3.253 \(\int (c e+d e x)^{5/2} (a+b \sinh ^{-1}(c+d x))^4 \, dx\)

Optimal. Leaf size=82 \[ \frac {2 (e (c+d x))^{7/2} \left (a+b \sinh ^{-1}(c+d x)\right )^4}{7 d e}-\frac {8 b \text {Int}\left (\frac {(e (c+d x))^{7/2} \left (a+b \sinh ^{-1}(c+d x)\right )^3}{\sqrt {(c+d x)^2+1}},x\right )}{7 e} \]

[Out]

2/7*(e*(d*x+c))^(7/2)*(a+b*arcsinh(d*x+c))^4/d/e-8/7*b*Unintegrable((e*(d*x+c))^(7/2)*(a+b*arcsinh(d*x+c))^3/(
1+(d*x+c)^2)^(1/2),x)/e

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Rubi [A]  time = 0.21, 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 = {} \[ \int (c e+d e x)^{5/2} \left (a+b \sinh ^{-1}(c+d x)\right )^4 \, dx \]

Verification is Not applicable to the result.

[In]

Int[(c*e + d*e*x)^(5/2)*(a + b*ArcSinh[c + d*x])^4,x]

[Out]

(2*(e*(c + d*x))^(7/2)*(a + b*ArcSinh[c + d*x])^4)/(7*d*e) - (8*b*Defer[Subst][Defer[Int][((e*x)^(7/2)*(a + b*
ArcSinh[x])^3)/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 )^4 \, dx &=\frac {\operatorname {Subst}\left (\int (e x)^{5/2} \left (a+b \sinh ^{-1}(x)\right )^4 \, dx,x,c+d x\right )}{d}\\ &=\frac {2 (e (c+d x))^{7/2} \left (a+b \sinh ^{-1}(c+d x)\right )^4}{7 d e}-\frac {(8 b) \operatorname {Subst}\left (\int \frac {(e x)^{7/2} \left (a+b \sinh ^{-1}(x)\right )^3}{\sqrt {1+x^2}} \, dx,x,c+d x\right )}{7 d e}\\ \end {align*}

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Mathematica [A]  time = 128.47, size = 0, normalized size = 0.00 \[ \int (c e+d e x)^{5/2} \left (a+b \sinh ^{-1}(c+d x)\right )^4 \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[(c*e + d*e*x)^(5/2)*(a + b*ArcSinh[c + d*x])^4,x]

[Out]

Integrate[(c*e + d*e*x)^(5/2)*(a + b*ArcSinh[c + d*x])^4, x]

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fricas [A]  time = 1.01, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (a^{4} d^{2} e^{2} x^{2} + 2 \, a^{4} c d e^{2} x + a^{4} c^{2} e^{2} + {\left (b^{4} d^{2} e^{2} x^{2} + 2 \, b^{4} c d e^{2} x + b^{4} c^{2} e^{2}\right )} \operatorname {arsinh}\left (d x + c\right )^{4} + 4 \, {\left (a b^{3} d^{2} e^{2} x^{2} + 2 \, a b^{3} c d e^{2} x + a b^{3} c^{2} e^{2}\right )} \operatorname {arsinh}\left (d x + c\right )^{3} + 6 \, {\left (a^{2} b^{2} d^{2} e^{2} x^{2} + 2 \, a^{2} b^{2} c d e^{2} x + a^{2} b^{2} c^{2} e^{2}\right )} \operatorname {arsinh}\left (d x + c\right )^{2} + 4 \, {\left (a^{3} b d^{2} e^{2} x^{2} + 2 \, a^{3} b c d e^{2} x + a^{3} b c^{2} e^{2}\right )} \operatorname {arsinh}\left (d x + c\right )\right )} \sqrt {d e x + c e}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*e*x+c*e)^(5/2)*(a+b*arcsinh(d*x+c))^4,x, algorithm="fricas")

[Out]

integral((a^4*d^2*e^2*x^2 + 2*a^4*c*d*e^2*x + a^4*c^2*e^2 + (b^4*d^2*e^2*x^2 + 2*b^4*c*d*e^2*x + b^4*c^2*e^2)*
arcsinh(d*x + c)^4 + 4*(a*b^3*d^2*e^2*x^2 + 2*a*b^3*c*d*e^2*x + a*b^3*c^2*e^2)*arcsinh(d*x + c)^3 + 6*(a^2*b^2
*d^2*e^2*x^2 + 2*a^2*b^2*c*d*e^2*x + a^2*b^2*c^2*e^2)*arcsinh(d*x + c)^2 + 4*(a^3*b*d^2*e^2*x^2 + 2*a^3*b*c*d*
e^2*x + a^3*b*c^2*e^2)*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 \[ \int {\left (d e x + c e\right )}^{\frac {5}{2}} {\left (b \operatorname {arsinh}\left (d x + c\right ) + a\right )}^{4}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*e*x+c*e)^(5/2)*(a+b*arcsinh(d*x+c))^4,x, algorithm="giac")

[Out]

integrate((d*e*x + c*e)^(5/2)*(b*arcsinh(d*x + c) + a)^4, x)

