\(\int \frac {x^2 (a+b \text {arctanh}(c \sqrt {x}))}{d+e x} \, dx\) [45]

Optimal result

Integrand size = 23, antiderivative size = 460 \[ \int \frac {x^2 \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )}{d+e x} \, dx=-\frac {b d \sqrt {x}}{c e^2}+\frac {b \sqrt {x}}{2 c^3 e}+\frac {b x^{3/2}}{6 c e}+\frac {b d \text {arctanh}\left (c \sqrt {x}\right )}{c^2 e^2}-\frac {b \text {arctanh}\left (c \sqrt {x}\right )}{2 c^4 e}-\frac {d x \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )}{e^2}+\frac {x^2 \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )}{2 e}-\frac {2 d^2 \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right ) \log \left (\frac {2}{1+c \sqrt {x}}\right )}{e^3}+\frac {d^2 \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right ) \log \left (\frac {2 c \left (\sqrt {-d}-\sqrt {e} \sqrt {x}\right )}{\left (c \sqrt {-d}-\sqrt {e}\right ) \left (1+c \sqrt {x}\right )}\right )}{e^3}+\frac {d^2 \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right ) \log \left (\frac {2 c \left (\sqrt {-d}+\sqrt {e} \sqrt {x}\right )}{\left (c \sqrt {-d}+\sqrt {e}\right ) \left (1+c \sqrt {x}\right )}\right )}{e^3}+\frac {b d^2 \operatorname {PolyLog}\left (2,1-\frac {2}{1+c \sqrt {x}}\right )}{e^3}-\frac {b d^2 \operatorname {PolyLog}\left (2,1-\frac {2 c \left (\sqrt {-d}-\sqrt {e} \sqrt {x}\right )}{\left (c \sqrt {-d}-\sqrt {e}\right ) \left (1+c \sqrt {x}\right )}\right )}{2 e^3}-\frac {b d^2 \operatorname {PolyLog}\left (2,1-\frac {2 c \left (\sqrt {-d}+\sqrt {e} \sqrt {x}\right )}{\left (c \sqrt {-d}+\sqrt {e}\right ) \left (1+c \sqrt {x}\right )}\right )}{2 e^3} \] Output:

-b*d*x^(1/2)/c/e^2+1/2*b*x^(1/2)/c^3/e+1/6*b*x^(3/2)/c/e+b*d*arctanh(c*x^( 
1/2))/c^2/e^2-1/2*b*arctanh(c*x^(1/2))/c^4/e-d*x*(a+b*arctanh(c*x^(1/2)))/ 
e^2+1/2*x^2*(a+b*arctanh(c*x^(1/2)))/e-2*d^2*(a+b*arctanh(c*x^(1/2)))*ln(2 
/(1+c*x^(1/2)))/e^3+d^2*(a+b*arctanh(c*x^(1/2)))*ln(2*c*((-d)^(1/2)-e^(1/2 
)*x^(1/2))/(c*(-d)^(1/2)-e^(1/2))/(1+c*x^(1/2)))/e^3+d^2*(a+b*arctanh(c*x^ 
(1/2)))*ln(2*c*((-d)^(1/2)+e^(1/2)*x^(1/2))/(c*(-d)^(1/2)+e^(1/2))/(1+c*x^ 
(1/2)))/e^3+b*d^2*polylog(2,1-2/(1+c*x^(1/2)))/e^3-1/2*b*d^2*polylog(2,1-2 
*c*((-d)^(1/2)-e^(1/2)*x^(1/2))/(c*(-d)^(1/2)-e^(1/2))/(1+c*x^(1/2)))/e^3- 
1/2*b*d^2*polylog(2,1-2*c*((-d)^(1/2)+e^(1/2)*x^(1/2))/(c*(-d)^(1/2)+e^(1/ 
2))/(1+c*x^(1/2)))/e^3
 

