Integrand size = 18, antiderivative size = 275 \[ \int (d+e x)^2 \left (a+b \arctan \left (c x^3\right )\right ) \, dx=-\frac {b d e \arctan \left (\sqrt [3]{c} x\right )}{c^{2/3}}-\frac {b d^3 \arctan \left (c x^3\right )}{3 e}+\frac {(d+e x)^3 \left (a+b \arctan \left (c x^3\right )\right )}{3 e}+\frac {b d e \arctan \left (\sqrt {3}-2 \sqrt [3]{c} x\right )}{2 c^{2/3}}-\frac {b d e \arctan \left (\sqrt {3}+2 \sqrt [3]{c} x\right )}{2 c^{2/3}}+\frac {\sqrt {3} b d^2 \arctan \left (\frac {1-2 c^{2/3} x^2}{\sqrt {3}}\right )}{2 \sqrt [3]{c}}+\frac {\sqrt {3} b d e \text {arctanh}\left (\frac {\sqrt {3} \sqrt [3]{c} x}{1+c^{2/3} x^2}\right )}{2 c^{2/3}}+\frac {b d^2 \log \left (1+c^{2/3} x^2\right )}{2 \sqrt [3]{c}}-\frac {b d^2 \log \left (1-c^{2/3} x^2+c^{4/3} x^4\right )}{4 \sqrt [3]{c}}-\frac {b e^2 \log \left (1+c^2 x^6\right )}{6 c} \] Output:
-b*d*e*arctan(c^(1/3)*x)/c^(2/3)-1/3*b*d^3*arctan(c*x^3)/e+1/3*(e*x+d)^3*( a+b*arctan(c*x^3))/e-1/2*b*d*e*arctan(-3^(1/2)+2*c^(1/3)*x)/c^(2/3)-1/2*b* d*e*arctan(3^(1/2)+2*c^(1/3)*x)/c^(2/3)+1/2*3^(1/2)*b*d^2*arctan(1/3*(1-2* c^(2/3)*x^2)*3^(1/2))/c^(1/3)+1/2*3^(1/2)*b*d*e*arctanh(3^(1/2)*c^(1/3)*x/ (1+c^(2/3)*x^2))/c^(2/3)+1/2*b*d^2*ln(1+c^(2/3)*x^2)/c^(1/3)-1/4*b*d^2*ln( 1-c^(2/3)*x^2+c^(4/3)*x^4)/c^(1/3)-1/6*b*e^2*ln(c^2*x^6+1)/c
Time = 125.21 (sec) , antiderivative size = 297, normalized size of antiderivative = 1.08 \[ \int (d+e x)^2 \left (a+b \arctan \left (c x^3\right )\right ) \, dx=\frac {12 a c d^2 x+12 a c d e x^2+4 a c e^2 x^3-12 b \sqrt [3]{c} d e \arctan \left (\sqrt [3]{c} x\right )+4 b c x \left (3 d^2+3 d e x+e^2 x^2\right ) \arctan \left (c x^3\right )+6 b \sqrt [3]{c} d \left (\sqrt {3} \sqrt [3]{c} d+e\right ) \arctan \left (\sqrt {3}-2 \sqrt [3]{c} x\right )+6 b \sqrt [3]{c} d \left (\sqrt {3} \sqrt [3]{c} d-e\right ) \arctan \left (\sqrt {3}+2 \sqrt [3]{c} x\right )+6 b c^{2/3} d^2 \log \left (1+c^{2/3} x^2\right )-3 b \sqrt [3]{c} d \left (\sqrt [3]{c} d+\sqrt {3} e\right ) \log \left (1-\sqrt {3} \sqrt [3]{c} x+c^{2/3} x^2\right )-3 b \sqrt [3]{c} d \left (\sqrt [3]{c} d-\sqrt {3} e\right ) \log \left (1+\sqrt {3} \sqrt [3]{c} x+c^{2/3} x^2\right )-2 b e^2 \log \left (1+c^2 x^6\right )}{12 c} \] Input:
Integrate[(d + e*x)^2*(a + b*ArcTan[c*x^3]),x]
Output:
(12*a*c*d^2*x + 12*a*c*d*e*x^2 + 4*a*c*e^2*x^3 - 12*b*c^(1/3)*d*e*ArcTan[c ^(1/3)*x] + 4*b*c*x*(3*d^2 + 3*d*e*x + e^2*x^2)*ArcTan[c*x^3] + 6*b*c^(1/3 )*d*(Sqrt[3]*c^(1/3)*d + e)*ArcTan[Sqrt[3] - 2*c^(1/3)*x] + 6*b*c^(1/3)*d* (Sqrt[3]*c^(1/3)*d - e)*ArcTan[Sqrt[3] + 2*c^(1/3)*x] + 6*b*c^(2/3)*d^2*Lo g[1 + c^(2/3)*x^2] - 3*b*c^(1/3)*d*(c^(1/3)*d + Sqrt[3]*e)*Log[1 - Sqrt[3] *c^(1/3)*x + c^(2/3)*x^2] - 3*b*c^(1/3)*d*(c^(1/3)*d - Sqrt[3]*e)*Log[1 + Sqrt[3]*c^(1/3)*x + c^(2/3)*x^2] - 2*b*e^2*Log[1 + c^2*x^6])/(12*c)
