3.2.46 \(\int (c e+d e x)^m (a+b \sinh ^{-1}(c+d x))^4 \, dx\) [146]

Optimal. Leaf size=87 \[ \frac {(e (c+d x))^{1+m} \left (a+b \sinh ^{-1}(c+d x)\right )^4}{d e (1+m)}-\frac {4 b \text {Int}\left (\frac {(e (c+d x))^{1+m} \left (a+b \sinh ^{-1}(c+d x)\right )^3}{\sqrt {1+(c+d x)^2}},x\right )}{e (1+m)} \]

[Out]

(e*(d*x+c))^(1+m)*(a+b*arcsinh(d*x+c))^4/d/e/(1+m)-4*b*Unintegrable((e*(d*x+c))^(1+m)*(a+b*arcsinh(d*x+c))^3/(
1+(d*x+c)^2)^(1/2),x)/e/(1+m)

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Rubi [A]
time = 0.12, 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 = {} \begin {gather*} \int (c e+d e x)^m \left (a+b \sinh ^{-1}(c+d x)\right )^4 \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(c*e + d*e*x)^m*(a + b*ArcSinh[c + d*x])^4,x]

[Out]

((e*(c + d*x))^(1 + m)*(a + b*ArcSinh[c + d*x])^4)/(d*e*(1 + m)) - (4*b*Defer[Subst][Defer[Int][((e*x)^(1 + m)
*(a + b*ArcSinh[x])^3)/Sqrt[1 + x^2], x], x, c + d*x])/(d*e*(1 + m))

Rubi steps

\begin {align*} \int (c e+d e x)^m \left (a+b \sinh ^{-1}(c+d x)\right )^4 \, dx &=\frac {\text {Subst}\left (\int (e x)^m \left (a+b \sinh ^{-1}(x)\right )^4 \, dx,x,c+d x\right )}{d}\\ &=\frac {(e (c+d x))^{1+m} \left (a+b \sinh ^{-1}(c+d x)\right )^4}{d e (1+m)}-\frac {(4 b) \text {Subst}\left (\int \frac {(e x)^{1+m} \left (a+b \sinh ^{-1}(x)\right )^3}{\sqrt {1+x^2}} \, dx,x,c+d x\right )}{d e (1+m)}\\ \end {align*}

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Mathematica [A]
time = 1.47, size = 0, normalized size = 0.00 \begin {gather*} \int (c e+d e x)^m \left (a+b \sinh ^{-1}(c+d x)\right )^4 \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[(c*e + d*e*x)^m*(a + b*ArcSinh[c + d*x])^4,x]

[Out]

Integrate[(c*e + d*e*x)^m*(a + b*ArcSinh[c + d*x])^4, x]

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Maple [A]
time = 180.00, size = 0, normalized size = 0.00 \[\int \left (d e x +c e \right )^{m} \left (a +b \arcsinh \left (d x +c \right )\right )^{4}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*e*x+c*e)^m*(a+b*arcsinh(d*x+c))^4,x)

[Out]

int((d*e*x+c*e)^m*(a+b*arcsinh(d*x+c))^4,x)

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Maxima [A]
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)^m*(a+b*arcsinh(d*x+c))^4,x, algorithm="maxima")

[Out]

