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

Optimal. Leaf size=82 \[ \frac {(d+e x)^{m+1} \left (a+b \cosh ^{-1}(c x)\right )^3}{e (m+1)}-\frac {3 b c \text {Int}\left (\frac {(d+e x)^{m+1} \left (a+b \cosh ^{-1}(c x)\right )^2}{\sqrt {c x-1} \sqrt {c x+1}},x\right )}{e (m+1)} \]

[Out]

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

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Rubi [A]  time = 0.38, 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 = {} \[ \int (d+e x)^m \left (a+b \cosh ^{-1}(c x)\right )^3 \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]  time = 6.70, size = 0, normalized size = 0.00 \[ \int (d+e x)^m \left (a+b \cosh ^{-1}(c x)\right )^3 \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.75, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (b^{3} \operatorname {arcosh}\left (c x\right )^{3} + 3 \, a b^{2} \operatorname {arcosh}\left (c x\right )^{2} + 3 \, a^{2} b \operatorname {arcosh}\left (c x\right ) + a^{3}\right )} {\left (e x + d\right )}^{m}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 4.04, size = 0, normalized size = 0.00 \[ \int \left (e x +d \right )^{m} \left (a +b \,\mathrm {arccosh}\left (c x \right )\right )^{3}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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