Optimal. Leaf size=106 \[ \frac {4 b (d x)^{3/2}}{15 c}+\frac {2 b d^{3/2} \text {ArcTan}\left (\frac {\sqrt {c} \sqrt {d x}}{\sqrt {d}}\right )}{5 c^{5/2}}+\frac {2 (d x)^{5/2} \left (a+b \tanh ^{-1}(c x)\right )}{5 d}-\frac {2 b d^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {d x}}{\sqrt {d}}\right )}{5 c^{5/2}} \]
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Rubi [A]
time = 0.05, antiderivative size = 106, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 6, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {6049, 327, 335,
304, 211, 214} \begin {gather*} \frac {2 (d x)^{5/2} \left (a+b \tanh ^{-1}(c x)\right )}{5 d}+\frac {2 b d^{3/2} \text {ArcTan}\left (\frac {\sqrt {c} \sqrt {d x}}{\sqrt {d}}\right )}{5 c^{5/2}}-\frac {2 b d^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {d x}}{\sqrt {d}}\right )}{5 c^{5/2}}+\frac {4 b (d x)^{3/2}}{15 c} \end {gather*}
Antiderivative was successfully verified.
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Rule 211
Rule 214
Rule 304
Rule 327
Rule 335
Rule 6049
Rubi steps
\begin {align*} \int (d x)^{3/2} \left (a+b \tanh ^{-1}(c x)\right ) \, dx &=\frac {2 (d x)^{5/2} \left (a+b \tanh ^{-1}(c x)\right )}{5 d}-\frac {(2 b c) \int \frac {(d x)^{5/2}}{1-c^2 x^2} \, dx}{5 d}\\ &=\frac {4 b (d x)^{3/2}}{15 c}+\frac {2 (d x)^{5/2} \left (a+b \tanh ^{-1}(c x)\right )}{5 d}-\frac {(2 b d) \int \frac {\sqrt {d x}}{1-c^2 x^2} \, dx}{5 c}\\ &=\frac {4 b (d x)^{3/2}}{15 c}+\frac {2 (d x)^{5/2} \left (a+b \tanh ^{-1}(c x)\right )}{5 d}-\frac {(4 b) \text {Subst}\left (\int \frac {x^2}{1-\frac {c^2 x^4}{d^2}} \, dx,x,\sqrt {d x}\right )}{5 c}\\ &=\frac {4 b (d x)^{3/2}}{15 c}+\frac {2 (d x)^{5/2} \left (a+b \tanh ^{-1}(c x)\right )}{5 d}-\frac {\left (2 b d^2\right ) \text {Subst}\left (\int \frac {1}{d-c x^2} \, dx,x,\sqrt {d x}\right )}{5 c^2}+\frac {\left (2 b d^2\right ) \text {Subst}\left (\int \frac {1}{d+c x^2} \, dx,x,\sqrt {d x}\right )}{5 c^2}\\ &=\frac {4 b (d x)^{3/2}}{15 c}+\frac {2 b d^{3/2} \tan ^{-1}\left (\frac {\sqrt {c} \sqrt {d x}}{\sqrt {d}}\right )}{5 c^{5/2}}+\frac {2 (d x)^{5/2} \left (a+b \tanh ^{-1}(c x)\right )}{5 d}-\frac {2 b d^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {d x}}{\sqrt {d}}\right )}{5 c^{5/2}}\\ \end {align*}
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Mathematica [A]
time = 0.04, size = 115, normalized size = 1.08 \begin {gather*} \frac {(d x)^{3/2} \left (4 b c^{3/2} x^{3/2}+6 a c^{5/2} x^{5/2}+6 b \text {ArcTan}\left (\sqrt {c} \sqrt {x}\right )+6 b c^{5/2} x^{5/2} \tanh ^{-1}(c x)+3 b \log \left (1-\sqrt {c} \sqrt {x}\right )-3 b \log \left (1+\sqrt {c} \sqrt {x}\right )\right )}{15 c^{5/2} x^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.04, size = 93, normalized size = 0.88
