3.4.5 \(\int \frac {(a+b \text {ArcSin}(c+d x))^4}{\sqrt {c e+d e x}} \, dx\) [305]
Optimal. Leaf size=80 \[ \frac {2 \sqrt {e (c+d x)} (a+b \text {ArcSin}(c+d x))^4}{d e}-\frac {8 b \text {Int}\left (\frac {\sqrt {e (c+d x)} (a+b \text {ArcSin}(c+d x))^3}{\sqrt {1-(c+d x)^2}},x\right )}{e} \]
[Out]
2*(a+b*arcsin(d*x+c))^4*(e*(d*x+c))^(1/2)/d/e-8*b*Unintegrable((a+b*arcsin(d*x+c))^3*(e*(d*x+c))^(1/2)/(1-(d*x
+c)^2)^(1/2),x)/e
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Rubi [A]
time = 0.12, 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 \frac {(a+b \text {ArcSin}(c+d x))^4}{\sqrt {c e+d e x}} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
Int[(a + b*ArcSin[c + d*x])^4/Sqrt[c*e + d*e*x],x]
[Out]
(2*Sqrt[e*(c + d*x)]*(a + b*ArcSin[c + d*x])^4)/(d*e) - (8*b*Defer[Subst][Defer[Int][(Sqrt[e*x]*(a + b*ArcSin[
x])^3)/Sqrt[1 - x^2], x], x, c + d*x])/(d*e)
Rubi steps
\begin {align*} \int \frac {\left (a+b \sin ^{-1}(c+d x)\right )^4}{\sqrt {c e+d e x}} \, dx &=\frac {\text {Subst}\left (\int \frac {\left (a+b \sin ^{-1}(x)\right )^4}{\sqrt {e x}} \, dx,x,c+d x\right )}{d}\\ &=\frac {2 \sqrt {e (c+d x)} \left (a+b \sin ^{-1}(c+d x)\right )^4}{d e}-\frac {(8 b) \text {Subst}\left (\int \frac {\sqrt {e x} \left (a+b \sin ^{-1}(x)\right )^3}{\sqrt {1-x^2}} \, dx,x,c+d x\right )}{d e}\\ \end {align*}
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Mathematica [A]
time = 56.61, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(a+b \text {ArcSin}(c+d x))^4}{\sqrt {c e+d e x}} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
Integrate[(a + b*ArcSin[c + d*x])^4/Sqrt[c*e + d*e*x],x]
[Out]
Integrate[(a + b*ArcSin[c + d*x])^4/Sqrt[c*e + d*e*x], x]
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Maple [A]
time = 180.00, size = 0, normalized size = 0.00 \[\int \frac {\left (a +b \arcsin \left (d x +c \right )\right )^{4}}{\sqrt {d e x +c e}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((a+b*arcsin(d*x+c))^4/(d*e*x+c*e)^(1/2),x)
[Out]
int((a+b*arcsin(d*x+c))^4/(d*e*x+c*e)^(1/2),x)
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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((a+b*arcsin(d*x+c))^4/(d*e*x+c*e)^(1/2),x, algorithm="maxima")
[Out]
1/2*(4*sqrt(d*x + c)*b^4*arctan2(d*x + c, sqrt(d*x + c + 1)*sqrt(-d*x - c + 1))^4*e^(1/2) + (8*a*b^3*d^2*e^(1/
