Optimal. Leaf size=107 \[ -\frac {4 b c}{3 d^2 \sqrt {d x}}-\frac {2 b c^{3/2} \text {ArcTan}\left (\frac {\sqrt {c} \sqrt {d x}}{\sqrt {d}}\right )}{3 d^{5/2}}-\frac {2 \left (a+b \tanh ^{-1}(c x)\right )}{3 d (d x)^{3/2}}+\frac {2 b c^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {d x}}{\sqrt {d}}\right )}{3 d^{5/2}} \]
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Rubi [A]
time = 0.05, antiderivative size = 107, 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, 331, 335,
304, 211, 214} \begin {gather*} -\frac {2 \left (a+b \tanh ^{-1}(c x)\right )}{3 d (d x)^{3/2}}-\frac {2 b c^{3/2} \text {ArcTan}\left (\frac {\sqrt {c} \sqrt {d x}}{\sqrt {d}}\right )}{3 d^{5/2}}+\frac {2 b c^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {d x}}{\sqrt {d}}\right )}{3 d^{5/2}}-\frac {4 b c}{3 d^2 \sqrt {d x}} \end {gather*}
Antiderivative was successfully verified.
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Rule 211
Rule 214
Rule 304
Rule 331
Rule 335
Rule 6049
Rubi steps
\begin {align*} \int \frac {a+b \tanh ^{-1}(c x)}{(d x)^{5/2}} \, dx &=-\frac {2 \left (a+b \tanh ^{-1}(c x)\right )}{3 d (d x)^{3/2}}+\frac {(2 b c) \int \frac {1}{(d x)^{3/2} \left (1-c^2 x^2\right )} \, dx}{3 d}\\ &=-\frac {4 b c}{3 d^2 \sqrt {d x}}-\frac {2 \left (a+b \tanh ^{-1}(c x)\right )}{3 d (d x)^{3/2}}+\frac {\left (2 b c^3\right ) \int \frac {\sqrt {d x}}{1-c^2 x^2} \, dx}{3 d^3}\\ &=-\frac {4 b c}{3 d^2 \sqrt {d x}}-\frac {2 \left (a+b \tanh ^{-1}(c x)\right )}{3 d (d x)^{3/2}}+\frac {\left (4 b c^3\right ) \text {Subst}\left (\int \frac {x^2}{1-\frac {c^2 x^4}{d^2}} \, dx,x,\sqrt {d x}\right )}{3 d^4}\\ &=-\frac {4 b c}{3 d^2 \sqrt {d x}}-\frac {2 \left (a+b \tanh ^{-1}(c x)\right )}{3 d (d x)^{3/2}}+\frac {\left (2 b c^2\right ) \text {Subst}\left (\int \frac {1}{d-c x^2} \, dx,x,\sqrt {d x}\right )}{3 d^2}-\frac {\left (2 b c^2\right ) \text {Subst}\left (\int \frac {1}{d+c x^2} \, dx,x,\sqrt {d x}\right )}{3 d^2}\\ &=-\frac {4 b c}{3 d^2 \sqrt {d x}}-\frac {2 b c^{3/2} \tan ^{-1}\left (\frac {\sqrt {c} \sqrt {d x}}{\sqrt {d}}\right )}{3 d^{5/2}}-\frac {2 \left (a+b \tanh ^{-1}(c x)\right )}{3 d (d x)^{3/2}}+\frac {2 b c^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {d x}}{\sqrt {d}}\right )}{3 d^{5/2}}\\ \end {align*}
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Mathematica [A]
time = 0.04, size = 107, normalized size = 1.00 \begin {gather*} -\frac {x \left (2 a+4 b c x+2 b c^{3/2} x^{3/2} \text {ArcTan}\left (\sqrt {c} \sqrt {x}\right )+2 b \tanh ^{-1}(c x)+b c^{3/2} x^{3/2} \log \left (1-\sqrt {c} \sqrt {x}\right )-b c^{3/2} x^{3/2} \log \left (1+\sqrt {c} \sqrt {x}\right )\right )}{3 (d x)^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.05, size = 93, normalized size = 0.87
method | result | size |
