3.227 \(\int (c e+d e x)^m (a+b \cosh ^{-1}(c+d x))^3 \, dx\)

Optimal. Leaf size=94 \[ \frac {(e (c+d x))^{m+1} \left (a+b \cosh ^{-1}(c+d x)\right )^3}{d e (m+1)}-\frac {3 b \text {Int}\left (\frac {(e (c+d x))^{m+1} \left (a+b \cosh ^{-1}(c+d x)\right )^2}{\sqrt {c+d x-1} \sqrt {c+d x+1}},x\right )}{e (m+1)} \]

[Out]

(e*(d*x+c))^(1+m)*(a+b*arccosh(d*x+c))^3/d/e/(1+m)-3*b*Unintegrable((e*(d*x+c))^(1+m)*(a+b*arccosh(d*x+c))^2/(
d*x+c-1)^(1/2)/(d*x+c+1)^(1/2),x)/e/(1+m)

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Rubi [A]  time = 0.29, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int (c e+d e x)^m \left (a+b \cosh ^{-1}(c+d x)\right )^3 \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

((e*(c + d*x))^(1 + m)*(a + b*ArcCosh[c + d*x])^3)/(d*e*(1 + m)) - (3*b*Defer[Subst][Defer[Int][((e*x)^(1 + m)
*(a + b*ArcCosh[x])^2)/(Sqrt[-1 + x]*Sqrt[1 + x]), x], x, c + d*x])/(d*e*(1 + m))

Rubi steps

\begin {align*} \int (c e+d e x)^m \left (a+b \cosh ^{-1}(c+d x)\right )^3 \, dx &=\frac {\operatorname {Subst}\left (\int (e x)^m \left (a+b \cosh ^{-1}(x)\right )^3 \, dx,x,c+d x\right )}{d}\\ &=\frac {(e (c+d x))^{1+m} \left (a+b \cosh ^{-1}(c+d x)\right )^3}{d e (1+m)}-\frac {(3 b) \operatorname {Subst}\left (\int \frac {(e x)^{1+m} \left (a+b \cosh ^{-1}(x)\right )^2}{\sqrt {-1+x} \sqrt {1+x}} \, dx,x,c+d x\right )}{d e (1+m)}\\ \end {align*}

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Mathematica [A]  time = 1.98, size = 0, normalized size = 0.00 \[ \int (c e+d e x)^m \left (a+b \cosh ^{-1}(c+d x)\right )^3 \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.64, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (b^{3} \operatorname {arcosh}\left (d x + c\right )^{3} + 3 \, a b^{2} \operatorname {arcosh}\left (d x + c\right )^{2} + 3 \, a^{2} b \operatorname {arcosh}\left (d x + c\right ) + a^{3}\right )} {\left (d e x + c e\right )}^{m}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (b \operatorname {arcosh}\left (d x + c\right ) + a\right )}^{3} {\left (d e x + c e\right )}^{m}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 2.54, size = 0, normalized size = 0.00 \[ \int \left (d e x +c e \right )^{m} \left (a +b \,\mathrm {arccosh}\left (d x +c \right )\right )^{3}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*e*x+c*e)^m*(a+b*arccosh(d*x+c))^3,x)

[Out]

int((d*e*x+c*e)^m*(a+b*arccosh(d*x+c))^3,x)

