Optimal. Leaf size=111 \[ \frac {1}{8} a^4 c^2 x^8 \tan ^{-1}(a x)-\frac {c^2 \tan ^{-1}(a x)}{24 a^4}-\frac {1}{56} a^3 c^2 x^7+\frac {c^2 x}{24 a^3}+\frac {1}{3} a^2 c^2 x^6 \tan ^{-1}(a x)-\frac {1}{24} a c^2 x^5+\frac {1}{4} c^2 x^4 \tan ^{-1}(a x)-\frac {c^2 x^3}{72 a} \]
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Rubi [A] time = 0.16, antiderivative size = 111, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 4, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {4948, 4852, 302, 203} \[ -\frac {1}{56} a^3 c^2 x^7+\frac {1}{8} a^4 c^2 x^8 \tan ^{-1}(a x)+\frac {1}{3} a^2 c^2 x^6 \tan ^{-1}(a x)+\frac {c^2 x}{24 a^3}-\frac {c^2 \tan ^{-1}(a x)}{24 a^4}-\frac {1}{24} a c^2 x^5-\frac {c^2 x^3}{72 a}+\frac {1}{4} c^2 x^4 \tan ^{-1}(a x) \]
Antiderivative was successfully verified.
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Rule 203
Rule 302
Rule 4852
Rule 4948
Rubi steps
\begin {align*} \int x^3 \left (c+a^2 c x^2\right )^2 \tan ^{-1}(a x) \, dx &=\int \left (c^2 x^3 \tan ^{-1}(a x)+2 a^2 c^2 x^5 \tan ^{-1}(a x)+a^4 c^2 x^7 \tan ^{-1}(a x)\right ) \, dx\\ &=c^2 \int x^3 \tan ^{-1}(a x) \, dx+\left (2 a^2 c^2\right ) \int x^5 \tan ^{-1}(a x) \, dx+\left (a^4 c^2\right ) \int x^7 \tan ^{-1}(a x) \, dx\\ &=\frac {1}{4} c^2 x^4 \tan ^{-1}(a x)+\frac {1}{3} a^2 c^2 x^6 \tan ^{-1}(a x)+\frac {1}{8} a^4 c^2 x^8 \tan ^{-1}(a x)-\frac {1}{4} \left (a c^2\right ) \int \frac {x^4}{1+a^2 x^2} \, dx-\frac {1}{3} \left (a^3 c^2\right ) \int \frac {x^6}{1+a^2 x^2} \, dx-\frac {1}{8} \left (a^5 c^2\right ) \int \frac {x^8}{1+a^2 x^2} \, dx\\ &=\frac {1}{4} c^2 x^4 \tan ^{-1}(a x)+\frac {1}{3} a^2 c^2 x^6 \tan ^{-1}(a x)+\frac {1}{8} a^4 c^2 x^8 \tan ^{-1}(a x)-\frac {1}{4} \left (a c^2\right ) \int \left (-\frac {1}{a^4}+\frac {x^2}{a^2}+\frac {1}{a^4 \left (1+a^2 x^2\right )}\right ) \, dx-\frac {1}{3} \left (a^3 c^2\right ) \int \left (\frac {1}{a^6}-\frac {x^2}{a^4}+\frac {x^4}{a^2}-\frac {1}{a^6 \left (1+a^2 x^2\right )}\right ) \, dx-\frac {1}{8} \left (a^5 c^2\right ) \int \left (-\frac {1}{a^8}+\frac {x^2}{a^6}-\frac {x^4}{a^4}+\frac {x^6}{a^2}+\frac {1}{a^8 \left (1+a^2 x^2\right )}\right ) \, dx\\ &=\frac {c^2 x}{24 a^3}-\frac {c^2 x^3}{72 a}-\frac {1}{24} a c^2 x^5-\frac {1}{56} a^3 c^2 x^7+\frac {1}{4} c^2 x^4 \tan ^{-1}(a x)+\frac {1}{3} a^2 c^2 x^6 \tan ^{-1}(a x)+\frac {1}{8} a^4 c^2 x^8 \tan ^{-1}(a x)-\frac {c^2 \int \frac {1}{1+a^2 x^2} \, dx}{8 a^3}-\frac {c^2 \int \frac {1}{1+a^2 x^2} \, dx}{4 a^3}+\frac {c^2 \int \frac {1}{1+a^2 x^2} \, dx}{3 a^3}\\ &=\frac {c^2 x}{24 a^3}-\frac {c^2 x^3}{72 a}-\frac {1}{24} a c^2 x^5-\frac {1}{56} a^3 c^2 x^7-\frac {c^2 \tan ^{-1}(a x)}{24 a^4}+\frac {1}{4} c^2 x^4 \tan ^{-1}(a x)+\frac {1}{3} a^2 c^2 x^6 \tan ^{-1}(a x)+\frac {1}{8} a^4 c^2 x^8 \tan ^{-1}(a x)\\ \end {align*}
