3.33 \(\int x (d+c d x)^4 (a+b \tanh ^{-1}(c x)) \, dx\)

Optimal. Leaf size=153 \[ \frac {d^4 (c x+1)^6 \left (a+b \tanh ^{-1}(c x)\right )}{6 c^2}-\frac {d^4 (c x+1)^5 \left (a+b \tanh ^{-1}(c x)\right )}{5 c^2}+\frac {b d^4 (c x+1)^5}{30 c^2}+\frac {b d^4 (c x+1)^4}{30 c^2}+\frac {4 b d^4 (c x+1)^3}{45 c^2}+\frac {4 b d^4 (c x+1)^2}{15 c^2}+\frac {32 b d^4 \log (1-c x)}{15 c^2}+\frac {16 b d^4 x}{15 c} \]

[Out]

16/15*b*d^4*x/c+4/15*b*d^4*(c*x+1)^2/c^2+4/45*b*d^4*(c*x+1)^3/c^2+1/30*b*d^4*(c*x+1)^4/c^2+1/30*b*d^4*(c*x+1)^
5/c^2-1/5*d^4*(c*x+1)^5*(a+b*arctanh(c*x))/c^2+1/6*d^4*(c*x+1)^6*(a+b*arctanh(c*x))/c^2+32/15*b*d^4*ln(-c*x+1)
/c^2

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Rubi [A]  time = 0.12, antiderivative size = 153, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {43, 5936, 12, 77} \[ \frac {d^4 (c x+1)^6 \left (a+b \tanh ^{-1}(c x)\right )}{6 c^2}-\frac {d^4 (c x+1)^5 \left (a+b \tanh ^{-1}(c x)\right )}{5 c^2}+\frac {b d^4 (c x+1)^5}{30 c^2}+\frac {b d^4 (c x+1)^4}{30 c^2}+\frac {4 b d^4 (c x+1)^3}{45 c^2}+\frac {4 b d^4 (c x+1)^2}{15 c^2}+\frac {32 b d^4 \log (1-c x)}{15 c^2}+\frac {16 b d^4 x}{15 c} \]

Antiderivative was successfully verified.

[In]

Int[x*(d + c*d*x)^4*(a + b*ArcTanh[c*x]),x]

[Out]

(16*b*d^4*x)/(15*c) + (4*b*d^4*(1 + c*x)^2)/(15*c^2) + (4*b*d^4*(1 + c*x)^3)/(45*c^2) + (b*d^4*(1 + c*x)^4)/(3
0*c^2) + (b*d^4*(1 + c*x)^5)/(30*c^2) - (d^4*(1 + c*x)^5*(a + b*ArcTanh[c*x]))/(5*c^2) + (d^4*(1 + c*x)^6*(a +
b*ArcTanh[c*x]))/(6*c^2) + (32*b*d^4*Log[1 - c*x])/(15*c^2)

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 77

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))

Rule 5936

Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))*((f_.)*(x_))^(m_.)*((d_.) + (e_.)*(x_))^(q_.), x_Symbol] :> With[{u =
IntHide[(f*x)^m*(d + e*x)^q, x]}, Dist[a + b*ArcTanh[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 (d+c d x)^4 \left (a+b \tanh ^{-1}(c x)\right ) \, dx &=-\frac {d^4 (1+c x)^5 \left (a+b \tanh ^{-1}(c x)\right )}{5 c^2}+\frac {d^4 (1+c x)^6 \left (a+b \tanh ^{-1}(c x)\right )}{6 c^2}-(b c) \int \frac {(-1+5 c x) (d+c d x)^4}{30 c^2 (1-c x)} \, dx\\ &=-\frac {d^4 (1+c x)^5 \left (a+b \tanh ^{-1}(c x)\right )}{5 c^2}+\frac {d^4 (1+c x)^6 \left (a+b \tanh ^{-1}(c x)\right )}{6 c^2}-\frac {b \int \frac {(-1+5 c x) (d+c d x)^4}{1-c x} \, dx}{30 c}\\ &=-\frac {d^4 (1+c x)^5 \left (a+b \tanh ^{-1}(c x)\right )}{5 c^2}+\frac {d^4 (1+c x)^6 \left (a+b \tanh ^{-1}(c x)\right )}{6 c^2}-\frac {b \int \left (-32 d^4-\frac {64 d^4}{-1+c x}-16 d^3 (d+c d x)-8 d^2 (d+c d x)^2-4 d (d+c d x)^3-5 (d+c d x)^4\right ) \, dx}{30 c}\\ &=\frac {16 b d^4 x}{15 c}+\frac {4 b d^4 (1+c x)^2}{15 c^2}+\frac {4 b d^4 (1+c x)^3}{45 c^2}+\frac {b d^4 (1+c x)^4}{30 c^2}+\frac {b d^4 (1+c x)^5}{30 c^2}-\frac {d^4 (1+c x)^5 \left (a+b \tanh ^{-1}(c x)\right )}{5 c^2}+\frac {d^4 (1+c x)^6 \left (a+b \tanh ^{-1}(c x)\right )}{6 c^2}+\frac {32 b d^4 \log (1-c x)}{15 c^2}\\ \end {align*}

