Optimal. Leaf size=170 \[ -\frac {\tanh ^{-1}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{a^6 c^{5/2}}+\frac {\sqrt {a^2 c x^2+c} \tan ^{-1}(a x)}{a^6 c^3}+\frac {5 \tan ^{-1}(a x)}{3 a^6 c^2 \sqrt {a^2 c x^2+c}}-\frac {5 x}{3 a^5 c^2 \sqrt {a^2 c x^2+c}}+\frac {x^2 \tan ^{-1}(a x)}{3 a^4 c \left (a^2 c x^2+c\right )^{3/2}}-\frac {x^3}{9 a^3 c \left (a^2 c x^2+c\right )^{3/2}} \]
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Rubi [A] time = 0.43, antiderivative size = 170, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 6, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {4964, 4930, 217, 206, 191, 4938} \[ -\frac {5 x}{3 a^5 c^2 \sqrt {a^2 c x^2+c}}+\frac {\sqrt {a^2 c x^2+c} \tan ^{-1}(a x)}{a^6 c^3}+\frac {5 \tan ^{-1}(a x)}{3 a^6 c^2 \sqrt {a^2 c x^2+c}}-\frac {\tanh ^{-1}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{a^6 c^{5/2}}-\frac {x^3}{9 a^3 c \left (a^2 c x^2+c\right )^{3/2}}+\frac {x^2 \tan ^{-1}(a x)}{3 a^4 c \left (a^2 c x^2+c\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 191
Rule 206
Rule 217
Rule 4930
Rule 4938
Rule 4964
Rubi steps
\begin {align*} \int \frac {x^5 \tan ^{-1}(a x)}{\left (c+a^2 c x^2\right )^{5/2}} \, dx &=-\frac {\int \frac {x^3 \tan ^{-1}(a x)}{\left (c+a^2 c x^2\right )^{5/2}} \, dx}{a^2}+\frac {\int \frac {x^3 \tan ^{-1}(a x)}{\left (c+a^2 c x^2\right )^{3/2}} \, dx}{a^2 c}\\ &=-\frac {x^3}{9 a^3 c \left (c+a^2 c x^2\right )^{3/2}}+\frac {x^2 \tan ^{-1}(a x)}{3 a^4 c \left (c+a^2 c x^2\right )^{3/2}}+\frac {\int \frac {x \tan ^{-1}(a x)}{\sqrt {c+a^2 c x^2}} \, dx}{a^4 c^2}-\frac {2 \int \frac {x \tan ^{-1}(a x)}{\left (c+a^2 c x^2\right )^{3/2}} \, dx}{3 a^4 c}-\frac {\int \frac {x \tan ^{-1}(a x)}{\left (c+a^2 c x^2\right )^{3/2}} \, dx}{a^4 c}\\ &=-\frac {x^3}{9 a^3 c \left (c+a^2 c x^2\right )^{3/2}}+\frac {x^2 \tan ^{-1}(a x)}{3 a^4 c \left (c+a^2 c x^2\right )^{3/2}}+\frac {5 \tan ^{-1}(a x)}{3 a^6 c^2 \sqrt {c+a^2 c x^2}}+\frac {\sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{a^6 c^3}-\frac {\int \frac {1}{\sqrt {c+a^2 c x^2}} \, dx}{a^5 c^2}-\frac {2 \int \frac {1}{\left (c+a^2 c x^2\right )^{3/2}} \, dx}{3 a^5 c}-\frac {\int \frac {1}{\left (c+a^2 c x^2\right )^{3/2}} \, dx}{a^5 c}\\ &=-\frac {x^3}{9 a^3 c \left (c+a^2 c x^2\right )^{3/2}}-\frac {5 x}{3 a^5 c^2 \sqrt {c+a^2 c x^2}}+\frac {x^2 \tan ^{-1}(a x)}{3 a^4 c \left (c+a^2 c x^2\right )^{3/2}}+\frac {5 \tan ^{-1}(a x)}{3 a^6 c^2 \sqrt {c+a^2 c x^2}}+\frac {\sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{a^6 c^3}-\frac {\operatorname {Subst}\left (\int \frac {1}{1-a^2 c x^2} \, dx,x,\frac {x}{\sqrt {c+a^2 c x^2}}\right )}{a^5 c^2}\\ &=-\frac {x^3}{9 a^3 c \left (c+a^2 c x^2\right )^{3/2}}-\frac {5 x}{3 a^5 c^2 \sqrt {c+a^2 c x^2}}+\frac {x^2 \tan ^{-1}(a x)}{3 a^4 c \left (c+a^2 c x^2\right )^{3/2}}+\frac {5 \tan ^{-1}(a x)}{3 a^6 c^2 \sqrt {c+a^2 c x^2}}+\frac {\sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{a^6 c^3}-\frac {\tanh ^{-1}\left (\frac {a \sqrt {c} x}{\sqrt {c+a^2 c x^2}}\right )}{a^6 c^{5/2}}\\ \end {align*}
