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

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

[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)

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Rubi [A]  time = 0.180548, antiderivative size = 171, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, 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 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 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]))

Rule 12

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

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]))

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*}

Mathematica [A]  time = 0.146447, 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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Maple [A]  time = 0.027, size = 225, normalized size = 1.3 \begin{align*}{\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{\it Artanh} \left ( cx \right ){x}^{7}}{7}}+{\frac{2\,{c}^{3}{d}^{4}b{\it Artanh} \left ( cx \right ){x}^{6}}{3}}+{\frac{6\,{c}^{2}{d}^{4}b{\it Artanh} \left ( cx \right ){x}^{5}}{5}}+c{d}^{4}b{\it Artanh} \left ( cx \right ){x}^{4}+{\frac{{d}^{4}b{\it Artanh} \left ( cx \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\,bc{d}^{4}{x}^{4}}{140}}+{\frac{5\,b{d}^{4}{x}^{3}}{9}}+{\frac{88\,{d}^{4}b{x}^{2}}{105\,c}}+{\frac{5\,b{d}^{4}x}{3\,{c}^{2}}}+{\frac{117\,{d}^{4}b\ln \left ( cx-1 \right ) }{70\,{c}^{3}}}+{\frac{{d}^{4}b\ln \left ( cx+1 \right ) }{210\,{c}^{3}}} \end{align*}

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 = 0.97381, size = 458, normalized size = 2.68 \begin{align*} \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} \end{align*}

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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Fricas [A]  time = 2.20207, size = 490, normalized size = 2.87 \begin{align*} \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}} \end{align*}

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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Sympy [A]  time = 5.84074, size = 279, normalized size = 1.63 \begin{align*} \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} \end{align*}

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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Giac [A]  time = 1.26854, size = 300, normalized size = 1.75 \begin{align*} \frac{1}{7} \, a c^{4} d^{4} x^{7} + \frac{1}{42} \,{\left (28 \, a c^{3} d^{4} + b c^{3} d^{4}\right )} x^{6} + \frac{88 \, b d^{4} x^{2}}{105 \, c} + \frac{2}{15} \,{\left (9 \, a c^{2} d^{4} + b c^{2} d^{4}\right )} x^{5} + \frac{1}{140} \,{\left (140 \, a c d^{4} + 47 \, b c d^{4}\right )} x^{4} + \frac{5 \, b d^{4} x}{3 \, c^{2}} + \frac{1}{9} \,{\left (3 \, a d^{4} + 5 \, b d^{4}\right )} x^{3} + \frac{b d^{4} \log \left (c x + 1\right )}{210 \, c^{3}} + \frac{117 \, b d^{4} \log \left (c x - 1\right )}{70 \, c^{3}} + \frac{1}{210} \,{\left (15 \, b c^{4} d^{4} x^{7} + 70 \, b c^{3} d^{4} x^{6} + 126 \, b c^{2} d^{4} x^{5} + 105 \, b c d^{4} x^{4} + 35 \, b d^{4} x^{3}\right )} \log \left (-\frac{c x + 1}{c x - 1}\right ) \end{align*}

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]

1/7*a*c^4*d^4*x^7 + 1/42*(28*a*c^3*d^4 + b*c^3*d^4)*x^6 + 88/105*b*d^4*x^2/c + 2/15*(9*a*c^2*d^4 + b*c^2*d^4)*
x^5 + 1/140*(140*a*c*d^4 + 47*b*c*d^4)*x^4 + 5/3*b*d^4*x/c^2 + 1/9*(3*a*d^4 + 5*b*d^4)*x^3 + 1/210*b*d^4*log(c
*x + 1)/c^3 + 117/70*b*d^4*log(c*x - 1)/c^3 + 1/210*(15*b*c^4*d^4*x^7 + 70*b*c^3*d^4*x^6 + 126*b*c^2*d^4*x^5 +
105*b*c*d^4*x^4 + 35*b*d^4*x^3)*log(-(c*x + 1)/(c*x - 1))