3.217 \(\int \frac {(a+b \sin ^{-1}(c x))^3}{\sqrt {d x}} \, dx\)

Optimal. Leaf size=65 \[ \frac {2 \sqrt {d x} \left (a+b \sin ^{-1}(c x)\right )^3}{d}-\frac {6 b c \text {Int}\left (\frac {\sqrt {d x} \left (a+b \sin ^{-1}(c x)\right )^2}{\sqrt {1-c^2 x^2}},x\right )}{d} \]

[Out]

2*(a+b*arcsin(c*x))^3*(d*x)^(1/2)/d-6*b*c*Unintegrable((a+b*arcsin(c*x))^2*(d*x)^(1/2)/(-c^2*x^2+1)^(1/2),x)/d

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Rubi [A]  time = 0.16, 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 = {} \[ \int \frac {\left (a+b \sin ^{-1}(c x)\right )^3}{\sqrt {d x}} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

(2*Sqrt[d*x]*(a + b*ArcSin[c*x])^3)/d - (6*b*c*Defer[Int][(Sqrt[d*x]*(a + b*ArcSin[c*x])^2)/Sqrt[1 - c^2*x^2],
x])/d

Rubi steps

\begin {align*} \int \frac {\left (a+b \sin ^{-1}(c x)\right )^3}{\sqrt {d x}} \, dx &=\frac {2 \sqrt {d x} \left (a+b \sin ^{-1}(c x)\right )^3}{d}-\frac {(6 b c) \int \frac {\sqrt {d x} \left (a+b \sin ^{-1}(c x)\right )^2}{\sqrt {1-c^2 x^2}} \, dx}{d}\\ \end {align*}

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Mathematica [A]  time = 9.01, size = 0, normalized size = 0.00 \[ \int \frac {\left (a+b \sin ^{-1}(c x)\right )^3}{\sqrt {d x}} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.47, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (b^{3} \arcsin \left (c x\right )^{3} + 3 \, a b^{2} \arcsin \left (c x\right )^{2} + 3 \, a^{2} b \arcsin \left (c x\right ) + a^{3}\right )} \sqrt {d x}}{d x}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arcsin(c*x))^3/(d*x)^(1/2),x, algorithm="fricas")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [F(-2)]  time = 180.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (a +b \arcsin \left (c x \right )\right )^{3}}{\sqrt {d x}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Timed out

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mupad [A]  time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^3}{\sqrt {d\,x}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*asin(c*x))**3/(d*x)**(1/2),x)

[Out]

Exception raised: TypeError

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