3.199 \(\int (c e+d e x)^{5/2} (a+b \cosh ^{-1}(c+d x)) \, dx\)

Optimal. Leaf size=169 \[ \frac {2 (e (c+d x))^{7/2} \left (a+b \cosh ^{-1}(c+d x)\right )}{7 d e}-\frac {20 b e^{5/2} \sqrt {-c-d x+1} F\left (\left .\sin ^{-1}\left (\frac {\sqrt {e (c+d x)}}{\sqrt {e}}\right )\right |-1\right )}{147 d \sqrt {c+d x-1}}-\frac {20 b e^2 \sqrt {c+d x-1} \sqrt {c+d x+1} \sqrt {e (c+d x)}}{147 d}-\frac {4 b \sqrt {c+d x-1} \sqrt {c+d x+1} (e (c+d x))^{5/2}}{49 d} \]

[Out]

2/7*(e*(d*x+c))^(7/2)*(a+b*arccosh(d*x+c))/d/e-20/147*b*e^(5/2)*EllipticF((e*(d*x+c))^(1/2)/e^(1/2),I)*(-d*x-c
+1)^(1/2)/d/(d*x+c-1)^(1/2)-4/49*b*(e*(d*x+c))^(5/2)*(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2)/d-20/147*b*e^2*(d*x+c-1)^
(1/2)*(e*(d*x+c))^(1/2)*(d*x+c+1)^(1/2)/d

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Rubi [A]  time = 0.13, antiderivative size = 169, normalized size of antiderivative = 1.00, 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, 117, 116} \[ \frac {2 (e (c+d x))^{7/2} \left (a+b \cosh ^{-1}(c+d x)\right )}{7 d e}-\frac {20 b e^2 \sqrt {c+d x-1} \sqrt {c+d x+1} \sqrt {e (c+d x)}}{147 d}-\frac {20 b e^{5/2} \sqrt {-c-d x+1} F\left (\left .\sin ^{-1}\left (\frac {\sqrt {e (c+d x)}}{\sqrt {e}}\right )\right |-1\right )}{147 d \sqrt {c+d x-1}}-\frac {4 b \sqrt {c+d x-1} \sqrt {c+d x+1} (e (c+d x))^{5/2}}{49 d} \]

Antiderivative was successfully verified.

[In]

Int[(c*e + d*e*x)^(5/2)*(a + b*ArcCosh[c + d*x]),x]

[Out]

(-20*b*e^2*Sqrt[-1 + c + d*x]*Sqrt[e*(c + d*x)]*Sqrt[1 + c + d*x])/(147*d) - (4*b*Sqrt[-1 + c + d*x]*(e*(c + d
*x))^(5/2)*Sqrt[1 + c + d*x])/(49*d) + (2*(e*(c + d*x))^(7/2)*(a + b*ArcCosh[c + d*x]))/(7*d*e) - (20*b*e^(5/2
)*Sqrt[1 - c - d*x]*EllipticF[ArcSin[Sqrt[e*(c + d*x)]/Sqrt[e]], -1])/(147*d*Sqrt[-1 + c + d*x])

Rule 12

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

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 116

Int[1/(Sqrt[(b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(e_) + (f_.)*(x_)]), x_Symbol] :> Simp[(2*Rt[-(b/d), 2]*E
llipticF[ArcSin[Sqrt[b*x]/(Sqrt[c]*Rt[-(b/d), 2])], (c*f)/(d*e)])/(b*Sqrt[e]), x] /; FreeQ[{b, c, d, e, f}, x]
&& GtQ[c, 0] && GtQ[e, 0] && (PosQ[-(b/d)] || NegQ[-(b/f)])

Rule 117

Int[1/(Sqrt[(b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(e_) + (f_.)*(x_)]), x_Symbol] :> Dist[(Sqrt[1 + (d*x)/c]
*Sqrt[1 + (f*x)/e])/(Sqrt[c + d*x]*Sqrt[e + f*x]), Int[1/(Sqrt[b*x]*Sqrt[1 + (d*x)/c]*Sqrt[1 + (f*x)/e]), x],
x] /; FreeQ[{b, c, d, e, f}, x] &&  !(GtQ[c, 0] && GtQ[e, 0])

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 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
]

