3.3.76 \(\int x (d+c^2 d x^2)^{5/2} (a+b \sinh ^{-1}(c x))^2 \, dx\) [276]

Optimal. Leaf size=366 \[ \frac {32 b^2 d^2 \sqrt {d+c^2 d x^2}}{245 c^2}+\frac {16 b^2 d^2 \left (1+c^2 x^2\right ) \sqrt {d+c^2 d x^2}}{735 c^2}+\frac {12 b^2 d^2 \left (1+c^2 x^2\right )^2 \sqrt {d+c^2 d x^2}}{1225 c^2}+\frac {2 b^2 d^2 \left (1+c^2 x^2\right )^3 \sqrt {d+c^2 d x^2}}{343 c^2}-\frac {2 b d^2 x \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{7 c \sqrt {1+c^2 x^2}}-\frac {2 b c d^2 x^3 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{7 \sqrt {1+c^2 x^2}}-\frac {6 b c^3 d^2 x^5 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{35 \sqrt {1+c^2 x^2}}-\frac {2 b c^5 d^2 x^7 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{49 \sqrt {1+c^2 x^2}}+\frac {\left (d+c^2 d x^2\right )^{7/2} \left (a+b \sinh ^{-1}(c x)\right )^2}{7 c^2 d} \]

[Out]

1/7*(c^2*d*x^2+d)^(7/2)*(a+b*arcsinh(c*x))^2/c^2/d+32/245*b^2*d^2*(c^2*d*x^2+d)^(1/2)/c^2+16/735*b^2*d^2*(c^2*
x^2+1)*(c^2*d*x^2+d)^(1/2)/c^2+12/1225*b^2*d^2*(c^2*x^2+1)^2*(c^2*d*x^2+d)^(1/2)/c^2+2/343*b^2*d^2*(c^2*x^2+1)
^3*(c^2*d*x^2+d)^(1/2)/c^2-2/7*b*d^2*x*(a+b*arcsinh(c*x))*(c^2*d*x^2+d)^(1/2)/c/(c^2*x^2+1)^(1/2)-2/7*b*c*d^2*
x^3*(a+b*arcsinh(c*x))*(c^2*d*x^2+d)^(1/2)/(c^2*x^2+1)^(1/2)-6/35*b*c^3*d^2*x^5*(a+b*arcsinh(c*x))*(c^2*d*x^2+
d)^(1/2)/(c^2*x^2+1)^(1/2)-2/49*b*c^5*d^2*x^7*(a+b*arcsinh(c*x))*(c^2*d*x^2+d)^(1/2)/(c^2*x^2+1)^(1/2)

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Rubi [A]
time = 0.20, antiderivative size = 366, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {5798, 200, 5784, 12, 1813, 1864} \begin {gather*} -\frac {2 b d^2 x \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )}{7 c \sqrt {c^2 x^2+1}}-\frac {2 b c d^2 x^3 \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )}{7 \sqrt {c^2 x^2+1}}+\frac {\left (c^2 d x^2+d\right )^{7/2} \left (a+b \sinh ^{-1}(c x)\right )^2}{7 c^2 d}-\frac {2 b c^5 d^2 x^7 \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )}{49 \sqrt {c^2 x^2+1}}-\frac {6 b c^3 d^2 x^5 \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )}{35 \sqrt {c^2 x^2+1}}+\frac {2 b^2 d^2 \left (c^2 x^2+1\right )^3 \sqrt {c^2 d x^2+d}}{343 c^2}+\frac {32 b^2 d^2 \sqrt {c^2 d x^2+d}}{245 c^2}+\frac {12 b^2 d^2 \left (c^2 x^2+1\right )^2 \sqrt {c^2 d x^2+d}}{1225 c^2}+\frac {16 b^2 d^2 \left (c^2 x^2+1\right ) \sqrt {c^2 d x^2+d}}{735 c^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x*(d + c^2*d*x^2)^(5/2)*(a + b*ArcSinh[c*x])^2,x]

[Out]