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Mathematica [A]  time = 0.14, size = 159, normalized size = 1.04 \[ \frac {d^4 \left (30 a c^6 x^6+144 a c^5 x^5+270 a c^4 x^4+240 a c^3 x^3+90 a c^2 x^2+6 b c^5 x^5+36 b c^4 x^4+100 b c^3 x^3+192 b c^2 x^2+6 b c^2 x^2 \left (5 c^4 x^4+24 c^3 x^3+45 c^2 x^2+40 c x+15\right ) \tanh ^{-1}(c x)+390 b c x+387 b \log (1-c x)-3 b \log (c x+1)\right )}{180 c^2} \]

Antiderivative was successfully verified.

[In]

Integrate[x*(d + c*d*x)^4*(a + b*ArcTanh[c*x]),x]

[Out]

(d^4*(390*b*c*x + 90*a*c^2*x^2 + 192*b*c^2*x^2 + 240*a*c^3*x^3 + 100*b*c^3*x^3 + 270*a*c^4*x^4 + 36*b*c^4*x^4
+ 144*a*c^5*x^5 + 6*b*c^5*x^5 + 30*a*c^6*x^6 + 6*b*c^2*x^2*(15 + 40*c*x + 45*c^2*x^2 + 24*c^3*x^3 + 5*c^4*x^4)
*ArcTanh[c*x] + 387*b*Log[1 - c*x] - 3*b*Log[1 + c*x]))/(180*c^2)

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fricas [A]  time = 0.50, size = 198, normalized size = 1.29 \[ \frac {30 \, a c^{6} d^{4} x^{6} + 6 \, {\left (24 \, a + b\right )} c^{5} d^{4} x^{5} + 18 \, {\left (15 \, a + 2 \, b\right )} c^{4} d^{4} x^{4} + 20 \, {\left (12 \, a + 5 \, b\right )} c^{3} d^{4} x^{3} + 6 \, {\left (15 \, a + 32 \, b\right )} c^{2} d^{4} x^{2} + 390 \, b c d^{4} x - 3 \, b d^{4} \log \left (c x + 1\right ) + 387 \, b d^{4} \log \left (c x - 1\right ) + 3 \, {\left (5 \, b c^{6} d^{4} x^{6} + 24 \, b c^{5} d^{4} x^{5} + 45 \, b c^{4} d^{4} x^{4} + 40 \, b c^{3} d^{4} x^{3} + 15 \, b c^{2} d^{4} x^{2}\right )} \log \left (-\frac {c x + 1}{c x - 1}\right )}{180 \, c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/180*(30*a*c^6*d^4*x^6 + 6*(24*a + b)*c^5*d^4*x^5 + 18*(15*a + 2*b)*c^4*d^4*x^4 + 20*(12*a + 5*b)*c^3*d^4*x^3
+ 6*(15*a + 32*b)*c^2*d^4*x^2 + 390*b*c*d^4*x - 3*b*d^4*log(c*x + 1) + 387*b*d^4*log(c*x - 1) + 3*(5*b*c^6*d^
4*x^6 + 24*b*c^5*d^4*x^5 + 45*b*c^4*d^4*x^4 + 40*b*c^3*d^4*x^3 + 15*b*c^2*d^4*x^2)*log(-(c*x + 1)/(c*x - 1)))/
c^2

