Integrand size = 23, antiderivative size = 460 \[ \int \frac {x^2 \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )}{d+e x} \, dx=-\frac {b d \sqrt {x}}{c e^2}+\frac {b \sqrt {x}}{2 c^3 e}+\frac {b x^{3/2}}{6 c e}+\frac {b d \text {arctanh}\left (c \sqrt {x}\right )}{c^2 e^2}-\frac {b \text {arctanh}\left (c \sqrt {x}\right )}{2 c^4 e}-\frac {d x \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )}{e^2}+\frac {x^2 \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )}{2 e}-\frac {2 d^2 \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right ) \log \left (\frac {2}{1+c \sqrt {x}}\right )}{e^3}+\frac {d^2 \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right ) \log \left (\frac {2 c \left (\sqrt {-d}-\sqrt {e} \sqrt {x}\right )}{\left (c \sqrt {-d}-\sqrt {e}\right ) \left (1+c \sqrt {x}\right )}\right )}{e^3}+\frac {d^2 \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right ) \log \left (\frac {2 c \left (\sqrt {-d}+\sqrt {e} \sqrt {x}\right )}{\left (c \sqrt {-d}+\sqrt {e}\right ) \left (1+c \sqrt {x}\right )}\right )}{e^3}+\frac {b d^2 \operatorname {PolyLog}\left (2,1-\frac {2}{1+c \sqrt {x}}\right )}{e^3}-\frac {b d^2 \operatorname {PolyLog}\left (2,1-\frac {2 c \left (\sqrt {-d}-\sqrt {e} \sqrt {x}\right )}{\left (c \sqrt {-d}-\sqrt {e}\right ) \left (1+c \sqrt {x}\right )}\right )}{2 e^3}-\frac {b d^2 \operatorname {PolyLog}\left (2,1-\frac {2 c \left (\sqrt {-d}+\sqrt {e} \sqrt {x}\right )}{\left (c \sqrt {-d}+\sqrt {e}\right ) \left (1+c \sqrt {x}\right )}\right )}{2 e^3} \]
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-b*d*x^(1/2)/c/e^ 2+1/2*b*x^(1/2)/c^3/e
Result contains complex when optimal does not.
Time = 2.11 (sec) , antiderivative size = 558, normalized size of antiderivative = 1.21 \[ \int \frac {x^2 \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )}{d+e x} \, dx=\frac {-6 a d e x+3 a e^2 x^2+6 a d^2 \log (d+e x)+\frac {b \left (2 c e \left (-3 c^2 d+2 e\right ) \sqrt {x}+c e^2 \sqrt {x} \left (-1+c^2 x\right )-6 \left (c^2 d-e\right ) e \left (-1+c^2 x\right ) \text {arctanh}\left (c \sqrt {x}\right )+3 e^2 \left (-1+c^2 x\right )^2 \text {arctanh}\left (c \sqrt {x}\right )-6 c^4 d^2 \left (\text {arctanh}\left (c \sqrt {x}\right ) \left (\text {arctanh}\left (c \sqrt {x}\right )+2 \log \left (1+e^{-2 \text {arctanh}\left (c \sqrt {x}\right )}\right )\right )-\operatorname {PolyLog}\left (2,-e^{-2 \text {arctanh}\left (c \sqrt {x}\right )}\right )\right )+3 c^4 d^2 \left (2 \text {arctanh}\left (c \sqrt {x}\right )^2-4 i \arcsin \left (\sqrt {\frac {c^2 d}{c^2 d+e}}\right ) \text {arctanh}\left (\frac {c e \sqrt {x}}{\sqrt {-c^2 d e}}\right )+2 \left (-i \arcsin \left (\sqrt {\frac {c^2 d}{c^2 d+e}}\right )+\text {arctanh}\left (c \sqrt {x}\right )\right ) \log \left (\frac {e^{-2 \text {arctanh}\left (c \sqrt {x}\right )} \left (-2 \sqrt {-c^2 d e}+e \left (-1+e^{2 \text {arctanh}\left (c \sqrt {x}\right )}\right )+c^2 d \left (1+e^{2 \text {arctanh}\left (c \sqrt {x}\right )}\right )\right )}{c^2 d+e}\right )+2 \left (i \arcsin \left (\sqrt {\frac {c^2 d}{c^2 d+e}}\right )+\text {arctanh}\left (c \sqrt {x}\right )\right ) \log \left (\frac {e^{-2 \text {arctanh}\left (c \sqrt {x}\right )} \left (2 \sqrt {-c^2 d