Integrand size = 18, antiderivative size = 188 \[ \int \left (d+e x^2\right )^3 (a+b \arctan (c x)) \, dx=-\frac {b e \left (35 c^4 d^2-21 c^2 d e+5 e^2\right ) x^2}{70 c^5}-\frac {b \left (21 c^2 d-5 e\right ) e^2 x^4}{140 c^3}-\frac {b e^3 x^6}{42 c}+d^3 x (a+b \arctan (c x))+d^2 e x^3 (a+b \arctan (c x))+\frac {3}{5} d e^2 x^5 (a+b \arctan (c x))+\frac {1}{7} e^3 x^7 (a+b \arctan (c x))-\frac {b \left (35 c^6 d^3-35 c^4 d^2 e+21 c^2 d e^2-5 e^3\right ) \log \left (1+c^2 x^2\right )}{70 c^7} \] Output:
-1/70*b*e*(35*c^4*d^2-21*c^2*d*e+5*e^2)*x^2/c^5-1/140*b*(21*c^2*d-5*e)*e^2 *x^4/c^3-1/42*b*e^3*x^6/c+d^3*x*(a+b*arctan(c*x))+d^2*e*x^3*(a+b*arctan(c* x))+3/5*d*e^2*x^5*(a+b*arctan(c*x))+1/7*e^3*x^7*(a+b*arctan(c*x))-1/70*b*( 35*c^6*d^3-35*c^4*d^2*e+21*c^2*d*e^2-5*e^3)*ln(c^2*x^2+1)/c^7
Time = 0.07 (sec) , antiderivative size = 192, normalized size of antiderivative = 1.02 \[ \int \left (d+e x^2\right )^3 (a+b \arctan (c x)) \, dx=\frac {c^2 x \left (12 a c^5 \left (35 d^3+35 d^2 e x^2+21 d e^2 x^4+5 e^3 x^6\right )-b e x \left (30 e^2-3 c^2 e \left (42 d+5 e x^2\right )+c^4 \left (210 d^2+63 d e x^2+10 e^2 x^4\right )\right )\right )+12 b c^7 x \left (35 d^3+35 d^2 e x^2+21 d e^2 x^4+5 e^3 x^6\right ) \arctan (c x)-6 b \left (35 c^6 d^3-35 c^4 d^2 e+21 c^2 d e^2-5 e^3\right ) \log \left (1+c^2 x^2\right )}{420 c^7} \] Input:
Integrate[(d + e*x^2)^3*(a + b*ArcTan[c*x]),x]
Output:
(c^2*x*(12*a*c^5*(35*d^3 + 35*d^2*e*x^2 + 21*d*e^2*x^4 + 5*e^3*x^6) - b*e* x*(30*e^2 - 3*c^2*e*(42*d + 5*e*x^2) + c^4*(210*d^2 + 63*d*e*x^2 + 10*e^2* x^4))) + 12*b*c^7*x*(35*d^3 + 35*d^2*e*x^2 + 21*d*e^2*x^4 + 5*e^3*x^6)*Arc Tan[c*x] - 6*b*(35*c^6*d^3 - 35*c^4*d^2*e + 21*c^2*d*e^2 - 5*e^3)*Log[1 + c^2*x^2])/(420*c^7)
Time = 0.58 (sec) , antiderivative size = 185, normalized size of antiderivative = 0.98, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.278, Rules used = {5447, 27, 2331, 2389, 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.