Mathematica [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 1.86 (sec) , antiderivative size = 558, normalized size of antiderivative = 1.21 \[ \int \frac {x^2 \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )}{d+e x} \, dx=\frac {-6 a d e x+3 a e^2 x^2+6 a d^2 \log (d+e x)+\frac {b \left (2 c e \left (-3 c^2 d+2 e\right ) \sqrt {x}+c e^2 \sqrt {x} \left (-1+c^2 x\right )-6 \left (c^2 d-e\right ) e \left (-1+c^2 x\right ) \text {arctanh}\left (c \sqrt {x}\right )+3 e^2 \left (-1+c^2 x\right )^2 \text {arctanh}\left (c \sqrt {x}\right )-6 c^4 d^2 \left (\text {arctanh}\left (c \sqrt {x}\right ) \left (\text {arctanh}\left (c \sqrt {x}\right )+2 \log \left (1+e^{-2 \text {arctanh}\left (c \sqrt {x}\right )}\right )\right )-\operatorname {PolyLog}\left (2,-e^{-2 \text {arctanh}\left (c \sqrt {x}\right )}\right )\right )+3 c^4 d^2 \left (2 \text {arctanh}\left (c \sqrt {x}\right )^2-4 i \arcsin \left (\sqrt {\frac {c^2 d}{c^2 d+e}}\right ) \text {arctanh}\left (\frac {c e \sqrt {x}}{\sqrt {-c^2 d e}}\right )+2 \left (-i \arcsin \left (\sqrt {\frac {c^2 d}{c^2 d+e}}\right )+\text {arctanh}\left (c \sqrt {x}\right )\right ) \log \left (\frac {e^{-2 \text {arctanh}\left (c \sqrt {x}\right )} \left (-2 \sqrt {-c^2 d e}+e \left (-1+e^{2 \text {arctanh}\left (c \sqrt {x}\right )}\right )+c^2 d \left (1+e^{2 \text {arctanh}\left (c \sqrt {x}\right )}\right )\right )}{c^2 d+e}\right )+2 \left (i \arcsin \left (\sqrt {\frac {c^2 d}{c^2 d+e}}\right )+\text {arctanh}\left (c \sqrt {x}\right )\right ) \log \left (\frac {e^{-2 \text {arctanh}\left (c \sqrt {x}\right )} \left (2 \sqrt {-c^2 d e}+e \left (-1+e^{2 \text {arctanh}\left (c \sqrt {x}\right )}\right )+c^2 d \left (1+e^{2 \text {arctanh}\left (c \sqrt {x}\right )}\right )\right )}{c^2 d+e}\right )-\operatorname {PolyLog}\left (2,\frac {\left (-c^2 d+e-2 \sqrt {-c^2 d e}\right ) e^{-2 \text {arctanh}\left (c \sqrt {x}\right )}}{c^2 d+e}\right )-\operatorname {PolyLog}\left (2,\frac {\left (-c^2 d+e+2 \sqrt {-c^2 d e}\right ) e^{-2 \text {arctanh}\left (c \sqrt {x}\right )}}{c^2 d+e}\right )\right )\right )}{c^4}}{6 e^3} \] Input:

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

Output:

(-6*a*d*e*x + 3*a*e^2*x^2 + 6*a*d^2*Log[d + e*x] + (b*(2*c*e*(-3*c^2*d + 2 
*e)*Sqrt[x] + c*e^2*Sqrt[x]*(-1 + c^2*x) - 6*(c^2*d - e)*e*(-1 + c^2*x)*Ar 
cTanh[c*Sqrt[x]] + 3*e^2*(-1 + c^2*x)^2*ArcTanh[c*Sqrt[x]] - 6*c^4*d^2*(Ar 
cTanh[c*Sqrt[x]]*(ArcTanh[c*Sqrt[x]] + 2*Log[1 + E^(-2*ArcTanh[c*Sqrt[x]]) 
]) - PolyLog[2, -E^(-2*ArcTanh[c*Sqrt[x]])]) + 3*c^4*d^2*(2*ArcTanh[c*Sqrt 
[x]]^2 - (4*I)*ArcSin[Sqrt[(c^2*d)/(c^2*d + e)]]*ArcTanh[(c*e*Sqrt[x])/Sqr 
t[-(c^2*d*e)]] + 2*((-I)*ArcSin[Sqrt[(c^2*d)/(c^2*d + e)]] + ArcTanh[c*Sqr 
t[x]])*Log[(-2*Sqrt[-(c^2*d*e)] + e*(-1 + E^(2*ArcTanh[c*Sqrt[x]])) + c^2* 
d*(1 + E^(2*ArcTanh[c*Sqrt[x]])))/((c^2*d + e)*E^(2*ArcTanh[c*Sqrt[x]]))] 
+ 2*(I*ArcSin[Sqrt[(c^2*d)/(c^2*d + e)]] + ArcTanh[c*Sqrt[x]])*Log[(2*Sqrt 
[-(c^2*d*e)] + e*(-1 + E^(2*ArcTanh[c*Sqrt[x]])) + c^2*d*(1 + E^(2*ArcTanh 
[c*Sqrt[x]])))/((c^2*d + e)*E^(2*ArcTanh[c*Sqrt[x]]))] - PolyLog[2, (-(c^2 
*d) + e - 2*Sqrt[-(c^2*d*e)])/((c^2*d + e)*E^(2*ArcTanh[c*Sqrt[x]]))] - Po 
lyLog[2, (-(c^2*d) + e + 2*Sqrt[-(c^2*d*e)])/((c^2*d + e)*E^(2*ArcTanh[c*S 
qrt[x]]))])))/c^4)/(6*e^3)
 

