3.136 \(\int x^3 (d+c^2 d x^2)^{5/2} (a+b \sinh ^{-1}(c x)) \, dx\)

Optimal. Leaf size=266 \[ \frac{\left (c^2 d x^2+d\right )^{9/2} \left (a+b \sinh ^{-1}(c x)\right )}{9 c^4 d^2}-\frac{\left (c^2 d x^2+d\right )^{7/2} \left (a+b \sinh ^{-1}(c x)\right )}{7 c^4 d}-\frac{b c^5 d^2 x^9 \sqrt{c^2 d x^2+d}}{81 \sqrt{c^2 x^2+1}}-\frac{19 b c^3 d^2 x^7 \sqrt{c^2 d x^2+d}}{441 \sqrt{c^2 x^2+1}}-\frac{b c d^2 x^5 \sqrt{c^2 d x^2+d}}{21 \sqrt{c^2 x^2+1}}-\frac{b d^2 x^3 \sqrt{c^2 d x^2+d}}{189 c \sqrt{c^2 x^2+1}}+\frac{2 b d^2 x \sqrt{c^2 d x^2+d}}{63 c^3 \sqrt{c^2 x^2+1}} \]

[Out]

(2*b*d^2*x*Sqrt[d + c^2*d*x^2])/(63*c^3*Sqrt[1 + c^2*x^2]) - (b*d^2*x^3*Sqrt[d + c^2*d*x^2])/(189*c*Sqrt[1 + c
^2*x^2]) - (b*c*d^2*x^5*Sqrt[d + c^2*d*x^2])/(21*Sqrt[1 + c^2*x^2]) - (19*b*c^3*d^2*x^7*Sqrt[d + c^2*d*x^2])/(
441*Sqrt[1 + c^2*x^2]) - (b*c^5*d^2*x^9*Sqrt[d + c^2*d*x^2])/(81*Sqrt[1 + c^2*x^2]) - ((d + c^2*d*x^2)^(7/2)*(
a + b*ArcSinh[c*x]))/(7*c^4*d) + ((d + c^2*d*x^2)^(9/2)*(a + b*ArcSinh[c*x]))/(9*c^4*d^2)

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Rubi [A]  time = 0.192028, antiderivative size = 266, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.192, Rules used = {266, 43, 5734, 12, 373} \[ \frac{\left (c^2 d x^2+d\right )^{9/2} \left (a+b \sinh ^{-1}(c x)\right )}{9 c^4 d^2}-\frac{\left (c^2 d x^2+d\right )^{7/2} \left (a+b \sinh ^{-1}(c x)\right )}{7 c^4 d}-\frac{b c^5 d^2 x^9 \sqrt{c^2 d x^2+d}}{81 \sqrt{c^2 x^2+1}}-\frac{19 b c^3 d^2 x^7 \sqrt{c^2 d x^2+d}}{441 \sqrt{c^2 x^2+1}}-\frac{b c d^2 x^5 \sqrt{c^2 d x^2+d}}{21 \sqrt{c^2 x^2+1}}-\frac{b d^2 x^3 \sqrt{c^2 d x^2+d}}{189 c \sqrt{c^2 x^2+1}}+\frac{2 b d^2 x \sqrt{c^2 d x^2+d}}{63 c^3 \sqrt{c^2 x^2+1}} \]

Antiderivative was successfully verified.