derivativedivides | \(\frac {-\frac {2 a}{3 \left (d x \right )^{\frac {3}{2}}}-\frac {2 b \arctanh \left (c x \right )}{3 \left (d x \right )^{\frac {3}{2}}}-\frac {2 b \,c^{2} \arctan \left (\frac {c \sqrt {d x}}{\sqrt {d c}}\right )}{3 d \sqrt {d c}}+\frac {2 b \,c^{2} \arctanh \left (\frac {c \sqrt {d x}}{\sqrt {d c}}\right )}{3 d \sqrt {d c}}-\frac {4 b c}{3 d \sqrt {d x}}}{d}\) | \(93\) |
default | \(\frac {-\frac {2 a}{3 \left (d x \right )^{\frac {3}{2}}}-\frac {2 b \arctanh \left (c x \right )}{3 \left (d x \right )^{\frac {3}{2}}}-\frac {2 b \,c^{2} \arctan \left (\frac {c \sqrt {d x}}{\sqrt {d c}}\right )}{3 d \sqrt {d c}}+\frac {2 b \,c^{2} \arctanh \left (\frac {c \sqrt {d x}}{\sqrt {d c}}\right )}{3 d \sqrt {d c}}-\frac {4 b c}{3 d \sqrt {d x}}}{d}\) | \(93\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.46, size = 101, normalized size = 0.94 \begin {gather*} -\frac {b {\left (\frac {{\left (\frac {2 \, c \arctan \left (\frac {\sqrt {d x} c}{\sqrt {c d}}\right )}{\sqrt {c d}} + \frac {c \log \left (\frac {\sqrt {d x} c - \sqrt {c d}}{\sqrt {d x} c + \sqrt {c d}}\right )}{\sqrt {c d}} + \frac {4}{\sqrt {d x}}\right )} c}{d} + \frac {2 \, \operatorname {artanh}\left (c x\right )}{\left (d x\right )^{\frac {3}{2}}}\right )} + \frac {2 \, a}{\left (d x\right )^{\frac {3}{2}}}}{3 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 243, normalized size = 2.27 \begin {gather*} \left [\frac {2 \, b c d x^{2} \sqrt {\frac {c}{d}} \arctan \left (\frac {\sqrt {d x} \sqrt {\frac {c}{d}}}{c x}\right ) + b c d x^{2} \sqrt {\frac {c}{d}} \log \left (\frac {c x + 2 \, \sqrt {d x} \sqrt {\frac {c}{d}} + 1}{c x - 1}\right ) - {\left (4 \, b c x + b \log \left (-\frac {c x + 1}{c x - 1}\right ) + 2 \, a\right )} \sqrt {d x}}{3 \, d^{3} x^{2}}, -\frac {2 \, b c d x^{2} \sqrt {-\frac {c}{d}} \arctan \left (\frac {\sqrt {d x} \sqrt {-\frac {c}{d}}}{c x}\right ) - b c d x^{2} \sqrt {-\frac {c}{d}} \log \left (\frac {c x - 2 \, \sqrt {d x} \sqrt {-\frac {c}{d}} - 1}{c x + 1}\right ) + {\left (4 \, b c x + b \log \left (-\frac {c x + 1}{c x - 1}\right ) + 2 \, a\right )} \sqrt {d x}}{3 \, d^{3} x^{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 \frac {a + b \operatorname {atanh}{\left (c x \right )}}{\left (d x\right )^{\frac {5}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.44, size = 117, normalized size = 1.09 \begin {gather*} -\frac {\frac {2 \, b c^{2} \arctan \left (\frac {\sqrt {d x} c}{\sqrt {c d}}\right )}{\sqrt {c d} d} + \frac {2 \, b c^{2} \arctan \left (\frac {\sqrt {d x} c}{\sqrt {-c d}}\right )}{\sqrt {-c d} d} + \frac {b \log \left (-\frac {c d x + d}{c d x - d}\right )}{\sqrt {d x} d x} + \frac {2 \, {\left (2 \, b c d x + a d\right )}}{\sqrt {d x} d^{2} x}}{3 \, d} \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 \frac {a+b\,\mathrm {atanh}\left (c\,x\right )}{{\left (d\,x\right )}^{5/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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