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

Optimal. Leaf size=27 \[ \frac {\text {Int}\left (\frac {1}{(c+d x) \left (a+b \sinh ^{-1}(c+d x)\right )^3},x\right )}{e} \]

[Out]

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

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Rubi [A]  time = 0.06, 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 = {} \[ \int \frac {1}{(c e+d e x) \left (a+b \sinh ^{-1}(c+d x)\right )^3} \, dx \]

Verification is Not applicable to the result.

[In]

Int[1/((c*e + d*e*x)*(a + b*ArcSinh[c + d*x])^3),x]

[Out]

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

Rubi steps

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

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Mathematica [A]  time = 1.06, size = 0, normalized size = 0.00 \[ \int \frac {1}{(c e+d e x) \left (a+b \sinh ^{-1}(c+d x)\right )^3} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[1/((c*e + d*e*x)*(a + b*ArcSinh[c + d*x])^3),x]

[Out]

Integrate[1/((c*e + d*e*x)*(a + b*ArcSinh[c + d*x])^3), x]

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fricas [A]  time = 0.62, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {1}{a^{3} d e x + a^{3} c e + {\left (b^{3} d e x + b^{3} c e\right )} \operatorname {arsinh}\left (d x + c\right )^{3} + 3 \, {\left (a b^{2} d e x + a b^{2} c e\right )} \operatorname {arsinh}\left (d x + c\right )^{2} + 3 \, {\left (a^{2} b d e x + a^{2} b c e\right )} \operatorname {arsinh}\left (d x + c\right )}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(d*e*x+c*e)/(a+b*arcsinh(d*x+c))^3,x, algorithm="fricas")

[Out]

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

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giac [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (d e x + c e\right )} {\left (b \operatorname {arsinh}\left (d x + c\right ) + a\right )}^{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(d*e*x+c*e)/(a+b*arcsinh(d*x+c))^3,x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.19, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (d e x +c e \right ) \left (a +b \arcsinh \left (d x +c \right )\right )^{3}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(1/(d*e*x+c*e)/(a+b*arcsinh(d*x+c))^3,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(1/(d*e*x+c*e)/(a+b*arcsinh(d*x+c))^3,x, algorithm="maxima")

[Out]