[In]

Int[x^3*(d + c^2*d*x^2)^(5/2)*(a + b*ArcSinh[c*x]),x]

[Out]

(2*b*d^2*x*Sqrt[d + c^2*d*x^2])/(63*c^3*Sqrt[1 + c^2*x^2]) - (b*d^2*x^3*Sqrt[d + c^2*d*x^2])/(189*c*Sqrt[1 + c
^2*x^2]) - (b*c*d^2*x^5*Sqrt[d + c^2*d*x^2])/(21*Sqrt[1 + c^2*x^2]) - (19*b*c^3*d^2*x^7*Sqrt[d + c^2*d*x^2])/(
441*Sqrt[1 + c^2*x^2]) - (b*c^5*d^2*x^9*Sqrt[d + c^2*d*x^2])/(81*Sqrt[1 + c^2*x^2]) - ((d + c^2*d*x^2)^(7/2)*(
a + b*ArcSinh[c*x]))/(7*c^4*d) + ((d + c^2*d*x^2)^(9/2)*(a + b*ArcSinh[c*x]))/(9*c^4*d^2)

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])

Rule 5734

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))*(x_)^(m_)*((d_) + (e_.)*(x_)^2)^(p_), x_Symbol] :> With[{u = IntHide[x
^m*(1 + c^2*x^2)^p, x]}, Dist[a + b*ArcSinh[c*x], Int[x^m*(d + e*x^2)^p, x], x] - Dist[(b*c*d^(p - 1/2)*Sqrt[d
+ e*x^2])/Sqrt[1 + c^2*x^2], 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 + 1/2, 0] && (IGtQ[(m + 1)/2, 0] || ILtQ[(m + 2*p + 3)/2, 0])

Rule 12

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

Rule 373

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

Rubi steps

\begin{align*} \int x^3 \left (d+c^2 d x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right ) \, dx &=-\frac{\left (b c d^2 \sqrt{d+c^2 d x^2}\right ) \int \frac{\left (1+c^2 x^2\right )^3 \left (-2+7 c^2 x^2\right )}{63 c^4} \, dx}{\sqrt{1+c^2 x^2}}+\left (a+b \sinh ^{-1}(c x)\right ) \int x^3 \left (d+c^2 d x^2\right )^{5/2} \, dx\\ &=-\frac{\left (b d^2 \sqrt{d+c^2 d x^2}\right ) \int \left (1+c^2 x^2\right )^3 \left (-2+7 c^2 x^2\right ) \, dx}{63 c^3 \sqrt{1+c^2 x^2}}+\frac{1}{2} \left (a+b \sinh ^{-1}(c x)\right ) \operatorname{Subst}\left (\int x \left (d+c^2 d x\right )^{5/2} \, dx,x,x^2\right )\\ &=-\frac{\left (b d^2 \sqrt{d+c^2 d x^2}\right ) \int \left (-2+c^2 x^2+15 c^4 x^4+19 c^6 x^6+7 c^8 x^8\right ) \, dx}{63 c^3 \sqrt{1+c^2 x^2}}+\frac{1}{2} \left (a+b \sinh ^{-1}(c x)\right ) \operatorname{Subst}\left (\int \left (-\frac{\left (d+c^2 d x\right )^{5/2}}{c^2}+\frac{\left (d+c^2 d x\right )^{7/2}}{c^2 d}\right ) \, dx,x,x^2\right )\\ &=\frac{2 b d^2 x \sqrt{d+c^2 d x^2}}{63 c^3 \sqrt{1+c^2 x^2}}-\frac{b d^2 x^3 \sqrt{d+c^2 d x^2}}{189 c \sqrt{1+c^2 x^2}}-\frac{b c d^2 x^5 \sqrt{d+c^2 d x^2}}{21 \sqrt{1+c^2 x^2}}-\frac{19 b c^3 d^2 x^7 \sqrt{d+c^2 d x^2}}{441 \sqrt{1+c^2 x^2}}-\frac{b c^5 d^2 x^9 \sqrt{d+c^2 d x^2}}{81 \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 )}{7 c^4 d}+\frac{\left (d+c^2 d x^2\right )^{9/2} \left (a+b \sinh ^{-1}(c x)\right )}{9 c^4 d^2}\\ \end{align*}

Mathematica [A]  time = 0.197109, size = 140, normalized size = 0.53 \[ \frac{d^2 \sqrt{c^2 d x^2+d} \left (63 a \left (7 c^2 x^2-2\right ) \left (c^2 x^2+1\right )^4-b c x \left (49 c^8 x^8+171 c^6 x^6+189 c^4 x^4+21 c^2 x^2-126\right ) \sqrt{c^2 x^2+1}+63 b \left (7 c^2 x^2-2\right ) \left (c^2 x^2+1\right )^4 \sinh ^{-1}(c x)\right )}{3969 c^4 \left (c^2 x^2+1\right )} \]

Antiderivative was successfully verified.

