3.3.46 \(\int (c e+d e x)^{3/2} (a+b \sinh ^{-1}(c+d x))^3 \, dx\) [246]

Optimal. Leaf size=82 \[ \frac {2 (e (c+d x))^{5/2} \left (a+b \sinh ^{-1}(c+d x)\right )^3}{5 d e}-\frac {6 b \text {Int}\left (\frac {(e (c+d x))^{5/2} \left (a+b \sinh ^{-1}(c+d x)\right )^2}{\sqrt {1+(c+d x)^2}},x\right )}{5 e} \]

[Out]

2/5*(e*(d*x+c))^(5/2)*(a+b*arcsinh(d*x+c))^3/d/e-6/5*b*Unintegrable((e*(d*x+c))^(5/2)*(a+b*arcsinh(d*x+c))^2/(
1+(d*x+c)^2)^(1/2),x)/e

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Rubi [A]
time = 0.13, 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)^{3/2} \left (a+b \sinh ^{-1}(c+d x)\right )^3 \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

(2*(e*(c + d*x))^(5/2)*(a + b*ArcSinh[c + d*x])^3)/(5*d*e) - (6*b*Defer[Subst][Defer[Int][((e*x)^(5/2)*(a + b*
ArcSinh[x])^2)/Sqrt[1 + x^2], x], x, c + d*x])/(5*d*e)

Rubi steps

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

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Mathematica [A]
time = 156.26, size = 0, normalized size = 0.00 \begin {gather*} \int (c e+d e x)^{3/2} \left (a+b \sinh ^{-1}(c+d x)\right )^3 \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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Maple [A]
time = 180.00, size = 0, normalized size = 0.00 \[\int \left (d e x +c e \right )^{\frac {3}{2}} \left (a +b \arcsinh \left (d x +c \right )\right )^{3}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

2/5*(d*x*e + c*e)^(5/2)*a^3*e^(-1)/d + 2/5*(b^3*d^2*x^2*e^(3/2) + 2*b^3*c*d*x*e^(3/2) + b^3*c^2*e^(3/2))*sqrt(
d*x + c)*log(d*x + c + sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1))^3/d + integrate(3/5*((((5*a*b^2*d^3 - 2*b^3*d^3)*x^3
*e^(3/2) + 3*(5*a*b^2*c*d^2 - 2*b^3*c*d^2)*x^2*e^(3/2) - (6*b^3*c^2*d - 5*(3*c^2*d + d)*a*b^2)*x*e^(3/2) - (2*
b^3*c^3 - 5*(c^3 + c)*a*b^2)*e^(3/2))*sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1)*sqrt(d*x + c) + ((5*a*b^2*d^4 - 2*b^3*
d^4)*x^4*e^(3/2) + 4*(5*a*b^2*c*d^3 - 2*b^3*c*d^3)*x^3*e^(3/2) + (5*(6*c^2*d^2 + d^2)*a*b^2 - 2*(6*c^2*d^2 + d
^2)*b^3)*x^2*e^(3/2) + 2*(5*(2*c^3*d + c*d)*a*b^2 - 2*(2*c^3*d + c*d)*b^3)*x*e^(3/2) + (5*(c^4 + c^2)*a*b^2 -
2*(c^4 + c^2)*b^3)*e^(3/2))*sqrt(d*x + c))*log(d*x + c + sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1))^2 + 5*((a^2*b*d^3*
x^3*e^(3/2) + 3*a^2*b*c*d^2*x^2*e^(3/2) + (3*c^2*d + d)*a^2*b*x*e^(3/2) + (c^3 + c)*a^2*b*e^(3/2))*sqrt(d^2*x^
2 + 2*c*d*x + c^2 + 1)*sqrt(d*x + c) + (a^2*b*d^4*x^4*e^(3/2) + 4*a^2*b*c*d^3*x^3*e^(3/2) + (6*c^2*d^2 + d^2)*
a^2*b*x^2*e^(3/2) + 2*(2*c^3*d + c*d)*a^2*b*x*e^(3/2) + (c^4 + c^2)*a^2*b*e^(3/2))*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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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)^(3/2)*(a+b*arcsinh(d*x+c))^3,x, algorithm="fricas")

[Out]

integral(((b^3*d*x + b^3*c)*arcsinh(d*x + c)^3*e + 3*(a*b^2*d*x + a*b^2*c)*arcsinh(d*x + c)^2*e + 3*(a^2*b*d*x
+ a^2*b*c)*arcsinh(d*x + c)*e + (a^3*d*x + a^3*c)*e)*sqrt(d*x + c)*e^(1/2), 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 )^{\frac {3}{2}} \left (a + b \operatorname {asinh}{\left (c + d x \right )}\right )^{3}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((e*(c + d*x))**(3/2)*(a + b*asinh(c + d*x))**3, 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)^(3/2)*(a+b*arcsinh(d*x+c))^3,x, algorithm="giac")

[Out]

integrate((d*e*x + c*e)^(3/2)*(b*arcsinh(d*x + c) + a)^3, 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 )}^{3/2}\,{\left (a+b\,\mathrm {asinh}\left (c+d\,x\right )\right )}^3 \,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*asinh(c + d*x))^3,x)

[Out]

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

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