Optimal. Leaf size=208 \[ -\frac {5 a^2 \sqrt {c+a^2 c x^2} \sqrt {\text {ArcTan}(a x)}}{8 c x}-\frac {5 a \sqrt {c+a^2 c x^2} \text {ArcTan}(a x)^{3/2}}{12 c x^2}-\frac {\sqrt {c+a^2 c x^2} \text {ArcTan}(a x)^{5/2}}{3 c x^3}+\frac {2 a^2 \sqrt {c+a^2 c x^2} \text {ArcTan}(a x)^{5/2}}{3 c x}+\frac {5}{16} a^3 \text {Int}\left (\frac {1}{x \sqrt {c+a^2 c x^2} \sqrt {\text {ArcTan}(a x)}},x\right )-\frac {25}{12} a^3 \text {Int}\left (\frac {\text {ArcTan}(a x)^{3/2}}{x \sqrt {c+a^2 c x^2}},x\right ) \]
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Rubi [A]
time = 0.48, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps
used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {}
\begin {gather*} \int \frac {\text {ArcTan}(a x)^{5/2}}{x^4 \sqrt {c+a^2 c x^2}} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {\tan ^{-1}(a x)^{5/2}}{x^4 \sqrt {c+a^2 c x^2}} \, dx &=-\frac {\sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^{5/2}}{3 c x^3}+\frac {1}{6} (5 a) \int \frac {\tan ^{-1}(a x)^{3/2}}{x^3 \sqrt {c+a^2 c x^2}} \, dx-\frac {1}{3} \left (2 a^2\right ) \int \frac {\tan ^{-1}(a x)^{5/2}}{x^2 \sqrt {c+a^2 c x^2}} \, dx\\ &=-\frac {5 a \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^{3/2}}{12 c x^2}-\frac {\sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^{5/2}}{3 c x^3}+\frac {2 a^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^{5/2}}{3 c x}+\frac {1}{8} \left (5 a^2\right ) \int \frac {\sqrt {\tan ^{-1}(a x)}}{x^2 \sqrt {c+a^2 c x^2}} \, dx-\frac {1}{12} \left (5 a^3\right ) \int \frac {\tan ^{-1}(a x)^{3/2}}{x \sqrt {c+a^2 c x^2}} \, dx-\frac {1}{3} \left (5 a^3\right ) \int \frac {\tan ^{-1}(a x)^{3/2}}{x \sqrt {c+a^2 c x^2}} \, dx\\ &=-\frac {5 a^2 \sqrt {c+a^2 c x^2} \sqrt {\tan ^{-1}(a x)}}{8 c x}-\frac {5 a \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^{3/2}}{12 c x^2}-\frac {\sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^{5/2}}{3 c x^3}+\frac {2 a^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^{5/2}}{3 c x}+\frac {1}{16} \left (5 a^3\right ) \int \frac {1}{x \sqrt {c+a^2 c x^2} \sqrt {\tan ^{-1}(a x)}} \, dx-\frac {1}{12} \left (5 a^3\right ) \int \frac {\tan ^{-1}(a x)^{3/2}}{x \sqrt {c+a^2 c x^2}} \, dx-\frac {1}{3} \left (5 a^3\right ) \int \frac {\tan ^{-1}(a x)^{3/2}}{x \sqrt {c+a^2 c x^2}} \, dx\\ \end {align*}
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Mathematica [A]
time = 11.88, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\text {ArcTan}(a x)^{5/2}}{x^4 \sqrt {c+a^2 c x^2}} \, dx \end {gather*}
Verification is not applicable to the result.
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Maple [A]
time = 5.42, size = 0, normalized size = 0.00 \[\int \frac {\arctan \left (a x \right )^{\frac {5}{2}}}{x^{4} \sqrt {a^{2} c \,x^{2}+c}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: RuntimeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
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 [A]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\mathrm {atan}\left (a\,x\right )}^{5/2}}{x^4\,\sqrt {c\,a^2\,x^2+c}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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