Integrand size = 28, antiderivative size = 291 \[ \int x^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2 \, dx=\frac {b^2 x \sqrt {d+c^2 d x^2}}{64 c^2}+\frac {1}{32} b^2 x^3 \sqrt {d+c^2 d x^2}-\frac {b^2 \sqrt {d+c^2 d x^2} \text {arcsinh}(c x)}{64 c^3 \sqrt {1+c^2 x^2}}-\frac {b x^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{8 c \sqrt {1+c^2 x^2}}-\frac {b c x^4 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{8 \sqrt {1+c^2 x^2}}+\frac {x \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{8 c^2}+\frac {1}{4} x^3 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2-\frac {\sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^3}{24 b c^3 \sqrt {1+c^2 x^2}} \] Output:
1/64*b^2*x*(c^2*d*x^2+d)^(1/2)/c^2+1/32*b^2*x^3*(c^2*d*x^2+d)^(1/2)-1/64*b ^2*(c^2*d*x^2+d)^(1/2)*arcsinh(c*x)/c^3/(c^2*x^2+1)^(1/2)-1/8*b*x^2*(c^2*d *x^2+d)^(1/2)*(a+b*arcsinh(c*x))/c/(c^2*x^2+1)^(1/2)-1/8*b*c*x^4*(c^2*d*x^ 2+d)^(1/2)*(a+b*arcsinh(c*x))/(c^2*x^2+1)^(1/2)+1/8*x*(c^2*d*x^2+d)^(1/2)* (a+b*arcsinh(c*x))^2/c^2+1/4*x^3*(c^2*d*x^2+d)^(1/2)*(a+b*arcsinh(c*x))^2- 1/24*(c^2*d*x^2+d)^(1/2)*(a+b*arcsinh(c*x))^3/b/c^3/(c^2*x^2+1)^(1/2)
Time = 1.24 (sec) , antiderivative size = 207, normalized size of antiderivative = 0.71 \[ \int x^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2 \, dx=-\frac {-96 a^2 c x \left (1+2 c^2 x^2\right ) \sqrt {d+c^2 d x^2}+96 a^2 \sqrt {d} \log \left (c d x+\sqrt {d} \sqrt {d+c^2 d x^2}\right )+\frac {12 a b \sqrt {d+c^2 d x^2} \left (8 \text {arcsinh}(c x)^2+\cosh (4 \text {arcsinh}(c x))-4 \text {arcsinh}(c x) \sinh (4 \text {arcsinh}(c x))\right )}{\sqrt {1+c^2 x^2}}+\frac {b^2 \sqrt {d+c^2 d x^2} \left (32 \text {arcsinh}(c x)^3+12 \text {arcsinh}(c x) \cosh (4 \text {arcsinh}(c x))-3 \left (1+8 \text {arcsinh}(c x)^2\right ) \sinh (4 \text {arcsinh}(c x))\right )}{\sqrt {1+c^2 x^2}}}{768 c^3} \] Input:
Integrate[x^2*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^2,x]
Output:
-1/768*(-96*a^2*c*x*(1 + 2*c^2*x^2)*Sqrt[d + c^2*d*x^2] + 96*a^2*Sqrt[d]*L og[c*d*x + Sqrt[d]*Sqrt[d + c^2*d*x^2]] + (12*a*b*Sqrt[d + c^2*d*x^2]*(8*A rcSinh[c*x]^2 + Cosh[4*ArcSinh[c*x]] - 4*ArcSinh[c*x]*Sinh[4*ArcSinh[c*x]] ))/Sqrt[1 + c^2*x^2] + (b^2*Sqrt[d + c^2*d*x^2]*(32*ArcSinh[c*x]^3 + 12*Ar cSinh[c*x]*Cosh[4*ArcSinh[c*x]] - 3*(1 + 8*ArcSinh[c*x]^2)*Sinh[4*ArcSinh[ c*x]]))/Sqrt[1 + c^2*x^2])/c^3
Time = 1.71 (sec) , antiderivative size = 295, normalized size of antiderivative = 1.01, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.357, Rules used = {6221, 6191, 262, 262, 222, 6227, 6191, 262, 222, 6198}
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.