e}+e \left (-1+e^{2 \text {arctanh}\left (c \sqrt {x}\right )}\right )+c^2 d \left (1+e^{2 \text {arctanh}\left (c \sqrt {x}\right )}\right )\right )}{c^2 d+e}\right )-\operatorname {PolyLog}\left (2,\frac {\left (-c^2 d+e-2 \sqrt {-c^2 d e}\right ) e^{-2 \text {arctanh}\left (c \sqrt {x}\right )}}{c^2 d+e}\right )-\operatorname {PolyLog}\left (2,\frac {\left (-c^2 d+e+2 \sqrt {-c^2 d e}\right ) e^{-2 \text {arctanh}\left (c \sqrt {x}\right )}}{c^2 d+e}\right )\right )\right )}{c^4}}{6 e^3} \]
(-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)
Time = 1.40 (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 |
\(\displaystyle 2 \left (\frac {\frac {1}{4} x^2 \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )-\frac {1}{4} b c \left (\frac {\text {arctanh}\left (c \sqrt {x}\right )}{c^5}-\frac {\sqrt {x}}{c^4}-\frac {x^{3/2}}{3 c^2}\right )}{e}-\frac {d \left (\frac {\frac {1}{2} x \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )-\frac {1}{2} b c \left (\frac {\text {arctanh}\left (c \sqrt {x}\right )}{c^3}-\frac {\sqrt {x}}{c^2}\right )}{e}-\frac {d \left (\frac {\left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right ) \log \left (\frac {2 c \left (\sqrt {-d}-\sqrt {e} \sqrt {x}\right )}{\left (c \sqrt {x}+1\right ) \left (c \sqrt {-d}-\sqrt {e}\right )}\right )}{2 e}+\frac {\left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right ) \log \left (\frac {2 c \left (\sqrt {-d}+\sqrt {e} \sqrt {x}\right )}{\left (c \sqrt {x}+1\right ) \left (c \sqrt {-d}+\sqrt {e}\right )}\right )}{2 e}-\frac {\log \left (\frac {2}{c \sqrt {x}+1}\right ) \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )}{e}-\frac {b \operatorname {PolyLog}\left (2,1-\frac {2 c \left (\sqrt {-d}-\sqrt {e} \sqrt {x}\right )}{\left (c \sqrt {-d}-\sqrt {e}\right ) \left (\sqrt {x} c+1\right )}\right )}{4 e}-\frac {b \operatorname {PolyLog}\left (2,1-\frac {2 c \left (\sqrt {-d}+\sqrt {e} \sqrt {x}\right )}{\left (\sqrt {-d} c+\sqrt {e}\right ) \left (\sqrt {x} c+1\right )}\right )}{4 e}+\frac {b \operatorname {PolyLog}\left (2,1-\frac {2}{\sqrt {x} c+1}\right )}{2 e}\right )}{e}\right )}{e}\right )\) |
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)
3.1.44.3.1 Defintions of rubi rules used
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])
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]
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]
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]
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]
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])
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]]
Time = 2.84 (sec) , antiderivative size = 564, normalized size of antiderivative = 1.23
method | result | size |
parts | \(\frac {a \,x^{2}}{2 e}-\frac {a d x}{e^{2}}+\frac {a \,d^{2} \ln \left (e x +d \right )}{e^{3}}+\frac {2 b \left (-\frac {c^{6} \operatorname {arctanh}\left (c \sqrt {x}\right ) x d}{2 e^{2}}+\frac {c^{6} \operatorname {arctanh}\left (c \sqrt {x}\right ) x^{2}}{4 e}+\frac {c^{6} \operatorname {arctanh}\left (c \sqrt {x}\right ) d^{2} \ln \left (c^{2} e x +c^{2} d \right )}{2 e^{3}}-\frac {c^{2} \left (-\frac {\frac {e \,c^{3} x^{\frac {3}{2}}}{3}-2 c^{3} d \sqrt {x}+e c \sqrt {x}+\frac {\left (-2 c^{2} d +e \right ) \ln \left (c \sqrt {x}-1\right )}{2}-\frac {\left (-2 c^{2} d +e \right ) \ln \left (1+c \sqrt {x}\right )}{2}}{2 e^{2}}-\frac {c^{4} d^{2} \left (-\frac {\ln \left (1+c \sqrt {x}\right ) \ln \left (c^{2} e x +c^{2} d \right )}{2}+e \left (\frac {\ln \left (1+c \sqrt {x}\right ) \left (\ln \left (\frac {c \sqrt {-d e}-e \left (1+c \sqrt {x}\right )+e}{c \sqrt {-d e}+e}\right )+\ln \left (\frac {c \sqrt {-d e}+e \left (1+c \sqrt {x}\right )-e}{c \sqrt {-d e}-e}\right )\right )}{2 e}+\frac {\operatorname {dilog}\left (\frac {c \sqrt {-d e}-e \left (1+c \sqrt {x}\right )+e}{c \sqrt {-d e}+e}\right )+\operatorname {dilog}\left (\frac {c \sqrt {-d e}+e \left (1+c \sqrt {x}\right )-e}{c \sqrt {-d e}-e}\right )}{2 e}\right )+\frac {\ln \left (c \sqrt {x}-1\right ) \ln \left (c^{2} e x +c^{2} d \right )}{2}-e \left (\frac {\ln \left (c \sqrt {x}-1\right ) \left (\ln \left (\frac {c \sqrt {-d e}-e \left (c \sqrt {x}-1\right )-e}{c \sqrt {-d e}-e}\right )+\ln \left (\frac {c \sqrt {-d e}+e \left (c \sqrt {x}-1\right )+e}{c \sqrt {-d e}+e}\right )\right )}{2 e}+\frac {\operatorname {dilog}\left (\frac {c \sqrt {-d e}-e \left (c \sqrt {x}-1\right )-e}{c \sqrt {-d e}-e}\right )+\operatorname {dilog}\left (\frac {c \sqrt {-d e}+e \left (c \sqrt {x}-1\right )+e}{c \sqrt {-d e}+e}\right )}{2 e}\right )\right )}{e^{3}}\right )}{2}\right )}{c^{6}}\) | \(564\) |
derivativedivides | \(\frac {-\frac {a \,c^{6} d x}{e^{2}}+\frac {a \,c^{6} x^{2}}{2 e}+\frac {a \,c^{6} d^{2} \ln \left (c^{2} e x +c^{2} d \right )}{e^{3}}+2 b \,c^{2} \left (-\frac {\operatorname {arctanh}\left (c \sqrt {x}\right ) c^{4} d x}{2 e^{2}}+\frac {\operatorname {arctanh}\left (c \sqrt {x}\right ) c^{4} x^{2}}{4 e}+\frac {\operatorname {arctanh}\left (c \sqrt {x}\right ) c^{4} d^{2} \ln \left (c^{2} e x +c^{2} d \right )}{2 e^{3}}-\frac {2 c^{3} d \sqrt {x}-\frac {e \,c^{3} x^{\frac {3}{2}}}{3}-e c \sqrt {x}+\frac {\left (2 c^{2} d -e \right ) \ln \left (c \sqrt {x}-1\right )}{2}-\frac {\left (2 c^{2} d -e \right ) \ln \left (1+c \sqrt {x}\right )}{2}}{4 e^{2}}+\frac {c^{4} d^{2} \left (-\frac {\ln \left (1+c \sqrt {x}\right ) \ln \left (c^{2} e x +c^{2} d \right )}{2}+e \left (\frac {\ln \left (1+c \sqrt {x}\right ) \left (\ln \left (\frac {c \sqrt {-d e}-e \left (1+c \sqrt {x}\right )+e}{c \sqrt {-d e}+e}\right )+\ln \left (\frac {c \sqrt {-d e}+e \left (1+c \sqrt {x}\right )-e}{c \sqrt {-d e}-e}\right )\right )}{2 e}+\frac {\operatorname {dilog}\left (\frac {c \sqrt {-d e}-e \left (1+c \sqrt {x}\right )+e}{c \sqrt {-d e}+e}\right )+\operatorname {dilog}\left (\frac {c \sqrt {-d e}+e \left (1+c \sqrt {x}\right )-e}{c \sqrt {-d e}-e}\right )}{2 e}\right )+\frac {\ln \left (c \sqrt {x}-1\right ) \ln \left (c^{2} e x +c^{2} d \right )}{2}-e \left (\frac {\ln \left (c \sqrt {x}-1\right ) \left (\ln \left (\frac {c \sqrt {-d e}-e \left (c \sqrt {x}-1\right )-e}{c \sqrt {-d e}-e}\right )+\ln \left (\frac {c \sqrt {-d e}+e \left (c \sqrt {x}-1\right )+e}{c \sqrt {-d e}+e}\right )\right )}{2 e}+\frac {\operatorname {dilog}\left (\frac {c \sqrt {-d e}-e \left (c \sqrt {x}-1\right )-e}{c \sqrt {-d e}-e}\right )+\operatorname {dilog}\left (\frac {c \sqrt {-d e}+e \left (c \sqrt {x}-1\right )+e}{c \sqrt {-d e}+e}\right )}{2 e}\right )\right )}{2 e^{3}}\right )}{c^{6}}\) | \(584\) |
