Optimal. Leaf size=315 \[ \frac {(d+e x)^3 \left (a+b \tan ^{-1}\left (c x^3\right )\right )}{3 e}+\frac {b d^2 \log \left (c^{2/3} x^2+1\right )}{2 \sqrt [3]{c}}+\frac {\sqrt {3} b d^2 \tan ^{-1}\left (\frac {1-2 c^{2/3} x^2}{\sqrt {3}}\right )}{2 \sqrt [3]{c}}-\frac {b d^2 \log \left (c^{4/3} x^4-c^{2/3} x^2+1\right )}{4 \sqrt [3]{c}}-\frac {\sqrt {3} b d e \log \left (c^{2/3} x^2-\sqrt {3} \sqrt [3]{c} x+1\right )}{4 c^{2/3}}+\frac {\sqrt {3} b d e \log \left (c^{2/3} x^2+\sqrt {3} \sqrt [3]{c} x+1\right )}{4 c^{2/3}}-\frac {b d e \tan ^{-1}\left (\sqrt [3]{c} x\right )}{c^{2/3}}+\frac {b d e \tan ^{-1}\left (\sqrt {3}-2 \sqrt [3]{c} x\right )}{2 c^{2/3}}-\frac {b d e \tan ^{-1}\left (2 \sqrt [3]{c} x+\sqrt {3}\right )}{2 c^{2/3}}-\frac {b e^2 \log \left (c^2 x^6+1\right )}{6 c}-\frac {b d^3 \tan ^{-1}\left (c x^3\right )}{3 e} \]
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Rubi [A] time = 0.71, antiderivative size = 331, normalized size of antiderivative = 1.05, number of steps used = 25, number of rules used = 14, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.778, Rules used = {6742, 5027, 275, 292, 31, 634, 617, 204, 628, 5033, 295, 618, 203, 260} \[ \frac {a (d+e x)^3}{3 e}+\frac {b d^2 \log \left (c^{2/3} x^2+1\right )}{2 \sqrt [3]{c}}-\frac {b d^2 \log \left (c^{4/3} x^4-c^{2/3} x^2+1\right )}{4 \sqrt [3]{c}}+\frac {\sqrt {3} b d^2 \tan ^{-1}\left (\frac {1-2 c^{2/3} x^2}{\sqrt {3}}\right )}{2 \sqrt [3]{c}}-\frac {\sqrt {3} b d e \log \left (c^{2/3} x^2-\sqrt {3} \sqrt [3]{c} x+1\right )}{4 c^{2/3}}+\frac {\sqrt {3} b d e \log \left (c^{2/3} x^2+\sqrt {3} \sqrt [3]{c} x+1\right )}{4 c^{2/3}}-\frac {b d e \tan ^{-1}\left (\sqrt [3]{c} x\right )}{c^{2/3}}+\frac {b d e \tan ^{-1}\left (\sqrt {3}-2 \sqrt [3]{c} x\right )}{2 c^{2/3}}-\frac {b d e \tan ^{-1}\left (2 \sqrt [3]{c} x+\sqrt {3}\right )}{2 c^{2/3}}-\frac {b e^2 \log \left (c^2 x^6+1\right )}{6 c}+b d^2 x \tan ^{-1}\left (c x^3\right )+b d e x^2 \tan ^{-1}\left (c x^3\right )+\frac {1}{3} b e^2 x^3 \tan ^{-1}\left (c x^3\right ) \]
Antiderivative was successfully verified.
