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

Optimal. Leaf size=134 \[ \frac{16 b^2 (e (c+d x))^{7/2} \text{HypergeometricPFQ}\left (\left \{1,\frac{7}{4},\frac{7}{4}\right \},\left \{\frac{9}{4},\frac{11}{4}\right \},-(c+d x)^2\right )}{105 d e^3}-\frac{8 b (e (c+d x))^{5/2} \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{5}{4},\frac{9}{4},-(c+d x)^2\right ) \left (a+b \sinh ^{-1}(c+d x)\right )}{15 d e^2}+\frac{2 (e (c+d x))^{3/2} \left (a+b \sinh ^{-1}(c+d x)\right )^2}{3 d e} \]

[Out]

(2*(e*(c + d*x))^(3/2)*(a + b*ArcSinh[c + d*x])^2)/(3*d*e) - (8*b*(e*(c + d*x))^(5/2)*(a + b*ArcSinh[c + d*x])
*Hypergeometric2F1[1/2, 5/4, 9/4, -(c + d*x)^2])/(15*d*e^2) + (16*b^2*(e*(c + d*x))^(7/2)*HypergeometricPFQ[{1
, 7/4, 7/4}, {9/4, 11/4}, -(c + d*x)^2])/(105*d*e^3)

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Rubi [A]  time = 0.209408, antiderivative size = 134, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.12, Rules used = {5865, 5661, 5762} \[ \frac{16 b^2 (e (c+d x))^{7/2} \, _3F_2\left (1,\frac{7}{4},\frac{7}{4};\frac{9}{4},\frac{11}{4};-(c+d x)^2\right )}{105 d e^3}-\frac{8 b (e (c+d x))^{5/2} \, _2F_1\left (\frac{1}{2},\frac{5}{4};\frac{9}{4};-(c+d x)^2\right ) \left (a+b \sinh ^{-1}(c+d x)\right )}{15 d e^2}+\frac{2 (e (c+d x))^{3/2} \left (a+b \sinh ^{-1}(c+d x)\right )^2}{3 d e} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(e*(c + d*x))^(3/2)*(a + b*ArcSinh[c + d*x])^2)/(3*d*e) - (8*b*(e*(c + d*x))^(5/2)*(a + b*ArcSinh[c + d*x])
*Hypergeometric2F1[1/2, 5/4, 9/4, -(c + d*x)^2])/(15*d*e^2) + (16*b^2*(e*(c + d*x))^(7/2)*HypergeometricPFQ[{1
, 7/4, 7/4}, {9/4, 11/4}, -(c + d*x)^2])/(105*d*e^3)

Rule 5865

Int[((a_.) + ArcSinh[(c_) + (d_.)*(x_)]*(b_.))^(n_.)*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> Dist[1/d, Subst[
Int[((d*e - c*f)/d + (f*x)/d)^m*(a + b*ArcSinh[x])^n, x], x, c + d*x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x
]

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 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]

Rubi steps

\begin{align*} \int \sqrt{c e+d e x} \left (a+b \sinh ^{-1}(c+d x)\right )^2 \, dx &=\frac{\operatorname{Subst}\left (\int \sqrt{e x} \left (a+b \sinh ^{-1}(x)\right )^2 \, dx,x,c+d x\right )}{d}\\ &=\frac{2 (e (c+d x))^{3/2} \left (a+b \sinh ^{-1}(c+d x)\right )^2}{3 d e}-\frac{(4 b) \operatorname{Subst}\left (\int \frac{(e x)^{3/2} \left (a+b \sinh ^{-1}(x)\right )}{\sqrt{1+x^2}} \, dx,x,c+d x\right )}{3 d e}\\ &=\frac{2 (e (c+d x))^{3/2} \left (a+b \sinh ^{-1}(c+d x)\right )^2}{3 d e}-\frac{8 b (e (c+d x))^{5/2} \left (a+b \sinh ^{-1}(c+d x)\right ) \, _2F_1\left (\frac{1}{2},\frac{5}{4};\frac{9}{4};-(c+d x)^2\right )}{15 d e^2}+\frac{16 b^2 (e (c+d x))^{7/2} \, _3F_2\left (1,\frac{7}{4},\frac{7}{4};\frac{9}{4},\frac{11}{4};-(c+d x)^2\right )}{105 d e^3}\\ \end{align*}

Mathematica [A]  time = 0.0848566, size = 110, normalized size = 0.82 \[ \frac{2 (e (c+d x))^{3/2} \left (8 b^2 (c+d x)^2 \text{HypergeometricPFQ}\left (\left \{1,\frac{7}{4},\frac{7}{4}\right \},\left \{\frac{9}{4},\frac{11}{4}\right \},-(c+d x)^2\right )-28 b (c+d x) \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{5}{4},\frac{9}{4},-(c+d x)^2\right ) \left (a+b \sinh ^{-1}(c+d x)\right )+35 \left (a+b \sinh ^{-1}(c+d x)\right )^2\right )}{105 d e} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(e*(c + d*x))^(3/2)*(35*(a + b*ArcSinh[c + d*x])^2 - 28*b*(c + d*x)*(a + b*ArcSinh[c + d*x])*Hypergeometric
2F1[1/2, 5/4, 9/4, -(c + d*x)^2] + 8*b^2*(c + d*x)^2*HypergeometricPFQ[{1, 7/4, 7/4}, {9/4, 11/4}, -(c + d*x)^
2]))/(105*d*e)

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Maple [F]  time = 0.275, size = 0, normalized size = 0. \begin{align*} \int \left ( a+b{\it Arcsinh} \left ( dx+c \right ) \right ) ^{2}\sqrt{dex+ce}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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