\(\int e^{\frac {5}{2} \text {arctanh}(a x)} x^3 \, dx\) [93]

Optimal result

Integrand size = 14, antiderivative size = 272 \[ \int e^{\frac {5}{2} \text {arctanh}(a x)} x^3 \, dx=\frac {475 (1-a x)^{3/4} \sqrt [4]{1+a x}}{64 a^4}+\frac {95 (1-a x)^{3/4} (1+a x)^{5/4}}{32 a^4}+\frac {2 (1+a x)^{9/4}}{a^4 \sqrt [4]{1-a x}}+\frac {17 (1-a x)^{3/4} (1+a x)^{9/4}}{24 a^4}-\frac {(1-a x)^{7/4} (1+a x)^{9/4}}{4 a^4}+\frac {475 \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{1+a x}}{\sqrt [4]{1-a x}}\right )}{64 \sqrt {2} a^4}-\frac {475 \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{1+a x}}{\sqrt [4]{1-a x}}\right )}{64 \sqrt {2} a^4}-\frac {475 \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{1+a x}}{\sqrt [4]{1-a x} \left (1+\frac {\sqrt {1+a x}}{\sqrt {1-a x}}\right )}\right )}{64 \sqrt {2} a^4} \] Output:

475/64*(-a*x+1)^(3/4)*(a*x+1)^(1/4)/a^4+95/32*(-a*x+1)^(3/4)*(a*x+1)^(5/4) 
/a^4+2*(a*x+1)^(9/4)/a^4/(-a*x+1)^(1/4)+17/24*(-a*x+1)^(3/4)*(a*x+1)^(9/4) 
/a^4-1/4*(-a*x+1)^(7/4)*(a*x+1)^(9/4)/a^4+475/128*arctan(1-2^(1/2)*(a*x+1) 
^(1/4)/(-a*x+1)^(1/4))*2^(1/2)/a^4-475/128*arctan(1+2^(1/2)*(a*x+1)^(1/4)/ 
(-a*x+1)^(1/4))*2^(1/2)/a^4-475/128*arctanh(2^(1/2)*(a*x+1)^(1/4)/(-a*x+1) 
^(1/4)/(1+(a*x+1)^(1/2)/(-a*x+1)^(1/2)))*2^(1/2)/a^4
 

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.

Time = 0.06 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.27 \[ \int e^{\frac {5}{2} \text {arctanh}(a x)} x^3 \, dx=\frac {(1+a x)^{9/4} \left (59-5 a x-6 a^2 x^2\right )-380 \sqrt [4]{2} (-1+a x) \operatorname {Hypergeometric2F1}\left (-\frac {5}{4},\frac {3}{4},\frac {7}{4},\frac {1}{2} (1-a x)\right )}{24 a^4 \sqrt [4]{1-a x}} \] Input:

Integrate[E^((5*ArcTanh[a*x])/2)*x^3,x]
 

Output:

((1 + a*x)^(9/4)*(59 - 5*a*x - 6*a^2*x^2) - 380*2^(1/4)*(-1 + a*x)*Hyperge 
ometric2F1[-5/4, 3/4, 7/4, (1 - a*x)/2])/(24*a^4*(1 - a*x)^(1/4))
 

Rubi [A] (warning: unable to verify)

Time = 0.93 (sec) , antiderivative size = 316, normalized size of antiderivative = 1.16, number of steps used = 18, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.214, Rules used = {6676, 108, 27, 170, 27, 164, 60, 73, 854, 826, 1476, 1082, 217, 1479, 25, 27, 1103}

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.

\(\displaystyle \int x^3 e^{\frac {5}{2} \text {arctanh}(a x)} \, dx\)

\(\Big \downarrow \) 6676

\(\displaystyle \int \frac {x^3 (a x+1)^{5/4}}{(1-a x)^{5/4}}dx\)

\(\Big \downarrow \) 108

\(\displaystyle \frac {4 x^3 (a x+1)^{5/4}}{a \sqrt [4]{1-a x}}-\frac {4 \int \frac {x^2 \sqrt [4]{a x+1} (17 a x+12)}{4 \sqrt [4]{1-a x}}dx}{a}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {4 x^3 (a x+1)^{5/4}}{a \sqrt [4]{1-a x}}-\frac {\int \frac {x^2 \sqrt [4]{a x+1} (17 a x+12)}{\sqrt [4]{1-a x}}dx}{a}\)