| \(\displaystyle \int x^2 \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2 \, dx\) |
| \(\Big \downarrow \) 6221 |
| \(\displaystyle \frac {\sqrt {c^2 d x^2+d} \int \frac {x^2 (a+b \text {arcsinh}(c x))^2}{\sqrt {c^2 x^2+1}}dx}{4 \sqrt {c^2 x^2+1}}-\frac {b c \sqrt {c^2 d x^2+d} \int x^3 (a+b \text {arcsinh}(c x))dx}{2 \sqrt {c^2 x^2+1}}+\frac {1}{4} x^3 \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2\) |
| \(\Big \downarrow \) 6191 |
| \(\displaystyle \frac {\sqrt {c^2 d x^2+d} \int \frac {x^2 (a+b \text {arcsinh}(c x))^2}{\sqrt {c^2 x^2+1}}dx}{4 \sqrt {c^2 x^2+1}}-\frac {b c \sqrt {c^2 d x^2+d} \left (\frac {1}{4} x^4 (a+b \text {arcsinh}(c x))-\frac {1}{4} b c \int \frac {x^4}{\sqrt {c^2 x^2+1}}dx\right )}{2 \sqrt {c^2 x^2+1}}+\frac {1}{4} x^3 \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2\) |
| \(\Big \downarrow \) 262 |
| \(\displaystyle \frac {\sqrt {c^2 d x^2+d} \int \frac {x^2 (a+b \text {arcsinh}(c x))^2}{\sqrt {c^2 x^2+1}}dx}{4 \sqrt {c^2 x^2+1}}-\frac {b c \sqrt {c^2 d x^2+d} \left (\frac {1}{4} x^4 (a+b \text {arcsinh}(c x))-\frac {1}{4} b c \left (\frac {x^3 \sqrt {c^2 x^2+1}}{4 c^2}-\frac {3 \int \frac {x^2}{\sqrt {c^2 x^2+1}}dx}{4 c^2}\right )\right )}{2 \sqrt {c^2 x^2+1}}+\frac {1}{4} x^3 \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2\) |
| \(\Big \downarrow \) 262 |
| \(\displaystyle \frac {\sqrt {c^2 d x^2+d} \int \frac {x^2 (a+b \text {arcsinh}(c x))^2}{\sqrt {c^2 x^2+1}}dx}{4 \sqrt {c^2 x^2+1}}-\frac {b c \sqrt {c^2 d x^2+d} \left (\frac {1}{4} x^4 (a+b \text {arcsinh}(c x))-\frac {1}{4} b c \left (\frac {x^3 \sqrt {c^2 x^2+1}}{4 c^2}-\frac {3 \left (\frac {x \sqrt {c^2 x^2+1}}{2 c^2}-\frac {\int \frac {1}{\sqrt {c^2 x^2+1}}dx}{2 c^2}\right )}{4 c^2}\right )\right )}{2 \sqrt {c^2 x^2+1}}+\frac {1}{4} x^3 \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2\) |
| \(\Big \downarrow \) 222 |
| \(\displaystyle \frac {\sqrt {c^2 d x^2+d} \int \frac {x^2 (a+b \text {arcsinh}(c x))^2}{\sqrt {c^2 x^2+1}}dx}{4 \sqrt {c^2 x^2+1}}+\frac {1}{4} x^3 \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2-\frac {b c \sqrt {c^2 d x^2+d} \left (\frac {1}{4} x^4 (a+b \text {arcsinh}(c x))-\frac {1}{4} b c \left (\frac {x^3 \sqrt {c^2 x^2+1}}{4 c^2}-\frac {3 \left (\frac {x \sqrt {c^2 x^2+1}}{2 c^2}-\frac {\text {arcsinh}(c x)}{2 c^3}\right )}{4 c^2}\right )\right )}{2 \sqrt {c^2 x^2+1}}\) |
| \(\Big \downarrow \) 6227 |
| \(\displaystyle \frac {\sqrt {c^2 d x^2+d} \left (-\frac {\int \frac {(a+b \text {arcsinh}(c x))^2}{\sqrt {c^2 x^2+1}}dx}{2 c^2}-\frac {b \int x (a+b \text {arcsinh}(c x))dx}{c}+\frac {x \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))^2}{2 c^2}\right )}{4 \sqrt {c^2 x^2+1}}+\frac {1}{4} x^3 \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2-\frac {b c \sqrt {c^2 d x^2+d} \left (\frac {1}{4} x^4 (a+b \text {arcsinh}(c x))-\frac {1}{4} b c \left (\frac {x^3 \sqrt {c^2 x^2+1}}{4 c^2}-\frac {3 \left (\frac {x \sqrt {c^2 x^2+1}}{2 c^2}-\frac {\text {arcsinh}(c x)}{2 c^3}\right )}{4 c^2}\right )\right )}{2 \sqrt {c^2 x^2+1}}\) |