default | \(\frac {-\frac {a \,c^{6} d x}{e^{2}}+\frac {a \,c^{6} x^{2}}{2 e}+\frac {a \,c^{6} d^{2} \ln \left (c^{2} e x +c^{2} d \right )}{e^{3}}+2 b \,c^{2} \left (-\frac {\operatorname {arctanh}\left (c \sqrt {x}\right ) c^{4} d x}{2 e^{2}}+\frac {\operatorname {arctanh}\left (c \sqrt {x}\right ) c^{4} x^{2}}{4 e}+\frac {\operatorname {arctanh}\left (c \sqrt {x}\right ) c^{4} d^{2} \ln \left (c^{2} e x +c^{2} d \right )}{2 e^{3}}-\frac {2 c^{3} d \sqrt {x}-\frac {e \,c^{3} x^{\frac {3}{2}}}{3}-e c \sqrt {x}+\frac {\left (2 c^{2} d -e \right ) \ln \left (c \sqrt {x}-1\right )}{2}-\frac {\left (2 c^{2} d -e \right ) \ln \left (1+c \sqrt {x}\right )}{2}}{4 e^{2}}+\frac {c^{4} d^{2} \left (-\frac {\ln \left (1+c \sqrt {x}\right ) \ln \left (c^{2} e x +c^{2} d \right )}{2}+e \left (\frac {\ln \left (1+c \sqrt {x}\right ) \left (\ln \left (\frac {c \sqrt {-d e}-e \left (1+c \sqrt {x}\right )+e}{c \sqrt {-d e}+e}\right )+\ln \left (\frac {c \sqrt {-d e}+e \left (1+c \sqrt {x}\right )-e}{c \sqrt {-d e}-e}\right )\right )}{2 e}+\frac {\operatorname {dilog}\left (\frac {c \sqrt {-d e}-e \left (1+c \sqrt {x}\right )+e}{c \sqrt {-d e}+e}\right )+\operatorname {dilog}\left (\frac {c \sqrt {-d e}+e \left (1+c \sqrt {x}\right )-e}{c \sqrt {-d e}-e}\right )}{2 e}\right )+\frac {\ln \left (c \sqrt {x}-1\right ) \ln \left (c^{2} e x +c^{2} d \right )}{2}-e \left (\frac {\ln \left (c \sqrt {x}-1\right ) \left (\ln \left (\frac {c \sqrt {-d e}-e \left (c \sqrt {x}-1\right )-e}{c \sqrt {-d e}-e}\right )+\ln \left (\frac {c \sqrt {-d e}+e \left (c \sqrt {x}-1\right )+e}{c \sqrt {-d e}+e}\right )\right )}{2 e}+\frac {\operatorname {dilog}\left (\frac {c \sqrt {-d e}-e \left (c \sqrt {x}-1\right )-e}{c \sqrt {-d e}-e}\right )+\operatorname {dilog}\left (\frac {c \sqrt {-d e}+e \left (c \sqrt {x}-1\right )+e}{c \sqrt {-d e}+e}\right )}{2 e}\right )\right )}{2 e^{3}}\right )}{c^{6}}\) | \(584\) |
1/2*a*x^2/e-a*d*x/e^2+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))*x^2/e+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(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)+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))))
\[ \int \frac {x^2 \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )}{d+e x} \, dx=\int { \frac {{\left (b \operatorname {artanh}\left (c \sqrt {x}\right ) + a\right )} x^{2}}{e x + d} \,d x } \]
Timed out. \[ \int \frac {x^2 \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )}{d+e x} \, dx=\text {Timed out} \]
\[ \int \frac {x^2 \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )}{d+e x} \, dx=\int { \frac {{\left (b \operatorname {artanh}\left (c \sqrt {x}\right ) + a\right )} x^{2}}{e x + d} \,d x } \]
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)
\[ \int \frac {x^2 \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )}{d+e x} \, dx=\int { \frac {{\left (b \operatorname {artanh}\left (c \sqrt {x}\right ) + a\right )} x^{2}}{e x + d} \,d x } \]
Timed out. \[ \int \frac {x^2 \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )}{d+e x} \, dx=\int \frac {x^2\,\left (a+b\,\mathrm {atanh}\left (c\,\sqrt {x}\right )\right )}{d+e\,x} \,d x \]