\(\Big \downarrow \) 170

\(\displaystyle \frac {4 x^3 (a x+1)^{5/4}}{a \sqrt [4]{1-a x}}-\frac {-\frac {\int -\frac {a x \sqrt [4]{a x+1} (113 a x+68)}{2 \sqrt [4]{1-a x}}dx}{4 a^2}-\frac {17 x^2 (1-a x)^{3/4} (a x+1)^{5/4}}{4 a}}{a}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {4 x^3 (a x+1)^{5/4}}{a \sqrt [4]{1-a x}}-\frac {\frac {\int \frac {x \sqrt [4]{a x+1} (113 a x+68)}{\sqrt [4]{1-a x}}dx}{8 a}-\frac {17 x^2 (1-a x)^{3/4} (a x+1)^{5/4}}{4 a}}{a}\)

\(\Big \downarrow \) 164

\(\displaystyle \frac {4 x^3 (a x+1)^{5/4}}{a \sqrt [4]{1-a x}}-\frac {\frac {\frac {475 \int \frac {\sqrt [4]{a x+1}}{\sqrt [4]{1-a x}}dx}{8 a}-\frac {(1-a x)^{3/4} (a x+1)^{5/4} (452 a x+521)}{12 a^2}}{8 a}-\frac {17 x^2 (1-a x)^{3/4} (a x+1)^{5/4}}{4 a}}{a}\)

\(\Big \downarrow \) 60

\(\displaystyle \frac {4 x^3 (a x+1)^{5/4}}{a \sqrt [4]{1-a x}}-\frac {\frac {\frac {475 \left (\frac {1}{2} \int \frac {1}{\sqrt [4]{1-a x} (a x+1)^{3/4}}dx-\frac {(1-a x)^{3/4} \sqrt [4]{a x+1}}{a}\right )}{8 a}-\frac {(1-a x)^{3/4} (a x+1)^{5/4} (452 a x+521)}{12 a^2}}{8 a}-\frac {17 x^2 (1-a x)^{3/4} (a x+1)^{5/4}}{4 a}}{a}\)

\(\Big \downarrow \) 73

\(\displaystyle \frac {4 x^3 (a x+1)^{5/4}}{a \sqrt [4]{1-a x}}-\frac {\frac {\frac {475 \left (-\frac {2 \int \frac {\sqrt {1-a x}}{(a x+1)^{3/4}}d\sqrt [4]{1-a x}}{a}-\frac {(1-a x)^{3/4} \sqrt [4]{a x+1}}{a}\right )}{8 a}-\frac {(1-a x)^{3/4} (a x+1)^{5/4} (452 a x+521)}{12 a^2}}{8 a}-\frac {17 x^2 (1-a x)^{3/4} (a x+1)^{5/4}}{4 a}}{a}\)

\(\Big \downarrow \) 854

\(\displaystyle \frac {4 x^3 (a x+1)^{5/4}}{a \sqrt [4]{1-a x}}-\frac {\frac {\frac {475 \left (-\frac {2 \int \frac {\sqrt {1-a x}}{2-a x}d\frac {\sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}}{a}-\frac {(1-a x)^{3/4} \sqrt [4]{a x+1}}{a}\right )}{8 a}-\frac {(1-a x)^{3/4} (a x+1)^{5/4} (452 a x+521)}{12 a^2}}{8 a}-\frac {17 x^2 (1-a x)^{3/4} (a x+1)^{5/4}}{4 a}}{a}\)

\(\Big \downarrow \) 826

\(\displaystyle \frac {4 x^3 (a x+1)^{5/4}}{a \sqrt [4]{1-a x}}-\frac {\frac {\frac {475 \left (-\frac {2 \left (\frac {1}{2} \int \frac {\sqrt {1-a x}+1}{2-a x}d\frac {\sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}-\frac {1}{2} \int \frac {1-\sqrt {1-a x}}{2-a x}d\frac {\sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}\right )}{a}-\frac {(1-a x)^{3/4} \sqrt [4]{a x+1}}{a}\right )}{8 a}-\frac {(1-a x)^{3/4} (a x+1)^{5/4} (452 a x+521)}{12 a^2}}{8 a}-\frac {17 x^2 (1-a x)^{3/4} (a x+1)^{5/4}}{4 a}}{a}\)