-1/2*(b*d^8*x^8 + 8*b*c*d^7*x^7 + (28*c^2*d^6 + 3*d^6)*b*x^6 + 2*(28*c^3*d^5 + 9*c*d^5)*b*x^5 + (70*c^4*d^4 +
45*c^2*d^4 + 3*d^4)*b*x^4 + 4*(14*c^5*d^3 + 15*c^3*d^3 + 3*c*d^3)*b*x^3 + (28*c^6*d^2 + 45*c^4*d^2 + 18*c^2*d^
2 + d^2)*b*x^2 + 2*(4*c^7*d + 9*c^5*d + 6*c^3*d + c*d)*b*x + (b*d^5*x^5 + 5*b*c*d^4*x^4 - (2*a*d^3 - (10*c^2*d
^3 + d^3)*b)*x^3 - (6*a*c*d^2 - (10*c^3*d^2 + 3*c*d^2)*b)*x^2 - 2*(c^3 + c)*a + (c^5 + c^3)*b - (2*(3*c^2*d +
d)*a - (5*c^4*d + 3*c^2*d)*b)*x)*(d^2*x^2 + 2*c*d*x + c^2 + 1)^(3/2) + (3*b*d^6*x^6 + 18*b*c*d^5*x^5 - (4*a*d^
4 - 5*(9*c^2*d^4 + d^4)*b)*x^4 - 4*(4*a*c*d^3 - 5*(3*c^3*d^3 + c*d^3)*b)*x^3 - ((24*c^2*d^2 + 5*d^2)*a - (45*c
^4*d^2 + 30*c^2*d^2 + 2*d^2)*b)*x^2 - (4*c^4 + 5*c^2 + 1)*a + (3*c^6 + 5*c^4 + 2*c^2)*b - 2*((8*c^3*d + 5*c*d)
*a - (9*c^5*d + 10*c^3*d + 2*c*d)*b)*x)*(d^2*x^2 + 2*c*d*x + c^2 + 1) + (c^8 + 3*c^6 + 3*c^4 + c^2)*b - (2*(b*
d^3*x^3 + 3*b*c*d^2*x^2 + (3*c^2*d + d)*b*x + (c^3 + c)*b)*(d^2*x^2 + 2*c*d*x + c^2 + 1)^(3/2) + (4*b*d^4*x^4
+ 16*b*c*d^3*x^3 + (24*c^2*d^2 + 5*d^2)*b*x^2 + 2*(8*c^3*d + 5*c*d)*b*x + (4*c^4 + 5*c^2 + 1)*b)*(d^2*x^2 + 2*
c*d*x + c^2 + 1) + (2*b*d^5*x^5 + 10*b*c*d^4*x^4 + (20*c^2*d^3 + 3*d^3)*b*x^3 + (20*c^3*d^2 + 9*c*d^2)*b*x^2 +
(10*c^4*d + 9*c^2*d + d)*b*x + (2*c^5 + 3*c^3 + c)*b)*sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1))*log(d*x + c + sqrt(d
^2*x^2 + 2*c*d*x + c^2 + 1)) + (3*b*d^7*x^7 + 21*b*c*d^6*x^6 - (2*a*d^5 - 7*(9*c^2*d^5 + d^5)*b)*x^5 - 5*(2*a*
c*d^4 - 7*(3*c^3*d^4 + c*d^4)*b)*x^4 - ((20*c^2*d^3 + 3*d^3)*a - 5*(21*c^4*d^3 + 14*c^2*d^3 + d^3)*b)*x^3 - ((
20*c^3*d^2 + 9*c*d^2)*a - (63*c^5*d^2 + 70*c^3*d^2 + 15*c*d^2)*b)*x^2 - (2*c^5 + 3*c^3 + c)*a + (3*c^7 + 7*c^5
+ 5*c^3 + c)*b - ((10*c^4*d + 9*c^2*d + d)*a - (21*c^6*d + 35*c^4*d + 15*c^2*d + d)*b)*x)*sqrt(d^2*x^2 + 2*c*
d*x + c^2 + 1))/(a^2*b^2*d^9*e*x^8 + 8*a^2*b^2*c*d^8*e*x^7 + (28*c^2*d^7*e + 3*d^7*e)*a^2*b^2*x^6 + 2*(28*c^3*
d^6*e + 9*c*d^6*e)*a^2*b^2*x^5 + (70*c^4*d^5*e + 45*c^2*d^5*e + 3*d^5*e)*a^2*b^2*x^4 + 4*(14*c^5*d^4*e + 15*c^
3*d^4*e + 3*c*d^4*e)*a^2*b^2*x^3 + (28*c^6*d^3*e + 45*c^4*d^3*e + 18*c^2*d^3*e + d^3*e)*a^2*b^2*x^2 + 2*(4*c^7
*d^2*e + 9*c^5*d^2*e + 6*c^3*d^2*e + c*d^2*e)*a^2*b^2*x + (c^8*d*e + 3*c^6*d*e + 3*c^4*d*e + c^2*d*e)*a^2*b^2