| \(\displaystyle \int \left (d+e x^2\right )^3 (a+b \arctan (c x)) \, dx\) |
| \(\Big \downarrow \) 5447 |
| \(\displaystyle -b c \int \frac {x \left (5 e^3 x^6+21 d e^2 x^4+35 d^2 e x^2+35 d^3\right )}{35 \left (c^2 x^2+1\right )}dx+d^3 x (a+b \arctan (c x))+d^2 e x^3 (a+b \arctan (c x))+\frac {3}{5} d e^2 x^5 (a+b \arctan (c x))+\frac {1}{7} e^3 x^7 (a+b \arctan (c x))\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {1}{35} b c \int \frac {x \left (5 e^3 x^6+21 d e^2 x^4+35 d^2 e x^2+35 d^3\right )}{c^2 x^2+1}dx+d^3 x (a+b \arctan (c x))+d^2 e x^3 (a+b \arctan (c x))+\frac {3}{5} d e^2 x^5 (a+b \arctan (c x))+\frac {1}{7} e^3 x^7 (a+b \arctan (c x))\) |
| \(\Big \downarrow \) 2331 |
| \(\displaystyle -\frac {1}{70} b c \int \frac {5 e^3 x^6+21 d e^2 x^4+35 d^2 e x^2+35 d^3}{c^2 x^2+1}dx^2+d^3 x (a+b \arctan (c x))+d^2 e x^3 (a+b \arctan (c x))+\frac {3}{5} d e^2 x^5 (a+b \arctan (c x))+\frac {1}{7} e^3 x^7 (a+b \arctan (c x))\) |
| \(\Big \downarrow \) 2389 |
| \(\displaystyle -\frac {1}{70} b c \int \left (\frac {5 e^3 x^4}{c^2}+\frac {\left (21 c^2 d-5 e\right ) e^2 x^2}{c^4}+\frac {e \left (35 d^2 c^4-21 d e c^2+5 e^2\right )}{c^6}+\frac {35 d^3 c^6-35 d^2 e c^4+21 d e^2 c^2-5 e^3}{c^6 \left (c^2 x^2+1\right )}\right )dx^2+d^3 x (a+b \arctan (c x))+d^2 e x^3 (a+b \arctan (c x))+\frac {3}{5} d e^2 x^5 (a+b \arctan (c x))+\frac {1}{7} e^3 x^7 (a+b \arctan (c x))\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle d^3 x (a+b \arctan (c x))+d^2 e x^3 (a+b \arctan (c x))+\frac {3}{5} d e^2 x^5 (a+b \arctan (c x))+\frac {1}{7} e^3 x^7 (a+b \arctan (c x))-\frac {1}{70} b c \left (\frac {5 e^3 x^6}{3 c^2}+\frac {e^2 x^4 \left (21 c^2 d-5 e\right )}{2 c^4}+\frac {e x^2 \left (35 c^4 d^2-21 c^2 d e+5 e^2\right )}{c^6}+\frac {\left (35 c^6 d^3-35 c^4 d^2 e+21 c^2 d e^2-5 e^3\right ) \log \left (c^2 x^2+1\right )}{c^8}\right )\) |
Input:
Int[(d + e*x^2)^3*(a + b*ArcTan[c*x]),x]
Output:
d^3*x*(a + b*ArcTan[c*x]) + d^2*e*x^3*(a + b*ArcTan[c*x]) + (3*d*e^2*x^5*( a + b*ArcTan[c*x]))/5 + (e^3*x^7*(a + b*ArcTan[c*x]))/7 - (b*c*((e*(35*c^4 *d^2 - 21*c^2*d*e + 5*e^2)*x^2)/c^6 + ((21*c^2*d - 5*e)*e^2*x^4)/(2*c^4) + (5*e^3*x^6)/(3*c^2) + ((35*c^6*d^3 - 35*c^4*d^2*e + 21*c^2*d*e^2 - 5*e^3) *Log[1 + c^2*x^2])/c^8))/70