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maple [F(-2)]  time = 180.00, size = 0, normalized size = 0.00 \[ \int \left (d e x +c e \right )^{\frac {5}{2}} \left (a +b \arcsinh \left (d x +c \right )\right )^{4}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*e*x+c*e)^(5/2)*(a+b*arcsinh(d*x+c))^4,x)

[Out]

int((d*e*x+c*e)^(5/2)*(a+b*arcsinh(d*x+c))^4,x)

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maxima [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {result too large to display} \]

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))^4,x, algorithm="maxima")

[Out]

2/7*(d*e*x + c*e)^(7/2)*a^4/(d*e) + 2/7*(b^4*d^3*e^(5/2)*x^3 + 3*b^4*c*d^2*e^(5/2)*x^2 + 3*b^4*c^2*d*e^(5/2)*x
+ b^4*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))^4/d + integrate(-2/7*(2*((2
*b^4*c^4*e^(5/2) - 7*(c^4*e^(5/2) + c^2*e^(5/2))*a*b^3 - (7*a*b^3*d^4*e^(5/2) - 2*b^4*d^4*e^(5/2))*x^4 - 4*(7*
a*b^3*c*d^3*e^(5/2) - 2*b^4*c*d^3*e^(5/2))*x^3 + (12*b^4*c^2*d^2*e^(5/2) - 7*(6*c^2*d^2*e^(5/2) + d^2*e^(5/2))
*a*b^3)*x^2 + 2*(4*b^4*c^3*d*e^(5/2) - 7*(2*c^3*d*e^(5/2) + c*d*e^(5/2))*a*b^3)*x)*sqrt(d^2*x^2 + 2*c*d*x + c^
2 + 1)*sqrt(d*x + c) - ((7*a*b^3*d^5*e^(5/2) - 2*b^4*d^5*e^(5/2))*x^5 + 7*(c^5*e^(5/2) + c^3*e^(5/2))*a*b^3 -
2*(c^5*e^(5/2) + c^3*e^(5/2))*b^4 + 5*(7*a*b^3*c*d^4*e^(5/2) - 2*b^4*c*d^4*e^(5/2))*x^4 + (7*(10*c^2*d^3*e^(5/
2) + d^3*e^(5/2))*a*b^3 - 2*(10*c^2*d^3*e^(5/2) + d^3*e^(5/2))*b^4)*x^3 + (7*(10*c^3*d^2*e^(5/2) + 3*c*d^2*e^(
5/2))*a*b^3 - 2*(10*c^3*d^2*e^(5/2) + 3*c*d^2*e^(5/2))*b^4)*x^2 + (7*(5*c^4*d*e^(5/2) + 3*c^2*d*e^(5/2))*a*b^3
- 2*(5*c^4*d*e^(5/2) + 3*c^2*d*e^(5/2))*b^4)*x)*sqrt(d*x + c))*log(d*x + c + sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1
))^3 - 21*((a^2*b^2*d^4*e^(5/2)*x^4 + 4*a^2*b^2*c*d^3*e^(5/2)*x^3 + (6*c^2*d^2*e^(5/2) + d^2*e^(5/2))*a^2*b^2*
x^2 + 2*(2*c^3*d*e^(5/2) + c*d*e^(5/2))*a^2*b^2*x + (c^4*e^(5/2) + c^2*e^(5/2))*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^5*e^(5/2)*x^5 + 5*a^2*b^2*c*d^4*e^(5/2)*x^4 + (10*c^2*d^3*e^(5/2) + d^
3*e^(5/2))*a^2*b^2*x^3 + (10*c^3*d^2*e^(5/2) + 3*c*d^2*e^(5/2))*a^2*b^2*x^2 + (5*c^4*d*e^(5/2) + 3*c^2*d*e^(5/
2))*a^2*b^2*x + (c^5*e^(5/2) + c^3*e^(5/2))*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 - 14*((a^3*b*d^4*e^(5/2)*x^4 + 4*a^3*b*c*d^3*e^(5/2)*x^3 + (6*c^2*d^2*e^(5/2) + d^2*e^(5/2))*a^3*b*x^
2 + 2*(2*c^3*d*e^(5/2) + c*d*e^(5/2))*a^3*b*x + (c^4*e^(5/2) + c^2*e^(5/2))*a^3*b)*sqrt(d^2*x^2 + 2*c*d*x + c^
2 + 1)*sqrt(d*x + c) + (a^3*b*d^5*e^(5/2)*x^5 + 5*a^3*b*c*d^4*e^(5/2)*x^4 + (10*c^2*d^3*e^(5/2) + d^3*e^(5/2))
*a^3*b*x^3 + (10*c^3*d^2*e^(5/2) + 3*c*d^2*e^(5/2))*a^3*b*x^2 + (5*c^4*d*e^(5/2) + 3*c^2*d*e^(5/2))*a^3*b*x +
(c^5*e^(5/2) + c^3*e^(5/2))*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 \[ \int {\left (c\,e+d\,e\,x\right )}^{5/2}\,{\left (a+b\,\mathrm {asinh}\left (c+d\,x\right )\right )}^4 \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*e + d*e*x)^(5/2)*(a + b*asinh(c + d*x))^4,x)

[Out]

int((c*e + d*e*x)^(5/2)*(a + b*asinh(c + d*x))^4, x)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

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))**4,x)

[Out]

Timed out

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