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

Optimal. Leaf size=170 \[ \frac{1}{7} c^6 d^3 x^7 \left (a+b \sinh ^{-1}(c x)\right )+\frac{3}{5} c^4 d^3 x^5 \left (a+b \sinh ^{-1}(c x)\right )+c^2 d^3 x^3 \left (a+b \sinh ^{-1}(c x)\right )+d^3 x \left (a+b \sinh ^{-1}(c x)\right )-\frac{b d^3 \left (c^2 x^2+1\right )^{7/2}}{49 c}-\frac{6 b d^3 \left (c^2 x^2+1\right )^{5/2}}{175 c}-\frac{8 b d^3 \left (c^2 x^2+1\right )^{3/2}}{105 c}-\frac{16 b d^3 \sqrt{c^2 x^2+1}}{35 c} \]

[Out]

(-16*b*d^3*Sqrt[1 + c^2*x^2])/(35*c) - (8*b*d^3*(1 + c^2*x^2)^(3/2))/(105*c) - (6*b*d^3*(1 + c^2*x^2)^(5/2))/(
175*c) - (b*d^3*(1 + c^2*x^2)^(7/2))/(49*c) + d^3*x*(a + b*ArcSinh[c*x]) + c^2*d^3*x^3*(a + b*ArcSinh[c*x]) +
(3*c^4*d^3*x^5*(a + b*ArcSinh[c*x]))/5 + (c^6*d^3*x^7*(a + b*ArcSinh[c*x]))/7

________________________________________________________________________________________

Rubi [A]  time = 0.16143, antiderivative size = 170, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.238, Rules used = {194, 5679, 12, 1799, 1850} \[ \frac{1}{7} c^6 d^3 x^7 \left (a+b \sinh ^{-1}(c x)\right )+\frac{3}{5} c^4 d^3 x^5 \left (a+b \sinh ^{-1}(c x)\right )+c^2 d^3 x^3 \left (a+b \sinh ^{-1}(c x)\right )+d^3 x \left (a+b \sinh ^{-1}(c x)\right )-\frac{b d^3 \left (c^2 x^2+1\right )^{7/2}}{49 c}-\frac{6 b d^3 \left (c^2 x^2+1\right )^{5/2}}{175 c}-\frac{8 b d^3 \left (c^2 x^2+1\right )^{3/2}}{105 c}-\frac{16 b d^3 \sqrt{c^2 x^2+1}}{35 c} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-16*b*d^3*Sqrt[1 + c^2*x^2])/(35*c) - (8*b*d^3*(1 + c^2*x^2)^(3/2))/(105*c) - (6*b*d^3*(1 + c^2*x^2)^(5/2))/(
175*c) - (b*d^3*(1 + c^2*x^2)^(7/2))/(49*c) + d^3*x*(a + b*ArcSinh[c*x]) + c^2*d^3*x^3*(a + b*ArcSinh[c*x]) +
(3*c^4*d^3*x^5*(a + b*ArcSinh[c*x]))/5 + (c^6*d^3*x^7*(a + b*ArcSinh[c*x]))/7

Rule 194

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 5679

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 12

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

Rule 1799

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 1850

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])