| \(\Big \downarrow \) 6191 |
| \(\displaystyle \frac {\sqrt {c^2 d x^2+d} \left (-\frac {b \left (\frac {1}{2} x^2 (a+b \text {arcsinh}(c x))-\frac {1}{2} b c \int \frac {x^2}{\sqrt {c^2 x^2+1}}dx\right )}{c}-\frac {\int \frac {(a+b \text {arcsinh}(c x))^2}{\sqrt {c^2 x^2+1}}dx}{2 c^2}+\frac {x \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))^2}{2 c^2}\right )}{4 \sqrt {c^2 x^2+1}}+\frac {1}{4} x^3 \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2-\frac {b c \sqrt {c^2 d x^2+d} \left (\frac {1}{4} x^4 (a+b \text {arcsinh}(c x))-\frac {1}{4} b c \left (\frac {x^3 \sqrt {c^2 x^2+1}}{4 c^2}-\frac {3 \left (\frac {x \sqrt {c^2 x^2+1}}{2 c^2}-\frac {\text {arcsinh}(c x)}{2 c^3}\right )}{4 c^2}\right )\right )}{2 \sqrt {c^2 x^2+1}}\) |
| \(\Big \downarrow \) 262 |
| \(\displaystyle \frac {\sqrt {c^2 d x^2+d} \left (-\frac {b \left (\frac {1}{2} x^2 (a+b \text {arcsinh}(c x))-\frac {1}{2} b c \left (\frac {x \sqrt {c^2 x^2+1}}{2 c^2}-\frac {\int \frac {1}{\sqrt {c^2 x^2+1}}dx}{2 c^2}\right )\right )}{c}-\frac {\int \frac {(a+b \text {arcsinh}(c x))^2}{\sqrt {c^2 x^2+1}}dx}{2 c^2}+\frac {x \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))^2}{2 c^2}\right )}{4 \sqrt {c^2 x^2+1}}+\frac {1}{4} x^3 \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2-\frac {b c \sqrt {c^2 d x^2+d} \left (\frac {1}{4} x^4 (a+b \text {arcsinh}(c x))-\frac {1}{4} b c \left (\frac {x^3 \sqrt {c^2 x^2+1}}{4 c^2}-\frac {3 \left (\frac {x \sqrt {c^2 x^2+1}}{2 c^2}-\frac {\text {arcsinh}(c x)}{2 c^3}\right )}{4 c^2}\right )\right )}{2 \sqrt {c^2 x^2+1}}\) |
| \(\Big \downarrow \) 222 |
| \(\displaystyle \frac {\sqrt {c^2 d x^2+d} \left (-\frac {\int \frac {(a+b \text {arcsinh}(c x))^2}{\sqrt {c^2 x^2+1}}dx}{2 c^2}+\frac {x \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))^2}{2 c^2}-\frac {b \left (\frac {1}{2} x^2 (a+b \text {arcsinh}(c x))-\frac {1}{2} b c \left (\frac {x \sqrt {c^2 x^2+1}}{2 c^2}-\frac {\text {arcsinh}(c x)}{2 c^3}\right )\right )}{c}\right )}{4 \sqrt {c^2 x^2+1}}+\frac {1}{4} x^3 \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2-\frac {b c \sqrt {c^2 d x^2+d} \left (\frac {1}{4} x^4 (a+b \text {arcsinh}(c x))-\frac {1}{4} b c \left (\frac {x^3 \sqrt {c^2 x^2+1}}{4 c^2}-\frac {3 \left (\frac {x \sqrt {c^2 x^2+1}}{2 c^2}-\frac {\text {arcsinh}(c x)}{2 c^3}\right )}{4 c^2}\right )\right )}{2 \sqrt {c^2 x^2+1}}\) |
| \(\Big \downarrow \) 6198 |