+ (b^4*d^9*e*x^8 + 8*b^4*c*d^8*e*x^7 + (28*c^2*d^7*e + 3*d^7*e)*b^4*x^6 + 2*(28*c^3*d^6*e + 9*c*d^6*e)*b^4*x^5
+ (70*c^4*d^5*e + 45*c^2*d^5*e + 3*d^5*e)*b^4*x^4 + 4*(14*c^5*d^4*e + 15*c^3*d^4*e + 3*c*d^4*e)*b^4*x^3 + (28
*c^6*d^3*e + 45*c^4*d^3*e + 18*c^2*d^3*e + d^3*e)*b^4*x^2 + 2*(4*c^7*d^2*e + 9*c^5*d^2*e + 6*c^3*d^2*e + c*d^2
*e)*b^4*x + (c^8*d*e + 3*c^6*d*e + 3*c^4*d*e + c^2*d*e)*b^4 + (b^4*d^6*e*x^5 + 5*b^4*c*d^5*e*x^4 + 10*b^4*c^2*
d^4*e*x^3 + 10*b^4*c^3*d^3*e*x^2 + 5*b^4*c^4*d^2*e*x + b^4*c^5*d*e)*(d^2*x^2 + 2*c*d*x + c^2 + 1)^(3/2) + 3*(b
^4*d^7*e*x^6 + 6*b^4*c*d^6*e*x^5 + (15*c^2*d^5*e + d^5*e)*b^4*x^4 + 4*(5*c^3*d^4*e + c*d^4*e)*b^4*x^3 + 3*(5*c
^4*d^3*e + 2*c^2*d^3*e)*b^4*x^2 + 2*(3*c^5*d^2*e + 2*c^3*d^2*e)*b^4*x + (c^6*d*e + c^4*d*e)*b^4)*(d^2*x^2 + 2*
c*d*x + c^2 + 1) + 3*(b^4*d^8*e*x^7 + 7*b^4*c*d^7*e*x^6 + (21*c^2*d^6*e + 2*d^6*e)*b^4*x^5 + 5*(7*c^3*d^5*e +
2*c*d^5*e)*b^4*x^4 + (35*c^4*d^4*e + 20*c^2*d^4*e + d^4*e)*b^4*x^3 + (21*c^5*d^3*e + 20*c^3*d^3*e + 3*c*d^3*e)
*b^4*x^2 + (7*c^6*d^2*e + 10*c^4*d^2*e + 3*c^2*d^2*e)*b^4*x + (c^7*d*e + 2*c^5*d*e + c^3*d*e)*b^4)*sqrt(d^2*x^
2 + 2*c*d*x + c^2 + 1))*log(d*x + c + sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1))^2 + (a^2*b^2*d^6*e*x^5 + 5*a^2*b^2*c*
d^5*e*x^4 + 10*a^2*b^2*c^2*d^4*e*x^3 + 10*a^2*b^2*c^3*d^3*e*x^2 + 5*a^2*b^2*c^4*d^2*e*x + a^2*b^2*c^5*d*e)*(d^
2*x^2 + 2*c*d*x + c^2 + 1)^(3/2) + 3*(a^2*b^2*d^7*e*x^6 + 6*a^2*b^2*c*d^6*e*x^5 + (15*c^2*d^5*e + d^5*e)*a^2*b
^2*x^4 + 4*(5*c^3*d^4*e + c*d^4*e)*a^2*b^2*x^3 + 3*(5*c^4*d^3*e + 2*c^2*d^3*e)*a^2*b^2*x^2 + 2*(3*c^5*d^2*e +
2*c^3*d^2*e)*a^2*b^2*x + (c^6*d*e + c^4*d*e)*a^2*b^2)*(d^2*x^2 + 2*c*d*x + c^2 + 1) + 2*(a*b^3*d^9*e*x^8 + 8*a
*b^3*c*d^8*e*x^7 + (28*c^2*d^7*e + 3*d^7*e)*a*b^3*x^6 + 2*(28*c^3*d^6*e + 9*c*d^6*e)*a*b^3*x^5 + (70*c^4*d^5*e
+ 45*c^2*d^5*e + 3*d^5*e)*a*b^3*x^4 + 4*(14*c^5*d^4*e + 15*c^3*d^4*e + 3*c*d^4*e)*a*b^3*x^3 + (28*c^6*d^3*e +
45*c^4*d^3*e + 18*c^2*d^3*e + d^3*e)*a*b^3*x^2 + 2*(4*c^7*d^2*e + 9*c^5*d^2*e + 6*c^3*d^2*e + c*d^2*e)*a*b^3*
x + (c^8*d*e + 3*c^6*d*e + 3*c^4*d*e + c^2*d*e)*a*b^3 + (a*b^3*d^6*e*x^5 + 5*a*b^3*c*d^5*e*x^4 + 10*a*b^3*c^2*