Rubi steps

\begin{align*} \int \left (d+c^2 d x^2\right )^3 \left (a+b \sinh ^{-1}(c x)\right ) \, dx &=d^3 x \left (a+b \sinh ^{-1}(c x)\right )+c^2 d^3 x^3 \left (a+b \sinh ^{-1}(c x)\right )+\frac{3}{5} c^4 d^3 x^5 \left (a+b \sinh ^{-1}(c x)\right )+\frac{1}{7} c^6 d^3 x^7 \left (a+b \sinh ^{-1}(c x)\right )-(b c) \int \frac{d^3 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\\ &=d^3 x \left (a+b \sinh ^{-1}(c x)\right )+c^2 d^3 x^3 \left (a+b \sinh ^{-1}(c x)\right )+\frac{3}{5} c^4 d^3 x^5 \left (a+b \sinh ^{-1}(c x)\right )+\frac{1}{7} c^6 d^3 x^7 \left (a+b \sinh ^{-1}(c x)\right )-\frac{1}{35} \left (b c d^3\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\\ &=d^3 x \left (a+b \sinh ^{-1}(c x)\right )+c^2 d^3 x^3 \left (a+b \sinh ^{-1}(c x)\right )+\frac{3}{5} c^4 d^3 x^5 \left (a+b \sinh ^{-1}(c x)\right )+\frac{1}{7} c^6 d^3 x^7 \left (a+b \sinh ^{-1}(c x)\right )-\frac{1}{70} \left (b c d^3\right ) \operatorname{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 )\\ &=d^3 x \left (a+b \sinh ^{-1}(c x)\right )+c^2 d^3 x^3 \left (a+b \sinh ^{-1}(c x)\right )+\frac{3}{5} c^4 d^3 x^5 \left (a+b \sinh ^{-1}(c x)\right )+\frac{1}{7} c^6 d^3 x^7 \left (a+b \sinh ^{-1}(c x)\right )-\frac{1}{70} \left (b c d^3\right ) \operatorname{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 )\\ &=-\frac{16 b d^3 \sqrt{1+c^2 x^2}}{35 c}-\frac{8 b d^3 \left (1+c^2 x^2\right )^{3/2}}{105 c}-\frac{6 b d^3 \left (1+c^2 x^2\right )^{5/2}}{175 c}-\frac{b d^3 \left (1+c^2 x^2\right )^{7/2}}{49 c}+d^3 x \left (a+b \sinh ^{-1}(c x)\right )+c^2 d^3 x^3 \left (a+b \sinh ^{-1}(c x)\right )+\frac{3}{5} c^4 d^3 x^5 \left (a+b \sinh ^{-1}(c x)\right )+\frac{1}{7} c^6 d^3 x^7 \left (a+b \sinh ^{-1}(c x)\right )\\ \end{align*}

Mathematica [A]  time = 0.165447, size = 119, normalized size = 0.7 \[ \frac{d^3 \left (105 a c x \left (5 c^6 x^6+21 c^4 x^4+35 c^2 x^2+35\right )-b \sqrt{c^2 x^2+1} \left (75 c^6 x^6+351 c^4 x^4+757 c^2 x^2+2161\right )+105 b c x \left (5 c^6 x^6+21 c^4 x^4+35 c^2 x^2+35\right ) \sinh ^{-1}(c x)\right )}{3675 c} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(d^3*(105*a*c*x*(35 + 35*c^2*x^2 + 21*c^4*x^4 + 5*c^6*x^6) - b*Sqrt[1 + c^2*x^2]*(2161 + 757*c^2*x^2 + 351*c^4
*x^4 + 75*c^6*x^6) + 105*b*c*x*(35 + 35*c^2*x^2 + 21*c^4*x^4 + 5*c^6*x^6)*ArcSinh[c*x]))/(3675*c)

________________________________________________________________________________________

Maple [A]  time = 0.006, size = 156, normalized size = 0.9 \begin{align*}{\frac{1}{c} \left ({d}^{3}a \left ({\frac{{c}^{7}{x}^{7}}{7}}+{\frac{3\,{c}^{5}{x}^{5}}{5}}+{c}^{3}{x}^{3}+cx \right ) +{d}^{3}b \left ({\frac{{\it Arcsinh} \left ( cx \right ){c}^{7}{x}^{7}}{7}}+{\frac{3\,{\it Arcsinh} \left ( cx \right ){c}^{5}{x}^{5}}{5}}+{\it Arcsinh} \left ( cx \right ){c}^{3}{x}^{3}+{\it Arcsinh} \left ( cx \right ) cx-{\frac{{c}^{6}{x}^{6}}{49}\sqrt{{c}^{2}{x}^{2}+1}}-{\frac{117\,{c}^{4}{x}^{4}}{1225}\sqrt{{c}^{2}{x}^{2}+1}}-{\frac{757\,{c}^{2}{x}^{2}}{3675}\sqrt{{c}^{2}{x}^{2}+1}}-{\frac{2161}{3675}\sqrt{{c}^{2}{x}^{2}+1}} \right ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/c*(d^3*a*(1/7*c^7*x^7+3/5*c^5*x^5+c^3*x^3+c*x)+d^3*b*(1/7*arcsinh(c*x)*c^7*x^7+3/5*arcsinh(c*x)*c^5*x^5+arcs
inh(c*x)*c^3*x^3+arcsinh(c*x)*c*x-1/49*c^6*x^6*(c^2*x^2+1)^(1/2)-117/1225*c^4*x^4*(c^2*x^2+1)^(1/2)-757/3675*c
^2*x^2*(c^2*x^2+1)^(1/2)-2161/3675*(c^2*x^2+1)^(1/2)))