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giac [B]  time = 0.28, size = 621, normalized size = 4.06 \[ -\frac {8}{45} \, {\left (\frac {12 \, b d^{4} \log \left (-\frac {c x + 1}{c x - 1} + 1\right )}{c^{3}} - \frac {12 \, b d^{4} \log \left (-\frac {c x + 1}{c x - 1}\right )}{c^{3}} - \frac {6 \, {\left (\frac {15 \, {\left (c x + 1\right )}^{5} b d^{4}}{{\left (c x - 1\right )}^{5}} - \frac {30 \, {\left (c x + 1\right )}^{4} b d^{4}}{{\left (c x - 1\right )}^{4}} + \frac {40 \, {\left (c x + 1\right )}^{3} b d^{4}}{{\left (c x - 1\right )}^{3}} - \frac {30 \, {\left (c x + 1\right )}^{2} b d^{4}}{{\left (c x - 1\right )}^{2}} + \frac {12 \, {\left (c x + 1\right )} b d^{4}}{c x - 1} - 2 \, b d^{4}\right )} \log \left (-\frac {c x + 1}{c x - 1}\right )}{\frac {{\left (c x + 1\right )}^{6} c^{3}}{{\left (c x - 1\right )}^{6}} - \frac {6 \, {\left (c x + 1\right )}^{5} c^{3}}{{\left (c x - 1\right )}^{5}} + \frac {15 \, {\left (c x + 1\right )}^{4} c^{3}}{{\left (c x - 1\right )}^{4}} - \frac {20 \, {\left (c x + 1\right )}^{3} c^{3}}{{\left (c x - 1\right )}^{3}} + \frac {15 \, {\left (c x + 1\right )}^{2} c^{3}}{{\left (c x - 1\right )}^{2}} - \frac {6 \, {\left (c x + 1\right )} c^{3}}{c x - 1} + c^{3}} - \frac {\frac {180 \, {\left (c x + 1\right )}^{5} a d^{4}}{{\left (c x - 1\right )}^{5}} - \frac {360 \, {\left (c x + 1\right )}^{4} a d^{4}}{{\left (c x - 1\right )}^{4}} + \frac {480 \, {\left (c x + 1\right )}^{3} a d^{4}}{{\left (c x - 1\right )}^{3}} - \frac {360 \, {\left (c x + 1\right )}^{2} a d^{4}}{{\left (c x - 1\right )}^{2}} + \frac {144 \, {\left (c x + 1\right )} a d^{4}}{c x - 1} - 24 \, a d^{4} + \frac {78 \, {\left (c x + 1\right )}^{5} b d^{4}}{{\left (c x - 1\right )}^{5}} - \frac {294 \, {\left (c x + 1\right )}^{4} b d^{4}}{{\left (c x - 1\right )}^{4}} + \frac {472 \, {\left (c x + 1\right )}^{3} b d^{4}}{{\left (c x - 1\right )}^{3}} - \frac {399 \, {\left (c x + 1\right )}^{2} b d^{4}}{{\left (c x - 1\right )}^{2}} + \frac {174 \, {\left (c x + 1\right )} b d^{4}}{c x - 1} - 31 \, b d^{4}}{\frac {{\left (c x + 1\right )}^{6} c^{3}}{{\left (c x - 1\right )}^{6}} - \frac {6 \, {\left (c x + 1\right )}^{5} c^{3}}{{\left (c x - 1\right )}^{5}} + \frac {15 \, {\left (c x + 1\right )}^{4} c^{3}}{{\left (c x - 1\right )}^{4}} - \frac {20 \, {\left (c x + 1\right )}^{3} c^{3}}{{\left (c x - 1\right )}^{3}} + \frac {15 \, {\left (c x + 1\right )}^{2} c^{3}}{{\left (c x - 1\right )}^{2}} - \frac {6 \, {\left (c x + 1\right )} c^{3}}{c x - 1} + c^{3}}\right )} c \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-8/45*(12*b*d^4*log(-(c*x + 1)/(c*x - 1) + 1)/c^3 - 12*b*d^4*log(-(c*x + 1)/(c*x - 1))/c^3 - 6*(15*(c*x + 1)^5
*b*d^4/(c*x - 1)^5 - 30*(c*x + 1)^4*b*d^4/(c*x - 1)^4 + 40*(c*x + 1)^3*b*d^4/(c*x - 1)^3 - 30*(c*x + 1)^2*b*d^
4/(c*x - 1)^2 + 12*(c*x + 1)*b*d^4/(c*x - 1) - 2*b*d^4)*log(-(c*x + 1)/(c*x - 1))/((c*x + 1)^6*c^3/(c*x - 1)^6
- 6*(c*x + 1)^5*c^3/(c*x - 1)^5 + 15*(c*x + 1)^4*c^3/(c*x - 1)^4 - 20*(c*x + 1)^3*c^3/(c*x - 1)^3 + 15*(c*x +
1)^2*c^3/(c*x - 1)^2 - 6*(c*x + 1)*c^3/(c*x - 1) + c^3) - (180*(c*x + 1)^5*a*d^4/(c*x - 1)^5 - 360*(c*x + 1)^
4*a*d^4/(c*x - 1)^4 + 480*(c*x + 1)^3*a*d^4/(c*x - 1)^3 - 360*(c*x + 1)^2*a*d^4/(c*x - 1)^2 + 144*(c*x + 1)*a*
d^4/(c*x - 1) - 24*a*d^4 + 78*(c*x + 1)^5*b*d^4/(c*x - 1)^5 - 294*(c*x + 1)^4*b*d^4/(c*x - 1)^4 + 472*(c*x + 1
)^3*b*d^4/(c*x - 1)^3 - 399*(c*x + 1)^2*b*d^4/(c*x - 1)^2 + 174*(c*x + 1)*b*d^4/(c*x - 1) - 31*b*d^4)/((c*x +
1)^6*c^3/(c*x - 1)^6 - 6*(c*x + 1)^5*c^3/(c*x - 1)^5 + 15*(c*x + 1)^4*c^3/(c*x - 1)^4 - 20*(c*x + 1)^3*c^3/(c*
x - 1)^3 + 15*(c*x + 1)^2*c^3/(c*x - 1)^2 - 6*(c*x + 1)*c^3/(c*x - 1) + c^3))*c