(32*b^2*d^2*Sqrt[d + c^2*d*x^2])/(245*c^2) + (16*b^2*d^2*(1 + c^2*x^2)*Sqrt[d + c^2*d*x^2])/(735*c^2) + (12*b^
2*d^2*(1 + c^2*x^2)^2*Sqrt[d + c^2*d*x^2])/(1225*c^2) + (2*b^2*d^2*(1 + c^2*x^2)^3*Sqrt[d + c^2*d*x^2])/(343*c
^2) - (2*b*d^2*x*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]))/(7*c*Sqrt[1 + c^2*x^2]) - (2*b*c*d^2*x^3*Sqrt[d + c
^2*d*x^2]*(a + b*ArcSinh[c*x]))/(7*Sqrt[1 + c^2*x^2]) - (6*b*c^3*d^2*x^5*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*
x]))/(35*Sqrt[1 + c^2*x^2]) - (2*b*c^5*d^2*x^7*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]))/(49*Sqrt[1 + c^2*x^2]
) + ((d + c^2*d*x^2)^(7/2)*(a + b*ArcSinh[c*x])^2)/(7*c^2*d)

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 200

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Int[ExpandIntegrand[(a + b*x^n)^p, x], x] /; FreeQ[{a, b}, x]
&& IGtQ[n, 0] && IGtQ[p, 0]

Rule 1813

Int[(Pq_)*(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Dist[1/2, Subst[Int[x^((m - 1)/2)*SubstFor[x^2,
Pq, x]*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, p}, x] && PolyQ[Pq, x^2] && IntegerQ[(m - 1)/2]

Rule 1864

Int[(Pq_)*((a_) + (b_.)*(x_)^(n_.))^(p_.), x_Symbol] :> Int[ExpandIntegrand[Pq*(a + b*x^n)^p, x], x] /; FreeQ[
{a, b, n}, x] && PolyQ[Pq, x] && (IGtQ[p, 0] || EqQ[n, 1])

Rule 5784

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> With[{u = IntHide[(d + e*x^2
)^p, x]}, Dist[a + b*ArcSinh[c*x], u, x] - Dist[b*c, Int[SimplifyIntegrand[u/Sqrt[1 + c^2*x^2], x], x], x]] /;
FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && IGtQ[p, 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]