2)*integrate(sqrt(d*x + c)*x^2*arctan(d*x/(sqrt(d*x + c + 1)*sqrt(-d*x - c + 1)) + c/(sqrt(d*x + c + 1)*sqrt(-
d*x - c + 1)))^3/(d^3*x^3*e + 3*c*d^2*x^2*e + 3*c^2*d*x*e + c^3*e - d*x*e - c*e), x) + 12*a^2*b^2*d^2*e^(1/2)*
integrate(sqrt(d*x + c)*x^2*arctan(d*x/(sqrt(d*x + c + 1)*sqrt(-d*x - c + 1)) + c/(sqrt(d*x + c + 1)*sqrt(-d*x
- c + 1)))^2/(d^3*x^3*e + 3*c*d^2*x^2*e + 3*c^2*d*x*e + c^3*e - d*x*e - c*e), x) + 16*a*b^3*c*d*e^(1/2)*integ
rate(sqrt(d*x + c)*x*arctan(d*x/(sqrt(d*x + c + 1)*sqrt(-d*x - c + 1)) + c/(sqrt(d*x + c + 1)*sqrt(-d*x - c +
1)))^3/(d^3*x^3*e + 3*c*d^2*x^2*e + 3*c^2*d*x*e + c^3*e - d*x*e - c*e), x) + 8*a^3*b*d^2*e^(1/2)*integrate(sqr
t(d*x + c)*x^2*arctan(d*x/(sqrt(d*x + c + 1)*sqrt(-d*x - c + 1)) + c/(sqrt(d*x + c + 1)*sqrt(-d*x - c + 1)))/(
d^3*x^3*e + 3*c*d^2*x^2*e + 3*c^2*d*x*e + c^3*e - d*x*e - c*e), x) + 24*a^2*b^2*c*d*e^(1/2)*integrate(sqrt(d*x
+ c)*x*arctan(d*x/(sqrt(d*x + c + 1)*sqrt(-d*x - c + 1)) + c/(sqrt(d*x + c + 1)*sqrt(-d*x - c + 1)))^2/(d^3*x
^3*e + 3*c*d^2*x^2*e + 3*c^2*d*x*e + c^3*e - d*x*e - c*e), x) + 8*a*b^3*c^2*e^(1/2)*integrate(sqrt(d*x + c)*ar
ctan(d*x/(sqrt(d*x + c + 1)*sqrt(-d*x - c + 1)) + c/(sqrt(d*x + c + 1)*sqrt(-d*x - c + 1)))^3/(d^3*x^3*e + 3*c
*d^2*x^2*e + 3*c^2*d*x*e + c^3*e - d*x*e - c*e), x) + 16*a^3*b*c*d*e^(1/2)*integrate(sqrt(d*x + c)*x*arctan(d*
x/(sqrt(d*x + c + 1)*sqrt(-d*x - c + 1)) + c/(sqrt(d*x + c + 1)*sqrt(-d*x - c + 1)))/(d^3*x^3*e + 3*c*d^2*x^2*
e + 3*c^2*d*x*e + c^3*e - d*x*e - c*e), x) + 12*a^2*b^2*c^2*e^(1/2)*integrate(sqrt(d*x + c)*arctan(d*x/(sqrt(d
*x + c + 1)*sqrt(-d*x - c + 1)) + c/(sqrt(d*x + c + 1)*sqrt(-d*x - c + 1)))^2/(d^3*x^3*e + 3*c*d^2*x^2*e + 3*c
^2*d*x*e + c^3*e - d*x*e - c*e), x) + 8*a^3*b*c^2*e^(1/2)*integrate(sqrt(d*x + c)*arctan(d*x/(sqrt(d*x + c + 1
)*sqrt(-d*x - c + 1)) + c/(sqrt(d*x + c + 1)*sqrt(-d*x - c + 1)))/(d^3*x^3*e + 3*c*d^2*x^2*e + 3*c^2*d*x*e + c
^3*e - d*x*e - c*e), x) - (2*arctan(sqrt(d*x + c))*e^(-1) + e^(-1)*log(sqrt(d*x + c) + 1) - e^(-1)*log(sqrt(d*
x + c) - 1))*a^4*c^2*e^(1/2)/d + 16*b^4*d*e^(1/2)*integrate(sqrt(d*x + c + 1)*sqrt(d*x + c)*sqrt(-d*x - c + 1)
*x*arctan(d*x/(sqrt(d*x + c + 1)*sqrt(-d*x - c + 1)) + c/(sqrt(d*x + c + 1)*sqrt(-d*x - c + 1)))^3/(d^3*x^3*e
+ 3*c*d^2*x^2*e + 3*c^2*d*x*e + c^3*e - d*x*e - c*e), x) + 16*b^4*c*e^(1/2)*integrate(sqrt(d*x + c + 1)*sqrt(d
*x + c)*sqrt(-d*x - c + 1)*arctan(d*x/(sqrt(d*x + c + 1)*sqrt(-d*x - c + 1)) + c/(sqrt(d*x + c + 1)*sqrt(-d*x
