Optimal. Leaf size=317 \[ \frac {8 b d (d x)^{3/2}}{21 c}+\frac {2 b d^{5/2} \text {ArcTan}\left (\frac {\sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}\right )}{7 c^{7/4}}+\frac {\sqrt {2} b d^{5/2} \text {ArcTan}\left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}\right )}{7 c^{7/4}}-\frac {\sqrt {2} b d^{5/2} \text {ArcTan}\left (1+\frac {\sqrt {2} \sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}\right )}{7 c^{7/4}}+\frac {2 (d x)^{7/2} \left (a+b \tanh ^{-1}\left (c x^2\right )\right )}{7 d}-\frac {2 b d^{5/2} \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}\right )}{7 c^{7/4}}-\frac {b d^{5/2} \log \left (\sqrt {d}+\sqrt {c} \sqrt {d} x-\sqrt {2} \sqrt [4]{c} \sqrt {d x}\right )}{7 \sqrt {2} c^{7/4}}+\frac {b d^{5/2} \log \left (\sqrt {d}+\sqrt {c} \sqrt {d} x+\sqrt {2} \sqrt [4]{c} \sqrt {d x}\right )}{7 \sqrt {2} c^{7/4}} \]
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Rubi [A]
time = 0.21, antiderivative size = 317, normalized size of antiderivative = 1.00, number of steps
used = 16, number of rules used = 13, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.722, Rules used = {6049, 327,
335, 306, 303, 1176, 631, 210, 1179, 642, 304, 211, 214} \begin {gather*} \frac {2 (d x)^{7/2} \left (a+b \tanh ^{-1}\left (c x^2\right )\right )}{7 d}+\frac {2 b d^{5/2} \text {ArcTan}\left (\frac {\sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}\right )}{7 c^{7/4}}+\frac {\sqrt {2} b d^{5/2} \text {ArcTan}\left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}\right )}{7 c^{7/4}}-\frac {\sqrt {2} b d^{5/2} \text {ArcTan}\left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}+1\right )}{7 c^{7/4}}-\frac {b d^{5/2} \log \left (\sqrt {c} \sqrt {d} x-\sqrt {2} \sqrt [4]{c} \sqrt {d x}+\sqrt {d}\right )}{7 \sqrt {2} c^{7/4}}+\frac {b d^{5/2} \log \left (\sqrt {c} \sqrt {d} x+\sqrt {2} \sqrt [4]{c} \sqrt {d x}+\sqrt {d}\right )}{7 \sqrt {2} c^{7/4}}-\frac {2 b d^{5/2} \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}\right )}{7 c^{7/4}}+\frac {8 b d (d x)^{3/2}}{21 c} \end {gather*}
Antiderivative was successfully verified.
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Rule 210
Rule 211
Rule 214
Rule 303
Rule 304
Rule 306
Rule 327
Rule 335
Rule 631
Rule 642
Rule 1176
Rule 1179
Rule 6049
Rubi steps
\begin {align*} \int (d x)^{5/2} \left (a+b \tanh ^{-1}\left (c x^2\right )\right ) \, dx &=\frac {2 (d x)^{7/2} \left (a+b \tanh ^{-1}\left (c x^2\right )\right )}{7 d}-\frac {(4 b c) \int \frac {x (d x)^{7/2}}{1-c^2 x^4} \, dx}{7 d}\\ &=\frac {2 (d x)^{7/2} \left (a+b \tanh ^{-1}\left (c x^2\right )\right )}{7 d}-\frac {(4 b c) \int \frac {(d x)^{9/2}}{1-c^2 x^4} \, dx}{7 d^2}\\ &=\frac {8 b d (d x)^{3/2}}{21 c}+\frac {2 (d x)^{7/2} \left (a+b \tanh ^{-1}\left (c x^2\right )\right )}{7 d}-\frac {\left (4 b d^2\right ) \int \frac {\sqrt {d x}}{1-c^2 x^4} \, dx}{7 c}\\ &=\frac {8 b d (d x)^{3/2}}{21 c}+\frac {2 (d x)^{7/2} \left (a+b \tanh ^{-1}\left (c x^2\right )\right )}{7 d}-\frac {(8 b d) \text {Subst}\left (\int \frac {x^2}{1-\frac {c^2 x^8}{d^4}} \, dx,x,\sqrt {d x}\right )}{7 c}\\ &=\frac {8 b d (d x)^{3/2}}{21 c}+\frac {2 (d x)^{7/2} \left (a+b \tanh ^{-1}\left (c x^2\right )\right )}{7 d}-\frac {\left (4 b d^3\right ) \text {Subst}\left (\int \frac {x^2}{d^2-c x^4} \, dx,x,\sqrt {d x}\right )}{7 c}-\frac {\left (4 b d^3\right ) \text {Subst}\left (\int \frac {x^2}{d^2+c x^4} \, dx,x,\sqrt {d x}\right )}{7 c}\\ &=\frac {8 b d (d x)^{3/2}}{21 c}+\frac {2 (d x)^{7/2} \left (a+b \tanh ^{-1}\left (c x^2\right )\right )}{7 d}-\frac {\left (2 b d^3\right ) \text {Subst}\left (\int \frac {1}{d-\sqrt {c} x^2} \, dx,x,\sqrt {d x}\right )}{7 c^{3/2}}+\frac {\left (2 b d^3\right ) \text {Subst}\left (\int \frac {1}{d+\sqrt {c} x^2} \, dx,x,\sqrt {d x}\right )}{7 c^{3/2}}+\frac {\left (2 b d^3\right ) \text {Subst}\left (\int \frac {d-\sqrt {c} x^2}{d^2+c x^4} \, dx,x,\sqrt {d x}\right )}{7 c^{3/2}}-\frac {\left (2 b d^3\right ) \text {Subst}\left (\int \frac {d+\sqrt {c} x^2}{d^2+c x^4} \, dx,x,\sqrt {d x}\right )}{7 c^{3/2}}\\ &=\frac {8 b d (d x)^{3/2}}{21 c}+\frac {2 b d^{5/2} \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}\right )}{7 c^{7/4}}+\frac {2 (d x)^{7/2} \left (a+b \tanh ^{-1}\left (c x^2\right )\right )}{7 d}-\frac {2 b d^{5/2} \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}\right )}{7 c^{7/4}}-\frac {\left (b d^{5/2}\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 )}{7 \sqrt {2} c^{7/4}}-\frac {\left (b d^{5/2}\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 )}{7 \sqrt {2} c^{7/4}}-\frac {\left (b d^3\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 )}{7 c^2}-\frac {\left (b d^3\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 )}{7 c^2}\\ &=\frac {8 b d (d x)^{3/2}}{21 c}+\frac {2 b d^{5/2} \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}\right )}{7 c^{7/4}}+\frac {2 (d x)^{7/2} \left (a+b \tanh ^{-1}\left (c x^2\right )\right )}{7 d}-\frac {2 b d^{5/2} \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}\right )}{7 c^{7/4}}-\frac {b d^{5/2} \log \left (\sqrt {d}+\sqrt {c} \sqrt {d} x-\sqrt {2} \sqrt [4]{c} \sqrt {d x}\right )}{7 \sqrt {2} c^{7/4}}+\frac {b d^{5/2} \log \left (\sqrt {d}+\sqrt {c} \sqrt {d} x+\sqrt {2} \sqrt [4]{c} \sqrt {d x}\right )}{7 \sqrt {2} c^{7/4}}-\frac {\left (\sqrt {2} b d^{5/2}\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 )}{7 c^{7/4}}+\frac {\left (\sqrt {2} b d^{5/2}\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 )}{7 c^{7/4}}\\ &=\frac {8 b d (d x)^{3/2}}{21 c}+\frac {2 b d^{5/2} \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}\right )}{7 c^{7/4}}+\frac {\sqrt {2} b d^{5/2} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}\right )}{7 c^{7/4}}-\frac {\sqrt {2} b d^{5/2} \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}\right )}{7 c^{7/4}}+\frac {2 (d x)^{7/2} \left (a+b \tanh ^{-1}\left (c x^2\right )\right )}{7 d}-\frac {2 b d^{5/2} \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}\right )}{7 c^{7/4}}-\frac {b d^{5/2} \log \left (\sqrt {d}+\sqrt {c} \sqrt {d} x-\sqrt {2} \sqrt [4]{c} \sqrt {d x}\right )}{7 \sqrt {2} c^{7/4}}+\frac {b d^{5/2} \log \left (\sqrt {d}+\sqrt {c} \sqrt {d} x+\sqrt {2} \sqrt [4]{c} \sqrt {d x}\right )}{7 \sqrt {2} c^{7/4}}\\ \end {align*}
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Mathematica [A]
