3.99 \(\int (d+e x) (f+g x+h x^2) (a+b \sin ^{-1}(c x)) \, dx\)

Optimal. Leaf size=223 \[ \frac{1}{2} x^2 (d g+e f) \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{3} x^3 (d h+e g) \left (a+b \sin ^{-1}(c x)\right )+d f x \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{4} e h x^4 \left (a+b \sin ^{-1}(c x)\right )+\frac{b \sqrt{1-c^2 x^2} \left (9 x \left (8 c^2 (d g+e f)+3 e h\right )+32 \left (9 c^2 d f+2 d h+2 e g\right )\right )}{288 c^3}-\frac{b \sin ^{-1}(c x) \left (8 c^2 (d g+e f)+3 e h\right )}{32 c^4}+\frac{b x^2 \sqrt{1-c^2 x^2} (d h+e g)}{9 c}+\frac{b e h x^3 \sqrt{1-c^2 x^2}}{16 c} \]

[Out]

(b*(e*g + d*h)*x^2*Sqrt[1 - c^2*x^2])/(9*c) + (b*e*h*x^3*Sqrt[1 - c^2*x^2])/(16*c) + (b*(32*(9*c^2*d*f + 2*e*g
+ 2*d*h) + 9*(8*c^2*(e*f + d*g) + 3*e*h)*x)*Sqrt[1 - c^2*x^2])/(288*c^3) - (b*(8*c^2*(e*f + d*g) + 3*e*h)*Arc
Sin[c*x])/(32*c^4) + d*f*x*(a + b*ArcSin[c*x]) + ((e*f + d*g)*x^2*(a + b*ArcSin[c*x]))/2 + ((e*g + d*h)*x^3*(a
+ b*ArcSin[c*x]))/3 + (e*h*x^4*(a + b*ArcSin[c*x]))/4

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Rubi [A]  time = 0.447616, antiderivative size = 223, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208, Rules used = {4749, 12, 1809, 780, 216} \[ \frac{1}{2} x^2 (d g+e f) \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{3} x^3 (d h+e g) \left (a+b \sin ^{-1}(c x)\right )+d f x \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{4} e h x^4 \left (a+b \sin ^{-1}(c x)\right )+\frac{b \sqrt{1-c^2 x^2} \left (9 x \left (8 c^2 (d g+e f)+3 e h\right )+32 \left (9 c^2 d f+2 d h+2 e g\right )\right )}{288 c^3}-\frac{b \sin ^{-1}(c x) \left (8 c^2 (d g+e f)+3 e h\right )}{32 c^4}+\frac{b x^2 \sqrt{1-c^2 x^2} (d h+e g)}{9 c}+\frac{b e h x^3 \sqrt{1-c^2 x^2}}{16 c} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(b*(e*g + d*h)*x^2*Sqrt[1 - c^2*x^2])/(9*c) + (b*e*h*x^3*Sqrt[1 - c^2*x^2])/(16*c) + (b*(32*(9*c^2*d*f + 2*e*g
+ 2*d*h) + 9*(8*c^2*(e*f + d*g) + 3*e*h)*x)*Sqrt[1 - c^2*x^2])/(288*c^3) - (b*(8*c^2*(e*f + d*g) + 3*e*h)*Arc
Sin[c*x])/(32*c^4) + d*f*x*(a + b*ArcSin[c*x]) + ((e*f + d*g)*x^2*(a + b*ArcSin[c*x]))/2 + ((e*g + d*h)*x^3*(a
+ b*ArcSin[c*x]))/3 + (e*h*x^4*(a + b*ArcSin[c*x]))/4

Rule 4749

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))*(Px_), x_Symbol] :> With[{u = IntHide[ExpandExpression[Px, x], x]}, Dis
t[a + b*ArcSin[c*x], u, x] - Dist[b*c, Int[SimplifyIntegrand[u/Sqrt[1 - c^2*x^2], x], x], x]] /; FreeQ[{a, b,
c}, x] && PolynomialQ[Px, x]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 1809

Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{q = Expon[Pq, x], f = Coeff[Pq, x,
Expon[Pq, x]]}, Simp[(f*(c*x)^(m + q - 1)*(a + b*x^2)^(p + 1))/(b*c^(q - 1)*(m + q + 2*p + 1)), x] + Dist[1/(
b*(m + q + 2*p + 1)), Int[(c*x)^m*(a + b*x^2)^p*ExpandToSum[b*(m + q + 2*p + 1)*Pq - b*f*(m + q + 2*p + 1)*x^q
- a*f*(m + q - 1)*x^(q - 2), x], x], x] /; GtQ[q, 1] && NeQ[m + q + 2*p + 1, 0]] /; FreeQ[{a, b, c, m, p}, x]
&& PolyQ[Pq, x] && ( !IGtQ[m, 0] || IGtQ[p + 1/2, -1])