(d*x*e + c*e)^(m + 1)*a^4*e^(-1)/(d*(m + 1)) + (b^4*d*x*e^m + b^4*c*e^m)*(d*x + c)^m*log(d*x + c + sqrt(d^2*x^
2 + 2*c*d*x + c^2 + 1))^4/(d*(m + 1)) + integrate(-2*(2*((b^4*c^2*e^m - ((m*e^m + e^m)*c^2 + m*e^m + e^m)*a*b^
3 - ((m*e^m + e^m)*a*b^3*d^2 - b^4*d^2*e^m)*x^2 - 2*((m*e^m + e^m)*a*b^3*c*d - b^4*c*d*e^m)*x)*sqrt(d^2*x^2 +
2*c*d*x + c^2 + 1)*(d*x + c)^m - (((m*e^m + e^m)*c^3 + (m*e^m + e^m)*c)*a*b^3 - (c^3*e^m + c*e^m)*b^4 + ((m*e^
m + e^m)*a*b^3*d^3 - b^4*d^3*e^m)*x^3 + 3*((m*e^m + e^m)*a*b^3*c*d^2 - b^4*c*d^2*e^m)*x^2 + ((3*(m*e^m + e^m)*
c^2*d + (m*e^m + e^m)*d)*a*b^3 - (3*c^2*d*e^m + d*e^m)*b^4)*x)*(d*x + c)^m)*log(d*x + c + sqrt(d^2*x^2 + 2*c*d
*x + c^2 + 1))^3 - 3*(((m*e^m + e^m)*a^2*b^2*d^2*x^2 + 2*(m*e^m + e^m)*a^2*b^2*c*d*x + ((m*e^m + e^m)*c^2 + m*
e^m + e^m)*a^2*b^2)*sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1)*(d*x + c)^m + ((m*e^m + e^m)*a^2*b^2*d^3*x^3 + 3*(m*e^m
+ e^m)*a^2*b^2*c*d^2*x^2 + (3*(m*e^m + e^m)*c^2*d + (m*e^m + e^m)*d)*a^2*b^2*x + ((m*e^m + e^m)*c^3 + (m*e^m +
e^m)*c)*a^2*b^2)*(d*x + c)^m)*log(d*x + c + sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1))^2 - 2*(((m*e^m + e^m)*a^3*b*d^
2*x^2 + 2*(m*e^m + e^m)*a^3*b*c*d*x + ((m*e^m + e^m)*c^2 + m*e^m + e^m)*a^3*b)*sqrt(d^2*x^2 + 2*c*d*x + c^2 +
1)*(d*x + c)^m + ((m*e^m + e^m)*a^3*b*d^3*x^3 + 3*(m*e^m + e^m)*a^3*b*c*d^2*x^2 + (3*(m*e^m + e^m)*c^2*d + (m*
e^m + e^m)*d)*a^3*b*x + ((m*e^m + e^m)*c^3 + (m*e^m + e^m)*c)*a^3*b)*(d*x + c)^m)*log(d*x + c + sqrt(d^2*x^2 +
2*c*d*x + c^2 + 1)))/(d^3*(m + 1)*x^3 + 3*c*d^2*(m + 1)*x^2 + c^3*(m + 1) + c*(m + 1) + (3*c^2*d*(m + 1) + d*
(m + 1))*x + (d^2*(m + 1)*x^2 + 2*c*d*(m + 1)*x + c^2*(m + 1) + m + 1)*sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1)), x)

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Fricas [A]
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)^m*(a+b*arcsinh(d*x+c))^4,x, algorithm="fricas")

[Out]

integral((b^4*arcsinh(d*x + c)^4 + 4*a*b^3*arcsinh(d*x + c)^3 + 6*a^2*b^2*arcsinh(d*x + c)^2 + 4*a^3*b*arcsinh
(d*x + c) + a^4)*((d*x + c)*e)^m, x)

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Sympy [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (e \left (c + d x\right )\right )^{m} \left (a + b \operatorname {asinh}{\left (c + d x \right )}\right )^{4}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*e*x+c*e)**m*(a+b*asinh(d*x+c))**4,x)

[Out]

Integral((e*(c + d*x))**m*(a + b*asinh(c + d*x))**4, x)

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Giac [A]
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)^m*(a+b*arcsinh(d*x+c))^4,x, algorithm="giac")

[Out]

integrate((b*arcsinh(d*x + c) + a)^4*(d*e*x + c*e)^m, x)

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Mupad [A]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int {\left (c\,e+d\,e\,x\right )}^m\,{\left (a+b\,\mathrm {asinh}\left (c+d\,x\right )\right )}^4 \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*e + d*e*x)^m*(a + b*asinh(c + d*x))^4,x)

[Out]

int((c*e + d*e*x)^m*(a + b*asinh(c + d*x))^4, x)

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