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[(Pq_)*(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[1/2 S ubst[Int[x^((m - 1)/2)*SubstFor[x^2, Pq, x]*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, p}, x] && PolyQ[Pq, x^2] && IntegerQ[(m - 1)/2]
Int[(Pq_)*((a_) + (b_.)*(x_)^(n_.))^(p_.), x_Symbol] :> Int[ExpandIntegrand [Pq*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, n}, x] && PolyQ[Pq, x] && (IGtQ[p , 0] || EqQ[n, 1])
Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))*((d_.) + (e_.)*(x_)^2)^(q_.), x_Symb ol] :> With[{u = IntHide[(d + e*x^2)^q, x]}, Simp[(a + b*ArcTan[c*x]) u, x] - Simp[b*c Int[SimplifyIntegrand[u/(1 + c^2*x^2), x], x], x]] /; FreeQ [{a, b, c, d, e}, x] && (IntegerQ[q] || ILtQ[q + 1/2, 0])
Time = 0.22 (sec) , antiderivative size = 210, normalized size of antiderivative = 1.12
| method | result | size |
| parts | \(a \left (\frac {1}{7} x^{7} e^{3}+\frac {3}{5} x^{5} e^{2} d +x^{3} e \,d^{2}+x \,d^{3}\right )+\frac {b \left (\frac {\arctan \left (c x \right ) c \,e^{3} x^{7}}{7}+\frac {3 \arctan \left (c x \right ) c \,e^{2} d \,x^{5}}{5}+\arctan \left (c x \right ) c \,d^{2} e \,x^{3}+\arctan \left (c x \right ) c x \,d^{3}-\frac {\frac {35 c^{6} d^{2} e \,x^{2}}{2}+\frac {21 c^{6} d \,e^{2} x^{4}}{4}+\frac {5 c^{6} e^{3} x^{6}}{6}-\frac {21 c^{4} d \,e^{2} x^{2}}{2}-\frac {5 c^{4} e^{3} x^{4}}{4}+\frac {5 c^{2} e^{3} x^{2}}{2}+\frac {\left (35 c^{6} d^{3}-35 c^{4} d^{2} e +21 e^{2} d \,c^{2}-5 e^{3}\right ) \ln \left (c^{2} x^{2}+1\right )}{2}}{35 c^{6}}\right )}{c}\) | \(210\) |
| derivativedivides | \(\frac {\frac {a \left (c^{7} x \,d^{3}+d^{2} c^{7} e \,x^{3}+\frac {3}{5} d \,c^{7} e^{2} x^{5}+\frac {1}{7} e^{3} c^{7} x^{7}\right )}{c^{6}}+\frac {b \left (\arctan \left (c x \right ) c^{7} x \,d^{3}+\arctan \left (c x \right ) d^{2} c^{7} e \,x^{3}+\frac {3 \arctan \left (c x \right ) d \,c^{7} e^{2} x^{5}}{5}+\frac {\arctan \left (c x \right ) e^{3} c^{7} x^{7}}{7}-\frac {c^{6} d^{2} e \,x^{2}}{2}-\frac {3 c^{6} d \,e^{2} x^{4}}{20}+\frac {3 c^{4} d \,e^{2} x^{2}}{10}-\frac {c^{6} e^{3} x^{6}}{42}+\frac {c^{4} e^{3} x^{4}}{28}-\frac {c^{2} e^{3} x^{2}}{14}-\frac {\left (35 c^{6} d^{3}-35 c^{4} d^{2} e +21 e^{2} d \,c^{2}-5 e^{3}\right ) \ln \left (c^{2} x^{2}+1\right )}{70}\right )}{c^{6}}}{c}\) | \(231\) |