\(\Big \downarrow \) 1476

\(\displaystyle \frac {4 x^3 (a x+1)^{5/4}}{a \sqrt [4]{1-a x}}-\frac {\frac {\frac {475 \left (-\frac {2 \left (\frac {1}{2} \left (\frac {1}{2} \int \frac {1}{\sqrt {1-a x}-\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1}d\frac {\sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+\frac {1}{2} \int \frac {1}{\sqrt {1-a x}+\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1}d\frac {\sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}\right )-\frac {1}{2} \int \frac {1-\sqrt {1-a x}}{2-a x}d\frac {\sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}\right )}{a}-\frac {(1-a x)^{3/4} \sqrt [4]{a x+1}}{a}\right )}{8 a}-\frac {(1-a x)^{3/4} (a x+1)^{5/4} (452 a x+521)}{12 a^2}}{8 a}-\frac {17 x^2 (1-a x)^{3/4} (a x+1)^{5/4}}{4 a}}{a}\)

\(\Big \downarrow \) 1082

\(\displaystyle \frac {4 x^3 (a x+1)^{5/4}}{a \sqrt [4]{1-a x}}-\frac {\frac {\frac {475 \left (-\frac {2 \left (\frac {1}{2} \left (\frac {\int \frac {1}{-\sqrt {1-a x}-1}d\left (1-\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}\right )}{\sqrt {2}}-\frac {\int \frac {1}{-\sqrt {1-a x}-1}d\left (\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1\right )}{\sqrt {2}}\right )-\frac {1}{2} \int \frac {1-\sqrt {1-a x}}{2-a x}d\frac {\sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}\right )}{a}-\frac {(1-a x)^{3/4} \sqrt [4]{a x+1}}{a}\right )}{8 a}-\frac {(1-a x)^{3/4} (a x+1)^{5/4} (452 a x+521)}{12 a^2}}{8 a}-\frac {17 x^2 (1-a x)^{3/4} (a x+1)^{5/4}}{4 a}}{a}\)

\(\Big \downarrow \) 217

\(\displaystyle \frac {4 x^3 (a x+1)^{5/4}}{a \sqrt [4]{1-a x}}-\frac {\frac {\frac {475 \left (-\frac {2 \left (\frac {1}{2} \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}\right )}{\sqrt {2}}\right )-\frac {1}{2} \int \frac {1-\sqrt {1-a x}}{2-a x}d\frac {\sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}\right )}{a}-\frac {(1-a x)^{3/4} \sqrt [4]{a x+1}}{a}\right )}{8 a}-\frac {(1-a x)^{3/4} (a x+1)^{5/4} (452 a x+521)}{12 a^2}}{8 a}-\frac {17 x^2 (1-a x)^{3/4} (a x+1)^{5/4}}{4 a}}{a}\)

