Optimal. Leaf size=301 \[ \frac {2 b c^{3/4} \text {ArcTan}\left (\frac {\sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}\right )}{3 d^{5/2}}-\frac {\sqrt {2} b c^{3/4} \text {ArcTan}\left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}\right )}{3 d^{5/2}}+\frac {\sqrt {2} b c^{3/4} \text {ArcTan}\left (1+\frac {\sqrt {2} \sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}\right )}{3 d^{5/2}}-\frac {2 \left (a+b \tanh ^{-1}\left (c x^2\right )\right )}{3 d (d x)^{3/2}}+\frac {2 b c^{3/4} \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}\right )}{3 d^{5/2}}-\frac {b c^{3/4} \log \left (\sqrt {d}+\sqrt {c} \sqrt {d} x-\sqrt {2} \sqrt [4]{c} \sqrt {d x}\right )}{3 \sqrt {2} d^{5/2}}+\frac {b c^{3/4} \log \left (\sqrt {d}+\sqrt {c} \sqrt {d} x+\sqrt {2} \sqrt [4]{c} \sqrt {d x}\right )}{3 \sqrt {2} d^{5/2}} \]
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Rubi [A]
time = 0.18, antiderivative size = 301, normalized size of antiderivative = 1.00, number of steps
used = 15, number of rules used = 12, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {6049, 335,
220, 218, 214, 211, 217, 1179, 642, 1176, 631, 210} \begin {gather*} -\frac {2 \left (a+b \tanh ^{-1}\left (c x^2\right )\right )}{3 d (d x)^{3/2}}+\frac {2 b c^{3/4} \text {ArcTan}\left (\frac {\sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}\right )}{3 d^{5/2}}-\frac {\sqrt {2} b c^{3/4} \text {ArcTan}\left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}\right )}{3 d^{5/2}}+\frac {\sqrt {2} b c^{3/4} \text {ArcTan}\left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}+1\right )}{3 d^{5/2}}-\frac {b c^{3/4} \log \left (\sqrt {c} \sqrt {d} x-\sqrt {2} \sqrt [4]{c} \sqrt {d x}+\sqrt {d}\right )}{3 \sqrt {2} d^{5/2}}+\frac {b c^{3/4} \log \left (\sqrt {c} \sqrt {d} x+\sqrt {2} \sqrt [4]{c} \sqrt {d x}+\sqrt {d}\right )}{3 \sqrt {2} d^{5/2}}+\frac {2 b c^{3/4} \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}\right )}{3 d^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 210
Rule 211
Rule 214
Rule 217
Rule 218
Rule 220
Rule 335
Rule 631
Rule 642
Rule 1176
Rule 1179
Rule 6049
Rubi steps
\begin {align*} \int \frac {a+b \tanh ^{-1}\left (c x^2\right )}{(d x)^{5/2}} \, dx &=-\frac {2 \left (a+b \tanh ^{-1}\left (c x^2\right )\right )}{3 d (d x)^{3/2}}+\frac {(4 b c) \int \frac {x}{(d x)^{3/2} \left (1-c^2 x^4\right )} \, dx}{3 d}\\ &=-\frac {2 \left (a+b \tanh ^{-1}\left (c x^2\right )\right )}{3 d (d x)^{3/2}}+\frac {(4 b c) \int \frac {1}{\sqrt {d x} \left (1-c^2 x^4\right )} \, dx}{3 d^2}\\ &=-\frac {2 \left (a+b \tanh ^{-1}\left (c x^2\right )\right )}{3 d (d x)^{3/2}}+\frac {(8 b c) \text {Subst}\left (\int \frac {1}{1-\frac {c^2 x^8}{d^4}} \, dx,x,\sqrt {d x}\right )}{3 d^3}\\ &=-\frac {2 \left (a+b \tanh ^{-1}\left (c x^2\right )\right )}{3 d (d x)^{3/2}}+\frac {(4 b c) \text {Subst}\left (\int \frac {1}{d^2-c x^4} \, dx,x,\sqrt {d x}\right )}{3 d}+\frac {(4 b c) \text {Subst}\left (\int \frac {1}{d^2+c x^4} \, dx,x,\sqrt {d x}\right )}{3 d}\\ &=-\frac {2 \left (a+b \tanh ^{-1}\left (c x^2\right )\right )}{3 d (d x)^{3/2}}+\frac {(2 b c) \text {Subst}\left (\int \frac {1}{d-\sqrt {c} x^2} \, dx,x,\sqrt {d x}\right )}{3 d^2}+\frac {(2 b c) \text {Subst}\left (\int \frac {1}{d+\sqrt {c} x^2} \, dx,x,\sqrt {d x}\right )}{3 d^2}+\frac {(2 b c) \text {Subst}\left (\int \frac {d-\sqrt {c} x^2}{d^2+c x^4} \, dx,x,\sqrt {d x}\right )}{3 d^2}+\frac {(2 b c) \text {Subst}\left (\int \frac {d+\sqrt {c} x^2}{d^2+c x^4} \, dx,x,\sqrt {d x}\right )}{3 d^2}\\ &=\frac {2 b c^{3/4} \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}\right )}{3 d^{5/2}}-\frac {2 \left (a+b \tanh ^{-1}\left (c x^2\right )\right )}{3 d (d x)^{3/2}}+\frac {2 b c^{3/4} \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}\right )}{3 d^{5/2}}-\frac {\left (b c^{3/4}\right ) \text {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt {d}}{\sqrt [4]{c}}+2 x}{-\frac {d}{\sqrt {c}}-\frac {\sqrt {2} \sqrt {d} x}{\sqrt [4]{c}}-x^2} \, dx,x,\sqrt {d x}\right )}{3 \sqrt {2} d^{5/2}}-\frac {\left (b c^{3/4}\right ) \text {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt {d}}{\sqrt [4]{c}}-2 x}{-\frac {d}{\sqrt {c}}+\frac {\sqrt {2} \sqrt {d} x}{\sqrt [4]{c}}-x^2} \, dx,x,\sqrt {d x}\right )}{3 \sqrt {2} d^{5/2}}+\frac {\left (b \sqrt {c}\right ) \text {Subst}\left (\int \frac {1}{\frac {d}{\sqrt {c}}-\frac {\sqrt {2} \sqrt {d} x}{\sqrt [4]{c}}+x^2} \, dx,x,\sqrt {d x}\right )}{3 d^2}+\frac {\left (b \sqrt {c}\right ) \text {Subst}\left (\int \frac {1}{\frac {d}{\sqrt {c}}+\frac {\sqrt {2} \sqrt {d} x}{\sqrt [4]{c}}+x^2} \, dx,x,\sqrt {d x}\right )}{3 d^2}\\ &=\frac {2 b c^{3/4} \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}\right )}{3 d^{5/2}}-\frac {2 \left (a+b \tanh ^{-1}\left (c x^2\right )\right )}{3 d (d x)^{3/2}}+\frac {2 b c^{3/4} \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}\right )}{3 d^{5/2}}-\frac {b c^{3/4} \log \left (\sqrt {d}+\sqrt {c} \sqrt {d} x-\sqrt {2} \sqrt [4]{c} \sqrt {d x}\right )}{3 \sqrt {2} d^{5/2}}+\frac {b c^{3/4} \log \left (\sqrt {d}+\sqrt {c} \sqrt {d} x+\sqrt {2} \sqrt [4]{c} \sqrt {d x}\right )}{3 \sqrt {2} d^{5/2}}+\frac {\left (\sqrt {2} b c^{3/4}\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}\right )}{3 d^{5/2}}-\frac {\left (\sqrt {2} b c^{3/4}\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}\right )}{3 d^{5/2}}\\ &=\frac {2 b c^{3/4} \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}\right )}{3 d^{5/2}}-\frac {\sqrt {2} b c^{3/4} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}\right )}{3 d^{5/2}}+\frac {\sqrt {2} b c^{3/4} \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}\right )}{3 d^{5/2}}-\frac {2 \left (a+b \tanh ^{-1}\left (c x^2\right )\right )}{3 d (d x)^{3/2}}+\frac {2 b c^{3/4} \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}\right )}{3 d^{5/2}}-\frac {b c^{3/4} \log \left (\sqrt {d}+\sqrt {c} \sqrt {d} x-\sqrt {2} \sqrt [4]{c} \sqrt {d x}\right )}{3 \sqrt {2} d^{5/2}}+\frac {b c^{3/4} \log \left (\sqrt {d}+\sqrt {c} \sqrt {d} x+\sqrt {2} \sqrt [4]{c} \sqrt {d x}\right )}{3 \sqrt {2} d^{5/2}}\\ \end {align*}
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Mathematica [A]
