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

Optimal. Leaf size=171 \[ \frac {d^4 (c x+1)^7 \left (a+b \tanh ^{-1}(c x)\right )}{7 c^3}-\frac {d^4 (c x+1)^6 \left (a+b \tanh ^{-1}(c x)\right )}{3 c^3}+\frac {d^4 (c x+1)^5 \left (a+b \tanh ^{-1}(c x)\right )}{5 c^3}+\frac {1}{42} b c^3 d^4 x^6+\frac {176 b d^4 \log (1-c x)}{105 c^3}+\frac {2}{15} b c^2 d^4 x^5+\frac {5 b d^4 x}{3 c^2}+\frac {47}{140} b c d^4 x^4+\frac {88 b d^4 x^2}{105 c}+\frac {5}{9} b d^4 x^3 \]

[Out]

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

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Rubi [A]  time = 0.18, antiderivative size = 171, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {43, 5936, 12, 893} \[ \frac {d^4 (c x+1)^7 \left (a+b \tanh ^{-1}(c x)\right )}{7 c^3}-\frac {d^4 (c x+1)^6 \left (a+b \tanh ^{-1}(c x)\right )}{3 c^3}+\frac {d^4 (c x+1)^5 \left (a+b \tanh ^{-1}(c x)\right )}{5 c^3}+\frac {1}{42} b c^3 d^4 x^6+\frac {2}{15} b c^2 d^4 x^5+\frac {5 b d^4 x}{3 c^2}+\frac {176 b d^4 \log (1-c x)}{105 c^3}+\frac {47}{140} b c d^4 x^4+\frac {88 b d^4 x^2}{105 c}+\frac {5}{9} b d^4 x^3 \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Mathematica [A]  time = 0.15, size = 168, normalized size = 0.98 \[ \frac {d^4 \left (180 a c^7 x^7+840 a c^6 x^6+1512 a c^5 x^5+1260 a c^4 x^4+420 a c^3 x^3+30 b c^6 x^6+168 b c^5 x^5+423 b c^4 x^4+700 b c^3 x^3+1056 b c^2 x^2+12 b c^3 x^3 \left (15 c^4 x^4+70 c^3 x^3+126 c^2 x^2+105 c x+35\right ) \tanh ^{-1}(c x)+2100 b c x+2106 b \log (1-c x)+6 b \log (c x+1)\right )}{1260 c^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(d^4*(2100*b*c*x + 1056*b*c^2*x^2 + 420*a*c^3*x^3 + 700*b*c^3*x^3 + 1260*a*c^4*x^4 + 423*b*c^4*x^4 + 1512*a*c^
5*x^5 + 168*b*c^5*x^5 + 840*a*c^6*x^6 + 30*b*c^6*x^6 + 180*a*c^7*x^7 + 12*b*c^3*x^3*(35 + 105*c*x + 126*c^2*x^
2 + 70*c^3*x^3 + 15*c^4*x^4)*ArcTanh[c*x] + 2106*b*Log[1 - c*x] + 6*b*Log[1 + c*x]))/(1260*c^3)

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fricas [A]  time = 0.52, size = 208, normalized size = 1.22 \[ \frac {180 \, a c^{7} d^{4} x^{7} + 30 \, {\left (28 \, a + b\right )} c^{6} d^{4} x^{6} + 168 \, {\left (9 \, a + b\right )} c^{5} d^{4} x^{5} + 9 \, {\left (140 \, a + 47 \, b\right )} c^{4} d^{4} x^{4} + 140 \, {\left (3 \, a + 5 \, b\right )} c^{3} d^{4} x^{3} + 1056 \, b c^{2} d^{4} x^{2} + 2100 \, b c d^{4} x + 6 \, b d^{4} \log \left (c x + 1\right ) + 2106 \, b d^{4} \log \left (c x - 1\right ) + 6 \, {\left (15 \, b c^{7} d^{4} x^{7} + 70 \, b c^{6} d^{4} x^{6} + 126 \, b c^{5} d^{4} x^{5} + 105 \, b c^{4} d^{4} x^{4} + 35 \, b c^{3} d^{4} x^{3}\right )} \log \left (-\frac {c x + 1}{c x - 1}\right )}{1260 \, c^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/1260*(180*a*c^7*d^4*x^7 + 30*(28*a + b)*c^6*d^4*x^6 + 168*(9*a + b)*c^5*d^4*x^5 + 9*(140*a + 47*b)*c^4*d^4*x
^4 + 140*(3*a + 5*b)*c^3*d^4*x^3 + 1056*b*c^2*d^4*x^2 + 2100*b*c*d^4*x + 6*b*d^4*log(c*x + 1) + 2106*b*d^4*log
(c*x - 1) + 6*(15*b*c^7*d^4*x^7 + 70*b*c^6*d^4*x^6 + 126*b*c^5*d^4*x^5 + 105*b*c^4*d^4*x^4 + 35*b*c^3*d^4*x^3)
*log(-(c*x + 1)/(c*x - 1)))/c^3

