Integrand size = 25, antiderivative size = 134 \[ \int \sqrt {c e+d e x} (a+b \text {arcsinh}(c+d x))^2 \, dx=\frac {2 (e (c+d x))^{3/2} (a+b \text {arcsinh}(c+d x))^2}{3 d e}-\frac {8 b (e (c+d x))^{5/2} (a+b \text {arcsinh}(c+d x)) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {5}{4},\frac {9}{4},-(c+d x)^2\right )}{15 d e^2}+\frac {16 b^2 (e (c+d x))^{7/2} \, _3F_2\left (1,\frac {7}{4},\frac {7}{4};\frac {9}{4},\frac {11}{4};-(c+d x)^2\right )}{105 d e^3} \] Output:
2/3*(e*(d*x+c))^(3/2)*(a+b*arcsinh(d*x+c))^2/d/e-8/15*b*(e*(d*x+c))^(5/2)* (a+b*arcsinh(d*x+c))*hypergeom([1/2, 5/4],[9/4],-(d*x+c)^2)/d/e^2+16/105*b ^2*(e*(d*x+c))^(7/2)*hypergeom([1, 7/4, 7/4],[9/4, 11/4],-(d*x+c)^2)/d/e^3
Time = 0.06 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.82 \[ \int \sqrt {c e+d e x} (a+b \text {arcsinh}(c+d x))^2 \, dx=\frac {2 (e (c+d x))^{3/2} \left (35 (a+b \text {arcsinh}(c+d x))^2-28 b (c+d x) (a+b \text {arcsinh}(c+d x)) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {5}{4},\frac {9}{4},-(c+d x)^2\right )+8 b^2 (c+d x)^2 \, _3F_2\left (1,\frac {7}{4},\frac {7}{4};\frac {9}{4},\frac {11}{4};-(c+d x)^2\right )\right )}{105 d e} \] Input:
Integrate[Sqrt[c*e + d*e*x]*(a + b*ArcSinh[c + d*x])^2,x]
Output:
(2*(e*(c + d*x))^(3/2)*(35*(a + b*ArcSinh[c + d*x])^2 - 28*b*(c + d*x)*(a + b*ArcSinh[c + d*x])*Hypergeometric2F1[1/2, 5/4, 9/4, -(c + d*x)^2] + 8*b ^2*(c + d*x)^2*HypergeometricPFQ[{1, 7/4, 7/4}, {9/4, 11/4}, -(c + d*x)^2] ))/(105*d*e)
Time = 0.64 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.01, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {6274, 6191, 6232}
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.
\(\displaystyle \int \sqrt {c e+d e x} (a+b \text {arcsinh}(c+d x))^2 \, dx\) |
\(\Big \downarrow \) 6274 |
\(\displaystyle \frac {\int \sqrt {e (c+d x)} (a+b \text {arcsinh}(c+d x))^2d(c+d x)}{d}\) |
\(\Big \downarrow \) 6191 |
\(\displaystyle \frac {\frac {2 (e (c+d x))^{3/2} (a+b \text {arcsinh}(c+d x))^2}{3 e}-\frac {4 b \int \frac {(e (c+d x))^{3/2} (a+b \text {arcsinh}(c+d x))}{\sqrt {(c+d x)^2+1}}d(c+d x)}{3 e}}{d}\) |
\(\Big \downarrow \) 6232 |
\(\displaystyle \frac {\frac {2 (e (c+d x))^{3/2} (a+b \text {arcsinh}(c+d x))^2}{3 e}-\frac {4 b \left (\frac {2 (e (c+d x))^{5/2} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {5}{4},\frac {9}{4},-(c+d x)^2\right ) (a+b \text {arcsinh}(c+d x))}{5 e}-\frac {4 b (e (c+d x))^{7/2} \, _3F_2\left (1,\frac {7}{4},\frac {7}{4};\frac {9}{4},\frac {11}{4};-(c+d x)^2\right )}{35 e^2}\right )}{3 e}}{d}\) |
Input:
Int[Sqrt[c*e + d*e*x]*(a + b*ArcSinh[c + d*x])^2,x]
Output:
((2*(e*(c + d*x))^(3/2)*(a + b*ArcSinh[c + d*x])^2)/(3*e) - (4*b*((2*(e*(c + d*x))^(5/2)*(a + b*ArcSinh[c + d*x])*Hypergeometric2F1[1/2, 5/4, 9/4, - (c + d*x)^2])/(5*e) - (4*b*(e*(c + d*x))^(7/2)*HypergeometricPFQ[{1, 7/4, 7/4}, {9/4, 11/4}, -(c + d*x)^2])/(35*e^2)))/(3*e))/d
