3.2.100 \(\int (c e+d e x)^{3/2} (a+b \cosh ^{-1}(c+d x)) \, dx\) [200]

Optimal. Leaf size=145 \[ -\frac {4 b \sqrt {-1+c+d x} (e (c+d x))^{3/2} \sqrt {1+c+d x}}{25 d}+\frac {2 (e (c+d x))^{5/2} \left (a+b \cosh ^{-1}(c+d x)\right )}{5 d e}-\frac {12 b e \sqrt {1-c-d x} \sqrt {e (c+d x)} E\left (\left .\text {ArcSin}\left (\frac {\sqrt {1+c+d x}}{\sqrt {2}}\right )\right |2\right )}{25 d \sqrt {-c-d x} \sqrt {-1+c+d x}} \]

[Out]

2/5*(e*(d*x+c))^(5/2)*(a+b*arccosh(d*x+c))/d/e-12/25*b*e*EllipticE(1/2*(d*x+c+1)^(1/2)*2^(1/2),2^(1/2))*(-d*x-
c+1)^(1/2)*(e*(d*x+c))^(1/2)/d/(-d*x-c)^(1/2)/(d*x+c-1)^(1/2)-4/25*b*(e*(d*x+c))^(3/2)*(d*x+c-1)^(1/2)*(d*x+c+
1)^(1/2)/d

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Rubi [A]
time = 0.08, antiderivative size = 145, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {5996, 5883, 104, 12, 115, 114} \begin {gather*} \frac {2 (e (c+d x))^{5/2} \left (a+b \cosh ^{-1}(c+d x)\right )}{5 d e}-\frac {12 b e \sqrt {-c-d x+1} \sqrt {e (c+d x)} E\left (\left .\text {ArcSin}\left (\frac {\sqrt {c+d x+1}}{\sqrt {2}}\right )\right |2\right )}{25 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))^{3/2}}{25 d} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(c*e + d*e*x)^(3/2)*(a + b*ArcCosh[c + d*x]),x]

[Out]

(-4*b*Sqrt[-1 + c + d*x]*(e*(c + d*x))^(3/2)*Sqrt[1 + c + d*x])/(25*d) + (2*(e*(c + d*x))^(5/2)*(a + b*ArcCosh
[c + d*x]))/(5*d*e) - (12*b*e*Sqrt[1 - c - d*x]*Sqrt[e*(c + d*x)]*EllipticE[ArcSin[Sqrt[1 + c + d*x]/Sqrt[2]],
2])/(25*d*Sqrt[-c - d*x]*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 104

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 114

Int[Sqrt[(e_.) + (f_.)*(x_)]/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Simp[(2/b)*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)))], 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])

Rule 115

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 5883

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)/(Sqrt
[1 + c*x]*Sqrt[-1 + c*x])), x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] && NeQ[m, -1]

Rule 5996

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)^{3/2} \left (a+b \cosh ^{-1}(c+d x)\right ) \, dx &=\frac {\text {Subst}\left (\int (e x)^{3/2} \left (a+b \cosh ^{-1}(x)\right ) \, dx,x,c+d x\right )}{d}\\ &=\frac {2 (e (c+d x))^{5/2} \left (a+b \cosh ^{-1}(c+d x)\right )}{5 d e}-\frac {(2 b) \text {Subst}\left (\int \frac {(e x)^{5/2}}{\sqrt {-1+x} \sqrt {1+x}} \, dx,x,c+d x\right )}{5 d e}\\ &=-\frac {4 b \sqrt {-1+c+d x} (e (c+d x))^{3/2} \sqrt {1+c+d x}}{25 d}+\frac {2 (e (c+d x))^{5/2} \left (a+b \cosh ^{-1}(c+d x)\right )}{5 d e}-\frac {(4 b) \text {Subst}\left (\int \frac {3 e^2 \sqrt {e x}}{2 \sqrt {-1+x} \sqrt {1+x}} \, dx,x,c+d x\right )}{25 d e}\\ &=-\frac {4 b \sqrt {-1+c+d x} (e (c+d x))^{3/2} \sqrt {1+c+d x}}{25 d}+\frac {2 (e (c+d x))^{5/2} \left (a+b \cosh ^{-1}(c+d x)\right )}{5 d e}-\frac {(6 b e) \text {Subst}\left (\int \frac {\sqrt {e x}}{\sqrt {-1+x} \sqrt {1+x}} \, dx,x,c+d x\right )}{25 d}\\ &=-\frac {4 b \sqrt {-1+c+d x} (e (c+d x))^{3/2} \sqrt {1+c+d x}}{25 d}+\frac {2 (e (c+d x))^{5/2} \left (a+b \cosh ^{-1}(c+d x)\right )}{5 d e}-\frac {\left (3 \sqrt {2} b e \sqrt {1-c-d x} \sqrt {e (c+d x)}\right ) \text {Subst}\left (\int \frac {\sqrt {-x}}{\sqrt {\frac {1}{2}-\frac {x}{2}} \sqrt {1+x}} \, dx,x,c+d x\right )}{25 d \sqrt {-c-d x} \sqrt {-1+c+d x}}\\ &=-\frac {4 b \sqrt {-1+c+d x} (e (c+d x))^{3/2} \sqrt {1+c+d x}}{25 d}+\frac {2 (e (c+d x))^{5/2} \left (a+b \cosh ^{-1}(c+d x)\right )}{5 d e}-\frac {12 b e \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 )}{25 d \sqrt {-c-d x} \sqrt {-1+c+d x}}\\ \end {align*}

