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

Optimal. Leaf size=198 \[ -\frac {d \left (a+b \sinh ^{-1}(c x)\right )^2}{24 c^4}-\frac {1}{18} b c d x^5 \sqrt {c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )+\frac {1}{6} d x^4 \left (c^2 x^2+1\right ) \left (a+b \sinh ^{-1}(c x)\right )^2-\frac {b d x^3 \sqrt {c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )}{18 c}+\frac {b d x \sqrt {c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )}{12 c^3}+\frac {1}{12} d x^4 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {1}{108} b^2 c^2 d x^6-\frac {b^2 d x^2}{24 c^2}+\frac {1}{72} b^2 d x^4 \]

[Out]

-1/24*b^2*d*x^2/c^2+1/72*b^2*d*x^4+1/108*b^2*c^2*d*x^6-1/24*d*(a+b*arcsinh(c*x))^2/c^4+1/12*d*x^4*(a+b*arcsinh
(c*x))^2+1/6*d*x^4*(c^2*x^2+1)*(a+b*arcsinh(c*x))^2+1/12*b*d*x*(a+b*arcsinh(c*x))*(c^2*x^2+1)^(1/2)/c^3-1/18*b
*d*x^3*(a+b*arcsinh(c*x))*(c^2*x^2+1)^(1/2)/c-1/18*b*c*d*x^5*(a+b*arcsinh(c*x))*(c^2*x^2+1)^(1/2)

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Rubi [A]  time = 0.57, antiderivative size = 198, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 6, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {5744, 5661, 5758, 5675, 30, 5742} \[ -\frac {1}{18} b c d x^5 \sqrt {c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )+\frac {1}{6} d x^4 \left (c^2 x^2+1\right ) \left (a+b \sinh ^{-1}(c x)\right )^2-\frac {b d x^3 \sqrt {c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )}{18 c}+\frac {b d x \sqrt {c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )}{12 c^3}-\frac {d \left (a+b \sinh ^{-1}(c x)\right )^2}{24 c^4}+\frac {1}{12} d x^4 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {1}{108} b^2 c^2 d x^6-\frac {b^2 d x^2}{24 c^2}+\frac {1}{72} b^2 d x^4 \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(b^2*d*x^2)/(24*c^2) + (b^2*d*x^4)/72 + (b^2*c^2*d*x^6)/108 + (b*d*x*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x]))/
(12*c^3) - (b*d*x^3*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x]))/(18*c) - (b*c*d*x^5*Sqrt[1 + c^2*x^2]*(a + b*ArcSi
nh[c*x]))/18 - (d*(a + b*ArcSinh[c*x])^2)/(24*c^4) + (d*x^4*(a + b*ArcSinh[c*x])^2)/12 + (d*x^4*(1 + c^2*x^2)*
(a + b*ArcSinh[c*x])^2)/6

Rule 30

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

Rule 5661

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 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 5742

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

Rule 5744

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Simp
[((f*x)^(m + 1)*(d + e*x^2)^p*(a + b*ArcSinh[c*x])^n)/(f*(m + 2*p + 1)), x] + (Dist[(2*d*p)/(m + 2*p + 1), Int
[(f*x)^m*(d + e*x^2)^(p - 1)*(a + b*ArcSinh[c*x])^n, x], x] - Dist[(b*c*n*d^IntPart[p]*(d + e*x^2)^FracPart[p]
)/(f*(m + 2*p + 1)*(1 + c^2*x^2)^FracPart[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, m}, x] && EqQ[e, c^2*d] && GtQ[n, 0] && GtQ[p, 0] &&  !LtQ[m, -1]
&& (RationalQ[m] || EqQ[n, 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]

