3.165 \(\int \frac{x^6 (a+b \sinh ^{-1}(c x))}{(d+c^2 d x^2)^{5/2}} \, dx\)

Optimal. Leaf size=281 \[ -\frac{5 x^3 \left (a+b \sinh ^{-1}(c x)\right )}{3 c^4 d^2 \sqrt{c^2 d x^2+d}}+\frac{5 x \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )}{2 c^6 d^3}-\frac{5 \sqrt{c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )^2}{4 b c^7 d^2 \sqrt{c^2 d x^2+d}}-\frac{x^5 \left (a+b \sinh ^{-1}(c x)\right )}{3 c^2 d \left (c^2 d x^2+d\right )^{3/2}}-\frac{b x^2 \sqrt{c^2 x^2+1}}{4 c^5 d^2 \sqrt{c^2 d x^2+d}}-\frac{b}{6 c^7 d^2 \sqrt{c^2 x^2+1} \sqrt{c^2 d x^2+d}}-\frac{7 b \sqrt{c^2 x^2+1} \log \left (c^2 x^2+1\right )}{6 c^7 d^2 \sqrt{c^2 d x^2+d}} \]

[Out]

-b/(6*c^7*d^2*Sqrt[1 + c^2*x^2]*Sqrt[d + c^2*d*x^2]) - (b*x^2*Sqrt[1 + c^2*x^2])/(4*c^5*d^2*Sqrt[d + c^2*d*x^2
]) - (x^5*(a + b*ArcSinh[c*x]))/(3*c^2*d*(d + c^2*d*x^2)^(3/2)) - (5*x^3*(a + b*ArcSinh[c*x]))/(3*c^4*d^2*Sqrt
[d + c^2*d*x^2]) + (5*x*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]))/(2*c^6*d^3) - (5*Sqrt[1 + c^2*x^2]*(a + b*Ar
cSinh[c*x])^2)/(4*b*c^7*d^2*Sqrt[d + c^2*d*x^2]) - (7*b*Sqrt[1 + c^2*x^2]*Log[1 + c^2*x^2])/(6*c^7*d^2*Sqrt[d
+ c^2*d*x^2])

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Rubi [A]  time = 0.434799, antiderivative size = 281, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 7, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.269, Rules used = {5751, 5758, 5677, 5675, 30, 266, 43} \[ -\frac{5 x^3 \left (a+b \sinh ^{-1}(c x)\right )}{3 c^4 d^2 \sqrt{c^2 d x^2+d}}+\frac{5 x \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )}{2 c^6 d^3}-\frac{5 \sqrt{c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )^2}{4 b c^7 d^2 \sqrt{c^2 d x^2+d}}-\frac{x^5 \left (a+b \sinh ^{-1}(c x)\right )}{3 c^2 d \left (c^2 d x^2+d\right )^{3/2}}-\frac{b x^2 \sqrt{c^2 x^2+1}}{4 c^5 d^2 \sqrt{c^2 d x^2+d}}-\frac{b}{6 c^7 d^2 \sqrt{c^2 x^2+1} \sqrt{c^2 d x^2+d}}-\frac{7 b \sqrt{c^2 x^2+1} \log \left (c^2 x^2+1\right )}{6 c^7 d^2 \sqrt{c^2 d x^2+d}} \]

Antiderivative was successfully verified.

[In]

Int[(x^6*(a + b*ArcSinh[c*x]))/(d + c^2*d*x^2)^(5/2),x]

[Out]

-b/(6*c^7*d^2*Sqrt[1 + c^2*x^2]*Sqrt[d + c^2*d*x^2]) - (b*x^2*Sqrt[1 + c^2*x^2])/(4*c^5*d^2*Sqrt[d + c^2*d*x^2
]) - (x^5*(a + b*ArcSinh[c*x]))/(3*c^2*d*(d + c^2*d*x^2)^(3/2)) - (5*x^3*(a + b*ArcSinh[c*x]))/(3*c^4*d^2*Sqrt
[d + c^2*d*x^2]) + (5*x*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]))/(2*c^6*d^3) - (5*Sqrt[1 + c^2*x^2]*(a + b*Ar
cSinh[c*x])^2)/(4*b*c^7*d^2*Sqrt[d + c^2*d*x^2]) - (7*b*Sqrt[1 + c^2*x^2]*Log[1 + c^2*x^2])/(6*c^7*d^2*Sqrt[d
+ c^2*d*x^2])

