3.7.13 \(\int (d+e x^2)^3 (a+b \sinh ^{-1}(c x))^2 \, dx\) [613]

Optimal. Leaf size=559 \[ 2 b^2 d^3 x-\frac {4 b^2 d^2 e x}{3 c^2}+\frac {16 b^2 d e^2 x}{25 c^4}-\frac {32 b^2 e^3 x}{245 c^6}+\frac {2}{9} b^2 d^2 e x^3-\frac {8 b^2 d e^2 x^3}{75 c^2}+\frac {16 b^2 e^3 x^3}{735 c^4}+\frac {6}{125} b^2 d e^2 x^5-\frac {12 b^2 e^3 x^5}{1225 c^2}+\frac {2}{343} b^2 e^3 x^7-\frac {2 b d^3 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{c}+\frac {4 b d^2 e \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{3 c^3}-\frac {16 b d e^2 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{25 c^5}+\frac {32 b e^3 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{245 c^7}-\frac {2 b d^2 e x^2 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{3 c}+\frac {8 b d e^2 x^2 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{25 c^3}-\frac {16 b e^3 x^2 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{245 c^5}-\frac {6 b d e^2 x^4 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{25 c}+\frac {12 b e^3 x^4 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{245 c^3}-\frac {2 b e^3 x^6 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{49 c}+d^3 x \left (a+b \sinh ^{-1}(c x)\right )^2+d^2 e x^3 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {3}{5} d e^2 x^5 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {1}{7} e^3 x^7 \left (a+b \sinh ^{-1}(c x)\right )^2 \]

[Out]

2*b^2*d^3*x-4/3*b^2*d^2*e*x/c^2+16/25*b^2*d*e^2*x/c^4-32/245*b^2*e^3*x/c^6+2/9*b^2*d^2*e*x^3-8/75*b^2*d*e^2*x^
3/c^2+16/735*b^2*e^3*x^3/c^4+6/125*b^2*d*e^2*x^5-12/1225*b^2*e^3*x^5/c^2+2/343*b^2*e^3*x^7+d^3*x*(a+b*arcsinh(
c*x))^2+d^2*e*x^3*(a+b*arcsinh(c*x))^2+3/5*d*e^2*x^5*(a+b*arcsinh(c*x))^2+1/7*e^3*x^7*(a+b*arcsinh(c*x))^2-2*b
*d^3*(a+b*arcsinh(c*x))*(c^2*x^2+1)^(1/2)/c+4/3*b*d^2*e*(a+b*arcsinh(c*x))*(c^2*x^2+1)^(1/2)/c^3-16/25*b*d*e^2
*(a+b*arcsinh(c*x))*(c^2*x^2+1)^(1/2)/c^5+32/245*b*e^3*(a+b*arcsinh(c*x))*(c^2*x^2+1)^(1/2)/c^7-2/3*b*d^2*e*x^
2*(a+b*arcsinh(c*x))*(c^2*x^2+1)^(1/2)/c+8/25*b*d*e^2*x^2*(a+b*arcsinh(c*x))*(c^2*x^2+1)^(1/2)/c^3-16/245*b*e^
3*x^2*(a+b*arcsinh(c*x))*(c^2*x^2+1)^(1/2)/c^5-6/25*b*d*e^2*x^4*(a+b*arcsinh(c*x))*(c^2*x^2+1)^(1/2)/c+12/245*
b*e^3*x^4*(a+b*arcsinh(c*x))*(c^2*x^2+1)^(1/2)/c^3-2/49*b*e^3*x^6*(a+b*arcsinh(c*x))*(c^2*x^2+1)^(1/2)/c

________________________________________________________________________________________

Rubi [A]
time = 0.63, antiderivative size = 559, normalized size of antiderivative = 1.00, number of steps used = 26, number of rules used = 7, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.350, Rules used = {5793, 5772, 5798, 8, 5776, 5812, 30} \begin {gather*} -\frac {2 b d^3 \sqrt {c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )}{c}-\frac {2 b d^2 e x^2 \sqrt {c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )}{3 c}-\frac {6 b d e^2 x^4 \sqrt {c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )}{25 c}-\frac {2 b e^3 x^6 \sqrt {c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )}{49 c}+\frac {32 b e^3 \sqrt {c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )}{245 c^7}-\frac {16 b d e^2 \sqrt {c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )}{25 c^5}-\frac {16 b e^3 x^2 \sqrt {c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )}{245 c^5}+\frac {4 b d^2 e \sqrt {c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )}{3 c^3}+\frac {8 b d e^2 x^2 \sqrt {c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )}{25 c^3}+\frac {12 b e^3 x^4 \sqrt {c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )}{245 c^3}+d^3 x \left (a+b \sinh ^{-1}(c x)\right )^2+d^2 e x^3 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {3}{5} d e^2 x^5 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {1}{7} e^3 x^7 \left (a+b \sinh ^{-1}(c x)\right )^2-\frac {32 b^2 e^3 x}{245 c^6}+\frac {16 b^2 d e^2 x}{25 c^4}+\frac {16 b^2 e^3 x^3}{735 c^4}-\frac {4 b^2 d^2 e x}{3 c^2}-\frac {8 b^2 d e^2 x^3}{75 c^2}-\frac {12 b^2 e^3 x^5}{1225 c^2}+2 b^2 d^3 x+\frac {2}{9} b^2 d^2 e x^3+\frac {6}{125} b^2 d e^2 x^5+\frac {2}{343} b^2 e^3 x^7 \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(d + e*x^2)^3*(a + b*ArcSinh[c*x])^2,x]

[Out]

2*b^2*d^3*x - (4*b^2*d^2*e*x)/(3*c^2) + (16*b^2*d*e^2*x)/(25*c^4) - (32*b^2*e^3*x)/(245*c^6) + (2*b^2*d^2*e*x^
3)/9 - (8*b^2*d*e^2*x^3)/(75*c^2) + (16*b^2*e^3*x^3)/(735*c^4) + (6*b^2*d*e^2*x^5)/125 - (12*b^2*e^3*x^5)/(122
5*c^2) + (2*b^2*e^3*x^7)/343 - (2*b*d^3*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x]))/c + (4*b*d^2*e*Sqrt[1 + c^2*x^
2]*(a + b*ArcSinh[c*x]))/(3*c^3) - (16*b*d*e^2*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x]))/(25*c^5) + (32*b*e^3*Sq
rt[1 + c^2*x^2]*(a + b*ArcSinh[c*x]))/(245*c^7) - (2*b*d^2*e*x^2*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x]))/(3*c)
+ (8*b*d*e^2*x^2*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x]))/(25*c^3) - (16*b*e^3*x^2*Sqrt[1 + c^2*x^2]*(a + b*Ar
cSinh[c*x]))/(245*c^5) - (6*b*d*e^2*x^4*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x]))/(25*c) + (12*b*e^3*x^4*Sqrt[1
+ c^2*x^2]*(a + b*ArcSinh[c*x]))/(245*c^3) - (2*b*e^3*x^6*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x]))/(49*c) + d^3
*x*(a + b*ArcSinh[c*x])^2 + d^2*e*x^3*(a + b*ArcSinh[c*x])^2 + (3*d*e^2*x^5*(a + b*ArcSinh[c*x])^2)/5 + (e^3*x
^7*(a + b*ArcSinh[c*x])^2)/7