Rubi steps

\begin {align*} \int x^3 \left (d+c^2 d x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2 \, dx &=\frac {1}{6} d x^4 \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {1}{3} d \int x^3 \left (a+b \sinh ^{-1}(c x)\right )^2 \, dx-\frac {1}{3} (b c d) \int x^4 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right ) \, dx\\ &=-\frac {1}{18} b c d x^5 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )+\frac {1}{12} d x^4 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {1}{6} d x^4 \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2-\frac {1}{18} (b c d) \int \frac {x^4 \left (a+b \sinh ^{-1}(c x)\right )}{\sqrt {1+c^2 x^2}} \, dx-\frac {1}{6} (b c d) \int \frac {x^4 \left (a+b \sinh ^{-1}(c x)\right )}{\sqrt {1+c^2 x^2}} \, dx+\frac {1}{18} \left (b^2 c^2 d\right ) \int x^5 \, dx\\ &=\frac {1}{108} b^2 c^2 d x^6-\frac {b d x^3 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{18 c}-\frac {1}{18} b c d x^5 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )+\frac {1}{12} d x^4 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {1}{6} d x^4 \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {1}{72} \left (b^2 d\right ) \int x^3 \, dx+\frac {1}{24} \left (b^2 d\right ) \int x^3 \, dx+\frac {(b d) \int \frac {x^2 \left (a+b \sinh ^{-1}(c x)\right )}{\sqrt {1+c^2 x^2}} \, dx}{24 c}+\frac {(b d) \int \frac {x^2 \left (a+b \sinh ^{-1}(c x)\right )}{\sqrt {1+c^2 x^2}} \, dx}{8 c}\\ &=\frac {1}{72} b^2 d x^4+\frac {1}{108} b^2 c^2 d x^6+\frac {b d x \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{12 c^3}-\frac {b d x^3 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{18 c}-\frac {1}{18} b c d x^5 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )+\frac {1}{12} d x^4 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {1}{6} d x^4 \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2-\frac {(b d) \int \frac {a+b \sinh ^{-1}(c x)}{\sqrt {1+c^2 x^2}} \, dx}{48 c^3}-\frac {(b d) \int \frac {a+b \sinh ^{-1}(c x)}{\sqrt {1+c^2 x^2}} \, dx}{16 c^3}-\frac {\left (b^2 d\right ) \int x \, dx}{48 c^2}-\frac {\left (b^2 d\right ) \int x \, dx}{16 c^2}\\ &=-\frac {b^2 d x^2}{24 c^2}+\frac {1}{72} b^2 d x^4+\frac {1}{108} b^2 c^2 d x^6+\frac {b d x \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{12 c^3}-\frac {b d x^3 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{18 c}-\frac {1}{18} b c d x^5 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )-\frac {d \left (a+b \sinh ^{-1}(c x)\right )^2}{24 c^4}+\frac {1}{12} d x^4 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {1}{6} d x^4 \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2\\ \end {align*}

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Mathematica [A]  time = 0.27, size = 186, normalized size = 0.94 \[ \frac {d \left (c x \left (18 a^2 c^3 x^3 \left (2 c^2 x^2+3\right )-6 a b \sqrt {c^2 x^2+1} \left (2 c^4 x^4+2 c^2 x^2-3\right )+b^2 c x \left (2 c^4 x^4+3 c^2 x^2-9\right )\right )+6 b \sinh ^{-1}(c x) \left (3 a \left (4 c^6 x^6+6 c^4 x^4-1\right )+b c x \sqrt {c^2 x^2+1} \left (-2 c^4 x^4-2 c^2 x^2+3\right )\right )+9 b^2 \left (4 c^6 x^6+6 c^4 x^4-1\right ) \sinh ^{-1}(c x)^2\right )}{216 c^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(d*(c*x*(18*a^2*c^3*x^3*(3 + 2*c^2*x^2) - 6*a*b*Sqrt[1 + c^2*x^2]*(-3 + 2*c^2*x^2 + 2*c^4*x^4) + b^2*c*x*(-9 +
3*c^2*x^2 + 2*c^4*x^4)) + 6*b*(b*c*x*Sqrt[1 + c^2*x^2]*(3 - 2*c^2*x^2 - 2*c^4*x^4) + 3*a*(-1 + 6*c^4*x^4 + 4*
c^6*x^6))*ArcSinh[c*x] + 9*b^2*(-1 + 6*c^4*x^4 + 4*c^6*x^6)*ArcSinh[c*x]^2))/(216*c^4)