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_.))*((f_.)*(x_))^(m_))/Sqrt[(d_) + (e_ .)*(x_)^2], x_Symbol] :> Simp[((f*x)^(m + 1)/(f*(m + 1)))*Simp[Sqrt[1 + c^2 *x^2]/Sqrt[d + e*x^2]]*(a + b*ArcSinh[c*x])*Hypergeometric2F1[1/2, (1 + m)/ 2, (3 + m)/2, (-c^2)*x^2], x] - Simp[b*c*((f*x)^(m + 2)/(f^2*(m + 1)*(m + 2 )))*Simp[Sqrt[1 + c^2*x^2]/Sqrt[d + e*x^2]]*HypergeometricPFQ[{1, 1 + m/2, 1 + m/2}, {3/2 + m/2, 2 + m/2}, (-c^2)*x^2], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[e, c^2*d] && !IntegerQ[m]
Int[((a_.) + ArcSinh[(c_) + (d_.)*(x_)]*(b_.))^(n_.)*((e_.) + (f_.)*(x_))^( m_.), x_Symbol] :> Simp[1/d Subst[Int[((d*e - c*f)/d + f*(x/d))^m*(a + b* ArcSinh[x])^n, x], x, c + d*x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x]
\[\int \sqrt {d e x +c e}\, \left (a +b \,\operatorname {arcsinh}\left (d x +c \right )\right )^{2}d x\]
Input:
int((d*e*x+c*e)^(1/2)*(a+b*arcsinh(d*x+c))^2,x)
Output:
int((d*e*x+c*e)^(1/2)*(a+b*arcsinh(d*x+c))^2,x)
\[ \int \sqrt {c e+d e x} (a+b \text {arcsinh}(c+d x))^2 \, dx=\int { \sqrt {d e x + c e} {\left (b \operatorname {arsinh}\left (d x + c\right ) + a\right )}^{2} \,d x } \] Input:
integrate((d*e*x+c*e)^(1/2)*(a+b*arcsinh(d*x+c))^2,x, algorithm="fricas")
Output:
integral((b^2*arcsinh(d*x + c)^2 + 2*a*b*arcsinh(d*x + c) + a^2)*sqrt(d*e* x + c*e), x)
\[ \int \sqrt {c e+d e x} (a+b \text {arcsinh}(c+d x))^2 \, dx=\int \sqrt {e \left (c + d x\right )} \left (a + b \operatorname {asinh}{\left (c + d x \right )}\right )^{2}\, dx \] Input:
integrate((d*e*x+c*e)**(1/2)*(a+b*asinh(d*x+c))**2,x)
Output:
Integral(sqrt(e*(c + d*x))*(a + b*asinh(c + d*x))**2, x)
Exception generated. \[ \int \sqrt {c e+d e x} (a+b \text {arcsinh}(c+d x))^2 \, dx=\text {Exception raised: ValueError} \] Input:
integrate((d*e*x+c*e)^(1/2)*(a+b*arcsinh(d*x+c))^2,x, algorithm="maxima")
Output:
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(e>0)', see `assume?` for more de tails)Is e
\[ \int \sqrt {c e+d e x} (a+b \text {arcsinh}(c+d x))^2 \, dx=\int { \sqrt {d e x + c e} {\left (b \operatorname {arsinh}\left (d x + c\right ) + a\right )}^{2} \,d x } \] Input:
integrate((d*e*x+c*e)^(1/2)*(a+b*arcsinh(d*x+c))^2,x, algorithm="giac")
Output:
integrate(sqrt(d*e*x + c*e)*(b*arcsinh(d*x + c) + a)^2, x)
Timed out. \[ \int \sqrt {c e+d e x} (a+b \text {arcsinh}(c+d x))^2 \, dx=\int \sqrt {c\,e+d\,e\,x}\,{\left (a+b\,\mathrm {asinh}\left (c+d\,x\right )\right )}^2 \,d x \] Input:
int((c*e + d*e*x)^(1/2)*(a + b*asinh(c + d*x))^2,x)
Output:
int((c*e + d*e*x)^(1/2)*(a + b*asinh(c + d*x))^2, x)
\[ \int \sqrt {c e+d e x} (a+b \text {arcsinh}(c+d x))^2 \, dx=\frac {\sqrt {e}\, \left (2 \sqrt {d x +c}\, a^{2} c +2 \sqrt {d x +c}\, a^{2} d x +6 \left (\int \sqrt {d x +c}\, \mathit {asinh} \left (d x +c \right )d x \right ) a b d +3 \left (\int \sqrt {d x +c}\, \mathit {asinh} \left (d x +c \right )^{2}d x \right ) b^{2} d \right )}{3 d} \] Input:
int((d*e*x+c*e)^(1/2)*(a+b*asinh(d*x+c))^2,x)
Output:
(sqrt(e)*(2*sqrt(c + d*x)*a**2*c + 2*sqrt(c + d*x)*a**2*d*x + 6*int(sqrt(c + d*x)*asinh(c + d*x),x)*a*b*d + 3*int(sqrt(c + d*x)*asinh(c + d*x)**2,x) *b**2*d))/(3*d)