Rule 780

Int[((d_.) + (e_.)*(x_))*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(((e*f + d*g)*(2*p
+ 3) + 2*e*g*(p + 1)*x)*(a + c*x^2)^(p + 1))/(2*c*(p + 1)*(2*p + 3)), x] - Dist[(a*e*g - c*d*f*(2*p + 3))/(c*
(2*p + 3)), Int[(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g, p}, x] &&  !LeQ[p, -1]

Rule 216

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSin[(Rt[-b, 2]*x)/Sqrt[a]]/Rt[-b, 2], x] /; FreeQ[{a, b}
, x] && GtQ[a, 0] && NegQ[b]

Rubi steps

\begin{align*} \int (d+e x) \left (f+g x+h x^2\right ) \left (a+b \sin ^{-1}(c x)\right ) \, dx &=d f x \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{2} (e f+d g) x^2 \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{3} (e g+d h) x^3 \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{4} e h x^4 \left (a+b \sin ^{-1}(c x)\right )-(b c) \int \frac{x \left (12 d f+6 (e f+d g) x+4 (e g+d h) x^2+3 e h x^3\right )}{12 \sqrt{1-c^2 x^2}} \, dx\\ &=d f x \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{2} (e f+d g) x^2 \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{3} (e g+d h) x^3 \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{4} e h x^4 \left (a+b \sin ^{-1}(c x)\right )-\frac{1}{12} (b c) \int \frac{x \left (12 d f+6 (e f+d g) x+4 (e g+d h) x^2+3 e h x^3\right )}{\sqrt{1-c^2 x^2}} \, dx\\ &=\frac{b e h x^3 \sqrt{1-c^2 x^2}}{16 c}+d f x \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{2} (e f+d g) x^2 \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{3} (e g+d h) x^3 \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{4} e h x^4 \left (a+b \sin ^{-1}(c x)\right )+\frac{b \int \frac{x \left (-48 c^2 d f-3 \left (8 c^2 (e f+d g)+3 e h\right ) x-16 c^2 (e g+d h) x^2\right )}{\sqrt{1-c^2 x^2}} \, dx}{48 c}\\ &=\frac{b (e g+d h) x^2 \sqrt{1-c^2 x^2}}{9 c}+\frac{b e h x^3 \sqrt{1-c^2 x^2}}{16 c}+d f x \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{2} (e f+d g) x^2 \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{3} (e g+d h) x^3 \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{4} e h x^4 \left (a+b \sin ^{-1}(c x)\right )-\frac{b \int \frac{x \left (16 c^2 \left (9 c^2 d f+2 e g+2 d h\right )+9 c^2 \left (8 c^2 (e f+d g)+3 e h\right ) x\right )}{\sqrt{1-c^2 x^2}} \, dx}{144 c^3}\\ &=\frac{b (e g+d h) x^2 \sqrt{1-c^2 x^2}}{9 c}+\frac{b e h x^3 \sqrt{1-c^2 x^2}}{16 c}+\frac{b \left (32 \left (9 c^2 d f+2 e g+2 d h\right )+9 \left (8 c^2 (e f+d g)+3 e h\right ) x\right ) \sqrt{1-c^2 x^2}}{288 c^3}+d f x \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{2} (e f+d g) x^2 \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{3} (e g+d h) x^3 \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{4} e h x^4 \left (a+b \sin ^{-1}(c x)\right )-\frac{\left (b \left (8 c^2 (e f+d g)+3 e h\right )\right ) \int \frac{1}{\sqrt{1-c^2 x^2}} \, dx}{32 c^3}\\ &=\frac{b (e g+d h) x^2 \sqrt{1-c^2 x^2}}{9 c}+\frac{b e h x^3 \sqrt{1-c^2 x^2}}{16 c}+\frac{b \left (32 \left (9 c^2 d f+2 e g+2 d h\right )+9 \left (8 c^2 (e f+d g)+3 e h\right ) x\right ) \sqrt{1-c^2 x^2}}{288 c^3}-\frac{b \left (8 c^2 (e f+d g)+3 e h\right ) \sin ^{-1}(c x)}{32 c^4}+d f x \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{2} (e f+d g) x^2 \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{3} (e g+d h) x^3 \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{4} e h x^4 \left (a+b \sin ^{-1}(c x)\right )\\ \end{align*}