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fricas [A]  time = 0.53, size = 240, normalized size = 1.21 \[ \frac {2 \, {\left (18 \, a^{2} + b^{2}\right )} c^{6} d x^{6} + 3 \, {\left (18 \, a^{2} + b^{2}\right )} c^{4} d x^{4} - 9 \, b^{2} c^{2} d x^{2} + 9 \, {\left (4 \, b^{2} c^{6} d x^{6} + 6 \, b^{2} c^{4} d x^{4} - b^{2} d\right )} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right )^{2} + 6 \, {\left (12 \, a b c^{6} d x^{6} + 18 \, a b c^{4} d x^{4} - 3 \, a b d - {\left (2 \, b^{2} c^{5} d x^{5} + 2 \, b^{2} c^{3} d x^{3} - 3 \, b^{2} c d x\right )} \sqrt {c^{2} x^{2} + 1}\right )} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right ) - 6 \, {\left (2 \, a b c^{5} d x^{5} + 2 \, a b c^{3} d x^{3} - 3 \, a b c d x\right )} \sqrt {c^{2} x^{2} + 1}}{216 \, c^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/216*(2*(18*a^2 + b^2)*c^6*d*x^6 + 3*(18*a^2 + b^2)*c^4*d*x^4 - 9*b^2*c^2*d*x^2 + 9*(4*b^2*c^6*d*x^6 + 6*b^2*
c^4*d*x^4 - b^2*d)*log(c*x + sqrt(c^2*x^2 + 1))^2 + 6*(12*a*b*c^6*d*x^6 + 18*a*b*c^4*d*x^4 - 3*a*b*d - (2*b^2*
c^5*d*x^5 + 2*b^2*c^3*d*x^3 - 3*b^2*c*d*x)*sqrt(c^2*x^2 + 1))*log(c*x + sqrt(c^2*x^2 + 1)) - 6*(2*a*b*c^5*d*x^
5 + 2*a*b*c^3*d*x^3 - 3*a*b*c*d*x)*sqrt(c^2*x^2 + 1))/c^4

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giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,x):;OUTPUT:sym2
poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

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maple [A]  time = 0.04, size = 267, normalized size = 1.35 \[ \frac {d \,a^{2} \left (\frac {1}{6} c^{6} x^{6}+\frac {1}{4} c^{4} x^{4}\right )+d \,b^{2} \left (\frac {\arcsinh \left (c x \right )^{2} c^{2} x^{2} \left (c^{2} x^{2}+1\right )^{2}}{6}-\frac {\arcsinh \left (c x \right )^{2} \left (c^{2} x^{2}+1\right )^{2}}{12}-\frac {\arcsinh \left (c x \right ) c x \left (c^{2} x^{2}+1\right )^{\frac {5}{2}}}{18}+\frac {\arcsinh \left (c x \right ) c x \left (c^{2} x^{2}+1\right )^{\frac {3}{2}}}{18}+\frac {\arcsinh \left (c x \right ) \sqrt {c^{2} x^{2}+1}\, c x}{12}+\frac {\arcsinh \left (c x \right )^{2}}{24}+\frac {\left (c^{2} x^{2}+1\right )^{3}}{108}-\frac {\left (c^{2} x^{2}+1\right )^{2}}{72}-\frac {c^{2} x^{2}}{24}-\frac {1}{24}\right )+2 d a b \left (\frac {\arcsinh \left (c x \right ) c^{6} x^{6}}{6}+\frac {\arcsinh \left (c x \right ) c^{4} x^{4}}{4}-\frac {c^{5} x^{5} \sqrt {c^{2} x^{2}+1}}{36}-\frac {c^{3} x^{3} \sqrt {c^{2} x^{2}+1}}{36}+\frac {c x \sqrt {c^{2} x^{2}+1}}{24}-\frac {\arcsinh \left (c x \right )}{24}\right )}{c^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/c^4*(d*a^2*(1/6*c^6*x^6+1/4*c^4*x^4)+d*b^2*(1/6*arcsinh(c*x)^2*c^2*x^2*(c^2*x^2+1)^2-1/12*arcsinh(c*x)^2*(c^
2*x^2+1)^2-1/18*arcsinh(c*x)*c*x*(c^2*x^2+1)^(5/2)+1/18*arcsinh(c*x)*c*x*(c^2*x^2+1)^(3/2)+1/12*arcsinh(c*x)*(
c^2*x^2+1)^(1/2)*c*x+1/24*arcsinh(c*x)^2+1/108*(c^2*x^2+1)^3-1/72*(c^2*x^2+1)^2-1/24*c^2*x^2-1/24)+2*d*a*b*(1/
6*arcsinh(c*x)*c^6*x^6+1/4*arcsinh(c*x)*c^4*x^4-1/36*c^5*x^5*(c^2*x^2+1)^(1/2)-1/36*c^3*x^3*(c^2*x^2+1)^(1/2)+
1/24*c*x*(c^2*x^2+1)^(1/2)-1/24*arcsinh(c*x)))