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rule 5772

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.), x_Symbol] :> Simp[x*(a + b*ArcSinh[c*x])^n, x] - Dist[b*c*n, In
t[x*((a + b*ArcSinh[c*x])^(n - 1)/Sqrt[1 + c^2*x^2]), x], x] /; FreeQ[{a, b, c}, x] && GtQ[n, 0]

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 5793

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(a
+ b*ArcSinh[c*x])^n, (d + e*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, n}, x] && NeQ[e, c^2*d] && IntegerQ[p] &&
(p > 0 || IGtQ[n, 0])

Rule 5798

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(d + e*x^2)^
(p + 1)*((a + b*ArcSinh[c*x])^n/(2*e*(p + 1))), x] - Dist[b*(n/(2*c*(p + 1)))*Simp[(d + e*x^2)^p/(1 + c^2*x^2)
^p], Int[(1 + c^2*x^2)^(p + 1/2)*(a + b*ArcSinh[c*x])^(n - 1), x], x] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[e
, c^2*d] && GtQ[n, 0] && NeQ[p, -1]

Rule 5812

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] + (-Dist[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] - Dist[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
]

Rubi steps

\begin {align*} \int \left (d+e x^2\right )^3 \left (a+b \sinh ^{-1}(c x)\right )^2 \, dx &=\int \left (d^3 \left (a+b \sinh ^{-1}(c x)\right )^2+3 d^2 e x^2 \left (a+b \sinh ^{-1}(c x)\right )^2+3 d e^2 x^4 \left (a+b \sinh ^{-1}(c x)\right )^2+e^3 x^6 \left (a+b \sinh ^{-1}(c x)\right )^2\right ) \, dx\\ &=d^3 \int \left (a+b \sinh ^{-1}(c x)\right )^2 \, dx+\left (3 d^2 e\right ) \int x^2 \left (a+b \sinh ^{-1}(c x)\right )^2 \, dx+\left (3 d e^2\right ) \int x^4 \left (a+b \sinh ^{-1}(c x)\right )^2 \, dx+e^3 \int x^6 \left (a+b \sinh ^{-1}(c x)\right )^2 \, dx\\ &=d^3 x \left (a+b \sinh ^{-1}(c x)\right )^2+d^2 e x^3 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {3}{5} d e^2 x^5 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {1}{7} e^3 x^7 \left (a+b \sinh ^{-1}(c x)\right )^2-\left (2 b c d^3\right ) \int \frac {x \left (a+b \sinh ^{-1}(c x)\right )}{\sqrt {1+c^2 x^2}} \, dx-\left (2 b c d^2 e\right ) \int \frac {x^3 \left (a+b \sinh ^{-1}(c x)\right )}{\sqrt {1+c^2 x^2}} \, dx-\frac {1}{5} \left (6 b c d e^2\right ) \int \frac {x^5 \left (a+b \sinh ^{-1}(c x)\right )}{\sqrt {1+c^2 x^2}} \, dx-\frac {1}{7} \left (2 b c e^3\right ) \int \frac {x^7 \left (a+b \sinh ^{-1}(c x)\right )}{\sqrt {1+c^2 x^2}} \, dx\\ &=-\frac {2 b d^3 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{c}-\frac {2 b d^2 e x^2 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{3 c}-\frac {6 b d e^2 x^4 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{25 c}-\frac {2 b e^3 x^6 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{49 c}+d^3 x \left (a+b \sinh ^{-1}(c x)\right )^2+d^2 e x^3 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {3}{5} d e^2 x^5 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {1}{7} e^3 x^7 \left (a+b \sinh ^{-1}(c x)\right )^2+\left (2 b^2 d^3\right ) \int 1 \, dx+\frac {1}{3} \left (2 b^2 d^2 e\right ) \int x^2 \, dx+\frac {\left (4 b d^2 e\right ) \int \frac {x \left (a+b \sinh ^{-1}(c x)\right )}{\sqrt {1+c^2 x^2}} \, dx}{3 c}+\frac {1}{25} \left (6 b^2 d e^2\right ) \int x^4 \, dx+\frac {\left (24 b d e^2\right ) \int \frac {x^3 \left (a+b \sinh ^{-1}(c x)\right )}{\sqrt {1+c^2 x^2}} \, dx}{25 c}+\frac {1}{49} \left (2 b^2 e^3\right ) \int x^6 \, dx+\frac {\left (12 b e^3\right ) \int \frac {x^5 \left (a+b \sinh ^{-1}(c x)\right )}{\sqrt {1+c^2 x^2}} \, dx}{49 c}\\ &=2 b^2 d^3 x+\frac {2}{9} b^2 d^2 e x^3+\frac {6}{125} b^2 d e^2 x^5+\frac {2}{343} b^2 e^3 x^7-\frac {2 b d^3 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{c}+\frac {4 b d^2 e \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{3 c^3}-\frac {2 b d^2 e x^2 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{3 c}+\frac {8 b d e^2 x^2 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{25 c^3}-\frac {6 b d e^2 x^4 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{25 c}+\frac {12 b e^3 x^4 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{245 c^3}-\frac {2 b e^3 x^6 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{49 c}+d^3 x \left (a+b \sinh ^{-1}(c x)\right )^2+d^2 e x^3 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {3}{5} d e^2 x^5 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {1}{7} e^3 x^7 \left (a+b \sinh ^{-1}(c x)\right )^2-\frac {\left (4 b^2 d^2 e\right ) \int 1 \, dx}{3 c^2}-\frac {\left (16 b d e^2\right ) \int \frac {x \left (a+b \sinh ^{-1}(c x)\right )}{\sqrt {1+c^2 x^2}} \, dx}{25 c^3}-\frac {\left (8 b^2 d e^2\right ) \int x^2 \, dx}{25 c^2}-\frac {\left (48 b e^3\right ) \int \frac {x^3 \left (a+b \sinh ^{-1}(c x)\right )}{\sqrt {1+c^2 x^2}} \, dx}{245 c^3}-\frac {\left (12 b^2 e^3\right ) \int x^4 \, dx}{245 c^2}\\ &=2 b^2 d^3 x-\frac {4 b^2 d^2 e x}{3 c^2}+\frac {2}{9} b^2 d^2 e x^3-\frac {8 b^2 d e^2 x^3}{75 c^2}+\frac {6}{125} b^2 d e^2 x^5-\frac {12 b^2 e^3 x^5}{1225 c^2}+\frac {2}{343} b^2 e^3 x^7-\frac {2 b d^3 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{c}+\frac {4 b d^2 e \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{3 c^3}-\frac {16 b d e^2 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{25 c^5}-\frac {2 b d^2 e x^2 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{3 c}+\frac {8 b d e^2 x^2 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{25 c^3}-\frac {16 b e^3 x^2 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{245 c^5}-\frac {6 b d e^2 x^4 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{25 c}+\frac {12 b e^3 x^4 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{245 c^3}-\frac {2 b e^3 x^6 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{49 c}+d^3 x \left (a+b \sinh ^{-1}(c x)\right )^2+d^2 e x^3 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {3}{5} d e^2 x^5 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {1}{7} e^3 x^7 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {\left (16 b^2 d e^2\right ) \int 1 \, dx}{25 c^4}+\frac {\left (32 b e^3\right ) \int \frac {x \left (a+b \sinh ^{-1}(c x)\right )}{\sqrt {1+c^2 x^2}} \, dx}{245 c^5}+\frac {\left (16 b^2 e^3\right ) \int x^2 \, dx}{245 c^4}\\ &=2 b^2 d^3 x-\frac {4 b^2 d^2 e x}{3 c^2}+\frac {16 b^2 d e^2 x}{25 c^4}+\frac {2}{9} b^2 d^2 e x^3-\frac {8 b^2 d e^2 x^3}{75 c^2}+\frac {16 b^2 e^3 x^3}{735 c^4}+\frac {6}{125} b^2 d e^2 x^5-\frac {12 b^2 e^3 x^5}{1225 c^2}+\frac {2}{343} b^2 e^3 x^7-\frac {2 b d^3 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{c}+\frac {4 b d^2 e \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{3 c^3}-\frac {16 b d e^2 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{25 c^5}+\frac {32 b e^3 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{245 c^7}-\frac {2 b d^2 e x^2 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{3 c}+\frac {8 b d e^2 x^2 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{25 c^3}-\frac {16 b e^3 x^2 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{245 c^5}-\frac {6 b d e^2 x^4 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{25 c}+\frac {12 b e^3 x^4 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{245 c^3}-\frac {2 b e^3 x^6 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{49 c}+d^3 x \left (a+b \sinh ^{-1}(c x)\right )^2+d^2 e x^3 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {3}{5} d e^2 x^5 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {1}{7} e^3 x^7 \left (a+b \sinh ^{-1}(c x)\right )^2-\frac {\left (32 b^2 e^3\right ) \int 1 \, dx}{245 c^6}\\ &=2 b^2 d^3 x-\frac {4 b^2 d^2 e x}{3 c^2}+\frac {16 b^2 d e^2 x}{25 c^4}-\frac {32 b^2 e^3 x}{245 c^6}+\frac {2}{9} b^2 d^2 e x^3-\frac {8 b^2 d e^2 x^3}{75 c^2}+\frac {16 b^2 e^3 x^3}{735 c^4}+\frac {6}{125} b^2 d e^2 x^5-\frac {12 b^2 e^3 x^5}{1225 c^2}+\frac {2}{343} b^2 e^3 x^7-\frac {2 b d^3 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{c}+\frac {4 b d^2 e \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{3 c^3}-\frac {16 b d e^2 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{25 c^5}+\frac {32 b e^3 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{245 c^7}-\frac {2 b d^2 e x^2 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{3 c}+\frac {8 b d e^2 x^2 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{25 c^3}-\frac {16 b e^3 x^2 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{245 c^5}-\frac {6 b d e^2 x^4 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{25 c}+\frac {12 b e^3 x^4 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{245 c^3}-\frac {2 b e^3 x^6 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{49 c}+d^3 x \left (a+b \sinh ^{-1}(c x)\right )^2+d^2 e x^3 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {3}{5} d e^2 x^5 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {1}{7} e^3 x^7 \left (a+b \sinh ^{-1}(c x)\right )^2\\ \end {align*}