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maple [A]  time = 0.03, size = 215, normalized size = 1.41 \[ \frac {c^{4} d^{4} a \,x^{6}}{6}+\frac {4 c^{3} d^{4} a \,x^{5}}{5}+\frac {3 c^{2} d^{4} a \,x^{4}}{2}+\frac {4 c \,d^{4} a \,x^{3}}{3}+\frac {d^{4} a \,x^{2}}{2}+\frac {c^{4} d^{4} b \arctanh \left (c x \right ) x^{6}}{6}+\frac {4 c^{3} d^{4} b \arctanh \left (c x \right ) x^{5}}{5}+\frac {3 c^{2} d^{4} b \arctanh \left (c x \right ) x^{4}}{2}+\frac {4 c \,d^{4} b \arctanh \left (c x \right ) x^{3}}{3}+\frac {d^{4} b \arctanh \left (c x \right ) x^{2}}{2}+\frac {c^{3} d^{4} b \,x^{5}}{30}+\frac {c^{2} d^{4} b \,x^{4}}{5}+\frac {5 c \,d^{4} b \,x^{3}}{9}+\frac {16 d^{4} b \,x^{2}}{15}+\frac {13 b \,d^{4} x}{6 c}+\frac {43 d^{4} b \ln \left (c x -1\right )}{20 c^{2}}-\frac {d^{4} b \ln \left (c x +1\right )}{60 c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*(c*d*x+d)^4*(a+b*arctanh(c*x)),x)

[Out]

1/6*c^4*d^4*a*x^6+4/5*c^3*d^4*a*x^5+3/2*c^2*d^4*a*x^4+4/3*c*d^4*a*x^3+1/2*d^4*a*x^2+1/6*c^4*d^4*b*arctanh(c*x)
*x^6+4/5*c^3*d^4*b*arctanh(c*x)*x^5+3/2*c^2*d^4*b*arctanh(c*x)*x^4+4/3*c*d^4*b*arctanh(c*x)*x^3+1/2*d^4*b*arct
anh(c*x)*x^2+1/30*c^3*d^4*b*x^5+1/5*c^2*d^4*b*x^4+5/9*c*d^4*b*x^3+16/15*d^4*b*x^2+13/6*b*d^4*x/c+43/20/c^2*d^4
*b*ln(c*x-1)-1/60/c^2*d^4*b*ln(c*x+1)

