3.3.45 \(\int (c e+d e x)^{5/2} (a+b \sinh ^{-1}(c+d x))^3 \, dx\) [245]

Optimal. Leaf size=82 \[ \frac {2 (e (c+d x))^{7/2} \left (a+b \sinh ^{-1}(c+d x)\right )^3}{7 d e}-\frac {6 b \text {Int}\left (\frac {(e (c+d x))^{7/2} \left (a+b \sinh ^{-1}(c+d x)\right )^2}{\sqrt {1+(c+d x)^2}},x\right )}{7 e} \]

[Out]

2/7*(e*(d*x+c))^(7/2)*(a+b*arcsinh(d*x+c))^3/d/e-6/7*b*Unintegrable((e*(d*x+c))^(7/2)*(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)^{5/2} \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)^(5/2)*(a + b*ArcSinh[c + d*x])^3,x]

[Out]

(2*(e*(c + d*x))^(7/2)*(a + b*ArcSinh[c + d*x])^3)/(7*d*e) - (6*b*Defer[Subst][Defer[Int][((e*x)^(7/2)*(a + b*
ArcSinh[x])^2)/Sqrt[1 + x^2], x], x, c + d*x])/(7*d*e)

Rubi steps

\begin {align*} \int (c e+d e x)^{5/2} \left (a+b \sinh ^{-1}(c+d x)\right )^3 \, dx &=\frac {\text {Subst}\left (\int (e x)^{5/2} \left (a+b \sinh ^{-1}(x)\right )^3 \, dx,x,c+d x\right )}{d}\\ &=\frac {2 (e (c+d x))^{7/2} \left (a+b \sinh ^{-1}(c+d x)\right )^3}{7 d e}-\frac {(6 b) \text {Subst}\left (\int \frac {(e x)^{7/2} \left (a+b \sinh ^{-1}(x)\right )^2}{\sqrt {1+x^2}} \, dx,x,c+d x\right )}{7 d e}\\ \end {align*}

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Mathematica [F]
time = 180.01, size = 0, normalized size = 0.00 \begin {gather*} \text {\$Aborted} \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[(c*e + d*e*x)^(5/2)*(a + b*ArcSinh[c + d*x])^3,x]

[Out]

$Aborted

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Maple [A]
time = 180.00, size = 0, normalized size = 0.00 \[\int \left (d e x +c e \right )^{\frac {5}{2}} \left (a +b \arcsinh \left (d x +c \right )\right )^{3}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*e*x+c*e)^(5/2)*(a+b*arcsinh(d*x+c))^3,x)

[Out]

int((d*e*x+c*e)^(5/2)*(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)^(5/2)*(a+b*arcsinh(d*x+c))^3,x, algorithm="maxima")

[Out]

2/7*(d*x*e + c*e)^(7/2)*a^3*e^(-1)/d + 2/7*(b^3*d^3*x^3*e^(5/2) + 3*b^3*c*d^2*x^2*e^(5/2) + 3*b^3*c^2*d*x*e^(5
/2) + b^3*c^3*e^(5/2))*sqrt(d*x + c)*log(d*x + c + sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1))^3/d + integrate(3/7*((((
7*a*b^2*d^4 - 2*b^3*d^4)*x^4*e^(5/2) + 4*(7*a*b^2*c*d^3 - 2*b^3*c*d^3)*x^3*e^(5/2) - (12*b^3*c^2*d^2 - 7*(6*c^
2*d^2 + d^2)*a*b^2)*x^2*e^(5/2) - 2*(4*b^3*c^3*d - 7*(2*c^3*d + c*d)*a*b^2)*x*e^(5/2) - (2*b^3*c^4 - 7*(c^4 +
c^2)*a*b^2)*e^(5/2))*sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1)*sqrt(d*x + c) + ((7*a*b^2*d^5 - 2*b^3*d^5)*x^5*e^(5/2)
+ 5*(7*a*b^2*c*d^4 - 2*b^3*c*d^4)*x^4*e^(5/2) + (7*(10*c^2*d^3 + d^3)*a*b^2 - 2*(10*c^2*d^3 + d^3)*b^3)*x^3*e^
(5/2) + (7*(10*c^3*d^2 + 3*c*d^2)*a*b^2 - 2*(10*c^3*d^2 + 3*c*d^2)*b^3)*x^2*e^(5/2) + (7*(5*c^4*d + 3*c^2*d)*a
*b^2 - 2*(5*c^4*d + 3*c^2*d)*b^3)*x*e^(5/2) + (7*(c^5 + c^3)*a*b^2 - 2*(c^5 + c^3)*b^3)*e^(5/2))*sqrt(d*x + c)
)*log(d*x + c + sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1))^2 + 7*((a^2*b*d^4*x^4*e^(5/2) + 4*a^2*b*c*d^3*x^3*e^(5/2) +
(6*c^2*d^2 + d^2)*a^2*b*x^2*e^(5/2) + 2*(2*c^3*d + c*d)*a^2*b*x*e^(5/2) + (c^4 + c^2)*a^2*b*e^(5/2))*sqrt(d^2
*x^2 + 2*c*d*x + c^2 + 1)*sqrt(d*x + c) + (a^2*b*d^5*x^5*e^(5/2) + 5*a^2*b*c*d^4*x^4*e^(5/2) + (10*c^2*d^3 + d
^3)*a^2*b*x^3*e^(5/2) + (10*c^3*d^2 + 3*c*d^2)*a^2*b*x^2*e^(5/2) + (5*c^4*d + 3*c^2*d)*a^2*b*x*e^(5/2) + (c^5
+ c^3)*a^2*b*e^(5/2))*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)^(5/2)*(a+b*arcsinh(d*x+c))^3,x, algorithm="fricas")

[Out]

integral(((b^3*d^2*x^2 + 2*b^3*c*d*x + b^3*c^2)*arcsinh(d*x + c)^3*e^2 + 3*(a*b^2*d^2*x^2 + 2*a*b^2*c*d*x + a*
b^2*c^2)*arcsinh(d*x + c)^2*e^2 + 3*(a^2*b*d^2*x^2 + 2*a^2*b*c*d*x + a^2*b*c^2)*arcsinh(d*x + c)*e^2 + (a^3*d^
2*x^2 + 2*a^3*c*d*x + a^3*c^2)*e^2)*sqrt(d*x + c)*e^(1/2), x)

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*e*x+c*e)**(5/2)*(a+b*asinh(d*x+c))**3,x)

[Out]

Timed out

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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)^(5/2)*(a+b*arcsinh(d*x+c))^3,x, algorithm="giac")

[Out]

integrate((d*e*x + c*e)^(5/2)*(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 )}^{5/2}\,{\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)^(5/2)*(a + b*asinh(c + d*x))^3,x)

[Out]

int((c*e + d*e*x)^(5/2)*(a + b*asinh(c + d*x))^3, x)

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