Time = 0.85 (sec) , antiderivative size = 325, normalized size of antiderivative = 1.18, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {5395, 2370, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int (d+e x)^2 \left (a+b \arctan \left (c x^3\right )\right ) \, dx\) |
| \(\Big \downarrow \) 5395 |
| \(\displaystyle \frac {(d+e x)^3 \left (a+b \arctan \left (c x^3\right )\right )}{3 e}-\frac {b c \int \frac {x^2 (d+e x)^3}{c^2 x^6+1}dx}{e}\) |
| \(\Big \downarrow \) 2370 |
| \(\displaystyle \frac {(d+e x)^3 \left (a+b \arctan \left (c x^3\right )\right )}{3 e}-\frac {b c \int \left (\frac {3 d e^2 x^4}{c^2 x^6+1}+\frac {3 d^2 e x^3}{c^2 x^6+1}+\frac {\left (d^3+e^3 x^3\right ) x^2}{c^2 x^6+1}\right )dx}{e}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {(d+e x)^3 \left (a+b \arctan \left (c x^3\right )\right )}{3 e}-\frac {b c \left (-\frac {\sqrt {3} d^2 e \arctan \left (\frac {1-2 c^{2/3} x^2}{\sqrt {3}}\right )}{2 c^{4/3}}+\frac {d e^2 \arctan \left (\sqrt [3]{c} x\right )}{c^{5/3}}-\frac {d e^2 \arctan \left (\sqrt {3}-2 \sqrt [3]{c} x\right )}{2 c^{5/3}}+\frac {d e^2 \arctan \left (2 \sqrt [3]{c} x+\sqrt {3}\right )}{2 c^{5/3}}+\frac {d^3 \arctan \left (c x^3\right )}{3 c}-\frac {d^2 e \log \left (c^{2/3} x^2+1\right )}{2 c^{4/3}}+\frac {d^2 e \log \left (c^{4/3} x^4-c^{2/3} x^2+1\right )}{4 c^{4/3}}+\frac {\sqrt {3} d e^2 \log \left (c^{2/3} x^2-\sqrt {3} \sqrt [3]{c} x+1\right )}{4 c^{5/3}}-\frac {\sqrt {3} d e^2 \log \left (c^{2/3} x^2+\sqrt {3} \sqrt [3]{c} x+1\right )}{4 c^{5/3}}+\frac {e^3 \log \left (c^2 x^6+1\right )}{6 c^2}\right )}{e}\) |
Input:
Int[(d + e*x)^2*(a + b*ArcTan[c*x^3]),x]
Output:
((d + e*x)^3*(a + b*ArcTan[c*x^3]))/(3*e) - (b*c*((d*e^2*ArcTan[c^(1/3)*x] )/c^(5/3) + (d^3*ArcTan[c*x^3])/(3*c) - (d*e^2*ArcTan[Sqrt[3] - 2*c^(1/3)* x])/(2*c^(5/3)) + (d*e^2*ArcTan[Sqrt[3] + 2*c^(1/3)*x])/(2*c^(5/3)) - (Sqr t[3]*d^2*e*ArcTan[(1 - 2*c^(2/3)*x^2)/Sqrt[3]])/(2*c^(4/3)) - (d^2*e*Log[1 + c^(2/3)*x^2])/(2*c^(4/3)) + (Sqrt[3]*d*e^2*Log[1 - Sqrt[3]*c^(1/3)*x + c^(2/3)*x^2])/(4*c^(5/3)) - (Sqrt[3]*d*e^2*Log[1 + Sqrt[3]*c^(1/3)*x + c^( 2/3)*x^2])/(4*c^(5/3)) + (d^2*e*Log[1 - c^(2/3)*x^2 + c^(4/3)*x^4])/(4*c^( 4/3)) + (e^3*Log[1 + c^2*x^6])/(6*c^2)))/e