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maxima [B]  time = 0.34, size = 326, normalized size = 2.13 \[ \frac {1}{6} \, a c^{4} d^{4} x^{6} + \frac {4}{5} \, a c^{3} d^{4} x^{5} + \frac {3}{2} \, a c^{2} d^{4} x^{4} + \frac {1}{180} \, {\left (30 \, x^{6} \operatorname {artanh}\left (c x\right ) + c {\left (\frac {2 \, {\left (3 \, c^{4} x^{5} + 5 \, c^{2} x^{3} + 15 \, x\right )}}{c^{6}} - \frac {15 \, \log \left (c x + 1\right )}{c^{7}} + \frac {15 \, \log \left (c x - 1\right )}{c^{7}}\right )}\right )} b c^{4} d^{4} + \frac {1}{5} \, {\left (4 \, x^{5} \operatorname {artanh}\left (c x\right ) + c {\left (\frac {c^{2} x^{4} + 2 \, x^{2}}{c^{4}} + \frac {2 \, \log \left (c^{2} x^{2} - 1\right )}{c^{6}}\right )}\right )} b c^{3} d^{4} + \frac {4}{3} \, a c d^{4} x^{3} + \frac {1}{4} \, {\left (6 \, x^{4} \operatorname {artanh}\left (c x\right ) + c {\left (\frac {2 \, {\left (c^{2} x^{3} + 3 \, x\right )}}{c^{4}} - \frac {3 \, \log \left (c x + 1\right )}{c^{5}} + \frac {3 \, \log \left (c x - 1\right )}{c^{5}}\right )}\right )} b c^{2} d^{4} + \frac {2}{3} \, {\left (2 \, x^{3} \operatorname {artanh}\left (c x\right ) + c {\left (\frac {x^{2}}{c^{2}} + \frac {\log \left (c^{2} x^{2} - 1\right )}{c^{4}}\right )}\right )} b c d^{4} + \frac {1}{2} \, a d^{4} x^{2} + \frac {1}{4} \, {\left (2 \, x^{2} \operatorname {artanh}\left (c x\right ) + c {\left (\frac {2 \, x}{c^{2}} - \frac {\log \left (c x + 1\right )}{c^{3}} + \frac {\log \left (c x - 1\right )}{c^{3}}\right )}\right )} b d^{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*a*c^4*d^4*x^6 + 4/5*a*c^3*d^4*x^5 + 3/2*a*c^2*d^4*x^4 + 1/180*(30*x^6*arctanh(c*x) + c*(2*(3*c^4*x^5 + 5*c
^2*x^3 + 15*x)/c^6 - 15*log(c*x + 1)/c^7 + 15*log(c*x - 1)/c^7))*b*c^4*d^4 + 1/5*(4*x^5*arctanh(c*x) + c*((c^2
*x^4 + 2*x^2)/c^4 + 2*log(c^2*x^2 - 1)/c^6))*b*c^3*d^4 + 4/3*a*c*d^4*x^3 + 1/4*(6*x^4*arctanh(c*x) + c*(2*(c^2
*x^3 + 3*x)/c^4 - 3*log(c*x + 1)/c^5 + 3*log(c*x - 1)/c^5))*b*c^2*d^4 + 2/3*(2*x^3*arctanh(c*x) + c*(x^2/c^2 +
log(c^2*x^2 - 1)/c^4))*b*c*d^4 + 1/2*a*d^4*x^2 + 1/4*(2*x^2*arctanh(c*x) + c*(2*x/c^2 - log(c*x + 1)/c^3 + lo
g(c*x - 1)/c^3))*b*d^4