Rubi steps

\begin {align*} \int x \left (d+c^2 d x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )^2 \, dx &=\frac {\left (d+c^2 d x^2\right )^{7/2} \left (a+b \sinh ^{-1}(c x)\right )^2}{7 c^2 d}-\frac {\left (2 b d^2 \sqrt {d+c^2 d x^2}\right ) \int \left (1+c^2 x^2\right )^3 \left (a+b \sinh ^{-1}(c x)\right ) \, dx}{7 c \sqrt {1+c^2 x^2}}\\ &=-\frac {2 b d^2 x \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{7 c \sqrt {1+c^2 x^2}}-\frac {2 b c d^2 x^3 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{7 \sqrt {1+c^2 x^2}}-\frac {6 b c^3 d^2 x^5 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{35 \sqrt {1+c^2 x^2}}-\frac {2 b c^5 d^2 x^7 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{49 \sqrt {1+c^2 x^2}}+\frac {\left (d+c^2 d x^2\right )^{7/2} \left (a+b \sinh ^{-1}(c x)\right )^2}{7 c^2 d}+\frac {\left (2 b^2 d^2 \sqrt {d+c^2 d x^2}\right ) \int \frac {x \left (35+35 c^2 x^2+21 c^4 x^4+5 c^6 x^6\right )}{35 \sqrt {1+c^2 x^2}} \, dx}{7 \sqrt {1+c^2 x^2}}\\ &=-\frac {2 b d^2 x \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{7 c \sqrt {1+c^2 x^2}}-\frac {2 b c d^2 x^3 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{7 \sqrt {1+c^2 x^2}}-\frac {6 b c^3 d^2 x^5 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{35 \sqrt {1+c^2 x^2}}-\frac {2 b c^5 d^2 x^7 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{49 \sqrt {1+c^2 x^2}}+\frac {\left (d+c^2 d x^2\right )^{7/2} \left (a+b \sinh ^{-1}(c x)\right )^2}{7 c^2 d}+\frac {\left (2 b^2 d^2 \sqrt {d+c^2 d x^2}\right ) \int \frac {x \left (35+35 c^2 x^2+21 c^4 x^4+5 c^6 x^6\right )}{\sqrt {1+c^2 x^2}} \, dx}{245 \sqrt {1+c^2 x^2}}\\ &=-\frac {2 b d^2 x \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{7 c \sqrt {1+c^2 x^2}}-\frac {2 b c d^2 x^3 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{7 \sqrt {1+c^2 x^2}}-\frac {6 b c^3 d^2 x^5 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{35 \sqrt {1+c^2 x^2}}-\frac {2 b c^5 d^2 x^7 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{49 \sqrt {1+c^2 x^2}}+\frac {\left (d+c^2 d x^2\right )^{7/2} \left (a+b \sinh ^{-1}(c x)\right )^2}{7 c^2 d}+\frac {\left (b^2 d^2 \sqrt {d+c^2 d x^2}\right ) \text {Subst}\left (\int \frac {35+35 c^2 x+21 c^4 x^2+5 c^6 x^3}{\sqrt {1+c^2 x}} \, dx,x,x^2\right )}{245 \sqrt {1+c^2 x^2}}\\ &=-\frac {2 b d^2 x \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{7 c \sqrt {1+c^2 x^2}}-\frac {2 b c d^2 x^3 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{7 \sqrt {1+c^2 x^2}}-\frac {6 b c^3 d^2 x^5 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{35 \sqrt {1+c^2 x^2}}-\frac {2 b c^5 d^2 x^7 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{49 \sqrt {1+c^2 x^2}}+\frac {\left (d+c^2 d x^2\right )^{7/2} \left (a+b \sinh ^{-1}(c x)\right )^2}{7 c^2 d}+\frac {\left (b^2 d^2 \sqrt {d+c^2 d x^2}\right ) \text {Subst}\left (\int \left (\frac {16}{\sqrt {1+c^2 x}}+8 \sqrt {1+c^2 x}+6 \left (1+c^2 x\right )^{3/2}+5 \left (1+c^2 x\right )^{5/2}\right ) \, dx,x,x^2\right )}{245 \sqrt {1+c^2 x^2}}\\ &=\frac {32 b^2 d^2 \sqrt {d+c^2 d x^2}}{245 c^2}+\frac {16 b^2 d^2 \left (1+c^2 x^2\right ) \sqrt {d+c^2 d x^2}}{735 c^2}+\frac {12 b^2 d^2 \left (1+c^2 x^2\right )^2 \sqrt {d+c^2 d x^2}}{1225 c^2}+\frac {2 b^2 d^2 \left (1+c^2 x^2\right )^3 \sqrt {d+c^2 d x^2}}{343 c^2}-\frac {2 b d^2 x \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{7 c \sqrt {1+c^2 x^2}}-\frac {2 b c d^2 x^3 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{7 \sqrt {1+c^2 x^2}}-\frac {6 b c^3 d^2 x^5 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{35 \sqrt {1+c^2 x^2}}-\frac {2 b c^5 d^2 x^7 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{49 \sqrt {1+c^2 x^2}}+\frac {\left (d+c^2 d x^2\right )^{7/2} \left (a+b \sinh ^{-1}(c x)\right )^2}{7 c^2 d}\\ \end {align*}

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Mathematica [A]
time = 0.25, size = 224, normalized size = 0.61 \begin {gather*} \frac {d^2 \sqrt {d+c^2 d x^2} \left (3675 a^2 \left (1+c^2 x^2\right )^4-210 a b c x \sqrt {1+c^2 x^2} \left (35+35 c^2 x^2+21 c^4 x^4+5 c^6 x^6\right )+2 b^2 \left (2161+2918 c^2 x^2+1108 c^4 x^4+426 c^6 x^6+75 c^8 x^8\right )+210 b \left (35 a \left (1+c^2 x^2\right )^4-b c x \sqrt {1+c^2 x^2} \left (35+35 c^2 x^2+21 c^4 x^4+5 c^6 x^6\right )\right ) \sinh ^{-1}(c x)+3675 b^2 \left (1+c^2 x^2\right )^4 \sinh ^{-1}(c x)^2\right )}{25725 c^2 \left (1+c^2 x^2\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x*(d + c^2*d*x^2)^(5/2)*(a + b*ArcSinh[c*x])^2,x]

