3.252 \(\int (c e+d e x)^{7/2} (a+b \sinh ^{-1}(c+d x))^4 \, dx\)
Optimal. Leaf size=82 \[ \frac {2 (e (c+d x))^{9/2} \left (a+b \sinh ^{-1}(c+d x)\right )^4}{9 d e}-\frac {8 b \text {Int}\left (\frac {(e (c+d x))^{9/2} \left (a+b \sinh ^{-1}(c+d x)\right )^3}{\sqrt {(c+d x)^2+1}},x\right )}{9 e} \]
[Out]
2/9*(e*(d*x+c))^(9/2)*(a+b*arcsinh(d*x+c))^4/d/e-8/9*b*Unintegrable((e*(d*x+c))^(9/2)*(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 =
{} \[ \int (c e+d e x)^{7/2} \left (a+b \sinh ^{-1}(c+d x)\right )^4 \, dx \]
Verification is Not applicable to the result.
[In]
Int[(c*e + d*e*x)^(7/2)*(a + b*ArcSinh[c + d*x])^4,x]
[Out]
(2*(e*(c + d*x))^(9/2)*(a + b*ArcSinh[c + d*x])^4)/(9*d*e) - (8*b*Defer[Subst][Defer[Int][((e*x)^(9/2)*(a + b*
ArcSinh[x])^3)/Sqrt[1 + x^2], x], x, c + d*x])/(9*d*e)
Rubi steps
\begin {align*} \int (c e+d e x)^{7/2} \left (a+b \sinh ^{-1}(c+d x)\right )^4 \, dx &=\frac {\operatorname {Subst}\left (\int (e x)^{7/2} \left (a+b \sinh ^{-1}(x)\right )^4 \, dx,x,c+d x\right )}{d}\\ &=\frac {2 (e (c+d x))^{9/2} \left (a+b \sinh ^{-1}(c+d x)\right )^4}{9 d e}-\frac {(8 b) \operatorname {Subst}\left (\int \frac {(e x)^{9/2} \left (a+b \sinh ^{-1}(x)\right )^3}{\sqrt {1+x^2}} \, dx,x,c+d x\right )}{9 d e}\\ \end {align*}
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Mathematica [A] time = 98.23, size = 0, normalized size = 0.00 \[ \int (c e+d e x)^{7/2} \left (a+b \sinh ^{-1}(c+d x)\right )^4 \, dx \]
Verification is Not applicable to the result.
[In]
Integrate[(c*e + d*e*x)^(7/2)*(a + b*ArcSinh[c + d*x])^4,x]
[Out]
Integrate[(c*e + d*e*x)^(7/2)*(a + b*ArcSinh[c + d*x])^4, x]
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fricas [A] time = 0.90, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (a^{4} d^{3} e^{3} x^{3} + 3 \, a^{4} c d^{2} e^{3} x^{2} + 3 \, a^{4} c^{2} d e^{3} x + a^{4} c^{3} e^{3} + {\left (b^{4} d^{3} e^{3} x^{3} + 3 \, b^{4} c d^{2} e^{3} x^{2} + 3 \, b^{4} c^{2} d e^{3} x + b^{4} c^{3} e^{3}\right )} \operatorname {arsinh}\left (d x + c\right )^{4} + 4 \, {\left (a b^{3} d^{3} e^{3} x^{3} + 3 \, a b^{3} c d^{2} e^{3} x^{2} + 3 \, a b^{3} c^{2} d e^{3} x + a b^{3} c^{3} e^{3}\right )} \operatorname {arsinh}\left (d x + c\right )^{3} + 6 \, {\left (a^{2} b^{2} d^{3} e^{3} x^{3} + 3 \, a^{2} b^{2} c d^{2} e^{3} x^{2} + 3 \, a^{2} b^{2} c^{2} d e^{3} x + a^{2} b^{2} c^{3} e^{3}\right )} \operatorname {arsinh}\left (d x + c\right )^{2} + 4 \, {\left (a^{3} b d^{3} e^{3} x^{3} + 3 \, a^{3} b c d^{2} e^{3} x^{2} + 3 \, a^{3} b c^{2} d e^{3} x + a^{3} b c^{3} e^{3}\right )} \operatorname {arsinh}\left (d x + c\right )\right )} \sqrt {d e x + c e}, x\right ) \]