| default | \(\frac {\frac {a \left (c^{7} x \,d^{3}+d^{2} c^{7} e \,x^{3}+\frac {3}{5} d \,c^{7} e^{2} x^{5}+\frac {1}{7} e^{3} c^{7} x^{7}\right )}{c^{6}}+\frac {b \left (\arctan \left (c x \right ) c^{7} x \,d^{3}+\arctan \left (c x \right ) d^{2} c^{7} e \,x^{3}+\frac {3 \arctan \left (c x \right ) d \,c^{7} e^{2} x^{5}}{5}+\frac {\arctan \left (c x \right ) e^{3} c^{7} x^{7}}{7}-\frac {c^{6} d^{2} e \,x^{2}}{2}-\frac {3 c^{6} d \,e^{2} x^{4}}{20}+\frac {3 c^{4} d \,e^{2} x^{2}}{10}-\frac {c^{6} e^{3} x^{6}}{42}+\frac {c^{4} e^{3} x^{4}}{28}-\frac {c^{2} e^{3} x^{2}}{14}-\frac {\left (35 c^{6} d^{3}-35 c^{4} d^{2} e +21 e^{2} d \,c^{2}-5 e^{3}\right ) \ln \left (c^{2} x^{2}+1\right )}{70}\right )}{c^{6}}}{c}\) | \(231\) |
| parallelrisch | \(-\frac {-60 x^{7} \arctan \left (c x \right ) b \,c^{7} e^{3}-60 a \,c^{7} e^{3} x^{7}-252 x^{5} \arctan \left (c x \right ) b \,c^{7} d \,e^{2}+10 b \,c^{6} e^{3} x^{6}-252 a \,c^{7} d \,e^{2} x^{5}-420 x^{3} \arctan \left (c x \right ) b \,c^{7} d^{2} e +63 b \,c^{6} d \,e^{2} x^{4}-420 a \,c^{7} d^{2} e \,x^{3}-420 x \arctan \left (c x \right ) b \,c^{7} d^{3}-15 b \,c^{4} e^{3} x^{4}+210 b \,c^{6} d^{2} e \,x^{2}-420 a \,c^{7} d^{3} x +210 \ln \left (c^{2} x^{2}+1\right ) b \,c^{6} d^{3}-126 b \,c^{4} d \,e^{2} x^{2}-210 \ln \left (c^{2} x^{2}+1\right ) b \,c^{4} d^{2} e +30 b \,c^{2} e^{3} x^{2}+126 \ln \left (c^{2} x^{2}+1\right ) b \,c^{2} d \,e^{2}-30 \ln \left (c^{2} x^{2}+1\right ) b \,e^{3}}{420 c^{7}}\) | \(269\) |
| risch | \(\frac {i b \,d^{3} x \ln \left (-i c x +1\right )}{2}-\frac {i b \left (5 x^{7} e^{3}+21 x^{5} e^{2} d +35 x^{3} e \,d^{2}+35 x \,d^{3}\right ) \ln \left (i c x +1\right )}{70}+\frac {3 i b d \,e^{2} x^{5} \ln \left (-i c x +1\right )}{10}+\frac {a \,e^{3} x^{7}}{7}+\frac {i b \,d^{2} e \,x^{3} \ln \left (-i c x +1\right )}{2}+\frac {3 a d \,e^{2} x^{5}}{5}-\frac {b \,e^{3} x^{6}}{42 c}+\frac {i b \,e^{3} x^{7} \ln \left (-i c x +1\right )}{14}+a \,d^{2} e \,x^{3}-\frac {3 b d \,e^{2} x^{4}}{20 c}+a \,d^{3} x -\frac {b \,d^{2} e \,x^{2}}{2 c}+\frac {b \,e^{3} x^{4}}{28 c^{3}}-\frac {\ln \left (-c^{2} x^{2}-1\right ) b \,d^{3}}{2 c}+\frac {3 b d \,e^{2} x^{2}}{10 c^{3}}+\frac {\ln \left (-c^{2} x^{2}-1\right ) b \,d^{2} e}{2 c^{3}}-\frac {b \,e^{3} x^{2}}{14 c^{5}}-\frac {3 \ln \left (-c^{2} x^{2}-1\right ) b d \,e^{2}}{10 c^{5}}+\frac {\ln \left (-c^{2} x^{2}-1\right ) b \,e^{3}}{14 c^{7}}\) | \(310\) |