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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
time = 0.35, size = 109, normalized size = 0.75 \begin {gather*} \frac {2 (e (c+d x))^{3/2} \left (5 (c+d x) \left (a+b \cosh ^{-1}(c+d x)\right )-\frac {2 b \left (-1+c^2+2 c d x+d^2 x^2+\sqrt {1-(c+d x)^2} \, _2F_1\left (\frac {1}{2},\frac {3}{4};\frac {7}{4};(c+d x)^2\right )\right )}{\sqrt {-1+c+d x} \sqrt {1+c+d x}}\right )}{25 d} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(c*e + d*e*x)^(3/2)*(a + b*ArcCosh[c + d*x]),x]

[Out]

(2*(e*(c + d*x))^(3/2)*(5*(c + d*x)*(a + b*ArcCosh[c + d*x]) - (2*b*(-1 + c^2 + 2*c*d*x + d^2*x^2 + Sqrt[1 - (
c + d*x)^2]*Hypergeometric2F1[1/2, 3/4, 7/4, (c + d*x)^2]))/(Sqrt[-1 + c + d*x]*Sqrt[1 + c + d*x])))/(25*d)

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Maple [C] Result contains complex when optimal does not.
time = 0.05, size = 253, normalized size = 1.74

method	result	size
derivativedivides	\(\frac {\frac {2 \left (d e x +c e \right )^{\frac {5}{2}} a}{5}+2 b \left (\frac {\left (d e x +c e \right )^{\frac {5}{2}} \mathrm {arccosh}\left (\frac {d e x +c e}{e}\right )}{5}-\frac {2 \left (\sqrt {-\frac {1}{e}}\, \left (d e x +c e \right )^{\frac {7}{2}}+3 \sqrt {\frac {d e x +c e +e}{e}}\, \sqrt {\frac {-d e x -c e +e}{e}}\, e^{3} \EllipticF \left (\sqrt {d e x +c e}\, \sqrt {-\frac {1}{e}}, i\right )-3 e^{3} \sqrt {\frac {d e x +c e +e}{e}}\, \sqrt {\frac {-d e x -c e +e}{e}}\, \EllipticE \left (\sqrt {d e x +c e}\, \sqrt {-\frac {1}{e}}, i\right )-\sqrt {-\frac {1}{e}}\, e^{2} \left (d e x +c e \right )^{\frac {3}{2}}\right )}{25 e \sqrt {-\frac {1}{e}}\, \sqrt {\frac {d e x +c e +e}{e}}\, \sqrt {-\frac {-d e x -c e +e}{e}}}\right )}{d e}\)	\(253\)
default	\(\frac {\frac {2 \left (d e x +c e \right )^{\frac {5}{2}} a}{5}+2 b \left (\frac {\left (d e x +c e \right )^{\frac {5}{2}} \mathrm {arccosh}\left (\frac {d e x +c e}{e}\right )}{5}-\frac {2 \left (\sqrt {-\frac {1}{e}}\, \left (d e x +c e \right )^{\frac {7}{2}}+3 \sqrt {\frac {d e x +c e +e}{e}}\, \sqrt {\frac {-d e x -c e +e}{e}}\, e^{3} \EllipticF \left (\sqrt {d e x +c e}\, \sqrt {-\frac {1}{e}}, i\right )-3 e^{3} \sqrt {\frac {d e x +c e +e}{e}}\, \sqrt {\frac {-d e x -c e +e}{e}}\, \EllipticE \left (\sqrt {d e x +c e}\, \sqrt {-\frac {1}{e}}, i\right )-\sqrt {-\frac {1}{e}}\, e^{2} \left (d e x +c e \right )^{\frac {3}{2}}\right )}{25 e \sqrt {-\frac {1}{e}}\, \sqrt {\frac {d e x +c e +e}{e}}\, \sqrt {-\frac {-d e x -c e +e}{e}}}\right )}{d e}\)	\(253\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*e*x+c*e)^(3/2)*(a+b*arccosh(d*x+c)),x,method=_RETURNVERBOSE)

[Out]

2/d/e*(1/5*(d*e*x+c*e)^(5/2)*a+b*(1/5*(d*e*x+c*e)^(5/2)*arccosh((d*e*x+c*e)/e)-2/25/e*((-1/e)^(1/2)*(d*e*x+c*e
)^(7/2)+3*((d*e*x+c*e+e)/e)^(1/2)*((-d*e*x-c*e+e)/e)^(1/2)*e^3*EllipticF((d*e*x+c*e)^(1/2)*(-1/e)^(1/2),I)-3*e
^3*((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)*e
^2*(d*e*x+c*e)^(3/2))/(-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 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*e*x+c*e)^(3/2)*(a+b*arccosh(d*x+c)),x, algorithm="maxima")