time = 0.07, size = 268, normalized size = 0.89 \begin {gather*} -\frac {x \left (4 a+2 \sqrt {2} b c^{3/4} x^{3/2} \text {ArcTan}\left (1-\sqrt {2} \sqrt [4]{c} \sqrt {x}\right )-2 \sqrt {2} b c^{3/4} x^{3/2} \text {ArcTan}\left (1+\sqrt {2} \sqrt [4]{c} \sqrt {x}\right )-4 b c^{3/4} x^{3/2} \text {ArcTan}\left (\sqrt [4]{c} \sqrt {x}\right )+4 b \tanh ^{-1}\left (c x^2\right )+2 b c^{3/4} x^{3/2} \log \left (1-\sqrt [4]{c} \sqrt {x}\right )-2 b c^{3/4} x^{3/2} \log \left (1+\sqrt [4]{c} \sqrt {x}\right )+\sqrt {2} b c^{3/4} x^{3/2} \log \left (1-\sqrt {2} \sqrt [4]{c} \sqrt {x}+\sqrt {c} x\right )-\sqrt {2} b c^{3/4} x^{3/2} \log \left (1+\sqrt {2} \sqrt [4]{c} \sqrt {x}+\sqrt {c} x\right )\right )}{6 (d x)^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.04, size = 279, normalized size = 0.93
method | result | size |
derivativedivides | \(\frac {-\frac {2 a}{3 \left (d x \right )^{\frac {3}{2}}}-\frac {2 b \arctanh \left (c \,x^{2}\right )}{3 \left (d x \right )^{\frac {3}{2}}}+\frac {b c \left (\frac {d^{2}}{c}\right )^{\frac {1}{4}} \sqrt {2}\, \ln \left (\frac {d x +\left (\frac {d^{2}}{c}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {d^{2}}{c}}}{d x -\left (\frac {d^{2}}{c}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {d^{2}}{c}}}\right )}{6 d^{2}}+\frac {b c \left (\frac {d^{2}}{c}\right )^{\frac {1}{4}} \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {d^{2}}{c}\right )^{\frac {1}{4}}}+1\right )}{3 d^{2}}+\frac {b c \left (\frac {d^{2}}{c}\right )^{\frac {1}{4}} \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {d^{2}}{c}\right )^{\frac {1}{4}}}-1\right )}{3 d^{2}}+\frac {b c \left (\frac {d^{2}}{c}\right )^{\frac {1}{4}} \ln \left (\frac {\sqrt {d x}+\left (\frac {d^{2}}{c}\right )^{\frac {1}{4}}}{\sqrt {d x}-\left (\frac {d^{2}}{c}\right )^{\frac {1}{4}}}\right )}{3 d^{2}}+\frac {2 b c \left (\frac {d^{2}}{c}\right )^{\frac {1}{4}} \arctan \left (\frac {\sqrt {d x}}{\left (\frac {d^{2}}{c}\right )^{\frac {1}{4}}}\right )}{3 d^{2}}}{d}\) | \(279\) |
default | \(\frac {-\frac {2 a}{3 \left (d x \right )^{\frac {3}{2}}}-\frac {2 b \arctanh \left (c \,x^{2}\right )}{3 \left (d x \right )^{\frac {3}{2}}}+\frac {b c \left (\frac {d^{2}}{c}\right )^{\frac {1}{4}} \sqrt {2}\, \ln \left (\frac {d x +\left (\frac {d^{2}}{c}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {d^{2}}{c}}}{d x -\left (\frac {d^{2}}{c}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {d^{2}}{c}}}\right )}{6 d^{2}}+\frac {b c \left (\frac {d^{2}}{c}\right )^{\frac {1}{4}} \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {d^{2}}{c}\right )^{\frac {1}{4}}}+1\right )}{3 d^{2}}+\frac {b c \left (\frac {d^{2}}{c}\right )^{\frac {1}{4}} \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {d^{2}}{c}\right )^{\frac {1}{4}}}-1\right )}{3 d^{2}}+\frac {b c \left (\frac {d^{2}}{c}\right )^{\frac {1}{4}} \ln \left (\frac {\sqrt {d x}+\left (\frac {d^{2}}{c}\right )^{\frac {1}{4}}}{\sqrt {d x}-\left (\frac {d^{2}}{c}\right )^{\frac {1}{4}}}\right )}{3 d^{2}}+\frac {2 b c \left (\frac {d^{2}}{c}\right )^{\frac {1}{4}} \arctan \left (\frac {\sqrt {d x}}{\left (\frac {d^{2}}{c}\right )^{\frac {1}{4}}}\right )}{3 