Optimal. Leaf size=136 \[ \frac {1}{9} a^6 c^3 x^9 \tan ^{-1}(a x)-\frac {1}{72} a^5 c^3 x^8+\frac {3}{7} a^4 c^3 x^7 \tan ^{-1}(a x)-\frac {10}{189} a^3 c^3 x^6+\frac {3}{5} a^2 c^3 x^5 \tan ^{-1}(a x)+\frac {8 c^3 \log \left (a^2 x^2+1\right )}{315 a^3}-\frac {89 a c^3 x^4}{1260}+\frac {1}{3} c^3 x^3 \tan ^{-1}(a x)-\frac {8 c^3 x^2}{315 a} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.23, antiderivative size = 136, normalized size of antiderivative = 1.00, number of steps used = 18, number of rules used = 4, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {4948, 4852, 266, 43} \[ -\frac {1}{72} a^5 c^3 x^8-\frac {10}{189} a^3 c^3 x^6+\frac {8 c^3 \log \left (a^2 x^2+1\right )}{315 a^3}+\frac {1}{9} a^6 c^3 x^9 \tan ^{-1}(a x)+\frac {3}{7} a^4 c^3 x^7 \tan ^{-1}(a x)+\frac {3}{5} a^2 c^3 x^5 \tan ^{-1}(a x)-\frac {89 a c^3 x^4}{1260}-\frac {8 c^3 x^2}{315 a}+\frac {1}{3} c^3 x^3 \tan ^{-1}(a x) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 43
Rule 266
Rule 4852
Rule 4948
Rubi steps
\begin {align*} \int x^2 \left (c+a^2 c x^2\right )^3 \tan ^{-1}(a x) \, dx &=\int \left (c^3 x^2 \tan ^{-1}(a x)+3 a^2 c^3 x^4 \tan ^{-1}(a x)+3 a^4 c^3 x^6 \tan ^{-1}(a x)+a^6 c^3 x^8 \tan ^{-1}(a x)\right ) \, dx\\ &=c^3 \int x^2 \tan ^{-1}(a x) \, dx+\left (3 a^2 c^3\right ) \int x^4 \tan ^{-1}(a x) \, dx+\left (3 a^4 c^3\right ) \int x^6 \tan ^{-1}(a x) \, dx+\left (a^6 c^3\right ) \int x^8 \tan ^{-1}(a x) \, dx\\ &=\frac {1}{3} c^3 x^3 \tan ^{-1}(a x)+\frac {3}{5} a^2 c^3 x^5 \tan ^{-1}(a x)+\frac {3}{7} a^4 c^3 x^7 \tan ^{-1}(a x)+\frac {1}{9} a^6 c^3 x^9 \tan ^{-1}(a x)-\frac {1}{3} \left (a c^3\right ) \int \frac {x^3}{1+a^2 x^2} \, dx-\frac {1}{5} \left (3 a^3 c^3\right ) \int \frac {x^5}{1+a^2 x^2} \, dx-\frac {1}{7} \left (3 a^5 c^3\right ) \int \frac {x^7}{1+a^2 x^2} \, dx-\frac {1}{9} \left (a^7 c^3\right ) \int \frac {x^9}{1+a^2 x^2} \, dx\\ &=\frac {1}{3} c^3 x^3 \tan ^{-1}(a x)+\frac {3}{5} a^2 c^3 x^5 \tan ^{-1}(a x)+\frac {3}{7} a^4 c^3 x^7 \tan ^{-1}(a x)+\frac {1}{9} a^6 c^3 x^9 \tan ^{-1}(a x)-\frac {1}{6} \left (a c^3\right ) \operatorname {Subst}\left (\int \frac {x}{1+a^2 x} \, dx,x,x^2\right )-\frac {1}{10} \left (3 a^3 c^3\right ) \operatorname {Subst}\left (\int \frac {x^2}{1+a^2 x} \, dx,x,x^2\right )-\frac {1}{14} \left (3 a^5 c^3\right ) \operatorname {Subst}\left (\int \frac {x^3}{1+a^2 x} \, dx,x,x^2\right )-\frac {1}{18} \left (a^7 c^3\right ) \operatorname {Subst}\left (\int \frac {x^4}{1+a^2 x} \, dx,x,x^2\right )\\ &=\frac {1}{3} c^3 x^3 \tan ^{-1}(a x)+\frac {3}{5} a^2 c^3 x^5 \tan ^{-1}(a x)+\frac {3}{7} a^4 c^3 x^7 \tan ^{-1}(a x)+\frac {1}{9} a^6 c^3 x^9 \tan ^{-1}(a x)-\frac {1}{6} \left (a c^3\right ) \operatorname {Subst}\left (\int \left (\frac {1}{a^2}-\frac {1}{a^2 \left (1+a^2 x\right )}\right ) \, dx,x,x^2\right )-\frac {1}{10} \left (3 a^3 c^3\right ) \operatorname {Subst}\left (\int \left (-\frac {1}{a^4}+\frac {x}{a^2}+\frac {1}{a^4 \left (1+a^2 x\right )}\right ) \, dx,x,x^2\right )-\frac {1}{14} \left (3 a^5 c^3\right ) \operatorname {Subst}\left (\int \left (\frac {1}{a^6}-\frac {x}{a^4}+\frac {x^2}{a^2}-\frac {1}{a^6 \left (1+a^2 x\right )}\right ) \, dx,x,x^2\right )-\frac {1}{18} \left (a^7 c^3\right ) \operatorname {Subst}\left (\int \left (-\frac {1}{a^8}+\frac {x}{a^6}-\frac {x^2}{a^4}+\frac {x^3}{a^2}+\frac {1}{a^8 \left (1+a^2 x\right )}\right ) \, dx,x,x^2\right )\\ &=-\frac {8 c^3 x^2}{315 a}-\frac {89 a c^3 x^4}{1260}-\frac {10}{189} a^3 c^3 x^6-\frac {1}{72} a^5 c^3 x^8+\frac {1}{3} c^3 x^3 \tan ^{-1}(a x)+\frac {3}{5} a^2 c^3 x^5 \tan ^{-1}(a x)+\frac {3}{7} a^4 c^3 x^7 \tan ^{-1}(a x)+\frac {1}{9} a^6 c^3 x^9 \tan ^{-1}(a x)+\frac {8 c^3 \log \left (1+a^2 x^2\right )}{315 a^3}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.09, size = 136, normalized size = 1.00 \[ \frac {1}{9} a^6 c^3 x^9 \tan ^{-1}(a x)-\frac {1}{72} a^5 c^3 x^8+\frac {3}{7} a^4 c^3 x^7 \tan ^{-1}(a x)-\frac {10}{189} a^3 c^3 x^6+\frac {3}{5} a^2 c^3 x^5 \tan ^{-1}(a x)+\frac {8 c^3 \log \left (a^2 x^2+1\right )}{315 a^3}-\frac {89 a c^3 x^4}{1260}+\frac {1}{3} c^3 x^3 \tan ^{-1}(a x)-\frac {8 c^3 x^2}{315 a} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.50, size = 116, normalized size = 0.85 \[ -\frac {105 \, a^{8} c^{3} x^{8} + 400 \, a^{6} c^{3} x^{6} + 534 \, a^{4} c^{3} x^{4} + 192 \, a^{2} c^{3} x^{2} - 192 \, c^{3} \log \left (a^{2} x^{2} + 1\right ) - 24 \, {\left (35 \, a^{9} c^{3} x^{9} + 135 \, a^{7} c^{3} x^{7} + 189 \, a^{5} c^{3} x^{5} + 105 \, a^{3} c^{3} x^{3}\right )} \arctan \left (a x\right )}{7560 \, a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.03, size = 119, normalized size = 0.88 \[ -\frac {8 c^{3} x^{2}}{315 a}-\frac {89 a \,c^{3} x^{4}}{1260}-\frac {10 a^{3} c^{3} x^{6}}{189}-\frac {a^{5} c^{3} x^{8}}{72}+\frac {c^{3} x^{3} \arctan \left (a x \right )}{3}+\frac {3 a^{2} c^{3} x^{5} \arctan \left (a x \right )}{5}+\frac {3 a^{4} c^{3} x^{7} \arctan \left (a x \right )}{7}+\frac {a^{6} c^{3} x^{9} \arctan \left (a x \right )}{9}+\frac {8 c^{3} \ln \left (a^{2} x^{2}+1\right )}{315 a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.32, size = 118, normalized size = 0.87 \[ \frac {1}{7560} \, a {\left (\frac {192 \, c^{3} \log \left (a^{2} x^{2} + 1\right )}{a^{4}} - \frac {105 \, a^{6} c^{3} x^{8} + 400 \, a^{4} c^{3} x^{6} + 534 \, a^{2} c^{3} x^{4} + 192 \, c^{3} x^{2}}{a^{2}}\right )} + \frac {1}{315} \, {\left (35 \, a^{6} c^{3} x^{9} + 135 \, a^{4} c^{3} x^{7} + 189 \, a^{2} c^{3} x^{5} + 105 \, c^{3} x^{3}\right )} \arctan \left (a x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.44, size = 108, normalized size = 0.79 \[ \mathrm {atan}\left (a\,x\right )\,\left (\frac {a^6\,c^3\,x^9}{9}+\frac {3\,a^4\,c^3\,x^7}{7}+\frac {3\,a^2\,c^3\,x^5}{5}+\frac {c^3\,x^3}{3}\right )-\frac {89\,a\,c^3\,x^4}{1260}+\frac {8\,c^3\,\ln \left (a^2\,x^2+1\right )}{315\,a^3}-\frac {8\,c^3\,x^2}{315\,a}-\frac {10\,a^3\,c^3\,x^6}{189}-\frac {a^5\,c^3\,x^8}{72} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 2.97, size = 138, normalized size = 1.01 \[ \begin {cases} \frac {a^{6} c^{3} x^{9} \operatorname {atan}{\left (a x \right )}}{9} - \frac {a^{5} c^{3} x^{8}}{72} + \frac {3 a^{4} c^{3} x^{7} \operatorname {atan}{\left (a x \right )}}{7} - \frac {10 a^{3} c^{3} x^{6}}{189} + \frac {3 a^{2} c^{3} x^{5} \operatorname {atan}{\left (a x \right )}}{5} - \frac {89 a c^{3} x^{4}}{1260} + \frac {c^{3} x^{3} \operatorname {atan}{\left (a x \right )}}{3} - \frac {8 c^{3} x^{2}}{315 a} + \frac {8 c^{3} \log {\left (x^{2} + \frac {1}{a^{2}} \right )}}{315 a^{3}} & \text {for}\: a \neq 0 \\0 & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________