Rule 5751

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)/(2*e*(p + 1)), x] + (-Dist[(f^2*(m - 1))/(2*e*(p
+ 1)), Int[(f*x)^(m - 2)*(d + e*x^2)^(p + 1)*(a + b*ArcSinh[c*x])^n, x], x] - Dist[(b*f*n*d^IntPart[p]*(d + e*
x^2)^FracPart[p])/(2*c*(p + 1)*(1 + c^2*x^2)^FracPart[p]), Int[(f*x)^(m - 1)*(1 + c^2*x^2)^(p + 1/2)*(a + b*Ar
cSinh[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[e, c^2*d] && GtQ[n, 0] && LtQ[p, -1] && Gt
Q[m, 1]

Rule 5758

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

Rule 5677

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

Rule 5675

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

Rule 30

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

Rule 266

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

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int \frac{x^6 \left (a+b \sinh ^{-1}(c x)\right )}{\left (d+c^2 d x^2\right )^{5/2}} \, dx &=-\frac{x^5 \left (a+b \sinh ^{-1}(c x)\right )}{3 c^2 d \left (d+c^2 d x^2\right )^{3/2}}+\frac{5 \int \frac{x^4 \left (a+b \sinh ^{-1}(c x)\right )}{\left (d+c^2 d x^2\right )^{3/2}} \, dx}{3 c^2 d}+\frac{\left (b \sqrt{1+c^2 x^2}\right ) \int \frac{x^5}{\left (1+c^2 x^2\right )^2} \, dx}{3 c d^2 \sqrt{d+c^2 d x^2}}\\ &=-\frac{x^5 \left (a+b \sinh ^{-1}(c x)\right )}{3 c^2 d \left (d+c^2 d x^2\right )^{3/2}}-\frac{5 x^3 \left (a+b \sinh ^{-1}(c x)\right )}{3 c^4 d^2 \sqrt{d+c^2 d x^2}}+\frac{5 \int \frac{x^2 \left (a+b \sinh ^{-1}(c x)\right )}{\sqrt{d+c^2 d x^2}} \, dx}{c^4 d^2}+\frac{\left (5 b \sqrt{1+c^2 x^2}\right ) \int \frac{x^3}{1+c^2 x^2} \, dx}{3 c^3 d^2 \sqrt{d+c^2 d x^2}}+\frac{\left (b \sqrt{1+c^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{x^2}{\left (1+c^2 x\right )^2} \, dx,x,x^2\right )}{6 c d^2 \sqrt{d+c^2 d x^2}}\\ &=-\frac{x^5 \left (a+b \sinh ^{-1}(c x)\right )}{3 c^2 d \left (d+c^2 d x^2\right )^{3/2}}-\frac{5 x^3 \left (a+b \sinh ^{-1}(c x)\right )}{3 c^4 d^2 \sqrt{d+c^2 d x^2}}+\frac{5 x \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{2 c^6 d^3}-\frac{5 \int \frac{a+b \sinh ^{-1}(c x)}{\sqrt{d+c^2 d x^2}} \, dx}{2 c^6 d^2}-\frac{\left (5 b \sqrt{1+c^2 x^2}\right ) \int x \, dx}{2 c^5 d^2 \sqrt{d+c^2 d x^2}}+\frac{\left (5 b \sqrt{1+c^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{x}{1+c^2 x} \, dx,x,x^2\right )}{6 c^3 d^2 \sqrt{d+c^2 d x^2}}+\frac{\left (b \sqrt{1+c^2 x^2}\right ) \operatorname{Subst}\left (\int \left (\frac{1}{c^4}+\frac{1}{c^4 \left (1+c^2 x\right )^2}-\frac{2}{c^4 \left (1+c^2 x\right )}\right ) \, dx,x,x^2\right )}{6 c d^2 \sqrt{d+c^2 d x^2}}\\ &=-\frac{b}{6 c^7 d^2 \sqrt{1+c^2 x^2} \sqrt{d+c^2 d x^2}}-\frac{13 b x^2 \sqrt{1+c^2 x^2}}{12 c^5 d^2 \sqrt{d+c^2 d x^2}}-\frac{x^5 \left (a+b \sinh ^{-1}(c x)\right )}{3 c^2 d \left (d+c^2 d x^2\right )^{3/2}}-\frac{5 x^3 \left (a+b \sinh ^{-1}(c x)\right )}{3 c^4 d^2 \sqrt{d+c^2 d x^2}}+\frac{5 x \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{2 c^6 d^3}-\frac{b \sqrt{1+c^2 x^2} \log \left (1+c^2 x^2\right )}{3 c^7 d^2 \sqrt{d+c^2 d x^2}}-\frac{\left (5 \sqrt{1+c^2 x^2}\right ) \int \frac{a+b \sinh ^{-1}(c x)}{\sqrt{1+c^2 x^2}} \, dx}{2 c^6 d^2 \sqrt{d+c^2 d x^2}}+\frac{\left (5 b \sqrt{1+c^2 x^2}\right ) \operatorname{Subst}\left (\int \left (\frac{1}{c^2}-\frac{1}{c^2 \left (1+c^2 x\right )}\right ) \, dx,x,x^2\right )}{6 c^3 d^2 \sqrt{d+c^2 d x^2}}\\ &=-\frac{b}{6 c^7 d^2 \sqrt{1+c^2 x^2} \sqrt{d+c^2 d x^2}}-\frac{b x^2 \sqrt{1+c^2 x^2}}{4 c^5 d^2 \sqrt{d+c^2 d x^2}}-\frac{x^5 \left (a+b \sinh ^{-1}(c x)\right )}{3 c^2 d \left (d+c^2 d x^2\right )^{3/2}}-\frac{5 x^3 \left (a+b \sinh ^{-1}(c x)\right )}{3 c^4 d^2 \sqrt{d+c^2 d x^2}}+\frac{5 x \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{2 c^6 d^3}-\frac{5 \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{4 b c^7 d^2 \sqrt{d+c^2 d x^2}}-\frac{7 b \sqrt{1+c^2 x^2} \log \left (1+c^2 x^2\right )}{6 c^7 d^2 \sqrt{d+c^2 d x^2}}\\ \end{align*}

