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

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

[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 = 0.21, 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 = {} \[ \int (c e+d e x)^{3/2} \left (a+b \sinh ^{-1}(c+d x)\right )^3 \, dx \]

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 {\operatorname {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) \operatorname {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 = 73.21, size = 0, normalized size = 0.00 \[ \int (c e+d e x)^{3/2} \left (a+b \sinh ^{-1}(c+d x)\right )^3 \, dx \]

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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fricas [A]  time = 1.06, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (a^{3} d e x + a^{3} c e + {\left (b^{3} d e x + b^{3} c e\right )} \operatorname {arsinh}\left (d x + c\right )^{3} + 3 \, {\left (a b^{2} d e x + a b^{2} c e\right )} \operatorname {arsinh}\left (d x + c\right )^{2} + 3 \, {\left (a^{2} b d e x + a^{2} b c e\right )} \operatorname {arsinh}\left (d x + c\right )\right )} \sqrt {d e x + c e}, x\right ) \]

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

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

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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maple [F(-2)]  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 \[ \frac {2 \, {\left (d e x + c e\right )}^{\frac {5}{2}} a^{3}}{5 \, d e} + \frac {2 \, {\left (b^{3} d^{2} e^{\frac {3}{2}} x^{2} + 2 \, b^{3} c d e^{\frac {3}{2}} x + b^{3} c^{2} e^{\frac {3}{2}}\right )} \sqrt {d x + c} \log \left (d x + c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1}\right )^{3}}{5 \, d} + \int -\frac {3 \, {\left ({\left ({\left (2 \, b^{3} c^{3} e^{\frac {3}{2}} - 5 \, {\left (c^{3} e^{\frac {3}{2}} + c e^{\frac {3}{2}}\right )} a b^{2} - {\left (5 \, a b^{2} d^{3} e^{\frac {3}{2}} - 2 \, b^{3} d^{3} e^{\frac {3}{2}}\right )} x^{3} - 3 \, {\left (5 \, a b^{2} c d^{2} e^{\frac {3}{2}} - 2 \, b^{3} c d^{2} e^{\frac {3}{2}}\right )} x^{2} + {\left (6 \, b^{3} c^{2} d e^{\frac {3}{2}} - 5 \, {\left (3 \, c^{2} d e^{\frac {3}{2}} + d e^{\frac {3}{2}}\right )} a b^{2}\right )} x\right )} \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1} \sqrt {d x + c} - {\left ({\left (5 \, a b^{2} d^{4} e^{\frac {3}{2}} - 2 \, b^{3} d^{4} e^{\frac {3}{2}}\right )} x^{4} + 5 \, {\left (c^{4} e^{\frac {3}{2}} + c^{2} e^{\frac {3}{2}}\right )} a b^{2} - 2 \, {\left (c^{4} e^{\frac {3}{2}} + c^{2} e^{\frac {3}{2}}\right )} b^{3} + 4 \, {\left (5 \, a b^{2} c d^{3} e^{\frac {3}{2}} - 2 \, b^{3} c d^{3} e^{\frac {3}{2}}\right )} x^{3} + {\left (5 \, {\left (6 \, c^{2} d^{2} e^{\frac {3}{2}} + d^{2} e^{\frac {3}{2}}\right )} a b^{2} - 2 \, {\left (6 \, c^{2} d^{2} e^{\frac {3}{2}} + d^{2} e^{\frac {3}{2}}\right )} b^{3}\right )} x^{2} + 2 \, {\left (5 \, {\left (2 \, c^{3} d e^{\frac {3}{2}} + c d e^{\frac {3}{2}}\right )} a b^{2} - 2 \, {\left (2 \, c^{3} d e^{\frac {3}{2}} + c d e^{\frac {3}{2}}\right )} b^{3}\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 )^{2} - 5 \, {\left ({\left (a^{2} b d^{3} e^{\frac {3}{2}} x^{3} + 3 \, a^{2} b c d^{2} e^{\frac {3}{2}} x^{2} + {\left (3 \, c^{2} d e^{\frac {3}{2}} + d e^{\frac {3}{2}}\right )} a^{2} b x + {\left (c^{3} e^{\frac {3}{2}} + c e^{\frac {3}{2}}\right )} a^{2} b\right )} \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1} \sqrt {d x + c} + {\left (a^{2} b d^{4} e^{\frac {3}{2}} x^{4} + 4 \, a^{2} b c d^{3} e^{\frac {3}{2}} x^{3} + {\left (6 \, c^{2} d^{2} e^{\frac {3}{2}} + d^{2} e^{\frac {3}{2}}\right )} a^{2} b x^{2} + 2 \, {\left (2 \, c^{3} d e^{\frac {3}{2}} + c d e^{\frac {3}{2}}\right )} a^{2} b x + {\left (c^{4} e^{\frac {3}{2}} + c^{2} e^{\frac {3}{2}}\right )} a^{2} 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 )}}{5 \, {\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((d*e*x+c*e)^(3/2)*(a+b*arcsinh(d*x+c))^3,x, algorithm="maxima")

[Out]

2/5*(d*e*x + c*e)^(5/2)*a^3/(d*e) + 2/5*(b^3*d^2*e^(3/2)*x^2 + 2*b^3*c*d*e^(3/2)*x + 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*(((2*b^3*c^3*e^(3/2) - 5*(c^3*e^(3
/2) + c*e^(3/2))*a*b^2 - (5*a*b^2*d^3*e^(3/2) - 2*b^3*d^3*e^(3/2))*x^3 - 3*(5*a*b^2*c*d^2*e^(3/2) - 2*b^3*c*d^
2*e^(3/2))*x^2 + (6*b^3*c^2*d*e^(3/2) - 5*(3*c^2*d*e^(3/2) + d*e^(3/2))*a*b^2)*x)*sqrt(d^2*x^2 + 2*c*d*x + c^2
+ 1)*sqrt(d*x + c) - ((5*a*b^2*d^4*e^(3/2) - 2*b^3*d^4*e^(3/2))*x^4 + 5*(c^4*e^(3/2) + c^2*e^(3/2))*a*b^2 - 2
*(c^4*e^(3/2) + c^2*e^(3/2))*b^3 + 4*(5*a*b^2*c*d^3*e^(3/2) - 2*b^3*c*d^3*e^(3/2))*x^3 + (5*(6*c^2*d^2*e^(3/2)
+ d^2*e^(3/2))*a*b^2 - 2*(6*c^2*d^2*e^(3/2) + d^2*e^(3/2))*b^3)*x^2 + 2*(5*(2*c^3*d*e^(3/2) + c*d*e^(3/2))*a*
b^2 - 2*(2*c^3*d*e^(3/2) + c*d*e^(3/2))*b^3)*x)*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*e^(3/2)*x^3 + 3*a^2*b*c*d^2*e^(3/2)*x^2 + (3*c^2*d*e^(3/2) + d*e^(3/2))*a^2*b*x + (c^3*e^(
3/2) + c*e^(3/2))*a^2*b)*sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1)*sqrt(d*x + c) + (a^2*b*d^4*e^(3/2)*x^4 + 4*a^2*b*c*
d^3*e^(3/2)*x^3 + (6*c^2*d^2*e^(3/2) + d^2*e^(3/2))*a^2*b*x^2 + 2*(2*c^3*d*e^(3/2) + c*d*e^(3/2))*a^2*b*x + (c
^4*e^(3/2) + c^2*e^(3/2))*a^2*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 \[ \int {\left (c\,e+d\,e\,x\right )}^{3/2}\,{\left (a+b\,\mathrm {asinh}\left (c+d\,x\right )\right )}^3 \,d x \]

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

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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