Int[((Pq_)*((c_.)*(x_))^(m_.))/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[ {v = Sum[(c*x)^(m + ii)*((Coeff[Pq, x, ii] + Coeff[Pq, x, n/2 + ii]*x^(n/2) )/(c^ii*(a + b*x^n))), {ii, 0, n/2 - 1}]}, Int[v, x] /; SumQ[v]] /; FreeQ[{ a, b, c, m}, x] && PolyQ[Pq, x] && IGtQ[n/2, 0] && Expon[Pq, x] < n
Int[((a_.) + ArcTan[(c_.)*(x_)^(n_)]*(b_.))*((d_) + (e_.)*(x_))^(m_.), x_Sy mbol] :> Simp[(d + e*x)^(m + 1)*((a + b*ArcTan[c*x^n])/(e*(m + 1))), x] - S imp[b*c*(n/(e*(m + 1))) Int[x^(n - 1)*((d + e*x)^(m + 1)/(1 + c^2*x^(2*n) )), x], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && NeQ[m, -1]
Leaf count of result is larger than twice the leaf count of optimal. \(502\) vs. \(2(216)=432\).
Time = 2.13 (sec) , antiderivative size = 503, normalized size of antiderivative = 1.83
| method | result | size |
| default | \(\frac {a \left (e x +d \right )^{3}}{3 e}+b \left (\frac {e^{2} \arctan \left (c \,x^{3}\right ) x^{3}}{3}+e \arctan \left (c \,x^{3}\right ) x^{2} d +\arctan \left (c \,x^{3}\right ) x \,d^{2}+\frac {\arctan \left (c \,x^{3}\right ) d^{3}}{3 e}-\frac {c \left (\frac {\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 \,e^{2}}{4}+\frac {\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} e}{4}+\frac {\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 ) e^{3}}{6 c^{2}}+\frac {\arctan \left (\frac {2 x}{\left (\frac {1}{c^{2}}\right )^{\frac {1}{6}}}-\sqrt {3}\right ) d \,e^{2}}{2 c^{2} \left (\frac {1}{c^{2}}\right )^{\frac {1}{6}}}+\frac {\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} e}{2}+\frac {\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}-\frac {\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 \,e^{2}}{4}+\frac {\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} e}{4}+\frac {\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 ) e^{3}}{6 c^{2}}+\frac {\arctan \left (\frac {2 x}{\left (\frac {1}{c^{2}}\right )^{\frac {1}{6}}}+\sqrt {3}\right ) d \,e^{2}}{2 c^{2} \left (\frac {1}{c^{2}}\right )^{\frac {1}{6}}}-\frac {\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} e}{2}+\frac {\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}-\frac {\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} e}{2}+\frac {\ln \left (x^{2}+\left (\frac {1}{c^{2}}\right )^{\frac {1}{3}}\right ) e^{3}}{6 c^{2}}-\frac {\sqrt {\frac {1}{c^{2}}}\, \arctan \left (\frac {x}{\left (\frac {1}{c^{2}}\right )^{\frac {1}{6}}}\right ) d^{3}}{3}+\frac {\arctan \left (\frac {x}{\left (\frac {1}{c^{2}}\right )^{\frac {1}{6}}}\right ) d \,e^{2}}{c^{2} \left (\frac {1}{c^{2}}\right )^{\frac {1}{6}}}\right )}{e}\right )\) | \(503\) |
