3.255 \(\int \sqrt {c e+d e x} (a+b \sinh ^{-1}(c+d x))^4 \, dx\)

Optimal. Leaf size=82 \[ \frac {2 (e (c+d x))^{3/2} \left (a+b \sinh ^{-1}(c+d x)\right )^4}{3 d e}-\frac {8 b \text {Int}\left (\frac {(e (c+d x))^{3/2} \left (a+b \sinh ^{-1}(c+d x)\right )^3}{\sqrt {(c+d x)^2+1}},x\right )}{3 e} \]

[Out]

2/3*(e*(d*x+c))^(3/2)*(a+b*arcsinh(d*x+c))^4/d/e-8/3*b*Unintegrable((e*(d*x+c))^(3/2)*(a+b*arcsinh(d*x+c))^3/(
1+(d*x+c)^2)^(1/2),x)/e

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Rubi [A]  time = 0.19, 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 = {} \[ \int \sqrt {c e+d e x} \left (a+b \sinh ^{-1}(c+d x)\right )^4 \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

(2*(e*(c + d*x))^(3/2)*(a + b*ArcSinh[c + d*x])^4)/(3*d*e) - (8*b*Defer[Subst][Defer[Int][((e*x)^(3/2)*(a + b*
ArcSinh[x])^3)/Sqrt[1 + x^2], x], x, c + d*x])/(3*d*e)

Rubi steps

\begin {align*} \int \sqrt {c e+d e x} \left (a+b \sinh ^{-1}(c+d x)\right )^4 \, dx &=\frac {\operatorname {Subst}\left (\int \sqrt {e x} \left (a+b \sinh ^{-1}(x)\right )^4 \, dx,x,c+d x\right )}{d}\\ &=\frac {2 (e (c+d x))^{3/2} \left (a+b \sinh ^{-1}(c+d x)\right )^4}{3 d e}-\frac {(8 b) \operatorname {Subst}\left (\int \frac {(e x)^{3/2} \left (a+b \sinh ^{-1}(x)\right )^3}{\sqrt {1+x^2}} \, dx,x,c+d x\right )}{3 d e}\\ \end {align*}

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Mathematica [A]  time = 120.92, size = 0, normalized size = 0.00 \[ \int \sqrt {c e+d e x} \left (a+b \sinh ^{-1}(c+d x)\right )^4 \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.89, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (b^{4} \operatorname {arsinh}\left (d x + c\right )^{4} + 4 \, a b^{3} \operatorname {arsinh}\left (d x + c\right )^{3} + 6 \, a^{2} b^{2} \operatorname {arsinh}\left (d x + c\right )^{2} + 4 \, a^{3} b \operatorname {arsinh}\left (d x + c\right ) + a^{4}\right )} \sqrt {d e x + c e}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arcsinh(d*x+c))^4*(d*e*x+c*e)^(1/2),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)*sqrt(d*e*x + c*e), x)

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giac [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {d e x + c e} {\left (b \operatorname {arsinh}\left (d x + c\right ) + a\right )}^{4}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arcsinh(d*x+c))^4*(d*e*x+c*e)^(1/2),x, algorithm="giac")

[Out]

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

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maple [F(-2)]  time = 180.00, size = 0, normalized size = 0.00 \[ \int \left (a +b \arcsinh \left (d x +c \right )\right )^{4} \sqrt {d e x +c e}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*arcsinh(d*x+c))^4*(d*e*x+c*e)^(1/2),x)

[Out]

int((a+b*arcsinh(d*x+c))^4*(d*e*x+c*e)^(1/2),x)

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maxima [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {2 \, {\left (b^{4} d \sqrt {e} x + b^{4} c \sqrt {e}\right )} \sqrt {d x + c} \log \left (d x + c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1}\right )^{4}}{3 \, d} + \frac {2 \, {\left (d e x + c e\right )}^{\frac {3}{2}} a^{4}}{3 \, d e} + \int -\frac {2 \, {\left (2 \, {\left ({\left (2 \, b^{4} c^{2} \sqrt {e} - 3 \, {\left (c^{2} \sqrt {e} + \sqrt {e}\right )} a b^{3} - {\left (3 \, a b^{3} d^{2} \sqrt {e} - 2 \, b^{4} d^{2} \sqrt {e}\right )} x^{2} - 2 \, {\left (3 \, a b^{3} c d \sqrt {e} - 2 \, b^{4} c d \sqrt {e}\right )} x\right )} \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1} \sqrt {d x + c} - {\left (3 \, {\left (c^{3} \sqrt {e} + c \sqrt {e}\right )} a b^{3} - 2 \, {\left (c^{3} \sqrt {e} + c \sqrt {e}\right )} b^{4} + {\left (3 \, a b^{3} d^{3} \sqrt {e} - 2 \, b^{4} d^{3} \sqrt {e}\right )} x^{3} + 3 \, {\left (3 \, a b^{3} c d^{2} \sqrt {e} - 2 \, b^{4} c d^{2} \sqrt {e}\right )} x^{2} + {\left (3 \, {\left (3 \, c^{2} d \sqrt {e} + d \sqrt {e}\right )} a b^{3} - 2 \, {\left (3 \, c^{2} d \sqrt {e} + d \sqrt {e}\right )} b^{4}\right )} x\right )} \sqrt {d x + c}\right )} \log \left (d x + c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1}\right )^{3} - 9 \, {\left ({\left (a^{2} b^{2} d^{2} \sqrt {e} x^{2} + 2 \, a^{2} b^{2} c d \sqrt {e} x + {\left (c^{2} \sqrt {e} + \sqrt {e}\right )} a^{2} b^{2}\right )} \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1} \sqrt {d x + c} + {\left (a^{2} b^{2} d^{3} \sqrt {e} x^{3} + 3 \, a^{2} b^{2} c d^{2} \sqrt {e} x^{2} + {\left (3 \, c^{2} d \sqrt {e} + d \sqrt {e}\right )} a^{2} b^{2} x + {\left (c^{3} \sqrt {e} + c \sqrt {e}\right )} a^{2} b^{2}\right )} \sqrt {d x + c}\right )} \log \left (d x + c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1}\right )^{2} - 6 \, {\left ({\left (a^{3} b d^{2} \sqrt {e} x^{2} + 2 \, a^{3} b c d \sqrt {e} x + {\left (c^{2} \sqrt {e} + \sqrt {e}\right )} a^{3} b\right )} \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1} \sqrt {d x + c} + {\left (a^{3} b d^{3} \sqrt {e} x^{3} + 3 \, a^{3} b c d^{2} \sqrt {e} x^{2} + {\left (3 \, c^{2} d \sqrt {e} + d \sqrt {e}\right )} a^{3} b x + {\left (c^{3} \sqrt {e} + c \sqrt {e}\right )} a^{3} b\right )} \sqrt {d x + c}\right )} \log \left (d x + c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1}\right )\right )}}{3 \, {\left (d^{3} x^{3} + 3 \, c d^{2} x^{2} + c^{3} + {\left (3 \, c^{2} d + d\right )} x + {\left (d^{2} x^{2} + 2 \, c d x + c^{2} + 1\right )}^{\frac {3}{2}} + c\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arcsinh(d*x+c))^4*(d*e*x+c*e)^(1/2),x, algorithm="maxima")