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giac [B]  time = 0.19, size = 723, normalized size = 4.23 \[ -\frac {4}{315} \, {\left (\frac {132 \, b d^{4} \log \left (-\frac {c x + 1}{c x - 1} + 1\right )}{c^{4}} - \frac {132 \, b d^{4} \log \left (-\frac {c x + 1}{c x - 1}\right )}{c^{4}} - \frac {12 \, {\left (\frac {105 \, {\left (c x + 1\right )}^{6} b d^{4}}{{\left (c x - 1\right )}^{6}} - \frac {210 \, {\left (c x + 1\right )}^{5} b d^{4}}{{\left (c x - 1\right )}^{5}} + \frac {385 \, {\left (c x + 1\right )}^{4} b d^{4}}{{\left (c x - 1\right )}^{4}} - \frac {385 \, {\left (c x + 1\right )}^{3} b d^{4}}{{\left (c x - 1\right )}^{3}} + \frac {231 \, {\left (c x + 1\right )}^{2} b d^{4}}{{\left (c x - 1\right )}^{2}} - \frac {77 \, {\left (c x + 1\right )} b d^{4}}{c x - 1} + 11 \, b d^{4}\right )} \log \left (-\frac {c x + 1}{c x - 1}\right )}{\frac {{\left (c x + 1\right )}^{7} c^{4}}{{\left (c x - 1\right )}^{7}} - \frac {7 \, {\left (c x + 1\right )}^{6} c^{4}}{{\left (c x - 1\right )}^{6}} + \frac {21 \, {\left (c x + 1\right )}^{5} c^{4}}{{\left (c x - 1\right )}^{5}} - \frac {35 \, {\left (c x + 1\right )}^{4} c^{4}}{{\left (c x - 1\right )}^{4}} + \frac {35 \, {\left (c x + 1\right )}^{3} c^{4}}{{\left (c x - 1\right )}^{3}} - \frac {21 \, {\left (c x + 1\right )}^{2} c^{4}}{{\left (c x - 1\right )}^{2}} + \frac {7 \, {\left (c x + 1\right )} c^{4}}{c x - 1} - c^{4}} - \frac {\frac {2520 \, {\left (c x + 1\right )}^{6} a d^{4}}{{\left (c x - 1\right )}^{6}} - \frac {5040 \, {\left (c x + 1\right )}^{5} a d^{4}}{{\left (c x - 1\right )}^{5}} + \frac {9240 \, {\left (c x + 1\right )}^{4} a d^{4}}{{\left (c x - 1\right )}^{4}} - \frac {9240 \, {\left (c x + 1\right )}^{3} a d^{4}}{{\left (c x - 1\right )}^{3}} + \frac {5544 \, {\left (c x + 1\right )}^{2} a d^{4}}{{\left (c x - 1\right )}^{2}} - \frac {1848 \, {\left (c x + 1\right )} a d^{4}}{c x - 1} + 264 \, a d^{4} + \frac {1128 \, {\left (c x + 1\right )}^{6} b d^{4}}{{\left (c x - 1\right )}^{6}} - \frac {4812 \, {\left (c x + 1\right )}^{5} b d^{4}}{{\left (c x - 1\right )}^{5}} + \frac {9476 \, {\left (c x + 1\right )}^{4} b d^{4}}{{\left (c x - 1\right )}^{4}} - \frac {10631 \, {\left (c x + 1\right )}^{3} b d^{4}}{{\left (c x - 1\right )}^{3}} + \frac {6933 \, {\left (c x + 1\right )}^{2} b d^{4}}{{\left (c x - 1\right )}^{2}} - \frac {2465 \, {\left (c x + 1\right )} b d^{4}}{c x - 1} + 371 \, b d^{4}}{\frac {{\left (c x + 1\right )}^{7} c^{4}}{{\left (c x - 1\right )}^{7}} - \frac {7 \, {\left (c x + 1\right )}^{6} c^{4}}{{\left (c x - 1\right )}^{6}} + \frac {21 \, {\left (c x + 1\right )}^{5} c^{4}}{{\left (c x - 1\right )}^{5}} - \frac {35 \, {\left (c x + 1\right )}^{4} c^{4}}{{\left (c x - 1\right )}^{4}} + \frac {35 \, {\left (c x + 1\right )}^{3} c^{4}}{{\left (c x - 1\right )}^{3}} - \frac {21 \, {\left (c x + 1\right )}^{2} c^{4}}{{\left (c x - 1\right )}^{2}} + \frac {7 \, {\left (c x + 1\right )} c^{4}}{c x - 1} - c^{4}}\right )} c \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-4/315*(132*b*d^4*log(-(c*x + 1)/(c*x - 1) + 1)/c^4 - 132*b*d^4*log(-(c*x + 1)/(c*x - 1))/c^4 - 12*(105*(c*x +
1)^6*b*d^4/(c*x - 1)^6 - 210*(c*x + 1)^5*b*d^4/(c*x - 1)^5 + 385*(c*x + 1)^4*b*d^4/(c*x - 1)^4 - 385*(c*x + 1
)^3*b*d^4/(c*x - 1)^3 + 231*(c*x + 1)^2*b*d^4/(c*x - 1)^2 - 77*(c*x + 1)*b*d^4/(c*x - 1) + 11*b*d^4)*log(-(c*x
+ 1)/(c*x - 1))/((c*x + 1)^7*c^4/(c*x - 1)^7 - 7*(c*x + 1)^6*c^4/(c*x - 1)^6 + 21*(c*x + 1)^5*c^4/(c*x - 1)^5
- 35*(c*x + 1)^4*c^4/(c*x - 1)^4 + 35*(c*x + 1)^3*c^4/(c*x - 1)^3 - 21*(c*x + 1)^2*c^4/(c*x - 1)^2 + 7*(c*x +
1)*c^4/(c*x - 1) - c^4) - (2520*(c*x + 1)^6*a*d^4/(c*x - 1)^6 - 5040*(c*x + 1)^5*a*d^4/(c*x - 1)^5 + 9240*(c*
x + 1)^4*a*d^4/(c*x - 1)^4 - 9240*(c*x + 1)^3*a*d^4/(c*x - 1)^3 + 5544*(c*x + 1)^2*a*d^4/(c*x - 1)^2 - 1848*(c
*x + 1)*a*d^4/(c*x - 1) + 264*a*d^4 + 1128*(c*x + 1)^6*b*d^4/(c*x - 1)^6 - 4812*(c*x + 1)^5*b*d^4/(c*x - 1)^5
+ 9476*(c*x + 1)^4*b*d^4/(c*x - 1)^4 - 10631*(c*x + 1)^3*b*d^4/(c*x - 1)^3 + 6933*(c*x + 1)^2*b*d^4/(c*x - 1)^
2 - 2465*(c*x + 1)*b*d^4/(c*x - 1) + 371*b*d^4)/((c*x + 1)^7*c^4/(c*x - 1)^7 - 7*(c*x + 1)^6*c^4/(c*x - 1)^6 +
21*(c*x + 1)^5*c^4/(c*x - 1)^5 - 35*(c*x + 1)^4*c^4/(c*x - 1)^4 + 35*(c*x + 1)^3*c^4/(c*x - 1)^3 - 21*(c*x +
1)^2*c^4/(c*x - 1)^2 + 7*(c*x + 1)*c^4/(c*x - 1) - c^4))*c