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Mathematica [A]
time = 0.30, size = 443, normalized size = 0.79 \begin {gather*} \frac {11025 a^2 c^7 x \left (35 d^3+35 d^2 e x^2+21 d e^2 x^4+5 e^3 x^6\right )-210 a b \sqrt {1+c^2 x^2} \left (-240 e^3+24 c^2 e^2 \left (49 d+5 e x^2\right )-2 c^4 e \left (1225 d^2+294 d e x^2+45 e^2 x^4\right )+c^6 \left (3675 d^3+1225 d^2 e x^2+441 d e^2 x^4+75 e^3 x^6\right )\right )+2 b^2 c x \left (-25200 e^3+840 c^2 e^2 \left (147 d+5 e x^2\right )-210 c^4 e \left (1225 d^2+98 d e x^2+9 e^2 x^4\right )+c^6 \left (385875 d^3+42875 d^2 e x^2+9261 d e^2 x^4+1125 e^3 x^6\right )\right )-210 b \left (-105 a c^7 x \left (35 d^3+35 d^2 e x^2+21 d e^2 x^4+5 e^3 x^6\right )+b \sqrt {1+c^2 x^2} \left (-240 e^3+24 c^2 e^2 \left (49 d+5 e x^2\right )-2 c^4 e \left (1225 d^2+294 d e x^2+45 e^2 x^4\right )+c^6 \left (3675 d^3+1225 d^2 e x^2+441 d e^2 x^4+75 e^3 x^6\right )\right )\right ) \sinh ^{-1}(c x)+11025 b^2 c^7 x \left (35 d^3+35 d^2 e x^2+21 d e^2 x^4+5 e^3 x^6\right ) \sinh ^{-1}(c x)^2}{385875 c^7} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x^2)^3*(a + b*ArcSinh[c*x])^2,x]

[Out]

(11025*a^2*c^7*x*(35*d^3 + 35*d^2*e*x^2 + 21*d*e^2*x^4 + 5*e^3*x^6) - 210*a*b*Sqrt[1 + c^2*x^2]*(-240*e^3 + 24
*c^2*e^2*(49*d + 5*e*x^2) - 2*c^4*e*(1225*d^2 + 294*d*e*x^2 + 45*e^2*x^4) + c^6*(3675*d^3 + 1225*d^2*e*x^2 + 4
41*d*e^2*x^4 + 75*e^3*x^6)) + 2*b^2*c*x*(-25200*e^3 + 840*c^2*e^2*(147*d + 5*e*x^2) - 210*c^4*e*(1225*d^2 + 98
*d*e*x^2 + 9*e^2*x^4) + c^6*(385875*d^3 + 42875*d^2*e*x^2 + 9261*d*e^2*x^4 + 1125*e^3*x^6)) - 210*b*(-105*a*c^
7*x*(35*d^3 + 35*d^2*e*x^2 + 21*d*e^2*x^4 + 5*e^3*x^6) + b*Sqrt[1 + c^2*x^2]*(-240*e^3 + 24*c^2*e^2*(49*d + 5*
e*x^2) - 2*c^4*e*(1225*d^2 + 294*d*e*x^2 + 45*e^2*x^4) + c^6*(3675*d^3 + 1225*d^2*e*x^2 + 441*d*e^2*x^4 + 75*e
^3*x^6)))*ArcSinh[c*x] + 11025*b^2*c^7*x*(35*d^3 + 35*d^2*e*x^2 + 21*d*e^2*x^4 + 5*e^3*x^6)*ArcSinh[c*x]^2)/(3
85875*c^7)

