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

Optimal. Leaf size=21 \[ \text {Int}\left (\frac {(d x)^{3/2}}{\left (a+b \sin ^{-1}(c x)\right )^2},x\right ) \]

[Out]

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

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Rubi [A]  time = 0.03, 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 = {} \[ \int \frac {(d x)^{3/2}}{\left (a+b \sin ^{-1}(c x)\right )^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]  time = 6.21, size = 0, normalized size = 0.00 \[ \int \frac {(d x)^{3/2}}{\left (a+b \sin ^{-1}(c x)\right )^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.48, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {d x} d x}{b^{2} \arcsin \left (c x\right )^{2} + 2 \, a b \arcsin \left (c x\right ) + a^{2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.12, size = 0, normalized size = 0.00 \[ \int \frac {\left (d x \right )^{\frac {3}{2}}}{\left (a +b \arcsin \left (c x \right )\right )^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.00, size = 0, normalized size = 0.00 \[ -\frac {\sqrt {c x + 1} \sqrt {-c x + 1} d^{\frac {3}{2}} x^{\frac {3}{2}} - \frac {1}{2} \, {\left (b^{2} c \arctan \left (c x, \sqrt {c x + 1} \sqrt {-c x + 1}\right ) + a b c\right )} \sqrt {d} \int \frac {{\left (5 \, c^{2} d x^{2} - 3 \, d\right )} \sqrt {c x + 1} \sqrt {-c x + 1} \sqrt {x}}{a b c^{3} x^{2} - a b c + {\left (b^{2} c^{3} x^{2} - b^{2} c\right )} \arctan \left (c x, \sqrt {c x + 1} \sqrt {-c x + 1}\right )}\,{d x}}{b^{2} c \arctan \left (c x, \sqrt {c x + 1} \sqrt {-c x + 1}\right ) + a b c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(sqrt(c*x + 1)*sqrt(-c*x + 1)*d^(3/2)*x^(3/2) - (b^2*c*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1)) + a*b*c)*sq
rt(d)*integrate(1/2*(5*c^2*d*x^2 - 3*d)*sqrt(c*x + 1)*sqrt(-c*x + 1)*sqrt(x)/(a*b*c^3*x^2 - a*b*c + (b^2*c^3*x
^2 - b^2*c)*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1))), x))/(b^2*c*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1))
+ a*b*c)

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mupad [A]  time = 0.00, size = -1, normalized size = -0.05 \[ \int \frac {{\left (d\,x\right )}^{3/2}}{{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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