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maxima [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {{\left (b^{3} d e^{m} x + b^{3} c e^{m}\right )} {\left (d x + c\right )}^{m} \log \left (d x + \sqrt {d x + c + 1} \sqrt {d x + c - 1} + c\right )^{3}}{d {\left (m + 1\right )}} + \frac {{\left (d e x + c e\right )}^{m + 1} a^{3}}{d e {\left (m + 1\right )}} + \int -\frac {3 \, {\left ({\left ({\left (b^{3} c^{2} e^{m} - {\left (c^{2} e^{m} {\left (m + 1\right )} - e^{m} {\left (m + 1\right )}\right )} a b^{2} - {\left (a b^{2} d^{2} e^{m} {\left (m + 1\right )} - b^{3} d^{2} e^{m}\right )} x^{2} - 2 \, {\left (a b^{2} c d e^{m} {\left (m + 1\right )} - b^{3} c d e^{m}\right )} x\right )} \sqrt {d x + c + 1} \sqrt {d x + c - 1} {\left (d x + c\right )}^{m} - {\left ({\left (c^{3} e^{m} {\left (m + 1\right )} - c e^{m} {\left (m + 1\right )}\right )} a b^{2} - {\left (c^{3} e^{m} - c e^{m}\right )} b^{3} + {\left (a b^{2} d^{3} e^{m} {\left (m + 1\right )} - b^{3} d^{3} e^{m}\right )} x^{3} + 3 \, {\left (a b^{2} c d^{2} e^{m} {\left (m + 1\right )} - b^{3} c d^{2} e^{m}\right )} x^{2} + {\left ({\left (3 \, c^{2} d e^{m} {\left (m + 1\right )} - d e^{m} {\left (m + 1\right )}\right )} a b^{2} - {\left (3 \, c^{2} d e^{m} - d e^{m}\right )} b^{3}\right )} x\right )} {\left (d x + c\right )}^{m}\right )} \log \left (d x + \sqrt {d x + c + 1} \sqrt {d x + c - 1} + c\right )^{2} - {\left ({\left (a^{2} b d^{2} e^{m} {\left (m + 1\right )} x^{2} + 2 \, a^{2} b c d e^{m} {\left (m + 1\right )} x + {\left (c^{2} e^{m} {\left (m + 1\right )} - e^{m} {\left (m + 1\right )}\right )} a^{2} b\right )} \sqrt {d x + c + 1} \sqrt {d x + c - 1} {\left (d x + c\right )}^{m} + {\left (a^{2} b d^{3} e^{m} {\left (m + 1\right )} x^{3} + 3 \, a^{2} b c d^{2} e^{m} {\left (m + 1\right )} x^{2} + {\left (3 \, c^{2} d e^{m} {\left (m + 1\right )} - d e^{m} {\left (m + 1\right )}\right )} a^{2} b x + {\left (c^{3} e^{m} {\left (m + 1\right )} - c e^{m} {\left (m + 1\right )}\right )} a^{2} b\right )} {\left (d x + c\right )}^{m}\right )} \log \left (d x + \sqrt {d x + c + 1} \sqrt {d x + c - 1} + c\right )\right )}}{d^{3} {\left (m + 1\right )} x^{3} + 3 \, c d^{2} {\left (m + 1\right )} x^{2} + c^{3} {\left (m + 1\right )} + {\left (d^{2} {\left (m + 1\right )} x^{2} + 2 \, c d {\left (m + 1\right )} x + c^{2} {\left (m + 1\right )} - m - 1\right )} \sqrt {d x + c + 1} \sqrt {d x + c - 1} - c {\left (m + 1\right )} + {\left (3 \, c^{2} d {\left (m + 1\right )} - d {\left (m + 1\right )}\right )} x}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(b^3*d*e^m*x + b^3*c*e^m)*(d*x + c)^m*log(d*x + sqrt(d*x + c + 1)*sqrt(d*x + c - 1) + c)^3/(d*(m + 1)) + (d*e*
x + c*e)^(m + 1)*a^3/(d*e*(m + 1)) + integrate(-3*(((b^3*c^2*e^m - (c^2*e^m*(m + 1) - e^m*(m + 1))*a*b^2 - (a*
b^2*d^2*e^m*(m + 1) - b^3*d^2*e^m)*x^2 - 2*(a*b^2*c*d*e^m*(m + 1) - b^3*c*d*e^m)*x)*sqrt(d*x + c + 1)*sqrt(d*x
+ c - 1)*(d*x + c)^m - ((c^3*e^m*(m + 1) - c*e^m*(m + 1))*a*b^2 - (c^3*e^m - c*e^m)*b^3 + (a*b^2*d^3*e^m*(m +
1) - b^3*d^3*e^m)*x^3 + 3*(a*b^2*c*d^2*e^m*(m + 1) - b^3*c*d^2*e^m)*x^2 + ((3*c^2*d*e^m*(m + 1) - d*e^m*(m +
1))*a*b^2 - (3*c^2*d*e^m - d*e^m)*b^3)*x)*(d*x + c)^m)*log(d*x + sqrt(d*x + c + 1)*sqrt(d*x + c - 1) + c)^2 -
((a^2*b*d^2*e^m*(m + 1)*x^2 + 2*a^2*b*c*d*e^m*(m + 1)*x + (c^2*e^m*(m + 1) - e^m*(m + 1))*a^2*b)*sqrt(d*x + c
+ 1)*sqrt(d*x + c - 1)*(d*x + c)^m + (a^2*b*d^3*e^m*(m + 1)*x^3 + 3*a^2*b*c*d^2*e^m*(m + 1)*x^2 + (3*c^2*d*e^m
*(m + 1) - d*e^m*(m + 1))*a^2*b*x + (c^3*e^m*(m + 1) - c*e^m*(m + 1))*a^2*b)*(d*x + c)^m)*log(d*x + sqrt(d*x +
c + 1)*sqrt(d*x + c - 1) + c))/(d^3*(m + 1)*x^3 + 3*c*d^2*(m + 1)*x^2 + c^3*(m + 1) + (d^2*(m + 1)*x^2 + 2*c*
d*(m + 1)*x + c^2*(m + 1) - m - 1)*sqrt(d*x + c + 1)*sqrt(d*x + c - 1) - c*(m + 1) + (3*c^2*d*(m + 1) - d*(m +
1))*x), x)

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mupad [A]  time = 0.00, size = -1, normalized size = -0.01 \[ \int {\left (c\,e+d\,e\,x\right )}^m\,{\left (a+b\,\mathrm {acosh}\left (c+d\,x\right )\right )}^3 \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (e \left (c + d x\right )\right )^{m} \left (a + b \operatorname {acosh}{\left (c + d x \right )}\right )^{3}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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