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Mathematica [A] time = 0.21, size = 131, normalized size = 0.77 \[ -\frac {a x \left (16 a^2 x^2+15\right ) \sqrt {a^2 c x^2+c}+9 \sqrt {c} \left (a^2 x^2+1\right )^2 \log \left (\sqrt {c} \sqrt {a^2 c x^2+c}+a c x\right )-3 \left (3 a^4 x^4+12 a^2 x^2+8\right ) \sqrt {a^2 c x^2+c} \tan ^{-1}(a x)}{9 a^6 c^3 \left (a^2 x^2+1\right )^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.53, size = 140, normalized size = 0.82 \[ \frac {9 \, {\left (a^{4} x^{4} + 2 \, a^{2} x^{2} + 1\right )} \sqrt {c} \log \left (-2 \, a^{2} c x^{2} + 2 \, \sqrt {a^{2} c x^{2} + c} a \sqrt {c} x - c\right ) - 2 \, {\left (16 \, a^{3} x^{3} + 15 \, a x - 3 \, {\left (3 \, a^{4} x^{4} + 12 \, a^{2} x^{2} + 8\right )} \arctan \left (a x\right )\right )} \sqrt {a^{2} c x^{2} + c}}{18 \, {\left (a^{10} c^{3} x^{4} + 2 \, a^{8} c^{3} x^{2} + a^{6} c^{3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 5.82, size = 386, normalized size = 2.27 \[ \frac {\left (i+3 \arctan \left (a x \right )\right ) \left (i x^{3} a^{3}+3 a^{2} x^{2}-3 i a x -1\right ) \sqrt {c \left (a x -i\right ) \left (a x +i\right )}}{72 \left (a^{2} x^{2}+1\right )^{2} c^{3} a^{6}}+\frac {7 \left (i+\arctan \left (a x \right )\right ) \left (i a x +1\right ) \sqrt {c \left (a x -i\right ) \left (a x +i\right )}}{8 a^{6} c^{3} \left (a^{2} x^{2}+1\right )}-\frac {7 \sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \left (i a x -1\right ) \left (\arctan \left (a x \right )-i\right )}{8 a^{6} c^{3} \left (a^{2} x^{2}+1\right )}-\frac {\sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \left (i x^{3} a^{3}-3 a^{2} x^{2}-3 i a x +1\right ) \left (-i+3 \arctan \left (a x \right )\right )}{72 a^{6} c^{3} \left (a^{4} x^{4}+2 a^{2} x^{2}+1\right )}+\frac {\arctan \left (a x \right ) \sqrt {c \left (a x -i\right ) \left (a x +i\right )}}{c^{3} a^{6}}-\frac {\ln \left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}+i\right ) \sqrt {c \left (a x -i\right ) \left (a x +i\right )}}{\sqrt {a^{2} x^{2}+1}\, a^{6} c^{3}}+\frac {\ln \left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}-i\right ) \sqrt {c \left (a x -i\right ) \left (a x +i\right )}}{\sqrt {a^{2} x^{2}+1}\, a^{6} c^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{5} \arctan \left (a x\right )}{{\left (a^{2} c x^{2} + c\right )}^{\frac {5}{2}}}\,{d x} \]
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 \[ \int \frac {x^5\,\mathrm {atan}\left (a\,x\right )}{{\left (c\,a^2\,x^2+c\right )}^{5/2}} \,d x \]
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 \[ \int \frac {x^{5} \operatorname {atan}{\left (a x \right )}}{\left (c \left (a^{2} x^{2} + 1\right )\right )^{\frac {5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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