d^{2}}}{d}\) | \(279\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.47, size = 277, normalized size = 0.92 \begin {gather*} \frac {b {\left (\frac {{\left (\frac {2 \, \sqrt {2} d \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} c^{\frac {1}{4}} \sqrt {d} + 2 \, \sqrt {d x} \sqrt {c}\right )}}{2 \, \sqrt {\sqrt {c} d}}\right )}{\sqrt {\sqrt {c} d}} + \frac {2 \, \sqrt {2} d \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} c^{\frac {1}{4}} \sqrt {d} - 2 \, \sqrt {d x} \sqrt {c}\right )}}{2 \, \sqrt {\sqrt {c} d}}\right )}{\sqrt {\sqrt {c} d}} + \frac {\sqrt {2} \sqrt {d} \log \left (\sqrt {c} d x + \sqrt {2} \sqrt {d x} c^{\frac {1}{4}} \sqrt {d} + d\right )}{c^{\frac {1}{4}}} - \frac {\sqrt {2} \sqrt {d} \log \left (\sqrt {c} d x - \sqrt {2} \sqrt {d x} c^{\frac {1}{4}} \sqrt {d} + d\right )}{c^{\frac {1}{4}}} + \frac {4 \, d \arctan \left (\frac {\sqrt {d x} \sqrt {c}}{\sqrt {\sqrt {c} d}}\right )}{\sqrt {\sqrt {c} d}} - \frac {2 \, d \log \left (\frac {\sqrt {d x} \sqrt {c} - \sqrt {\sqrt {c} d}}{\sqrt {d x} \sqrt {c} + \sqrt {\sqrt {c} d}}\right )}{\sqrt {\sqrt {c} d}}\right )} c}{d^{2}} - \frac {4 \, \operatorname {artanh}\left (c x^{2}\right )}{\left (d x\right )^{\frac {3}{2}}}\right )} - \frac {4 \, a}{\left (d x\right )^{\frac {3}{2}}}}{6 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 428 vs.
\(2 (200) = 400\).
time = 0.40, size = 428, normalized size = 1.42 \begin {gather*} -\frac {4 \, d^{3} x^{2} \left (\frac {b^{4} c^{3}}{d^{10}}\right )^{\frac {1}{4}} \arctan \left (-\frac {\sqrt {d x} b c d^{7} \left (\frac {b^{4} c^{3}}{d^{10}}\right )^{\frac {3}{4}} - \sqrt {d^{6} \sqrt {\frac {b^{4} c^{3}}{d^{10}}} + b^{2} c^{2} d x} d^{7} \left (\frac {b^{4} c^{3}}{d^{10}}\right )^{\frac {3}{4}}}{b^{4} c^{3}}\right ) - 4 \, d^{3} x^{2} \left (-\frac {b^{4} c^{3}}{d^{10}}\right )^{\frac {1}{4}} \arctan \left (-\frac {\sqrt {d x} b c d^{7} \left (-\frac {b^{4} c^{3}}{d^{10}}\right )^{\frac {3}{4}} - \sqrt {d^{6} \sqrt {-\frac {b^{4} c^{3}}{d^{10}}} + b^{2} c^{2} d x} d^{7} \left (-\frac {b^{4} c^{3}}{d^{10}}\right )^{\frac {3}{4}}}{b^{4} c^{3}}\right ) - d^{3} x^{2} \left (\frac {b^{4} c^{3}}{d^{10}}\right )^{\frac {1}{4}} \log \left (d^{3} \left (\frac {b^{4} c^{3}}{d^{10}}\right )^{\frac {1}{4}} + \sqrt {d x} b c\right ) + d^{3} x^{2} \left (\frac {b^{4} c^{3}}{d^{10}}\right )^{\frac {1}{4}} \log \left (-d^{3} \left (\frac {b^{4} c^{3}}{d^{10}}\right )^{\frac {1}{4}} + \sqrt {d x} b c\right ) - d^{3} x^{2} \left (-\frac {b^{4} c^{3}}{d^{10}}\right )^{\frac {1}{4}} \log \left (d^{3} \left (-\frac {b^{4} c^{3}}{d^{10}}\right )^{\frac {1}{4}} + \sqrt {d x} b c\right ) + d^{3} x^{2} \left (-\frac {b^{4} c^{3}}{d^{10}}\right )^{\frac {1}{4}} \log \left (-d^{3} \left (-\frac {b^{4} c^{3}}{d^{10}}\right )^{\frac {1}{4}} + \sqrt {d x} b c\right ) + \sqrt {d x} {\left (b \log \left (-\frac {c x^{2} + 1}{c x^{2} - 1}\right ) + 2 \, a\right )}}{3 \, d^{3} x^{2}} \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^{2} \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 [B] Leaf count of result is larger than twice the leaf count of optimal. 516 vs.