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Maple [A]
time = 2.24, size = 752, normalized size = 1.35 \[\text {Expression too large to display}\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x^2+d)^3*(a+b*arcsinh(c*x))^2,x)

[Out]

1/c*(a^2/c^6*(d^3*c^7*x+d^2*c^7*e*x^3+3/5*d*c^7*e^2*x^5+1/7*e^3*c^7*x^7)+1/c^6*b^2*(3/2000*c^2*d*e^2*(25*arcsi
nh(c*x)^2*sinh(5*arcsinh(c*x))-10*arcsinh(c*x)*cosh(5*arcsinh(c*x))+2*sinh(5*arcsinh(c*x)))-1/1600*e^3*(25*arc
sinh(c*x)^2*sinh(5*arcsinh(c*x))-10*arcsinh(c*x)*cosh(5*arcsinh(c*x))+2*sinh(5*arcsinh(c*x)))+1/36*c^4*d^2*e*(
9*arcsinh(c*x)^2*sinh(3*arcsinh(c*x))-6*arcsinh(c*x)*cosh(3*arcsinh(c*x))+2*sinh(3*arcsinh(c*x)))-1/48*c^2*d*e
^2*(9*arcsinh(c*x)^2*sinh(3*arcsinh(c*x))-6*arcsinh(c*x)*cosh(3*arcsinh(c*x))+2*sinh(3*arcsinh(c*x)))+1/192*e^
3*(9*arcsinh(c*x)^2*sinh(3*arcsinh(c*x))-6*arcsinh(c*x)*cosh(3*arcsinh(c*x))+2*sinh(3*arcsinh(c*x)))+d^3*c^6*(
arcsinh(c*x)^2*c*x-2*arcsinh(c*x)*(c^2*x^2+1)^(1/2)+2*c*x)-3/4*c^4*d^2*e*(arcsinh(c*x)^2*c*x-2*arcsinh(c*x)*(c
^2*x^2+1)^(1/2)+2*c*x)+3/8*c^2*d*e^2*(arcsinh(c*x)^2*c*x-2*arcsinh(c*x)*(c^2*x^2+1)^(1/2)+2*c*x)-5/64*e^3*(arc
sinh(c*x)^2*c*x-2*arcsinh(c*x)*(c^2*x^2+1)^(1/2)+2*c*x)+1/21952*e^3*(49*arcsinh(c*x)^2*sinh(7*arcsinh(c*x))-14
*arcsinh(c*x)*cosh(7*arcsinh(c*x))+2*sinh(7*arcsinh(c*x))))+2*a*b/c^6*(1/7*arcsinh(c*x)*e^3*x^7*c^7+3/5*arcsin
h(c*x)*c^7*d*e^2*x^5+arcsinh(c*x)*c^7*d^2*e*x^3+arcsinh(c*x)*x*c^7*d^3-1/7*e^3*(1/7*(c^2*x^2+1)^(1/2)*c^6*x^6-
6/35*(c^2*x^2+1)^(1/2)*c^4*x^4+8/35*c^2*x^2*(c^2*x^2+1)^(1/2)-16/35*(c^2*x^2+1)^(1/2))-3/5*d*c^2*e^2*(1/5*(c^2
*x^2+1)^(1/2)*c^4*x^4-4/15*c^2*x^2*(c^2*x^2+1)^(1/2)+8/15*(c^2*x^2+1)^(1/2))-d^2*c^4*e*(1/3*c^2*x^2*(c^2*x^2+1
)^(1/2)-2/3*(c^2*x^2+1)^(1/2))-d^3*c^6*(c^2*x^2+1)^(1/2)))

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Maxima [A]
time = 0.29, size = 680, normalized size = 1.22 \begin {gather*} \frac {1}{7} \, b^{2} x^{7} \operatorname {arsinh}\left (c x\right )^{2} e^{3} + \frac {3}{5} \, b^{2} d x^{5} \operatorname {arsinh}\left (c x\right )^{2} e^{2} + \frac {1}{7} \, a^{2} x^{7} e^{3} + b^{2} d^{2} x^{3} \operatorname {arsinh}\left (c x\right )^{2} e + \frac {3}{5} \, a^{2} d x^{5} e^{2} + b^{2} d^{3} x \operatorname {arsinh}\left (c x\right )^{2} + a^{2} d^{2} x^{3} e + 2 \, b^{2} d^{3} {\left (x - \frac {\sqrt {c^{2} x^{2} + 1} \operatorname {arsinh}\left (c x\right )}{c}\right )} + a^{2} d^{3} x + \frac {2}{3} \, {\left (3 \, x^{3} \operatorname {arsinh}\left (c x\right ) - c {\left (\frac {\sqrt {c^{2} x^{2} + 1} x^{2}}{c^{2}} - \frac {2 \, \sqrt {c^{2} x^{2} + 1}}{c^{4}}\right )}\right )} a b d^{2} e - \frac {2}{9} \, {\left (3 \, c {\left (\frac {\sqrt {c^{2} x^{2} + 1} x^{2}}{c^{2}} - \frac {2 \, \sqrt {c^{2} x^{2} + 1}}{c^{4}}\right )} \operatorname {arsinh}\left (c x\right ) - \frac {c^{2} x^{3} - 6 \, x}{c^{2}}\right )} b^{2} d^{2} e + \frac {2 \, {\left (c x \operatorname {arsinh}\left (c x\right ) - \sqrt {c^{2} x^{2} + 1}\right )} a b d^{3}}{c} + \frac {2}{25} \, {\left (15 \, x^{5} \operatorname {arsinh}\left (c x\right ) - {\left (\frac {3 \, \sqrt {c^{2} x^{2} + 1} x^{4}}{c^{2}} - \frac {4 \, \sqrt {c^{2} x^{2} + 1} x^{2}}{c^{4}} + \frac {8 \, \sqrt {c^{2} x^{2} + 1}}{c^{6}}\right )} c\right )} a b d e^{2} - \frac {2}{375} \, {\left (15 \, {\left (\frac {3 \, \sqrt {c^{2} x^{2} + 1} x^{4}}{c^{2}} - \frac {4 \, \sqrt {c^{2} x^{2} + 1} x^{2}}{c^{4}} + \frac {8 \, \sqrt {c^{2} x^{2} + 1}}{c^{6}}\right )} c \operatorname {arsinh}\left (c x\right ) - \frac {9 \, c^{4} x^{5} - 20 \, c^{2} x^{3} + 120 \, x}{c^{4}}\right )} b^{2} d e^{2} + \frac {2}{245} \, {\left (35 \, x^{7} \operatorname {arsinh}\left (c x\right ) - {\left (\frac {5 \, \sqrt {c^{2} x^{2} + 1} x^{6}}{c^{2}} - \frac {6 \, \sqrt {c^{2} x^{2} + 1} x^{4}}{c^{4}} + \frac {8 \, \sqrt {c^{2} x^{2} + 1} x^{2}}{c^{6}} - \frac {16 \, \sqrt {c^{2} x^{2} + 1}}{c^{8}}\right )} c\right )} a b e^{3} - \frac {2}{25725} \, {\left (105 \, {\left (\frac {5 \, \sqrt {c^{2} x^{2} + 1} x^{6}}{c^{2}} - \frac {6 \, \sqrt {c^{2} x^{2} + 1} x^{4}}{c^{4}} + \frac {8 \, \sqrt {c^{2} x^{2} + 1} x^{2}}{c^{6}} - \frac {16 \, \sqrt {c^{2} x^{2} + 1}}{c^{8}}\right )} c \operatorname {arsinh}\left (c x\right ) - \frac {75 \, c^{6} x^{7} - 126 \, c^{4} x^{5} + 280 \, c^{2} x^{3} - 1680 \, x}{c^{6}}\right )} b^{2} e^{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x^2+d)^3*(a+b*arcsinh(c*x))^2,x, algorithm="maxima")