d^4*e*x^3 + 10*a*b^3*c^3*d^3*e*x^2 + 5*a*b^3*c^4*d^2*e*x + a*b^3*c^5*d*e)*(d^2*x^2 + 2*c*d*x + c^2 + 1)^(3/2)
+ 3*(a*b^3*d^7*e*x^6 + 6*a*b^3*c*d^6*e*x^5 + (15*c^2*d^5*e + d^5*e)*a*b^3*x^4 + 4*(5*c^3*d^4*e + c*d^4*e)*a*b^
3*x^3 + 3*(5*c^4*d^3*e + 2*c^2*d^3*e)*a*b^3*x^2 + 2*(3*c^5*d^2*e + 2*c^3*d^2*e)*a*b^3*x + (c^6*d*e + c^4*d*e)*
a*b^3)*(d^2*x^2 + 2*c*d*x + c^2 + 1) + 3*(a*b^3*d^8*e*x^7 + 7*a*b^3*c*d^7*e*x^6 + (21*c^2*d^6*e + 2*d^6*e)*a*b
^3*x^5 + 5*(7*c^3*d^5*e + 2*c*d^5*e)*a*b^3*x^4 + (35*c^4*d^4*e + 20*c^2*d^4*e + d^4*e)*a*b^3*x^3 + (21*c^5*d^3
*e + 20*c^3*d^3*e + 3*c*d^3*e)*a*b^3*x^2 + (7*c^6*d^2*e + 10*c^4*d^2*e + 3*c^2*d^2*e)*a*b^3*x + (c^7*d*e + 2*c
^5*d*e + c^3*d*e)*a*b^3)*sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1))*log(d*x + c + sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1)) +
3*(a^2*b^2*d^8*e*x^7 + 7*a^2*b^2*c*d^7*e*x^6 + (21*c^2*d^6*e + 2*d^6*e)*a^2*b^2*x^5 + 5*(7*c^3*d^5*e + 2*c*d^
5*e)*a^2*b^2*x^4 + (35*c^4*d^4*e + 20*c^2*d^4*e + d^4*e)*a^2*b^2*x^3 + (21*c^5*d^3*e + 20*c^3*d^3*e + 3*c*d^3*
e)*a^2*b^2*x^2 + (7*c^6*d^2*e + 10*c^4*d^2*e + 3*c^2*d^2*e)*a^2*b^2*x + (c^7*d*e + 2*c^5*d*e + c^3*d*e)*a^2*b^
2)*sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1)) + integrate(1/2*(4*(d^4*x^4 + 4*c*d^3*x^3 + c^4 + 2*(3*c^2*d^2 + d^2)*x^
2 + 2*c^2 + 4*(c^3*d + c*d)*x)*(d^2*x^2 + 2*c*d*x + c^2 + 1)^2 + (12*d^5*x^5 + 60*c*d^4*x^4 + 12*c^5 + 2*(60*c
^2*d^3 + 11*d^3)*x^3 + 22*c^3 + 6*(20*c^3*d^2 + 11*c*d^2)*x^2 + (60*c^4*d + 66*c^2*d + 7*d)*x + 7*c)*(d^2*x^2
+ 2*c*d*x + c^2 + 1)^(3/2) + 2*(6*d^6*x^6 + 36*c*d^5*x^5 + 6*c^6 + 10*(9*c^2*d^4 + d^4)*x^4 + 10*c^4 + 40*(3*c
^3*d^3 + c*d^3)*x^3 + 5*(18*c^4*d^2 + 12*c^2*d^2 + d^2)*x^2 + 5*c^2 + 2*(18*c^5*d + 20*c^3*d + 5*c*d)*x + 1)*(
d^2*x^2 + 2*c*d*x + c^2 + 1) + (4*d^7*x^7 + 28*c*d^6*x^6 + 4*c^7 + 6*(14*c^2*d^5 + d^5)*x^5 + 6*c^5 + 10*(14*c
^3*d^4 + 3*c*d^4)*x^4 + (140*c^4*d^3 + 60*c^2*d^3 + 3*d^3)*x^3 + 3*c^3 + 3*(28*c^5*d^2 + 20*c^3*d^2 + 3*c*d^2)
*x^2 + (28*c^6*d + 30*c^4*d + 9*c^2*d + d)*x + c)*sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1))/(a*b^2*d^11*e*x^11 + 11*a
*b^2*c*d^10*e*x^10 + (55*c^2*d^9*e + 4*d^9*e)*a*b^2*x^9 + 3*(55*c^3*d^8*e + 12*c*d^8*e)*a*b^2*x^8 + 6*(55*c^4*