Mathematica [A]  time = 1.00256, size = 222, normalized size = 0.79 \[ \frac{4 a c d x \left (3 c^4 x^4+20 c^2 x^2+15\right )-60 a \sqrt{d} \left (c^2 x^2+1\right ) \sqrt{c^2 d x^2+d} \log \left (\sqrt{d} \sqrt{c^2 d x^2+d}+c d x\right )+b d \left (-\sqrt{c^2 x^2+1} \left (6 c^4 x^4+9 c^2 x^2+28 \left (c^2 x^2+1\right ) \log \left (c^2 x^2+1\right )+7\right )-30 \left (c^2 x^2+1\right )^{3/2} \sinh ^{-1}(c x)^2+4 c x \left (3 c^4 x^4+20 c^2 x^2+15\right ) \sinh ^{-1}(c x)\right )}{24 c^7 d^3 \left (c^2 x^2+1\right ) \sqrt{c^2 d x^2+d}} \]

Antiderivative was successfully verified.

[In]

Integrate[(x^6*(a + b*ArcSinh[c*x]))/(d + c^2*d*x^2)^(5/2),x]

[Out]

(4*a*c*d*x*(15 + 20*c^2*x^2 + 3*c^4*x^4) + b*d*(4*c*x*(15 + 20*c^2*x^2 + 3*c^4*x^4)*ArcSinh[c*x] - 30*(1 + c^2
*x^2)^(3/2)*ArcSinh[c*x]^2 - Sqrt[1 + c^2*x^2]*(7 + 9*c^2*x^2 + 6*c^4*x^4 + 28*(1 + c^2*x^2)*Log[1 + c^2*x^2])
) - 60*a*Sqrt[d]*(1 + c^2*x^2)*Sqrt[d + c^2*d*x^2]*Log[c*d*x + Sqrt[d]*Sqrt[d + c^2*d*x^2]])/(24*c^7*d^3*(1 +
c^2*x^2)*Sqrt[d + c^2*d*x^2])

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Maple [B]  time = 0.341, size = 1607, normalized size = 5.7 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^6*(a+b*arcsinh(c*x))/(c^2*d*x^2+d)^(5/2),x)

[Out]

-5/2*a/c^6/d^2*ln(x*c^2*d/(c^2*d)^(1/2)+(c^2*d*x^2+d)^(1/2))/(c^2*d)^(1/2)-1/8*b*(d*(c^2*x^2+1))^(1/2)/c^7/d^3
/(c^2*x^2+1)^(1/2)+1/2*a*x^5/c^2/d/(c^2*d*x^2+d)^(3/2)+5/6*a/c^4*x^3/d/(c^2*d*x^2+d)^(3/2)+5/2*a/c^6/d^2*x/(c^
2*d*x^2+d)^(1/2)+49/6*b*(d*(c^2*x^2+1))^(1/2)/(63*c^8*x^8+237*c^6*x^6+334*c^4*x^4+209*c^2*x^2+49)/d^3*x^7+1009
/3*b*(d*(c^2*x^2+1))^(1/2)/(63*c^8*x^8+237*c^6*x^6+334*c^4*x^4+209*c^2*x^2+49)/c^4/d^3*arcsinh(c*x)*x^3-37/2*b
*(d*(c^2*x^2+1))^(1/2)/(63*c^8*x^8+237*c^6*x^6+334*c^4*x^4+209*c^2*x^2+49)/c^5/d^3*x^2*(c^2*x^2+1)^(1/2)-7*b*(
d*(c^2*x^2+1))^(1/2)/(63*c^8*x^8+237*c^6*x^6+334*c^4*x^4+209*c^2*x^2+49)/c^6/d^3*(c^2*x^2+1)*x+98*b*(d*(c^2*x^
