3.198 \(\int (c e+d e x)^{7/2} (a+b \cosh ^{-1}(c+d x)) \, dx\)

Optimal. Leaf size=189 \[ \frac{2 (e (c+d x))^{9/2} \left (a+b \cosh ^{-1}(c+d x)\right )}{9 d e}-\frac{28 b e^2 \sqrt{c+d x-1} \sqrt{c+d x+1} (e (c+d x))^{3/2}}{405 d}-\frac{28 b e^3 \sqrt{-c-d x+1} \sqrt{e (c+d x)} E\left (\left .\sin ^{-1}\left (\frac{\sqrt{c+d x+1}}{\sqrt{2}}\right )\right |2\right )}{135 d \sqrt{-c-d x} \sqrt{c+d x-1}}-\frac{4 b \sqrt{c+d x-1} \sqrt{c+d x+1} (e (c+d x))^{7/2}}{81 d} \]

[Out]

(-28*b*e^2*Sqrt[-1 + c + d*x]*(e*(c + d*x))^(3/2)*Sqrt[1 + c + d*x])/(405*d) - (4*b*Sqrt[-1 + c + d*x]*(e*(c +
d*x))^(7/2)*Sqrt[1 + c + d*x])/(81*d) + (2*(e*(c + d*x))^(9/2)*(a + b*ArcCosh[c + d*x]))/(9*d*e) - (28*b*e^3*
Sqrt[1 - c - d*x]*Sqrt[e*(c + d*x)]*EllipticE[ArcSin[Sqrt[1 + c + d*x]/Sqrt[2]], 2])/(135*d*Sqrt[-c - d*x]*Sqr
t[-1 + c + d*x])

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Rubi [A]  time = 0.143705, antiderivative size = 189, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.261, Rules used = {5866, 5662, 102, 12, 114, 113} \[ \frac{2 (e (c+d x))^{9/2} \left (a+b \cosh ^{-1}(c+d x)\right )}{9 d e}-\frac{28 b e^2 \sqrt{c+d x-1} \sqrt{c+d x+1} (e (c+d x))^{3/2}}{405 d}-\frac{28 b e^3 \sqrt{-c-d x+1} \sqrt{e (c+d x)} E\left (\left .\sin ^{-1}\left (\frac{\sqrt{c+d x+1}}{\sqrt{2}}\right )\right |2\right )}{135 d \sqrt{-c-d x} \sqrt{c+d x-1}}-\frac{4 b \sqrt{c+d x-1} \sqrt{c+d x+1} (e (c+d x))^{7/2}}{81 d} \]

Antiderivative was successfully verified.

[In]

Int[(c*e + d*e*x)^(7/2)*(a + b*ArcCosh[c + d*x]),x]

[Out]

(-28*b*e^2*Sqrt[-1 + c + d*x]*(e*(c + d*x))^(3/2)*Sqrt[1 + c + d*x])/(405*d) - (4*b*Sqrt[-1 + c + d*x]*(e*(c +
d*x))^(7/2)*Sqrt[1 + c + d*x])/(81*d) + (2*(e*(c + d*x))^(9/2)*(a + b*ArcCosh[c + d*x]))/(9*d*e) - (28*b*e^3*
Sqrt[1 - c - d*x]*Sqrt[e*(c + d*x)]*EllipticE[ArcSin[Sqrt[1 + c + d*x]/Sqrt[2]], 2])/(135*d*Sqrt[-c - d*x]*Sqr
t[-1 + c + d*x])

Rule 5866

Int[((a_.) + ArcCosh[(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*ArcCosh[x])^n, x], x, c + d*x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x
]

Rule 5662

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

Rule 102

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*(a +
b*x)^(m - 1)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/(d*f*(m + n + p + 1)), x] + Dist[1/(d*f*(m + n + p + 1)), I
nt[(a + b*x)^(m - 2)*(c + d*x)^n*(e + f*x)^p*Simp[a^2*d*f*(m + n + p + 1) - b*(b*c*e*(m - 1) + a*(d*e*(n + 1)
+ c*f*(p + 1))) + b*(a*d*f*(2*m + n + p) - b*(d*e*(m + n) + c*f*(m + p)))*x, x], x], x] /; FreeQ[{a, b, c, d,
e, f, n, p}, x] && GtQ[m, 1] && NeQ[m + n + p + 1, 0] && IntegersQ[2*m, 2*n, 2*p]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 114