[Out]

(d^2*Sqrt[d + c^2*d*x^2]*(3675*a^2*(1 + c^2*x^2)^4 - 210*a*b*c*x*Sqrt[1 + c^2*x^2]*(35 + 35*c^2*x^2 + 21*c^4*x
^4 + 5*c^6*x^6) + 2*b^2*(2161 + 2918*c^2*x^2 + 1108*c^4*x^4 + 426*c^6*x^6 + 75*c^8*x^8) + 210*b*(35*a*(1 + c^2
*x^2)^4 - b*c*x*Sqrt[1 + c^2*x^2]*(35 + 35*c^2*x^2 + 21*c^4*x^4 + 5*c^6*x^6))*ArcSinh[c*x] + 3675*b^2*(1 + c^2
*x^2)^4*ArcSinh[c*x]^2))/(25725*c^2*(1 + c^2*x^2))

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(1772\) vs. \(2(322)=644\).
time = 0.95, size = 1773, normalized size = 4.84

method	result	size
default	\(\text {Expression too large to display}\)	\(1773\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*(c^2*d*x^2+d)^(5/2)*(a+b*arcsinh(c*x))^2,x,method=_RETURNVERBOSE)

[Out]

1/7*a^2/c^2/d*(c^2*d*x^2+d)^(7/2)+b^2*(1/43904*(d*(c^2*x^2+1))^(1/2)*(64*x^8*c^8+64*(c^2*x^2+1)^(1/2)*x^7*c^7+
144*x^6*c^6+112*(c^2*x^2+1)^(1/2)*x^5*c^5+104*c^4*x^4+56*(c^2*x^2+1)^(1/2)*x^3*c^3+25*c^2*x^2+7*(c^2*x^2+1)^(1
/2)*c*x+1)*(49*arcsinh(c*x)^2-14*arcsinh(c*x)+2)*d^2/c^2/(c^2*x^2+1)+1/3200*(d*(c^2*x^2+1))^(1/2)*(16*x^6*c^6+
16*(c^2*x^2+1)^(1/2)*x^5*c^5+28*c^4*x^4+20*(c^2*x^2+1)^(1/2)*x^3*c^3+13*c^2*x^2+5*(c^2*x^2+1)^(1/2)*c*x+1)*(25
*arcsinh(c*x)^2-10*arcsinh(c*x)+2)*d^2/c^2/(c^2*x^2+1)+1/384*(d*(c^2*x^2+1))^(1/2)*(4*c^4*x^4+4*(c^2*x^2+1)^(1
/2)*x^3*c^3+5*c^2*x^2+3*(c^2*x^2+1)^(1/2)*c*x+1)*(9*arcsinh(c*x)^2-6*arcsinh(c*x)+2)*d^2/c^2/(c^2*x^2+1)+5/128
*(d*(c^2*x^2+1))^(1/2)*(c^2*x^2+(c^2*x^2+1)^(1/2)*c*x+1)*(arcsinh(c*x)^2-2*arcsinh(c*x)+2)*d^2/c^2/(c^2*x^2+1)
+5/128*(d*(c^2*x^2+1))^(1/2)*(c^2*x^2-(c^2*x^2+1)^(1/2)*c*x+1)*(arcsinh(c*x)^2+2*arcsinh(c*x)+2)*d^2/c^2/(c^2*
x^2+1)+1/384*(d*(c^2*x^2+1))^(1/2)*(4*c^4*x^4-4*(c^2*x^2+1)^(1/2)*x^3*c^3+5*c^2*x^2-3*(c^2*x^2+1)^(1/2)*c*x+1)
*(9*arcsinh(c*x)^2+6*arcsinh(c*x)+2)*d^2/c^2/(c^2*x^2+1)+1/3200*(d*(c^2*x^2+1))^(1/2)*(16*x^6*c^6-16*(c^2*x^2+