| \(\displaystyle \frac {1}{4} x^3 \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2+\frac {\sqrt {c^2 d x^2+d} \left (-\frac {(a+b \text {arcsinh}(c x))^3}{6 b c^3}+\frac {x \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))^2}{2 c^2}-\frac {b \left (\frac {1}{2} x^2 (a+b \text {arcsinh}(c x))-\frac {1}{2} b c \left (\frac {x \sqrt {c^2 x^2+1}}{2 c^2}-\frac {\text {arcsinh}(c x)}{2 c^3}\right )\right )}{c}\right )}{4 \sqrt {c^2 x^2+1}}-\frac {b c \sqrt {c^2 d x^2+d} \left (\frac {1}{4} x^4 (a+b \text {arcsinh}(c x))-\frac {1}{4} b c \left (\frac {x^3 \sqrt {c^2 x^2+1}}{4 c^2}-\frac {3 \left (\frac {x \sqrt {c^2 x^2+1}}{2 c^2}-\frac {\text {arcsinh}(c x)}{2 c^3}\right )}{4 c^2}\right )\right )}{2 \sqrt {c^2 x^2+1}}\) |
Input:
Int[x^2*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^2,x]
Output:
(x^3*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^2)/4 - (b*c*Sqrt[d + c^2*d*x ^2]*((x^4*(a + b*ArcSinh[c*x]))/4 - (b*c*((x^3*Sqrt[1 + c^2*x^2])/(4*c^2) - (3*((x*Sqrt[1 + c^2*x^2])/(2*c^2) - ArcSinh[c*x]/(2*c^3)))/(4*c^2)))/4)) /(2*Sqrt[1 + c^2*x^2]) + (Sqrt[d + c^2*d*x^2]*((x*Sqrt[1 + c^2*x^2]*(a + b *ArcSinh[c*x])^2)/(2*c^2) - (a + b*ArcSinh[c*x])^3/(6*b*c^3) - (b*((x^2*(a + b*ArcSinh[c*x]))/2 - (b*c*((x*Sqrt[1 + c^2*x^2])/(2*c^2) - ArcSinh[c*x] /(2*c^3)))/2))/c))/(4*Sqrt[1 + c^2*x^2])
Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSinh[Rt[b, 2]*(x/Sqrt [a])]/Rt[b, 2], x] /; FreeQ[{a, b}, x] && GtQ[a, 0] && PosQ[b]
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_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[(d*x)^(m + 1)*((a + b*ArcSinh[c*x])^n/(d*(m + 1))), x] - Simp[b*c* (n/(d*(m + 1))) Int[(d*x)^(m + 1)*((a + b*ArcSinh[c*x])^(n - 1)/Sqrt[1 + c^2*x^2]), x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] && NeQ[m, -1]
Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)/Sqrt[(d_) + (e_.)*(x_)^2], x_ Symbol] :> Simp[(1/(b*c*(n + 1)))*Simp[Sqrt[1 + c^2*x^2]/Sqrt[d + e*x^2]]*( a + b*ArcSinh[c*x])^(n + 1), x] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[e, c ^2*d] && NeQ[n, -1]
Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[(f*x)^(m + 1)*Sqrt[d + e*x^2]*((a + b*Arc Sinh[c*x])^n/(f*(m + 2))), x] + (Simp[(1/(m + 2))*Simp[Sqrt[d + e*x^2]/Sqrt [1 + c^2*x^2]] Int[(f*x)^m*((a + b*ArcSinh[c*x])^n/Sqrt[1 + c^2*x^2]), x] , x] - Simp[b*c*(n/(f*(m + 2)))*Simp[Sqrt[d + e*x^2]/Sqrt[1 + c^2*x^2]] I nt[(f*x)^(m + 1)*(a + b*ArcSinh[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d , e, f, m}, x] && EqQ[e, c^2*d] && IGtQ[n, 0] && (IGtQ[m, -2] || EqQ[n, 1])
Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d_) + (e_ .)*(x_)^2)^(p_), x_Symbol] :> Simp[f*(f*x)^(m - 1)*(d + e*x^2)^(p + 1)*((a + b*ArcSinh[c*x])^n/(e*(m + 2*p + 1))), x] + (-Simp[f^2*((m - 1)/(c^2*(m + 2*p + 1))) Int[(f*x)^(m - 2)*(d + e*x^2)^p*(a + b*ArcSinh[c*x])^n, x], x] - Simp[b*f*(n/(c*(m + 2*p + 1)))*Simp[(d + e*x^2)^p/(1 + c^2*x^2)^p] Int [(f*x)^(m - 1)*(1 + c^2*x^2)^(p + 1/2)*(a + b*ArcSinh[c*x])^(n - 1), x], x] ) /; FreeQ[{a, b, c, d, e, f, p}, x] && EqQ[e, c^2*d] && GtQ[n, 0] && IGtQ[ m, 1] && NeQ[m + 2*p + 1, 0]
Leaf count of result is larger than twice the leaf count of optimal. \(617\) vs. \(2(251)=502\).
