3.7.8 \(\int (d+e x^2)^2 (a+b \sinh ^{-1}(c x)) \, dx\) [608]

Optimal. Leaf size=147 \[ -\frac {b \left (15 c^4 d^2-10 c^2 d e+3 e^2\right ) \sqrt {1+c^2 x^2}}{15 c^5}-\frac {2 b \left (5 c^2 d-3 e\right ) e \left (1+c^2 x^2\right )^{3/2}}{45 c^5}-\frac {b e^2 \left (1+c^2 x^2\right )^{5/2}}{25 c^5}+d^2 x \left (a+b \sinh ^{-1}(c x)\right )+\frac {2}{3} d e x^3 \left (a+b \sinh ^{-1}(c x)\right )+\frac {1}{5} e^2 x^5 \left (a+b \sinh ^{-1}(c x)\right ) \]

[Out]

-2/45*b*(5*c^2*d-3*e)*e*(c^2*x^2+1)^(3/2)/c^5-1/25*b*e^2*(c^2*x^2+1)^(5/2)/c^5+d^2*x*(a+b*arcsinh(c*x))+2/3*d*
e*x^3*(a+b*arcsinh(c*x))+1/5*e^2*x^5*(a+b*arcsinh(c*x))-1/15*b*(15*c^4*d^2-10*c^2*d*e+3*e^2)*(c^2*x^2+1)^(1/2)
/c^5

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Rubi [A]
time = 0.10, antiderivative size = 147, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.278, Rules used = {200, 5792, 12, 1261, 712} \begin {gather*} d^2 x \left (a+b \sinh ^{-1}(c x)\right )+\frac {2}{3} d e x^3 \left (a+b \sinh ^{-1}(c x)\right )+\frac {1}{5} e^2 x^5 \left (a+b \sinh ^{-1}(c x)\right )-\frac {2 b e \left (c^2 x^2+1\right )^{3/2} \left (5 c^2 d-3 e\right )}{45 c^5}-\frac {b e^2 \left (c^2 x^2+1\right )^{5/2}}{25 c^5}-\frac {b \sqrt {c^2 x^2+1} \left (15 c^4 d^2-10 c^2 d e+3 e^2\right )}{15 c^5} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/15*(b*(15*c^4*d^2 - 10*c^2*d*e + 3*e^2)*Sqrt[1 + c^2*x^2])/c^5 - (2*b*(5*c^2*d - 3*e)*e*(1 + c^2*x^2)^(3/2)
)/(45*c^5) - (b*e^2*(1 + c^2*x^2)^(5/2))/(25*c^5) + d^2*x*(a + b*ArcSinh[c*x]) + (2*d*e*x^3*(a + b*ArcSinh[c*x
]))/3 + (e^2*x^5*(a + b*ArcSinh[c*x]))/5

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 712

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d +
e*x)^m*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*
e + a*e^2, 0] && NeQ[2*c*d - b*e, 0] && IntegerQ[p] && (GtQ[p, 0] || (EqQ[a, 0] && IntegerQ[m]))

Rule 1261

Int[(x_)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Dist[1/2, Subst[
Int[(d + e*x)^q*(a + b*x + c*x^2)^p, x], x, x^2], x] /; FreeQ[{a, b, c, d, e, p, q}, x]

Rule 5792

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] && NeQ[e, c^2*d] && (IGtQ[p, 0] || ILtQ[p + 1/2, 0])