1)^(1/2)*x^5*c^5+28*c^4*x^4-20*(c^2*x^2+1)^(1/2)*x^3*c^3+13*c^2*x^2-5*(c^2*x^2+1)^(1/2)*c*x+1)*(25*arcsinh(c*x
)^2+10*arcsinh(c*x)+2)*d^2/c^2/(c^2*x^2+1)+1/43904*(d*(c^2*x^2+1))^(1/2)*(64*x^8*c^8-64*(c^2*x^2+1)^(1/2)*x^7*
c^7+144*x^6*c^6-112*(c^2*x^2+1)^(1/2)*x^5*c^5+104*c^4*x^4-56*(c^2*x^2+1)^(1/2)*x^3*c^3+25*c^2*x^2-7*(c^2*x^2+1
)^(1/2)*c*x+1)*(49*arcsinh(c*x)^2+14*arcsinh(c*x)+2)*d^2/c^2/(c^2*x^2+1))+2*a*b*(1/6272*(d*(c^2*x^2+1))^(1/2)*
(64*x^8*c^8+64*(c^2*x^2+1)^(1/2)*x^7*c^7+144*x^6*c^6+112*(c^2*x^2+1)^(1/2)*x^5*c^5+104*c^4*x^4+56*(c^2*x^2+1)^
(1/2)*x^3*c^3+25*c^2*x^2+7*(c^2*x^2+1)^(1/2)*c*x+1)*(-1+7*arcsinh(c*x))*d^2/c^2/(c^2*x^2+1)+1/640*(d*(c^2*x^2+
1))^(1/2)*(16*x^6*c^6+16*(c^2*x^2+1)^(1/2)*x^5*c^5+28*c^4*x^4+20*(c^2*x^2+1)^(1/2)*x^3*c^3+13*c^2*x^2+5*(c^2*x
^2+1)^(1/2)*c*x+1)*(-1+5*arcsinh(c*x))*d^2/c^2/(c^2*x^2+1)+1/128*(d*(c^2*x^2+1))^(1/2)*(4*c^4*x^4+4*(c^2*x^2+1
)^(1/2)*x^3*c^3+5*c^2*x^2+3*(c^2*x^2+1)^(1/2)*c*x+1)*(-1+3*arcsinh(c*x))*d^2/c^2/(c^2*x^2+1)+5/128*(d*(c^2*x^2
+1))^(1/2)*(c^2*x^2+(c^2*x^2+1)^(1/2)*c*x+1)*(arcsinh(c*x)-1)*d^2/c^2/(c^2*x^2+1)+5/128*(d*(c^2*x^2+1))^(1/2)*
(c^2*x^2-(c^2*x^2+1)^(1/2)*c*x+1)*(1+arcsinh(c*x))*d^2/c^2/(c^2*x^2+1)+1/128*(d*(c^2*x^2+1))^(1/2)*(4*c^4*x^4-
4*(c^2*x^2+1)^(1/2)*x^3*c^3+5*c^2*x^2-3*(c^2*x^2+1)^(1/2)*c*x+1)*(1+3*arcsinh(c*x))*d^2/c^2/(c^2*x^2+1)+1/640*
(d*(c^2*x^2+1))^(1/2)*(16*x^6*c^6-16*(c^2*x^2+1)^(1/2)*x^5*c^5+28*c^4*x^4-20*(c^2*x^2+1)^(1/2)*x^3*c^3+13*c^2*
x^2-5*(c^2*x^2+1)^(1/2)*c*x+1)*(1+5*arcsinh(c*x))*d^2/c^2/(c^2*x^2+1)+1/6272*(d*(c^2*x^2+1))^(1/2)*(64*x^8*c^8
-64*(c^2*x^2+1)^(1/2)*x^7*c^7+144*x^6*c^6-112*(c^2*x^2+1)^(1/2)*x^5*c^5+104*c^4*x^4-56*(c^2*x^2+1)^(1/2)*x^3*c
^3+25*c^2*x^2-7*(c^2*x^2+1)^(1/2)*c*x+1)*(1+7*arcsinh(c*x))*d^2/c^2/(c^2*x^2+1))