[Out]

1/7*b^2*x^7*arcsinh(c*x)^2*e^3 + 3/5*b^2*d*x^5*arcsinh(c*x)^2*e^2 + 1/7*a^2*x^7*e^3 + b^2*d^2*x^3*arcsinh(c*x)
^2*e + 3/5*a^2*d*x^5*e^2 + b^2*d^3*x*arcsinh(c*x)^2 + a^2*d^2*x^3*e + 2*b^2*d^3*(x - sqrt(c^2*x^2 + 1)*arcsinh
(c*x)/c) + a^2*d^3*x + 2/3*(3*x^3*arcsinh(c*x) - c*(sqrt(c^2*x^2 + 1)*x^2/c^2 - 2*sqrt(c^2*x^2 + 1)/c^4))*a*b*
d^2*e - 2/9*(3*c*(sqrt(c^2*x^2 + 1)*x^2/c^2 - 2*sqrt(c^2*x^2 + 1)/c^4)*arcsinh(c*x) - (c^2*x^3 - 6*x)/c^2)*b^2
*d^2*e + 2*(c*x*arcsinh(c*x) - sqrt(c^2*x^2 + 1))*a*b*d^3/c + 2/25*(15*x^5*arcsinh(c*x) - (3*sqrt(c^2*x^2 + 1)
*x^4/c^2 - 4*sqrt(c^2*x^2 + 1)*x^2/c^4 + 8*sqrt(c^2*x^2 + 1)/c^6)*c)*a*b*d*e^2 - 2/375*(15*(3*sqrt(c^2*x^2 + 1
)*x^4/c^2 - 4*sqrt(c^2*x^2 + 1)*x^2/c^4 + 8*sqrt(c^2*x^2 + 1)/c^6)*c*arcsinh(c*x) - (9*c^4*x^5 - 20*c^2*x^3 +
120*x)/c^4)*b^2*d*e^2 + 2/245*(35*x^7*arcsinh(c*x) - (5*sqrt(c^2*x^2 + 1)*x^6/c^2 - 6*sqrt(c^2*x^2 + 1)*x^4/c^
4 + 8*sqrt(c^2*x^2 + 1)*x^2/c^6 - 16*sqrt(c^2*x^2 + 1)/c^8)*c)*a*b*e^3 - 2/25725*(105*(5*sqrt(c^2*x^2 + 1)*x^6
/c^2 - 6*sqrt(c^2*x^2 + 1)*x^4/c^4 + 8*sqrt(c^2*x^2 + 1)*x^2/c^6 - 16*sqrt(c^2*x^2 + 1)/c^8)*c*arcsinh(c*x) -
(75*c^6*x^7 - 126*c^4*x^5 + 280*c^2*x^3 - 1680*x)/c^6)*b^2*e^3