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maxima [B]  time = 0.47, size = 442, normalized size = 2.23 \[ \frac {1}{6} \, b^{2} c^{2} d x^{6} \operatorname {arsinh}\left (c x\right )^{2} + \frac {1}{6} \, a^{2} c^{2} d x^{6} + \frac {1}{4} \, b^{2} d x^{4} \operatorname {arsinh}\left (c x\right )^{2} + \frac {1}{4} \, a^{2} d x^{4} + \frac {1}{144} \, {\left (48 \, x^{6} \operatorname {arsinh}\left (c x\right ) - {\left (\frac {8 \, \sqrt {c^{2} x^{2} + 1} x^{5}}{c^{2}} - \frac {10 \, \sqrt {c^{2} x^{2} + 1} x^{3}}{c^{4}} + \frac {15 \, \sqrt {c^{2} x^{2} + 1} x}{c^{6}} - \frac {15 \, \operatorname {arsinh}\left (c x\right )}{c^{7}}\right )} c\right )} a b c^{2} d + \frac {1}{864} \, {\left ({\left (\frac {8 \, x^{6}}{c^{2}} - \frac {15 \, x^{4}}{c^{4}} + \frac {45 \, x^{2}}{c^{6}} - \frac {45 \, \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right )^{2}}{c^{8}}\right )} c^{2} - 6 \, {\left (\frac {8 \, \sqrt {c^{2} x^{2} + 1} x^{5}}{c^{2}} - \frac {10 \, \sqrt {c^{2} x^{2} + 1} x^{3}}{c^{4}} + \frac {15 \, \sqrt {c^{2} x^{2} + 1} x}{c^{6}} - \frac {15 \, \operatorname {arsinh}\left (c x\right )}{c^{7}}\right )} c \operatorname {arsinh}\left (c x\right )\right )} b^{2} c^{2} d + \frac {1}{16} \, {\left (8 \, x^{4} \operatorname {arsinh}\left (c x\right ) - {\left (\frac {2 \, \sqrt {c^{2} x^{2} + 1} x^{3}}{c^{2}} - \frac {3 \, \sqrt {c^{2} x^{2} + 1} x}{c^{4}} + \frac {3 \, \operatorname {arsinh}\left (c x\right )}{c^{5}}\right )} c\right )} a b d + \frac {1}{32} \, {\left ({\left (\frac {x^{4}}{c^{2}} - \frac {3 \, x^{2}}{c^{4}} + \frac {3 \, \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right )^{2}}{c^{6}}\right )} c^{2} - 2 \, {\left (\frac {2 \, \sqrt {c^{2} x^{2} + 1} x^{3}}{c^{2}} - \frac {3 \, \sqrt {c^{2} x^{2} + 1} x}{c^{4}} + \frac {3 \, \operatorname {arsinh}\left (c x\right )}{c^{5}}\right )} c \operatorname {arsinh}\left (c x\right )\right )} b^{2} d \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*b^2*c^2*d*x^6*arcsinh(c*x)^2 + 1/6*a^2*c^2*d*x^6 + 1/4*b^2*d*x^4*arcsinh(c*x)^2 + 1/4*a^2*d*x^4 + 1/144*(4
8*x^6*arcsinh(c*x) - (8*sqrt(c^2*x^2 + 1)*x^5/c^2 - 10*sqrt(c^2*x^2 + 1)*x^3/c^4 + 15*sqrt(c^2*x^2 + 1)*x/c^6
- 15*arcsinh(c*x)/c^7)*c)*a*b*c^2*d + 1/864*((8*x^6/c^2 - 15*x^4/c^4 + 45*x^2/c^6 - 45*log(c*x + sqrt(c^2*x^2
+ 1))^2/c^8)*c^2 - 6*(8*sqrt(c^2*x^2 + 1)*x^5/c^2 - 10*sqrt(c^2*x^2 + 1)*x^3/c^4 + 15*sqrt(c^2*x^2 + 1)*x/c^6
- 15*arcsinh(c*x)/c^7)*c*arcsinh(c*x))*b^2*c^2*d + 1/16*(8*x^4*arcsinh(c*x) - (2*sqrt(c^2*x^2 + 1)*x^3/c^2 - 3
*sqrt(c^2*x^2 + 1)*x/c^4 + 3*arcsinh(c*x)/c^5)*c)*a*b*d + 1/32*((x^4/c^2 - 3*x^2/c^4 + 3*log(c*x + sqrt(c^2*x^
2 + 1))^2/c^6)*c^2 - 2*(2*sqrt(c^2*x^2 + 1)*x^3/c^2 - 3*sqrt(c^2*x^2 + 1)*x/c^4 + 3*arcsinh(c*x)/c^5)*c*arcsin
h(c*x))*b^2*d