Input:
int((e*x^2+d)^3*(a+b*arctan(c*x)),x,method=_RETURNVERBOSE)
Output:
a*(1/7*x^7*e^3+3/5*x^5*e^2*d+x^3*e*d^2+x*d^3)+b/c*(1/7*arctan(c*x)*c*e^3*x ^7+3/5*arctan(c*x)*c*e^2*d*x^5+arctan(c*x)*c*d^2*e*x^3+arctan(c*x)*c*x*d^3 -1/35/c^6*(35/2*c^6*d^2*e*x^2+21/4*c^6*d*e^2*x^4+5/6*c^6*e^3*x^6-21/2*c^4* d*e^2*x^2-5/4*c^4*e^3*x^4+5/2*c^2*e^3*x^2+1/2*(35*c^6*d^3-35*c^4*d^2*e+21* c^2*d*e^2-5*e^3)*ln(c^2*x^2+1)))
Time = 0.12 (sec) , antiderivative size = 229, normalized size of antiderivative = 1.22 \[ \int \left (d+e x^2\right )^3 (a+b \arctan (c x)) \, dx=\frac {60 \, a c^{7} e^{3} x^{7} + 252 \, a c^{7} d e^{2} x^{5} - 10 \, b c^{6} e^{3} x^{6} + 420 \, a c^{7} d^{2} e x^{3} + 420 \, a c^{7} d^{3} x - 3 \, {\left (21 \, b c^{6} d e^{2} - 5 \, b c^{4} e^{3}\right )} x^{4} - 6 \, {\left (35 \, b c^{6} d^{2} e - 21 \, b c^{4} d e^{2} + 5 \, b c^{2} e^{3}\right )} x^{2} + 12 \, {\left (5 \, b c^{7} e^{3} x^{7} + 21 \, b c^{7} d e^{2} x^{5} + 35 \, b c^{7} d^{2} e x^{3} + 35 \, b c^{7} d^{3} x\right )} \arctan \left (c x\right ) - 6 \, {\left (35 \, b c^{6} d^{3} - 35 \, b c^{4} d^{2} e + 21 \, b c^{2} d e^{2} - 5 \, b e^{3}\right )} \log \left (c^{2} x^{2} + 1\right )}{420 \, c^{7}} \] Input:
integrate((e*x^2+d)^3*(a+b*arctan(c*x)),x, algorithm="fricas")
Output:
1/420*(60*a*c^7*e^3*x^7 + 252*a*c^7*d*e^2*x^5 - 10*b*c^6*e^3*x^6 + 420*a*c ^7*d^2*e*x^3 + 420*a*c^7*d^3*x - 3*(21*b*c^6*d*e^2 - 5*b*c^4*e^3)*x^4 - 6* (35*b*c^6*d^2*e - 21*b*c^4*d*e^2 + 5*b*c^2*e^3)*x^2 + 12*(5*b*c^7*e^3*x^7 + 21*b*c^7*d*e^2*x^5 + 35*b*c^7*d^2*e*x^3 + 35*b*c^7*d^3*x)*arctan(c*x) - 6*(35*b*c^6*d^3 - 35*b*c^4*d^2*e + 21*b*c^2*d*e^2 - 5*b*e^3)*log(c^2*x^2 + 1))/c^7
Time = 0.48 (sec) , antiderivative size = 306, normalized size of antiderivative = 1.63 \[ \int \left (d+e x^2\right )^3 (a+b \arctan (c x)) \, dx=\begin {cases} a d^{3} x + a d^{2} e x^{3} + \frac {3 a d e^{2} x^{5}}{5} + \frac {a e^{3} x^{7}}{7} + b d^{3} x \operatorname {atan}{\left (c x \right )} + b d^{2} e x^{3} \operatorname {atan}{\left (c x \right )} + \frac {3 b d e^{2} x^{5} \operatorname {atan}{\left (c x \right )}}{5} + \frac {b e^{3} x^{7} \operatorname {atan}{\left (c x \right )}}{7} - \frac {b d^{3} \log {\left (x^{2} + \frac {1}{c^{2}} \right )}}{2 c} - \frac {b d^{2} e x^{2}}{2 c} - \frac {3 b d e^{2} x^{4}}{20 c} - \frac {b e^{3} x^{6}}{42 c} + \frac {b d^{2} e \log {\left (x^{2} + \frac {1}{c^{2}} \right )}}{2 c^{3}} + \frac {3 b d e^{2} x^{2}}{10 c^{3}} + \frac {b e^{3} x^{4}}{28 c^{3}} - \frac {3 b d e^{2} \log {\left (x^{2} + \frac {1}{c^{2}} \right )}}{10 c^{5}} - \frac {b e^{3} x^{2}}{14 c^{5}} + \frac {b e^{3} \log {\left (x^{2} + \frac {1}{c^{2}} \right )}}{14 c^{7}} & \text {for}\: c \neq 0 \\a \left (d^{3} x + d^{2} e x^{3} + \frac {3 d e^{2} x^{5}}{5} + \frac {e^{3} x^{7}}{7}\right ) & \text {otherwise} \end {cases} \] Input:
integrate((e*x**2+d)**3*(a+b*atan(c*x)),x)
Output:
Piecewise((a*d**3*x + a*d**2*e*x**3 + 3*a*d*e**2*x**5/5 + a*e**3*x**7/7 + b*d**3*x*atan(c*x) + b*d**2*e*x**3*atan(c*x) + 3*b*d*e**2*x**5*atan(c*x)/5 + b*e**3*x**7*atan(c*x)/7 - b*d**3*log(x**2 + c**(-2))/(2*c) - b*d**2*e*x **2/(2*c) - 3*b*d*e**2*x**4/(20*c) - b*e**3*x**6/(42*c) + b*d**2*e*log(x** 2 + c**(-2))/(2*c**3) + 3*b*d*e**2*x**2/(10*c**3) + b*e**3*x**4/(28*c**3) - 3*b*d*e**2*log(x**2 + c**(-2))/(10*c**5) - b*e**3*x**2/(14*c**5) + b*e** 3*log(x**2 + c**(-2))/(14*c**7), Ne(c, 0)), (a*(d**3*x + d**2*e*x**3 + 3*d *e**2*x**5/5 + e**3*x**7/7), True))
Time = 0.04 (sec) , antiderivative size = 222, normalized size of antiderivative = 1.18 \[ \int \left (d+e x^2\right )^3 (a+b \arctan (c x)) \, dx=\frac {1}{7} \, a e^{3} x^{7} + \frac {3}{5} \, a d e^{2} x^{5} + a d^{2} e x^{3} + \frac {1}{2} \, {\left (2 \, x^{3} \arctan \left (c x\right ) - c {\left (\frac {x^{2}}{c^{2}} - \frac {\log \left (c^{2} x^{2} + 1\right )}{c^{4}}\right )}\right )} b d^{2} e + \frac {3}{20} \, {\left (4 \, x^{5} \arctan \left (c x\right ) - c {\left (\frac {c^{2} x^{4} - 2 \, x^{2}}{c^{4}} + \frac {2 \, \log \left (c^{2} x^{2} + 1\right )}{c^{6}}\right )}\right )} b d e^{2} + \frac {1}{84} \, {\left (12 \, x^{7} \arctan \left (c x\right ) - c {\left (\frac {2 \, c^{4} x^{6} - 3 \, c^{2} x^{4} + 6 \, x^{2}}{c^{6}} - \frac {6 \, \log \left (c^{2} x^{2} + 1\right )}{c^{8}}\right )}\right )} b e^{3} + a d^{3} x + \frac {{\left (2 \, c x \arctan \left (c x\right ) - \log \left (c^{2} x^{2} + 1\right )\right )} b d^{3}}{2 \, c} \] Input:
integrate((e*x^2+d)^3*(a+b*arctan(c*x)),x, algorithm="maxima")
Output:
1/7*a*e^3*x^7 + 3/5*a*d*e^2*x^5 + a*d^2*e*x^3 + 1/2*(2*x^3*arctan(c*x) - c *(x^2/c^2 - log(c^2*x^2 + 1)/c^4))*b*d^2*e + 3/20*(4*x^5*arctan(c*x) - c*( (c^2*x^4 - 2*x^2)/c^4 + 2*log(c^2*x^2 + 1)/c^6))*b*d*e^2 + 1/84*(12*x^7*ar ctan(c*x) - c*((2*c^4*x^6 - 3*c^2*x^4 + 6*x^2)/c^6 - 6*log(c^2*x^2 + 1)/c^ 8))*b*e^3 + a*d^3*x + 1/2*(2*c*x*arctan(c*x) - log(c^2*x^2 + 1))*b*d^3/c
Time = 0.21 (sec) , antiderivative size = 268, normalized size of antiderivative = 1.43 \[ \int \left (d+e x^2\right )^3 (a+b \arctan (c x)) \, dx=\frac {60 \, b c^{7} e^{3} x^{7} \arctan \left (c x\right ) + 60 \, a c^{7} e^{3} x^{7} + 252 \, b c^{7} d e^{2} x^{5} \arctan \left (c x\right ) + 252 \, a c^{7} d e^{2} x^{5} - 10 \, b c^{6} e^{3} x^{6} + 420 \, b c^{7} d^{2} e x^{3} \arctan \left (c x\right ) + 420 \, a c^{7} d^{2} e x^{3} - 63 \, b c^{6} d e^{2} x^{4} + 420 \, b c^{7} d^{3} x \arctan \left (c x\right ) + 420 \, a c^{7} d^{3} x - 210 \, b c^{6} d^{2} e x^{2} + 15 \, b c^{4} e^{3} x^{4} - 210 \, b c^{6} d^{3} \log \left (c^{2} x^{2} + 1\right ) + 126 \, b c^{4} d e^{2} x^{2} + 210 \, b c^{4} d^{2} e \log \left (c^{2} x^{2} + 1\right ) - 30 \, b c^{2} e^{3} x^{2} - 126 \, b c^{2} d e^{2} \log \left (c^{2} x^{2} + 1\right ) + 30 \, b e^{3} \log \left (c^{2} x^{2} + 1\right )}{420 \, c^{7}} \] Input:
integrate((e*x^2+d)^3*(a+b*arctan(c*x)),x, algorithm="giac")
Output:
1/420*(60*b*c^7*e^3*x^7*arctan(c*x) + 60*a*c^7*e^3*x^7 + 252*b*c^7*d*e^2*x ^5*arctan(c*x) + 252*a*c^7*d*e^2*x^5 - 10*b*c^6*e^3*x^6 + 420*b*c^7*d^2*e* x^3*arctan(c*x) + 420*a*c^7*d^2*e*x^3 - 63*b*c^6*d*e^2*x^4 + 420*b*c^7*d^3 *x*arctan(c*x) + 420*a*c^7*d^3*x - 210*b*c^6*d^2*e*x^2 + 15*b*c^4*e^3*x^4 - 210*b*c^6*d^3*log(c^2*x^2 + 1) + 126*b*c^4*d*e^2*x^2 + 210*b*c^4*d^2*e*l og(c^2*x^2 + 1) - 30*b*c^2*e^3*x^2 - 126*b*c^2*d*e^2*log(c^2*x^2 + 1) + 30 *b*e^3*log(c^2*x^2 + 1))/c^7
Time = 0.48 (sec) , antiderivative size = 238, normalized size of antiderivative = 1.27 \[ \int \left (d+e x^2\right )^3 (a+b \arctan (c x)) \, dx=\frac {a\,e^3\,x^7}{7}+a\,d^3\,x-\frac {b\,d^3\,\ln \left (c^2\,x^2+1\right )}{2\,c}+\frac {b\,e^3\,\ln \left (c^2\,x^2+1\right )}{14\,c^7}-\frac {b\,e^3\,x^6}{42\,c}+\frac {b\,e^3\,x^4}{28\,c^3}-\frac {b\,e^3\,x^2}{14\,c^5}+b\,d^3\,x\,\mathrm {atan}\left (c\,x\right )+a\,d^2\,e\,x^3+\frac {3\,a\,d\,e^2\,x^5}{5}+\frac {b\,e^3\,x^7\,\mathrm {atan}\left (c\,x\right )}{7}+b\,d^2\,e\,x^3\,\mathrm {atan}\left (c\,x\right )+\frac {3\,b\,d\,e^2\,x^5\,\mathrm {atan}\left (c\,x\right )}{5}+\frac {b\,d^2\,e\,\ln \left (c^2\,x^2+1\right )}{2\,c^3}-\frac {3\,b\,d\,e^2\,\ln \left (c^2\,x^2+1\right )}{10\,c^5}-\frac {b\,d^2\,e\,x^2}{2\,c}-\frac {3\,b\,d\,e^2\,x^4}{20\,c}+\frac {3\,b\,d\,e^2\,x^2}{10\,c^3} \] Input:
int((a + b*atan(c*x))*(d + e*x^2)^3,x)
Output:
(a*e^3*x^7)/7 + a*d^3*x - (b*d^3*log(c^2*x^2 + 1))/(2*c) + (b*e^3*log(c^2* x^2 + 1))/(14*c^7) - (b*e^3*x^6)/(42*c) + (b*e^3*x^4)/(28*c^3) - (b*e^3*x^ 2)/(14*c^5) + b*d^3*x*atan(c*x) + a*d^2*e*x^3 + (3*a*d*e^2*x^5)/5 + (b*e^3 *x^7*atan(c*x))/7 + b*d^2*e*x^3*atan(c*x) + (3*b*d*e^2*x^5*atan(c*x))/5 + (b*d^2*e*log(c^2*x^2 + 1))/(2*c^3) - (3*b*d*e^2*log(c^2*x^2 + 1))/(10*c^5) - (b*d^2*e*x^2)/(2*c) - (3*b*d*e^2*x^4)/(20*c) + (3*b*d*e^2*x^2)/(10*c^3)
Time = 0.20 (sec) , antiderivative size = 268, normalized size of antiderivative = 1.43 \[ \int \left (d+e x^2\right )^3 (a+b \arctan (c x)) \, dx=\frac {420 \mathit {atan} \left (c x \right ) b \,c^{7} d^{3} x +420 \mathit {atan} \left (c x \right ) b \,c^{7} d^{2} e \,x^{3}+252 \mathit {atan} \left (c x \right ) b \,c^{7} d \,e^{2} x^{5}+60 \mathit {atan} \left (c x \right ) b \,c^{7} e^{3} x^{7}-210 \,\mathrm {log}\left (c^{2} x^{2}+1\right ) b \,c^{6} d^{3}+210 \,\mathrm {log}\left (c^{2} x^{2}+1\right ) b \,c^{4} d^{2} e -126 \,\mathrm {log}\left (c^{2} x^{2}+1\right ) b \,c^{2} d \,e^{2}+30 \,\mathrm {log}\left (c^{2} x^{2}+1\right ) b \,e^{3}+420 a \,c^{7} d^{3} x +420 a \,c^{7} d^{2} e \,x^{3}+252 a \,c^{7} d \,e^{2} x^{5}+60 a \,c^{7} e^{3} x^{7}-210 b \,c^{6} d^{2} e \,x^{2}-63 b \,c^{6} d \,e^{2} x^{4}-10 b \,c^{6} e^{3} x^{6}+126 b \,c^{4} d \,e^{2} x^{2}+15 b \,c^{4} e^{3} x^{4}-30 b \,c^{2} e^{3} x^{2}}{420 c^{7}} \] Input:
int((e*x^2+d)^3*(a+b*atan(c*x)),x)
Output:
(420*atan(c*x)*b*c**7*d**3*x + 420*atan(c*x)*b*c**7*d**2*e*x**3 + 252*atan (c*x)*b*c**7*d*e**2*x**5 + 60*atan(c*x)*b*c**7*e**3*x**7 - 210*log(c**2*x* *2 + 1)*b*c**6*d**3 + 210*log(c**2*x**2 + 1)*b*c**4*d**2*e - 126*log(c**2* x**2 + 1)*b*c**2*d*e**2 + 30*log(c**2*x**2 + 1)*b*e**3 + 420*a*c**7*d**3*x + 420*a*c**7*d**2*e*x**3 + 252*a*c**7*d*e**2*x**5 + 60*a*c**7*e**3*x**7 - 210*b*c**6*d**2*e*x**2 - 63*b*c**6*d*e**2*x**4 - 10*b*c**6*e**3*x**6 + 12 6*b*c**4*d*e**2*x**2 + 15*b*c**4*e**3*x**4 - 30*b*c**2*e**3*x**2)/(420*c** 7)