Time = 0.72 (sec) , antiderivative size = 618, normalized size of antiderivative = 2.12
| method | result | size |
| default | \(\frac {a^{2} x \left (c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}{4 c^{2} d}-\frac {a^{2} x \sqrt {c^{2} d \,x^{2}+d}}{8 c^{2}}-\frac {a^{2} d \ln \left (\frac {x \,c^{2} d}{\sqrt {c^{2} d}}+\sqrt {c^{2} d \,x^{2}+d}\right )}{8 c^{2} \sqrt {c^{2} d}}+b^{2} \left (-\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {arcsinh}\left (x c \right )^{3}}{24 \sqrt {c^{2} x^{2}+1}\, c^{3}}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (8 x^{5} c^{5}+8 x^{4} c^{4} \sqrt {c^{2} x^{2}+1}+12 x^{3} c^{3}+8 x^{2} c^{2} \sqrt {c^{2} x^{2}+1}+4 x c +\sqrt {c^{2} x^{2}+1}\right ) \left (8 \operatorname {arcsinh}\left (x c \right )^{2}-4 \,\operatorname {arcsinh}\left (x c \right )+1\right )}{512 c^{3} \left (c^{2} x^{2}+1\right )}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (8 x^{5} c^{5}-8 x^{4} c^{4} \sqrt {c^{2} x^{2}+1}+12 x^{3} c^{3}-8 x^{2} c^{2} \sqrt {c^{2} x^{2}+1}+4 x c -\sqrt {c^{2} x^{2}+1}\right ) \left (8 \operatorname {arcsinh}\left (x c \right )^{2}+4 \,\operatorname {arcsinh}\left (x c \right )+1\right )}{512 c^{3} \left (c^{2} x^{2}+1\right )}\right )+2 a b \left (-\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {arcsinh}\left (x c \right )^{2}}{16 \sqrt {c^{2} x^{2}+1}\, c^{3}}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (8 x^{5} c^{5}+8 x^{4} c^{4} \sqrt {c^{2} x^{2}+1}+12 x^{3} c^{3}+8 x^{2} c^{2} \sqrt {c^{2} x^{2}+1}+4 x c +\sqrt {c^{2} x^{2}+1}\right ) \left (-1+4 \,\operatorname {arcsinh}\left (x c \right )\right )}{256 c^{3} \left (c^{2} x^{2}+1\right )}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (8 x^{5} c^{5}-8 x^{4} c^{4} \sqrt {c^{2} x^{2}+1}+12 x^{3} c^{3}-8 x^{2} c^{2} \sqrt {c^{2} x^{2}+1}+4 x c -\sqrt {c^{2} x^{2}+1}\right ) \left (1+4 \,\operatorname {arcsinh}\left (x c \right )\right )}{256 c^{3} \left (c^{2} x^{2}+1\right )}\right )\) | \(618\) |
| parts | \(\frac {a^{2} x \left (c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}{4 c^{2} d}-\frac {a^{2} x \sqrt {c^{2} d \,x^{2}+d}}{8 c^{2}}-\frac {a^{2} d \ln \left (\frac {x \,c^{2} d}{\sqrt {c^{2} d}}+\sqrt {c^{2} d \,x^{2}+d}\right )}{8 c^{2} \sqrt {c^{2} d}}+b^{2} \left (-\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {arcsinh}\left (x c \right )^{3}}{24 \sqrt {c^{2} x^{2}+1}\, c^{3}}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (8 x^{5} c^{5}+8 