3.32 \(\int x^2 (d+i c d x)^4 (a+b \tan ^{-1}(c x)) \, dx\)

Optimal. Leaf size=193 \[ \frac {i d^4 (1+i c x)^7 \left (a+b \tan ^{-1}(c x)\right )}{7 c^3}-\frac {i d^4 (1+i c x)^6 \left (a+b \tan ^{-1}(c x)\right )}{3 c^3}+\frac {i d^4 (1+i c x)^5 \left (a+b \tan ^{-1}(c x)\right )}{5 c^3}-\frac {1}{42} b c^3 d^4 x^6+\frac {176 b d^4 \log (c x+i)}{105 c^3}+\frac {2}{15} i b c^2 d^4 x^5+\frac {5 i b d^4 x}{3 c^2}+\frac {47}{140} b c d^4 x^4-\frac {88 b d^4 x^2}{105 c}-\frac {5}{9} i b d^4 x^3 \]

[Out]

5/3*I*b*d^4*x/c^2-88/105*b*d^4*x^2/c-5/9*I*b*d^4*x^3+47/140*b*c*d^4*x^4+2/15*I*b*c^2*d^4*x^5-1/42*b*c^3*d^4*x^
6+1/5*I*d^4*(1+I*c*x)^5*(a+b*arctan(c*x))/c^3-1/3*I*d^4*(1+I*c*x)^6*(a+b*arctan(c*x))/c^3+1/7*I*d^4*(1+I*c*x)^
7*(a+b*arctan(c*x))/c^3+176/105*b*d^4*ln(I+c*x)/c^3

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Rubi [A]  time = 0.17, antiderivative size = 193, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {43, 4872, 12, 893} \[ \frac {i d^4 (1+i c x)^7 \left (a+b \tan ^{-1}(c x)\right )}{7 c^3}-\frac {i d^4 (1+i c x)^6 \left (a+b \tan ^{-1}(c x)\right )}{3 c^3}+\frac {i d^4 (1+i c x)^5 \left (a+b \tan ^{-1}(c x)\right )}{5 c^3}-\frac {1}{42} b c^3 d^4 x^6+\frac {2}{15} i b c^2 d^4 x^5+\frac {5 i b d^4 x}{3 c^2}+\frac {176 b d^4 \log (c x+i)}{105 c^3}+\frac {47}{140} b c d^4 x^4-\frac {88 b d^4 x^2}{105 c}-\frac {5}{9} i b d^4 x^3 \]

Antiderivative was successfully verified.

[In]

Int[x^2*(d + I*c*d*x)^4*(a + b*ArcTan[c*x]),x]

[Out]

(((5*I)/3)*b*d^4*x)/c^2 - (88*b*d^4*x^2)/(105*c) - ((5*I)/9)*b*d^4*x^3 + (47*b*c*d^4*x^4)/140 + ((2*I)/15)*b*c
^2*d^4*x^5 - (b*c^3*d^4*x^6)/42 + ((I/5)*d^4*(1 + I*c*x)^5*(a + b*ArcTan[c*x]))/c^3 - ((I/3)*d^4*(1 + I*c*x)^6
*(a + b*ArcTan[c*x]))/c^3 + ((I/7)*d^4*(1 + I*c*x)^7*(a + b*ArcTan[c*x]))/c^3 + (176*b*d^4*Log[I + c*x])/(105*
c^3)

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 893

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :
> Int[ExpandIntegrand[(d + e*x)^m*(f + g*x)^n*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] &
& NeQ[e*f - d*g, 0] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IntegerQ[p] && ((EqQ[p, 1] && I
ntegersQ[m, n]) || (ILtQ[m, 0] && ILtQ[n, 0]))

Rule 4872

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))*((f_.)*(x_))^(m_.)*((d_.) + (e_.)*(x_))^(q_.), x_Symbol] :> With[{u = I
ntHide[(f*x)^m*(d + e*x)^q, x]}, Dist[a + b*ArcTan[c*x], u, x] - Dist[b*c, Int[SimplifyIntegrand[u/(1 + c^2*x^
2), x], x], x]] /; FreeQ[{a, b, c, d, e, f, q}, x] && NeQ[q, -1] && IntegerQ[2*m] && ((IGtQ[m, 0] && IGtQ[q, 0
]) || (ILtQ[m + q + 1, 0] && LtQ[m*q, 0]))