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))^4,x, algorithm="fricas")
[Out]
integral((a^4*d^3*e^3*x^3 + 3*a^4*c*d^2*e^3*x^2 + 3*a^4*c^2*d*e^3*x + a^4*c^3*e^3 + (b^4*d^3*e^3*x^3 + 3*b^4*c
*d^2*e^3*x^2 + 3*b^4*c^2*d*e^3*x + b^4*c^3*e^3)*arcsinh(d*x + c)^4 + 4*(a*b^3*d^3*e^3*x^3 + 3*a*b^3*c*d^2*e^3*
x^2 + 3*a*b^3*c^2*d*e^3*x + a*b^3*c^3*e^3)*arcsinh(d*x + c)^3 + 6*(a^2*b^2*d^3*e^3*x^3 + 3*a^2*b^2*c*d^2*e^3*x
^2 + 3*a^2*b^2*c^2*d*e^3*x + a^2*b^2*c^3*e^3)*arcsinh(d*x + c)^2 + 4*(a^3*b*d^3*e^3*x^3 + 3*a^3*b*c*d^2*e^3*x^
2 + 3*a^3*b*c^2*d*e^3*x + a^3*b*c^3*e^3)*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 \[ \int {\left (d e x + c e\right )}^{\frac {7}{2}} {\left (b \operatorname {arsinh}\left (d x + c\right ) + a\right )}^{4}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((d*e*x+c*e)^(7/2)*(a+b*arcsinh(d*x+c))^4,x, algorithm="giac")
[Out]
integrate((d*e*x + c*e)^(7/2)*(b*arcsinh(d*x + c) + a)^4, x)
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maple [F(-2)] time = 180.00, size = 0, normalized size = 0.00 \[ \int \left (d e x +c e \right )^{\frac {7}{2}} \left (a +b \arcsinh \left (d x +c \right )\right )^{4}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((d*e*x+c*e)^(7/2)*(a+b*arcsinh(d*x+c))^4,x)
[Out]
int((d*e*x+c*e)^(7/2)*(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)^(7/2)*(a+b*arcsinh(d*x+c))^4,x, algorithm="maxima")
[Out]
2/9*(d*e*x + c*e)^(9/2)*a^4/(d*e) + 2/9*(b^4*d^4*e^(7/2)*x^4 + 4*b^4*c*d^3*e^(7/2)*x^3 + 6*b^4*c^2*d^2*e^(7/2)
*x^2 + 4*b^4*c^3*d*e^(7/2)*x + b^4*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))
^4/d + integrate(-2/9*(2*((2*b^4*c^5*e^(7/2) - (9*a*b^3*d^5*e^(7/2) - 2*b^4*d^5*e^(7/2))*x^5 - 9*(c^5*e^(7/2)
+ c^3*e^(7/2))*a*b^3 - 5*(9*a*b^3*c*d^4*e^(7/2) - 2*b^4*c*d^4*e^(7/2))*x^4 + (20*b^4*c^2*d^3*e^(7/2) - 9*(10*c
^2*d^3*e^(7/2) + d^3*e^(7/2))*a*b^3)*x^3 + (20*b^4*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^3)*x^2 + (10*b^4*c^4*d*e^(7/2) - 9*(5*c^4*d*e^(7/2) + 3*c^2*d*e^(7/2))*a*b^3)*x)*sqrt(d^2*x^2 + 2*c*d*x +