[Out]

2/5*(d*x*e + c*e)^(5/2)*a*e^(-1)/d + 1/25*(10*(d^2*x^2*e^(3/2) + 2*c*d*x*e^(3/2) + c^2*e^(3/2))*sqrt(d*x + c)*
log(d*x + sqrt(d*x + c + 1)*sqrt(d*x + c - 1) + c)/d + (-5*I*(log(I*sqrt(d*x + c) + 1) - log(-I*sqrt(d*x + c)
+ 1))*e^(3/2) + 5*e^(3/2)*log(sqrt(d*x + c) + 1) - 5*e^(3/2)*log(sqrt(d*x + c) - 1) - 4*e^(5/2*log(d*x + c) +
3/2) - 20*e^(1/2*log(d*x + c) + 3/2))/d + 25*integrate(2/5*(d^2*x^2*e^(3/2) + 2*c*d*x*e^(3/2) + c^2*e^(3/2))*s
qrt(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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Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.10, size = 279, normalized size = 1.92 \begin {gather*} \frac {2 \, {\left (5 \, {\left ({\left (b d^{3} x^{2} + 2 \, b c d^{2} x + b c^{2} d\right )} \cosh \left (1\right ) + {\left (b d^{3} x^{2} + 2 \, b c d^{2} x + b c^{2} d\right )} \sinh \left (1\right )\right )} \sqrt {{\left (d x + c\right )} \cosh \left (1\right ) + {\left (d x + c\right )} \sinh \left (1\right )} \log \left (d x + c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1}\right ) + 6 \, \sqrt {d^{3} \cosh \left (1\right ) + d^{3} \sinh \left (1\right )} {\left (b \cosh \left (1\right ) + b \sinh \left (1\right )\right )} {\rm weierstrassZeta}\left (\frac {4}{d^{2}}, 0, {\rm weierstrassPInverse}\left (\frac {4}{d^{2}}, 0, \frac {d x + c}{d}\right )\right ) + {\left (5 \, {\left (a d^{3} x^{2} + 2 \, a c d^{2} x + a c^{2} d\right )} \cosh \left (1\right ) + 5 \, {\left (a d^{3} x^{2} + 2 \, a c d^{2} x + a c^{2} d\right )} \sinh \left (1\right ) - 2 \, \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1} {\left ({\left (b d^{2} x + b c d\right )} \cosh \left (1\right ) + {\left (b d^{2} x + b c d\right )} \sinh \left (1\right )\right )}\right )} \sqrt {{\left (d x + c\right )} \cosh \left (1\right ) + {\left (d x + c\right )} \sinh \left (1\right )}\right )}}{25 \, d^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*e*x+c*e)^(3/2)*(a+b*arccosh(d*x+c)),x, algorithm="fricas")

[Out]

2/25*(5*((b*d^3*x^2 + 2*b*c*d^2*x + b*c^2*d)*cosh(1) + (b*d^3*x^2 + 2*b*c*d^2*x + b*c^2*d)*sinh(1))*sqrt((d*x
+ c)*cosh(1) + (d*x + c)*sinh(1))*log(d*x + c + sqrt(d^2*x^2 + 2*c*d*x + c^2 - 1)) + 6*sqrt(d^3*cosh(1) + d^3*
sinh(1))*(b*cosh(1) + b*sinh(1))*weierstrassZeta(4/d^2, 0, weierstrassPInverse(4/d^2, 0, (d*x + c)/d)) + (5*(a
*d^3*x^2 + 2*a*c*d^2*x + a*c^2*d)*cosh(1) + 5*(a*d^3*x^2 + 2*a*c*d^2*x + a*c^2*d)*sinh(1) - 2*sqrt(d^2*x^2 + 2
*c*d*x + c^2 - 1)*((b*d^2*x + b*c*d)*cosh(1) + (b*d^2*x + b*c*d)*sinh(1)))*sqrt((d*x + c)*cosh(1) + (d*x + c)*
sinh(1)))/d^2

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (e \left (c + d x\right )\right )^{\frac {3}{2}} \left (a + b \operatorname {acosh}{\left (c + d x \right )}\right )\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*e*x+c*e)**(3/2)*(a+b*acosh(d*x+c)),x)

[Out]

Integral((e*(c + d*x))**(3/2)*(a + b*acosh(c + d*x)), x)

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*e*x+c*e)^(3/2)*(a+b*arccosh(d*x+c)),x, algorithm="giac")

[Out]

integrate((d*e*x + c*e)^(3/2)*(b*arccosh(d*x + c) + a), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int {\left (c\,e+d\,e\,x\right )}^{3/2}\,\left (a+b\,\mathrm {acosh}\left (c+d\,x\right )\right ) \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*e + d*e*x)^(3/2)*(a + b*acosh(c + d*x)),x)

[Out]

int((c*e + d*e*x)^(3/2)*(a + b*acosh(c + d*x)), x)

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