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Rule 31
Rule 203
Rule 204
Rule 260
Rule 275
Rule 292
Rule 295
Rule 617
Rule 618
Rule 628
Rule 634
Rule 5027
Rule 5033
Rule 6742
Rubi steps
\begin {align*} \int (d+e x)^2 \left (a+b \tan ^{-1}\left (c x^3\right )\right ) \, dx &=\int \left (a (d+e x)^2+b (d+e x)^2 \tan ^{-1}\left (c x^3\right )\right ) \, dx\\ &=\frac {a (d+e x)^3}{3 e}+b \int (d+e x)^2 \tan ^{-1}\left (c x^3\right ) \, dx\\ &=\frac {a (d+e x)^3}{3 e}+b \int \left (d^2 \tan ^{-1}\left (c x^3\right )+2 d e x \tan ^{-1}\left (c x^3\right )+e^2 x^2 \tan ^{-1}\left (c x^3\right )\right ) \, dx\\ &=\frac {a (d+e x)^3}{3 e}+\left (b d^2\right ) \int \tan ^{-1}\left (c x^3\right ) \, dx+(2 b d e) \int x \tan ^{-1}\left (c x^3\right ) \, dx+\left (b e^2\right ) \int x^2 \tan ^{-1}\left (c x^3\right ) \, dx\\ &=\frac {a (d+e x)^3}{3 e}+b d^2 x \tan ^{-1}\left (c x^3\right )+b d e x^2 \tan ^{-1}\left (c x^3\right )+\frac {1}{3} b e^2 x^3 \tan ^{-1}\left (c x^3\right )-\left (3 b c d^2\right ) \int \frac {x^3}{1+c^2 x^6} \, dx-(3 b c d e) \int \frac {x^4}{1+c^2 x^6} \, dx-\left (b c e^2\right ) \int \frac {x^5}{1+c^2 x^6} \, dx\\ &=\frac {a (d+e x)^3}{3 e}+b d^2 x \tan ^{-1}\left (c x^3\right )+b d e x^2 \tan ^{-1}\left (c x^3\right )+\frac {1}{3} b e^2 x^3 \tan ^{-1}\left (c x^3\right )-\frac {b e^2 \log \left (1+c^2 x^6\right )}{6 c}-\frac {1}{2} \left (3 b c d^2\right ) \operatorname {Subst}\left (\int \frac {x}{1+c^2 x^3} \, dx,x,x^2\right )-\frac {(b d e) \int \frac {1}{1+c^{2/3} x^2} \, dx}{\sqrt [3]{c}}-\frac {(b d e) \int \frac {-\frac {1}{2}+\frac {1}{2} \sqrt {3} \sqrt [3]{c} x}{1-\sqrt {3} \sqrt [3]{c} x+c^{2/3} x^2} \, dx}{\sqrt [3]{c}}-\frac {(b d e) \int \frac {-\frac {1}{2}-\frac {1}{2} \sqrt {3} \sqrt [3]{c} x}{1+\sqrt {3} \sqrt [3]{c} x+c^{2/3} x^2} \, dx}{\sqrt [3]{c}}\\ &=\frac {a (d+e x)^3}{3 e}-\frac {b d e \tan ^{-1}\left (\sqrt [3]{c} x\right )}{c^{2/3}}+b d^2 x \tan ^{-1}\left (c x^3\right )+b d e x^2 \tan ^{-1}\left (c x^3\right )+\frac {1}{3} b e^2 x^3 \tan ^{-1}\left (c x^3\right )-\frac {b e^2 \log \left (1+c^2 x^6\right )}{6 c}+\frac {1}{2} \left (b \sqrt [3]{c} d^2\right ) \operatorname {Subst}\left (\int \frac {1}{1+c^{2/3} x} \, dx,x,x^2\right )-\frac {1}{2} \left (b \sqrt [3]{c} d^2\right ) \operatorname {Subst}\left (\int \frac {1+c^{2/3} x}{1-c^{2/3} x+c^{4/3} x^2} \, dx,x,x^2\right )-\frac {\left (\sqrt {3} b d e\right ) \int \frac {-\sqrt {3} \sqrt [3]{c}+2 c^{2/3} x}{1-\sqrt {3} \sqrt [3]{c} x+c^{2/3} x^2} \, dx}{4 c^{2/3}}+\frac {\left (\sqrt {3} b d e\right ) \int \frac {\sqrt {3} \sqrt [3]{c}+2 c^{2/3} x}{1+\sqrt {3} \sqrt [3]{c} x+c^{2/3} x^2} \, dx}{4 