x^{4} c^{4} \sqrt {c^{2} x^{2}+1}+12 x^{3} c^{3}+8 x^{2} c^{2} \sqrt {c^{2} x^{2}+1}+4 x c +\sqrt {c^{2} x^{2}+1}\right ) \left (8 \operatorname {arcsinh}\left (x c \right )^{2}-4 \,\operatorname {arcsinh}\left (x c \right )+1\right )}{512 c^{3} \left (c^{2} x^{2}+1\right )}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (8 x^{5} c^{5}-8 x^{4} c^{4} \sqrt {c^{2} x^{2}+1}+12 x^{3} c^{3}-8 x^{2} c^{2} \sqrt {c^{2} x^{2}+1}+4 x c -\sqrt {c^{2} x^{2}+1}\right ) \left (8 \operatorname {arcsinh}\left (x c \right )^{2}+4 \,\operatorname {arcsinh}\left (x c \right )+1\right )}{512 c^{3} \left (c^{2} x^{2}+1\right )}\right )+2 a b \left (-\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {arcsinh}\left (x c \right )^{2}}{16 \sqrt {c^{2} x^{2}+1}\, c^{3}}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (8 x^{5} c^{5}+8 x^{4} c^{4} \sqrt {c^{2} x^{2}+1}+12 x^{3} c^{3}+8 x^{2} c^{2} \sqrt {c^{2} x^{2}+1}+4 x c +\sqrt {c^{2} x^{2}+1}\right ) \left (-1+4 \,\operatorname {arcsinh}\left (x c \right )\right )}{256 c^{3} \left (c^{2} x^{2}+1\right )}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (8 x^{5} c^{5}-8 x^{4} c^{4} \sqrt {c^{2} x^{2}+1}+12 x^{3} c^{3}-8 x^{2} c^{2} \sqrt {c^{2} x^{2}+1}+4 x c -\sqrt {c^{2} x^{2}+1}\right ) \left (1+4 \,\operatorname {arcsinh}\left (x c \right )\right )}{256 c^{3} \left (c^{2} x^{2}+1\right )}\right )\) | \(618\) |
Input:
int(x^2*(c^2*d*x^2+d)^(1/2)*(a+b*arcsinh(x*c))^2,x,method=_RETURNVERBOSE)
Output:
1/4*a^2*x*(c^2*d*x^2+d)^(3/2)/c^2/d-1/8*a^2/c^2*x*(c^2*d*x^2+d)^(1/2)-1/8* a^2/c^2*d*ln(x*c^2*d/(c^2*d)^(1/2)+(c^2*d*x^2+d)^(1/2))/(c^2*d)^(1/2)+b^2* (-1/24*(d*(c^2*x^2+1))^(1/2)/(c^2*x^2+1)^(1/2)/c^3*arcsinh(x*c)^3+1/512*(d *(c^2*x^2+1))^(1/2)*(8*x^5*c^5+8*x^4*c^4*(c^2*x^2+1)^(1/2)+12*x^3*c^3+8*x^ 2*c^2*(c^2*x^2+1)^(1/2)+4*x*c+(c^2*x^2+1)^(1/2))*(8*arcsinh(x*c)^2-4*arcsi nh(x*c)+1)/c^3/(c^2*x^2+1)+1/512*(d*(c^2*x^2+1))^(1/2)*(8*x^5*c^5-8*x^4*c^ 4*(c^2*x^2+1)^(1/2)+12*x^3*c^3-8*x^2*c^2*(c^2*x^2+1)^(1/2)+4*x*c-(c^2*x^2+ 1)^(1/2))*(8*arcsinh(x*c)^2+4*arcsinh(x*c)+1)/c^3/(c^2*x^2+1))+2*a*b*(-1/1 6*(d*(c^2*x^2+1))^(1/2)/(c^2*x^2+1)^(1/2)/c^3*arcsinh(x*c)^2+1/256*(d*(c^2 *x^2+1))^(1/2)*(8*x^5*c^5+8*x^4*c^4*(c^2*x^2+1)^(1/2)+12*x^3*c^3+8*x^2*c^2 *(c^2*x^2+1)^(1/2)+4*x*c+(c^2*x^2+1)^(1/2))*(-1+4*arcsinh(x*c))/c^3/(c^2*x ^2+1)+1/256*(d*(c^2*x^2+1))^(1/2)*(8*x^5*c^5-8*x^4*c^4*(c^2*x^2+1)^(1/2)+1 2*x^3*c^3-8*x^2*c^2*(c^2*x^2+1)^(1/2)+4*x*c-(c^2*x^2+1)^(1/2))*(1+4*arcsin h(x*c))/c^3/(c^2*x^2+1))