[In]

Integrate[x^3*(d + c^2*d*x^2)^(5/2)*(a + b*ArcSinh[c*x]),x]

[Out]

(d^2*Sqrt[d + c^2*d*x^2]*(63*a*(1 + c^2*x^2)^4*(-2 + 7*c^2*x^2) - b*c*x*Sqrt[1 + c^2*x^2]*(-126 + 21*c^2*x^2 +
189*c^4*x^4 + 171*c^6*x^6 + 49*c^8*x^8) + 63*b*(1 + c^2*x^2)^4*(-2 + 7*c^2*x^2)*ArcSinh[c*x]))/(3969*c^4*(1 +
c^2*x^2))

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Maple [B]  time = 0.216, size = 996, normalized size = 3.7 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*(c^2*d*x^2+d)^(5/2)*(a+b*arcsinh(c*x)),x)

[Out]

a*(1/9*x^2*(c^2*d*x^2+d)^(7/2)/c^2/d-2/63/d/c^4*(c^2*d*x^2+d)^(7/2))+b*(1/41472*(d*(c^2*x^2+1))^(1/2)*(256*c^1
0*x^10+256*c^9*x^9*(c^2*x^2+1)^(1/2)+704*c^8*x^8+576*c^7*x^7*(c^2*x^2+1)^(1/2)+688*c^6*x^6+432*c^5*x^5*(c^2*x^
2+1)^(1/2)+280*c^4*x^4+120*c^3*x^3*(c^2*x^2+1)^(1/2)+41*c^2*x^2+9*c*x*(c^2*x^2+1)^(1/2)+1)*(-1+9*arcsinh(c*x))
*d^2/c^4/(c^2*x^2+1)+3/25088*(d*(c^2*x^2+1))^(1/2)*(64*c^8*x^8+64*c^7*x^7*(c^2*x^2+1)^(1/2)+144*c^6*x^6+112*c^
5*x^5*(c^2*x^2+1)^(1/2)+104*c^4*x^4+56*c^3*x^3*(c^2*x^2+1)^(1/2)+25*c^2*x^2+7*c*x*(c^2*x^2+1)^(1/2)+1)*(-1+7*a
rcsinh(c*x))*d^2/c^4/(c^2*x^2+1)-1/576*(d*(c^2*x^2+1))^(1/2)*(4*c^4*x^4+4*c^3*x^3*(c^2*x^2+1)^(1/2)+5*c^2*x^2+
3*c*x*(c^2*x^2+1)^(1/2)+1)*(-1+3*arcsinh(c*x))*d^2/c^4/(c^2*x^2+1)-3/256*(d*(c^2*x^2+1))^(1/2)*(c^2*x^2+c*x*(c
^2*x^2+1)^(1/2)+1)*(-1+arcsinh(c*x))*d^2/c^4/(c^2*x^2+1)-3/256*(d*(c^2*x^2+1))^(1/2)*(c^2*x^2-c*x*(c^2*x^2+1)^
(1/2)+1)*(1+arcsinh(c*x))*d^2/c^4/(c^2*x^2+1)-1/576*(d*(c^2*x^2+1))^(1/2)*(4*c^4*x^4-4*c^3*x^3*(c^2*x^2+1)^(1/
2)+5*c^2*x^2-3*c*x*(c^2*x^2+1)^(1/2)+1)*(1+3*arcsinh(c*x))*d^2/c^4/(c^2*x^2+1)+3/25088*(d*(c^2*x^2+1))^(1/2)*(
64*c^8*x^8-64*c^7*x^7*(c^2*x^2+1)^(1/2)+144*c^6*x^6-112*c^5*x^5*(c^2*x^2+1)^(1/2)+104*c^4*x^4-56*c^3*x^3*(c^2*
x^2+1)^(1/2)+25*c^2*x^2-7*c*x*(c^2*x^2+1)^(1/2)+1)*(1+7*arcsinh(c*x))*d^2/c^4/(c^2*x^2+1)+1/41472*(d*(c^2*x^2+
1))^(1/2)*(256*c^10*x^10-256*c^9*x^9*(c^2*x^2+1)^(1/2)+704*c^8*x^8-576*c^7*x^7*(c^2*x^2+1)^(1/2)+688*c^6*x^6-4
32*c^5*x^5*(c^2*x^2+1)^(1/2)+280*c^4*x^4-120*c^3*x^3*(c^2*x^2+1)^(1/2)+41*c^2*x^2-9*c*x*(c^2*x^2+1)^(1/2)+1)*(
1+9*arcsinh(c*x))*d^2/c^4/(c^2*x^2+1))