Rubi steps

\begin {align*} \int (c e+d e x)^{5/2} \left (a+b \cosh ^{-1}(c+d x)\right ) \, dx &=\frac {\operatorname {Subst}\left (\int (e x)^{5/2} \left (a+b \cosh ^{-1}(x)\right ) \, dx,x,c+d x\right )}{d}\\ &=\frac {2 (e (c+d x))^{7/2} \left (a+b \cosh ^{-1}(c+d x)\right )}{7 d e}-\frac {(2 b) \operatorname {Subst}\left (\int \frac {(e x)^{7/2}}{\sqrt {-1+x} \sqrt {1+x}} \, dx,x,c+d x\right )}{7 d e}\\ &=-\frac {4 b \sqrt {-1+c+d x} (e (c+d x))^{5/2} \sqrt {1+c+d x}}{49 d}+\frac {2 (e (c+d x))^{7/2} \left (a+b \cosh ^{-1}(c+d x)\right )}{7 d e}-\frac {(4 b) \operatorname {Subst}\left (\int \frac {5 e^2 (e x)^{3/2}}{2 \sqrt {-1+x} \sqrt {1+x}} \, dx,x,c+d x\right )}{49 d e}\\ &=-\frac {4 b \sqrt {-1+c+d x} (e (c+d x))^{5/2} \sqrt {1+c+d x}}{49 d}+\frac {2 (e (c+d x))^{7/2} \left (a+b \cosh ^{-1}(c+d x)\right )}{7 d e}-\frac {(10 b e) \operatorname {Subst}\left (\int \frac {(e x)^{3/2}}{\sqrt {-1+x} \sqrt {1+x}} \, dx,x,c+d x\right )}{49 d}\\ &=-\frac {20 b e^2 \sqrt {-1+c+d x} \sqrt {e (c+d x)} \sqrt {1+c+d x}}{147 d}-\frac {4 b \sqrt {-1+c+d x} (e (c+d x))^{5/2} \sqrt {1+c+d x}}{49 d}+\frac {2 (e (c+d x))^{7/2} \left (a+b \cosh ^{-1}(c+d x)\right )}{7 d e}-\frac {(20 b e) \operatorname {Subst}\left (\int \frac {e^2}{2 \sqrt {-1+x} \sqrt {e x} \sqrt {1+x}} \, dx,x,c+d x\right )}{147 d}\\ &=-\frac {20 b e^2 \sqrt {-1+c+d x} \sqrt {e (c+d x)} \sqrt {1+c+d x}}{147 d}-\frac {4 b \sqrt {-1+c+d x} (e (c+d x))^{5/2} \sqrt {1+c+d x}}{49 d}+\frac {2 (e (c+d x))^{7/2} \left (a+b \cosh ^{-1}(c+d x)\right )}{7 d e}-\frac {\left (10 b e^3\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {-1+x} \sqrt {e x} \sqrt {1+x}} \, dx,x,c+d x\right )}{147 d}\\ &=-\frac {20 b e^2 \sqrt {-1+c+d x} \sqrt {e (c+d x)} \sqrt {1+c+d x}}{147 d}-\frac {4 b \sqrt {-1+c+d x} (e (c+d x))^{5/2} \sqrt {1+c+d x}}{49 d}+\frac {2 (e (c+d x))^{7/2} \left (a+b \cosh ^{-1}(c+d x)\right )}{7 d e}-\frac {\left (10 b e^3 \sqrt {1-c-d x}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-x} \sqrt {e x} \sqrt {1+x}} \, dx,x,c+d x\right )}{147 d \sqrt {-1+c+d x}}\\ &=-\frac {20 b e^2 \sqrt {-1+c+d x} \sqrt {e (c+d x)} \sqrt {1+c+d x}}{147 d}-\frac {4 b \sqrt {-1+c+d x} (e (c+d x))^{5/2} \sqrt {1+c+d x}}{49 d}+\frac {2 (e (c+d x))^{7/2} \left (a+b \cosh ^{-1}(c+d x)\right )}{7 d e}-\frac {20 b e^{5/2} \sqrt {1-c-d x} F\left (\left .\sin ^{-1}\left (\frac {\sqrt {e (c+d x)}}{\sqrt {e}}\right )\right |-1\right )}{147 d \sqrt {-1+c+d x}}\\ \end {align*}