time = 0.08, size = 241, normalized size = 0.76 \begin {gather*} \frac {(d x)^{5/2} \left (16 b c^{3/4} x^{3/2}+12 a c^{7/4} x^{7/2}+6 \sqrt {2} b \text {ArcTan}\left (1-\sqrt {2} \sqrt [4]{c} \sqrt {x}\right )-6 \sqrt {2} b \text {ArcTan}\left (1+\sqrt {2} \sqrt [4]{c} \sqrt {x}\right )+12 b \text {ArcTan}\left (\sqrt [4]{c} \sqrt {x}\right )+12 b c^{7/4} x^{7/2} \tanh ^{-1}\left (c x^2\right )+6 b \log \left (1-\sqrt [4]{c} \sqrt {x}\right )-6 b \log \left (1+\sqrt [4]{c} \sqrt {x}\right )-3 \sqrt {2} b \log \left (1-\sqrt {2} \sqrt [4]{c} \sqrt {x}+\sqrt {c} x\right )+3 \sqrt {2} b \log \left (1+\sqrt {2} \sqrt [4]{c} \sqrt {x}+\sqrt {c} x\right )\right )}{42 c^{7/4} x^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.08, size = 303, normalized size = 0.96
method | result | size |
derivativedivides | \(\frac {\frac {2 \left (d x \right )^{\frac {7}{2}} a}{7}+\frac {2 b \left (d x \right )^{\frac {7}{2}} \arctanh \left (c \,x^{2}\right )}{7}+\frac {8 b \,d^{2} \left (d x \right )^{\frac {3}{2}}}{21 c}-\frac {b \,d^{4} \sqrt {2}\, \ln \left (\frac {d x -\left (\frac {d^{2}}{c}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {d^{2}}{c}}}{d x +\left (\frac {d^{2}}{c}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {d^{2}}{c}}}\right )}{14 c^{2} \left (\frac {d^{2}}{c}\right )^{\frac {1}{4}}}-\frac {b \,d^{4} \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {d^{2}}{c}\right )^{\frac {1}{4}}}+1\right )}{7 c^{2} \left (\frac {d^{2}}{c}\right )^{\frac {1}{4}}}-\frac {b \,d^{4} \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {d^{2}}{c}\right )^{\frac {1}{4}}}-1\right )}{7 c^{2} \left (\frac {d^{2}}{c}\right )^{\frac {1}{4}}}+\frac {2 b \,d^{4} \arctan \left (\frac {\sqrt {d x}}{\left (\frac {d^{2}}{c}\right )^{\frac {1}{4}}}\right )}{7 c^{2} \left (\frac {d^{2}}{c}\right )^{\frac {1}{4}}}-\frac {b \,d^{4} \ln \left (\frac {\sqrt {d x}+\left (\frac {d^{2}}{c}\right )^{\frac {1}{4}}}{\sqrt {d x}-\left (\frac {d^{2}}{c}\right )^{\frac {1}{4}}}\right )}{7 c^{2} \left (\frac {d^{2}}{c}\right )^{\frac {1}{4}}}}{d}\) | \(303\) |
default | \(\frac {\frac {2 \left (d x \right )^{\frac {7}{2}} a}{7}+\frac {2 b \left (d x \right )^{\frac {7}{2}} \arctanh \left (c \,x^{2}\right )}{7}+\frac {8 b \,d^{2} \left (d x \right )^{\frac {3}{2}}}{21 c}-\frac {b \,d^{4} \sqrt {2}\, \ln \left (\frac {d x -\left (\frac {d^{2}}{c}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {d^{2}}{c}}}{d x +\left (\frac {d^{2}}{c}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {d^{2}}{c}}}\right )}{14 c^{2} \left (\frac {d^{2}}{c}\right )^{\frac {1}{4}}}-\frac {b \,d^{4} \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {d^{2}}{c}\right )^{\frac {1}{4}}}+1\right )}{7 c^{2} \left (\frac {d^{2}}{c}\right )^{\frac {1}{4}}}-\frac {b \,d^{4} \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {d^{2}}{c}\right )^{\frac {1}{4}}}-1\right )}{7 c^{2} \left (\frac {d^{2}}{c}\right )^{\frac {1}{4}}}+\frac {2 b \,d^{4} \arctan \left (\frac {\sqrt {d x}}{\left (\frac {d^{2}}{c}\right )^{\frac {1}{4}}}\right )}{7 c^{2} \left (\frac {d^{2}}{c}\right )^{\frac {1}{4}}}-\frac {b \,d^{4} \ln \left (\frac {\sqrt {d x}+\left (\frac {d^{2}}{c}\right )^{\frac {1}{4}}}{\sqrt {d x}-\left (\frac {d^{2}}{c}\right )^{\frac {1}{4}}}\right )}{7 c^{2} \left (\frac {d^{2}}{c}\right )^{\frac {1}{4}}}}{d}\) | \(303\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.47, size = 316, normalized size = 1.00 \begin {gather*} \frac {12 \, \left (d x\right )^{\frac {7}{2}} a + {\left (12 \, \left (d x\right )^{\frac {7}{2}} \operatorname {artanh}\left (c x^{2}\right ) - \frac {{\left (\frac {3 \, d^{6} {\left (\frac {2 \, \sqrt {2} \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} \sqrt {c}} + \frac {2 \, \sqrt {2} \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} \sqrt {c}} - \frac {\sqrt {2} \log \left (\sqrt {c} d x + \sqrt {2} \sqrt {d x} c^{\frac {1}{4}} \sqrt {d} + d\right )}{c^{\frac {3}{4}} \sqrt {d}} + \frac {\sqrt {2} \log \left (\sqrt {c} d x - \sqrt {2} \sqrt {d x} c^{\frac {1}{4}} \sqrt {d} + d\right )}{c^{\frac {3}{4}} \sqrt {d}}\right )}}{c^{2}} - \frac {6 \, d^{6} {\left (\frac {2 \, \arctan \left (\frac {\sqrt {d x} \sqrt {c}}{\sqrt {\sqrt {c} d}}\right )}{\sqrt {\sqrt {c} d} \sqrt {c}} + \frac {\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} \sqrt {c}}\right )}}{c^{2}} - \frac {16 \, \left (d x\right )^{\frac {3}{2}} d^{4}}{c^{2}}\right )} c}{d^{2}}\right )} b}{42 \, 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. 455 vs.
\(2 (212) = 424\).
time = 0.40, size = 455, normalized size = 1.44 \begin {gather*} -\frac {12 \, \left (\frac {b^{4} d^{10}}{c^{7}}\right )^{\frac {1}{4}} c \arctan \left (-\frac {\left (\frac {b^{4} d^{10}}{c^{7}}\right )^{\frac {1}{4}} \sqrt {d x} b^{3} c^{2} d^{7} - \sqrt {b^{6} d^{15} x + \sqrt {\frac {b^{4} d^{10}}{c^{7}}} b^{4} c^{3} d^{10}} \left (\frac {b^{4} d^{10}}{c^{7}}\right )^{\frac {1}{4}} c^{2}}{b^{4} d^{10}}\right ) - 12 \, \left (-\frac {b^{4} d^{10}}{c^{7}}\right )^{\frac {1}{4}} c \arctan \left (-\frac {\left (-\frac {b^{4} d^{10}}{c^{7}}\right )^{\frac {1}{4}} \sqrt {d x} b^{3} c^{2} d^{7} - \sqrt {b^{6} d^{15} x - \sqrt {-\frac {b^{4} d^{10}}{c^{7}}} b^{4} c^{3} d^{10}} \left (-\frac {b^{4} d^{10}}{c^{7}}\right )^{\frac {1}{4}} c^{2}}{b^{4} d^{10}}\right ) + 3 \, \left (\frac {b^{4} d^{10}}{c^{7}}\right )^{\frac {1}{4}} c \log \left (\sqrt {d x} b^{3} d^{7} + \left (\frac {b^{4} d^{10}}{c^{7}}\right )^{\frac {3}{4}} c^{5}\right ) - 3 \, \left (\frac {b^{4} d^{10}}{c^{7}}\right )^{\frac {1}{4}} c \log \left (\sqrt {d x} b^{3} d^{7} - \left (\frac {b^{4} d^{10}}{c^{7}}\right )^{\frac {3}{4}} c^{5}\right ) + 3 \, \left (-\frac {b^{4} d^{10}}{c^{7}}\right )^{\frac {1}{4}} c \log \left (\sqrt {d x} b^{3} d^{7} + \left (-\frac {b^{4} d^{10}}{c^{7}}\right )^{\frac {3}{4}} c^{5}\right ) - 3 \, \left (-\frac {b^{4} d^{10}}{c^{7}}\right )^{\frac {1}{4}} c \log \left (\sqrt {d x} b^{3} d^{7} - \left (-\frac {b^{4} d^{10}}{c^{7}}\right )^{\frac {3}{4}} c^{5}\right ) - {\left (3 \, b c d^{2} x^{3} \log \left (-\frac {c x^{2} + 1}{c x^{2} - 1}\right ) + 6 \, a c d^{2} x^{3} + 8 \, b d^{2} x\right )} \sqrt {d x}}{21 \, c} \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 {5}{2}} \left (a + b \operatorname {atanh}{\left (c x^{2} \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 = -0.00 \begin {gather*} \int {\left (d\,x\right )}^{5/2}\,\left (a+b\,\mathrm {atanh}\left (c\,x^2\right )\right ) \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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