Rubi steps

\begin {align*} \int \left (d+e x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right ) \, dx &=d^2 x \left (a+b \sinh ^{-1}(c x)\right )+\frac {2}{3} d e x^3 \left (a+b \sinh ^{-1}(c x)\right )+\frac {1}{5} e^2 x^5 \left (a+b \sinh ^{-1}(c x)\right )-(b c) \int \frac {x \left (15 d^2+10 d e x^2+3 e^2 x^4\right )}{15 \sqrt {1+c^2 x^2}} \, dx\\ &=d^2 x \left (a+b \sinh ^{-1}(c x)\right )+\frac {2}{3} d e x^3 \left (a+b \sinh ^{-1}(c x)\right )+\frac {1}{5} e^2 x^5 \left (a+b \sinh ^{-1}(c x)\right )-\frac {1}{15} (b c) \int \frac {x \left (15 d^2+10 d e x^2+3 e^2 x^4\right )}{\sqrt {1+c^2 x^2}} \, dx\\ &=d^2 x \left (a+b \sinh ^{-1}(c x)\right )+\frac {2}{3} d e x^3 \left (a+b \sinh ^{-1}(c x)\right )+\frac {1}{5} e^2 x^5 \left (a+b \sinh ^{-1}(c x)\right )-\frac {1}{30} (b c) \text {Subst}\left (\int \frac {15 d^2+10 d e x+3 e^2 x^2}{\sqrt {1+c^2 x}} \, dx,x,x^2\right )\\ &=d^2 x \left (a+b \sinh ^{-1}(c x)\right )+\frac {2}{3} d e x^3 \left (a+b \sinh ^{-1}(c x)\right )+\frac {1}{5} e^2 x^5 \left (a+b \sinh ^{-1}(c x)\right )-\frac {1}{30} (b c) \text {Subst}\left (\int \left (\frac {15 c^4 d^2-10 c^2 d e+3 e^2}{c^4 \sqrt {1+c^2 x}}+\frac {2 \left (5 c^2 d-3 e\right ) e \sqrt {1+c^2 x}}{c^4}+\frac {3 e^2 \left (1+c^2 x\right )^{3/2}}{c^4}\right ) \, dx,x,x^2\right )\\ &=-\frac {b \left (15 c^4 d^2-10 c^2 d e+3 e^2\right ) \sqrt {1+c^2 x^2}}{15 c^5}-\frac {2 b \left (5 c^2 d-3 e\right ) e \left (1+c^2 x^2\right )^{3/2}}{45 c^5}-\frac {b e^2 \left (1+c^2 x^2\right )^{5/2}}{25 c^5}+d^2 x \left (a+b \sinh ^{-1}(c x)\right )+\frac {2}{3} d e x^3 \left (a+b \sinh ^{-1}(c x)\right )+\frac {1}{5} e^2 x^5 \left (a+b \sinh ^{-1}(c x)\right )\\ \end {align*}

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Mathematica [A]
time = 0.10, size = 125, normalized size = 0.85 \begin {gather*} \frac {1}{225} \left (15 a x \left (15 d^2+10 d e x^2+3 e^2 x^4\right )-\frac {b \sqrt {1+c^2 x^2} \left (24 e^2-4 c^2 e \left (25 d+3 e x^2\right )+c^4 \left (225 d^2+50 d e x^2+9 e^2 x^4\right )\right )}{c^5}+15 b x \left (15 d^2+10 d e x^2+3 e^2 x^4\right ) \sinh ^{-1}(c x)\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(15*a*x*(15*d^2 + 10*d*e*x^2 + 3*e^2*x^4) - (b*Sqrt[1 + c^2*x^2]*(24*e^2 - 4*c^2*e*(25*d + 3*e*x^2) + c^4*(225
*d^2 + 50*d*e*x^2 + 9*e^2*x^4)))/c^5 + 15*b*x*(15*d^2 + 10*d*e*x^2 + 3*e^2*x^4)*ArcSinh[c*x])/225

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Maple [A]
time = 0.64, size = 204, normalized size = 1.39