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Mathematica [C]  time = 0.28, size = 180, normalized size = 1.07 \[ \frac {2 (e (c+d x))^{5/2} \left (21 a \sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)^3-10 b \sqrt {1-(c+d x)^2} \, _2F_1\left (\frac {1}{4},\frac {1}{2};\frac {5}{4};(c+d x)^2\right )-6 b (c+d x)^4-4 b (c+d x)^2+21 b \sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)^3 \cosh ^{-1}(c+d x)+10 b\right )}{147 d \sqrt {\frac {c+d x-1}{c+d x}} (c+d x)^{5/2} \sqrt {c+d x+1}} \]

Antiderivative was successfully verified.

[In]

Integrate[(c*e + d*e*x)^(5/2)*(a + b*ArcCosh[c + d*x]),x]

[Out]

(2*(e*(c + d*x))^(5/2)*(10*b - 4*b*(c + d*x)^2 - 6*b*(c + d*x)^4 + 21*a*Sqrt[-1 + c + d*x]*(c + d*x)^3*Sqrt[1
+ c + d*x] + 21*b*Sqrt[-1 + c + d*x]*(c + d*x)^3*Sqrt[1 + c + d*x]*ArcCosh[c + d*x] - 10*b*Sqrt[1 - (c + d*x)^
2]*Hypergeometric2F1[1/4, 1/2, 5/4, (c + d*x)^2]))/(147*d*Sqrt[(-1 + c + d*x)/(c + d*x)]*(c + d*x)^(5/2)*Sqrt[
1 + c + d*x])

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fricas [F]  time = 1.44, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (a d^{2} e^{2} x^{2} + 2 \, a c d e^{2} x + a c^{2} e^{2} + {\left (b d^{2} e^{2} x^{2} + 2 \, b c d e^{2} x + b c^{2} e^{2}\right )} \operatorname {arcosh}\left (d x + c\right )\right )} \sqrt {d e x + c e}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*e*x+c*e)^(5/2)*(a+b*arccosh(d*x+c)),x, algorithm="fricas")

[Out]

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

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (d e x + c e\right )}^{\frac {5}{2}} {\left (b \operatorname {arcosh}\left (d x + c\right ) + a\right )}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*e*x+c*e)^(5/2)*(a+b*arccosh(d*x+c)),x, algorithm="giac")

[Out]

integrate((d*e*x + c*e)^(5/2)*(b*arccosh(d*x + c) + a), x)

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maple [A]  time = 0.03, size = 218, normalized size = 1.29 \[ \frac {\frac {2 \left (d e x +c e \right )^{\frac {7}{2}} a}{7}+2 b \left (\frac {\left (d e x +c e \right )^{\frac {7}{2}} \mathrm {arccosh}\left (\frac {d e x +c e}{e}\right )}{7}-\frac {2 \left (3 \sqrt {-\frac {1}{e}}\, \left (d e x +c e \right )^{\frac {9}{2}}+2 \sqrt {-\frac {1}{e}}\, \left (d e x +c e \right )^{\frac {5}{2}} e^{2}+5 e^{4} \sqrt {\frac {d e x +c e +e}{e}}\, \sqrt {-\frac {d e x +c e -e}{e}}\, \EllipticF \left (\sqrt {d e x +c e}\, \sqrt {-\frac {1}{e}}, i\right )-5 \sqrt {-\frac {1}{e}}\, \sqrt {d e x +c e}\, e^{4}\right )}{147 e \sqrt {-\frac {1}{e}}\, \sqrt {\frac {d e x +c e +e}{e}}\, \sqrt {\frac {d e x +c e -e}{e}}}\right )}{d e} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*e*x+c*e)^(5/2)*(a+b*arccosh(d*x+c)),x)

[Out]