| parts | \(\frac {a \left (e x +d \right )^{3}}{3 e}+b \left (\frac {e^{2} \arctan \left (c \,x^{3}\right ) x^{3}}{3}+e \arctan \left (c \,x^{3}\right ) x^{2} d +\arctan \left (c \,x^{3}\right ) x \,d^{2}+\frac {\arctan \left (c \,x^{3}\right ) d^{3}}{3 e}-\frac {c \left (\frac {\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 \,e^{2}}{4}+\frac {\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} e}{4}+\frac {\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 ) e^{3}}{6 c^{2}}+\frac {\arctan \left (\frac {2 x}{\left (\frac {1}{c^{2}}\right )^{\frac {1}{6}}}-\sqrt {3}\right ) d \,e^{2}}{2 c^{2} \left (\frac {1}{c^{2}}\right )^{\frac {1}{6}}}+\frac {\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} e}{2}+\frac {\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}-\frac {\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 \,e^{2}}{4}+\frac {\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} e}{4}+\frac {\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 ) e^{3}}{6 c^{2}}+\frac {\arctan \left (\frac {2 x}{\left (\frac {1}{c^{2}}\right )^{\frac {1}{6}}}+\sqrt {3}\right ) d \,e^{2}}{2 c^{2} \left (\frac {1}{c^{2}}\right )^{\frac {1}{6}}}-\frac {\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} e}{2}+\frac {\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}-\frac {\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} e}{2}+\frac {\ln \left (x^{2}+\left (\frac {1}{c^{2}}\right )^{\frac {1}{3}}\right ) e^{3}}{6 c^{2}}-\frac {\sqrt {\frac {1}{c^{2}}}\, \arctan \left (\frac {x}{\left (\frac {1}{c^{2}}\right )^{\frac {1}{6}}}\right ) d^{3}}{3}+\frac {\arctan \left (\frac {x}{\left (\frac {1}{c^{2}}\right )^{\frac {1}{6}}}\right ) d \,e^{2}}{c^{2} \left (\frac {1}{c^{2}}\right )^{\frac {1}{6}}}\right )}{e}\right )\) | \(503\) |
Input:
int((e*x+d)^2*(a+b*arctan(c*x^3)),x,method=_RETURNVERBOSE)
Output:
1/3*a*(e*x+d)^3/e+b*(1/3*e^2*arctan(c*x^3)*x^3+e*arctan(c*x^3)*x^2*d+arcta n(c*x^3)*x*d^2+1/3/e*arctan(c*x^3)*d^3-1/e*c*(1/4*ln(x^2-3^(1/2)*(1/c^2)^( 1/6)*x+(1/c^2)^(1/3))*3^(1/2)*(1/c^2)^(5/6)*d*e^2+1/4*ln(x^2-3^(1/2)*(1/c^ 2)^(1/6)*x+(1/c^2)^(1/3))*(1/c^2)^(2/3)*d^2*e+1/6/c^2*ln(x^2-3^(1/2)*(1/c^ 2)^(1/6)*x+(1/c^2)^(1/3))*e^3+1/2/c^2/(1/c^2)^(1/6)*arctan(2*x/(1/c^2)^(1/ 6)-3^(1/2))*d*e^2+1/2*(1/c^2)^(2/3)*arctan(2*x/(1/c^2)^(1/6)-3^(1/2))*3^(1 /2)*d^2*e+1/3*(1/c^2)^(1/2)*arctan(2*x/(1/c^2)^(1/6)-3^(1/2))*d^3-1/4*ln(x ^2+3^(1/2)*(1/c^2)^(1/6)*x+(1/c^2)^(1/3))*3^(1/2)*(1/c^2)^(5/6)*d*e^2+1/4* ln(x^2+3^(1/2)*(1/c^2)^(1/6)*x+(1/c^2)^(1/3))*(1/c^2)^(2/3)*d^2*e+1/6/c^2* ln(x^2+3^(1/2)*(1/c^2)^(1/6)*x+(1/c^2)^(1/3))*e^3+1/2/c^2/(1/c^2)^(1/6)*ar