c^2 + 1)*sqrt(d*x + c) - ((9*a*b^3*d^6*e^(7/2) - 2*b^4*d^6*e^(7/2))*x^6 + 6*(9*a*b^3*c*d^5*e^(7/2) - 2*b^4*c*d
^5*e^(7/2))*x^5 + 9*(c^6*e^(7/2) + c^4*e^(7/2))*a*b^3 - 2*(c^6*e^(7/2) + c^4*e^(7/2))*b^4 + (9*(15*c^2*d^4*e^(
7/2) + d^4*e^(7/2))*a*b^3 - 2*(15*c^2*d^4*e^(7/2) + d^4*e^(7/2))*b^4)*x^4 + 4*(9*(5*c^3*d^3*e^(7/2) + c*d^3*e^
(7/2))*a*b^3 - 2*(5*c^3*d^3*e^(7/2) + c*d^3*e^(7/2))*b^4)*x^3 + 3*(9*(5*c^4*d^2*e^(7/2) + 2*c^2*d^2*e^(7/2))*a
*b^3 - 2*(5*c^4*d^2*e^(7/2) + 2*c^2*d^2*e^(7/2))*b^4)*x^2 + 2*(9*(3*c^5*d*e^(7/2) + 2*c^3*d*e^(7/2))*a*b^3 - 2
*(3*c^5*d*e^(7/2) + 2*c^3*d*e^(7/2))*b^4)*x)*sqrt(d*x + c))*log(d*x + c + sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1))^3
- 27*((a^2*b^2*d^5*e^(7/2)*x^5 + 5*a^2*b^2*c*d^4*e^(7/2)*x^4 + (10*c^2*d^3*e^(7/2) + d^3*e^(7/2))*a^2*b^2*x^3
+ (10*c^3*d^2*e^(7/2) + 3*c*d^2*e^(7/2))*a^2*b^2*x^2 + (5*c^4*d*e^(7/2) + 3*c^2*d*e^(7/2))*a^2*b^2*x + (c^5*e
^(7/2) + c^3*e^(7/2))*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^6*e^(7/2)*x^6 + 6*
a^2*b^2*c*d^5*e^(7/2)*x^5 + (15*c^2*d^4*e^(7/2) + d^4*e^(7/2))*a^2*b^2*x^4 + 4*(5*c^3*d^3*e^(7/2) + c*d^3*e^(7
/2))*a^2*b^2*x^3 + 3*(5*c^4*d^2*e^(7/2) + 2*c^2*d^2*e^(7/2))*a^2*b^2*x^2 + 2*(3*c^5*d*e^(7/2) + 2*c^3*d*e^(7/2
))*a^2*b^2*x + (c^6*e^(7/2) + c^4*e^(7/2))*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 - 18*((a^3*b*d^5*e^(7/2)*x^5 + 5*a^3*b*c*d^4*e^(7/2)*x^4 + (10*c^2*d^3*e^(7/2) + d^3*e^(7/2))*a^3*b*x^
3 + (10*c^3*d^2*e^(7/2) + 3*c*d^2*e^(7/2))*a^3*b*x^2 + (5*c^4*d*e^(7/2) + 3*c^2*d*e^(7/2))*a^3*b*x + (c^5*e^(7
/2) + c^3*e^(7/2))*a^3*b)*sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1)*sqrt(d*x + c) + (a^3*b*d^6*e^(7/2)*x^6 + 6*a^3*b*c
*d^5*e^(7/2)*x^5 + (15*c^2*d^4*e^(7/2) + d^4*e^(7/2))*a^3*b*x^4 + 4*(5*c^3*d^3*e^(7/2) + c*d^3*e^(7/2))*a^3*b*
x^3 + 3*(5*c^4*d^2*e^(7/2) + 2*c^2*d^2*e^(7/2))*a^3*b*x^2 + 2*(3*c^5*d*e^(7/2) + 2*c^3*d*e^(7/2))*a^3*b*x + (c
^6*e^(7/2) + c^4*e^(7/2))*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 \[ \int {\left (c\,e+d\,e\,x\right )}^{7/2}\,{\left (a+b\,\mathrm {asinh}\left (c+d\,x\right )\right )}^4 \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((c*e + d*e*x)^(7/2)*(a + b*asinh(c + d*x))^4,x)
[Out]
int((c*e + d*e*x)^(7/2)*(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)**(7/2)*(a+b*asinh(d*x+c))**4,x)
[Out]
Timed out
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