c^{2/3}}-\frac {(b d e) \int \frac {1}{1-\sqrt {3} \sqrt [3]{c} x+c^{2/3} x^2} \, dx}{4 \sqrt [3]{c}}-\frac {(b d e) \int \frac {1}{1+\sqrt {3} \sqrt [3]{c} x+c^{2/3} x^2} \, dx}{4 \sqrt [3]{c}}\\ &=\frac {a (d+e x)^3}{3 e}-\frac {b d e \tan ^{-1}\left (\sqrt [3]{c} x\right )}{c^{2/3}}+b d^2 x \tan ^{-1}\left (c x^3\right )+b d e x^2 \tan ^{-1}\left (c x^3\right )+\frac {1}{3} b e^2 x^3 \tan ^{-1}\left (c x^3\right )+\frac {b d^2 \log \left (1+c^{2/3} x^2\right )}{2 \sqrt [3]{c}}-\frac {\sqrt {3} b d e \log \left (1-\sqrt {3} \sqrt [3]{c} x+c^{2/3} x^2\right )}{4 c^{2/3}}+\frac {\sqrt {3} b d e \log \left (1+\sqrt {3} \sqrt [3]{c} x+c^{2/3} x^2\right )}{4 c^{2/3}}-\frac {b e^2 \log \left (1+c^2 x^6\right )}{6 c}-\frac {\left (b d^2\right ) \operatorname {Subst}\left (\int \frac {-c^{2/3}+2 c^{4/3} x}{1-c^{2/3} x+c^{4/3} x^2} \, dx,x,x^2\right )}{4 \sqrt [3]{c}}-\frac {1}{4} \left (3 b \sqrt [3]{c} d^2\right ) \operatorname {Subst}\left (\int \frac {1}{1-c^{2/3} x+c^{4/3} x^2} \, dx,x,x^2\right )-\frac {(b d e) \operatorname {Subst}\left (\int \frac {1}{-\frac {1}{3}-x^2} \, dx,x,1-\frac {2 \sqrt [3]{c} x}{\sqrt {3}}\right )}{2 \sqrt {3} c^{2/3}}+\frac {(b d e) \operatorname {Subst}\left (\int \frac {1}{-\frac {1}{3}-x^2} \, dx,x,1+\frac {2 \sqrt [3]{c} x}{\sqrt {3}}\right )}{2 \sqrt {3} c^{2/3}}\\ &=\frac {a (d+e x)^3}{3 e}-\frac {b d e \tan ^{-1}\left (\sqrt [3]{c} x\right )}{c^{2/3}}+b d^2 x \tan ^{-1}\left (c x^3\right )+b d e x^2 \tan ^{-1}\left (c x^3\right )+\frac {1}{3} b e^2 x^3 \tan ^{-1}\left (c x^3\right )+\frac {b d e \tan ^{-1}\left (\sqrt {3}-2 \sqrt [3]{c} x\right )}{2 c^{2/3}}-\frac {b d e \tan ^{-1}\left (\sqrt {3}+2 \sqrt [3]{c} x\right )}{2 c^{2/3}}+\frac {b d^2 \log \left (1+c^{2/3} x^2\right )}{2 \sqrt [3]{c}}-\frac {\sqrt {3} b d e \log \left (1-\sqrt {3} \sqrt [3]{c} x+c^{2/3} x^2\right )}{4 c^{2/3}}+\frac {\sqrt {3} b d e \log \left (1+\sqrt {3} \sqrt [3]{c} x+c^{2/3} x^2\right )}{4 c^{2/3}}-\frac {b d^2 \log \left (1-c^{2/3} x^2+c^{4/3} x^4\right )}{4 \sqrt [3]{c}}-\frac {b e^2 \log \left (1+c^2 x^6\right )}{6 c}-\frac {\left (3 b d^2\right ) \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-2 c^{2/3} x^2\right )}{2 \sqrt [3]{c}}\\ &=\frac {a (d+e x)^3}{3 e}-\frac {b d e \tan ^{-1}\left (\sqrt [3]{c} x\right )}{c^{2/3}}+b d^2 x \tan ^{-1}\left (c x^3\right )+b d e x^2 \tan ^{-1}\left (c x^3\right )+\frac {1}{3} b e^2 x^3 \tan ^{-1}\left (c x^3\right )+\frac {b d e \tan ^{-1}\left (\sqrt {3}-2 \sqrt [3]{c} x\right )}{2 c^{2/3}}-\frac {b d e \tan ^{-1}\left (\sqrt {3}+2 \sqrt [3]{c} x\right )}{2 c^{2/3}}+\frac {\sqrt {3} b d^2 \tan ^{-1}\left (\frac {1-2 c^{2/3} x^2}{\sqrt {3}}\right )}{2 \sqrt [3]{c}}+\frac {b d^2 \log \left (1+c^{2/3} x^2\right )}{2 \sqrt [3]{c}}-\frac {\sqrt {3} b d e \log \left (1-\sqrt {3} \sqrt [3]{c} x+c^{2/3} x^2\right )}{4 c^{2/3}}+\frac {\sqrt {3} b d e \log \left (1+\sqrt {3} \sqrt [3]{c} x+c^{2/3} x^2\right )}{4 c^{2/3}}-\frac {b d^2 \log \left (1-c^{2/3} x^2+c^{4/3} x^4\right )}{4 \sqrt [3]{c}}-\frac {b e^2 \log \left (1+c^2 x^6\right )}{6 c}\\ \end {align*}
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Mathematica [A] time = 168.71, size = 297, normalized size = 0.94 \[ \frac {12 a c d^2 x+12 a c d e x^2+4 a c e^2 x^3+6 b c^{2/3} d^2 \log \left (c^{2/3} x^2+1\right )-3 b \sqrt [3]{c} d \left (\sqrt [3]{c} d+\sqrt {3} e\right ) \log \left (c^{2/3} x^2-\sqrt {3} \sqrt [3]{c} x+1\right )-3 b \sqrt [3]{c} d \left (\sqrt [3]{c} d-\sqrt {3} e\right ) \log \left (c^{2/3} x^2+\sqrt {3} \sqrt [3]{c} x+1\right )-2 b e^2 \log \left (c^2 x^6+1\right )+4 b c x \tan ^{-1}\left (c x^3\right ) \left (3 d^2+3 d e x+e^2 x^2\right )-12 b \sqrt [3]{c} d e \tan ^{-1}\left (\sqrt [3]{c} x\right )+6 b \sqrt [3]{c} d \left (\sqrt {3} \sqrt [3]{c} d+e\right ) \tan ^{-1}\left (\sqrt {3}-2 \sqrt [3]{c} x\right )+6 b \sqrt [3]{c} d \left (\sqrt {3} \sqrt [3]{c} d-e\right ) \tan ^{-1}\left (2 \sqrt [3]{c} x+\sqrt {3}\right )}{12 c} \]
Antiderivative was successfully verified.
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fricas [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 2.45, size = 326, normalized size = 1.03 \[ \frac {b d^{2} {\left | c \right |}^{\frac {2}{3}} \log \left (x^{2} + \frac {1}{{\left | c \right |}^{\frac {2}{3}}}\right )}{2 \, c} - \frac {b c d \arctan \left (x {\left | c \right |}^{\frac {1}{3}}\right ) e}{{\left | c \right |}^{\frac {5}{3}}} + \frac {2 \, b c x^{3} \arctan \left (c x^{3}\right ) e^{2} + 6 \, b c d x^{2} \arctan \left (c x^{3}\right ) e + 6 \, b c d^{2} x \arctan \left (c x^{3}\right ) + 2 \, a c x^{3} e^{2} + 6 \, a c d x^{2} e + 6 \, a c d^{2} x - b e^{2} \log \left (c^{2} x^{6} + 1\right )}{6 \, c} + \frac {{\left (\sqrt {3} b c d^{2} {\left | c \right |}^{\frac {2}{3}} - b c d {\left | c \right |}^{\frac {1}{3}} e\right )} \arctan \left ({\left (2 \, x + \frac {\sqrt {3}}{{\left | c \right |}^{\frac {1}{3}}}\right )} {\left | c \right |}^{\frac {1}{3}}\right )}{2 \, c^{2}} - \frac {{\left (\sqrt {3} b c d^{2} {\left | c \right |}^{\frac {2}{3}} + b c d {\left | c \right |}^{\frac {1}{3}} e\right )} \arctan \left ({\left (2 \, x - \frac {\sqrt {3}}{{\left | c \right |}^{\frac {1}{3}}}\right )} {\left | c \right |}^{\frac {1}{3}}\right )}{2 \, c^{2}} + \frac {{\left (\sqrt {3} b c d {\left | c \right |}^{\frac {1}{3}} e - b c d^{2} {\left | c \right |}^{\frac {2}{3}}\right )} \log \left (x^{2} + \frac {\sqrt {3} x}{{\left | c \right |}^{\frac {1}{3}}} + \frac {1}{{\left | c \right |}^{\frac {2}{3}}}\right )}{4 \, c^{2}} - \frac {{\left (\sqrt {3} b c d {\left | c \right |}^{\frac {1}{3}} e + b c d^{2} {\left | c \right |}^{\frac {2}{3}}\right )} \log \left (x^{2} - \frac {\sqrt {3} x}{{\left | c \right |}^{\frac {1}{3}}} + \frac {1}{{\left | c \right |}^{\frac {2}{3}}}\right )}{4 \, c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.18, size = 536, normalized size = 1.70 \[ \frac {a \,e^{2} x^{3}}{3}+a e d \,x^{2}+a x \,d^{2}+\frac {a \,d^{3}}{3 e}+\frac {b \,e^{2} \arctan \left (c \,x^{3}\right ) x^{3}}{3}+b e \arctan \left (c \,x^{3}\right ) x^{2} d +b \arctan \left (c \,x^{3}\right ) x \,d^{2}+\frac {b \,d^{3} \arctan \left (c \,x^{3}\right )}{3 e}+\frac {b c \ln \left (x^{2}+\left (\frac {1}{c^{2}}\right )^{\frac {1}{3}}\right ) \left (\frac {1}{c^{2}}\right )^{\frac {2}{3}} d^{2}}{2}-\frac {b \,e^{2} \ln \left (x^{2}+\left (\frac {1}{c^{2}}\right )^{\frac {1}{3}}\right )}{6 c}+\frac {b c \sqrt {\frac {1}{c^{2}}}\, \arctan \left (\frac {x}{\left (\frac {1}{c^{2}}\right )^{\frac {1}{6}}}\right ) d^{3}}{3 e}-\frac {b e \arctan \left (\frac {x}{\left (\frac {1}{c^{2}}\right )^{\frac {1}{6}}}\right ) d}{c \left (\frac {1}{c^{2}}\right )^{\frac {1}{6}}}-\frac {b e c \ln \left (x^{2}-\sqrt {3}\, \left (\frac {1}{c^{2}}\right )^{\frac {1}{6}} x +\left (\frac {1}{c^{2}}\right )^{\frac {1}{3}}\right ) \sqrt {3}\, \left (\frac {1}{c^{2}}\right )^{\frac {5}{6}} d}{4}-\frac {b c \ln \left (x^{2}-\sqrt {3}\, \left (\frac {1}{c^{2}}\right )^{\frac {1}{6}} x +\left (\frac {1}{c^{2}}\right )^{\frac {1}{3}}\right ) \left (\frac {1}{c^{2}}\right )^{\frac {2}{3}} d^{2}}{4}-\frac {b \,e^{2} \ln \left (x^{2}-\sqrt {3}\, \left (\frac {1}{c^{2}}\right )^{\frac {1}{6}} x +\left (\frac {1}{c^{2}}\right )^{\frac {1}{3}}\right )}{6 c}-\frac {b e \arctan \left (\frac {2 x}{\left (\frac {1}{c^{2}}\right )^{\frac {1}{6}}}-\sqrt {3}\right ) d}{2 c \left (\frac {1}{c^{2}}\right )^{\frac {1}{6}}}-\frac {b c \left (\frac {1}{c^{2}}\right )^{\frac {2}{3}} \arctan \left (\frac {2 x}{\left (\frac {1}{c^{2}}\right )^{\frac {1}{6}}}-\sqrt {3}\right ) \sqrt {3}\, d^{2}}{2}-\frac {b c \sqrt {\frac {1}{c^{2}}}\, \arctan \left (\frac {2 x}{\left (\frac {1}{c^{2}}\right )^{\frac {1}{6}}}-\sqrt {3}\right ) d^{3}}{3 e}+\frac {b e c \ln \left (x^{2}+\sqrt {3}\, \left (\frac {1}{c^{2}}\right )^{\frac {1}{6}} x +\left (\frac {1}{c^{2}}\right )^{\frac {1}{3}}\right ) \sqrt {3}\, \left (\frac {1}{c^{2}}\right )^{\frac {5}{6}} d}{4}-\frac {b c \ln \left (x^{2}+\sqrt {3}\, \left (\frac {1}{c^{2}}\right )^{\frac {1}{6}} x +\left (\frac {1}{c^{2}}\right )^{\frac {1}{3}}\right ) \left (\frac {1}{c^{2}}\right )^{\frac {2}{3}} d^{2}}{4}-\frac {b \,e^{2} \ln \left (x^{2}+\sqrt {3}\, \left (\frac {1}{c^{2}}\right )^{\frac {1}{6}} x +\left (\frac {1}{c^{2}}\right )^{\frac {1}{3}}\right )}{6 c}-\frac {b e \arctan \left (\frac {2 x}{\left (\frac {1}{c^{2}}\right )^{\frac {1}{6}}}+\sqrt {3}\right ) d}{2 c \left (\frac {1}{c^{2}}\right )^{\frac {1}{6}}}+\frac {b c \left (\frac {1}{c^{2}}\right )^{\frac {2}{3}} \arctan \left (\frac {2 x}{\left (\frac {1}{c^{2}}\right )^{\frac {1}{6}}}+\sqrt {3}\right ) \sqrt {3}\, d^{2}}{2}-\frac {b c \sqrt {\frac {1}{c^{2}}}\, \arctan \left (\frac {2 x}{\left (\frac {1}{c^{2}}\right )^{\frac {1}{6}}}+\sqrt {3}\right ) d^{3}}{3 e} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.41, size = 280, normalized size = 0.89 \[ \frac {1}{3} \, a e^{2} x^{3} + a d e x^{2} - \frac {1}{4} \, {\left (c {\left (\frac {2 \, \sqrt {3} \arctan \left (\frac {\sqrt {3} {\left (2 \, c^{\frac {4}{3}} x^{2} - c^{\frac {2}{3}}\right )}}{3 \, c^{\frac {2}{3}}}\right )}{c^{\frac {4}{3}}} + \frac {\log \left (c^{\frac {4}{3}} x^{4} - c^{\frac {2}{3}} x^{2} + 1\right )}{c^{\frac {4}{3}}} - \frac {2 \, \log \left (\frac {c^{\frac {2}{3}} x^{2} + 1}{c^{\frac {2}{3}}}\right )}{c^{\frac {4}{3}}}\right )} - 4 \, x \arctan \left (c x^{3}\right )\right )} b d^{2} + \frac {1}{4} \, {\left (4 \, x^{2} \arctan \left (c x^{3}\right ) + c {\left (\frac {\sqrt {3} \log \left (c^{\frac {2}{3}} x^{2} + \sqrt {3} c^{\frac {1}{3}} x + 1\right )}{c^{\frac {5}{3}}} - \frac {\sqrt {3} \log \left (c^{\frac {2}{3}} x^{2} - \sqrt {3} c^{\frac {1}{3}} x + 1\right )}{c^{\frac {5}{3}}} - \frac {4 \, \arctan \left (c^{\frac {1}{3}} x\right )}{c^{\frac {5}{3}}} - \frac {2 \, \arctan \left (\frac {2 \, c^{\frac {2}{3}} x + \sqrt {3} c^{\frac {1}{3}}}{c^{\frac {1}{3}}}\right )}{c^{\frac {5}{3}}} - \frac {2 \, \arctan \left (\frac {2 \, c^{\frac {2}{3}} x - \sqrt {3} c^{\frac {1}{3}}}{c^{\frac {1}{3}}}\right )}{c^{\frac {5}{3}}}\right )}\right )} b d e + a d^{2} x + \frac {{\left (2 \, c x^{3} \arctan \left (c x^{3}\right ) - \log \left (c^{2} x^{6} + 1\right )\right )} b e^{2}}{6 \, c} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.52, size = 988, normalized size = 3.14 \[ \mathrm {atan}\left (c\,x^3\right )\,\left (b\,d^2\,x+b\,d\,e\,x^2+\frac {b\,e^2\,x^3}{3}\right )+\left (\sum _{k=1}^6\ln \left (x\,\left (6\,b^5\,c^7\,d^2\,e^8-162\,b^5\,c^9\,d^8\,e^2\right )+\mathrm {root}\left (46656\,a^6\,c^6+46656\,a^5\,b\,c^5\,e^2+19440\,a^4\,b^2\,c^4\,e^4+4320\,a^3\,b^3\,c^3\,e^6-11664\,a^3\,b^3\,c^5\,d^6+20412\,a^2\,b^4\,c^4\,d^6\,e^2+540\,a^2\,b^4\,c^2\,e^8-972\,a\,b^5\,c^3\,d^6\,e^4+36\,a\,b^5\,c\,e^{10}-54\,b^6\,c^2\,d^6\,e^6+729\,b^6\,c^4\,d^{12}+b^6\,e^{12},a,k\right )\,\left (x\,\left (486\,b^4\,c^{10}\,d^8+90\,b^4\,c^8\,d^2\,e^6\right )-\mathrm {root}\left (46656\,a^6\,c^6+46656\,a^5\,b\,c^5\,e^2+19440\,a^4\,b^2\,c^4\,e^4+4320\,a^3\,b^3\,c^3\,e^6-11664\,a^3\,b^3\,c^5\,d^6+20412\,a^2\,b^4\,c^4\,d^6\,e^2+540\,a^2\,b^4\,c^2\,e^8-972\,a\,b^5\,c^3\,d^6\,e^4+36\,a\,b^5\,c\,e^{10}-54\,b^6\,c^2\,d^6\,e^6+729\,b^6\,c^4\,d^{12}+b^6\,e^{12},a,k\right )\,\left (\mathrm {root}\left (46656\,a^6\,c^6+46656\,a^5\,b\,c^5\,e^2+19440\,a^4\,b^2\,c^4\,e^4+4320\,a^3\,b^3\,c^3\,e^6-11664\,a^3\,b^3\,c^5\,d^6+20412\,a^2\,b^4\,c^4\,d^6\,e^2+540\,a^2\,b^4\,c^2\,e^8-972\,a\,b^5\,c^3\,d^6\,e^4+36\,a\,b^5\,c\,e^{10}-54\,b^6\,c^2\,d^6\,e^6+729\,b^6\,c^4\,d^{12}+b^6\,e^{12},a,k\right )\,\left (3888\,b^2\,c^{10}\,d^3\,e+\mathrm {root}\left (46656\,a^6\,c^6+46656\,a^5\,b\,c^5\,e^2+19440\,a^4\,b^2\,c^4\,e^4+4320\,a^3\,b^3\,c^3\,e^6-11664\,a^3\,b^3\,c^5\,d^6+20412\,a^2\,b^4\,c^4\,d^6\,e^2+540\,a^2\,b^4\,c^2\,e^8-972\,a\,b^5\,c^3\,d^6\,e^4+36\,a\,b^5\,c\,e^{10}-54\,b^6\,c^2\,d^6\,e^6+729\,b^6\,c^4\,d^{12}+b^6\,e^{12},a,k\right )\,b\,c^{11}\,d^2\,x\,3888+648\,b^2\,c^{10}\,d^2\,e^2\,x\right )+972\,b^3\,c^9\,d^3\,e^3-324\,b^3\,c^9\,d^2\,e^4\,x\right )\right )-243\,b^5\,c^9\,d^9\,e+9\,b^5\,c^7\,d^3\,e^7\right )\,\mathrm {root}\left (46656\,a^6\,c^6+46656\,a^5\,b\,c^5\,e^2+19440\,a^4\,b^2\,c^4\,e^4+4320\,a^3\,b^3\,c^3\,e^6-11664\,a^3\,b^3\,c^5\,d^6+20412\,a^2\,b^4\,c^4\,d^6\,e^2+540\,a^2\,b^4\,c^2\,e^8-972\,a\,b^5\,c^3\,d^6\,e^4+36\,a\,b^5\,c\,e^{10}-54\,b^6\,c^2\,d^6\,e^6+729\,b^6\,c^4\,d^{12}+b^6\,e^{12},a,k\right )\right )+\frac {a\,e^2\,x^3}{3}+a\,d^2\,x+a\,d\,e\,x^2 \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 50.14, size = 151, normalized size = 0.48 \[ a d^{2} x + a d e x^{2} + \frac {a e^{2} x^{3}}{3} - 3 b c d^{2} \operatorname {RootSum} {\left (216 t^{3} c^{4} + 1, \left (t \mapsto t \log {\left (36 t^{2} c^{2} + x^{2} \right )} \right )\right )} - 3 b c d e \operatorname {RootSum} {\left (46656 t^{6} c^{10} + 1, \left (t \mapsto t \log {\left (7776 t^{5} c^{8} + x \right )} \right )\right )} + b d^{2} x \operatorname {atan}{\left (c x^{3} \right )} + b d e x^{2} \operatorname {atan}{\left (c x^{3} \right )} + b e^{2} \left (\begin {cases} 0 & \text {for}\: c = 0 \\\frac {x^{3} \operatorname {atan}{\left (c x^{3} \right )}}{3} - \frac {\log {\left (c^{2} x^{6} + 1 \right )}}{6 c} & \text {otherwise} \end {cases}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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