3.1.5 \(\int (d+e x)^2 (a+b \sinh ^{-1}(c x)) \, dx\) [5]

Optimal. Leaf size=124 \[ -\frac {b (d+e x)^2 \sqrt {1+c^2 x^2}}{9 c}-\frac {b \left (4 \left (4 c^2 d^2-e^2\right )+5 c^2 d e x\right ) \sqrt {1+c^2 x^2}}{18 c^3}-\frac {b d \left (2 d^2-\frac {3 e^2}{c^2}\right ) \sinh ^{-1}(c x)}{6 e}+\frac {(d+e x)^3 \left (a+b \sinh ^{-1}(c x)\right )}{3 e} \]

[Out]

-1/6*b*d*(2*d^2-3*e^2/c^2)*arcsinh(c*x)/e+1/3*(e*x+d)^3*(a+b*arcsinh(c*x))/e-1/9*b*(e*x+d)^2*(c^2*x^2+1)^(1/2)
/c-1/18*b*(5*c^2*d*e*x+16*c^2*d^2-4*e^2)*(c^2*x^2+1)^(1/2)/c^3

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Rubi [A]
time = 0.06, antiderivative size = 124, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {5828, 757, 794, 221} \begin {gather*} \frac {(d+e x)^3 \left (a+b \sinh ^{-1}(c x)\right )}{3 e}-\frac {b d \left (2 d^2-\frac {3 e^2}{c^2}\right ) \sinh ^{-1}(c x)}{6 e}-\frac {b \sqrt {c^2 x^2+1} (d+e x)^2}{9 c}-\frac {b \sqrt {c^2 x^2+1} \left (4 \left (4 c^2 d^2-e^2\right )+5 c^2 d e x\right )}{18 c^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/9*(b*(d + e*x)^2*Sqrt[1 + c^2*x^2])/c - (b*(4*(4*c^2*d^2 - e^2) + 5*c^2*d*e*x)*Sqrt[1 + c^2*x^2])/(18*c^3)
- (b*d*(2*d^2 - (3*e^2)/c^2)*ArcSinh[c*x])/(6*e) + ((d + e*x)^3*(a + b*ArcSinh[c*x]))/(3*e)

Rule 221

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSinh[Rt[b, 2]*(x/Sqrt[a])]/Rt[b, 2], x] /; FreeQ[{a, b},
x] && GtQ[a, 0] && PosQ[b]

Rule 757

Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[e*(d + e*x)^(m - 1)*((a + c*x^2)^(p
+ 1)/(c*(m + 2*p + 1))), x] + Dist[1/(c*(m + 2*p + 1)), Int[(d + e*x)^(m - 2)*Simp[c*d^2*(m + 2*p + 1) - a*e^
2*(m - 1) + 2*c*d*e*(m + p)*x, x]*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, m, p}, x] && NeQ[c*d^2 + a*e^2,
0] && If[RationalQ[m], GtQ[m, 1], SumSimplerQ[m, -2]] && NeQ[m + 2*p + 1, 0] && IntQuadraticQ[a, 0, c, d, e, m
, p, x]

Rule 794

Int[((d_.) + (e_.)*(x_))*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((e*f + d*g)*(2*p
+ 3) + 2*e*g*(p + 1)*x)*((a + c*x^2)^(p + 1)/(2*c*(p + 1)*(2*p + 3))), x] - Dist[(a*e*g - c*d*f*(2*p + 3))/(c*
(2*p + 3)), Int[(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g, p}, x] &&  !LeQ[p, -1]

Rule 5828

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

Rubi steps

\begin {align*} \int (d+e x)^2 \left (a+b \sinh ^{-1}(c x)\right ) \, dx &=\frac {(d+e x)^3 \left (a+b \sinh ^{-1}(c x)\right )}{3 e}-\frac {(b c) \int \frac {(d+e x)^3}{\sqrt {1+c^2 x^2}} \, dx}{3 e}\\ &=-\frac {b (d+e x)^2 \sqrt {1+c^2 x^2}}{9 c}+\frac {(d+e x)^3 \left (a+b \sinh ^{-1}(c x)\right )}{3 e}-\frac {b \int \frac {(d+e x) \left (3 c^2 d^2-2 e^2+5 c^2 d e x\right )}{\sqrt {1+c^2 x^2}} \, dx}{9 c e}\\ &=-\frac {b (d+e x)^2 \sqrt {1+c^2 x^2}}{9 c}-\frac {b \left (4 \left (4 c^2 d^2-e^2\right )+5 c^2 d e x\right ) \sqrt {1+c^2 x^2}}{18 c^3}+\frac {(d+e x)^3 \left (a+b \sinh ^{-1}(c x)\right )}{3 e}-\frac {1}{6} \left (b d \left (\frac {2 c d^2}{e}-\frac {3 e}{c}\right )\right ) \int \frac {1}{\sqrt {1+c^2 x^2}} \, dx\\ &=-\frac {b (d+e x)^2 \sqrt {1+c^2 x^2}}{9 c}-\frac {b \left (4 \left (4 c^2 d^2-e^2\right )+5 c^2 d e x\right ) \sqrt {1+c^2 x^2}}{18 c^3}-\frac {b d \left (2 d^2-\frac {3 e^2}{c^2}\right ) \sinh ^{-1}(c x)}{6 e}+\frac {(d+e x)^3 \left (a+b \sinh ^{-1}(c x)\right )}{3 e}\\ \end {align*}

