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

Optimal. Leaf size=153 \[ -\frac{\left (d-c^2 d x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )}{5 c^2 d}+\frac{b c^3 d x^5 \sqrt{d-c^2 d x^2}}{25 \sqrt{1-c^2 x^2}}-\frac{2 b c d x^3 \sqrt{d-c^2 d x^2}}{15 \sqrt{1-c^2 x^2}}+\frac{b d x \sqrt{d-c^2 d x^2}}{5 c \sqrt{1-c^2 x^2}} \]

[Out]

(b*d*x*Sqrt[d - c^2*d*x^2])/(5*c*Sqrt[1 - c^2*x^2]) - (2*b*c*d*x^3*Sqrt[d - c^2*d*x^2])/(15*Sqrt[1 - c^2*x^2])
+ (b*c^3*d*x^5*Sqrt[d - c^2*d*x^2])/(25*Sqrt[1 - c^2*x^2]) - ((d - c^2*d*x^2)^(5/2)*(a + b*ArcSin[c*x]))/(5*c
^2*d)

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Rubi [A]  time = 0.087443, antiderivative size = 153, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.08, Rules used = {4677, 194} \[ -\frac{\left (d-c^2 d x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )}{5 c^2 d}+\frac{b c^3 d x^5 \sqrt{d-c^2 d x^2}}{25 \sqrt{1-c^2 x^2}}-\frac{2 b c d x^3 \sqrt{d-c^2 d x^2}}{15 \sqrt{1-c^2 x^2}}+\frac{b d x \sqrt{d-c^2 d x^2}}{5 c \sqrt{1-c^2 x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(b*d*x*Sqrt[d - c^2*d*x^2])/(5*c*Sqrt[1 - c^2*x^2]) - (2*b*c*d*x^3*Sqrt[d - c^2*d*x^2])/(15*Sqrt[1 - c^2*x^2])
+ (b*c^3*d*x^5*Sqrt[d - c^2*d*x^2])/(25*Sqrt[1 - c^2*x^2]) - ((d - c^2*d*x^2)^(5/2)*(a + b*ArcSin[c*x]))/(5*c
^2*d)

Rule 4677

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*(x_)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[((d + e*x^2)^
(p + 1)*(a + b*ArcSin[c*x])^n)/(2*e*(p + 1)), x] + Dist[(b*n*d^IntPart[p]*(d + e*x^2)^FracPart[p])/(2*c*(p + 1
)*(1 - c^2*x^2)^FracPart[p]), Int[(1 - c^2*x^2)^(p + 1/2)*(a + b*ArcSin[c*x])^(n - 1), x], x] /; FreeQ[{a, b,
c, d, e, p}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && NeQ[p, -1]

Rule 194

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Int[ExpandIntegrand[(a + b*x^n)^p, x], x] /; FreeQ[{a, b}, x]
&& IGtQ[n, 0] && IGtQ[p, 0]

Rubi steps

\begin{align*} \int x \left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right ) \, dx &=-\frac{\left (d-c^2 d x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )}{5 c^2 d}+\frac{\left (b d \sqrt{d-c^2 d x^2}\right ) \int \left (1-c^2 x^2\right )^2 \, dx}{5 c \sqrt{1-c^2 x^2}}\\ &=-\frac{\left (d-c^2 d x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )}{5 c^2 d}+\frac{\left (b d \sqrt{d-c^2 d x^2}\right ) \int \left (1-2 c^2 x^2+c^4 x^4\right ) \, dx}{5 c \sqrt{1-c^2 x^2}}\\ &=\frac{b d x \sqrt{d-c^2 d x^2}}{5 c \sqrt{1-c^2 x^2}}-\frac{2 b c d x^3 \sqrt{d-c^2 d x^2}}{15 \sqrt{1-c^2 x^2}}+\frac{b c^3 d x^5 \sqrt{d-c^2 d x^2}}{25 \sqrt{1-c^2 x^2}}-\frac{\left (d-c^2 d x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )}{5 c^2 d}\\ \end{align*}