\(2 (200) = 400\).
time = 0.76, size = 516, normalized size = 1.71 \begin {gather*} \frac {b c d^{2} {\left (\frac {2 \, \sqrt {2} \left (c^{3} d^{2}\right )^{\frac {1}{4}} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {d^{2}}{c}\right )^{\frac {1}{4}} + 2 \, \sqrt {d x}\right )}}{2 \, \left (\frac {d^{2}}{c}\right )^{\frac {1}{4}}}\right )}{c d^{4}} + \frac {2 \, \sqrt {2} \left (c^{3} d^{2}\right )^{\frac {1}{4}} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {d^{2}}{c}\right )^{\frac {1}{4}} - 2 \, \sqrt {d x}\right )}}{2 \, \left (\frac {d^{2}}{c}\right )^{\frac {1}{4}}}\right )}{c d^{4}} + \frac {2 \, \sqrt {2} \left (-c^{3} d^{2}\right )^{\frac {1}{4}} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (-\frac {d^{2}}{c}\right )^{\frac {1}{4}} + 2 \, \sqrt {d x}\right )}}{2 \, \left (-\frac {d^{2}}{c}\right )^{\frac {1}{4}}}\right )}{c d^{4}} + \frac {2 \, \sqrt {2} \left (-c^{3} d^{2}\right )^{\frac {1}{4}} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (-\frac {d^{2}}{c}\right )^{\frac {1}{4}} - 2 \, \sqrt {d x}\right )}}{2 \, \left (-\frac {d^{2}}{c}\right )^{\frac {1}{4}}}\right )}{c d^{4}} + \frac {\sqrt {2} \left (c^{3} d^{2}\right )^{\frac {1}{4}} \log \left (d x + \sqrt {2} \sqrt {d x} \left (\frac {d^{2}}{c}\right )^{\frac {1}{4}} + \sqrt {\frac {d^{2}}{c}}\right )}{c d^{4}} - \frac {\sqrt {2} \left (c^{3} d^{2}\right )^{\frac {1}{4}} \log \left (d x - \sqrt {2} \sqrt {d x} \left (\frac {d^{2}}{c}\right )^{\frac {1}{4}} + \sqrt {\frac {d^{2}}{c}}\right )}{c d^{4}} + \frac {\sqrt {2} \left (-c^{3} d^{2}\right )^{\frac {1}{4}} \log \left (d x + \sqrt {2} \sqrt {d x} \left (-\frac {d^{2}}{c}\right )^{\frac {1}{4}} + \sqrt {-\frac {d^{2}}{c}}\right )}{c d^{4}} - \frac {\sqrt {2} \left (-c^{3} d^{2}\right )^{\frac {1}{4}} \log \left (d x - \sqrt {2} \sqrt {d x} \left (-\frac {d^{2}}{c}\right )^{\frac {1}{4}} + \sqrt {-\frac {d^{2}}{c}}\right )}{c d^{4}}\right )} - \frac {2 \, b \log \left (-\frac {c d^{2} x^{2} + d^{2}}{c d^{2} x^{2} - d^{2}}\right )}{\sqrt {d x} d x} - \frac {4 \, a}{\sqrt {d x} d x}}{6 \, 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 = -0.00 \begin {gather*} \int \frac {a+b\,\mathrm {atanh}\left (c\,x^2\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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