________________________________________________________________________________________

Maxima [A]  time = 1.14275, size = 406, normalized size = 2.39 \begin{align*} \frac{1}{7} \, a c^{6} d^{3} x^{7} + \frac{3}{5} \, a c^{4} d^{3} x^{5} + \frac{1}{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 )} b c^{6} d^{3} + \frac{1}{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 )} b c^{4} d^{3} + a c^{2} d^{3} x^{3} + \frac{1}{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 )} b c^{2} d^{3} + a d^{3} x + \frac{{\left (c x \operatorname{arsinh}\left (c x\right ) - \sqrt{c^{2} x^{2} + 1}\right )} b d^{3}}{c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/7*a*c^6*d^3*x^7 + 3/5*a*c^4*d^3*x^5 + 1/245*(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)*b*c^6*d^3 + 1/25*(15*x^5*arcsin
h(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)*b*c^4*d^3 +
a*c^2*d^3*x^3 + 1/3*(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))*b*c^2*d^3 +
a*d^3*x + (c*x*arcsinh(c*x) - sqrt(c^2*x^2 + 1))*b*d^3/c

________________________________________________________________________________________

Fricas [A]  time = 2.3234, size = 387, normalized size = 2.28 \begin{align*} \frac{525 \, a c^{7} d^{3} x^{7} + 2205 \, a c^{5} d^{3} x^{5} + 3675 \, a c^{3} d^{3} x^{3} + 3675 \, a c d^{3} x + 105 \,{\left (5 \, b c^{7} d^{3} x^{7} + 21 \, b c^{5} d^{3} x^{5} + 35 \, b c^{3} d^{3} x^{3} + 35 \, b c d^{3} x\right )} \log \left (c x + \sqrt{c^{2} x^{2} + 1}\right ) -{\left (75 \, b c^{6} d^{3} x^{6} + 351 \, b c^{4} d^{3} x^{4} + 757 \, b c^{2} d^{3} x^{2} + 2161 \, b d^{3}\right )} \sqrt{c^{2} x^{2} + 1}}{3675 \, c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3675*(525*a*c^7*d^3*x^7 + 2205*a*c^5*d^3*x^5 + 3675*a*c^3*d^3*x^3 + 3675*a*c*d^3*x + 105*(5*b*c^7*d^3*x^7 +
21*b*c^5*d^3*x^5 + 35*b*c^3*d^3*x^3 + 35*b*c*d^3*x)*log(c*x + sqrt(c^2*x^2 + 1)) - (75*b*c^6*d^3*x^6 + 351*b*c
^4*d^3*x^4 + 757*b*c^2*d^3*x^2 + 2161*b*d^3)*sqrt(c^2*x^2 + 1))/c