Rubi steps

\begin {align*} \int x^2 (d+i c d x)^4 \left (a+b \tan ^{-1}(c x)\right ) \, dx &=\frac {i d^4 (1+i c x)^5 \left (a+b \tan ^{-1}(c x)\right )}{5 c^3}-\frac {i d^4 (1+i c x)^6 \left (a+b \tan ^{-1}(c x)\right )}{3 c^3}+\frac {i d^4 (1+i c x)^7 \left (a+b \tan ^{-1}(c x)\right )}{7 c^3}-(b c) \int \frac {d^4 (i-c x)^4 \left (-1+5 i c x+15 c^2 x^2\right )}{105 c^3 (i+c x)} \, dx\\ &=\frac {i d^4 (1+i c x)^5 \left (a+b \tan ^{-1}(c x)\right )}{5 c^3}-\frac {i d^4 (1+i c x)^6 \left (a+b \tan ^{-1}(c x)\right )}{3 c^3}+\frac {i d^4 (1+i c x)^7 \left (a+b \tan ^{-1}(c x)\right )}{7 c^3}-\frac {\left (b d^4\right ) \int \frac {(i-c x)^4 \left (-1+5 i c x+15 c^2 x^2\right )}{i+c x} \, dx}{105 c^2}\\ &=\frac {i d^4 (1+i c x)^5 \left (a+b \tan ^{-1}(c x)\right )}{5 c^3}-\frac {i d^4 (1+i c x)^6 \left (a+b \tan ^{-1}(c x)\right )}{3 c^3}+\frac {i d^4 (1+i c x)^7 \left (a+b \tan ^{-1}(c x)\right )}{7 c^3}-\frac {\left (b d^4\right ) \int \left (-175 i+176 c x+175 i c^2 x^2-141 c^3 x^3-70 i c^4 x^4+15 c^5 x^5-\frac {176}{i+c x}\right ) \, dx}{105 c^2}\\ &=\frac {5 i b d^4 x}{3 c^2}-\frac {88 b d^4 x^2}{105 c}-\frac {5}{9} i b d^4 x^3+\frac {47}{140} b c d^4 x^4+\frac {2}{15} i b c^2 d^4 x^5-\frac {1}{42} b c^3 d^4 x^6+\frac {i d^4 (1+i c x)^5 \left (a+b \tan ^{-1}(c x)\right )}{5 c^3}-\frac {i d^4 (1+i c x)^6 \left (a+b \tan ^{-1}(c x)\right )}{3 c^3}+\frac {i d^4 (1+i c x)^7 \left (a+b \tan ^{-1}(c x)\right )}{7 c^3}+\frac {176 b d^4 \log (i+c x)}{105 c^3}\\ \end {align*}

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Mathematica [A]  time = 0.13, size = 276, normalized size = 1.43 \[ \frac {1}{7} a c^4 d^4 x^7-\frac {2}{3} i a c^3 d^4 x^6-\frac {6}{5} a c^2 d^4 x^5+i a c d^4 x^4+\frac {1}{3} a d^4 x^3+\frac {1}{7} b c^4 d^4 x^7 \tan ^{-1}(c x)-\frac {1}{42} b c^3 d^4 x^6-\frac {2}{3} i b c^3 d^4 x^6 \tan ^{-1}(c x)-\frac {5 i b d^4 \tan ^{-1}(c x)}{3 c^3}+\frac {2}{15} i b c^2 d^4 x^5-\frac {6}{5} b c^2 d^4 x^5 \tan ^{-1}(c x)+\frac {5 i b d^4 x}{3 c^2}+\frac {88 b d^4 \log \left (c^2 x^2+1\right )}{105 c^3}+\frac {47}{140} b c d^4 x^4+i b c d^4 x^4 \tan ^{-1}(c x)+\frac {1}{3} b d^4 x^3 \tan ^{-1}(c x)-\frac {88 b d^4 x^2}{105 c}-\frac {5}{9} i b d^4 x^3 \]

Antiderivative was successfully verified.

[In]

Integrate[x^2*(d + I*c*d*x)^4*(a + b*ArcTan[c*x]),x]

[Out]

(((5*I)/3)*b*d^4*x)/c^2 - (88*b*d^4*x^2)/(105*c) + (a*d^4*x^3)/3 - ((5*I)/9)*b*d^4*x^3 + I*a*c*d^4*x^4 + (47*b
*c*d^4*x^4)/140 - (6*a*c^2*d^4*x^5)/5 + ((2*I)/15)*b*c^2*d^4*x^5 - ((2*I)/3)*a*c^3*d^4*x^6 - (b*c^3*d^4*x^6)/4
2 + (a*c^4*d^4*x^7)/7 - (((5*I)/3)*b*d^4*ArcTan[c*x])/c^3 + (b*d^4*x^3*ArcTan[c*x])/3 + I*b*c*d^4*x^4*ArcTan[c
*x] - (6*b*c^2*d^4*x^5*ArcTan[c*x])/5 - ((2*I)/3)*b*c^3*d^4*x^6*ArcTan[c*x] + (b*c^4*d^4*x^7*ArcTan[c*x])/7 +
(88*b*d^4*Log[1 + c^2*x^2])/(105*c^3)