2/d/e*(1/7*(d*e*x+c*e)^(7/2)*a+b*(1/7*(d*e*x+c*e)^(7/2)*arccosh((d*e*x+c*e)/e)-2/147/e*(3*(-1/e)^(1/2)*(d*e*x+
c*e)^(9/2)+2*(-1/e)^(1/2)*(d*e*x+c*e)^(5/2)*e^2+5*e^4*((d*e*x+c*e+e)/e)^(1/2)*(-(d*e*x+c*e-e)/e)^(1/2)*Ellipti
cF((d*e*x+c*e)^(1/2)*(-1/e)^(1/2),I)-5*(-1/e)^(1/2)*(d*e*x+c*e)^(1/2)*e^4)/(-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]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {2 \, {\left (d e x + c e\right )}^{\frac {7}{2}} a}{7 \, d e} + \frac {1}{147} \, {\left (\frac {42 \, {\left (d^{3} e^{\frac {5}{2}} x^{3} + 3 \, c d^{2} e^{\frac {5}{2}} x^{2} + 3 \, c^{2} d e^{\frac {5}{2}} x + c^{3} e^{\frac {5}{2}}\right )} \sqrt {d x + c} \log \left (d x + \sqrt {d x + c + 1} \sqrt {d x + c - 1} + c\right )}{d} - \frac {12 \, {\left (d x + c\right )}^{\frac {7}{2}} e^{\frac {5}{2}} + 28 \, {\left (d x + c\right )}^{\frac {3}{2}} e^{\frac {5}{2}} - 21 i \, e^{\frac {5}{2}} {\left (\log \left (i \, \sqrt {d x + c} + 1\right ) - \log \left (-i \, \sqrt {d x + c} + 1\right )\right )} - 21 \, e^{\frac {5}{2}} \log \left (\sqrt {d x + c} + 1\right ) + 21 \, e^{\frac {5}{2}} \log \left (\sqrt {d x + c} - 1\right )}{d} + 147 \, \int \frac {2 \, {\left (d^{3} e^{\frac {5}{2}} x^{3} + 3 \, c d^{2} e^{\frac {5}{2}} x^{2} + 3 \, c^{2} d e^{\frac {5}{2}} x + c^{3} e^{\frac {5}{2}}\right )} \sqrt {d x + c}}{7 \, {\left (d^{3} x^{3} + 3 \, c d^{2} x^{2} + c^{3} + {\left (d^{2} x^{2} + 2 \, c d x + c^{2} - 1\right )} \sqrt {d x + c + 1} \sqrt {d x + c - 1} + {\left (3 \, c^{2} d - d\right )} x - c\right )}}\,{d x}\right )} b \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*e*x+c*e)^(5/2)*(a+b*arccosh(d*x+c)),x, algorithm="maxima")

[Out]

2/7*(d*e*x + c*e)^(7/2)*a/(d*e) + 1/147*(42*(d^3*e^(5/2)*x^3 + 3*c*d^2*e^(5/2)*x^2 + 3*c^2*d*e^(5/2)*x + c^3*e
^(5/2))*sqrt(d*x + c)*log(d*x + sqrt(d*x + c + 1)*sqrt(d*x + c - 1) + c)/d - (12*(d*x + c)^(7/2)*e^(5/2) + 28*
(d*x + c)^(3/2)*e^(5/2) - 21*I*e^(5/2)*(log(I*sqrt(d*x + c) + 1) - log(-I*sqrt(d*x + c) + 1)) - 21*e^(5/2)*log
(sqrt(d*x + c) + 1) + 21*e^(5/2)*log(sqrt(d*x + c) - 1))/d + 147*integrate(2/7*(d^3*e^(5/2)*x^3 + 3*c*d^2*e^(5
/2)*x^2 + 3*c^2*d*e^(5/2)*x + c^3*e^(5/2))*sqrt(d*x + c)/(d^3*x^3 + 3*c*d^2*x^2 + c^3 + (d^2*x^2 + 2*c*d*x + c
^2 - 1)*sqrt(d*x + c + 1)*sqrt(d*x + c - 1) + (3*c^2*d - d)*x - c), x))*b

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int {\left (c\,e+d\,e\,x\right )}^{5/2}\,\left (a+b\,\mathrm {acosh}\left (c+d\,x\right )\right ) \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*e + d*e*x)^(5/2)*(a + b*acosh(c + d*x)),x)

[Out]

int((c*e + d*e*x)^(5/2)*(a + b*acosh(c + d*x)), x)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*e*x+c*e)**(5/2)*(a+b*acosh(d*x+c)),x)

[Out]

Timed out

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