method	result	size
derivativedivides	\(\frac {\frac {a \left (d^{2} c^{5} x +\frac {2}{3} d \,c^{5} e \,x^{3}+\frac {1}{5} e^{2} c^{5} x^{5}\right )}{c^{4}}+\frac {b \left (\arcsinh \left (c x \right ) d^{2} c^{5} x +\frac {2 \arcsinh \left (c x \right ) d \,c^{5} e \,x^{3}}{3}+\frac {\arcsinh \left (c x \right ) e^{2} c^{5} x^{5}}{5}-d^{2} c^{4} \sqrt {c^{2} x^{2}+1}-\frac {2 d \,c^{2} e \left (\frac {c^{2} x^{2} \sqrt {c^{2} x^{2}+1}}{3}-\frac {2 \sqrt {c^{2} x^{2}+1}}{3}\right )}{3}-\frac {e^{2} \left (\frac {\sqrt {c^{2} x^{2}+1}\, c^{4} x^{4}}{5}-\frac {4 c^{2} x^{2} \sqrt {c^{2} x^{2}+1}}{15}+\frac {8 \sqrt {c^{2} x^{2}+1}}{15}\right )}{5}\right )}{c^{4}}}{c}\)	\(204\)
default	\(\frac {\frac {a \left (d^{2} c^{5} x +\frac {2}{3} d \,c^{5} e \,x^{3}+\frac {1}{5} e^{2} c^{5} x^{5}\right )}{c^{4}}+\frac {b \left (\arcsinh \left (c x \right ) d^{2} c^{5} x +\frac {2 \arcsinh \left (c x \right ) d \,c^{5} e \,x^{3}}{3}+\frac {\arcsinh \left (c x \right ) e^{2} c^{5} x^{5}}{5}-d^{2} c^{4} \sqrt {c^{2} x^{2}+1}-\frac {2 d \,c^{2} e \left (\frac {c^{2} x^{2} \sqrt {c^{2} x^{2}+1}}{3}-\frac {2 \sqrt {c^{2} x^{2}+1}}{3}\right )}{3}-\frac {e^{2} \left (\frac {\sqrt {c^{2} x^{2}+1}\, c^{4} x^{4}}{5}-\frac {4 c^{2} x^{2} \sqrt {c^{2} x^{2}+1}}{15}+\frac {8 \sqrt {c^{2} x^{2}+1}}{15}\right )}{5}\right )}{c^{4}}}{c}\)	\(204\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/c*(a/c^4*(d^2*c^5*x+2/3*d*c^5*e*x^3+1/5*e^2*c^5*x^5)+b/c^4*(arcsinh(c*x)*d^2*c^5*x+2/3*arcsinh(c*x)*d*c^5*e*
x^3+1/5*arcsinh(c*x)*e^2*c^5*x^5-d^2*c^4*(c^2*x^2+1)^(1/2)-2/3*d*c^2*e*(1/3*c^2*x^2*(c^2*x^2+1)^(1/2)-2/3*(c^2
*x^2+1)^(1/2))-1/5*e^2*(1/5*(c^2*x^2+1)^(1/2)*c^4*x^4-4/15*c^2*x^2*(c^2*x^2+1)^(1/2)+8/15*(c^2*x^2+1)^(1/2))))

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Maxima [A]
time = 0.27, size = 180, normalized size = 1.22 \begin {gather*} \frac {1}{5} \, a x^{5} e^{2} + \frac {2}{3} \, a d x^{3} e + a d^{2} x + \frac {2}{9} \, {\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 d e + \frac {{\left (c x \operatorname {arsinh}\left (c x\right ) - \sqrt {c^{2} x^{2} + 1}\right )} b d^{2}}{c} + \frac {1}{75} \, {\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 e^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/5*a*x^5*e^2 + 2/3*a*d*x^3*e + a*d^2*x + 2/9*(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*d*e + (c*x*arcsinh(c*x) - sqrt(c^2*x^2 + 1))*b*d^2/c + 1/75*(15*x^5*arcsinh(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*e^2