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Maxima [A]
time = 0.29, size = 274, normalized size = 0.75 \begin {gather*} \frac {{\left (c^{2} d x^{2} + d\right )}^{\frac {7}{2}} b^{2} \operatorname {arsinh}\left (c x\right )^{2}}{7 \, c^{2} d} + \frac {2 \, {\left (c^{2} d x^{2} + d\right )}^{\frac {7}{2}} a b \operatorname {arsinh}\left (c x\right )}{7 \, c^{2} d} + \frac {2}{25725} \, b^{2} {\left (\frac {75 \, \sqrt {c^{2} x^{2} + 1} c^{4} d^{\frac {7}{2}} x^{6} + 351 \, \sqrt {c^{2} x^{2} + 1} c^{2} d^{\frac {7}{2}} x^{4} + 757 \, \sqrt {c^{2} x^{2} + 1} d^{\frac {7}{2}} x^{2} + \frac {2161 \, \sqrt {c^{2} x^{2} + 1} d^{\frac {7}{2}}}{c^{2}}}{d} - \frac {105 \, {\left (5 \, c^{6} d^{\frac {7}{2}} x^{7} + 21 \, c^{4} d^{\frac {7}{2}} x^{5} + 35 \, c^{2} d^{\frac {7}{2}} x^{3} + 35 \, d^{\frac {7}{2}} x\right )} \operatorname {arsinh}\left (c x\right )}{c d}\right )} + \frac {{\left (c^{2} d x^{2} + d\right )}^{\frac {7}{2}} a^{2}}{7 \, c^{2} d} - \frac {2 \, {\left (5 \, c^{6} d^{\frac {7}{2}} x^{7} + 21 \, c^{4} d^{\frac {7}{2}} x^{5} + 35 \, c^{2} d^{\frac {7}{2}} x^{3} + 35 \, d^{\frac {7}{2}} x\right )} a b}{245 \, c d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(c^2*d*x^2+d)^(5/2)*(a+b*arcsinh(c*x))^2,x, algorithm="maxima")

[Out]

1/7*(c^2*d*x^2 + d)^(7/2)*b^2*arcsinh(c*x)^2/(c^2*d) + 2/7*(c^2*d*x^2 + d)^(7/2)*a*b*arcsinh(c*x)/(c^2*d) + 2/
25725*b^2*((75*sqrt(c^2*x^2 + 1)*c^4*d^(7/2)*x^6 + 351*sqrt(c^2*x^2 + 1)*c^2*d^(7/2)*x^4 + 757*sqrt(c^2*x^2 +
1)*d^(7/2)*x^2 + 2161*sqrt(c^2*x^2 + 1)*d^(7/2)/c^2)/d - 105*(5*c^6*d^(7/2)*x^7 + 21*c^4*d^(7/2)*x^5 + 35*c^2*
d^(7/2)*x^3 + 35*d^(7/2)*x)*arcsinh(c*x)/(c*d)) + 1/7*(c^2*d*x^2 + d)^(7/2)*a^2/(c^2*d) - 2/245*(5*c^6*d^(7/2)
*x^7 + 21*c^4*d^(7/2)*x^5 + 35*c^2*d^(7/2)*x^3 + 35*d^(7/2)*x)*a*b/(c*d)