Rubi [A] (verified)

Time = 1.46 (sec) , antiderivative size = 459, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.478, Rules used = {7267, 6542, 6452, 254, 2009, 6542, 6452, 262, 219, 6606, 2009}

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.

\(\displaystyle \int \frac {x^2 \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )}{d+e x} \, dx\)

\(\Big \downarrow \) 7267

\(\displaystyle 2 \int \frac {x^{5/2} \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )}{d+e x}d\sqrt {x}\)

\(\Big \downarrow \) 6542

\(\displaystyle 2 \left (\frac {\int x^{3/2} \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )d\sqrt {x}}{e}-\frac {d \int \frac {x^{3/2} \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )}{d+e x}d\sqrt {x}}{e}\right )\)

\(\Big \downarrow \) 6452

\(\displaystyle 2 \left (\frac {\frac {1}{4} x^2 \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )-\frac {1}{4} b c \int \frac {x^2}{1-c^2 x}d\sqrt {x}}{e}-\frac {d \int \frac {x^{3/2} \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )}{d+e x}d\sqrt {x}}{e}\right )\)

\(\Big \downarrow \) 254

\(\displaystyle 2 \left (\frac {\frac {1}{4} x^2 \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )-\frac {1}{4} b c \int \left (-\frac {x}{c^2}+\frac {1}{c^4 \left (1-c^2 x\right )}-\frac {1}{c^4}\right )d\sqrt {x}}{e}-\frac {d \int \frac {x^{3/2} \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )}{d+e x}d\sqrt {x}}{e}\right )\)

\(\Big \downarrow \) 2009

\(\displaystyle 2 \left (\frac {\frac {1}{4} x^2 \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )-\frac {1}{4} b c \left (\frac {\text {arctanh}\left (c \sqrt {x}\right )}{c^5}-\frac {\sqrt {x}}{c^4}-\frac {x^{3/2}}{3 c^2}\right )}{e}-\frac {d \int \frac {x^{3/2} \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )}{d+e x}d\sqrt {x}}{e}\right )\)

\(\Big \downarrow \) 6542

\(\displaystyle 2 \left (\frac {\frac {1}{4} x^2 \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )-\frac {1}{4} b c \left (\frac {\text {arctanh}\left (c \sqrt {x}\right )}{c^5}-\frac {\sqrt {x}}{c^4}-\frac {x^{3/2}}{3 c^2}\right )}{e}-\frac {d \left (\frac {\int \sqrt {x} \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )d\sqrt {x}}{e}-\frac {d \int \frac {\sqrt {x} \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )}{d+e x}d\sqrt {x}}{e}\right )}{e}\right )\)

\(\Big \downarrow \) 6452

\(\displaystyle 2 \left (\frac {\frac {1}{4} x^2 \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )-\frac {1}{4} b c \left (\frac {\text {arctanh}\left (c \sqrt {x}\right )}{c^5}-\frac {\sqrt {x}}{c^4}-\frac {x^{3/2}}{3 c^2}\right )}{e}-\frac {d \left (\frac {\frac {1}{2} x \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )-\frac {1}{2} b c \int \frac {x}{1-c^2 x}d\sqrt {x}}{e}-\frac {d \int \frac {\sqrt {x} \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )}{d+e x}d\sqrt {x}}{e}\right )}{e}\right )\)

\(\Big \downarrow \) 262

\(\displaystyle 2 \left (\frac {\frac {1}{4} x^2 \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )-\frac {1}{4} b c \left (\frac {\text {arctanh}\left (c \sqrt {x}\right )}{c^5}-\frac {\sqrt {x}}{c^4}-\frac {x^{3/2}}{3 c^2}\right )}{e}-\frac {d \left (\frac {\frac {1}{2} x \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )-\frac {1}{2} b c \left (\frac {\int \frac {1}{1-c^2 x}d\sqrt {x}}{c^2}-\frac {\sqrt {x}}{c^2}\right )}{e}-\frac {d \int \frac {\sqrt {x} \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )}{d+e x}d\sqrt {x}}{e}\right )}{e}\right )\)