Mathematica [A]  time = 0.0558955, size = 84, normalized size = 0.55 \[ \frac{d \sqrt{d-c^2 d x^2} \left (\frac{b c \left (\frac{c^4 x^5}{5}-\frac{2 c^2 x^3}{3}+x\right )}{\sqrt{1-c^2 x^2}}-\left (c^2 x^2-1\right )^2 \left (a+b \sin ^{-1}(c x)\right )\right )}{5 c^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(d*Sqrt[d - c^2*d*x^2]*((b*c*(x - (2*c^2*x^3)/3 + (c^4*x^5)/5))/Sqrt[1 - c^2*x^2] - (-1 + c^2*x^2)^2*(a + b*Ar
cSin[c*x])))/(5*c^2)

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Maple [C]  time = 0.175, size = 597, normalized size = 3.9 \begin{align*} -{\frac{a}{5\,{c}^{2}d} \left ( -{c}^{2}d{x}^{2}+d \right ) ^{{\frac{5}{2}}}}+b \left ( -{\frac{ \left ( i+5\,\arcsin \left ( cx \right ) \right ) d}{800\,{c}^{2} \left ({c}^{2}{x}^{2}-1 \right ) }\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) } \left ( 16\,{c}^{6}{x}^{6}-28\,{c}^{4}{x}^{4}-16\,i\sqrt{-{c}^{2}{x}^{2}+1}{x}^{5}{c}^{5}+13\,{c}^{2}{x}^{2}+20\,i\sqrt{-{c}^{2}{x}^{2}+1}{x}^{3}{c}^{3}-5\,i\sqrt{-{c}^{2}{x}^{2}+1}xc-1 \right ) }+{\frac{ \left ( i+3\,\arcsin \left ( cx \right ) \right ) d}{96\,{c}^{2} \left ({c}^{2}{x}^{2}-1 \right ) }\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) } \left ( 4\,{c}^{4}{x}^{4}-5\,{c}^{2}{x}^{2}-4\,i\sqrt{-{c}^{2}{x}^{2}+1}{x}^{3}{c}^{3}+3\,i\sqrt{-{c}^{2}{x}^{2}+1}xc+1 \right ) }-{\frac{ \left ( \arcsin \left ( cx \right ) +i \right ) d}{16\,{c}^{2} \left ({c}^{2}{x}^{2}-1 \right ) }\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) } \left ({c}^{2}{x}^{2}-i\sqrt{-{c}^{2}{x}^{2}+1}xc-1 \right ) }-{\frac{ \left ( \arcsin \left ( cx \right ) -i \right ) d}{16\,{c}^{2} \left ({c}^{2}{x}^{2}-1 \right ) }\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) } \left ( i\sqrt{-{c}^{2}{x}^{2}+1}xc+{c}^{2}{x}^{2}-1 \right ) }+{\frac{ \left ( -i+3\,\arcsin \left ( cx \right ) \right ) d}{96\,{c}^{2} \left ({c}^{2}{x}^{2}-1 \right ) }\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) } \left ( 4\,i\sqrt{-{c}^{2}{x}^{2}+1}{x}^{3}{c}^{3}+4\,{c}^{4}{x}^{4}-3\,i\sqrt{-{c}^{2}{x}^{2}+1}xc-5\,{c}^{2}{x}^{2}+1 \right ) }-{\frac{ \left ( -i+5\,\arcsin \left ( cx \right ) \right ) d}{800\,{c}^{2} \left ({c}^{2}{x}^{2}-1 \right ) }\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) } \left ( 16\,i\sqrt{-{c}^{2}{x}^{2}+1}{x}^{5}{c}^{5}+16\,{c}^{6}{x}^{6}-20\,i\sqrt{-{c}^{2}{x}^{2}+1}{x}^{3}{c}^{3}-28\,{c}^{4}{x}^{4}+5\,i\sqrt{-{c}^{2}{x}^{2}+1}xc+13\,{c}^{2}{x}^{2}-1 \right ) } \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/5*a/c^2/d*(-c^2*d*x^2+d)^(5/2)+b*(-1/800*(-d*(c^2*x^2-1))^(1/2)*(16*c^6*x^6-28*c^4*x^4-16*I*(-c^2*x^2+1)^(1
/2)*x^5*c^5+13*c^2*x^2+20*I*(-c^2*x^2+1)^(1/2)*x^3*c^3-5*I*(-c^2*x^2+1)^(1/2)*x*c-1)*(I+5*arcsin(c*x))*d/c^2/(
c^2*x^2-1)+1/96*(-d*(c^2*x^2-1))^(1/2)*(4*c^4*x^4-5*c^2*x^2-4*I*(-c^2*x^2+1)^(1/2)*x^3*c^3+3*I*(-c^2*x^2+1)^(1
/2)*x*c+1)*(I+3*arcsin(c*x))*d/c^2/(c^2*x^2-1)-1/16*(-d*(c^2*x^2-1))^(1/2)*(c^2*x^2-I*(-c^2*x^2+1)^(1/2)*x*c-1
)*(arcsin(c*x)+I)*d/c^2/(c^2*x^2-1)-1/16*(-d*(c^2*x^2-1))^(1/2)*(I*(-c^2*x^2+1)^(1/2)*x*c+c^2*x^2-1)*(arcsin(c
*x)-I)*d/c^2/(c^2*x^2-1)+1/96*(-d*(c^2*x^2-1))^(1/2)*(4*I*(-c^2*x^2+1)^(1/2)*x^3*c^3+4*c^4*x^4-3*I*(-c^2*x^2+1
)^(1/2)*x*c-5*c^2*x^2+1)*(-I+3*arcsin(c*x))*d/c^2/(c^2*x^2-1)-1/800*(-d*(c^2*x^2-1))^(1/2)*(16*I*(-c^2*x^2+1)^
(1/2)*x^5*c^5+16*c^6*x^6-20*I*(-c^2*x^2+1)^(1/2)*x^3*c^3-28*c^4*x^4+5*I*(-c^2*x^2+1)^(1/2)*x*c+13*c^2*x^2-1)*(
-I+5*arcsin(c*x))*d/c^2/(c^2*x^2-1))