- c + 1)))^3/(d^3*x^3*e + 3*c*d^2*x^2*e + 3*c^2*d*x*e + c^3*e - d*x*e - c*e), x) + 2*(2*(c + 1)*arctan(sqrt(d*
x + c))*e^(-1) + (c - 1)*e^(-1)*log(sqrt(d*x + c) + 1) - (c - 1)*e^(-1)*log(sqrt(d*x + c) - 1))*a^4*c*e^(1/2)/
d - 8*a*b^3*e^(1/2)*integrate(sqrt(d*x + c)*arctan(d*x/(sqrt(d*x + c + 1)*sqrt(-d*x - c + 1)) + c/(sqrt(d*x +
c + 1)*sqrt(-d*x - c + 1)))^3/(d^3*x^3*e + 3*c*d^2*x^2*e + 3*c^2*d*x*e + c^3*e - d*x*e - c*e), x) - 12*a^2*b^2
*e^(1/2)*integrate(sqrt(d*x + c)*arctan(d*x/(sqrt(d*x + c + 1)*sqrt(-d*x - c + 1)) + c/(sqrt(d*x + c + 1)*sqrt
(-d*x - c + 1)))^2/(d^3*x^3*e + 3*c*d^2*x^2*e + 3*c^2*d*x*e + c^3*e - d*x*e - c*e), x) - 8*a^3*b*e^(1/2)*integ
rate(sqrt(d*x + c)*arctan(d*x/(sqrt(d*x + c + 1)*sqrt(-d*x - c + 1)) + c/(sqrt(d*x + c + 1)*sqrt(-d*x - c + 1)
))/(d^3*x^3*e + 3*c*d^2*x^2*e + 3*c^2*d*x*e + c^3*e - d*x*e - c*e), x) - (2*(c^2 + 2*c + 1)*arctan(sqrt(d*x +
c))*e^(-1) + (c^2 - 2*c + 1)*e^(-1)*log(sqrt(d*x + c) + 1) - (c^2 - 2*c + 1)*e^(-1)*log(sqrt(d*x + c) - 1) - 4
*sqrt(d*x + c)*e^(-1))*a^4*e^(1/2)/d + (2*arctan(sqrt(d*x + c))*e^(-1) + e^(-1)*log(sqrt(d*x + c) + 1) - e^(-1
)*log(sqrt(d*x + c) - 1))*a^4*e^(1/2)/d)*d*e)*e^(-1)/d
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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((a+b*arcsin(d*x+c))^4/(d*e*x+c*e)^(1/2),x, algorithm="fricas")
[Out]
integral((b^4*arcsin(d*x + c)^4 + 4*a*b^3*arcsin(d*x + c)^3 + 6*a^2*b^2*arcsin(d*x + c)^2 + 4*a^3*b*arcsin(d*x
+ c) + a^4)*e^(-1/2)/sqrt(d*x + c), x)
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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((a+b*asin(d*x+c))**4/(d*e*x+c*e)**(1/2),x)
[Out]
Exception raised: TypeError >> Invalid comparison of non-real zoo
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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((a+b*arcsin(d*x+c))^4/(d*e*x+c*e)^(1/2),x, algorithm="giac")
[Out]
integrate((b*arcsin(d*x + c) + a)^4/sqrt(d*e*x + c*e), x)
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Mupad [A]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (a+b\,\mathrm {asin}\left (c+d\,x\right )\right )}^4}{\sqrt {c\,e+d\,e\,x}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((a + b*asin(c + d*x))^4/(c*e + d*e*x)^(1/2),x)
[Out]
int((a + b*asin(c + d*x))^4/(c*e + d*e*x)^(1/2), x)
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