3.193 \(\int x^m \sqrt{d+c^2 d x^2} (a+b \sinh ^{-1}(c x)) \, dx\)

Optimal. Leaf size=240 \[ -\frac{b c x^{m+2} \sqrt{c^2 d x^2+d} \text{HypergeometricPFQ}\left (\left \{1,\frac{m}{2}+1,\frac{m}{2}+1\right \},\left \{\frac{m}{2}+\frac{3}{2},\frac{m}{2}+2\right \},-c^2 x^2\right )}{(m+1) (m+2)^2 \sqrt{c^2 x^2+1}}+\frac{x^{m+1} \sqrt{c^2 d x^2+d} \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{m+1}{2},\frac{m+3}{2},-c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )}{\left (m^2+3 m+2\right ) \sqrt{c^2 x^2+1}}+\frac{x^{m+1} \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )}{m+2}-\frac{b c x^{m+2} \sqrt{c^2 d x^2+d}}{(m+2)^2 \sqrt{c^2 x^2+1}} \]

[Out]

-((b*c*x^(2 + m)*Sqrt[d + c^2*d*x^2])/((2 + m)^2*Sqrt[1 + c^2*x^2])) + (x^(1 + m)*Sqrt[d + c^2*d*x^2]*(a + b*A
rcSinh[c*x]))/(2 + m) + (x^(1 + m)*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])*Hypergeometric2F1[1/2, (1 + m)/2,
(3 + m)/2, -(c^2*x^2)])/((2 + 3*m + m^2)*Sqrt[1 + c^2*x^2]) - (b*c*x^(2 + m)*Sqrt[d + c^2*d*x^2]*Hypergeometri
cPFQ[{1, 1 + m/2, 1 + m/2}, {3/2 + m/2, 2 + m/2}, -(c^2*x^2)])/((1 + m)*(2 + m)^2*Sqrt[1 + c^2*x^2])

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Rubi [A]  time = 0.202771, antiderivative size = 240, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.115, Rules used = {5742, 5762, 30} \[ -\frac{b c x^{m+2} \sqrt{c^2 d x^2+d} \, _3F_2\left (1,\frac{m}{2}+1,\frac{m}{2}+1;\frac{m}{2}+\frac{3}{2},\frac{m}{2}+2;-c^2 x^2\right )}{(m+1) (m+2)^2 \sqrt{c^2 x^2+1}}+\frac{x^{m+1} \sqrt{c^2 d x^2+d} \, _2F_1\left (\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};-c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )}{\left (m^2+3 m+2\right ) \sqrt{c^2 x^2+1}}+\frac{x^{m+1} \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )}{m+2}-\frac{b c x^{m+2} \sqrt{c^2 d x^2+d}}{(m+2)^2 \sqrt{c^2 x^2+1}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((b*c*x^(2 + m)*Sqrt[d + c^2*d*x^2])/((2 + m)^2*Sqrt[1 + c^2*x^2])) + (x^(1 + m)*Sqrt[d + c^2*d*x^2]*(a + b*A
rcSinh[c*x]))/(2 + m) + (x^(1 + m)*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])*Hypergeometric2F1[1/2, (1 + m)/2,
(3 + m)/2, -(c^2*x^2)])/((2 + 3*m + m^2)*Sqrt[1 + c^2*x^2]) - (b*c*x^(2 + m)*Sqrt[d + c^2*d*x^2]*Hypergeometri
cPFQ[{1, 1 + m/2, 1 + m/2}, {3/2 + m/2, 2 + m/2}, -(c^2*x^2)])/((1 + m)*(2 + m)^2*Sqrt[1 + c^2*x^2])

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 5762

Int[(((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))*((f_.)*(x_))^(m_))/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[((f*x
)^(m + 1)*(a + b*ArcSinh[c*x])*Hypergeometric2F1[1/2, (1 + m)/2, (3 + m)/2, -(c^2*x^2)])/(Sqrt[d]*f*(m + 1)),
x] - Simp[(b*c*(f*x)^(m + 2)*HypergeometricPFQ[{1, 1 + m/2, 1 + m/2}, {3/2 + m/2, 2 + m/2}, -(c^2*x^2)])/(Sqrt
[d]*f^2*(m + 1)*(m + 2)), x] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[e, c^2*d] && GtQ[d, 0] &&  !IntegerQ[m]

