\(\int (c e+d e x)^m (a+b \text {arcsinh}(c+d x))^2 \, dx\) [126]

Optimal result

Integrand size = 23, antiderivative size = 187 \[ \int (c e+d e x)^m (a+b \text {arcsinh}(c+d x))^2 \, dx=\frac {(e (c+d x))^{1+m} (a+b \text {arcsinh}(c+d x))^2}{d e (1+m)}-\frac {2 b (e (c+d x))^{2+m} (a+b \text {arcsinh}(c+d x)) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {2+m}{2},\frac {4+m}{2},-(c+d x)^2\right )}{d e^2 (1+m) (2+m)}+\frac {2 b^2 (e (c+d x))^{3+m} \, _3F_2\left (1,\frac {3}{2}+\frac {m}{2},\frac {3}{2}+\frac {m}{2};2+\frac {m}{2},\frac {5}{2}+\frac {m}{2};-(c+d x)^2\right )}{d e^3 (1+m) (2+m) (3+m)} \]

[Out]

(e*(d*x+c))^(1+m)*(a+b*arcsinh(d*x+c))^2/d/e/(1+m)-2*b*(e*(d*x+c))^(2+m)*(a+b*arcsinh(d*x+c))*hypergeom([1/2,
1+1/2*m],[2+1/2*m],-(d*x+c)^2)/d/e^2/(1+m)/(2+m)+2*b^2*(e*(d*x+c))^(3+m)*hypergeom([1, 3/2+1/2*m, 3/2+1/2*m],[
2+1/2*m, 5/2+1/2*m],-(d*x+c)^2)/d/e^3/(3+m)/(m^2+3*m+2)

Rubi [A] (verified)

Time = 0.15 (sec) , antiderivative size = 187, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {5859, 5776, 5817} \[ \int (c e+d e x)^m (a+b \text {arcsinh}(c+d x))^2 \, dx=\frac {2 b^2 (e (c+d x))^{m+3} \, _3F_2\left (1,\frac {m}{2}+\frac {3}{2},\frac {m}{2}+\frac {3}{2};\frac {m}{2}+2,\frac {m}{2}+\frac {5}{2};-(c+d x)^2\right )}{d e^3 (m+1) (m+2) (m+3)}-\frac {2 b (e (c+d x))^{m+2} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {m+2}{2},\frac {m+4}{2},-(c+d x)^2\right ) (a+b \text {arcsinh}(c+d x))}{d e^2 (m+1) (m+2)}+\frac {(e (c+d x))^{m+1} (a+b \text {arcsinh}(c+d x))^2}{d e (m+1)} \]

[In]

Int[(c*e + d*e*x)^m*(a + b*ArcSinh[c + d*x])^2,x]

[Out]

((e*(c + d*x))^(1 + m)*(a + b*ArcSinh[c + d*x])^2)/(d*e*(1 + m)) - (2*b*(e*(c + d*x))^(2 + m)*(a + b*ArcSinh[c
+ d*x])*Hypergeometric2F1[1/2, (2 + m)/2, (4 + m)/2, -(c + d*x)^2])/(d*e^2*(1 + m)*(2 + m)) + (2*b^2*(e*(c +
d*x))^(3 + m)*HypergeometricPFQ[{1, 3/2 + m/2, 3/2 + m/2}, {2 + m/2, 5/2 + m/2}, -(c + d*x)^2])/(d*e^3*(1 + m)
*(2 + m)*(3 + m))

Rule 5776

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[(d*x)^(m + 1)*((a + b*ArcS
inh[c*x])^n/(d*(m + 1))), x] - Dist[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]

Rule 5817

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]/Sqr
t[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]

Rule 5859

Int[((a_.) + ArcSinh[(c_) + (d_.)*(x_)]*(b_.))^(n_.)*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> Dist[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
]

Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int (e x)^m (a+b \text {arcsinh}(x))^2 \, dx,x,c+d x\right )}{d} \\ & = \frac {(e (c+d x))^{1+m} (a+b \text {arcsinh}(c+d x))^2}{d e (1+m)}-\frac {(2 b) \text {Subst}\left (\int \frac {(e x)^{1+m} (a+b \text {arcsinh}(x))}{\sqrt {1+x^2}} \, dx,x,c+d x\right )}{d e (1+m)} \\ & = \frac {(e (c+d x))^{1+m} (a+b \text {arcsinh}(c+d x))^2}{d e (1+m)}-\frac {2 b (e (c+d x))^{2+m} (a+b \text {arcsinh}(c+d x)) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {2+m}{2},\frac {4+m}{2},-(c+d x)^2\right )}{d e^2 (1+m) (2+m)}+\frac {2 b^2 (e (c+d x))^{3+m} \, _3F_2\left (1,\frac {3}{2}+\frac {m}{2},\frac {3}{2}+\frac {m}{2};2+\frac {m}{2},\frac {5}{2}+\frac {m}{2};-(c+d x)^2\right )}{d e^3 (1+m) (2+m) (3+m)} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.07 (sec) , antiderivative size = 155, normalized size of antiderivative = 0.83 \[ \int (c e+d e x)^m (a+b \text {arcsinh}(c+d x))^2 \, dx=\frac {(c+d x) (e (c+d x))^m \left ((a+b \text {arcsinh}(c+d x))^2-\frac {2 b (c+d x) (a+b \text {arcsinh}(c+d x)) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {2+m}{2},\frac {4+m}{2},-(c+d x)^2\right )}{2+m}+\frac {2 b^2 (c+d x)^2 \, _3F_2\left (1,\frac {3}{2}+\frac {m}{2},\frac {3}{2}+\frac {m}{2};2+\frac {m}{2},\frac {5}{2}+\frac {m}{2};-(c+d x)^2\right )}{(2+m) (3+m)}\right )}{d (1+m)} \]

[In]

Integrate[(c*e + d*e*x)^m*(a + b*ArcSinh[c + d*x])^2,x]

[Out]

((c + d*x)*(e*(c + d*x))^m*((a + b*ArcSinh[c + d*x])^2 - (2*b*(c + d*x)*(a + b*ArcSinh[c + d*x])*Hypergeometri
c2F1[1/2, (2 + m)/2, (4 + m)/2, -(c + d*x)^2])/(2 + m) + (2*b^2*(c + d*x)^2*HypergeometricPFQ[{1, 3/2 + m/2, 3
/2 + m/2}, {2 + m/2, 5/2 + m/2}, -(c + d*x)^2])/((2 + m)*(3 + m))))/(d*(1 + m))

Maple [F]

\[\int \left (d e x +c e \right )^{m} \left (a +b \,\operatorname {arcsinh}\left (d x +c \right )\right )^{2}d x\]

[In]

int((d*e*x+c*e)^m*(a+b*arcsinh(d*x+c))^2,x)

[Out]

int((d*e*x+c*e)^m*(a+b*arcsinh(d*x+c))^2,x)

Fricas [F]

\[ \int (c e+d e x)^m (a+b \text {arcsinh}(c+d x))^2 \, dx=\int { {\left (b \operatorname {arsinh}\left (d x + c\right ) + a\right )}^{2} {\left (d e x + c e\right )}^{m} \,d x } \]

[In]

integrate((d*e*x+c*e)^m*(a+b*arcsinh(d*x+c))^2,x, algorithm="fricas")

[Out]

integral((b^2*arcsinh(d*x + c)^2 + 2*a*b*arcsinh(d*x + c) + a^2)*(d*e*x + c*e)^m, x)

Sympy [F]

\[ \int (c e+d e x)^m (a+b \text {arcsinh}(c+d x))^2 \, dx=\int \left (e \left (c + d x\right )\right )^{m} \left (a + b \operatorname {asinh}{\left (c + d x \right )}\right )^{2}\, dx \]

[In]

integrate((d*e*x+c*e)**m*(a+b*asinh(d*x+c))**2,x)

[Out]

Integral((e*(c + d*x))**m*(a + b*asinh(c + d*x))**2, x)

Maxima [F]

\[ \int (c e+d e x)^m (a+b \text {arcsinh}(c+d x))^2 \, dx=\int { {\left (b \operatorname {arsinh}\left (d x + c\right ) + a\right )}^{2} {\left (d e x + c e\right )}^{m} \,d x } \]

[In]

integrate((d*e*x+c*e)^m*(a+b*arcsinh(d*x+c))^2,x, algorithm="maxima")

[Out]

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

Giac [F]

\[ \int (c e+d e x)^m (a+b \text {arcsinh}(c+d x))^2 \, dx=\int { {\left (b \operatorname {arsinh}\left (d x + c\right ) + a\right )}^{2} {\left (d e x + c e\right )}^{m} \,d x } \]

[In]

integrate((d*e*x+c*e)^m*(a+b*arcsinh(d*x+c))^2,x, algorithm="giac")

[Out]

integrate((b*arcsinh(d*x + c) + a)^2*(d*e*x + c*e)^m, x)

Mupad [F(-1)]

Timed out. \[ \int (c e+d e x)^m (a+b \text {arcsinh}(c+d x))^2 \, dx=\int {\left (c\,e+d\,e\,x\right )}^m\,{\left (a+b\,\mathrm {asinh}\left (c+d\,x\right )\right )}^2 \,d x \]

[In]

int((c*e + d*e*x)^m*(a + b*asinh(c + d*x))^2,x)

[Out]

int((c*e + d*e*x)^m*(a + b*asinh(c + d*x))^2, x)