Int[Sqrt[(e_.) + (f_.)*(x_)]/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Dist[(Sqrt[e + f*
x]*Sqrt[(b*(c + d*x))/(b*c - a*d)])/(Sqrt[c + d*x]*Sqrt[(b*(e + f*x))/(b*e - a*f)]), Int[Sqrt[(b*e)/(b*e - a*f
) + (b*f*x)/(b*e - a*f)]/(Sqrt[a + b*x]*Sqrt[(b*c)/(b*c - a*d) + (b*d*x)/(b*c - a*d)]), x], x] /; FreeQ[{a, b,
c, d, e, f}, x] &&  !(GtQ[b/(b*c - a*d), 0] && GtQ[b/(b*e - a*f), 0]) &&  !LtQ[-((b*c - a*d)/d), 0]

Rule 113

Int[Sqrt[(e_.) + (f_.)*(x_)]/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Simp[(2*Rt[-((b*e
- a*f)/d), 2]*EllipticE[ArcSin[Sqrt[a + b*x]/Rt[-((b*c - a*d)/d), 2]], (f*(b*c - a*d))/(d*(b*e - a*f))])/b, x
] /; FreeQ[{a, b, c, d, e, f}, x] && GtQ[b/(b*c - a*d), 0] && GtQ[b/(b*e - a*f), 0] &&  !LtQ[-((b*c - a*d)/d),
0] &&  !(SimplerQ[c + d*x, a + b*x] && GtQ[-(d/(b*c - a*d)), 0] && GtQ[d/(d*e - c*f), 0] &&  !LtQ[(b*c - a*d)
/b, 0])

