3.303 \(\int \frac {(a+b \sin ^{-1}(c+d x))^3}{(c e+d e x)^{5/2}} \, dx\)

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

[Out]

-2/3*(a+b*arcsin(d*x+c))^3/d/e/(e*(d*x+c))^(3/2)+2*b*Unintegrable((a+b*arcsin(d*x+c))^2/(e*(d*x+c))^(3/2)/(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 = {} \[ \int \frac {\left (a+b \sin ^{-1}(c+d x)\right )^3}{(c e+d e x)^{5/2}} \, dx \]

Verification is Not applicable to the result.

[In]

Int[(a + b*ArcSin[c + d*x])^3/(c*e + d*e*x)^(5/2),x]

[Out]

(-2*(a + b*ArcSin[c + d*x])^3)/(3*d*e*(e*(c + d*x))^(3/2)) + (2*b*Defer[Subst][Defer[Int][(a + b*ArcSin[x])^2/
((e*x)^(3/2)*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}{(c e+d e x)^{5/2}} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {\left (a+b \sin ^{-1}(x)\right )^3}{(e x)^{5/2}} \, dx,x,c+d x\right )}{d}\\ &=-\frac {2 \left (a+b \sin ^{-1}(c+d x)\right )^3}{3 d e (e (c+d x))^{3/2}}+\frac {(2 b) \operatorname {Subst}\left (\int \frac {\left (a+b \sin ^{-1}(x)\right )^2}{(e x)^{3/2} \sqrt {1-x^2}} \, dx,x,c+d x\right )}{d e}\\ \end {align*}

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Mathematica [A]  time = 28.80, size = 0, normalized size = 0.00 \[ \int \frac {\left (a+b \sin ^{-1}(c+d x)\right )^3}{(c e+d e x)^{5/2}} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[(a + b*ArcSin[c + d*x])^3/(c*e + d*e*x)^(5/2),x]

[Out]

Integrate[(a + b*ArcSin[c + d*x])^3/(c*e + d*e*x)^(5/2), x]

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fricas [A]  time = 0.50, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (b^{3} \arcsin \left (d x + c\right )^{3} + 3 \, a b^{2} \arcsin \left (d x + c\right )^{2} + 3 \, a^{2} b \arcsin \left (d x + c\right ) + a^{3}\right )} \sqrt {d e x + c e}}{d^{3} e^{3} x^{3} + 3 \, c d^{2} e^{3} x^{2} + 3 \, c^{2} d e^{3} x + c^{3} e^{3}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arcsin(d*x+c))^3/(d*e*x+c*e)^(5/2),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)*sqrt(d*e*x + c*e)
/(d^3*e^3*x^3 + 3*c*d^2*e^3*x^2 + 3*c^2*d*e^3*x + c^3*e^3), x)

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giac [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b \arcsin \left (d x + c\right ) + a\right )}^{3}}{{\left (d e x + c e\right )}^{\frac {5}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arcsin(d*x+c))^3/(d*e*x+c*e)^(5/2),x, algorithm="giac")

[Out]

integrate((b*arcsin(d*x + c) + a)^3/(d*e*x + c*e)^(5/2), x)

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maple [A]  time = 0.38, size = 0, normalized size = 0.00 \[ \int \frac {\left (a +b \arcsin \left (d x +c \right )\right )^{3}}{\left (d e x +c e \right )^{\frac {5}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*arcsin(d*x+c))^3/(d*e*x+c*e)^(5/2),x)

[Out]

int((a+b*arcsin(d*x+c))^3/(d*e*x+c*e)^(5/2),x)

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maxima [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((a+b*arcsin(d*x+c))^3/(d*e*x+c*e)^(5/2),x, algorithm="maxima")

[Out]

Timed out

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mupad [A]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (a+b\,\mathrm {asin}\left (c+d\,x\right )\right )}^3}{{\left (c\,e+d\,e\,x\right )}^{5/2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*asin(c + d*x))^3/(c*e + d*e*x)^(5/2),x)

[Out]

int((a + b*asin(c + d*x))^3/(c*e + d*e*x)^(5/2), x)

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sympy [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (a + b \operatorname {asin}{\left (c + d x \right )}\right )^{3}}{\left (e \left (c + d x\right )\right )^{\frac {5}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*asin(d*x+c))**3/(d*e*x+c*e)**(5/2),x)

[Out]

Integral((a + b*asin(c + d*x))**3/(e*(c + d*x))**(5/2), x)

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