3.3.44 \(\int (c e+d e x)^{7/2} (a+b \sinh ^{-1}(c+d x))^3 \, dx\) [244]

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

[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 = 0.13, 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)^{7/2} \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)^(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 {\text {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) \text {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 = 172.82, size = 0, normalized size = 0.00 \begin {gather*} \int (c e+d e x)^{7/2} \left (a+b \sinh ^{-1}(c+d x)\right )^3 \, dx \end {gather*}

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]

________________________________________________________________________________________

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

[Out]

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

________________________________________________________________________________________

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

[Out]

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

________________________________________________________________________________________

Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}

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]

Exception raised: SystemError >> excessive stack use: stack is 5985 deep

________________________________________________________________________________________

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)^(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)

________________________________________________________________________________________

Mupad [A]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int {\left (c\,e+d\,e\,x\right )}^{7/2}\,{\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)^(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)

________________________________________________________________________________________