d^7*e + 24*c^2*d^7*e + d^7*e)*a*b^2*x^7 + 42*(11*c^5*d^6*e + 8*c^3*d^6*e + c*d^6*e)*a*b^2*x^6 + 2*(231*c^6*d^5
*e + 252*c^4*d^5*e + 63*c^2*d^5*e + 2*d^5*e)*a*b^2*x^5 + 2*(165*c^7*d^4*e + 252*c^5*d^4*e + 105*c^3*d^4*e + 10
*c*d^4*e)*a*b^2*x^4 + (165*c^8*d^3*e + 336*c^6*d^3*e + 210*c^4*d^3*e + 40*c^2*d^3*e + d^3*e)*a*b^2*x^3 + (55*c
^9*d^2*e + 144*c^7*d^2*e + 126*c^5*d^2*e + 40*c^3*d^2*e + 3*c*d^2*e)*a*b^2*x^2 + (11*c^10*d*e + 36*c^8*d*e + 4
2*c^6*d*e + 20*c^4*d*e + 3*c^2*d*e)*a*b^2*x + (c^11*e + 4*c^9*e + 6*c^7*e + 4*c^5*e + c^3*e)*a*b^2 + (a*b^2*d^
7*e*x^7 + 7*a*b^2*c*d^6*e*x^6 + 21*a*b^2*c^2*d^5*e*x^5 + 35*a*b^2*c^3*d^4*e*x^4 + 35*a*b^2*c^4*d^3*e*x^3 + 21*
a*b^2*c^5*d^2*e*x^2 + 7*a*b^2*c^6*d*e*x + a*b^2*c^7*e)*(d^2*x^2 + 2*c*d*x + c^2 + 1)^2 + 4*(a*b^2*d^8*e*x^8 +
8*a*b^2*c*d^7*e*x^7 + (28*c^2*d^6*e + d^6*e)*a*b^2*x^6 + 2*(28*c^3*d^5*e + 3*c*d^5*e)*a*b^2*x^5 + 5*(14*c^4*d^
4*e + 3*c^2*d^4*e)*a*b^2*x^4 + 4*(14*c^5*d^3*e + 5*c^3*d^3*e)*a*b^2*x^3 + (28*c^6*d^2*e + 15*c^4*d^2*e)*a*b^2*
x^2 + 2*(4*c^7*d*e + 3*c^5*d*e)*a*b^2*x + (c^8*e + c^6*e)*a*b^2)*(d^2*x^2 + 2*c*d*x + c^2 + 1)^(3/2) + 6*(a*b^
2*d^9*e*x^9 + 9*a*b^2*c*d^8*e*x^8 + 2*(18*c^2*d^7*e + d^7*e)*a*b^2*x^7 + 14*(6*c^3*d^6*e + c*d^6*e)*a*b^2*x^6
+ (126*c^4*d^5*e + 42*c^2*d^5*e + d^5*e)*a*b^2*x^5 + (126*c^5*d^4*e + 70*c^3*d^4*e + 5*c*d^4*e)*a*b^2*x^4 + 2*
(42*c^6*d^3*e + 35*c^4*d^3*e + 5*c^2*d^3*e)*a*b^2*x^3 + 2*(18*c^7*d^2*e + 21*c^5*d^2*e + 5*c^3*d^2*e)*a*b^2*x^
2 + (9*c^8*d*e + 14*c^6*d*e + 5*c^4*d*e)*a*b^2*x + (c^9*e + 2*c^7*e + c^5*e)*a*b^2)*(d^2*x^2 + 2*c*d*x + c^2 +
1) + (b^3*d^11*e*x^11 + 11*b^3*c*d^10*e*x^10 + (55*c^2*d^9*e + 4*d^9*e)*b^3*x^9 + 3*(55*c^3*d^8*e + 12*c*d^8*
e)*b^3*x^8 + 6*(55*c^4*d^7*e + 24*c^2*d^7*e + d^7*e)*b^3*x^7 + 42*(11*c^5*d^6*e + 8*c^3*d^6*e + c*d^6*e)*b^3*x
^6 + 2*(231*c^6*d^5*e + 252*c^4*d^5*e + 63*c^2*d^5*e + 2*d^5*e)*b^3*x^5 + 2*(165*c^7*d^4*e + 252*c^5*d^4*e + 1
05*c^3*d^4*e + 10*c*d^4*e)*b^3*x^4 + (165*c^8*d^3*e + 336*c^6*d^3*e + 210*c^4*d^3*e + 40*c^2*d^3*e + d^3*e)*b^
3*x^3 + (55*c^9*d^2*e + 144*c^7*d^2*e + 126*c^5*d^2*e + 40*c^3*d^2*e + 3*c*d^2*e)*b^3*x^2 + (11*c^10*d*e + 36*
c^8*d*e + 42*c^6*d*e + 20*c^4*d*e + 3*c^2*d*e)*b^3*x + (c^11*e + 4*c^9*e + 6*c^7*e + 4*c^5*e + c^3*e)*b^3 + (b