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Mathematica [A] time = 0.09, size = 111, normalized size = 1.00 \[ \frac {1}{8} a^4 c^2 x^8 \tan ^{-1}(a x)-\frac {c^2 \tan ^{-1}(a x)}{24 a^4}-\frac {1}{56} a^3 c^2 x^7+\frac {c^2 x}{24 a^3}+\frac {1}{3} a^2 c^2 x^6 \tan ^{-1}(a x)-\frac {1}{24} a c^2 x^5+\frac {1}{4} c^2 x^4 \tan ^{-1}(a x)-\frac {c^2 x^3}{72 a} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.47, size = 91, normalized size = 0.82 \[ -\frac {9 \, a^{7} c^{2} x^{7} + 21 \, a^{5} c^{2} x^{5} + 7 \, a^{3} c^{2} x^{3} - 21 \, a c^{2} x - 21 \, {\left (3 \, a^{8} c^{2} x^{8} + 8 \, a^{6} c^{2} x^{6} + 6 \, a^{4} c^{2} x^{4} - c^{2}\right )} \arctan \left (a x\right )}{504 \, a^{4}} \]
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 \[ \mathit {sage}_{0} x \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 96, normalized size = 0.86 \[ \frac {c^{2} x}{24 a^{3}}-\frac {c^{2} x^{3}}{72 a}-\frac {a \,c^{2} x^{5}}{24}-\frac {a^{3} c^{2} x^{7}}{56}-\frac {c^{2} \arctan \left (a x \right )}{24 a^{4}}+\frac {c^{2} x^{4} \arctan \left (a x \right )}{4}+\frac {a^{2} c^{2} x^{6} \arctan \left (a x \right )}{3}+\frac {a^{4} c^{2} x^{8} \arctan \left (a x \right )}{8} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.44, size = 98, normalized size = 0.88 \[ -\frac {1}{504} \, a {\left (\frac {21 \, c^{2} \arctan \left (a x\right )}{a^{5}} + \frac {9 \, a^{6} c^{2} x^{7} + 21 \, a^{4} c^{2} x^{5} + 7 \, a^{2} c^{2} x^{3} - 21 \, c^{2} x}{a^{4}}\right )} + \frac {1}{24} \, {\left (3 \, a^{4} c^{2} x^{8} + 8 \, a^{2} c^{2} x^{6} + 6 \, c^{2} x^{4}\right )} \arctan \left (a x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.42, size = 89, normalized size = 0.80 \[ \mathrm {atan}\left (a\,x\right )\,\left (\frac {a^4\,c^2\,x^8}{8}+\frac {a^2\,c^2\,x^6}{3}+\frac {c^2\,x^4}{4}\right )+\frac {c^2\,x}{24\,a^3}-\frac {a\,c^2\,x^5}{24}-\frac {c^2\,\mathrm {atan}\left (a\,x\right )}{24\,a^4}-\frac {c^2\,x^3}{72\,a}-\frac {a^3\,c^2\,x^7}{56} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 2.50, size = 104, normalized size = 0.94 \[ \begin {cases} \frac {a^{4} c^{2} x^{8} \operatorname {atan}{\left (a x \right )}}{8} - \frac {a^{3} c^{2} x^{7}}{56} + \frac {a^{2} c^{2} x^{6} \operatorname {atan}{\left (a x \right )}}{3} - \frac {a c^{2} x^{5}}{24} + \frac {c^{2} x^{4} \operatorname {atan}{\left (a x \right )}}{4} - \frac {c^{2} x^{3}}{72 a} + \frac {c^{2} x}{24 a^{3}} - \frac {c^{2} \operatorname {atan}{\left (a x \right )}}{24 a^{4}} & \text {for}\: a \neq 0 \\0 & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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