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1528 vs. \(2 (488) = 976\).
time = 0.39, size = 1528, normalized size = 2.73 \begin {gather*} \frac {385875 \, {\left (a^{2} + 2 \, b^{2}\right )} c^{7} d^{3} x + 15 \, {\left (75 \, {\left (49 \, a^{2} + 2 \, b^{2}\right )} c^{7} x^{7} - 252 \, b^{2} c^{5} x^{5} + 560 \, b^{2} c^{3} x^{3} - 3360 \, b^{2} c x\right )} \cosh \left (1\right )^{3} + 15 \, {\left (75 \, {\left (49 \, a^{2} + 2 \, b^{2}\right )} c^{7} x^{7} - 252 \, b^{2} c^{5} x^{5} + 560 \, b^{2} c^{3} x^{3} - 3360 \, b^{2} c x\right )} \sinh \left (1\right )^{3} + 1029 \, {\left (9 \, {\left (25 \, a^{2} + 2 \, b^{2}\right )} c^{7} d x^{5} - 40 \, b^{2} c^{5} d x^{3} + 240 \, b^{2} c^{3} d x\right )} \cosh \left (1\right )^{2} + 11025 \, {\left (5 \, b^{2} c^{7} x^{7} \cosh \left (1\right )^{3} + 5 \, b^{2} c^{7} x^{7} \sinh \left (1\right )^{3} + 21 \, b^{2} c^{7} d x^{5} \cosh \left (1\right )^{2} + 35 \, b^{2} c^{7} d^{2} x^{3} \cosh \left (1\right ) + 35 \, b^{2} c^{7} d^{3} x + 3 \, {\left (5 \, b^{2} c^{7} x^{7} \cosh \left (1\right ) + 7 \, b^{2} c^{7} d x^{5}\right )} \sinh \left (1\right )^{2} + {\left (15 \, b^{2} c^{7} x^{7} \cosh \left (1\right )^{2} + 42 \, b^{2} c^{7} d x^{5} \cosh \left (1\right ) + 35 \, b^{2} c^{7} d^{2} x^{3}\right )} \sinh \left (1\right )\right )} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right )^{2} + 3 \, {\left (3087 \, {\left (25 \, a^{2} + 2 \, b^{2}\right )} c^{7} d x^{5} - 13720 \, b^{2} c^{5} d x^{3} + 82320 \, b^{2} c^{3} d x + 15 \, {\left (75 \, {\left (49 \, a^{2} + 2 \, b^{2}\right )} c^{7} x^{7} - 252 \, b^{2} c^{5} x^{5} + 560 \, b^{2} c^{3} x^{3} - 3360 \, b^{2} c x\right )} \cosh \left (1\right )\right )} \sinh \left (1\right )^{2} + 42875 \, {\left ({\left (9 \, a^{2} + 2 \, b^{2}\right )} c^{7} d^{2} x^{3} - 12 \, b^{2} c^{5} d^{2} x\right )} \cosh \left (1\right ) + 210 \, {\left (525 \, a b c^{7} x^{7} \cosh \left (1\right )^{3} + 525 \, a b c^{7} x^{7} \sinh \left (1\right )^{3} + 2205 \, a b c^{7} d x^{5} \cosh \left (1\right )^{2} + 3675 \, a b c^{7} d^{2} x^{3} \cosh \left (1\right ) + 3675 \, a b c^{7} d^{3} x + 315 \, {\left (5 \, a b c^{7} x^{7} \cosh \left (1\right ) + 7 \, a b c^{7} d x^{5}\right )} \sinh \left (1\right )^{2} + 105 \, {\left (15 \, a b c^{7} x^{7} \cosh \left (1\right )^{2} + 42 \, a b c^{7} d x^{5} \cosh \left (1\right ) + 35 \, a b c^{7} d^{2} x^{3}\right )} \sinh \left (1\right ) - {\left (3675 \, b^{2} c^{6} d^{3} + 15 \, {\left (5 \, b^{2} c^{6} x^{6} - 6 \, b^{2} c^{4} x^{4} + 8 \, b^{2} c^{2} x^{2} - 16 \, b^{2}\right )} \cosh \left (1\right )^{3} + 15 \, {\left (5 \, b^{2} c^{6} x^{6} - 6 \, b^{2} c^{4} x^{4} + 8 \, b^{2} c^{2} x^{2} - 16 \, b^{2}\right )} \sinh \left (1\right )^{3} + 147 \, {\left (3 \, b^{2} c^{6} d x^{4} - 4 \, b^{2} c^{4} d x^{2} + 8 \, b^{2} c^{2} d\right )} \cosh \left (1\right )^{2} + 3 \, {\left (147 \, b^{2} c^{6} d x^{4} - 196 \, b^{2} c^{4} d x^{2} + 392 \, b^{2} c^{2} d + 15 \, {\left (5 \, b^{2} c^{6} x^{6} - 6 \, b^{2} c^{4} x^{4} + 8 \, b^{2} c^{2} x^{2} - 16 \, b^{2}\right )} \cosh \left (1\right )\right )} \sinh \left (1\right )^{2} + 1225 \, {\left (b^{2} c^{6} d^{2} x^{2} - 2 \, b^{2} c^{4} d^{2}\right )} \cosh \left (1\right ) + {\left (1225 \, b^{2} c^{6} d^{2} x^{2} - 2450 \, b^{2} c^{4} d^{2} + 45 \, {\left (5 \, b^{2} c^{6} x^{6} - 6 \, b^{2} c^{4} x^{4} + 8 \, b^{2} c^{2} x^{2} - 16 \, b^{2}\right )} \cosh \left (1\right )^{2} + 294 \, {\left (3 \, b^{2} c^{6} d x^{4} - 4 \, b^{2} c^{4} d x^{2} + 8 \, b^{2} c^{2} d\right )} \cosh \left (1\right )\right )} \sinh \left (1\right )\right )} \sqrt {c^{2} x^{2} + 1}\right )} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right ) + {\left (42875 \, {\left (9 \, a^{2} + 2 \, b^{2}\right )} c^{7} d^{2} x^{3} - 514500 \, b^{2} c^{5} d^{2} x + 45 \, {\left (75 \, {\left (49 \, a^{2} + 2 \, b^{2}\right )} c^{7} x^{7} - 252 \, b^{2} c^{5} x^{5} + 560 \, b^{2} c^{3} x^{3} - 3360 \, b^{2} c x\right )} \cosh \left (1\right )^{2} + 2058 \, {\left (9 \, {\left (25 \, a^{2} + 2 \, b^{2}\right )} c^{7} d x^{5} - 40 \, b^{2} c^{5} d x^{3} + 240 \, b^{2} c^{3} d x\right )} \cosh \left (1\right )\right )} \sinh \left (1\right ) - 210 \, {\left (3675 \, a b c^{6} d^{3} + 15 \, {\left (5 \, a b c^{6} x^{6} - 6 \, a b c^{4} x^{4} + 8 \, a b c^{2} x^{2} - 16 \, a b\right )} \cosh \left (1\right )^{3} + 15 \, {\left (5 \, a b c^{6} x^{6} - 6 \, a b c^{4} x^{4} + 8 \, a b c^{2} x^{2} - 16 \, a b\right )} \sinh \left (1\right )^{3} + 147 \, {\left (3 \, a b c^{6} d x^{4} - 4 \, a b c^{4} d x^{2} + 8 \, a b c^{2} d\right )} \cosh \left (1\right )^{2} + 3 \, {\left (147 \, a b c^{6} d x^{4} - 196 \, a b c^{4} d x^{2} + 392 \, a b c^{2} d + 15 \, {\left (5 \, a b c^{6} x^{6} - 6 \, a b c^{4} x^{4} + 8 \, a b c^{2} x^{2} - 16 \, a b\right )} \cosh \left (1\right )\right )} \sinh \left (1\right )^{2} + 1225 \, {\left (a b c^{6} d^{2} x^{2} - 2 \, a b c^{4} d^{2}\right )} \cosh \left (1\right ) + {\left (1225 \, a b c^{6} d^{2} x^{2} - 2450 \, a b c^{4} d^{2} + 45 \, {\left (5 \, a b c^{6} x^{6} - 6 \, a b c^{4} x^{4} + 8 \, a b c^{2} x^{2} - 16 \, a b\right )} \cosh \left (1\right )^{2} + 294 \, {\left (3 \, a b c^{6} d x^{4} - 4 \, a b c^{4} d x^{2} + 8 \, a b c^{2} d\right )} \cosh \left (1\right )\right )} \sinh \left (1\right )\right )} \sqrt {c^{2} x^{2} + 1}}{385875 \, c^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x^2+d)^3*(a+b*arcsinh(c*x))^2,x, algorithm="fricas")

[Out]