\(\Big \downarrow \) 1479

\(\displaystyle \frac {4 x^3 (a x+1)^{5/4}}{a \sqrt [4]{1-a x}}-\frac {\frac {\frac {475 \left (-\frac {2 \left (\frac {1}{2} \left (\frac {\int -\frac {\sqrt {2}-\frac {2 \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}}{\sqrt {1-a x}-\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1}d\frac {\sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}}{2 \sqrt {2}}+\frac {\int -\frac {\sqrt {2} \left (\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1\right )}{\sqrt {1-a x}+\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1}d\frac {\sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}}{2 \sqrt {2}}\right )+\frac {1}{2} \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}\right )}{\sqrt {2}}\right )\right )}{a}-\frac {(1-a x)^{3/4} \sqrt [4]{a x+1}}{a}\right )}{8 a}-\frac {(1-a x)^{3/4} (a x+1)^{5/4} (452 a x+521)}{12 a^2}}{8 a}-\frac {17 x^2 (1-a x)^{3/4} (a x+1)^{5/4}}{4 a}}{a}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {4 x^3 (a x+1)^{5/4}}{a \sqrt [4]{1-a x}}-\frac {\frac {\frac {475 \left (-\frac {2 \left (\frac {1}{2} \left (-\frac {\int \frac {\sqrt {2}-\frac {2 \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}}{\sqrt {1-a x}-\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1}d\frac {\sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}}{2 \sqrt {2}}-\frac {\int \frac {\sqrt {2} \left (\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1\right )}{\sqrt {1-a x}+\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1}d\frac {\sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}}{2 \sqrt {2}}\right )+\frac {1}{2} \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}\right )}{\sqrt {2}}\right )\right )}{a}-\frac {(1-a x)^{3/4} \sqrt [4]{a x+1}}{a}\right )}{8 a}-\frac {(1-a x)^{3/4} (a x+1)^{5/4} (452 a x+521)}{12 a^2}}{8 a}-\frac {17 x^2 (1-a x)^{3/4} (a x+1)^{5/4}}{4 a}}{a}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {4 x^3 (a x+1)^{5/4}}{a \sqrt [4]{1-a x}}-\frac {\frac {\frac {475 \left (-\frac {2 \left (\frac {1}{2} \left (-\frac {\int \frac {\sqrt {2}-\frac {2 \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}}{\sqrt {1-a x}-\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1}d\frac {\sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}}{2 \sqrt {2}}-\frac {1}{2} \int \frac {\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1}{\sqrt {1-a x}+\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1}d\frac {\sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}\right )+\frac {1}{2} \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}\right )}{\sqrt {2}}\right )\right )}{a}-\frac {(1-a x)^{3/4} \sqrt [4]{a x+1}}{a}\right )}{8 a}-\frac {(1-a x)^{3/4} (a x+1)^{5/4} (452 a x+521)}{12 a^2}}{8 a}-\frac {17 x^2 (1-a x)^{3/4} (a x+1)^{5/4}}{4 a}}{a}\)

\(\Big \downarrow \) 1103

\(\displaystyle \frac {4 x^3 (a x+1)^{5/4}}{a \sqrt [4]{1-a x}}-\frac {\frac {\frac {475 \left (-\frac {2 \left (\frac {1}{2} \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}\right )}{\sqrt {2}}\right )+\frac {1}{2} \left (\frac {\log \left (\sqrt {1-a x}-\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1\right )}{2 \sqrt {2}}-\frac {\log \left (\sqrt {1-a x}+\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1\right )}{2 \sqrt {2}}\right )\right )}{a}-\frac {(1-a x)^{3/4} \sqrt [4]{a x+1}}{a}\right )}{8 a}-\frac {(1-a x)^{3/4} (a x+1)^{5/4} (452 a x+521)}{12 a^2}}{8 a}-\frac {17 x^2 (1-a x)^{3/4} (a x+1)^{5/4}}{4 a}}{a}\)

Input:

Int[E^((5*ArcTanh[a*x])/2)*x^3,x]
 

Output:

(4*x^3*(1 + a*x)^(5/4))/(a*(1 - a*x)^(1/4)) - ((-17*x^2*(1 - a*x)^(3/4)*(1 
 + a*x)^(5/4))/(4*a) + (-1/12*((1 - a*x)^(3/4)*(1 + a*x)^(5/4)*(521 + 452* 
a*x))/a^2 + (475*(-(((1 - a*x)^(3/4)*(1 + a*x)^(1/4))/a) - (2*((-(ArcTan[1 
 - (Sqrt[2]*(1 - a*x)^(1/4))/(1 + a*x)^(1/4)]/Sqrt[2]) + ArcTan[1 + (Sqrt[ 
2]*(1 - a*x)^(1/4))/(1 + a*x)^(1/4)]/Sqrt[2])/2 + (Log[1 + Sqrt[1 - a*x] - 
 (Sqrt[2]*(1 - a*x)^(1/4))/(1 + a*x)^(1/4)]/(2*Sqrt[2]) - Log[1 + Sqrt[1 - 
 a*x] + (Sqrt[2]*(1 - a*x)^(1/4))/(1 + a*x)^(1/4)]/(2*Sqrt[2]))/2))/a))/(8 
*a))/(8*a))/a
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 60