2+1))^(1/2)/(63*c^8*x^8+237*c^6*x^6+334*c^4*x^4+209*c^2*x^2+49)/c^6/d^3*arcsinh(c*x)*x-343/3*b*(d*(c^2*x^2+1))
^(1/2)/(63*c^8*x^8+237*c^6*x^6+334*c^4*x^4+209*c^2*x^2+49)/c^7/d^3*arcsinh(c*x)*(c^2*x^2+1)^(1/2)+1/2*b*(d*(c^
2*x^2+1))^(1/2)/c^4/d^3/(c^2*x^2+1)*arcsinh(c*x)*x^3+1/2*b*(d*(c^2*x^2+1))^(1/2)/c^6/d^3/(c^2*x^2+1)*arcsinh(c
*x)*x-49/6*b*(d*(c^2*x^2+1))^(1/2)/(63*c^8*x^8+237*c^6*x^6+334*c^4*x^4+209*c^2*x^2+49)/c^2/d^3*(c^2*x^2+1)*x^5
+385*b*(d*(c^2*x^2+1))^(1/2)/(63*c^8*x^8+237*c^6*x^6+334*c^4*x^4+209*c^2*x^2+49)/c^2/d^3*arcsinh(c*x)*x^5-21/2
*b*(d*(c^2*x^2+1))^(1/2)/(63*c^8*x^8+237*c^6*x^6+334*c^4*x^4+209*c^2*x^2+49)/c^3/d^3*(c^2*x^2+1)^(1/2)*x^4-91/
6*b*(d*(c^2*x^2+1))^(1/2)/(63*c^8*x^8+237*c^6*x^6+334*c^4*x^4+209*c^2*x^2+49)/c^4/d^3*(c^2*x^2+1)*x^3-147*b*(d
*(c^2*x^2+1))^(1/2)/(63*c^8*x^8+237*c^6*x^6+334*c^4*x^4+209*c^2*x^2+49)/c/d^3*arcsinh(c*x)*(c^2*x^2+1)^(1/2)*x
^6-406*b*(d*(c^2*x^2+1))^(1/2)/(63*c^8*x^8+237*c^6*x^6+334*c^4*x^4+209*c^2*x^2+49)/c^3/d^3*arcsinh(c*x)*(c^2*x
^2+1)^(1/2)*x^4-1120/3*b*(d*(c^2*x^2+1))^(1/2)/(63*c^8*x^8+237*c^6*x^6+334*c^4*x^4+209*c^2*x^2+49)/c^5/d^3*arc
sinh(c*x)*(c^2*x^2+1)^(1/2)*x^2-1/4*b*(d*(c^2*x^2+1))^(1/2)/c^5/d^3/(c^2*x^2+1)^(1/2)*x^2+7*b*(d*(c^2*x^2+1))^
(1/2)/(63*c^8*x^8+237*c^6*x^6+334*c^4*x^4+209*c^2*x^2+49)/c^6/d^3*x-49/6*b*(d*(c^2*x^2+1))^(1/2)/(63*c^8*x^8+2
37*c^6*x^6+334*c^4*x^4+209*c^2*x^2+49)/c^7/d^3*(c^2*x^2+1)^(1/2)-5/4*b*(d*(c^2*x^2+1))^(1/2)/(c^2*x^2+1)^(1/2)
/c^7/d^3*arcsinh(c*x)^2+147*b*(d*(c^2*x^2+1))^(1/2)/(63*c^8*x^8+237*c^6*x^6+334*c^4*x^4+209*c^2*x^2+49)/d^3*ar
csinh(c*x)*x^7+14/3*b*(d*(c^2*x^2+1))^(1/2)/(c^2*x^2+1)^(1/2)/c^7/d^3*arcsinh(c*x)-7/3*b*(d*(c^2*x^2+1))^(1/2)
/(c^2*x^2+1)^(1/2)/c^7/d^3*ln(1+(c*x+(c^2*x^2+1)^(1/2))^2)+70/3*b*(d*(c^2*x^2+1))^(1/2)/(63*c^8*x^8+237*c^6*x^
6+334*c^4*x^4+209*c^2*x^2+49)/c^2/d^3*x^5+133/6*b*(d*(c^2*x^2+1))^(1/2)/(63*c^8*x^8+237*c^6*x^6+334*c^4*x^4+20
9*c^2*x^2+49)/c^4/d^3*x^3