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Fricas [A]
time = 0.38, size = 446, normalized size = 1.22 \begin {gather*} \frac {3675 \, {\left (b^{2} c^{8} d^{2} x^{8} + 4 \, b^{2} c^{6} d^{2} x^{6} + 6 \, b^{2} c^{4} d^{2} x^{4} + 4 \, b^{2} c^{2} d^{2} x^{2} + b^{2} d^{2}\right )} \sqrt {c^{2} d x^{2} + d} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right )^{2} + 210 \, {\left (35 \, a b c^{8} d^{2} x^{8} + 140 \, a b c^{6} d^{2} x^{6} + 210 \, a b c^{4} d^{2} x^{4} + 140 \, a b c^{2} d^{2} x^{2} + 35 \, a b d^{2} - {\left (5 \, b^{2} c^{7} d^{2} x^{7} + 21 \, b^{2} c^{5} d^{2} x^{5} + 35 \, b^{2} c^{3} d^{2} x^{3} + 35 \, b^{2} c d^{2} x\right )} \sqrt {c^{2} x^{2} + 1}\right )} \sqrt {c^{2} d x^{2} + d} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right ) + {\left (75 \, {\left (49 \, a^{2} + 2 \, b^{2}\right )} c^{8} d^{2} x^{8} + 12 \, {\left (1225 \, a^{2} + 71 \, b^{2}\right )} c^{6} d^{2} x^{6} + 2 \, {\left (11025 \, a^{2} + 1108 \, b^{2}\right )} c^{4} d^{2} x^{4} + 4 \, {\left (3675 \, a^{2} + 1459 \, b^{2}\right )} c^{2} d^{2} x^{2} + {\left (3675 \, a^{2} + 4322 \, b^{2}\right )} d^{2} - 210 \, {\left (5 \, a b c^{7} d^{2} x^{7} + 21 \, a b c^{5} d^{2} x^{5} + 35 \, a b c^{3} d^{2} x^{3} + 35 \, a b c d^{2} x\right )} \sqrt {c^{2} x^{2} + 1}\right )} \sqrt {c^{2} d x^{2} + d}}{25725 \, {\left (c^{4} x^{2} + c^{2}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(c^2*d*x^2+d)^(5/2)*(a+b*arcsinh(c*x))^2,x, algorithm="fricas")

[Out]

1/25725*(3675*(b^2*c^8*d^2*x^8 + 4*b^2*c^6*d^2*x^6 + 6*b^2*c^4*d^2*x^4 + 4*b^2*c^2*d^2*x^2 + b^2*d^2)*sqrt(c^2
*d*x^2 + d)*log(c*x + sqrt(c^2*x^2 + 1))^2 + 210*(35*a*b*c^8*d^2*x^8 + 140*a*b*c^6*d^2*x^6 + 210*a*b*c^4*d^2*x
^4 + 140*a*b*c^2*d^2*x^2 + 35*a*b*d^2 - (5*b^2*c^7*d^2*x^7 + 21*b^2*c^5*d^2*x^5 + 35*b^2*c^3*d^2*x^3 + 35*b^2*
c*d^2*x)*sqrt(c^2*x^2 + 1))*sqrt(c^2*d*x^2 + d)*log(c*x + sqrt(c^2*x^2 + 1)) + (75*(49*a^2 + 2*b^2)*c^8*d^2*x^
8 + 12*(1225*a^2 + 71*b^2)*c^6*d^2*x^6 + 2*(11025*a^2 + 1108*b^2)*c^4*d^2*x^4 + 4*(3675*a^2 + 1459*b^2)*c^2*d^
2*x^2 + (3675*a^2 + 4322*b^2)*d^2 - 210*(5*a*b*c^7*d^2*x^7 + 21*a*b*c^5*d^2*x^5 + 35*a*b*c^3*d^2*x^3 + 35*a*b*
c*d^2*x)*sqrt(c^2*x^2 + 1))*sqrt(c^2*d*x^2 + d))/(c^4*x^2 + c^2)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int x \left (d \left (c^{2} x^{2} + 1\right )\right )^{\frac {5}{2}} \left (a + b \operatorname {asinh}{\left (c x \right )}\right )^{2}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(c**2*d*x**2+d)**(5/2)*(a+b*asinh(c*x))**2,x)

[Out]

Integral(x*(d*(c**2*x**2 + 1))**(5/2)*(a + b*asinh(c*x))**2, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(c^2*d*x^2+d)^(5/2)*(a+b*arcsinh(c*x))^2,x, algorithm="giac")

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:sym2poly/r2sym(const 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 x\,{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2\,{\left (d\,c^2\,x^2+d\right )}^{5/2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*(a + b*asinh(c*x))^2*(d + c^2*d*x^2)^(5/2),x)

[Out]

int(x*(a + b*asinh(c*x))^2*(d + c^2*d*x^2)^(5/2), x)

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