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ 
(a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + n + 1))), x] + Simp[n*((b*c - a*d)/( 
b*(m + n + 1)))   Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, 
 c, d}, x] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !Integer 
Q[n] || (GtQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinear 
Q[a, b, c, d, m, n, x]
 

rule 73

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ 
{p = Denominator[m]}, Simp[p/b   Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + 
 d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt 
Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL 
inearQ[a, b, c, d, m, n, x]
 

rule 108

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_), x_] :> Simp[(a + b*x)^(m + 1)*(c + d*x)^n*((e + f*x)^p/(b*(m + 1))) 
, x] - Simp[1/(b*(m + 1))   Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f* 
x)^(p - 1)*Simp[d*e*n + c*f*p + d*f*(n + p)*x, x], x], x] /; FreeQ[{a, b, c 
, d, e, f}, x] && LtQ[m, -1] && GtQ[n, 0] && GtQ[p, 0] && (IntegersQ[2*m, 2 
*n, 2*p] || IntegersQ[m, n + p] || IntegersQ[p, m + n])
 

rule 164

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_) + (f_.)*(x_ 
))*((g_.) + (h_.)*(x_)), x_] :> Simp[(-(a*d*f*h*(n + 2) + b*c*f*h*(m + 2) - 
 b*d*(f*g + e*h)*(m + n + 3) - b*d*f*h*(m + n + 2)*x))*(a + b*x)^(m + 1)*(( 
c + d*x)^(n + 1)/(b^2*d^2*(m + n + 2)*(m + n + 3))), x] + Simp[(a^2*d^2*f*h 
*(n + 1)*(n + 2) + a*b*d*(n + 1)*(2*c*f*h*(m + 1) - d*(f*g + e*h)*(m + n + 
3)) + b^2*(c^2*f*h*(m + 1)*(m + 2) - c*d*(f*g + e*h)*(m + 1)*(m + n + 3) + 
d^2*e*g*(m + n + 2)*(m + n + 3)))/(b^2*d^2*(m + n + 2)*(m + n + 3))   Int[( 
a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, e, f, g, h, m, n}, x] 
&& NeQ[m + n + 2, 0] && NeQ[m + n + 3, 0]
 

rule 170

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_)*((g_.) + (h_.)*(x_)), x_] :> Simp[h*(a + b*x)^m*(c + d*x)^(n + 1)*(( 
e + f*x)^(p + 1)/(d*f*(m + n + p + 2))), x] + Simp[1/(d*f*(m + n + p + 2)) 
  Int[(a + b*x)^(m - 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*g*(m + n + p + 2 
) - h*(b*c*e*m + a*(d*e*(n + 1) + c*f*(p + 1))) + (b*d*f*g*(m + n + p + 2) 
+ h*(a*d*f*m - b*(d*e*(m + n + 1) + c*f*(m + p + 1))))*x, x], x], x] /; Fre 
eQ[{a, b, c, d, e, f, g, h, n, p}, x] && GtQ[m, 0] && NeQ[m + n + p + 2, 0] 
 && IntegerQ[m]
 

rule 217

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( 
-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & 
& (LtQ[a, 0] || LtQ[b, 0])
 

rule 826

Int[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[a/b, 
2]], s = Denominator[Rt[a/b, 2]]}, Simp[1/(2*s)   Int[(r + s*x^2)/(a + b*x^ 
4), x], x] - Simp[1/(2*s)   Int[(r - s*x^2)/(a + b*x^4), x], x]] /; FreeQ[{ 
a, b}, x] && (GtQ[a/b, 0] || (PosQ[a/b] && AtomQ[SplitProduct[SumBaseQ, a]] 
 && AtomQ[SplitProduct[SumBaseQ, b]]))
 

rule 854

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^(p + (m + 
 1)/n)   Subst[Int[x^m/(1 - b*x^n)^(p + (m + 1)/n + 1), x], x, x/(a + b*x^n 
)^(1/n)], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && LtQ[-1, p, 0] && NeQ[p, - 
2^(-1)] && IntegersQ[m, p + (m + 1)/n]
 