1/385875*(385875*(a^2 + 2*b^2)*c^7*d^3*x + 15*(75*(49*a^2 + 2*b^2)*c^7*x^7 - 252*b^2*c^5*x^5 + 560*b^2*c^3*x^3
- 3360*b^2*c*x)*cosh(1)^3 + 15*(75*(49*a^2 + 2*b^2)*c^7*x^7 - 252*b^2*c^5*x^5 + 560*b^2*c^3*x^3 - 3360*b^2*c*
x)*sinh(1)^3 + 1029*(9*(25*a^2 + 2*b^2)*c^7*d*x^5 - 40*b^2*c^5*d*x^3 + 240*b^2*c^3*d*x)*cosh(1)^2 + 11025*(5*b
^2*c^7*x^7*cosh(1)^3 + 5*b^2*c^7*x^7*sinh(1)^3 + 21*b^2*c^7*d*x^5*cosh(1)^2 + 35*b^2*c^7*d^2*x^3*cosh(1) + 35*
b^2*c^7*d^3*x + 3*(5*b^2*c^7*x^7*cosh(1) + 7*b^2*c^7*d*x^5)*sinh(1)^2 + (15*b^2*c^7*x^7*cosh(1)^2 + 42*b^2*c^7
*d*x^5*cosh(1) + 35*b^2*c^7*d^2*x^3)*sinh(1))*log(c*x + sqrt(c^2*x^2 + 1))^2 + 3*(3087*(25*a^2 + 2*b^2)*c^7*d*
x^5 - 13720*b^2*c^5*d*x^3 + 82320*b^2*c^3*d*x + 15*(75*(49*a^2 + 2*b^2)*c^7*x^7 - 252*b^2*c^5*x^5 + 560*b^2*c^
3*x^3 - 3360*b^2*c*x)*cosh(1))*sinh(1)^2 + 42875*((9*a^2 + 2*b^2)*c^7*d^2*x^3 - 12*b^2*c^5*d^2*x)*cosh(1) + 21
0*(525*a*b*c^7*x^7*cosh(1)^3 + 525*a*b*c^7*x^7*sinh(1)^3 + 2205*a*b*c^7*d*x^5*cosh(1)^2 + 3675*a*b*c^7*d^2*x^3
*cosh(1) + 3675*a*b*c^7*d^3*x + 315*(5*a*b*c^7*x^7*cosh(1) + 7*a*b*c^7*d*x^5)*sinh(1)^2 + 105*(15*a*b*c^7*x^7*
cosh(1)^2 + 42*a*b*c^7*d*x^5*cosh(1) + 35*a*b*c^7*d^2*x^3)*sinh(1) - (3675*b^2*c^6*d^3 + 15*(5*b^2*c^6*x^6 - 6
*b^2*c^4*x^4 + 8*b^2*c^2*x^2 - 16*b^2)*cosh(1)^3 + 15*(5*b^2*c^6*x^6 - 6*b^2*c^4*x^4 + 8*b^2*c^2*x^2 - 16*b^2)
*sinh(1)^3 + 147*(3*b^2*c^6*d*x^4 - 4*b^2*c^4*d*x^2 + 8*b^2*c^2*d)*cosh(1)^2 + 3*(147*b^2*c^6*d*x^4 - 196*b^2*
c^4*d*x^2 + 392*b^2*c^2*d + 15*(5*b^2*c^6*x^6 - 6*b^2*c^4*x^4 + 8*b^2*c^2*x^2 - 16*b^2)*cosh(1))*sinh(1)^2 + 1
225*(b^2*c^6*d^2*x^2 - 2*b^2*c^4*d^2)*cosh(1) + (1225*b^2*c^6*d^2*x^2 - 2450*b^2*c^4*d^2 + 45*(5*b^2*c^6*x^6 -
6*b^2*c^4*x^4 + 8*b^2*c^2*x^2 - 16*b^2)*cosh(1)^2 + 294*(3*b^2*c^6*d*x^4 - 4*b^2*c^4*d*x^2 + 8*b^2*c^2*d)*cos
h(1))*sinh(1))*sqrt(c^2*x^2 + 1))*log(c*x + sqrt(c^2*x^2 + 1)) + (42875*(9*a^2 + 2*b^2)*c^7*d^2*x^3 - 514500*b
^2*c^5*d^2*x + 45*(75*(49*a^2 + 2*b^2)*c^7*x^7 - 252*b^2*c^5*x^5 + 560*b^2*c^3*x^3 - 3360*b^2*c*x)*cosh(1)^2 +
2058*(9*(25*a^2 + 2*b^2)*c^7*d*x^5 - 40*b^2*c^5*d*x^3 + 240*b^2*c^3*d*x)*cosh(1))*sinh(1) - 210*(3675*a*b*c^6
*d^3 + 15*(5*a*b*c^6*x^6 - 6*a*b*c^4*x^4 + 8*a*b*c^2*x^2 - 16*a*b)*cosh(1)^3 + 15*(5*a*b*c^6*x^6 - 6*a*b*c^4*x
^4 + 8*a*b*c^2*x^2 - 16*a*b)*sinh(1)^3 + 147*(3*a*b*c^6*d*x^4 - 4*a*b*c^4*d*x^2 + 8*a*b*c^2*d)*cosh(1)^2 + 3*(
147*a*b*c^6*d*x^4 - 196*a*b*c^4*d*x^2 + 392*a*b*c^2*d + 15*(5*a*b*c^6*x^6 - 6*a*b*c^4*x^4 + 8*a*b*c^2*x^2 - 16
*a*b)*cosh(1))*sinh(1)^2 + 1225*(a*b*c^6*d^2*x^2 - 2*a*b*c^4*d^2)*cosh(1) + (1225*a*b*c^6*d^2*x^2 - 2450*a*b*c
^4*d^2 + 45*(5*a*b*c^6*x^6 - 6*a*b*c^4*x^4 + 8*a*b*c^2*x^2 - 16*a*b)*cosh(1)^2 + 294*(3*a*b*c^6*d*x^4 - 4*a*b*
c^4*d*x^2 + 8*a*b*c^2*d)*cosh(1))*sinh(1))*sqrt(c^2*x^2 + 1))/c^7