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Maxima [A]  time = 1.57588, size = 117, normalized size = 0.76 \begin{align*} -\frac{{\left (-c^{2} d x^{2} + d\right )}^{\frac{5}{2}} b \arcsin \left (c x\right )}{5 \, c^{2} d} - \frac{{\left (-c^{2} d x^{2} + d\right )}^{\frac{5}{2}} a}{5 \, c^{2} d} + \frac{{\left (3 \, c^{4} d^{\frac{5}{2}} x^{5} - 10 \, c^{2} d^{\frac{5}{2}} x^{3} + 15 \, d^{\frac{5}{2}} x\right )} b}{75 \, c d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/5*(-c^2*d*x^2 + d)^(5/2)*b*arcsin(c*x)/(c^2*d) - 1/5*(-c^2*d*x^2 + d)^(5/2)*a/(c^2*d) + 1/75*(3*c^4*d^(5/2)
*x^5 - 10*c^2*d^(5/2)*x^3 + 15*d^(5/2)*x)*b/(c*d)

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Fricas [A]  time = 2.27427, size = 344, normalized size = 2.25 \begin{align*} -\frac{{\left (3 \, b c^{5} d x^{5} - 10 \, b c^{3} d x^{3} + 15 \, b c d x\right )} \sqrt{-c^{2} d x^{2} + d} \sqrt{-c^{2} x^{2} + 1} + 15 \,{\left (a c^{6} d x^{6} - 3 \, a c^{4} d x^{4} + 3 \, a c^{2} d x^{2} - a d +{\left (b c^{6} d x^{6} - 3 \, b c^{4} d x^{4} + 3 \, b c^{2} d x^{2} - b d\right )} \arcsin \left (c x\right )\right )} \sqrt{-c^{2} d x^{2} + d}}{75 \,{\left (c^{4} x^{2} - c^{2}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/75*((3*b*c^5*d*x^5 - 10*b*c^3*d*x^3 + 15*b*c*d*x)*sqrt(-c^2*d*x^2 + d)*sqrt(-c^2*x^2 + 1) + 15*(a*c^6*d*x^6
- 3*a*c^4*d*x^4 + 3*a*c^2*d*x^2 - a*d + (b*c^6*d*x^6 - 3*b*c^4*d*x^4 + 3*b*c^2*d*x^2 - b*d)*arcsin(c*x))*sqrt
(-c^2*d*x^2 + d))/(c^4*x^2 - c^2)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (-c^{2} d x^{2} + d\right )}^{\frac{3}{2}}{\left (b \arcsin \left (c x\right ) + a\right )} x\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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