3.244 \(\int (c e+d e x)^{7/2} (a+b \sinh ^{-1}(c+d x))^3 \, dx\)

Optimal. Leaf size=82 \[ \frac {2 (e (c+d x))^{9/2} \left (a+b \sinh ^{-1}(c+d x)\right )^3}{9 d e}-\frac {2 b \text {Int}\left (\frac {(e (c+d x))^{9/2} \left (a+b \sinh ^{-1}(c+d x)\right )^2}{\sqrt {(c+d x)^2+1}},x\right )}{3 e} \]

[Out]

2/9*(e*(d*x+c))^(9/2)*(a+b*arcsinh(d*x+c))^3/d/e-2/3*b*Unintegrable((e*(d*x+c))^(9/2)*(a+b*arcsinh(d*x+c))^2/(
1+(d*x+c)^2)^(1/2),x)/e

________________________________________________________________________________________

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 = {} \[ \int (c e+d e x)^{7/2} \left (a+b \sinh ^{-1}(c+d x)\right )^3 \, dx \]

Verification is Not applicable to the result.

[In]

Int[(c*e + d*e*x)^(7/2)*(a + b*ArcSinh[c + d*x])^3,x]

[Out]

(2*(e*(c + d*x))^(9/2)*(a + b*ArcSinh[c + d*x])^3)/(9*d*e) - (2*b*Defer[Subst][Defer[Int][((e*x)^(9/2)*(a + b*
ArcSinh[x])^2)/Sqrt[1 + x^2], x], x, c + d*x])/(3*d*e)

Rubi steps

\begin {align*} \int (c e+d e x)^{7/2} \left (a+b \sinh ^{-1}(c+d x)\right )^3 \, dx &=\frac {\operatorname {Subst}\left (\int (e x)^{7/2} \left (a+b \sinh ^{-1}(x)\right )^3 \, dx,x,c+d x\right )}{d}\\ &=\frac {2 (e (c+d x))^{9/2} \left (a+b \sinh ^{-1}(c+d x)\right )^3}{9 d e}-\frac {(2 b) \operatorname {Subst}\left (\int \frac {(e x)^{9/2} \left (a+b \sinh ^{-1}(x)\right )^2}{\sqrt {1+x^2}} \, dx,x,c+d x\right )}{3 d e}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 89.80, size = 0, normalized size = 0.00 \[ \int (c e+d e x)^{7/2} \left (a+b \sinh ^{-1}(c+d x)\right )^3 \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[(c*e + d*e*x)^(7/2)*(a + b*ArcSinh[c + d*x])^3,x]

[Out]

Integrate[(c*e + d*e*x)^(7/2)*(a + b*ArcSinh[c + d*x])^3, x]

________________________________________________________________________________________

fricas [A]  time = 2.03, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (a^{3} d^{3} e^{3} x^{3} + 3 \, a^{3} c d^{2} e^{3} x^{2} + 3 \, a^{3} c^{2} d e^{3} x + a^{3} c^{3} e^{3} + {\left (b^{3} d^{3} e^{3} x^{3} + 3 \, b^{3} c d^{2} e^{3} x^{2} + 3 \, b^{3} c^{2} d e^{3} x + b^{3} c^{3} e^{3}\right )} \operatorname {arsinh}\left (d x + c\right )^{3} + 3 \, {\left (a b^{2} d^{3} e^{3} x^{3} + 3 \, a b^{2} c d^{2} e^{3} x^{2} + 3 \, a b^{2} c^{2} d e^{3} x + a b^{2} c^{3} e^{3}\right )} \operatorname {arsinh}\left (d x + c\right )^{2} + 3 \, {\left (a^{2} b d^{3} e^{3} x^{3} + 3 \, a^{2} b c d^{2} e^{3} x^{2} + 3 \, a^{2} b c^{2} d e^{3} x + a^{2} b c^{3} e^{3}\right )} \operatorname {arsinh}\left (d x + c\right )\right )} \sqrt {d e x + c e}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*e*x+c*e)^(7/2)*(a+b*arcsinh(d*x+c))^3,x, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

giac [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (d e x + c e\right )}^{\frac {7}{2}} {\left (b \operatorname {arsinh}\left (d x + c\right ) + a\right )}^{3}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*e*x+c*e)^(7/2)*(a+b*arcsinh(d*x+c))^3,x, algorithm="giac")

[Out]

integrate((d*e*x + c*e)^(7/2)*(b*arcsinh(d*x + c) + a)^3, x)

________________________________________________________________________________________

maple [F(-2)]  time = 180.00, size = 0, normalized size = 0.00 \[ \int \left (d e x +c e \right )^{\frac {7}{2}} \left (a +b \arcsinh \left (d x +c \right )\right )^{3}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*e*x+c*e)^(7/2)*(a+b*arcsinh(d*x+c))^3,x)

[Out]

int((d*e*x+c*e)^(7/2)*(a+b*arcsinh(d*x+c))^3,x)

