Optimal. Leaf size=107 \[ -\frac {2 \left (a+b \tanh ^{-1}(c x)\right )}{5 d (d x)^{5/2}}+\frac {2 b c^{5/2} \tan ^{-1}\left (\frac {\sqrt {c} \sqrt {d x}}{\sqrt {d}}\right )}{5 d^{7/2}}+\frac {2 b c^{5/2} \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {d x}}{\sqrt {d}}\right )}{5 d^{7/2}}-\frac {4 b c}{15 d^2 (d x)^{3/2}} \]
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Rubi [A] time = 0.07, 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 = {5916, 325, 329, 212, 208, 205} \[ -\frac {2 \left (a+b \tanh ^{-1}(c x)\right )}{5 d (d x)^{5/2}}+\frac {2 b c^{5/2} \tan ^{-1}\left (\frac {\sqrt {c} \sqrt {d x}}{\sqrt {d}}\right )}{5 d^{7/2}}+\frac {2 b c^{5/2} \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {d x}}{\sqrt {d}}\right )}{5 d^{7/2}}-\frac {4 b c}{15 d^2 (d x)^{3/2}} \]
Antiderivative was successfully verified.
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Rule 205
Rule 208
Rule 212
Rule 325
Rule 329
Rule 5916
Rubi steps
\begin {align*} \int \frac {a+b \tanh ^{-1}(c x)}{(d x)^{7/2}} \, dx &=-\frac {2 \left (a+b \tanh ^{-1}(c x)\right )}{5 d (d x)^{5/2}}+\frac {(2 b c) \int \frac {1}{(d x)^{5/2} \left (1-c^2 x^2\right )} \, dx}{5 d}\\ &=-\frac {4 b c}{15 d^2 (d x)^{3/2}}-\frac {2 \left (a+b \tanh ^{-1}(c x)\right )}{5 d (d x)^{5/2}}+\frac {\left (2 b c^3\right ) \int \frac {1}{\sqrt {d x} \left (1-c^2 x^2\right )} \, dx}{5 d^3}\\ &=-\frac {4 b c}{15 d^2 (d x)^{3/2}}-\frac {2 \left (a+b \tanh ^{-1}(c x)\right )}{5 d (d x)^{5/2}}+\frac {\left (4 b c^3\right ) \operatorname {Subst}\left (\int \frac {1}{1-\frac {c^2 x^4}{d^2}} \, dx,x,\sqrt {d x}\right )}{5 d^4}\\ &=-\frac {4 b c}{15 d^2 (d x)^{3/2}}-\frac {2 \left (a+b \tanh ^{-1}(c x)\right )}{5 d (d x)^{5/2}}+\frac {\left (2 b c^3\right ) \operatorname {Subst}\left (\int \frac {1}{d-c x^2} \, dx,x,\sqrt {d x}\right )}{5 d^3}+\frac {\left (2 b c^3\right ) \operatorname {Subst}\left (\int \frac {1}{d+c x^2} \, dx,x,\sqrt {d x}\right )}{5 d^3}\\ &=-\frac {4 b c}{15 d^2 (d x)^{3/2}}+\frac {2 b c^{5/2} \tan ^{-1}\left (\frac {\sqrt {c} \sqrt {d x}}{\sqrt {d}}\right )}{5 d^{7/2}}-\frac {2 \left (a+b \tanh ^{-1}(c x)\right )}{5 d (d x)^{5/2}}+\frac {2 b c^{5/2} \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {d x}}{\sqrt {d}}\right )}{5 d^{7/2}}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 108, normalized size = 1.01 \[ \frac {x \left (-6 a-3 b c^{5/2} x^{5/2} \log \left (1-\sqrt {c} \sqrt {x}\right )+3 b c^{5/2} x^{5/2} \log \left (\sqrt {c} \sqrt {x}+1\right )+6 b c^{5/2} x^{5/2} \tan ^{-1}\left (\sqrt {c} \sqrt {x}\right )-4 b c x-6 b \tanh ^{-1}(c x)\right )}{15 (d x)^{7/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.65, size = 253, normalized size = 2.36 \[ \left [-\frac {6 \, b c^{2} d x^{3} \sqrt {\frac {c}{d}} \arctan \left (\frac {\sqrt {d x} \sqrt {\frac {c}{d}}}{c x}\right ) - 3 \, b c^{2} d x^{3} \sqrt {\frac {c}{d}} \log \left (\frac {c x + 2 \, \sqrt {d x} \sqrt {\frac {c}{d}} + 1}{c x - 1}\right ) + {\left (4 \, b c x + 3 \, b \log \left (-\frac {c x + 1}{c x - 1}\right ) + 6 \, a\right )} \sqrt {d x}}{15 \, d^{4} x^{3}}, -\frac {6 \, b c^{2} d x^{3} \sqrt {-\frac {c}{d}} \arctan \left (\frac {\sqrt {d x} \sqrt {-\frac {c}{d}}}{c x}\right ) - 3 \, b c^{2} d x^{3} \sqrt {-\frac {c}{d}} \log \left (\frac {c x + 2 \, \sqrt {d x} \sqrt {-\frac {c}{d}} - 1}{c x + 1}\right ) + {\left (4 \, b c x + 3 \, b \log \left (-\frac {c x + 1}{c x - 1}\right ) + 6 \, a\right )} \sqrt {d x}}{15 \, d^{4} x^{3}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.20, size = 117, normalized size = 1.09 \[ \frac {6 \, b c^{3} {\left (\frac {\arctan \left (\frac {\sqrt {d x} c}{\sqrt {c d}}\right )}{\sqrt {c d} d^{2}} - \frac {\arctan \left (\frac {\sqrt {d x} c}{\sqrt {-c d}}\right )}{\sqrt {-c d} d^{2}}\right )} - \frac {3 \, b \log \left (-\frac {c d x + d}{c d x - d}\right )}{\sqrt {d x} d^{2} x^{2}} - \frac {2 \, {\left (2 \, b c d x + 3 \, a d\right )}}{\sqrt {d x} d^{3} x^{2}}}{15 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 94, normalized size = 0.88 \[ -\frac {2 a}{5 d \left (d x \right )^{\frac {5}{2}}}-\frac {2 b \arctanh \left (c x \right )}{5 d \left (d x \right )^{\frac {5}{2}}}-\frac {4 b c}{15 d^{2} \left (d x \right )^{\frac {3}{2}}}+\frac {2 b \,c^{3} \arctan \left (\frac {c \sqrt {d x}}{\sqrt {c d}}\right )}{5 d^{3} \sqrt {c d}}+\frac {2 b \,c^{3} \arctanh \left (\frac {c \sqrt {d x}}{\sqrt {c d}}\right )}{5 d^{3} \sqrt {c d}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.43, size = 112, normalized size = 1.05 \[ \frac {b {\left (\frac {{\left (\frac {6 \, c^{2} \arctan \left (\frac {\sqrt {d x} c}{\sqrt {c d}}\right )}{\sqrt {c d} d} - \frac {3 \, c^{2} \log \left (\frac {\sqrt {d x} c - \sqrt {c d}}{\sqrt {d x} c + \sqrt {c d}}\right )}{\sqrt {c d} d} - \frac {4}{\left (d x\right )^{\frac {3}{2}}}\right )} c}{d} - \frac {6 \, \operatorname {artanh}\left (c x\right )}{\left (d x\right )^{\frac {5}{2}}}\right )} - \frac {6 \, a}{\left (d x\right )^{\frac {5}{2}}}}{15 \, d} \]
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 \[ \int \frac {a+b\,\mathrm {atanh}\left (c\,x\right )}{{\left (d\,x\right )}^{7/2}} \,d x \]
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 \[ \int \frac {a + b \operatorname {atanh}{\left (c x \right )}}{\left (d x\right )^{\frac {7}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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