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int x^3\,{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2\,\left (d\,c^2\,x^2+d\right ) \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 7.86, size = 332, normalized size = 1.68 \[ \begin {cases} \frac {a^{2} c^{2} d x^{6}}{6} + \frac {a^{2} d x^{4}}{4} + \frac {a b c^{2} d x^{6} \operatorname {asinh}{\left (c x \right )}}{3} - \frac {a b c d x^{5} \sqrt {c^{2} x^{2} + 1}}{18} + \frac {a b d x^{4} \operatorname {asinh}{\left (c x \right )}}{2} - \frac {a b d x^{3} \sqrt {c^{2} x^{2} + 1}}{18 c} + \frac {a b d x \sqrt {c^{2} x^{2} + 1}}{12 c^{3}} - \frac {a b d \operatorname {asinh}{\left (c x \right )}}{12 c^{4}} + \frac {b^{2} c^{2} d x^{6} \operatorname {asinh}^{2}{\left (c x \right )}}{6} + \frac {b^{2} c^{2} d x^{6}}{108} - \frac {b^{2} c d x^{5} \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{18} + \frac {b^{2} d x^{4} \operatorname {asinh}^{2}{\left (c x \right )}}{4} + \frac {b^{2} d x^{4}}{72} - \frac {b^{2} d x^{3} \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{18 c} - \frac {b^{2} d x^{2}}{24 c^{2}} + \frac {b^{2} d x \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{12 c^{3}} - \frac {b^{2} d \operatorname {asinh}^{2}{\left (c x \right )}}{24 c^{4}} & \text {for}\: c \neq 0 \\\frac {a^{2} d x^{4}}{4} & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((a**2*c**2*d*x**6/6 + a**2*d*x**4/4 + a*b*c**2*d*x**6*asinh(c*x)/3 - a*b*c*d*x**5*sqrt(c**2*x**2 + 1
)/18 + a*b*d*x**4*asinh(c*x)/2 - a*b*d*x**3*sqrt(c**2*x**2 + 1)/(18*c) + a*b*d*x*sqrt(c**2*x**2 + 1)/(12*c**3)
- a*b*d*asinh(c*x)/(12*c**4) + b**2*c**2*d*x**6*asinh(c*x)**2/6 + b**2*c**2*d*x**6/108 - b**2*c*d*x**5*sqrt(c
**2*x**2 + 1)*asinh(c*x)/18 + b**2*d*x**4*asinh(c*x)**2/4 + b**2*d*x**4/72 - b**2*d*x**3*sqrt(c**2*x**2 + 1)*a
sinh(c*x)/(18*c) - b**2*d*x**2/(24*c**2) + b**2*d*x*sqrt(c**2*x**2 + 1)*asinh(c*x)/(12*c**3) - b**2*d*asinh(c*
x)**2/(24*c**4), Ne(c, 0)), (a**2*d*x**4/4, True))

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