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maple [A]  time = 0.03, size = 225, normalized size = 1.32 \[ \frac {c^{4} d^{4} a \,x^{7}}{7}+\frac {2 c^{3} d^{4} a \,x^{6}}{3}+\frac {6 c^{2} d^{4} a \,x^{5}}{5}+c \,d^{4} a \,x^{4}+\frac {d^{4} a \,x^{3}}{3}+\frac {c^{4} d^{4} b \arctanh \left (c x \right ) x^{7}}{7}+\frac {2 c^{3} d^{4} b \arctanh \left (c x \right ) x^{6}}{3}+\frac {6 c^{2} d^{4} b \arctanh \left (c x \right ) x^{5}}{5}+c \,d^{4} b \arctanh \left (c x \right ) x^{4}+\frac {d^{4} b \arctanh \left (c x \right ) x^{3}}{3}+\frac {b \,c^{3} d^{4} x^{6}}{42}+\frac {2 b \,c^{2} d^{4} x^{5}}{15}+\frac {47 b c \,d^{4} x^{4}}{140}+\frac {5 b \,d^{4} x^{3}}{9}+\frac {88 b \,d^{4} x^{2}}{105 c}+\frac {5 b \,d^{4} x}{3 c^{2}}+\frac {117 d^{4} b \ln \left (c x -1\right )}{70 c^{3}}+\frac {d^{4} b \ln \left (c x +1\right )}{210 c^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/7*c^4*d^4*a*x^7+2/3*c^3*d^4*a*x^6+6/5*c^2*d^4*a*x^5+c*d^4*a*x^4+1/3*d^4*a*x^3+1/7*c^4*d^4*b*arctanh(c*x)*x^7
+2/3*c^3*d^4*b*arctanh(c*x)*x^6+6/5*c^2*d^4*b*arctanh(c*x)*x^5+c*d^4*b*arctanh(c*x)*x^4+1/3*d^4*b*arctanh(c*x)
*x^3+1/42*b*c^3*d^4*x^6+2/15*b*c^2*d^4*x^5+47/140*b*c*d^4*x^4+5/9*b*d^4*x^3+88/105*b*d^4*x^2/c+5/3*b*d^4*x/c^2
+117/70/c^3*d^4*b*ln(c*x-1)+1/210/c^3*d^4*b*ln(c*x+1)