^3*d^7*e*x^7 + 7*b^3*c*d^6*e*x^6 + 21*b^3*c^2*d^5*e*x^5 + 35*b^3*c^3*d^4*e*x^4 + 35*b^3*c^4*d^3*e*x^3 + 21*b^3
*c^5*d^2*e*x^2 + 7*b^3*c^6*d*e*x + b^3*c^7*e)*(d^2*x^2 + 2*c*d*x + c^2 + 1)^2 + 4*(b^3*d^8*e*x^8 + 8*b^3*c*d^7
*e*x^7 + (28*c^2*d^6*e + d^6*e)*b^3*x^6 + 2*(28*c^3*d^5*e + 3*c*d^5*e)*b^3*x^5 + 5*(14*c^4*d^4*e + 3*c^2*d^4*e
)*b^3*x^4 + 4*(14*c^5*d^3*e + 5*c^3*d^3*e)*b^3*x^3 + (28*c^6*d^2*e + 15*c^4*d^2*e)*b^3*x^2 + 2*(4*c^7*d*e + 3*
c^5*d*e)*b^3*x + (c^8*e + c^6*e)*b^3)*(d^2*x^2 + 2*c*d*x + c^2 + 1)^(3/2) + 6*(b^3*d^9*e*x^9 + 9*b^3*c*d^8*e*x
^8 + 2*(18*c^2*d^7*e + d^7*e)*b^3*x^7 + 14*(6*c^3*d^6*e + c*d^6*e)*b^3*x^6 + (126*c^4*d^5*e + 42*c^2*d^5*e + d
^5*e)*b^3*x^5 + (126*c^5*d^4*e + 70*c^3*d^4*e + 5*c*d^4*e)*b^3*x^4 + 2*(42*c^6*d^3*e + 35*c^4*d^3*e + 5*c^2*d^
3*e)*b^3*x^3 + 2*(18*c^7*d^2*e + 21*c^5*d^2*e + 5*c^3*d^2*e)*b^3*x^2 + (9*c^8*d*e + 14*c^6*d*e + 5*c^4*d*e)*b^
3*x + (c^9*e + 2*c^7*e + c^5*e)*b^3)*(d^2*x^2 + 2*c*d*x + c^2 + 1) + 4*(b^3*d^10*e*x^10 + 10*b^3*c*d^9*e*x^9 +
3*(15*c^2*d^8*e + d^8*e)*b^3*x^8 + 24*(5*c^3*d^7*e + c*d^7*e)*b^3*x^7 + 3*(70*c^4*d^6*e + 28*c^2*d^6*e + d^6*
e)*b^3*x^6 + 6*(42*c^5*d^5*e + 28*c^3*d^5*e + 3*c*d^5*e)*b^3*x^5 + (210*c^6*d^4*e + 210*c^4*d^4*e + 45*c^2*d^4
*e + d^4*e)*b^3*x^4 + 4*(30*c^7*d^3*e + 42*c^5*d^3*e + 15*c^3*d^3*e + c*d^3*e)*b^3*x^3 + 3*(15*c^8*d^2*e + 28*
c^6*d^2*e + 15*c^4*d^2*e + 2*c^2*d^2*e)*b^3*x^2 + 2*(5*c^9*d*e + 12*c^7*d*e + 9*c^5*d*e + 2*c^3*d*e)*b^3*x + (
c^10*e + 3*c^8*e + 3*c^6*e + c^4*e)*b^3)*sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1))*log(d*x + c + sqrt(d^2*x^2 + 2*c*d
*x + c^2 + 1)) + 4*(a*b^2*d^10*e*x^10 + 10*a*b^2*c*d^9*e*x^9 + 3*(15*c^2*d^8*e + d^8*e)*a*b^2*x^8 + 24*(5*c^3*
d^7*e + c*d^7*e)*a*b^2*x^7 + 3*(70*c^4*d^6*e + 28*c^2*d^6*e + d^6*e)*a*b^2*x^6 + 6*(42*c^5*d^5*e + 28*c^3*d^5*
e + 3*c*d^5*e)*a*b^2*x^5 + (210*c^6*d^4*e + 210*c^4*d^4*e + 45*c^2*d^4*e + d^4*e)*a*b^2*x^4 + 4*(30*c^7*d^3*e
+ 42*c^5*d^3*e + 15*c^3*d^3*e + c*d^3*e)*a*b^2*x^3 + 3*(15*c^8*d^2*e + 28*c^6*d^2*e + 15*c^4*d^2*e + 2*c^2*d^2
*e)*a*b^2*x^2 + 2*(5*c^9*d*e + 12*c^7*d*e + 9*c^5*d*e + 2*c^3*d*e)*a*b^2*x + (c^10*e + 3*c^8*e + 3*c^6*e + c^4
*e)*a*b^2)*sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1)), x)