________________________________________________________________________________________

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)^(7/2)*(a+b*arcsinh(d*x+c))^3,x, algorithm="maxima")

[Out]

2/9*(d*e*x + c*e)^(9/2)*a^3/(d*e) + 2/9*(b^3*d^4*e^(7/2)*x^4 + 4*b^3*c*d^3*e^(7/2)*x^3 + 6*b^3*c^2*d^2*e^(7/2)
*x^2 + 4*b^3*c^3*d*e^(7/2)*x + b^3*c^4*e^(7/2))*sqrt(d*x + c)*log(d*x + c + sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1))
^3/d + integrate(-1/3*(((2*b^3*c^5*e^(7/2) - (9*a*b^2*d^5*e^(7/2) - 2*b^3*d^5*e^(7/2))*x^5 - 5*(9*a*b^2*c*d^4*
e^(7/2) - 2*b^3*c*d^4*e^(7/2))*x^4 - 9*(c^5*e^(7/2) + c^3*e^(7/2))*a*b^2 + (20*b^3*c^2*d^3*e^(7/2) - 9*(10*c^2
*d^3*e^(7/2) + d^3*e^(7/2))*a*b^2)*x^3 + (20*b^3*c^3*d^2*e^(7/2) - 9*(10*c^3*d^2*e^(7/2) + 3*c*d^2*e^(7/2))*a*
b^2)*x^2 + (10*b^3*c^4*d*e^(7/2) - 9*(5*c^4*d*e^(7/2) + 3*c^2*d*e^(7/2))*a*b^2)*x)*sqrt(d^2*x^2 + 2*c*d*x + c^
2 + 1)*sqrt(d*x + c) - ((9*a*b^2*d^6*e^(7/2) - 2*b^3*d^6*e^(7/2))*x^6 + 6*(9*a*b^2*c*d^5*e^(7/2) - 2*b^3*c*d^5
*e^(7/2))*x^5 + (9*(15*c^2*d^4*e^(7/2) + d^4*e^(7/2))*a*b^2 - 2*(15*c^2*d^4*e^(7/2) + d^4*e^(7/2))*b^3)*x^4 +
9*(c^6*e^(7/2) + c^4*e^(7/2))*a*b^2 - 2*(c^6*e^(7/2) + c^4*e^(7/2))*b^3 + 4*(9*(5*c^3*d^3*e^(7/2) + c*d^3*e^(7
/2))*a*b^2 - 2*(5*c^3*d^3*e^(7/2) + c*d^3*e^(7/2))*b^3)*x^3 + 3*(9*(5*c^4*d^2*e^(7/2) + 2*c^2*d^2*e^(7/2))*a*b
^2 - 2*(5*c^4*d^2*e^(7/2) + 2*c^2*d^2*e^(7/2))*b^3)*x^2 + 2*(9*(3*c^5*d*e^(7/2) + 2*c^3*d*e^(7/2))*a*b^2 - 2*(
3*c^5*d*e^(7/2) + 2*c^3*d*e^(7/2))*b^3)*x)*sqrt(d*x + c))*log(d*x + c + sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1))^2 -
9*((a^2*b*d^5*e^(7/2)*x^5 + 5*a^2*b*c*d^4*e^(7/2)*x^4 + (10*c^2*d^3*e^(7/2) + d^3*e^(7/2))*a^2*b*x^3 + (10*c^
3*d^2*e^(7/2) + 3*c*d^2*e^(7/2))*a^2*b*x^2 + (5*c^4*d*e^(7/2) + 3*c^2*d*e^(7/2))*a^2*b*x + (c^5*e^(7/2) + c^3*
e^(7/2))*a^2*b)*sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1)*sqrt(d*x + c) + (a^2*b*d^6*e^(7/2)*x^6 + 6*a^2*b*c*d^5*e^(7/
2)*x^5 + (15*c^2*d^4*e^(7/2) + d^4*e^(7/2))*a^2*b*x^4 + 4*(5*c^3*d^3*e^(7/2) + c*d^3*e^(7/2))*a^2*b*x^3 + 3*(5
*c^4*d^2*e^(7/2) + 2*c^2*d^2*e^(7/2))*a^2*b*x^2 + 2*(3*c^5*d*e^(7/2) + 2*c^3*d*e^(7/2))*a^2*b*x + (c^6*e^(7/2)
+ c^4*e^(7/2))*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)

________________________________________________________________________________________

mupad [A]  time = 0.00, size = -1, normalized size = -0.01 \[ \int {\left (c\,e+d\,e\,x\right )}^{7/2}\,{\left (a+b\,\mathrm {asinh}\left (c+d\,x\right )\right )}^3 \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*e + d*e*x)^(7/2)*(a + b*asinh(c + d*x))^3,x)

[Out]

int((c*e + d*e*x)^(7/2)*(a + b*asinh(c + d*x))^3, x)

________________________________________________________________________________________

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)**(7/2)*(a+b*asinh(d*x+c))**3,x)

[Out]

Timed out

________________________________________________________________________________________