\(\Big \downarrow \) 219

\(\displaystyle 2 \left (\frac {\frac {1}{4} x^2 \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )-\frac {1}{4} b c \left (\frac {\text {arctanh}\left (c \sqrt {x}\right )}{c^5}-\frac {\sqrt {x}}{c^4}-\frac {x^{3/2}}{3 c^2}\right )}{e}-\frac {d \left (\frac {\frac {1}{2} x \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )-\frac {1}{2} b c \left (\frac {\text {arctanh}\left (c \sqrt {x}\right )}{c^3}-\frac {\sqrt {x}}{c^2}\right )}{e}-\frac {d \int \frac {\sqrt {x} \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )}{d+e x}d\sqrt {x}}{e}\right )}{e}\right )\)

\(\Big \downarrow \) 6606

\(\displaystyle 2 \left (\frac {\frac {1}{4} x^2 \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )-\frac {1}{4} b c \left (\frac {\text {arctanh}\left (c \sqrt {x}\right )}{c^5}-\frac {\sqrt {x}}{c^4}-\frac {x^{3/2}}{3 c^2}\right )}{e}-\frac {d \left (\frac {\frac {1}{2} x \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )-\frac {1}{2} b c \left (\frac {\text {arctanh}\left (c \sqrt {x}\right )}{c^3}-\frac {\sqrt {x}}{c^2}\right )}{e}-\frac {d \int \left (\frac {a+b \text {arctanh}\left (c \sqrt {x}\right )}{2 \sqrt {e} \left (\sqrt {-d}+\sqrt {e} \sqrt {x}\right )}-\frac {a+b \text {arctanh}\left (c \sqrt {x}\right )}{2 \sqrt {e} \left (\sqrt {-d}-\sqrt {e} \sqrt {x}\right )}\right )d\sqrt {x}}{e}\right )}{e}\right )\)

\(\Big \downarrow \) 2009

\(\displaystyle 2 \left (\frac {\frac {1}{4} x^2 \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )-\frac {1}{4} b c \left (\frac {\text {arctanh}\left (c \sqrt {x}\right )}{c^5}-\frac {\sqrt {x}}{c^4}-\frac {x^{3/2}}{3 c^2}\right )}{e}-\frac {d \left (\frac {\frac {1}{2} x \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )-\frac {1}{2} b c \left (\frac {\text {arctanh}\left (c \sqrt {x}\right )}{c^3}-\frac {\sqrt {x}}{c^2}\right )}{e}-\frac {d \left (\frac {\left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right ) \log \left (\frac {2 c \left (\sqrt {-d}-\sqrt {e} \sqrt {x}\right )}{\left (c \sqrt {x}+1\right ) \left (c \sqrt {-d}-\sqrt {e}\right )}\right )}{2 e}+\frac {\left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right ) \log \left (\frac {2 c \left (\sqrt {-d}+\sqrt {e} \sqrt {x}\right )}{\left (c \sqrt {x}+1\right ) \left (c \sqrt {-d}+\sqrt {e}\right )}\right )}{2 e}-\frac {\log \left (\frac {2}{c \sqrt {x}+1}\right ) \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )}{e}-\frac {b \operatorname {PolyLog}\left (2,1-\frac {2 c \left (\sqrt {-d}-\sqrt {e} \sqrt {x}\right )}{\left (c \sqrt {-d}-\sqrt {e}\right ) \left (\sqrt {x} c+1\right )}\right )}{4 e}-\frac {b \operatorname {PolyLog}\left (2,1-\frac {2 c \left (\sqrt {-d}+\sqrt {e} \sqrt {x}\right )}{\left (\sqrt {-d} c+\sqrt {e}\right ) \left (\sqrt {x} c+1\right )}\right )}{4 e}+\frac {b \operatorname {PolyLog}\left (2,1-\frac {2}{\sqrt {x} c+1}\right )}{2 e}\right )}{e}\right )}{e}\right )\)

Input:

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

Output:

2*(((x^2*(a + b*ArcTanh[c*Sqrt[x]]))/4 - (b*c*(-(Sqrt[x]/c^4) - x^(3/2)/(3 
*c^2) + ArcTanh[c*Sqrt[x]]/c^5))/4)/e - (d*(((x*(a + b*ArcTanh[c*Sqrt[x]]) 
)/2 - (b*c*(-(Sqrt[x]/c^2) + ArcTanh[c*Sqrt[x]]/c^3))/2)/e - (d*(-(((a + b 
*ArcTanh[c*Sqrt[x]])*Log[2/(1 + c*Sqrt[x])])/e) + ((a + b*ArcTanh[c*Sqrt[x 
]])*Log[(2*c*(Sqrt[-d] - Sqrt[e]*Sqrt[x]))/((c*Sqrt[-d] - Sqrt[e])*(1 + c* 
Sqrt[x]))])/(2*e) + ((a + b*ArcTanh[c*Sqrt[x]])*Log[(2*c*(Sqrt[-d] + Sqrt[ 
e]*Sqrt[x]))/((c*Sqrt[-d] + Sqrt[e])*(1 + c*Sqrt[x]))])/(2*e) + (b*PolyLog 
[2, 1 - 2/(1 + c*Sqrt[x])])/(2*e) - (b*PolyLog[2, 1 - (2*c*(Sqrt[-d] - Sqr 
t[e]*Sqrt[x]))/((c*Sqrt[-d] - Sqrt[e])*(1 + c*Sqrt[x]))])/(4*e) - (b*PolyL 
og[2, 1 - (2*c*(Sqrt[-d] + Sqrt[e]*Sqrt[x]))/((c*Sqrt[-d] + Sqrt[e])*(1 + 
c*Sqrt[x]))])/(4*e)))/e))/e)
 

Defintions of rubi rules used

rule 219

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

rule 254

Int[(x_)^(m_)/((a_) + (b_.)*(x_)^2), x_Symbol] :> Int[PolynomialDivide[x^m, 
 a + b*x^2, x], x] /; FreeQ[{a, b}, x] && IGtQ[m, 3]
 

rule 262

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[c*(c*x) 
^(m - 1)*((a + b*x^2)^(p + 1)/(b*(m + 2*p + 1))), x] - Simp[a*c^2*((m - 1)/ 
(b*(m + 2*p + 1)))   Int[(c*x)^(m - 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b 
, c, p}, x] && GtQ[m, 2 - 1] && NeQ[m + 2*p + 1, 0] && IntBinomialQ[a, b, c 
, 2, m, p, x]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 6452

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

rule 6542

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

rule 6606

Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_.)*((d_) + (e 
_.)*(x_)^2)^(q_.), x_Symbol] :> With[{u = ExpandIntegrand[(a + b*ArcTanh[c* 
x])^p, (f*x)^m*(d + e*x^2)^q, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c, 
 d, e, f, m}, x] && IntegerQ[q] && IGtQ[p, 0] && (GtQ[q, 0] || IntegerQ[m])
 

rule 7267

Int[u_, x_Symbol] :> With[{lst = SubstForFractionalPowerOfLinear[u, x]}, Si 
mp[lst[[2]]*lst[[4]]   Subst[Int[lst[[1]], x], x, lst[[3]]^(1/lst[[2]])], x 
] /;  !FalseQ[lst] && SubstForFractionalPowerQ[u, lst[[3]], x]]
 

Maple [A] (verified)

Time = 0.18 (sec) , antiderivative size = 564, normalized size of antiderivative = 1.23