Rule 30

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

Rubi steps

\begin{align*} \int x^m \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right ) \, dx &=\frac{x^{1+m} \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{2+m}+\frac{\sqrt{d+c^2 d x^2} \int \frac{x^m \left (a+b \sinh ^{-1}(c x)\right )}{\sqrt{1+c^2 x^2}} \, dx}{(2+m) \sqrt{1+c^2 x^2}}-\frac{\left (b c \sqrt{d+c^2 d x^2}\right ) \int x^{1+m} \, dx}{(2+m) \sqrt{1+c^2 x^2}}\\ &=-\frac{b c x^{2+m} \sqrt{d+c^2 d x^2}}{(2+m)^2 \sqrt{1+c^2 x^2}}+\frac{x^{1+m} \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{2+m}+\frac{x^{1+m} \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right ) \, _2F_1\left (\frac{1}{2},\frac{1+m}{2};\frac{3+m}{2};-c^2 x^2\right )}{\left (2+3 m+m^2\right ) \sqrt{1+c^2 x^2}}-\frac{b c x^{2+m} \sqrt{d+c^2 d x^2} \, _3F_2\left (1,1+\frac{m}{2},1+\frac{m}{2};\frac{3}{2}+\frac{m}{2},2+\frac{m}{2};-c^2 x^2\right )}{(1+m) (2+m)^2 \sqrt{1+c^2 x^2}}\\ \end{align*}

Mathematica [A]  time = 0.0709386, size = 179, normalized size = 0.75 \[ \frac{x^{m+1} \sqrt{c^2 d x^2+d} \left (-b c x \text{HypergeometricPFQ}\left (\left \{1,\frac{m}{2}+1,\frac{m}{2}+1\right \},\left \{\frac{m}{2}+\frac{3}{2},\frac{m}{2}+2\right \},-c^2 x^2\right )+(m+2) \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{m+1}{2},\frac{m+3}{2},-c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )+(m+1) \left (a (m+2) \sqrt{c^2 x^2+1}+b (m+2) \sqrt{c^2 x^2+1} \sinh ^{-1}(c x)-b c x\right )\right )}{(m+1) (m+2)^2 \sqrt{c^2 x^2+1}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(x^(1 + m)*Sqrt[d + c^2*d*x^2]*((1 + m)*(-(b*c*x) + a*(2 + m)*Sqrt[1 + c^2*x^2] + b*(2 + m)*Sqrt[1 + c^2*x^2]*
ArcSinh[c*x]) + (2 + m)*(a + b*ArcSinh[c*x])*Hypergeometric2F1[1/2, (1 + m)/2, (3 + m)/2, -(c^2*x^2)] - b*c*x*
HypergeometricPFQ[{1, 1 + m/2, 1 + m/2}, {3/2 + m/2, 2 + m/2}, -(c^2*x^2)]))/((1 + m)*(2 + m)^2*Sqrt[1 + c^2*x
^2])

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Maple [F]  time = 1.042, size = 0, normalized size = 0. \begin{align*} \int{x}^{m}\sqrt{{c}^{2}d{x}^{2}+d} \left ( a+b{\it Arcsinh} \left ( cx \right ) \right ) \, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^m*(c^2*d*x^2+d)^(1/2)*(a+b*arcsinh(c*x)),x)

[Out]

int(x^m*(c^2*d*x^2+d)^(1/2)*(a+b*arcsinh(c*x)),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{c^{2} d x^{2} + d}{\left (b \operatorname{arsinh}\left (c x\right ) + a\right )} x^{m}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^m*(c^2*d*x^2+d)^(1/2)*(a+b*arcsinh(c*x)),x, algorithm="maxima")

[Out]

integrate(sqrt(c^2*d*x^2 + d)*(b*arcsinh(c*x) + a)*x^m, x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\sqrt{c^{2} d x^{2} + d}{\left (b \operatorname{arsinh}\left (c x\right ) + a\right )} x^{m}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^m*(c^2*d*x^2+d)^(1/2)*(a+b*arcsinh(c*x)),x, algorithm="fricas")

[Out]

integral(sqrt(c^2*d*x^2 + d)*(b*arcsinh(c*x) + a)*x^m, x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{m} \sqrt{d \left (c^{2} x^{2} + 1\right )} \left (a + b \operatorname{asinh}{\left (c x \right )}\right )\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**m*(c**2*d*x**2+d)**(1/2)*(a+b*asinh(c*x)),x)

[Out]

Integral(x**m*sqrt(d*(c**2*x**2 + 1))*(a + b*asinh(c*x)), x)

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Giac [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^m*(c^2*d*x^2+d)^(1/2)*(a+b*arcsinh(c*x)),x, algorithm="giac")

[Out]

Timed out