ctan(2*x/(1/c^2)^(1/6)+3^(1/2))*d*e^2-1/2*(1/c^2)^(2/3)*arctan(2*x/(1/c^2) ^(1/6)+3^(1/2))*3^(1/2)*d^2*e+1/3*(1/c^2)^(1/2)*arctan(2*x/(1/c^2)^(1/6)+3 ^(1/2))*d^3-1/2*ln(x^2+(1/c^2)^(1/3))*(1/c^2)^(2/3)*d^2*e+1/6/c^2*ln(x^2+( 1/c^2)^(1/3))*e^3-1/3*(1/c^2)^(1/2)*arctan(x/(1/c^2)^(1/6))*d^3+1/c^2/(1/c ^2)^(1/6)*arctan(x/(1/c^2)^(1/6))*d*e^2))
Exception generated. \[ \int (d+e x)^2 \left (a+b \arctan \left (c x^3\right )\right ) \, dx=\text {Exception raised: RuntimeError} \] Input:
integrate((e*x+d)^2*(a+b*arctan(c*x^3)),x, algorithm="fricas")
Output:
Exception raised: RuntimeError >> no explicit roots found
Time = 27.10 (sec) , antiderivative size = 151, normalized size of antiderivative = 0.55 \[ \int (d+e x)^2 \left (a+b \arctan \left (c x^3\right )\right ) \, dx=a d^{2} x + a d e x^{2} + \frac {a e^{2} x^{3}}{3} - 3 b c d^{2} \operatorname {RootSum} {\left (216 t^{3} c^{4} + 1, \left ( t \mapsto t \log {\left (36 t^{2} c^{2} + x^{2} \right )} \right )\right )} - 3 b c d e \operatorname {RootSum} {\left (46656 t^{6} c^{10} + 1, \left ( t \mapsto t \log {\left (7776 t^{5} c^{8} + x \right )} \right )\right )} + b d^{2} x \operatorname {atan}{\left (c x^{3} \right )} + b d e x^{2} \operatorname {atan}{\left (c x^{3} \right )} + b e^{2} \left (\begin {cases} 0 & \text {for}\: c = 0 \\\frac {x^{3} \operatorname {atan}{\left (c x^{3} \right )}}{3} - \frac {\log {\left (c^{2} x^{6} + 1 \right )}}{6 c} & \text {otherwise} \end {cases}\right ) \] Input:
integrate((e*x+d)**2*(a+b*atan(c*x**3)),x)
Output:
a*d**2*x + a*d*e*x**2 + a*e**2*x**3/3 - 3*b*c*d**2*RootSum(216*_t**3*c**4 + 1, Lambda(_t, _t*log(36*_t**2*c**2 + x**2))) - 3*b*c*d*e*RootSum(46656*_ t**6*c**10 + 1, Lambda(_t, _t*log(7776*_t**5*c**8 + x))) + b*d**2*x*atan(c *x**3) + b*d*e*x**2*atan(c*x**3) + b*e**2*Piecewise((0, Eq(c, 0)), (x**3*a tan(c*x**3)/3 - log(c**2*x**6 + 1)/(6*c), True))
Time = 0.12 (sec) , antiderivative size = 280, normalized size of antiderivative = 1.02 \[ \int (d+e x)^2 \left (a+b \arctan \left (c x^3\right )\right ) \, dx=\frac {1}{3} \, a e^{2} x^{3} + a d e x^{2} - \frac {1}{4} \, {\left (c {\left (\frac {2 \, \sqrt {3} \arctan \left (\frac {\sqrt {3} {\left (2 \, c^{\frac {4}{3}} x^{2} - c^{\frac {2}{3}}\right )}}{3 \, c^{\frac {2}{3}}}\right )}{c^{\frac {4}{3}}} + \frac {\log \left (c^{\frac {4}{3}} x^{4} - c^{\frac {2}{3}} x^{2} + 1\right )}{c^{\frac {4}{3}}} - \frac {2 \, \log \left (\frac {c^{\frac {2}{3}} x^{2} + 1}{c^{\frac {2}{3}}}\right )}{c^{\frac {4}{3}}}\right )} - 4 \, x \arctan \left (c x^{3}\right )\right )} b d^{2} + \frac {1}{4} \, {\left (4 \, x^{2} \arctan \left (c x^{3}\right ) + c {\left (\frac {\sqrt {3} \log \left (c^{\frac {2}{3}} x^{2} + \sqrt {3} c^{\frac {1}{3}} x + 1\right )}{c^{\frac {5}{3}}} - \frac {\sqrt {3} \log \left (c^{\frac {2}{3}} x^{2} - \sqrt {3} c^{\frac {1}{3}} x + 1\right )}{c^{\frac {5}{3}}} - \frac {4 \, \arctan \left (c^{\frac {1}{3}} x\right )}{c^{\frac {5}{3}}} - \frac {2 \, \arctan \left (\frac {2 \, c^{\frac {2}{3}} x + \sqrt {3} c^{\frac {1}{3}}}{c^{\frac {1}{3}}}\right )}{c^{\frac {5}{3}}} - \frac {2 \, \arctan \left (\frac {2 \, c^{\frac {2}{3}} x - \sqrt {3} c^{\frac {1}{3}}}{c^{\frac {1}{3}}}\right )}{c^{\frac {5}{3}}}\right )}\right )} b d e + a d^{2} x + \frac {{\left (2 \, c x^{3} \arctan \left (c x^{3}\right ) - \log \left (c^{2} x^{6} + 1\right )\right )} b e^{2}}{6 \, c} \] Input:
integrate((e*x+d)^2*(a+b*arctan(c*x^3)),x, algorithm="maxima")
Output:
1/3*a*e^2*x^3 + a*d*e*x^2 - 1/4*(c*(2*sqrt(3)*arctan(1/3*sqrt(3)*(2*c^(4/3 )*x^2 - c^(2/3))/c^(2/3))/c^(4/3) + log(c^(4/3)*x^4 - c^(2/3)*x^2 + 1)/c^( 4/3) - 2*log((c^(2/3)*x^2 + 1)/c^(2/3))/c^(4/3)) - 4*x*arctan(c*x^3))*b*d^ 2 + 1/4*(4*x^2*arctan(c*x^3) + c*(sqrt(3)*log(c^(2/3)*x^2 + sqrt(3)*c^(1/3 )*x + 1)/c^(5/3) - sqrt(3)*log(c^(2/3)*x^2 - sqrt(3)*c^(1/3)*x + 1)/c^(5/3 ) - 4*arctan(c^(1/3)*x)/c^(5/3) - 2*arctan((2*c^(2/3)*x + sqrt(3)*c^(1/3)) /c^(1/3))/c^(5/3) - 2*arctan((2*c^(2/3)*x - sqrt(3)*c^(1/3))/c^(1/3))/c^(5 /3)))*b*d*e + a*d^2*x + 1/6*(2*c*x^3*arctan(c*x^3) - log(c^2*x^6 + 1))*b*e ^2/c
Time = 8.81 (sec) , antiderivative size = 312, normalized size of antiderivative = 1.13 \[ \int (d+e x)^2 \left (a+b \arctan \left (c x^3\right )\right ) \, dx=\frac {1}{3} \, b e^{2} x^{3} \arctan \left (c x^{3}\right ) + \frac {1}{3} \, a e^{2} x^{3} + b d e x^{2} \arctan \left (c x^{3}\right ) + a d e x^{2} + b d^{2} x \arctan \left (c x^{3}\right ) + a d^{2} x - \frac {b c d e \arctan \left (x {\left | c \right |}^{\frac {1}{3}}\right )}{{\left | c \right |}^{\frac {5}{3}}} + \frac {{\left (\sqrt {3} b c d^{2} {\left | c \right |}^{\frac {1}{3}} - b c d e\right )} \arctan \left ({\left (2 \, x + \frac {\sqrt {3}}{{\left | c \right |}^{\frac {1}{3}}}\right )} {\left | c \right |}^{\frac {1}{3}}\right )}{2 \, {\left | c \right |}^{\frac {5}{3}}} - \frac {{\left (\sqrt {3} b c d^{2} {\left | c \right |}^{\frac {1}{3}} + b c d e\right )} \arctan \left ({\left (2 \, x - \frac {\sqrt {3}}{{\left | c \right |}^{\frac {1}{3}}}\right )} {\left | c \right |}^{\frac {1}{3}}\right )}{2 \, {\left | c \right |}^{\frac {5}{3}}} + \frac {{\left (3 \, \sqrt {3} b c d e {\left | c \right |}^{\frac {1}{3}} - 3 \, b c d^{2} {\left | c \right |}^{\frac {2}{3}} - 2 \, b c e^{2}\right )} \log \left (x^{2} + \frac {\sqrt {3} x}{{\left | c \right |}^{\frac {1}{3}}} + \frac {1}{{\left | c \right |}^{\frac {2}{3}}}\right )}{12 \, c^{2}} - \frac {{\left (3 \, \sqrt {3} b c d e {\left | c \right |}^{\frac {1}{3}} + 3 \, b c d^{2} {\left | c \right |}^{\frac {2}{3}} + 2 \, b c e^{2}\right )} \log \left (x^{2} - \frac {\sqrt {3} x}{{\left | c \right |}^{\frac {1}{3}}} + \frac {1}{{\left | c \right |}^{\frac {2}{3}}}\right )}{12 \, c^{2}} + \frac {{\left (3 \, b c d^{2} {\left | c \right |}^{\frac {2}{3}} - b c e^{2}\right )} \log \left (x^{2} + \frac {1}{{\left | c \right |}^{\frac {2}{3}}}\right )}{6 \, c^{2}} \] Input:
integrate((e*x+d)^2*(a+b*arctan(c*x^3)),x, algorithm="giac")
Output:
1/3*b*e^2*x^3*arctan(c*x^3) + 1/3*a*e^2*x^3 + b*d*e*x^2*arctan(c*x^3) + a* d*e*x^2 + b*d^2*x*arctan(c*x^3) + a*d^2*x - b*c*d*e*arctan(x*abs(c)^(1/3)) /abs(c)^(5/3) + 1/2*(sqrt(3)*b*c*d^2*abs(c)^(1/3) - b*c*d*e)*arctan((2*x + sqrt(3)/abs(c)^(1/3))*abs(c)^(1/3))/abs(c)^(5/3) - 1/2*(sqrt(3)*b*c*d^2*a bs(c)^(1/3) + b*c*d*e)*arctan((2*x - sqrt(3)/abs(c)^(1/3))*abs(c)^(1/3))/a bs(c)^(5/3) + 1/12*(3*sqrt(3)*b*c*d*e*abs(c)^(1/3) - 3*b*c*d^2*abs(c)^(2/3 ) - 2*b*c*e^2)*log(x^2 + sqrt(3)*x/abs(c)^(1/3) + 1/abs(c)^(2/3))/c^2 - 1/ 12*(3*sqrt(3)*b*c*d*e*abs(c)^(1/3) + 3*b*c*d^2*abs(c)^(2/3) + 2*b*c*e^2)*l og(x^2 - sqrt(3)*x/abs(c)^(1/3) + 1/abs(c)^(2/3))/c^2 + 1/6*(3*b*c*d^2*abs (c)^(2/3) - b*c*e^2)*log(x^2 + 1/abs(c)^(2/3))/c^2
Time = 0.85 (sec) , antiderivative size = 988, normalized size of antiderivative = 3.59 \[ \int (d+e x)^2 \left (a+b \arctan \left (c x^3\right )\right ) \, dx =\text {Too large to display} \] Input:
int((a + b*atan(c*x^3))*(d + e*x)^2,x)
Output:
atan(c*x^3)*((b*e^2*x^3)/3 + b*d^2*x + b*d*e*x^2) + symsum(log(x*(6*b^5*c^ 7*d^2*e^8 - 162*b^5*c^9*d^8*e^2) + root(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) *(x*(486*b^4*c^10*d^8 + 90*b^4*c^8*d^2*e^6) - root(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)*(root(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)*(3888*b^2*c^10*d^3*e + 3888*root(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)*b*c^11*d^2*x + 648*b^2*c^10 *d^2*e^2*x) + 972*b^3*c^9*d^3*e^3 - 324*b^3*c^9*d^2*e^4*x)) - 243*b^5*c^9* d^9*e + 9*b^5*c^7*d^3*e^7)*root(46656*a^6*c^6 + 46656*a^5*b*c^5*e^2 + 1944 0*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 + 2041...