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fricas [A]  time = 0.45, size = 217, normalized size = 1.12 \[ \frac {180 \, a c^{7} d^{4} x^{7} + {\left (-840 i \, a - 30 \, b\right )} c^{6} d^{4} x^{6} - 168 \, {\left (9 \, a - i \, b\right )} c^{5} d^{4} x^{5} + {\left (1260 i \, a + 423 \, b\right )} c^{4} d^{4} x^{4} + 140 \, {\left (3 \, a - 5 i \, b\right )} c^{3} d^{4} x^{3} - 1056 \, b c^{2} d^{4} x^{2} + 2100 i \, b c d^{4} x + 2106 \, b d^{4} \log \left (\frac {c x + i}{c}\right ) + 6 \, b d^{4} \log \left (\frac {c x - i}{c}\right ) + {\left (90 i \, b c^{7} d^{4} x^{7} + 420 \, b c^{6} d^{4} x^{6} - 756 i \, b c^{5} d^{4} x^{5} - 630 \, b c^{4} d^{4} x^{4} + 210 i \, b c^{3} d^{4} x^{3}\right )} \log \left (-\frac {c x + i}{c x - i}\right )}{1260 \, c^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(d+I*c*d*x)^4*(a+b*arctan(c*x)),x, algorithm="fricas")

[Out]

1/1260*(180*a*c^7*d^4*x^7 + (-840*I*a - 30*b)*c^6*d^4*x^6 - 168*(9*a - I*b)*c^5*d^4*x^5 + (1260*I*a + 423*b)*c
^4*d^4*x^4 + 140*(3*a - 5*I*b)*c^3*d^4*x^3 - 1056*b*c^2*d^4*x^2 + 2100*I*b*c*d^4*x + 2106*b*d^4*log((c*x + I)/
c) + 6*b*d^4*log((c*x - I)/c) + (90*I*b*c^7*d^4*x^7 + 420*b*c^6*d^4*x^6 - 756*I*b*c^5*d^4*x^5 - 630*b*c^4*d^4*
x^4 + 210*I*b*c^3*d^4*x^3)*log(-(c*x + I)/(c*x - I)))/c^3

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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]

integrate(x^2*(d+I*c*d*x)^4*(a+b*arctan(c*x)),x, algorithm="giac")

[Out]

sage0*x

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maple [A]  time = 0.05, size = 237, normalized size = 1.23 \[ \frac {c^{4} d^{4} a \,x^{7}}{7}-\frac {5 i d^{4} b \arctan \left (c x \right )}{3 c^{3}}-\frac {6 c^{2} d^{4} a \,x^{5}}{5}-\frac {2 i c^{3} d^{4} b \arctan \left (c x \right ) x^{6}}{3}+\frac {d^{4} a \,x^{3}}{3}+\frac {c^{4} d^{4} b \arctan \left (c x \right ) x^{7}}{7}+i c \,d^{4} a \,x^{4}-\frac {6 c^{2} d^{4} b \arctan \left (c x \right ) x^{5}}{5}-\frac {5 i b \,d^{4} x^{3}}{9}+\frac {d^{4} b \arctan \left (c x \right ) x^{3}}{3}+i c \,d^{4} b \arctan \left (c x \right ) x^{4}-\frac {b \,c^{3} d^{4} x^{6}}{42}+\frac {5 i b \,d^{4} x}{3 c^{2}}+\frac {47 b c \,d^{4} x^{4}}{140}-\frac {2 i c^{3} d^{4} a \,x^{6}}{3}-\frac {88 b \,d^{4} x^{2}}{105 c}+\frac {88 d^{4} b \ln \left (c^{2} x^{2}+1\right )}{105 c^{3}}+\frac {2 i b \,c^{2} d^{4} x^{5}}{15} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*(d+I*c*d*x)^4*(a+b*arctan(c*x)),x)

[Out]