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 321 vs. \(2 (132) = 264\).
time = 0.40, size = 321, normalized size = 2.18 \begin {gather*} \frac {45 \, a c^{5} x^{5} \cosh \left (1\right )^{2} + 45 \, a c^{5} x^{5} \sinh \left (1\right )^{2} + 150 \, a c^{5} d x^{3} \cosh \left (1\right ) + 225 \, a c^{5} d^{2} x + 15 \, {\left (3 \, b c^{5} x^{5} \cosh \left (1\right )^{2} + 3 \, b c^{5} x^{5} \sinh \left (1\right )^{2} + 10 \, b c^{5} d x^{3} \cosh \left (1\right ) + 15 \, b c^{5} d^{2} x + 2 \, {\left (3 \, b c^{5} x^{5} \cosh \left (1\right ) + 5 \, b c^{5} d x^{3}\right )} \sinh \left (1\right )\right )} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right ) + 30 \, {\left (3 \, a c^{5} x^{5} \cosh \left (1\right ) + 5 \, a c^{5} d x^{3}\right )} \sinh \left (1\right ) - {\left (225 \, b c^{4} d^{2} + 3 \, {\left (3 \, b c^{4} x^{4} - 4 \, b c^{2} x^{2} + 8 \, b\right )} \cosh \left (1\right )^{2} + 3 \, {\left (3 \, b c^{4} x^{4} - 4 \, b c^{2} x^{2} + 8 \, b\right )} \sinh \left (1\right )^{2} + 50 \, {\left (b c^{4} d x^{2} - 2 \, b c^{2} d\right )} \cosh \left (1\right ) + 2 \, {\left (25 \, b c^{4} d x^{2} - 50 \, b c^{2} d + 3 \, {\left (3 \, b c^{4} x^{4} - 4 \, b c^{2} x^{2} + 8 \, b\right )} \cosh \left (1\right )\right )} \sinh \left (1\right )\right )} \sqrt {c^{2} x^{2} + 1}}{225 \, c^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/225*(45*a*c^5*x^5*cosh(1)^2 + 45*a*c^5*x^5*sinh(1)^2 + 150*a*c^5*d*x^3*cosh(1) + 225*a*c^5*d^2*x + 15*(3*b*c
^5*x^5*cosh(1)^2 + 3*b*c^5*x^5*sinh(1)^2 + 10*b*c^5*d*x^3*cosh(1) + 15*b*c^5*d^2*x + 2*(3*b*c^5*x^5*cosh(1) +
5*b*c^5*d*x^3)*sinh(1))*log(c*x + sqrt(c^2*x^2 + 1)) + 30*(3*a*c^5*x^5*cosh(1) + 5*a*c^5*d*x^3)*sinh(1) - (225
*b*c^4*d^2 + 3*(3*b*c^4*x^4 - 4*b*c^2*x^2 + 8*b)*cosh(1)^2 + 3*(3*b*c^4*x^4 - 4*b*c^2*x^2 + 8*b)*sinh(1)^2 + 5
0*(b*c^4*d*x^2 - 2*b*c^2*d)*cosh(1) + 2*(25*b*c^4*d*x^2 - 50*b*c^2*d + 3*(3*b*c^4*x^4 - 4*b*c^2*x^2 + 8*b)*cos
h(1))*sinh(1))*sqrt(c^2*x^2 + 1))/c^5

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Sympy [A]
time = 0.36, size = 240, normalized size = 1.63 \begin {gather*} \begin {cases} a d^{2} x + \frac {2 a d e x^{3}}{3} + \frac {a e^{2} x^{5}}{5} + b d^{2} x \operatorname {asinh}{\left (c x \right )} + \frac {2 b d e x^{3} \operatorname {asinh}{\left (c x \right )}}{3} + \frac {b e^{2} x^{5} \operatorname {asinh}{\left (c x \right )}}{5} - \frac {b d^{2} \sqrt {c^{2} x^{2} + 1}}{c} - \frac {2 b d e x^{2} \sqrt {c^{2} x^{2} + 1}}{9 c} - \frac {b e^{2} x^{4} \sqrt {c^{2} x^{2} + 1}}{25 c} + \frac {4 b d e \sqrt {c^{2} x^{2} + 1}}{9 c^{3}} + \frac {4 b e^{2} x^{2} \sqrt {c^{2} x^{2} + 1}}{75 c^{3}} - \frac {8 b e^{2} \sqrt {c^{2} x^{2} + 1}}{75 c^{5}} & \text {for}\: c \neq 0 \\a \left (d^{2} x + \frac {2 d e x^{3}}{3} + \frac {e^{2} x^{5}}{5}\right ) & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((a*d**2*x + 2*a*d*e*x**3/3 + a*e**2*x**5/5 + b*d**2*x*asinh(c*x) + 2*b*d*e*x**3*asinh(c*x)/3 + b*e**
2*x**5*asinh(c*x)/5 - b*d**2*sqrt(c**2*x**2 + 1)/c - 2*b*d*e*x**2*sqrt(c**2*x**2 + 1)/(9*c) - b*e**2*x**4*sqrt
(c**2*x**2 + 1)/(25*c) + 4*b*d*e*sqrt(c**2*x**2 + 1)/(9*c**3) + 4*b*e**2*x**2*sqrt(c**2*x**2 + 1)/(75*c**3) -
8*b*e**2*sqrt(c**2*x**2 + 1)/(75*c**5), Ne(c, 0)), (a*(d**2*x + 2*d*e*x**3/3 + e**2*x**5/5), True))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: RuntimeError >> An error occurred running a Giac command:INPUT:sage2OUTPUT:sym2poly/r2sym(co
nst 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 = -0.01 \begin {gather*} \int \left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )\,{\left (e\,x^2+d\right )}^2 \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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