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

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

[Out]

-(b*(d + e*x)^2*Sqrt[1 + c^2*x^2])/(9*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)

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Rubi [A]  time = 0.0957302, antiderivative size = 124, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {5801, 743, 780, 215} \[ \frac{(d+e x)^3 \left (a+b \sinh ^{-1}(c x)\right )}{3 e}-\frac{b \sqrt{c^2 x^2+1} \left (4 \left (4 c^2 d^2-e^2\right )+5 c^2 d e x\right )}{18 c^3}-\frac{b d \left (2 d^2-\frac{3 e^2}{c^2}\right ) \sinh ^{-1}(c x)}{6 e}-\frac{b \sqrt{c^2 x^2+1} (d+e x)^2}{9 c} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(b*(d + e*x)^2*Sqrt[1 + c^2*x^2])/(9*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 5801

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]

Rule 743

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 780

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 215

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]

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*}

Mathematica [A]  time = 0.0854894, size = 121, normalized size = 0.98 \[ \frac{6 a c^3 x \left (3 d^2+3 d e x+e^2 x^2\right )-b \sqrt{c^2 x^2+1} \left (c^2 \left (18 d^2+9 d e x+2 e^2 x^2\right )-4 e^2\right )+3 b c \sinh ^{-1}(c x) \left (6 c^2 d^2 x+3 d \left (2 c^2 e x^2+e\right )+2 c^2 e^2 x^3\right )}{18 c^3} \]

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 = 0.005, size = 189, normalized size = 1.5 \begin{align*}{\frac{1}{c} \left ({\frac{ \left ( cex+cd \right ) ^{3}a}{3\,{c}^{2}e}}+{\frac{b}{{c}^{2}} \left ({\frac{{e}^{2}{\it Arcsinh} \left ( cx \right ){c}^{3}{x}^{3}}{3}}+e{\it Arcsinh} \left ( cx \right ){c}^{3}{x}^{2}d+{\it Arcsinh} \left ( cx \right ){c}^{3}x{d}^{2}+{\frac{{c}^{3}{d}^{3}{\it Arcsinh} \left ( cx \right ) }{3\,e}}-{\frac{1}{3\,e} \left ({e}^{3} \left ({\frac{{c}^{2}{x}^{2}}{3}\sqrt{{c}^{2}{x}^{2}+1}}-{\frac{2}{3}\sqrt{{c}^{2}{x}^{2}+1}} \right ) +3\,cd{e}^{2} \left ( 1/2\,cx\sqrt{{c}^{2}{x}^{2}+1}-1/2\,{\it Arcsinh} \left ( cx \right ) \right ) +3\,{c}^{2}{d}^{2}e\sqrt{{c}^{2}{x}^{2}+1}+{c}^{3}{d}^{3}{\it Arcsinh} \left ( cx \right ) \right ) } \right ) } \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.05731, size = 219, normalized size = 1.77 \begin{align*} \frac{1}{3} \, a e^{2} x^{3} + a d e x^{2} + \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 (\frac{c^{2} x}{\sqrt{c^{2}}}\right )}{\sqrt{c^{2}} c^{2}}\right )}\right )} b d e + \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} + a d^{2} x + \frac{{\left (c x \operatorname{arsinh}\left (c x\right ) - \sqrt{c^{2} x^{2} + 1}\right )} b d^{2}}{c} \end{align*}

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

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Fricas [A]  time = 2.54437, size = 324, normalized size = 2.61 \begin{align*} \frac{6 \, a c^{3} e^{2} x^{3} + 18 \, a c^{3} d e x^{2} + 18 \, a c^{3} d^{2} x + 3 \,{\left (2 \, b c^{3} e^{2} x^{3} + 6 \, b c^{3} d e x^{2} + 6 \, b c^{3} d^{2} x + 3 \, b c d e\right )} \log \left (c x + \sqrt{c^{2} x^{2} + 1}\right ) -{\left (2 \, b c^{2} e^{2} x^{2} + 9 \, b c^{2} d e x + 18 \, b c^{2} d^{2} - 4 \, b e^{2}\right )} \sqrt{c^{2} x^{2} + 1}}{18 \, c^{3}} \end{align*}

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

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Sympy [A]  time = 0.963745, size = 190, normalized size = 1.53 \begin{align*} \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{align*}

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 [A]  time = 1.56095, size = 265, normalized size = 2.14 \begin{align*}{\left (x \log \left (c x + \sqrt{c^{2} x^{2} + 1}\right ) - \frac{\sqrt{c^{2} x^{2} + 1}}{c}\right )} b d^{2} + a d^{2} x + \frac{1}{9} \,{\left (3 \, a x^{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\right )} e^{2} + \frac{1}{2} \,{\left (2 \, a d x^{2} +{\left (2 \, x^{2} \log \left (c x + \sqrt{c^{2} x^{2} + 1}\right ) - c{\left (\frac{\sqrt{c^{2} x^{2} + 1} x}{c^{2}} + \frac{\log \left ({\left | -x{\left | c \right |} + \sqrt{c^{2} x^{2} + 1} \right |}\right )}{c^{2}{\left | c \right |}}\right )}\right )} b d\right )} e \end{align*}

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]

(x*log(c*x + sqrt(c^2*x^2 + 1)) - sqrt(c^2*x^2 + 1)/c)*b*d^2 + a*d^2*x + 1/9*(3*a*x^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)*e^2 + 1/2*(2*a*d*x^2 + (2*x^2*log(c*x + sq
rt(c^2*x^2 + 1)) - c*(sqrt(c^2*x^2 + 1)*x/c^2 + log(abs(-x*abs(c) + sqrt(c^2*x^2 + 1)))/(c^2*abs(c))))*b*d)*e