1/7*c^4*d^4*a*x^7-5/3*I/c^3*d^4*b*arctan(c*x)-6/5*c^2*d^4*a*x^5-2/3*I*c^3*d^4*b*arctan(c*x)*x^6+1/3*d^4*a*x^3+
1/7*c^4*d^4*b*arctan(c*x)*x^7+I*c*d^4*a*x^4-6/5*c^2*d^4*b*arctan(c*x)*x^5-5/9*I*b*d^4*x^3+1/3*d^4*b*arctan(c*x
)*x^3+I*c*d^4*b*arctan(c*x)*x^4-1/42*b*c^3*d^4*x^6+5/3*I*b*d^4*x/c^2+47/140*b*c*d^4*x^4-2/3*I*c^3*d^4*a*x^6-88
/105*b*d^4*x^2/c+88/105/c^3*d^4*b*ln(c^2*x^2+1)+2/15*I*b*c^2*d^4*x^5

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maxima [B]  time = 0.41, size = 318, normalized size = 1.65 \[ \frac {1}{7} \, a c^{4} d^{4} x^{7} - \frac {2}{3} i \, a c^{3} d^{4} x^{6} - \frac {6}{5} \, a c^{2} d^{4} x^{5} + \frac {1}{84} \, {\left (12 \, x^{7} \arctan \left (c x\right ) - c {\left (\frac {2 \, c^{4} x^{6} - 3 \, c^{2} x^{4} + 6 \, x^{2}}{c^{6}} - \frac {6 \, \log \left (c^{2} x^{2} + 1\right )}{c^{8}}\right )}\right )} b c^{4} d^{4} + i \, a c d^{4} x^{4} - \frac {2}{45} i \, {\left (15 \, x^{6} \arctan \left (c x\right ) - c {\left (\frac {3 \, c^{4} x^{5} - 5 \, c^{2} x^{3} + 15 \, x}{c^{6}} - \frac {15 \, \arctan \left (c x\right )}{c^{7}}\right )}\right )} b c^{3} d^{4} - \frac {3}{10} \, {\left (4 \, x^{5} \arctan \left (c x\right ) - c {\left (\frac {c^{2} x^{4} - 2 \, x^{2}}{c^{4}} + \frac {2 \, \log \left (c^{2} x^{2} + 1\right )}{c^{6}}\right )}\right )} b c^{2} d^{4} + \frac {1}{3} \, a d^{4} x^{3} + \frac {1}{3} i \, {\left (3 \, x^{4} \arctan \left (c x\right ) - c {\left (\frac {c^{2} x^{3} - 3 \, x}{c^{4}} + \frac {3 \, \arctan \left (c x\right )}{c^{5}}\right )}\right )} b c d^{4} + \frac {1}{6} \, {\left (2 \, x^{3} \arctan \left (c x\right ) - c {\left (\frac {x^{2}}{c^{2}} - \frac {\log \left (c^{2} x^{2} + 1\right )}{c^{4}}\right )}\right )} b d^{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(d+I*c*d*x)^4*(a+b*arctan(c*x)),x, algorithm="maxima")

[Out]

1/7*a*c^4*d^4*x^7 - 2/3*I*a*c^3*d^4*x^6 - 6/5*a*c^2*d^4*x^5 + 1/84*(12*x^7*arctan(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*c^4*d^4 + I*a*c*d^4*x^4 - 2/45*I*(15*x^6*arctan(c*x) - c*((3*c^
4*x^5 - 5*c^2*x^3 + 15*x)/c^6 - 15*arctan(c*x)/c^7))*b*c^3*d^4 - 3/10*(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*c^2*d^4 + 1/3*a*d^4*x^3 + 1/3*I*(3*x^4*arctan(c*x) - c*((c^2*x^3 - 3*x)/c^4
+ 3*arctan(c*x)/c^5))*b*c*d^4 + 1/6*(2*x^3*arctan(c*x) - c*(x^2/c^2 - log(c^2*x^2 + 1)/c^4))*b*d^4

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mupad [B]  time = 0.64, size = 205, normalized size = 1.06 \[ \frac {c^4\,d^4\,\left (180\,a\,x^7+180\,b\,x^7\,\mathrm {atan}\left (c\,x\right )\right )}{1260}+\frac {d^4\,\left (420\,a\,x^3+420\,b\,x^3\,\mathrm {atan}\left (c\,x\right )-b\,x^3\,700{}\mathrm {i}\right )}{1260}-\frac {\frac {d^4\,\left (-1056\,b\,\ln \left (c^2\,x^2+1\right )+b\,\mathrm {atan}\left (c\,x\right )\,2100{}\mathrm {i}\right )}{1260}+\frac {88\,b\,c^2\,d^4\,x^2}{105}-\frac {b\,c\,d^4\,x\,5{}\mathrm {i}}{3}}{c^3}+\frac {c\,d^4\,\left (a\,x^4\,1260{}\mathrm {i}+423\,b\,x^4+b\,x^4\,\mathrm {atan}\left (c\,x\right )\,1260{}\mathrm {i}\right )}{1260}-\frac {c^3\,d^4\,\left (a\,x^6\,840{}\mathrm {i}+30\,b\,x^6+b\,x^6\,\mathrm {atan}\left (c\,x\right )\,840{}\mathrm {i}\right )}{1260}-\frac {c^2\,d^4\,\left (1512\,a\,x^5+1512\,b\,x^5\,\mathrm {atan}\left (c\,x\right )-b\,x^5\,168{}\mathrm {i}\right )}{1260} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*(a + b*atan(c*x))*(d + c*d*x*1i)^4,x)