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Mathematica [A]
time = 0.07, size = 121, normalized size = 0.98 \begin {gather*} \frac {6 a c^3 x \left (3 d^2+3 d e x+e^2 x^2\right )-b \sqrt {1+c^2 x^2} \left (-4 e^2+c^2 \left (18 d^2+9 d e x+2 e^2 x^2\right )\right )+3 b c \left (6 c^2 d^2 x+2 c^2 e^2 x^3+3 d \left (e+2 c^2 e x^2\right )\right ) \sinh ^{-1}(c x)}{18 c^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]
time = 0.67, size = 189, normalized size = 1.52

method	result	size
derivativedivides	\(\frac {\frac {\left (c e x +c d \right )^{3} a}{3 c^{2} e}+\frac {b \left (\frac {\arcsinh \left (c x \right ) c^{3} d^{3}}{3 e}+\arcsinh \left (c x \right ) c^{3} d^{2} x +e \arcsinh \left (c x \right ) c^{3} d \,x^{2}+\frac {e^{2} \arcsinh \left (c x \right ) c^{3} x^{3}}{3}-\frac {c^{3} d^{3} \arcsinh \left (c x \right )+3 d^{2} c^{2} e \sqrt {c^{2} x^{2}+1}+3 d c \,e^{2} \left (\frac {\sqrt {c^{2} x^{2}+1}\, c x}{2}-\frac {\arcsinh \left (c x \right )}{2}\right )+e^{3} \left (\frac {c^{2} x^{2} \sqrt {c^{2} x^{2}+1}}{3}-\frac {2 \sqrt {c^{2} x^{2}+1}}{3}\right )}{3 e}\right )}{c^{2}}}{c}\)	\(189\)
default	\(\frac {\frac {\left (c e x +c d \right )^{3} a}{3 c^{2} e}+\frac {b \left (\frac {\arcsinh \left (c x \right ) c^{3} d^{3}}{3 e}+\arcsinh \left (c x \right ) c^{3} d^{2} x +e \arcsinh \left (c x \right ) c^{3} d \,x^{2}+\frac {e^{2} \arcsinh \left (c x \right ) c^{3} x^{3}}{3}-\frac {c^{3} d^{3} \arcsinh \left (c x \right )+3 d^{2} c^{2} e \sqrt {c^{2} x^{2}+1}+3 d c \,e^{2} \left (\frac {\sqrt {c^{2} x^{2}+1}\, c x}{2}-\frac {\arcsinh \left (c x \right )}{2}\right )+e^{3} \left (\frac {c^{2} x^{2} \sqrt {c^{2} x^{2}+1}}{3}-\frac {2 \sqrt {c^{2} x^{2}+1}}{3}\right )}{3 e}\right )}{c^{2}}}{c}\)	\(189\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.26, size = 150, normalized size = 1.21 \begin {gather*} \frac {1}{3} \, a x^{3} e^{2} + a d x^{2} e + a d^{2} x + \frac {1}{2} \, {\left (2 \, x^{2} \operatorname {arsinh}\left (c x\right ) - c {\left (\frac {\sqrt {c^{2} x^{2} + 1} x}{c^{2}} - \frac {\operatorname {arsinh}\left (c x\right )}{c^{3}}\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}{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 e^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 281 vs. \(2 (108) = 216\).
time = 0.40, size = 281, normalized size = 2.27 \begin {gather*} \frac {6 \, a c^{3} x^{3} \cosh \left (1\right )^{2} + 6 \, a c^{3} x^{3} \sinh \left (1\right )^{2} + 18 \, a c^{3} d x^{2} \cosh \left (1\right ) + 18 \, a c^{3} d^{2} x + 3 \, {\left (2 \, b c^{3} x^{3} \cosh \left (1\right )^{2} + 2 \, b c^{3} x^{3} \sinh \left (1\right )^{2} + 6 \, b c^{3} d^{2} x + 3 \, {\left (2 \, b c^{3} d x^{2} + b c d\right )} \cosh \left (1\right ) + {\left (4 \, b c^{3} x^{3} \cosh \left (1\right ) + 6 \, b c^{3} d x^{2} + 3 \, b c d\right )} \sinh \left (1\right )\right )} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right ) + 6 \, {\left (2 \, a c^{3} x^{3} \cosh \left (1\right ) + 3 \, a c^{3} d x^{2}\right )} \sinh \left (1\right ) - {\left (9 \, b c^{2} d x \cosh \left (1\right ) + 18 \, b c^{2} d^{2} + 2 \, {\left (b c^{2} x^{2} - 2 \, b\right )} \cosh \left (1\right )^{2} + 2 \, {\left (b c^{2} x^{2} - 2 \, b\right )} \sinh \left (1\right )^{2} + {\left (9 \, b c^{2} d x + 4 \, {\left (b c^{2} x^{2} - 2 \, b\right )} \cosh \left (1\right )\right )} \sinh \left (1\right )\right )} \sqrt {c^{2} x^{2} + 1}}{18 \, c^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/18*(6*a*c^3*x^3*cosh(1)^2 + 6*a*c^3*x^3*sinh(1)^2 + 18*a*c^3*d*x^2*cosh(1) + 18*a*c^3*d^2*x + 3*(2*b*c^3*x^3
*cosh(1)^2 + 2*b*c^3*x^3*sinh(1)^2 + 6*b*c^3*d^2*x + 3*(2*b*c^3*d*x^2 + b*c*d)*cosh(1) + (4*b*c^3*x^3*cosh(1)
+ 6*b*c^3*d*x^2 + 3*b*c*d)*sinh(1))*log(c*x + sqrt(c^2*x^2 + 1)) + 6*(2*a*c^3*x^3*cosh(1) + 3*a*c^3*d*x^2)*sin
h(1) - (9*b*c^2*d*x*cosh(1) + 18*b*c^2*d^2 + 2*(b*c^2*x^2 - 2*b)*cosh(1)^2 + 2*(b*c^2*x^2 - 2*b)*sinh(1)^2 + (
9*b*c^2*d*x + 4*(b*c^2*x^2 - 2*b)*cosh(1))*sinh(1))*sqrt(c^2*x^2 + 1))/c^3

