3.4.1 \(\int \frac {(a+b \text {ArcSin}(c+d x))^3}{\sqrt {c e+d e x}} \, dx\) [301]

Optimal. Leaf size=80 \[ \frac {2 \sqrt {e (c+d x)} (a+b \text {ArcSin}(c+d x))^3}{d e}-\frac {6 b \text {Int}\left (\frac {\sqrt {e (c+d x)} (a+b \text {ArcSin}(c+d x))^2}{\sqrt {1-(c+d x)^2}},x\right )}{e} \]

[Out]

2*(a+b*arcsin(d*x+c))^3*(e*(d*x+c))^(1/2)/d/e-6*b*Unintegrable((a+b*arcsin(d*x+c))^2*(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))^3}{\sqrt {c e+d e x}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

(2*Sqrt[e*(c + d*x)]*(a + b*ArcSin[c + d*x])^3)/(d*e) - (6*b*Defer[Subst][Defer[Int][(Sqrt[e*x]*(a + b*ArcSin[
x])^2)/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 )^3}{\sqrt {c e+d e x}} \, dx &=\frac {\text {Subst}\left (\int \frac {\left (a+b \sin ^{-1}(x)\right )^3}{\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 )^3}{d e}-\frac {(6 b) \text {Subst}\left (\int \frac {\sqrt {e x} \left (a+b \sin ^{-1}(x)\right )^2}{\sqrt {1-x^2}} \, dx,x,c+d x\right )}{d e}\\ \end {align*}

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Mathematica [A]
time = 127.54, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(a+b \text {ArcSin}(c+d x))^3}{\sqrt {c e+d e x}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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Maple [A]
time = 180.00, size = 0, normalized size = 0.00 \[\int \frac {\left (a +b \arcsin \left (d x +c \right )\right )^{3}}{\sqrt {d e x +c e}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*arcsin(d*x+c))^3/(d*e*x+c*e)^(1/2),x)

[Out]

int((a+b*arcsin(d*x+c))^3/(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))^3/(d*e*x+c*e)^(1/2),x, algorithm="maxima")

[Out]

1/2*(4*sqrt(d*x + c)*b^3*arctan2(d*x + c, sqrt(d*x + c + 1)*sqrt(-d*x - c + 1))^3*e^(1/2) + (6*a*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) + 6*a^2*b*d^2*e^(1/2)*int
egrate(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)))/(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*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) + 12*a^2*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) + 6*a*b^2*c^2*e^(1/2)*integrate(sqrt(d*x + c)*arc
tan(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) + 6*a^2*b*c^2*e^(1/2)*integrate(sqrt(d*x + c)*arctan(d*x/(s
qrt(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^3*c^2*e^(1/2)/d + 12*b^3*d*e^(1/2)*integrate(sqrt(d*x + c + 1)*sqrt(d*x + c)*sqr
t(-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))
)^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) + 12*b^3*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)))^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) + 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^3*c*e^(1/2)/d - 6*a*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) - 6*a^2*b*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*(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^3*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^3*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))^3/(d*e*x+c*e)^(1/2),x, algorithm="fricas")

[Out]

integral((b^3*arcsin(d*x + c)^3 + 3*a*b^2*arcsin(d*x + c)^2 + 3*a^2*b*arcsin(d*x + c) + a^3)*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))**3/(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))^3/(d*e*x+c*e)^(1/2),x, algorithm="giac")

[Out]

integrate((b*arcsin(d*x + c) + a)^3/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 )}^3}{\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))^3/(c*e + d*e*x)^(1/2),x)

[Out]

int((a + b*asin(c + d*x))^3/(c*e + d*e*x)^(1/2), x)

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