rule 1082

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*S 
implify[a*(c/b^2)]}, Simp[-2/b   Subst[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b 
)], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /; Fre 
eQ[{a, b, c}, x]
 

rule 1103

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S 
imp[d*(Log[RemoveContent[a + b*x + c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, 
e}, x] && EqQ[2*c*d - b*e, 0]
 

rule 1476

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ 
2*(d/e), 2]}, Simp[e/(2*c)   Int[1/Simp[d/e + q*x + x^2, x], x], x] + Simp[ 
e/(2*c)   Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; FreeQ[{a, c, d, e}, x] 
 && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]
 

rule 1479

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ 
-2*(d/e), 2]}, Simp[e/(2*c*q)   Int[(q - 2*x)/Simp[d/e + q*x - x^2, x], x], 
 x] + Simp[e/(2*c*q)   Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /; F 
reeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]
 

rule 6676

Int[E^(ArcTanh[(a_.)*(x_)]*(n_))*((c_.)*(x_))^(m_.), x_Symbol] :> Int[(c*x) 
^m*((1 + a*x)^(n/2)/(1 - a*x)^(n/2)), x] /; FreeQ[{a, c, m, n}, x] &&  !Int 
egerQ[(n - 1)/2]
 

Maple [F]

\[\int {\left (\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )}^{\frac {5}{2}} x^{3}d x\]

Input:

int(((a*x+1)/(-a^2*x^2+1)^(1/2))^(5/2)*x^3,x)
 

Output:

int(((a*x+1)/(-a^2*x^2+1)^(1/2))^(5/2)*x^3,x)
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 262, normalized size of antiderivative = 0.96 \[ \int e^{\frac {5}{2} \text {arctanh}(a x)} x^3 \, dx=-\frac {4 \, {\left (48 \, a^{4} x^{4} + 136 \, a^{3} x^{3} + 226 \, a^{2} x^{2} + 521 \, a x - 2467\right )} \sqrt {-\frac {\sqrt {-a^{2} x^{2} + 1}}{a x - 1}} + 2850 \, \sqrt {2} \arctan \left (\sqrt {2} \sqrt {-\frac {\sqrt {-a^{2} x^{2} + 1}}{a x - 1}} + 1\right ) + 2850 \, \sqrt {2} \arctan \left (\sqrt {2} \sqrt {-\frac {\sqrt {-a^{2} x^{2} + 1}}{a x - 1}} - 1\right ) + 1425 \, \sqrt {2} \log \left (\frac {a x + \sqrt {2} {\left (a x - 1\right )} \sqrt {-\frac {\sqrt {-a^{2} x^{2} + 1}}{a x - 1}} - \sqrt {-a^{2} x^{2} + 1} - 1}{a x - 1}\right ) - 1425 \, \sqrt {2} \log \left (\frac {a x - \sqrt {2} {\left (a x - 1\right )} \sqrt {-\frac {\sqrt {-a^{2} x^{2} + 1}}{a x - 1}} - \sqrt {-a^{2} x^{2} + 1} - 1}{a x - 1}\right )}{768 \, a^{4}} \] Input:

integrate(((a*x+1)/(-a^2*x^2+1)^(1/2))^(5/2)*x^3,x, algorithm="fricas")
 

Output:

-1/768*(4*(48*a^4*x^4 + 136*a^3*x^3 + 226*a^2*x^2 + 521*a*x - 2467)*sqrt(- 
sqrt(-a^2*x^2 + 1)/(a*x - 1)) + 2850*sqrt(2)*arctan(sqrt(2)*sqrt(-sqrt(-a^ 
2*x^2 + 1)/(a*x - 1)) + 1) + 2850*sqrt(2)*arctan(sqrt(2)*sqrt(-sqrt(-a^2*x 
^2 + 1)/(a*x - 1)) - 1) + 1425*sqrt(2)*log((a*x + sqrt(2)*(a*x - 1)*sqrt(- 
sqrt(-a^2*x^2 + 1)/(a*x - 1)) - sqrt(-a^2*x^2 + 1) - 1)/(a*x - 1)) - 1425* 
sqrt(2)*log((a*x - sqrt(2)*(a*x - 1)*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1)) - 
 sqrt(-a^2*x^2 + 1) - 1)/(a*x - 1)))/a^4
 