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Sympy [A]
time = 1.22, size = 989, normalized size = 1.77 \begin {gather*} \begin {cases} a^{2} d^{3} x + a^{2} d^{2} e x^{3} + \frac {3 a^{2} d e^{2} x^{5}}{5} + \frac {a^{2} e^{3} x^{7}}{7} + 2 a b d^{3} x \operatorname {asinh}{\left (c x \right )} + 2 a b d^{2} e x^{3} \operatorname {asinh}{\left (c x \right )} + \frac {6 a b d e^{2} x^{5} \operatorname {asinh}{\left (c x \right )}}{5} + \frac {2 a b e^{3} x^{7} \operatorname {asinh}{\left (c x \right )}}{7} - \frac {2 a b d^{3} \sqrt {c^{2} x^{2} + 1}}{c} - \frac {2 a b d^{2} e x^{2} \sqrt {c^{2} x^{2} + 1}}{3 c} - \frac {6 a b d e^{2} x^{4} \sqrt {c^{2} x^{2} + 1}}{25 c} - \frac {2 a b e^{3} x^{6} \sqrt {c^{2} x^{2} + 1}}{49 c} + \frac {4 a b d^{2} e \sqrt {c^{2} x^{2} + 1}}{3 c^{3}} + \frac {8 a b d e^{2} x^{2} \sqrt {c^{2} x^{2} + 1}}{25 c^{3}} + \frac {12 a b e^{3} x^{4} \sqrt {c^{2} x^{2} + 1}}{245 c^{3}} - \frac {16 a b d e^{2} \sqrt {c^{2} x^{2} + 1}}{25 c^{5}} - \frac {16 a b e^{3} x^{2} \sqrt {c^{2} x^{2} + 1}}{245 c^{5}} + \frac {32 a b e^{3} \sqrt {c^{2} x^{2} + 1}}{245 c^{7}} + b^{2} d^{3} x \operatorname {asinh}^{2}{\left (c x \right )} + 2 b^{2} d^{3} x + b^{2} d^{2} e x^{3} \operatorname {asinh}^{2}{\left (c x \right )} + \frac {2 b^{2} d^{2} e x^{3}}{9} + \frac {3 b^{2} d e^{2} x^{5} \operatorname {asinh}^{2}{\left (c x \right )}}{5} + \frac {6 b^{2} d e^{2} x^{5}}{125} + \frac {b^{2} e^{3} x^{7} \operatorname {asinh}^{2}{\left (c x \right )}}{7} + \frac {2 b^{2} e^{3} x^{7}}{343} - \frac {2 b^{2} d^{3} \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{c} - \frac {2 b^{2} d^{2} e x^{2} \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{3 c} - \frac {6 b^{2} d e^{2} x^{4} \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{25 c} - \frac {2 b^{2} e^{3} x^{6} \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{49 c} - \frac {4 b^{2} d^{2} e x}{3 c^{2}} - \frac {8 b^{2} d e^{2} x^{3}}{75 c^{2}} - \frac {12 b^{2} e^{3} x^{5}}{1225 c^{2}} + \frac {4 b^{2} d^{2} e \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{3 c^{3}} + \frac {8 b^{2} d e^{2} x^{2} \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{25 c^{3}} + \frac {12 b^{2} e^{3} x^{4} \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{245 c^{3}} + \frac {16 b^{2} d e^{2} x}{25 c^{4}} + \frac {16 b^{2} e^{3} x^{3}}{735 c^{4}} - \frac {16 b^{2} d e^{2} \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{25 c^{5}} - \frac {16 b^{2} e^{3} x^{2} \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{245 c^{5}} - \frac {32 b^{2} e^{3} x}{245 c^{6}} + \frac {32 b^{2} e^{3} \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{245 c^{7}} & \text {for}\: c \neq 0 \\a^{2} \left (d^{3} x + d^{2} e x^{3} + \frac {3 d e^{2} x^{5}}{5} + \frac {e^{3} x^{7}}{7}\right ) & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x**2+d)**3*(a+b*asinh(c*x))**2,x)

[Out]

Piecewise((a**2*d**3*x + a**2*d**2*e*x**3 + 3*a**2*d*e**2*x**5/5 + a**2*e**3*x**7/7 + 2*a*b*d**3*x*asinh(c*x)
+ 2*a*b*d**2*e*x**3*asinh(c*x) + 6*a*b*d*e**2*x**5*asinh(c*x)/5 + 2*a*b*e**3*x**7*asinh(c*x)/7 - 2*a*b*d**3*sq
rt(c**2*x**2 + 1)/c - 2*a*b*d**2*e*x**2*sqrt(c**2*x**2 + 1)/(3*c) - 6*a*b*d*e**2*x**4*sqrt(c**2*x**2 + 1)/(25*
c) - 2*a*b*e**3*x**6*sqrt(c**2*x**2 + 1)/(49*c) + 4*a*b*d**2*e*sqrt(c**2*x**2 + 1)/(3*c**3) + 8*a*b*d*e**2*x**
2*sqrt(c**2*x**2 + 1)/(25*c**3) + 12*a*b*e**3*x**4*sqrt(c**2*x**2 + 1)/(245*c**3) - 16*a*b*d*e**2*sqrt(c**2*x*
*2 + 1)/(25*c**5) - 16*a*b*e**3*x**2*sqrt(c**2*x**2 + 1)/(245*c**5) + 32*a*b*e**3*sqrt(c**2*x**2 + 1)/(245*c**
7) + b**2*d**3*x*asinh(c*x)**2 + 2*b**2*d**3*x + b**2*d**2*e*x**3*asinh(c*x)**2 + 2*b**2*d**2*e*x**3/9 + 3*b**
2*d*e**2*x**5*asinh(c*x)**2/5 + 6*b**2*d*e**2*x**5/125 + b**2*e**3*x**7*asinh(c*x)**2/7 + 2*b**2*e**3*x**7/343
- 2*b**2*d**3*sqrt(c**2*x**2 + 1)*asinh(c*x)/c - 2*b**2*d**2*e*x**2*sqrt(c**2*x**2 + 1)*asinh(c*x)/(3*c) - 6*
b**2*d*e**2*x**4*sqrt(c**2*x**2 + 1)*asinh(c*x)/(25*c) - 2*b**2*e**3*x**6*sqrt(c**2*x**2 + 1)*asinh(c*x)/(49*c
) - 4*b**2*d**2*e*x/(3*c**2) - 8*b**2*d*e**2*x**3/(75*c**2) - 12*b**2*e**3*x**5/(1225*c**2) + 4*b**2*d**2*e*sq
rt(c**2*x**2 + 1)*asinh(c*x)/(3*c**3) + 8*b**2*d*e**2*x**2*sqrt(c**2*x**2 + 1)*asinh(c*x)/(25*c**3) + 12*b**2*
e**3*x**4*sqrt(c**2*x**2 + 1)*asinh(c*x)/(245*c**3) + 16*b**2*d*e**2*x/(25*c**4) + 16*b**2*e**3*x**3/(735*c**4
) - 16*b**2*d*e**2*sqrt(c**2*x**2 + 1)*asinh(c*x)/(25*c**5) - 16*b**2*e**3*x**2*sqrt(c**2*x**2 + 1)*asinh(c*x)
/(245*c**5) - 32*b**2*e**3*x/(245*c**6) + 32*b**2*e**3*sqrt(c**2*x**2 + 1)*asinh(c*x)/(245*c**7), Ne(c, 0)), (
a**2*(d**3*x + d**2*e*x**3 + 3*d*e**2*x**5/5 + e**3*x**7/7), True))

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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: RuntimeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x^2+d)^3*(a+b*arcsinh(c*x))^2,x, algorithm="giac")

[Out]

Exception raised: RuntimeError >> An error occurred running a Giac command:INPUT:sage2OUTPUT:sym2poly/r2sym(co
nst gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int {\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2\,{\left (e\,x^2+d\right )}^3 \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*asinh(c*x))^2*(d + e*x^2)^3,x)

[Out]

int((a + b*asinh(c*x))^2*(d + e*x^2)^3, x)

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