method | result | size |
derivativedivides | \(\frac {\frac {2 \left (d x \right )^{\frac {5}{2}} a}{5}+\frac {2 b \left (d x \right )^{\frac {5}{2}} \arctanh \left (c x \right )}{5}+\frac {4 b d \left (d x \right )^{\frac {3}{2}}}{15 c}+\frac {2 b \,d^{3} \arctan \left (\frac {c \sqrt {d x}}{\sqrt {d c}}\right )}{5 c^{2} \sqrt {d c}}-\frac {2 b \,d^{3} \arctanh \left (\frac {c \sqrt {d x}}{\sqrt {d c}}\right )}{5 c^{2} \sqrt {d c}}}{d}\) | \(93\) |
default | \(\frac {\frac {2 \left (d x \right )^{\frac {5}{2}} a}{5}+\frac {2 b \left (d x \right )^{\frac {5}{2}} \arctanh \left (c x \right )}{5}+\frac {4 b d \left (d x \right )^{\frac {3}{2}}}{15 c}+\frac {2 b \,d^{3} \arctan \left (\frac {c \sqrt {d x}}{\sqrt {d c}}\right )}{5 c^{2} \sqrt {d c}}-\frac {2 b \,d^{3} \arctanh \left (\frac {c \sqrt {d x}}{\sqrt {d c}}\right )}{5 c^{2} \sqrt {d c}}}{d}\) | \(93\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.47, size = 118, normalized size = 1.11 \begin {gather*} \frac {6 \, \left (d x\right )^{\frac {5}{2}} a + {\left (6 \, \left (d x\right )^{\frac {5}{2}} \operatorname {artanh}\left (c x\right ) + \frac {{\left (\frac {4 \, \left (d x\right )^{\frac {3}{2}} d^{2}}{c^{2}} + \frac {6 \, d^{4} \arctan \left (\frac {\sqrt {d x} c}{\sqrt {c d}}\right )}{\sqrt {c d} c^{3}} + \frac {3 \, d^{4} \log \left (\frac {\sqrt {d x} c - \sqrt {c d}}{\sqrt {d x} c + \sqrt {c d}}\right )}{\sqrt {c d} c^{3}}\right )} c}{d}\right )} b}{15 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.39, size = 255, normalized size = 2.41 \begin {gather*} \left [\frac {6 \, b d \sqrt {\frac {d}{c}} \arctan \left (\frac {\sqrt {d x} c \sqrt {\frac {d}{c}}}{d}\right ) + 3 \, b d \sqrt {\frac {d}{c}} \log \left (\frac {c d x - 2 \, \sqrt {d x} c \sqrt {\frac {d}{c}} + d}{c x - 1}\right ) + {\left (3 \, b c^{2} d x^{2} \log \left (-\frac {c x + 1}{c x - 1}\right ) + 6 \, a c^{2} d x^{2} + 4 \, b c d x\right )} \sqrt {d x}}{15 \, c^{2}}, \frac {6 \, b d \sqrt {-\frac {d}{c}} \arctan \left (\frac {\sqrt {d x} c \sqrt {-\frac {d}{c}}}{d}\right ) + 3 \, b d \sqrt {-\frac {d}{c}} \log \left (\frac {c d x + 2 \, \sqrt {d x} c \sqrt {-\frac {d}{c}} - d}{c x + 1}\right ) + {\left (3 \, b c^{2} d x^{2} \log \left (-\frac {c x + 1}{c x - 1}\right ) + 6 \, a c^{2} d x^{2} + 4 \, b c d x\right )} \sqrt {d x}}{15 \, c^{2}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (d x\right )^{\frac {3}{2}} \left (a + b \operatorname {atanh}{\left (c x \right )}\right )\, dx \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 [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \left (a+b\,\mathrm {atanh}\left (c\,x\right )\right )\,{\left (d\,x\right )}^{3/2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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