3.27 \(\int (d+e x)^m (a+b \sin ^{-1}(c x))^2 \, dx\)

Optimal. Leaf size=76 \[ \frac {(d+e x)^{m+1} \left (a+b \sin ^{-1}(c x)\right )^2}{e (m+1)}-\frac {2 b c \text {Int}\left (\frac {(d+e x)^{m+1} \left (a+b \sin ^{-1}(c x)\right )}{\sqrt {1-c^2 x^2}},x\right )}{e (m+1)} \]

[Out]

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

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Rubi [A]  time = 0.20, 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 = {} \[ \int (d+e x)^m \left (a+b \sin ^{-1}(c x)\right )^2 \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]  time = 5.29, size = 0, normalized size = 0.00 \[ \int (d+e x)^m \left (a+b \sin ^{-1}(c x)\right )^2 \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 1.13, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (b^{2} \arcsin \left (c x\right )^{2} + 2 \, a b \arcsin \left (c x\right ) + a^{2}\right )} {\left (e x + d\right )}^{m}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((b^2*arcsin(c*x)^2 + 2*a*b*arcsin(c*x) + a^2)*(e*x + d)^m, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((b*arcsin(c*x) + a)^2*(e*x + d)^m, x)

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maple [A]  time = 0.45, size = 0, normalized size = 0.00 \[ \int \left (e x +d \right )^{m} \left (a +b \arcsin \left (c x \right )\right )^{2}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^m*(a+b*arcsin(c*x))^2,x)

[Out]

int((e*x+d)^m*(a+b*arcsin(c*x))^2,x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(e*x + d)^(m + 1)*a^2/(e*(m + 1)) + ((b^2*e*x + b^2*d)*(e*x + d)^m*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1))^
2 + (e*m + e)*integrate(2*((b^2*c*e*x + b^2*c*d)*sqrt(c*x + 1)*sqrt(-c*x + 1)*(e*x + d)^m*arctan2(c*x, sqrt(c*
x + 1)*sqrt(-c*x + 1)) - (a*b*e*m + a*b*e - (a*b*c^2*e*m + a*b*c^2*e)*x^2)*(e*x + d)^m*arctan2(c*x, sqrt(c*x +
1)*sqrt(-c*x + 1)))/((c^2*e*m + c^2*e)*x^2 - e*m - e), x))/(e*m + e)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*asin(c*x))^2*(d + e*x)^m,x)

[Out]

int((a + b*asin(c*x))^2*(d + e*x)^m, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)**m*(a+b*asin(c*x))**2,x)

[Out]

Integral((a + b*asin(c*x))**2*(d + e*x)**m, x)

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