Optimal. Leaf size=159 \[ \frac {1}{3} x^3 \left (a+b \tan ^{-1}\left (c x^2\right )\right )-\frac {b \log \left (c x^2-\sqrt {2} \sqrt {c} x+1\right )}{6 \sqrt {2} c^{3/2}}+\frac {b \log \left (c x^2+\sqrt {2} \sqrt {c} x+1\right )}{6 \sqrt {2} c^{3/2}}-\frac {b \tan ^{-1}\left (1-\sqrt {2} \sqrt {c} x\right )}{3 \sqrt {2} c^{3/2}}+\frac {b \tan ^{-1}\left (\sqrt {2} \sqrt {c} x+1\right )}{3 \sqrt {2} c^{3/2}}-\frac {2 b x}{3 c} \]
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Rubi [A] time = 0.10, antiderivative size = 159, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 8, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.571, Rules used = {5033, 321, 211, 1165, 628, 1162, 617, 204} \[ \frac {1}{3} x^3 \left (a+b \tan ^{-1}\left (c x^2\right )\right )-\frac {b \log \left (c x^2-\sqrt {2} \sqrt {c} x+1\right )}{6 \sqrt {2} c^{3/2}}+\frac {b \log \left (c x^2+\sqrt {2} \sqrt {c} x+1\right )}{6 \sqrt {2} c^{3/2}}-\frac {b \tan ^{-1}\left (1-\sqrt {2} \sqrt {c} x\right )}{3 \sqrt {2} c^{3/2}}+\frac {b \tan ^{-1}\left (\sqrt {2} \sqrt {c} x+1\right )}{3 \sqrt {2} c^{3/2}}-\frac {2 b x}{3 c} \]
Antiderivative was successfully verified.
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Rule 204
Rule 211
Rule 321
Rule 617
Rule 628
Rule 1162
Rule 1165
Rule 5033
Rubi steps
\begin {align*} \int x^2 \left (a+b \tan ^{-1}\left (c x^2\right )\right ) \, dx &=\frac {1}{3} x^3 \left (a+b \tan ^{-1}\left (c x^2\right )\right )-\frac {1}{3} (2 b c) \int \frac {x^4}{1+c^2 x^4} \, dx\\ &=-\frac {2 b x}{3 c}+\frac {1}{3} x^3 \left (a+b \tan ^{-1}\left (c x^2\right )\right )+\frac {(2 b) \int \frac {1}{1+c^2 x^4} \, dx}{3 c}\\ &=-\frac {2 b x}{3 c}+\frac {1}{3} x^3 \left (a+b \tan ^{-1}\left (c x^2\right )\right )+\frac {b \int \frac {1-c x^2}{1+c^2 x^4} \, dx}{3 c}+\frac {b \int \frac {1+c x^2}{1+c^2 x^4} \, dx}{3 c}\\ &=-\frac {2 b x}{3 c}+\frac {1}{3} x^3 \left (a+b \tan ^{-1}\left (c x^2\right )\right )+\frac {b \int \frac {1}{\frac {1}{c}-\frac {\sqrt {2} x}{\sqrt {c}}+x^2} \, dx}{6 c^2}+\frac {b \int \frac {1}{\frac {1}{c}+\frac {\sqrt {2} x}{\sqrt {c}}+x^2} \, dx}{6 c^2}-\frac {b \int \frac {\frac {\sqrt {2}}{\sqrt {c}}+2 x}{-\frac {1}{c}-\frac {\sqrt {2} x}{\sqrt {c}}-x^2} \, dx}{6 \sqrt {2} c^{3/2}}-\frac {b \int \frac {\frac {\sqrt {2}}{\sqrt {c}}-2 x}{-\frac {1}{c}+\frac {\sqrt {2} x}{\sqrt {c}}-x^2} \, dx}{6 \sqrt {2} c^{3/2}}\\ &=-\frac {2 b x}{3 c}+\frac {1}{3} x^3 \left (a+b \tan ^{-1}\left (c x^2\right )\right )-\frac {b \log \left (1-\sqrt {2} \sqrt {c} x+c x^2\right )}{6 \sqrt {2} c^{3/2}}+\frac {b \log \left (1+\sqrt {2} \sqrt {c} x+c x^2\right )}{6 \sqrt {2} c^{3/2}}+\frac {b \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\sqrt {2} \sqrt {c} x\right )}{3 \sqrt {2} c^{3/2}}-\frac {b \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\sqrt {2} \sqrt {c} x\right )}{3 \sqrt {2} c^{3/2}}\\ &=-\frac {2 b x}{3 c}+\frac {1}{3} x^3 \left (a+b \tan ^{-1}\left (c x^2\right )\right )-\frac {b \tan ^{-1}\left (1-\sqrt {2} \sqrt {c} x\right )}{3 \sqrt {2} c^{3/2}}+\frac {b \tan ^{-1}\left (1+\sqrt {2} \sqrt {c} x\right )}{3 \sqrt {2} c^{3/2}}-\frac {b \log \left (1-\sqrt {2} \sqrt {c} x+c x^2\right )}{6 \sqrt {2} c^{3/2}}+\frac {b \log \left (1+\sqrt {2} \sqrt {c} x+c x^2\right )}{6 \sqrt {2} c^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 177, normalized size = 1.11 \[ \frac {a x^3}{3}-\frac {b \log \left (c x^2-\sqrt {2} \sqrt {c} x+1\right )}{6 \sqrt {2} c^{3/2}}+\frac {b \log \left (c x^2+\sqrt {2} \sqrt {c} x+1\right )}{6 \sqrt {2} c^{3/2}}+\frac {b \tan ^{-1}\left (\frac {2 \sqrt {c} x-\sqrt {2}}{\sqrt {2}}\right )}{3 \sqrt {2} c^{3/2}}+\frac {b \tan ^{-1}\left (\frac {2 \sqrt {c} x+\sqrt {2}}{\sqrt {2}}\right )}{3 \sqrt {2} c^{3/2}}+\frac {1}{3} b x^3 \tan ^{-1}\left (c x^2\right )-\frac {2 b x}{3 c} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.44, size = 337, normalized size = 2.12 \[ \frac {4 \, b c x^{3} \arctan \left (c x^{2}\right ) + 4 \, a c x^{3} - 4 \, \sqrt {2} c \left (\frac {b^{4}}{c^{6}}\right )^{\frac {1}{4}} \arctan \left (-\frac {\sqrt {2} b c^{5} x \left (\frac {b^{4}}{c^{6}}\right )^{\frac {3}{4}} - \sqrt {2} \sqrt {b^{2} x^{2} + \sqrt {2} b c x \left (\frac {b^{4}}{c^{6}}\right )^{\frac {1}{4}} + c^{2} \sqrt {\frac {b^{4}}{c^{6}}}} c^{5} \left (\frac {b^{4}}{c^{6}}\right )^{\frac {3}{4}} + b^{4}}{b^{4}}\right ) - 4 \, \sqrt {2} c \left (\frac {b^{4}}{c^{6}}\right )^{\frac {1}{4}} \arctan \left (-\frac {\sqrt {2} b c^{5} x \left (\frac {b^{4}}{c^{6}}\right )^{\frac {3}{4}} - \sqrt {2} \sqrt {b^{2} x^{2} - \sqrt {2} b c x \left (\frac {b^{4}}{c^{6}}\right )^{\frac {1}{4}} + c^{2} \sqrt {\frac {b^{4}}{c^{6}}}} c^{5} \left (\frac {b^{4}}{c^{6}}\right )^{\frac {3}{4}} - b^{4}}{b^{4}}\right ) + \sqrt {2} c \left (\frac {b^{4}}{c^{6}}\right )^{\frac {1}{4}} \log \left (b^{2} x^{2} + \sqrt {2} b c x \left (\frac {b^{4}}{c^{6}}\right )^{\frac {1}{4}} + c^{2} \sqrt {\frac {b^{4}}{c^{6}}}\right ) - \sqrt {2} c \left (\frac {b^{4}}{c^{6}}\right )^{\frac {1}{4}} \log \left (b^{2} x^{2} - \sqrt {2} b c x \left (\frac {b^{4}}{c^{6}}\right )^{\frac {1}{4}} + c^{2} \sqrt {\frac {b^{4}}{c^{6}}}\right ) - 8 \, b x}{12 \, c} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 3.07, size = 165, normalized size = 1.04 \[ \frac {1}{12} \, b c^{5} {\left (\frac {2 \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (2 \, x + \frac {\sqrt {2}}{\sqrt {{\left | c \right |}}}\right )} \sqrt {{\left | c \right |}}\right )}{c^{6} \sqrt {{\left | c \right |}}} + \frac {2 \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (2 \, x - \frac {\sqrt {2}}{\sqrt {{\left | c \right |}}}\right )} \sqrt {{\left | c \right |}}\right )}{c^{6} \sqrt {{\left | c \right |}}} + \frac {\sqrt {2} \log \left (x^{2} + \frac {\sqrt {2} x}{\sqrt {{\left | c \right |}}} + \frac {1}{{\left | c \right |}}\right )}{c^{6} \sqrt {{\left | c \right |}}} - \frac {\sqrt {2} \log \left (x^{2} - \frac {\sqrt {2} x}{\sqrt {{\left | c \right |}}} + \frac {1}{{\left | c \right |}}\right )}{c^{6} \sqrt {{\left | c \right |}}}\right )} + \frac {b c x^{3} \arctan \left (c x^{2}\right ) + a c x^{3} - 2 \, b x}{3 \, c} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 138, normalized size = 0.87 \[ \frac {x^{3} a}{3}+\frac {b \,x^{3} \arctan \left (c \,x^{2}\right )}{3}-\frac {2 b x}{3 c}+\frac {b \left (\frac {1}{c^{2}}\right )^{\frac {1}{4}} \sqrt {2}\, \ln \left (\frac {x^{2}+\left (\frac {1}{c^{2}}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {1}{c^{2}}}}{x^{2}-\left (\frac {1}{c^{2}}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {1}{c^{2}}}}\right )}{12 c}+\frac {b \left (\frac {1}{c^{2}}\right )^{\frac {1}{4}} \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {1}{c^{2}}\right )^{\frac {1}{4}}}+1\right )}{6 c}+\frac {b \left (\frac {1}{c^{2}}\right )^{\frac {1}{4}} \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {1}{c^{2}}\right )^{\frac {1}{4}}}-1\right )}{6 c} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.43, size = 145, normalized size = 0.91 \[ \frac {1}{3} \, a x^{3} + \frac {1}{12} \, {\left (4 \, x^{3} \arctan \left (c x^{2}\right ) - c {\left (\frac {8 \, x}{c^{2}} - \frac {\frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (2 \, c x + \sqrt {2} \sqrt {c}\right )}}{2 \, \sqrt {c}}\right )}{\sqrt {c}} + \frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (2 \, c x - \sqrt {2} \sqrt {c}\right )}}{2 \, \sqrt {c}}\right )}{\sqrt {c}} + \frac {\sqrt {2} \log \left (c x^{2} + \sqrt {2} \sqrt {c} x + 1\right )}{\sqrt {c}} - \frac {\sqrt {2} \log \left (c x^{2} - \sqrt {2} \sqrt {c} x + 1\right )}{\sqrt {c}}}{c^{2}}\right )}\right )} b \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.39, size = 62, normalized size = 0.39 \[ \frac {a\,x^3}{3}+\frac {b\,x^3\,\mathrm {atan}\left (c\,x^2\right )}{3}-\frac {2\,b\,x}{3\,c}-\frac {{\left (-1\right )}^{1/4}\,b\,\mathrm {atan}\left ({\left (-1\right )}^{1/4}\,\sqrt {c}\,x\right )\,1{}\mathrm {i}}{3\,c^{3/2}}-\frac {{\left (-1\right )}^{1/4}\,b\,\mathrm {atan}\left ({\left (-1\right )}^{1/4}\,\sqrt {c}\,x\,1{}\mathrm {i}\right )}{3\,c^{3/2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 17.68, size = 173, normalized size = 1.09 \[ \begin {cases} \frac {a x^{3}}{3} + \frac {b x^{3} \operatorname {atan}{\left (c x^{2} \right )}}{3} + \frac {\left (-1\right )^{\frac {3}{4}} b \left (\frac {1}{c^{2}}\right )^{\frac {3}{4}} \operatorname {atan}{\left (c x^{2} \right )}}{3} - \frac {2 b x}{3 c} - \frac {\sqrt [4]{-1} b \sqrt [4]{\frac {1}{c^{2}}} \log {\left (x - \sqrt [4]{-1} \sqrt [4]{\frac {1}{c^{2}}} \right )}}{3 c} + \frac {\sqrt [4]{-1} b \sqrt [4]{\frac {1}{c^{2}}} \log {\left (x^{2} + i \sqrt {\frac {1}{c^{2}}} \right )}}{6 c} - \frac {\sqrt [4]{-1} b \sqrt [4]{\frac {1}{c^{2}}} \operatorname {atan}{\left (\frac {\left (-1\right )^{\frac {3}{4}} x}{\sqrt [4]{\frac {1}{c^{2}}}} \right )}}{3 c} & \text {for}\: c \neq 0 \\\frac {a x^{3}}{3} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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