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Sympy [A]
time = 0.17, size = 190, normalized size = 1.53 \begin {gather*} \begin {cases} a d^{2} x + a d e x^{2} + \frac {a e^{2} x^{3}}{3} + b d^{2} x \operatorname {asinh}{\left (c x \right )} + b d e x^{2} \operatorname {asinh}{\left (c x \right )} + \frac {b e^{2} x^{3} \operatorname {asinh}{\left (c x \right )}}{3} - \frac {b d^{2} \sqrt {c^{2} x^{2} + 1}}{c} - \frac {b d e x \sqrt {c^{2} x^{2} + 1}}{2 c} - \frac {b e^{2} x^{2} \sqrt {c^{2} x^{2} + 1}}{9 c} + \frac {b d e \operatorname {asinh}{\left (c x \right )}}{2 c^{2}} + \frac {2 b e^{2} \sqrt {c^{2} x^{2} + 1}}{9 c^{3}} & \text {for}\: c \neq 0 \\a \left (d^{2} x + d e x^{2} + \frac {e^{2} x^{3}}{3}\right ) & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((a*d**2*x + a*d*e*x**2 + a*e**2*x**3/3 + b*d**2*x*asinh(c*x) + b*d*e*x**2*asinh(c*x) + b*e**2*x**3*a
sinh(c*x)/3 - b*d**2*sqrt(c**2*x**2 + 1)/c - b*d*e*x*sqrt(c**2*x**2 + 1)/(2*c) - b*e**2*x**2*sqrt(c**2*x**2 +
1)/(9*c) + b*d*e*asinh(c*x)/(2*c**2) + 2*b*e**2*sqrt(c**2*x**2 + 1)/(9*c**3), Ne(c, 0)), (a*(d**2*x + d*e*x**2
+ e**2*x**3/3), 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+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 (d+e\,x\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,x)

[Out]

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

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