[Out]

2/3*(b^4*d*sqrt(e)*x + b^4*c*sqrt(e))*sqrt(d*x + c)*log(d*x + c + sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1))^4/d + 2/3
*(d*e*x + c*e)^(3/2)*a^4/(d*e) + integrate(-2/3*(2*((2*b^4*c^2*sqrt(e) - 3*(c^2*sqrt(e) + sqrt(e))*a*b^3 - (3*
a*b^3*d^2*sqrt(e) - 2*b^4*d^2*sqrt(e))*x^2 - 2*(3*a*b^3*c*d*sqrt(e) - 2*b^4*c*d*sqrt(e))*x)*sqrt(d^2*x^2 + 2*c
*d*x + c^2 + 1)*sqrt(d*x + c) - (3*(c^3*sqrt(e) + c*sqrt(e))*a*b^3 - 2*(c^3*sqrt(e) + c*sqrt(e))*b^4 + (3*a*b^
3*d^3*sqrt(e) - 2*b^4*d^3*sqrt(e))*x^3 + 3*(3*a*b^3*c*d^2*sqrt(e) - 2*b^4*c*d^2*sqrt(e))*x^2 + (3*(3*c^2*d*sqr
t(e) + d*sqrt(e))*a*b^3 - 2*(3*c^2*d*sqrt(e) + d*sqrt(e))*b^4)*x)*sqrt(d*x + c))*log(d*x + c + sqrt(d^2*x^2 +
2*c*d*x + c^2 + 1))^3 - 9*((a^2*b^2*d^2*sqrt(e)*x^2 + 2*a^2*b^2*c*d*sqrt(e)*x + (c^2*sqrt(e) + sqrt(e))*a^2*b^
2)*sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1)*sqrt(d*x + c) + (a^2*b^2*d^3*sqrt(e)*x^3 + 3*a^2*b^2*c*d^2*sqrt(e)*x^2 +
(3*c^2*d*sqrt(e) + d*sqrt(e))*a^2*b^2*x + (c^3*sqrt(e) + c*sqrt(e))*a^2*b^2)*sqrt(d*x + c))*log(d*x + c + sqrt
(d^2*x^2 + 2*c*d*x + c^2 + 1))^2 - 6*((a^3*b*d^2*sqrt(e)*x^2 + 2*a^3*b*c*d*sqrt(e)*x + (c^2*sqrt(e) + sqrt(e))
*a^3*b)*sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1)*sqrt(d*x + c) + (a^3*b*d^3*sqrt(e)*x^3 + 3*a^3*b*c*d^2*sqrt(e)*x^2 +
(3*c^2*d*sqrt(e) + d*sqrt(e))*a^3*b*x + (c^3*sqrt(e) + c*sqrt(e))*a^3*b)*sqrt(d*x + c))*log(d*x + c + sqrt(d^
2*x^2 + 2*c*d*x + c^2 + 1)))/(d^3*x^3 + 3*c*d^2*x^2 + c^3 + (3*c^2*d + d)*x + (d^2*x^2 + 2*c*d*x + c^2 + 1)^(3
/2) + c), x)

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mupad [A]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \sqrt {c\,e+d\,e\,x}\,{\left (a+b\,\mathrm {asinh}\left (c+d\,x\right )\right )}^4 \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*e + d*e*x)^(1/2)*(a + b*asinh(c + d*x))^4,x)

[Out]

int((c*e + d*e*x)^(1/2)*(a + b*asinh(c + d*x))^4, x)

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sympy [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {e \left (c + d x\right )} \left (a + b \operatorname {asinh}{\left (c + d x \right )}\right )^{4}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*asinh(d*x+c))**4*(d*e*x+c*e)**(1/2),x)

[Out]

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

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