Rubi steps

\begin{align*} \int (c e+d e x)^{7/2} \left (a+b \cosh ^{-1}(c+d x)\right ) \, dx &=\frac{\operatorname{Subst}\left (\int (e x)^{7/2} \left (a+b \cosh ^{-1}(x)\right ) \, dx,x,c+d x\right )}{d}\\ &=\frac{2 (e (c+d x))^{9/2} \left (a+b \cosh ^{-1}(c+d x)\right )}{9 d e}-\frac{(2 b) \operatorname{Subst}\left (\int \frac{(e x)^{9/2}}{\sqrt{-1+x} \sqrt{1+x}} \, dx,x,c+d x\right )}{9 d e}\\ &=-\frac{4 b \sqrt{-1+c+d x} (e (c+d x))^{7/2} \sqrt{1+c+d x}}{81 d}+\frac{2 (e (c+d x))^{9/2} \left (a+b \cosh ^{-1}(c+d x)\right )}{9 d e}-\frac{(4 b) \operatorname{Subst}\left (\int \frac{7 e^2 (e x)^{5/2}}{2 \sqrt{-1+x} \sqrt{1+x}} \, dx,x,c+d x\right )}{81 d e}\\ &=-\frac{4 b \sqrt{-1+c+d x} (e (c+d x))^{7/2} \sqrt{1+c+d x}}{81 d}+\frac{2 (e (c+d x))^{9/2} \left (a+b \cosh ^{-1}(c+d x)\right )}{9 d e}-\frac{(14 b e) \operatorname{Subst}\left (\int \frac{(e x)^{5/2}}{\sqrt{-1+x} \sqrt{1+x}} \, dx,x,c+d x\right )}{81 d}\\ &=-\frac{28 b e^2 \sqrt{-1+c+d x} (e (c+d x))^{3/2} \sqrt{1+c+d x}}{405 d}-\frac{4 b \sqrt{-1+c+d x} (e (c+d x))^{7/2} \sqrt{1+c+d x}}{81 d}+\frac{2 (e (c+d x))^{9/2} \left (a+b \cosh ^{-1}(c+d x)\right )}{9 d e}-\frac{(28 b e) \operatorname{Subst}\left (\int \frac{3 e^2 \sqrt{e x}}{2 \sqrt{-1+x} \sqrt{1+x}} \, dx,x,c+d x\right )}{405 d}\\ &=-\frac{28 b e^2 \sqrt{-1+c+d x} (e (c+d x))^{3/2} \sqrt{1+c+d x}}{405 d}-\frac{4 b \sqrt{-1+c+d x} (e (c+d x))^{7/2} \sqrt{1+c+d x}}{81 d}+\frac{2 (e (c+d x))^{9/2} \left (a+b \cosh ^{-1}(c+d x)\right )}{9 d e}-\frac{\left (14 b e^3\right ) \operatorname{Subst}\left (\int \frac{\sqrt{e x}}{\sqrt{-1+x} \sqrt{1+x}} \, dx,x,c+d x\right )}{135 d}\\ &=-\frac{28 b e^2 \sqrt{-1+c+d x} (e (c+d x))^{3/2} \sqrt{1+c+d x}}{405 d}-\frac{4 b \sqrt{-1+c+d x} (e (c+d x))^{7/2} \sqrt{1+c+d x}}{81 d}+\frac{2 (e (c+d x))^{9/2} \left (a+b \cosh ^{-1}(c+d x)\right )}{9 d e}-\frac{\left (7 \sqrt{2} b e^3 \sqrt{1-c-d x} \sqrt{e (c+d x)}\right ) \operatorname{Subst}\left (\int \frac{\sqrt{-x}}{\sqrt{\frac{1}{2}-\frac{x}{2}} \sqrt{1+x}} \, dx,x,c+d x\right )}{135 d \sqrt{-c-d x} \sqrt{-1+c+d x}}\\ &=-\frac{28 b e^2 \sqrt{-1+c+d x} (e (c+d x))^{3/2} \sqrt{1+c+d x}}{405 d}-\frac{4 b \sqrt{-1+c+d x} (e (c+d x))^{7/2} \sqrt{1+c+d x}}{81 d}+\frac{2 (e (c+d x))^{9/2} \left (a+b \cosh ^{-1}(c+d x)\right )}{9 d e}-\frac{28 b e^3 \sqrt{1-c-d x} \sqrt{e (c+d x)} E\left (\left .\sin ^{-1}\left (\frac{\sqrt{1+c+d x}}{\sqrt{2}}\right )\right |2\right )}{135 d \sqrt{-c-d x} \sqrt{-1+c+d x}}\\ \end{align*}

Mathematica [C]  time = 0.327371, size = 150, normalized size = 0.79 \[ \frac{2 (e (c+d x))^{7/2} \left (\frac{2 b (c+d x)^{3/2} \left (-7 \sqrt{1-(c+d x)^2} \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{3}{4},\frac{7}{4},(c+d x)^2\right )+5 \left (1-(c+d x)^2\right ) (c+d x)^2+7 \left (1-(c+d x)^2\right )\right )}{45 \sqrt{c+d x-1} \sqrt{c+d x+1}}+(c+d x)^{9/2} \left (a+b \cosh ^{-1}(c+d x)\right )\right )}{9 d (c+d x)^{7/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(c*e + d*e*x)^(7/2)*(a + b*ArcCosh[c + d*x]),x]

[Out]

(2*(e*(c + d*x))^(7/2)*((c + d*x)^(9/2)*(a + b*ArcCosh[c + d*x]) + (2*b*(c + d*x)^(3/2)*(7*(1 - (c + d*x)^2) +
5*(c + d*x)^2*(1 - (c + d*x)^2) - 7*Sqrt[1 - (c + d*x)^2]*Hypergeometric2F1[1/2, 3/4, 7/4, (c + d*x)^2]))/(45
*Sqrt[-1 + c + d*x]*Sqrt[1 + c + d*x])))/(9*d*(c + d*x)^(7/2))