[Out]

(d^4*(420*a*x^3 - b*x^3*700i + 420*b*x^3*atan(c*x)))/1260 - ((d^4*(b*atan(c*x)*2100i - 1056*b*log(c^2*x^2 + 1)
))/1260 + (88*b*c^2*d^4*x^2)/105 - (b*c*d^4*x*5i)/3)/c^3 + (c^4*d^4*(180*a*x^7 + 180*b*x^7*atan(c*x)))/1260 +
(c*d^4*(a*x^4*1260i + 423*b*x^4 + b*x^4*atan(c*x)*1260i))/1260 - (c^3*d^4*(a*x^6*840i + 30*b*x^6 + b*x^6*atan(
c*x)*840i))/1260 - (c^2*d^4*(1512*a*x^5 - b*x^5*168i + 1512*b*x^5*atan(c*x)))/1260

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sympy [A]  time = 5.99, size = 367, normalized size = 1.90 \[ \frac {a c^{4} d^{4} x^{7}}{7} - \frac {88 b d^{4} x^{2}}{105 c} + \frac {5 i b d^{4} x}{3 c^{2}} + \frac {b d^{4} \left (\frac {\log {\left (2299 b c d^{4} x - 2299 i b d^{4} \right )}}{210} + \frac {769 \log {\left (2299 b c d^{4} x + 2299 i b d^{4} \right )}}{560}\right )}{c^{3}} + x^{6} \left (- \frac {2 i a c^{3} d^{4}}{3} - \frac {b c^{3} d^{4}}{42}\right ) + x^{5} \left (- \frac {6 a c^{2} d^{4}}{5} + \frac {2 i b c^{2} d^{4}}{15}\right ) + x^{4} \left (i a c d^{4} + \frac {47 b c d^{4}}{140}\right ) + x^{3} \left (\frac {a d^{4}}{3} - \frac {5 i b d^{4}}{9}\right ) + \left (- \frac {i b c^{4} d^{4} x^{7}}{14} - \frac {b c^{3} d^{4} x^{6}}{3} + \frac {3 i b c^{2} d^{4} x^{5}}{5} + \frac {b c d^{4} x^{4}}{2} - \frac {i b d^{4} x^{3}}{6}\right ) \log {\left (i c x + 1 \right )} - \frac {\left (- 120 i b c^{7} d^{4} x^{7} - 560 b c^{6} d^{4} x^{6} + 1008 i b c^{5} d^{4} x^{5} + 840 b c^{4} d^{4} x^{4} - 280 i b c^{3} d^{4} x^{3} - 501 b d^{4}\right ) \log {\left (- i c x + 1 \right )}}{1680 c^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**2*(d+I*c*d*x)**4*(a+b*atan(c*x)),x)

[Out]

a*c**4*d**4*x**7/7 - 88*b*d**4*x**2/(105*c) + 5*I*b*d**4*x/(3*c**2) + b*d**4*(log(2299*b*c*d**4*x - 2299*I*b*d
**4)/210 + 769*log(2299*b*c*d**4*x + 2299*I*b*d**4)/560)/c**3 + x**6*(-2*I*a*c**3*d**4/3 - b*c**3*d**4/42) + x
**5*(-6*a*c**2*d**4/5 + 2*I*b*c**2*d**4/15) + x**4*(I*a*c*d**4 + 47*b*c*d**4/140) + x**3*(a*d**4/3 - 5*I*b*d**
4/9) + (-I*b*c**4*d**4*x**7/14 - b*c**3*d**4*x**6/3 + 3*I*b*c**2*d**4*x**5/5 + b*c*d**4*x**4/2 - I*b*d**4*x**3
/6)*log(I*c*x + 1) - (-120*I*b*c**7*d**4*x**7 - 560*b*c**6*d**4*x**6 + 1008*I*b*c**5*d**4*x**5 + 840*b*c**4*d*
*4*x**4 - 280*I*b*c**3*d**4*x**3 - 501*b*d**4)*log(-I*c*x + 1)/(1680*c**3)

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