method	result	size
parts	\(\frac {a \,x^{2}}{2 e}-\frac {a d x}{e^{2}}+\frac {a \,d^{2} \ln \left (e x +d \right )}{e^{3}}+\frac {2 b \left (-\frac {c^{6} \operatorname {arctanh}\left (c \sqrt {x}\right ) x d}{2 e^{2}}+\frac {c^{6} \operatorname {arctanh}\left (c \sqrt {x}\right ) x^{2}}{4 e}+\frac {c^{6} \operatorname {arctanh}\left (c \sqrt {x}\right ) d^{2} \ln \left (c^{2} e x +c^{2} d \right )}{2 e^{3}}-\frac {c^{2} \left (-\frac {\frac {e \,c^{3} x^{\frac {3}{2}}}{3}-2 c^{3} d \sqrt {x}+e c \sqrt {x}+\frac {\left (-2 c^{2} d +e \right ) \ln \left (c \sqrt {x}-1\right )}{2}-\frac {\left (-2 c^{2} d +e \right ) \ln \left (1+c \sqrt {x}\right )}{2}}{2 e^{2}}-\frac {c^{4} d^{2} \left (\frac {\ln \left (c \sqrt {x}-1\right ) \ln \left (c^{2} e x +c^{2} d \right )}{2}-e \left (\frac {\ln \left (c \sqrt {x}-1\right ) \left (\ln \left (\frac {c \sqrt {-d e}-e \left (c \sqrt {x}-1\right )-e}{c \sqrt {-d e}-e}\right )+\ln \left (\frac {c \sqrt {-d e}+e \left (c \sqrt {x}-1\right )+e}{c \sqrt {-d e}+e}\right )\right )}{2 e}+\frac {\operatorname {dilog}\left (\frac {c \sqrt {-d e}-e \left (c \sqrt {x}-1\right )-e}{c \sqrt {-d e}-e}\right )+\operatorname {dilog}\left (\frac {c \sqrt {-d e}+e \left (c \sqrt {x}-1\right )+e}{c \sqrt {-d e}+e}\right )}{2 e}\right )-\frac {\ln \left (1+c \sqrt {x}\right ) \ln \left (c^{2} e x +c^{2} d \right )}{2}+e \left (\frac {\ln \left (1+c \sqrt {x}\right ) \left (\ln \left (\frac {c \sqrt {-d e}-e \left (1+c \sqrt {x}\right )+e}{c \sqrt {-d e}+e}\right )+\ln \left (\frac {c \sqrt {-d e}+e \left (1+c \sqrt {x}\right )-e}{c \sqrt {-d e}-e}\right )\right )}{2 e}+\frac {\operatorname {dilog}\left (\frac {c \sqrt {-d e}-e \left (1+c \sqrt {x}\right )+e}{c \sqrt {-d e}+e}\right )+\operatorname {dilog}\left (\frac {c \sqrt {-d e}+e \left (1+c \sqrt {x}\right )-e}{c \sqrt {-d e}-e}\right )}{2 e}\right )\right )}{e^{3}}\right )}{2}\right )}{c^{6}}\)	\(564\)
derivativedivides	\(\frac {-\frac {a \,c^{6} d x}{e^{2}}+\frac {a \,c^{6} x^{2}}{2 e}+\frac {a \,c^{6} d^{2} \ln \left (c^{2} e x +c^{2} d \right )}{e^{3}}+2 b \,c^{2} \left (-\frac {\operatorname {arctanh}\left (c \sqrt {x}\right ) c^{4} d x}{2 e^{2}}+\frac {\operatorname {arctanh}\left (c \sqrt {x}\right ) c^{4} x^{2}}{4 e}+\frac {\operatorname {arctanh}\left (c \sqrt {x}\right ) c^{4} d^{2} \ln \left (c^{2} e x +c^{2} d \right )}{2 e^{3}}-\frac {2 c^{3} d \sqrt {x}-\frac {e \,c^{3} x^{\frac {3}{2}}}{3}-e c \sqrt {x}+\frac {\left (2 c^{2} d -e \right ) \ln \left (c \sqrt {x}-1\right )}{2}-\frac {\left (2 c^{2} d -e \right ) \ln \left (1+c \sqrt {x}\right )}{2}}{4 e^{2}}+\frac {c^{4} d^{2} \left (\frac {\ln \left (c \sqrt {x}-1\right ) \ln \left (c^{2} e x +c^{2} d \right )}{2}-e \left (\frac {\ln \left (c \sqrt {x}-1\right ) \left (\ln \left (\frac {c \sqrt {-d e}-e \left (c \sqrt {x}-1\right )-e}{c \sqrt {-d e}-e}\right )+\ln \left (\frac {c \sqrt {-d e}+e \left (c \sqrt {x}-1\right )+e}{c \sqrt {-d e}+e}\right )\right )}{2 e}+\frac {\operatorname {dilog}\left (\frac {c \sqrt {-d e}-e \left (c \sqrt {x}-1\right )-e}{c \sqrt {-d e}-e}\right )+\operatorname {dilog}\left (\frac {c \sqrt {-d e}+e \left (c \sqrt {x}-1\right )+e}{c \sqrt {-d e}+e}\right )}{2 e}\right )-\frac {\ln \left (1+c \sqrt {x}\right ) \ln \left (c^{2} e x +c^{2} d \right )}{2}+e \left (\frac {\ln \left (1+c \sqrt {x}\right ) \left (\ln \left (\frac {c \sqrt {-d e}-e \left (1+c \sqrt {x}\right )+e}{c \sqrt {-d e}+e}\right )+\ln \left (\frac {c \sqrt {-d e}+e \left (1+c \sqrt {x}\right )-e}{c \sqrt {-d e}-e}\right )\right )}{2 e}+\frac {\operatorname {dilog}\left (\frac {c \sqrt {-d e}-e \left (1+c \sqrt {x}\right )+e}{c \sqrt {-d e}+e}\right )+\operatorname {dilog}\left (\frac {c \sqrt {-d e}+e \left (1+c \sqrt {x}\right )-e}{c \sqrt {-d e}-e}\right )}{2 e}\right )\right )}{2 e^{3}}\right )}{c^{6}}\)	\(584\)
default	\(\frac {-\frac {a \,c^{6} d x}{e^{2}}+\frac {a \,c^{6} x^{2}}{2 e}+\frac {a \,c^{6} d^{2} \ln \left (c^{2} e x +c^{2} d \right )}{e^{3}}+2 b \,c^{2} \left (-\frac {\operatorname {arctanh}\left (c \sqrt {x}\right ) c^{4} d x}{2 e^{2}}+\frac {\operatorname {arctanh}\left (c \sqrt {x}\right ) c^{4} x^{2}}{4 e}+\frac {\operatorname {arctanh}\left (c \sqrt {x}\right ) c^{4} d^{2} \ln \left (c^{2} e x +c^{2} d \right )}{2 e^{3}}-\frac {2 c^{3} d \sqrt {x}-\frac {e \,c^{3} x^{\frac {3}{2}}}{3}-e c \sqrt {x}+\frac {\left (2 c^{2} d -e \right ) \ln \left (c \sqrt {x}-1\right )}{2}-\frac {\left (2 c^{2} d -e \right ) \ln \left (1+c \sqrt {x}\right )}{2}}{4 e^{2}}+\frac {c^{4} d^{2} \left (\frac {\ln \left (c \sqrt {x}-1\right ) \ln \left (c^{2} e x +c^{2} d \right )}{2}-e \left (\frac {\ln \left (c \sqrt {x}-1\right ) \left (\ln \left (\frac {c \sqrt {-d e}-e \left (c \sqrt {x}-1\right )-e}{c \sqrt {-d e}-e}\right )+\ln \left (\frac {c \sqrt {-d e}+e \left (c \sqrt {x}-1\right )+e}{c \sqrt {-d e}+e}\right )\right )}{2 e}+\frac {\operatorname {dilog}\left (\frac {c \sqrt {-d e}-e \left (c \sqrt {x}-1\right )-e}{c \sqrt {-d e}-e}\right )+\operatorname {dilog}\left (\frac {c \sqrt {-d e}+e \left (c \sqrt {x}-1\right )+e}{c \sqrt {-d e}+e}\right )}{2 e}\right )-\frac {\ln \left (1+c \sqrt {x}\right ) \ln \left (c^{2} e x +c^{2} d \right )}{2}+e \left (\frac {\ln \left (1+c \sqrt {x}\right ) \left (\ln \left (\frac {c \sqrt {-d e}-e \left (1+c \sqrt {x}\right )+e}{c \sqrt {-d e}+e}\right )+\ln \left (\frac {c \sqrt {-d e}+e \left (1+c \sqrt {x}\right )-e}{c \sqrt {-d e}-e}\right )\right )}{2 e}+\frac {\operatorname {dilog}\left (\frac {c \sqrt {-d e}-e \left (1+c \sqrt {x}\right )+e}{c \sqrt {-d e}+e}\right )+\operatorname {dilog}\left (\frac {c \sqrt {-d e}+e \left (1+c \sqrt {x}\right )-e}{c \sqrt {-d e}-e}\right )}{2 e}\right )\right )}{2 e^{3}}\right )}{c^{6}}\)	\(584\)