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Maple [C]  time = 0.064, size = 277, normalized size = 1.5 \begin{align*} 2\,{\frac{1}{de} \left ( 1/9\, \left ( dex+ce \right ) ^{9/2}a+b \left ( 1/9\, \left ( dex+ce \right ) ^{9/2}{\rm arccosh} \left ({\frac{dex+ce}{e}}\right )-{\frac{2}{405\,e} \left ( 5\,\sqrt{-{e}^{-1}} \left ( dex+ce \right ) ^{11/2}+2\,\sqrt{-{e}^{-1}} \left ( dex+ce \right ) ^{7/2}{e}^{2}-7\,\sqrt{-{e}^{-1}} \left ( dex+ce \right ) ^{3/2}{e}^{4}+21\,{e}^{5}\sqrt{{\frac{dex+ce+e}{e}}}\sqrt{-{\frac{dex+ce-e}{e}}}{\it EllipticF} \left ( \sqrt{dex+ce}\sqrt{-{e}^{-1}},i \right ) -21\,{e}^{5}\sqrt{{\frac{dex+ce+e}{e}}}\sqrt{-{\frac{dex+ce-e}{e}}}{\it EllipticE} \left ( \sqrt{dex+ce}\sqrt{-{e}^{-1}},i \right ) \right ){\frac{1}{\sqrt{-{e}^{-1}}}}{\frac{1}{\sqrt{{\frac{dex+ce+e}{e}}}}}{\frac{1}{\sqrt{{\frac{dex+ce-e}{e}}}}}} \right ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*e*x+c*e)^(7/2)*(a+b*arccosh(d*x+c)),x)

[Out]

2/d/e*(1/9*(d*e*x+c*e)^(9/2)*a+b*(1/9*(d*e*x+c*e)^(9/2)*arccosh(1/e*(d*e*x+c*e))-2/405/e*(5*(-1/e)^(1/2)*(d*e*
x+c*e)^(11/2)+2*(-1/e)^(1/2)*(d*e*x+c*e)^(7/2)*e^2-7*(-1/e)^(1/2)*(d*e*x+c*e)^(3/2)*e^4+21*e^5*((d*e*x+c*e+e)/
e)^(1/2)*(-(d*e*x+c*e-e)/e)^(1/2)*EllipticF((d*e*x+c*e)^(1/2)*(-1/e)^(1/2),I)-21*e^5*((d*e*x+c*e+e)/e)^(1/2)*(
-(d*e*x+c*e-e)/e)^(1/2)*EllipticE((d*e*x+c*e)^(1/2)*(-1/e)^(1/2),I))/(-1/e)^(1/2)/((d*e*x+c*e+e)/e)^(1/2)/((d*
e*x+c*e-e)/e)^(1/2)))

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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((d*e*x+c*e)^(7/2)*(a+b*arccosh(d*x+c)),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 (a d^{3} e^{3} x^{3} + 3 \, a c d^{2} e^{3} x^{2} + 3 \, a c^{2} d e^{3} x + a c^{3} e^{3} +{\left (b d^{3} e^{3} x^{3} + 3 \, b c d^{2} e^{3} x^{2} + 3 \, b c^{2} d e^{3} x + b c^{3} e^{3}\right )} \operatorname{arcosh}\left (d x + c\right )\right )} \sqrt{d e x + c e}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*e*x+c*e)^(7/2)*(a+b*arccosh(d*x+c)),x, algorithm="fricas")

[Out]

integral((a*d^3*e^3*x^3 + 3*a*c*d^2*e^3*x^2 + 3*a*c^2*d*e^3*x + a*c^3*e^3 + (b*d^3*e^3*x^3 + 3*b*c*d^2*e^3*x^2
+ 3*b*c^2*d*e^3*x + b*c^3*e^3)*arccosh(d*x + c))*sqrt(d*e*x + c*e), x)

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Sympy [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((d*e*x+c*e)**(7/2)*(a+b*acosh(d*x+c)),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (d e x + c e\right )}^{\frac{7}{2}}{\left (b \operatorname{arcosh}\left (d x + c\right ) + a\right )}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*e*x+c*e)^(7/2)*(a+b*arccosh(d*x+c)),x, algorithm="giac")

[Out]

integrate((d*e*x + c*e)^(7/2)*(b*arccosh(d*x + c) + a), x)