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^6*(a+b*arcsinh(c*x))/(c^2*d*x^2+d)^(5/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (b x^{6} \operatorname{arsinh}\left (c x\right ) + a x^{6}\right )} \sqrt{c^{2} d x^{2} + d}}{c^{6} d^{3} x^{6} + 3 \, c^{4} d^{3} x^{4} + 3 \, c^{2} d^{3} x^{2} + d^{3}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^6*(a+b*arcsinh(c*x))/(c^2*d*x^2+d)^(5/2),x, algorithm="fricas")

[Out]

integral((b*x^6*arcsinh(c*x) + a*x^6)*sqrt(c^2*d*x^2 + d)/(c^6*d^3*x^6 + 3*c^4*d^3*x^4 + 3*c^2*d^3*x^2 + d^3),
x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**6*(a+b*asinh(c*x))/(c**2*d*x**2+d)**(5/2),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b \operatorname{arsinh}\left (c x\right ) + a\right )} x^{6}}{{\left (c^{2} d x^{2} + d\right )}^{\frac{5}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^6*(a+b*arcsinh(c*x))/(c^2*d*x^2+d)^(5/2),x, algorithm="giac")

[Out]

integrate((b*arcsinh(c*x) + a)*x^6/(c^2*d*x^2 + d)^(5/2), x)