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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^3*(c^2*d*x^2+d)^(5/2)*(a+b*arcsinh(c*x)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 2.43565, size = 585, normalized size = 2.2 \begin{align*} \frac{63 \,{\left (7 \, b c^{10} d^{2} x^{10} + 26 \, b c^{8} d^{2} x^{8} + 34 \, b c^{6} d^{2} x^{6} + 16 \, b c^{4} d^{2} x^{4} - b c^{2} d^{2} x^{2} - 2 \, b d^{2}\right )} \sqrt{c^{2} d x^{2} + d} \log \left (c x + \sqrt{c^{2} x^{2} + 1}\right ) +{\left (441 \, a c^{10} d^{2} x^{10} + 1638 \, a c^{8} d^{2} x^{8} + 2142 \, a c^{6} d^{2} x^{6} + 1008 \, a c^{4} d^{2} x^{4} - 63 \, a c^{2} d^{2} x^{2} - 126 \, a d^{2} -{\left (49 \, b c^{9} d^{2} x^{9} + 171 \, b c^{7} d^{2} x^{7} + 189 \, b c^{5} d^{2} x^{5} + 21 \, b c^{3} d^{2} x^{3} - 126 \, b c d^{2} x\right )} \sqrt{c^{2} x^{2} + 1}\right )} \sqrt{c^{2} d x^{2} + d}}{3969 \,{\left (c^{6} x^{2} + c^{4}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(c^2*d*x^2+d)^(5/2)*(a+b*arcsinh(c*x)),x, algorithm="fricas")

[Out]

1/3969*(63*(7*b*c^10*d^2*x^10 + 26*b*c^8*d^2*x^8 + 34*b*c^6*d^2*x^6 + 16*b*c^4*d^2*x^4 - b*c^2*d^2*x^2 - 2*b*d
^2)*sqrt(c^2*d*x^2 + d)*log(c*x + sqrt(c^2*x^2 + 1)) + (441*a*c^10*d^2*x^10 + 1638*a*c^8*d^2*x^8 + 2142*a*c^6*
d^2*x^6 + 1008*a*c^4*d^2*x^4 - 63*a*c^2*d^2*x^2 - 126*a*d^2 - (49*b*c^9*d^2*x^9 + 171*b*c^7*d^2*x^7 + 189*b*c^
5*d^2*x^5 + 21*b*c^3*d^2*x^3 - 126*b*c*d^2*x)*sqrt(c^2*x^2 + 1))*sqrt(c^2*d*x^2 + d))/(c^6*x^2 + c^4)

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(c^2*d*x^2+d)^(5/2)*(a+b*arcsinh(c*x)),x, algorithm="giac")

[Out]

Exception raised: NotImplementedError