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mupad [A]  time = 0.00, size = -1, normalized size = -0.04 \[ \int \frac {1}{\left (c\,e+d\,e\,x\right )\,{\left (a+b\,\mathrm {asinh}\left (c+d\,x\right )\right )}^3} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((c*e + d*e*x)*(a + b*asinh(c + d*x))^3),x)

[Out]

int(1/((c*e + d*e*x)*(a + b*asinh(c + d*x))^3), x)

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sympy [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {\int \frac {1}{a^{3} c + a^{3} d x + 3 a^{2} b c \operatorname {asinh}{\left (c + d x \right )} + 3 a^{2} b d x \operatorname {asinh}{\left (c + d x \right )} + 3 a b^{2} c \operatorname {asinh}^{2}{\left (c + d x \right )} + 3 a b^{2} d x \operatorname {asinh}^{2}{\left (c + d x \right )} + b^{3} c \operatorname {asinh}^{3}{\left (c + d x \right )} + b^{3} d x \operatorname {asinh}^{3}{\left (c + d x \right )}}\, dx}{e} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(d*e*x+c*e)/(a+b*asinh(d*x+c))**3,x)

[Out]

Integral(1/(a**3*c + a**3*d*x + 3*a**2*b*c*asinh(c + d*x) + 3*a**2*b*d*x*asinh(c + d*x) + 3*a*b**2*c*asinh(c +
d*x)**2 + 3*a*b**2*d*x*asinh(c + d*x)**2 + b**3*c*asinh(c + d*x)**3 + b**3*d*x*asinh(c + d*x)**3), x)/e

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