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mupad [B]  time = 1.05, size = 185, normalized size = 1.21 \[ \frac {d^4\,\left (45\,a\,x^2+96\,b\,x^2+45\,b\,x^2\,\mathrm {atanh}\left (c\,x\right )\right )}{90}-\frac {\frac {d^4\,\left (195\,b\,\mathrm {atanh}\left (c\,x\right )-96\,b\,\ln \left (c^2\,x^2-1\right )\right )}{90}-\frac {13\,b\,c\,d^4\,x}{6}}{c^2}+\frac {c^4\,d^4\,\left (15\,a\,x^6+15\,b\,x^6\,\mathrm {atanh}\left (c\,x\right )\right )}{90}+\frac {c\,d^4\,\left (120\,a\,x^3+50\,b\,x^3+120\,b\,x^3\,\mathrm {atanh}\left (c\,x\right )\right )}{90}+\frac {c^3\,d^4\,\left (72\,a\,x^5+3\,b\,x^5+72\,b\,x^5\,\mathrm {atanh}\left (c\,x\right )\right )}{90}+\frac {c^2\,d^4\,\left (135\,a\,x^4+18\,b\,x^4+135\,b\,x^4\,\mathrm {atanh}\left (c\,x\right )\right )}{90} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*(a + b*atanh(c*x))*(d + c*d*x)^4,x)

[Out]

(d^4*(45*a*x^2 + 96*b*x^2 + 45*b*x^2*atanh(c*x)))/90 - ((d^4*(195*b*atanh(c*x) - 96*b*log(c^2*x^2 - 1)))/90 -
(13*b*c*d^4*x)/6)/c^2 + (c^4*d^4*(15*a*x^6 + 15*b*x^6*atanh(c*x)))/90 + (c*d^4*(120*a*x^3 + 50*b*x^3 + 120*b*x
^3*atanh(c*x)))/90 + (c^3*d^4*(72*a*x^5 + 3*b*x^5 + 72*b*x^5*atanh(c*x)))/90 + (c^2*d^4*(135*a*x^4 + 18*b*x^4
+ 135*b*x^4*atanh(c*x)))/90

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sympy [A]  time = 2.36, size = 269, normalized size = 1.76 \[ \begin {cases} \frac {a c^{4} d^{4} x^{6}}{6} + \frac {4 a c^{3} d^{4} x^{5}}{5} + \frac {3 a c^{2} d^{4} x^{4}}{2} + \frac {4 a c d^{4} x^{3}}{3} + \frac {a d^{4} x^{2}}{2} + \frac {b c^{4} d^{4} x^{6} \operatorname {atanh}{\left (c x \right )}}{6} + \frac {4 b c^{3} d^{4} x^{5} \operatorname {atanh}{\left (c x \right )}}{5} + \frac {b c^{3} d^{4} x^{5}}{30} + \frac {3 b c^{2} d^{4} x^{4} \operatorname {atanh}{\left (c x \right )}}{2} + \frac {b c^{2} d^{4} x^{4}}{5} + \frac {4 b c d^{4} x^{3} \operatorname {atanh}{\left (c x \right )}}{3} + \frac {5 b c d^{4} x^{3}}{9} + \frac {b d^{4} x^{2} \operatorname {atanh}{\left (c x \right )}}{2} + \frac {16 b d^{4} x^{2}}{15} + \frac {13 b d^{4} x}{6 c} + \frac {32 b d^{4} \log {\left (x - \frac {1}{c} \right )}}{15 c^{2}} - \frac {b d^{4} \operatorname {atanh}{\left (c x \right )}}{30 c^{2}} & \text {for}\: c \neq 0 \\\frac {a d^{4} x^{2}}{2} & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(c*d*x+d)**4*(a+b*atanh(c*x)),x)

[Out]

Piecewise((a*c**4*d**4*x**6/6 + 4*a*c**3*d**4*x**5/5 + 3*a*c**2*d**4*x**4/2 + 4*a*c*d**4*x**3/3 + a*d**4*x**2/
2 + b*c**4*d**4*x**6*atanh(c*x)/6 + 4*b*c**3*d**4*x**5*atanh(c*x)/5 + b*c**3*d**4*x**5/30 + 3*b*c**2*d**4*x**4
*atanh(c*x)/2 + b*c**2*d**4*x**4/5 + 4*b*c*d**4*x**3*atanh(c*x)/3 + 5*b*c*d**4*x**3/9 + b*d**4*x**2*atanh(c*x)
/2 + 16*b*d**4*x**2/15 + 13*b*d**4*x/(6*c) + 32*b*d**4*log(x - 1/c)/(15*c**2) - b*d**4*atanh(c*x)/(30*c**2), N
e(c, 0)), (a*d**4*x**2/2, True))

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