Optimal. Leaf size=188 \[ -\frac {a+b \tanh ^{-1}\left (c x^{3/2}\right )}{2 x^2}-\frac {1}{8} b c^{4/3} \log \left (c^{2/3} x-\sqrt [3]{c} \sqrt {x}+1\right )+\frac {1}{8} b c^{4/3} \log \left (c^{2/3} x+\sqrt [3]{c} \sqrt {x}+1\right )+\frac {1}{4} \sqrt {3} b c^{4/3} \tan ^{-1}\left (\frac {1-2 \sqrt [3]{c} \sqrt {x}}{\sqrt {3}}\right )-\frac {1}{4} \sqrt {3} b c^{4/3} \tan ^{-1}\left (\frac {2 \sqrt [3]{c} \sqrt {x}+1}{\sqrt {3}}\right )+\frac {1}{2} b c^{4/3} \tanh ^{-1}\left (\sqrt [3]{c} \sqrt {x}\right )-\frac {3 b c}{2 \sqrt {x}} \]
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Rubi [A] time = 0.29, antiderivative size = 188, 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, 325, 329, 296, 634, 618, 204, 628, 206} \[ -\frac {a+b \tanh ^{-1}\left (c x^{3/2}\right )}{2 x^2}-\frac {1}{8} b c^{4/3} \log \left (c^{2/3} x-\sqrt [3]{c} \sqrt {x}+1\right )+\frac {1}{8} b c^{4/3} \log \left (c^{2/3} x+\sqrt [3]{c} \sqrt {x}+1\right )+\frac {1}{4} \sqrt {3} b c^{4/3} \tan ^{-1}\left (\frac {1-2 \sqrt [3]{c} \sqrt {x}}{\sqrt {3}}\right )-\frac {1}{4} \sqrt {3} b c^{4/3} \tan ^{-1}\left (\frac {2 \sqrt [3]{c} \sqrt {x}+1}{\sqrt {3}}\right )+\frac {1}{2} b c^{4/3} \tanh ^{-1}\left (\sqrt [3]{c} \sqrt {x}\right )-\frac {3 b c}{2 \sqrt {x}} \]
Antiderivative was successfully verified.
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Rule 204
Rule 206
Rule 296
Rule 325
Rule 329
Rule 618
Rule 628
Rule 634
Rule 6097
Rubi steps
\begin {align*} \int \frac {a+b \tanh ^{-1}\left (c x^{3/2}\right )}{x^3} \, dx &=-\frac {a+b \tanh ^{-1}\left (c x^{3/2}\right )}{2 x^2}+\frac {1}{4} (3 b c) \int \frac {1}{x^{3/2} \left (1-c^2 x^3\right )} \, dx\\ &=-\frac {3 b c}{2 \sqrt {x}}-\frac {a+b \tanh ^{-1}\left (c x^{3/2}\right )}{2 x^2}+\frac {1}{4} \left (3 b c^3\right ) \int \frac {x^{3/2}}{1-c^2 x^3} \, dx\\ &=-\frac {3 b c}{2 \sqrt {x}}-\frac {a+b \tanh ^{-1}\left (c x^{3/2}\right )}{2 x^2}+\frac {1}{2} \left (3 b c^3\right ) \operatorname {Subst}\left (\int \frac {x^4}{1-c^2 x^6} \, dx,x,\sqrt {x}\right )\\ &=-\frac {3 b c}{2 \sqrt {x}}-\frac {a+b \tanh ^{-1}\left (c x^{3/2}\right )}{2 x^2}+\frac {1}{2} \left (b c^{5/3}\right ) \operatorname {Subst}\left (\int \frac {1}{1-c^{2/3} x^2} \, dx,x,\sqrt {x}\right )+\frac {1}{2} \left (b c^{5/3}\right ) \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 )+\frac {1}{2} \left (b c^{5/3}\right ) \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 )\\ &=-\frac {3 b c}{2 \sqrt {x}}+\frac {1}{2} b c^{4/3} \tanh ^{-1}\left (\sqrt [3]{c} \sqrt {x}\right )-\frac {a+b \tanh ^{-1}\left (c x^{3/2}\right )}{2 x^2}-\frac {1}{8} \left (b c^{4/3}\right ) \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 )+\frac {1}{8} \left (b c^{4/3}\right ) \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 )-\frac {1}{8} \left (3 b c^{5/3}\right ) \operatorname {Subst}\left (\int \frac {1}{1-\sqrt [3]{c} x+c^{2/3} x^2} \, dx,x,\sqrt {x}\right )-\frac {1}{8} \left (3 b c^{5/3}\right ) \operatorname {Subst}\left (\int \frac {1}{1+\sqrt [3]{c} x+c^{2/3} x^2} \, dx,x,\sqrt {x}\right )\\ &=-\frac {3 b c}{2 \sqrt {x}}+\frac {1}{2} b c^{4/3} \tanh ^{-1}\left (\sqrt [3]{c} \sqrt {x}\right )-\frac {a+b \tanh ^{-1}\left (c x^{3/2}\right )}{2 x^2}-\frac {1}{8} b c^{4/3} \log \left (1-\sqrt [3]{c} \sqrt {x}+c^{2/3} x\right )+\frac {1}{8} b c^{4/3} \log \left (1+\sqrt [3]{c} \sqrt {x}+c^{2/3} x\right )-\frac {1}{4} \left (3 b c^{4/3}\right ) \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-2 \sqrt [3]{c} \sqrt {x}\right )+\frac {1}{4} \left (3 b c^{4/3}\right ) \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+2 \sqrt [3]{c} \sqrt {x}\right )\\ &=-\frac {3 b c}{2 \sqrt {x}}+\frac {1}{4} \sqrt {3} b c^{4/3} \tan ^{-1}\left (\frac {1-2 \sqrt [3]{c} \sqrt {x}}{\sqrt {3}}\right )-\frac {1}{4} \sqrt {3} b c^{4/3} \tan ^{-1}\left (\frac {1+2 \sqrt [3]{c} \sqrt {x}}{\sqrt {3}}\right )+\frac {1}{2} b c^{4/3} \tanh ^{-1}\left (\sqrt [3]{c} \sqrt {x}\right )-\frac {a+b \tanh ^{-1}\left (c x^{3/2}\right )}{2 x^2}-\frac {1}{8} b c^{4/3} \log \left (1-\sqrt [3]{c} \sqrt {x}+c^{2/3} x\right )+\frac {1}{8} b c^{4/3} \log \left (1+\sqrt [3]{c} \sqrt {x}+c^{2/3} x\right )\\ \end {align*}
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Mathematica [A] time = 0.06, size = 220, normalized size = 1.17 \[ -\frac {a}{2 x^2}-\frac {1}{4} b c^{4/3} \log \left (1-\sqrt [3]{c} \sqrt {x}\right )+\frac {1}{4} b c^{4/3} \log \left (\sqrt [3]{c} \sqrt {x}+1\right )-\frac {1}{8} b c^{4/3} \log \left (c^{2/3} x-\sqrt [3]{c} \sqrt {x}+1\right )+\frac {1}{8} b c^{4/3} \log \left (c^{2/3} x+\sqrt [3]{c} \sqrt {x}+1\right )-\frac {1}{4} \sqrt {3} b c^{4/3} \tan ^{-1}\left (\frac {2 \sqrt [3]{c} \sqrt {x}-1}{\sqrt {3}}\right )-\frac {1}{4} \sqrt {3} b c^{4/3} \tan ^{-1}\left (\frac {2 \sqrt [3]{c} \sqrt {x}+1}{\sqrt {3}}\right )-\frac {b \tanh ^{-1}\left (c x^{3/2}\right )}{2 x^2}-\frac {3 b c}{2 \sqrt {x}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.63, size = 214, normalized size = 1.14 \[ -\frac {2 \, \sqrt {3} b \left (-c\right )^{\frac {1}{3}} c x^{2} \arctan \left (\frac {2}{3} \, \sqrt {3} \left (-c\right )^{\frac {1}{3}} \sqrt {x} - \frac {1}{3} \, \sqrt {3}\right ) + 2 \, \sqrt {3} b c^{\frac {4}{3}} x^{2} \arctan \left (\frac {2}{3} \, \sqrt {3} c^{\frac {1}{3}} \sqrt {x} - \frac {1}{3} \, \sqrt {3}\right ) + b \left (-c\right )^{\frac {1}{3}} c x^{2} \log \left (c x + \left (-c\right )^{\frac {2}{3}} \sqrt {x} - \left (-c\right )^{\frac {1}{3}}\right ) + b c^{\frac {4}{3}} x^{2} \log \left (c x - c^{\frac {2}{3}} \sqrt {x} + c^{\frac {1}{3}}\right ) - 2 \, b \left (-c\right )^{\frac {1}{3}} c x^{2} \log \left (c \sqrt {x} - \left (-c\right )^{\frac {2}{3}}\right ) - 2 \, b c^{\frac {4}{3}} x^{2} \log \left (c \sqrt {x} + c^{\frac {2}{3}}\right ) + 12 \, b c x^{\frac {3}{2}} + 2 \, b \log \left (-\frac {c^{2} x^{3} + 2 \, c x^{\frac {3}{2}} + 1}{c^{2} x^{3} - 1}\right ) + 4 \, a}{8 \, x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.34, size = 194, normalized size = 1.03 \[ -\frac {1}{4} \, \sqrt {3} b c {\left | c \right |}^{\frac {1}{3}} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, \sqrt {x} + \frac {1}{{\left | c \right |}^{\frac {1}{3}}}\right )} {\left | c \right |}^{\frac {1}{3}}\right ) - \frac {1}{4} \, \sqrt {3} b c {\left | c \right |}^{\frac {1}{3}} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, \sqrt {x} - \frac {1}{{\left | c \right |}^{\frac {1}{3}}}\right )} {\left | c \right |}^{\frac {1}{3}}\right ) + \frac {b c^{3} \log \left (x + \frac {\sqrt {x}}{{\left | c \right |}^{\frac {1}{3}}} + \frac {1}{{\left | c \right |}^{\frac {2}{3}}}\right )}{8 \, {\left | c \right |}^{\frac {5}{3}}} - \frac {b c^{3} \log \left (x - \frac {\sqrt {x}}{{\left | c \right |}^{\frac {1}{3}}} + \frac {1}{{\left | c \right |}^{\frac {2}{3}}}\right )}{8 \, {\left | c \right |}^{\frac {5}{3}}} + \frac {1}{4} \, b c {\left | c \right |}^{\frac {1}{3}} \log \left (\sqrt {x} + \frac {1}{{\left | c \right |}^{\frac {1}{3}}}\right ) - \frac {b c^{3} \log \left ({\left | \sqrt {x} - \frac {1}{{\left | c \right |}^{\frac {1}{3}}} \right |}\right )}{4 \, {\left | c \right |}^{\frac {5}{3}}} - \frac {b \log \left (-\frac {c x^{\frac {3}{2}} + 1}{c x^{\frac {3}{2}} - 1}\right )}{4 \, x^{2}} - \frac {3 \, b c x^{\frac {3}{2}} + a}{2 \, x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 180, normalized size = 0.96 \[ -\frac {a}{2 x^{2}}-\frac {b \arctanh \left (c \,x^{\frac {3}{2}}\right )}{2 x^{2}}-\frac {3 b c}{2 \sqrt {x}}-\frac {b c \ln \left (\sqrt {x}-\left (\frac {1}{c}\right )^{\frac {1}{3}}\right )}{4 \left (\frac {1}{c}\right )^{\frac {1}{3}}}+\frac {b c \ln \left (x +\left (\frac {1}{c}\right )^{\frac {1}{3}} \sqrt {x}+\left (\frac {1}{c}\right )^{\frac {2}{3}}\right )}{8 \left (\frac {1}{c}\right )^{\frac {1}{3}}}-\frac {b c \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \sqrt {x}}{\left (\frac {1}{c}\right )^{\frac {1}{3}}}+1\right )}{3}\right )}{4 \left (\frac {1}{c}\right )^{\frac {1}{3}}}+\frac {b c \ln \left (\sqrt {x}+\left (\frac {1}{c}\right )^{\frac {1}{3}}\right )}{4 \left (\frac {1}{c}\right )^{\frac {1}{3}}}-\frac {b c \ln \left (x -\left (\frac {1}{c}\right )^{\frac {1}{3}} \sqrt {x}+\left (\frac {1}{c}\right )^{\frac {2}{3}}\right )}{8 \left (\frac {1}{c}\right )^{\frac {1}{3}}}-\frac {b c \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \sqrt {x}}{\left (\frac {1}{c}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{4 \left (\frac {1}{c}\right )^{\frac {1}{3}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.42, size = 168, normalized size = 0.89 \[ -\frac {1}{8} \, {\left ({\left (2 \, \sqrt {3} c^{\frac {1}{3}} \arctan \left (\frac {\sqrt {3} {\left (2 \, c^{\frac {2}{3}} \sqrt {x} + c^{\frac {1}{3}}\right )}}{3 \, c^{\frac {1}{3}}}\right ) + 2 \, \sqrt {3} c^{\frac {1}{3}} \arctan \left (\frac {\sqrt {3} {\left (2 \, c^{\frac {2}{3}} \sqrt {x} - c^{\frac {1}{3}}\right )}}{3 \, c^{\frac {1}{3}}}\right ) - c^{\frac {1}{3}} \log \left (c^{\frac {2}{3}} x + c^{\frac {1}{3}} \sqrt {x} + 1\right ) + c^{\frac {1}{3}} \log \left (c^{\frac {2}{3}} x - c^{\frac {1}{3}} \sqrt {x} + 1\right ) - 2 \, c^{\frac {1}{3}} \log \left (\frac {c^{\frac {1}{3}} \sqrt {x} + 1}{c^{\frac {1}{3}}}\right ) + 2 \, c^{\frac {1}{3}} \log \left (\frac {c^{\frac {1}{3}} \sqrt {x} - 1}{c^{\frac {1}{3}}}\right ) + \frac {12}{\sqrt {x}}\right )} c + \frac {4 \, \operatorname {artanh}\left (c x^{\frac {3}{2}}\right )}{x^{2}}\right )} b - \frac {a}{2 \, x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 7.37, size = 228, normalized size = 1.21 \[ \frac {b\,c^{4/3}\,\ln \left (\frac {c^{1/3}\,\sqrt {x}+1}{c^{1/3}\,\sqrt {x}-1}\right )}{4}-\frac {a}{2\,x^2}+\frac {\ln \left (1-c\,x^{3/2}\right )\,\left (\frac {b\,x}{2}-\frac {b\,c^2\,x^4}{2}\right )}{2\,x^3-2\,c^2\,x^6}-\frac {3\,b\,c}{2\,\sqrt {x}}-\frac {b\,\ln \left (c\,x^{3/2}+1\right )}{4\,x^2}+\frac {b\,c^{4/3}\,\ln \left (\frac {\sqrt {3}+c^{2/3}\,x\,1{}\mathrm {i}+c^{1/3}\,\sqrt {x}\,4{}\mathrm {i}-\sqrt {3}\,c^{2/3}\,x+1{}\mathrm {i}}{2\,c^{2/3}\,x+1+\sqrt {3}\,1{}\mathrm {i}}\right )\,\sqrt {-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}{4}+\frac {b\,c^{4/3}\,\ln \left (\frac {\sqrt {3}\,c^{2/3}\,x+c^{2/3}\,x\,1{}\mathrm {i}-c^{1/3}\,\sqrt {x}\,4{}\mathrm {i}-\sqrt {3}+1{}\mathrm {i}}{2\,c^{2/3}\,x+1-\sqrt {3}\,1{}\mathrm {i}}\right )\,\sqrt {\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}\,1{}\mathrm {i}}{4} \]
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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