mathrm {i}-\sqrt {3}+1{}\mathrm {i}}{2\,c^{2/3}\,x+1-\sqrt {3}\,1{}\mathrm {i}}\right )\,\sqrt {1+\sqrt {3}\,1{}\mathrm {i}}\,1{}\mathrm {i}}{16\,c^{8/3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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