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maxima [B]  time = 0.33, size = 339, normalized size = 1.98 \[ \frac {1}{7} \, a c^{4} d^{4} x^{7} + \frac {2}{3} \, a c^{3} d^{4} x^{6} + \frac {6}{5} \, a c^{2} d^{4} x^{5} + \frac {1}{84} \, {\left (12 \, x^{7} \operatorname {artanh}\left (c x\right ) + c {\left (\frac {2 \, c^{4} x^{6} + 3 \, c^{2} x^{4} + 6 \, x^{2}}{c^{6}} + \frac {6 \, \log \left (c^{2} x^{2} - 1\right )}{c^{8}}\right )}\right )} b c^{4} d^{4} + a c d^{4} x^{4} + \frac {1}{45} \, {\left (30 \, x^{6} \operatorname {artanh}\left (c x\right ) + c {\left (\frac {2 \, {\left (3 \, c^{4} x^{5} + 5 \, c^{2} x^{3} + 15 \, x\right )}}{c^{6}} - \frac {15 \, \log \left (c x + 1\right )}{c^{7}} + \frac {15 \, \log \left (c x - 1\right )}{c^{7}}\right )}\right )} b c^{3} d^{4} + \frac {3}{10} \, {\left (4 \, x^{5} \operatorname {artanh}\left (c x\right ) + c {\left (\frac {c^{2} x^{4} + 2 \, x^{2}}{c^{4}} + \frac {2 \, \log \left (c^{2} x^{2} - 1\right )}{c^{6}}\right )}\right )} b c^{2} d^{4} + \frac {1}{3} \, a d^{4} x^{3} + \frac {1}{6} \, {\left (6 \, x^{4} \operatorname {artanh}\left (c x\right ) + c {\left (\frac {2 \, {\left (c^{2} x^{3} + 3 \, x\right )}}{c^{4}} - \frac {3 \, \log \left (c x + 1\right )}{c^{5}} + \frac {3 \, \log \left (c x - 1\right )}{c^{5}}\right )}\right )} b c d^{4} + \frac {1}{6} \, {\left (2 \, x^{3} \operatorname {artanh}\left (c x\right ) + c {\left (\frac {x^{2}}{c^{2}} + \frac {\log \left (c^{2} x^{2} - 1\right )}{c^{4}}\right )}\right )} b d^{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/7*a*c^4*d^4*x^7 + 2/3*a*c^3*d^4*x^6 + 6/5*a*c^2*d^4*x^5 + 1/84*(12*x^7*arctanh(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 + a*c*d^4*x^4 + 1/45*(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^3*d^4 + 3/10*(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^2*d^4 + 1/3*a*d^4*x^3 + 1/6*(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*d^4 + 1/6*(2*x^3*arctanh(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 = 1.08, size = 196, normalized size = 1.15 \[ \frac {\frac {88\,b\,c^2\,d^4\,x^2}{105}-\frac {d^4\,\left (2100\,b\,\mathrm {atanh}\left (c\,x\right )-1056\,b\,\ln \left (c^2\,x^2-1\right )\right )}{1260}+\frac {5\,b\,c\,d^4\,x}{3}}{c^3}+\frac {d^4\,\left (420\,a\,x^3+700\,b\,x^3+420\,b\,x^3\,\mathrm {atanh}\left (c\,x\right )\right )}{1260}+\frac {c^4\,d^4\,\left (180\,a\,x^7+180\,b\,x^7\,\mathrm {atanh}\left (c\,x\right )\right )}{1260}+\frac {c\,d^4\,\left (1260\,a\,x^4+423\,b\,x^4+1260\,b\,x^4\,\mathrm {atanh}\left (c\,x\right )\right )}{1260}+\frac {c^3\,d^4\,\left (840\,a\,x^6+30\,b\,x^6+840\,b\,x^6\,\mathrm {atanh}\left (c\,x\right )\right )}{1260}+\frac {c^2\,d^4\,\left (1512\,a\,x^5+168\,b\,x^5+1512\,b\,x^5\,\mathrm {atanh}\left (c\,x\right )\right )}{1260} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 2.99, size = 279, normalized size = 1.63 \[ \begin {cases} \frac {a c^{4} d^{4} x^{7}}{7} + \frac {2 a c^{3} d^{4} x^{6}}{3} + \frac {6 a c^{2} d^{4} x^{5}}{5} + a c d^{4} x^{4} + \frac {a d^{4} x^{3}}{3} + \frac {b c^{4} d^{4} x^{7} \operatorname {atanh}{\left (c x \right )}}{7} + \frac {2 b c^{3} d^{4} x^{6} \operatorname {atanh}{\left (c x \right )}}{3} + \frac {b c^{3} d^{4} x^{6}}{42} + \frac {6 b c^{2} d^{4} x^{5} \operatorname {atanh}{\left (c x \right )}}{5} + \frac {2 b c^{2} d^{4} x^{5}}{15} + b c d^{4} x^{4} \operatorname {atanh}{\left (c x \right )} + \frac {47 b c d^{4} x^{4}}{140} + \frac {b d^{4} x^{3} \operatorname {atanh}{\left (c x \right )}}{3} + \frac {5 b d^{4} x^{3}}{9} + \frac {88 b d^{4} x^{2}}{105 c} + \frac {5 b d^{4} x}{3 c^{2}} + \frac {176 b d^{4} \log {\left (x - \frac {1}{c} \right )}}{105 c^{3}} + \frac {b d^{4} \operatorname {atanh}{\left (c x \right )}}{105 c^{3}} & \text {for}\: c \neq 0 \\\frac {a d^{4} x^{3}}{3} & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((a*c**4*d**4*x**7/7 + 2*a*c**3*d**4*x**6/3 + 6*a*c**2*d**4*x**5/5 + a*c*d**4*x**4 + a*d**4*x**3/3 +
b*c**4*d**4*x**7*atanh(c*x)/7 + 2*b*c**3*d**4*x**6*atanh(c*x)/3 + b*c**3*d**4*x**6/42 + 6*b*c**2*d**4*x**5*ata
nh(c*x)/5 + 2*b*c**2*d**4*x**5/15 + b*c*d**4*x**4*atanh(c*x) + 47*b*c*d**4*x**4/140 + b*d**4*x**3*atanh(c*x)/3
+ 5*b*d**4*x**3/9 + 88*b*d**4*x**2/(105*c) + 5*b*d**4*x/(3*c**2) + 176*b*d**4*log(x - 1/c)/(105*c**3) + b*d**
4*atanh(c*x)/(105*c**3), Ne(c, 0)), (a*d**4*x**3/3, True))

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