Sympy [F(-1)]

Timed out. \[ \int e^{\frac {5}{2} \text {arctanh}(a x)} x^3 \, dx=\text {Timed out} \] Input:

integrate(((a*x+1)/(-a**2*x**2+1)**(1/2))**(5/2)*x**3,x)
 

Output:

Timed out
 

Maxima [F]

\[ \int e^{\frac {5}{2} \text {arctanh}(a x)} x^3 \, dx=\int { x^{3} \left (\frac {a x + 1}{\sqrt {-a^{2} x^{2} + 1}}\right )^{\frac {5}{2}} \,d x } \] Input:

integrate(((a*x+1)/(-a^2*x^2+1)^(1/2))^(5/2)*x^3,x, algorithm="maxima")
 

Output:

integrate(x^3*((a*x + 1)/sqrt(-a^2*x^2 + 1))^(5/2), x)
 

Giac [F(-2)]

Exception generated. \[ \int e^{\frac {5}{2} \text {arctanh}(a x)} x^3 \, dx=\text {Exception raised: TypeError} \] Input:

integrate(((a*x+1)/(-a^2*x^2+1)^(1/2))^(5/2)*x^3,x, algorithm="giac")
 

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:sym2poly/r2sym(const gen & e,const 
index_m & i,const vecteur & l) Error: Bad Argument Value
 

Mupad [F(-1)]

Timed out. \[ \int e^{\frac {5}{2} \text {arctanh}(a x)} x^3 \, dx=\int x^3\,{\left (\frac {a\,x+1}{\sqrt {1-a^2\,x^2}}\right )}^{5/2} \,d x \] Input:

int(x^3*((a*x + 1)/(1 - a^2*x^2)^(1/2))^(5/2),x)
 

Output:

int(x^3*((a*x + 1)/(1 - a^2*x^2)^(1/2))^(5/2), x)
 

Reduce [F]

\[ \int e^{\frac {5}{2} \text {arctanh}(a x)} x^3 \, dx=\frac {48 \sqrt {a x +1}\, \left (-a^{2} x^{2}+1\right )^{\frac {3}{4}} a^{3} x^{3}+88 \sqrt {a x +1}\, \left (-a^{2} x^{2}+1\right )^{\frac {3}{4}} a^{2} x^{2}+138 \sqrt {a x +1}\, \left (-a^{2} x^{2}+1\right )^{\frac {3}{4}} a x -92 \sqrt {a x +1}\, \left (-a^{2} x^{2}+1\right )^{\frac {3}{4}}+475 \left (\int \frac {\sqrt {a x +1}\, \left (-a^{2} x^{2}+1\right )^{\frac {3}{4}} x}{a^{3} x^{3}-a^{2} x^{2}-a x +1}d x \right ) a^{3} x -475 \left (\int \frac {\sqrt {a x +1}\, \left (-a^{2} x^{2}+1\right )^{\frac {3}{4}} x}{a^{3} x^{3}-a^{2} x^{2}-a x +1}d x \right ) a^{2}}{192 a^{4} \left (a x -1\right )} \] Input:

int(((a*x+1)/(-a^2*x^2+1)^(1/2))^(5/2)*x^3,x)
                                                                                    
                                                                                    
 

Output:

(48*sqrt(a*x + 1)*( - a**2*x**2 + 1)**(3/4)*a**3*x**3 + 88*sqrt(a*x + 1)*( 
 - a**2*x**2 + 1)**(3/4)*a**2*x**2 + 138*sqrt(a*x + 1)*( - a**2*x**2 + 1)* 
*(3/4)*a*x - 92*sqrt(a*x + 1)*( - a**2*x**2 + 1)**(3/4) + 475*int((sqrt(a* 
x + 1)*( - a**2*x**2 + 1)**(3/4)*x)/(a**3*x**3 - a**2*x**2 - a*x + 1),x)*a 
**3*x - 475*int((sqrt(a*x + 1)*( - a**2*x**2 + 1)**(3/4)*x)/(a**3*x**3 - a 
**2*x**2 - a*x + 1),x)*a**2)/(192*a**4*(a*x - 1))