Input:

int(x^2*(a+b*arctanh(c*x^(1/2)))/(e*x+d),x,method=_RETURNVERBOSE)
 

Output:

1/2*a/e*x^2-a/e^2*d*x+a*d^2/e^3*ln(e*x+d)+2*b/c^6*(-1/2*c^6*arctanh(c*x^(1 
/2))/e^2*x*d+1/4*c^6*arctanh(c*x^(1/2))/e*x^2+1/2*c^6*arctanh(c*x^(1/2))*d 
^2/e^3*ln(c^2*e*x+c^2*d)-1/2*c^2*(-1/2/e^2*(1/3*e*c^3*x^(3/2)-2*c^3*d*x^(1 
/2)+e*c*x^(1/2)+1/2*(-2*c^2*d+e)*ln(c*x^(1/2)-1)-1/2*(-2*c^2*d+e)*ln(1+c*x 
^(1/2)))-c^4*d^2/e^3*(1/2*ln(c*x^(1/2)-1)*ln(c^2*e*x+c^2*d)-e*(1/2*ln(c*x^ 
(1/2)-1)*(ln((c*(-d*e)^(1/2)-e*(c*x^(1/2)-1)-e)/(c*(-d*e)^(1/2)-e))+ln((c* 
(-d*e)^(1/2)+e*(c*x^(1/2)-1)+e)/(c*(-d*e)^(1/2)+e)))/e+1/2*(dilog((c*(-d*e 
)^(1/2)-e*(c*x^(1/2)-1)-e)/(c*(-d*e)^(1/2)-e))+dilog((c*(-d*e)^(1/2)+e*(c* 
x^(1/2)-1)+e)/(c*(-d*e)^(1/2)+e)))/e)-1/2*ln(1+c*x^(1/2))*ln(c^2*e*x+c^2*d 
)+e*(1/2*ln(1+c*x^(1/2))*(ln((c*(-d*e)^(1/2)-e*(1+c*x^(1/2))+e)/(c*(-d*e)^ 
(1/2)+e))+ln((c*(-d*e)^(1/2)+e*(1+c*x^(1/2))-e)/(c*(-d*e)^(1/2)-e)))/e+1/2 
*(dilog((c*(-d*e)^(1/2)-e*(1+c*x^(1/2))+e)/(c*(-d*e)^(1/2)+e))+dilog((c*(- 
d*e)^(1/2)+e*(1+c*x^(1/2))-e)/(c*(-d*e)^(1/2)-e)))/e))))
 

Fricas [F]

\[ \int \frac {x^2 \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )}{d+e x} \, dx=\int { \frac {{\left (b \operatorname {artanh}\left (c \sqrt {x}\right ) + a\right )} x^{2}}{e x + d} \,d x } \] Input:

integrate(x^2*(a+b*arctanh(c*x^(1/2)))/(e*x+d),x, algorithm="fricas")
 

Output:

integral((b*x^2*arctanh(c*sqrt(x)) + a*x^2)/(e*x + d), x)
 

Sympy [F(-1)]

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

integrate(x**2*(a+b*atanh(c*x**(1/2)))/(e*x+d),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {x^2 \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )}{d+e x} \, dx=\int { \frac {{\left (b \operatorname {artanh}\left (c \sqrt {x}\right ) + a\right )} x^{2}}{e x + d} \,d x } \] Input:

integrate(x^2*(a+b*arctanh(c*x^(1/2)))/(e*x+d),x, algorithm="maxima")
 

Output:

1/2*a*(2*d^2*log(e*x + d)/e^3 + (e*x^2 - 2*d*x)/e^2) + b*integrate(1/2*x^2 
*log(c*sqrt(x) + 1)/(e*x + d), x) - b*integrate(1/2*x^2*log(-c*sqrt(x) + 1 
)/(e*x + d), x)
 

Giac [F]

\[ \int \frac {x^2 \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )}{d+e x} \, dx=\int { \frac {{\left (b \operatorname {artanh}\left (c \sqrt {x}\right ) + a\right )} x^{2}}{e x + d} \,d x } \] Input:

integrate(x^2*(a+b*arctanh(c*x^(1/2)))/(e*x+d),x, algorithm="giac")
 

Output:

integrate((b*arctanh(c*sqrt(x)) + a)*x^2/(e*x + d), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {x^2 \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )}{d+e x} \, dx=\int \frac {x^2\,\left (a+b\,\mathrm {atanh}\left (c\,\sqrt {x}\right )\right )}{d+e\,x} \,d x \] Input:

int((x^2*(a + b*atanh(c*x^(1/2))))/(d + e*x),x)
                                                                                    
                                                                                    
 

Output:

int((x^2*(a + b*atanh(c*x^(1/2))))/(d + e*x), x)
 

Reduce [F]

\[ \int \frac {x^2 \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )}{d+e x} \, dx=\frac {2 \left (\int \frac {\mathit {atanh} \left (\sqrt {x}\, c \right ) x^{2}}{e x +d}d x \right ) b \,e^{3}+2 \,\mathrm {log}\left (e x +d \right ) a \,d^{2}-2 a d e x +a \,e^{2} x^{2}}{2 e^{3}} \] Input:

int(x^2*(a+b*atanh(c*x^(1/2)))/(e*x+d),x)
 

Output:

(2*int((atanh(sqrt(x)*c)*x**2)/(d + e*x),x)*b*e**3 + 2*log(d + e*x)*a*d**2 
 - 2*a*d*e*x + a*e**2*x**2)/(2*e**3)