\[ \int x^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2 \, dx=\int { \sqrt {c^{2} d x^{2} + d} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2} x^{2} \,d x } \] Input:
integrate(x^2*(c^2*d*x^2+d)^(1/2)*(a+b*arcsinh(c*x))^2,x, algorithm="frica s")
Output:
integral((b^2*x^2*arcsinh(c*x)^2 + 2*a*b*x^2*arcsinh(c*x) + a^2*x^2)*sqrt( c^2*d*x^2 + d), x)
\[ \int x^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2 \, dx=\int x^{2} \sqrt {d \left (c^{2} x^{2} + 1\right )} \left (a + b \operatorname {asinh}{\left (c x \right )}\right )^{2}\, dx \] Input:
integrate(x**2*(c**2*d*x**2+d)**(1/2)*(a+b*asinh(c*x))**2,x)
Output:
Integral(x**2*sqrt(d*(c**2*x**2 + 1))*(a + b*asinh(c*x))**2, x)
Exception generated. \[ \int x^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2 \, dx=\text {Exception raised: RuntimeError} \] Input:
integrate(x^2*(c^2*d*x^2+d)^(1/2)*(a+b*arcsinh(c*x))^2,x, algorithm="maxim a")
Output:
Exception raised: RuntimeError >> ECL says: expt: undefined: 0 to a negati ve exponent.
\[ \int x^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2 \, dx=\int { \sqrt {c^{2} d x^{2} + d} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2} x^{2} \,d x } \] Input:
integrate(x^2*(c^2*d*x^2+d)^(1/2)*(a+b*arcsinh(c*x))^2,x, algorithm="giac" )
Output:
integrate(sqrt(c^2*d*x^2 + d)*(b*arcsinh(c*x) + a)^2*x^2, x)
Timed out. \[ \int x^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2 \, dx=\int x^2\,{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2\,\sqrt {d\,c^2\,x^2+d} \,d x \] Input:
int(x^2*(a + b*asinh(c*x))^2*(d + c^2*d*x^2)^(1/2),x)
Output:
int(x^2*(a + b*asinh(c*x))^2*(d + c^2*d*x^2)^(1/2), x)
\[ \int x^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2 \, dx=\frac {\sqrt {d}\, \left (2 \sqrt {c^{2} x^{2}+1}\, a^{2} c^{3} x^{3}+\sqrt {c^{2} x^{2}+1}\, a^{2} c x +16 \left (\int \sqrt {c^{2} x^{2}+1}\, \mathit {asinh} \left (c x \right ) x^{2}d x \right ) a b \,c^{3}+8 \left (\int \sqrt {c^{2} x^{2}+1}\, \mathit {asinh} \left (c x \right )^{2} x^{2}d x \right ) b^{2} c^{3}-\mathrm {log}\left (\sqrt {c^{2} x^{2}+1}+c x \right ) a^{2}\right )}{8 c^{3}} \] Input:
int(x^2*(c^2*d*x^2+d)^(1/2)*(a+b*asinh(c*x))^2,x)
Output:
(sqrt(d)*(2*sqrt(c**2*x**2 + 1)*a**2*c**3*x**3 + sqrt(c**2*x**2 + 1)*a**2* c*x + 16*int(sqrt(c**2*x**2 + 1)*asinh(c*x)*x**2,x)*a*b*c**3 + 8*int(sqrt( c**2*x**2 + 1)*asinh(c*x)**2*x**2,x)*b**2*c**3 - log(sqrt(c**2*x**2 + 1) + c*x)*a**2))/(8*c**3)