Time = 0.18 (sec) , antiderivative size = 362, normalized size of antiderivative = 1.32 \[ \int (d+e x)^2 \left (a+b \arctan \left (c x^3\right )\right ) \, dx=\frac {-12 \sqrt {3}\, \mathit {atan} \left (2 c^{\frac {1}{3}} x -\sqrt {3}\right ) b c \,d^{2}-18 c^{\frac {2}{3}} \mathit {atan} \left (c^{\frac {1}{3}} x \right ) b d e +6 \sqrt {3}\, \mathit {atan} \left (c^{\frac {1}{3}} x \right ) b c \,d^{2}-6 c^{\frac {2}{3}} \mathit {atan} \left (c \,x^{3}\right ) b d e +12 c^{\frac {4}{3}} \mathit {atan} \left (c \,x^{3}\right ) b \,d^{2} x +12 c^{\frac {4}{3}} \mathit {atan} \left (c \,x^{3}\right ) b d e \,x^{2}+4 c^{\frac {4}{3}} \mathit {atan} \left (c \,x^{3}\right ) b \,e^{2} x^{3}+6 \sqrt {3}\, \mathit {atan} \left (c \,x^{3}\right ) b c \,d^{2}-3 c^{\frac {2}{3}} \sqrt {3}\, \mathrm {log}\left (c^{\frac {2}{3}} x^{2}-c^{\frac {1}{3}} \sqrt {3}\, x +1\right ) b d e +3 c^{\frac {2}{3}} \sqrt {3}\, \mathrm {log}\left (c^{\frac {2}{3}} x^{2}+c^{\frac {1}{3}} \sqrt {3}\, x +1\right ) b d e -2 c^{\frac {1}{3}} \mathrm {log}\left (c^{\frac {2}{3}} x^{2}-c^{\frac {1}{3}} \sqrt {3}\, x +1\right ) b \,e^{2}-2 c^{\frac {1}{3}} \mathrm {log}\left (c^{\frac {2}{3}} x^{2}+c^{\frac {1}{3}} \sqrt {3}\, x +1\right ) b \,e^{2}-2 c^{\frac {1}{3}} \mathrm {log}\left (c^{\frac {2}{3}} x^{2}+1\right ) b \,e^{2}+12 c^{\frac {4}{3}} a \,d^{2} x +12 c^{\frac {4}{3}} a d e \,x^{2}+4 c^{\frac {4}{3}} a \,e^{2} x^{3}-3 \,\mathrm {log}\left (c^{\frac {2}{3}} x^{2}-c^{\frac {1}{3}} \sqrt {3}\, x +1\right ) b c \,d^{2}-3 \,\mathrm {log}\left (c^{\frac {2}{3}} x^{2}+c^{\frac {1}{3}} \sqrt {3}\, x +1\right ) b c \,d^{2}+6 \,\mathrm {log}\left (c^{\frac {2}{3}} x^{2}+1\right ) b c \,d^{2}}{12 c^{\frac {4}{3}}} \] Input:
int((e*x+d)^2*(a+b*atan(c*x^3)),x)
Output:
( - 12*sqrt(3)*atan(2*c**(1/3)*x - sqrt(3))*b*c*d**2 - 18*c**(2/3)*atan(c* *(1/3)*x)*b*d*e + 6*sqrt(3)*atan(c**(1/3)*x)*b*c*d**2 - 6*c**(2/3)*atan(c* x**3)*b*d*e + 12*c**(1/3)*atan(c*x**3)*b*c*d**2*x + 12*c**(1/3)*atan(c*x** 3)*b*c*d*e*x**2 + 4*c**(1/3)*atan(c*x**3)*b*c*e**2*x**3 + 6*sqrt(3)*atan(c *x**3)*b*c*d**2 - 3*c**(2/3)*sqrt(3)*log(c**(2/3)*x**2 - c**(1/3)*sqrt(3)* x + 1)*b*d*e + 3*c**(2/3)*sqrt(3)*log(c**(2/3)*x**2 + c**(1/3)*sqrt(3)*x + 1)*b*d*e - 2*c**(1/3)*log(c**(2/3)*x**2 - c**(1/3)*sqrt(3)*x + 1)*b*e**2 - 2*c**(1/3)*log(c**(2/3)*x**2 + c**(1/3)*sqrt(3)*x + 1)*b*e**2 - 2*c**(1/ 3)*log(c**(2/3)*x**2 + 1)*b*e**2 + 12*c**(1/3)*a*c*d**2*x + 12*c**(1/3)*a* c*d*e*x**2 + 4*c**(1/3)*a*c*e**2*x**3 - 3*log(c**(2/3)*x**2 - c**(1/3)*sqr t(3)*x + 1)*b*c*d**2 - 3*log(c**(2/3)*x**2 + c**(1/3)*sqrt(3)*x + 1)*b*c*d **2 + 6*log(c**(2/3)*x**2 + 1)*b*c*d**2)/(12*c**(1/3)*c)