________________________________________________________________________________________

Sympy [A]  time = 8.66608, size = 221, normalized size = 1.3 \begin{align*} \begin{cases} \frac{a c^{6} d^{3} x^{7}}{7} + \frac{3 a c^{4} d^{3} x^{5}}{5} + a c^{2} d^{3} x^{3} + a d^{3} x + \frac{b c^{6} d^{3} x^{7} \operatorname{asinh}{\left (c x \right )}}{7} - \frac{b c^{5} d^{3} x^{6} \sqrt{c^{2} x^{2} + 1}}{49} + \frac{3 b c^{4} d^{3} x^{5} \operatorname{asinh}{\left (c x \right )}}{5} - \frac{117 b c^{3} d^{3} x^{4} \sqrt{c^{2} x^{2} + 1}}{1225} + b c^{2} d^{3} x^{3} \operatorname{asinh}{\left (c x \right )} - \frac{757 b c d^{3} x^{2} \sqrt{c^{2} x^{2} + 1}}{3675} + b d^{3} x \operatorname{asinh}{\left (c x \right )} - \frac{2161 b d^{3} \sqrt{c^{2} x^{2} + 1}}{3675 c} & \text{for}\: c \neq 0 \\a d^{3} x & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((a*c**6*d**3*x**7/7 + 3*a*c**4*d**3*x**5/5 + a*c**2*d**3*x**3 + a*d**3*x + b*c**6*d**3*x**7*asinh(c*
x)/7 - b*c**5*d**3*x**6*sqrt(c**2*x**2 + 1)/49 + 3*b*c**4*d**3*x**5*asinh(c*x)/5 - 117*b*c**3*d**3*x**4*sqrt(c
**2*x**2 + 1)/1225 + b*c**2*d**3*x**3*asinh(c*x) - 757*b*c*d**3*x**2*sqrt(c**2*x**2 + 1)/3675 + b*d**3*x*asinh
(c*x) - 2161*b*d**3*sqrt(c**2*x**2 + 1)/(3675*c), Ne(c, 0)), (a*d**3*x, True))

________________________________________________________________________________________

Giac [B]  time = 1.67599, size = 416, normalized size = 2.45 \begin{align*} \frac{1}{7} \, a c^{6} d^{3} x^{7} + \frac{3}{5} \, a c^{4} d^{3} x^{5} + \frac{1}{245} \,{\left (35 \, x^{7} \log \left (c x + \sqrt{c^{2} x^{2} + 1}\right ) - \frac{5 \,{\left (c^{2} x^{2} + 1\right )}^{\frac{7}{2}} - 21 \,{\left (c^{2} x^{2} + 1\right )}^{\frac{5}{2}} + 35 \,{\left (c^{2} x^{2} + 1\right )}^{\frac{3}{2}} - 35 \, \sqrt{c^{2} x^{2} + 1}}{c^{7}}\right )} b c^{6} d^{3} + \frac{1}{25} \,{\left (15 \, x^{5} \log \left (c x + \sqrt{c^{2} x^{2} + 1}\right ) - \frac{3 \,{\left (c^{2} x^{2} + 1\right )}^{\frac{5}{2}} - 10 \,{\left (c^{2} x^{2} + 1\right )}^{\frac{3}{2}} + 15 \, \sqrt{c^{2} x^{2} + 1}}{c^{5}}\right )} b c^{4} d^{3} + a c^{2} d^{3} x^{3} + \frac{1}{3} \,{\left (3 \, x^{3} \log \left (c x + \sqrt{c^{2} x^{2} + 1}\right ) - \frac{{\left (c^{2} x^{2} + 1\right )}^{\frac{3}{2}} - 3 \, \sqrt{c^{2} x^{2} + 1}}{c^{3}}\right )} b c^{2} d^{3} +{\left (x \log \left (c x + \sqrt{c^{2} x^{2} + 1}\right ) - \frac{\sqrt{c^{2} x^{2} + 1}}{c}\right )} b d^{3} + a d^{3} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/7*a*c^6*d^3*x^7 + 3/5*a*c^4*d^3*x^5 + 1/245*(35*x^7*log(c*x + sqrt(c^2*x^2 + 1)) - (5*(c^2*x^2 + 1)^(7/2) -
21*(c^2*x^2 + 1)^(5/2) + 35*(c^2*x^2 + 1)^(3/2) - 35*sqrt(c^2*x^2 + 1))/c^7)*b*c^6*d^3 + 1/25*(15*x^5*log(c*x
+ sqrt(c^2*x^2 + 1)) - (3*(c^2*x^2 + 1)^(5/2) - 10*(c^2*x^2 + 1)^(3/2) + 15*sqrt(c^2*x^2 + 1))/c^5)*b*c^4*d^3
+ a*c^2*d^3*x^3 + 1/3*(3*x^3*log(c*x + sqrt(c^2*x^2 + 1)) - ((c^2*x^2 + 1)^(3/2) - 3*sqrt(c^2*x^2 + 1))/c^3)*b
*c^2*d^3 + (x*log(c*x + sqrt(c^2*x^2 + 1)) - sqrt(c^2*x^2 + 1)/c)*b*d^3 + a*d^3*x