Mathematica [A]  time = 0.299223, size = 186, normalized size = 0.83 \[ \frac{24 a c^4 x (2 d (6 f+x (3 g+2 h x))+e x (6 f+x (4 g+3 h x)))+b c \sqrt{1-c^2 x^2} \left (2 c^2 \left (4 d \left (36 f+9 g x+4 h x^2\right )+e x \left (36 f+16 g x+9 h x^2\right )\right )+64 d h+64 e g+27 e h x\right )+3 b \sin ^{-1}(c x) \left (8 c^4 x \left (2 d \left (6 f+3 g x+2 h x^2\right )+e x \left (6 f+4 g x+3 h x^2\right )\right )-24 c^2 (d g+e f)-9 e h\right )}{288 c^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(24*a*c^4*x*(2*d*(6*f + x*(3*g + 2*h*x)) + e*x*(6*f + x*(4*g + 3*h*x))) + b*c*Sqrt[1 - c^2*x^2]*(64*e*g + 64*d
*h + 27*e*h*x + 2*c^2*(4*d*(36*f + 9*g*x + 4*h*x^2) + e*x*(36*f + 16*g*x + 9*h*x^2))) + 3*b*(-24*c^2*(e*f + d*
g) - 9*e*h + 8*c^4*x*(2*d*(6*f + 3*g*x + 2*h*x^2) + e*x*(6*f + 4*g*x + 3*h*x^2)))*ArcSin[c*x])/(288*c^4)

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Maple [A]  time = 0.005, size = 307, normalized size = 1.4 \begin{align*}{\frac{1}{c} \left ({\frac{a}{{c}^{3}} \left ({\frac{eh{c}^{4}{x}^{4}}{4}}+{\frac{ \left ( dch+ecg \right ){c}^{3}{x}^{3}}{3}}+{\frac{ \left ( d{c}^{2}g+ef{c}^{2} \right ){c}^{2}{x}^{2}}{2}}+{c}^{4}fdx \right ) }+{\frac{b}{{c}^{3}} \left ({\frac{\arcsin \left ( cx \right ) eh{c}^{4}{x}^{4}}{4}}+{\frac{\arcsin \left ( cx \right ){c}^{4}{x}^{3}dh}{3}}+{\frac{\arcsin \left ( cx \right ){c}^{4}{x}^{3}eg}{3}}+{\frac{\arcsin \left ( cx \right ){c}^{4}{x}^{2}dg}{2}}+{\frac{\arcsin \left ( cx \right ){c}^{4}{x}^{2}ef}{2}}+\arcsin \left ( cx \right ){c}^{4}fdx-{\frac{eh}{4} \left ( -{\frac{{c}^{3}{x}^{3}}{4}\sqrt{-{c}^{2}{x}^{2}+1}}-{\frac{3\,cx}{8}\sqrt{-{c}^{2}{x}^{2}+1}}+{\frac{3\,\arcsin \left ( cx \right ) }{8}} \right ) }-{\frac{4\,dch+4\,ecg}{12} \left ( -{\frac{{c}^{2}{x}^{2}}{3}\sqrt{-{c}^{2}{x}^{2}+1}}-{\frac{2}{3}\sqrt{-{c}^{2}{x}^{2}+1}} \right ) }-{\frac{6\,d{c}^{2}g+6\,ef{c}^{2}}{12} \left ( -{\frac{cx}{2}\sqrt{-{c}^{2}{x}^{2}+1}}+{\frac{\arcsin \left ( cx \right ) }{2}} \right ) }+d{c}^{3}f\sqrt{-{c}^{2}{x}^{2}+1} \right ) } \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)*(h*x^2+g*x+f)*(a+b*arcsin(c*x)),x)

[Out]

1/c*(a/c^3*(1/4*e*h*c^4*x^4+1/3*(c*d*h+c*e*g)*c^3*x^3+1/2*(c^2*d*g+c^2*e*f)*c^2*x^2+c^4*f*d*x)+b/c^3*(1/4*arcs
in(c*x)*e*h*c^4*x^4+1/3*arcsin(c*x)*c^4*x^3*d*h+1/3*arcsin(c*x)*c^4*x^3*e*g+1/2*arcsin(c*x)*c^4*x^2*d*g+1/2*ar
csin(c*x)*c^4*x^2*e*f+arcsin(c*x)*c^4*f*d*x-1/4*e*h*(-1/4*c^3*x^3*(-c^2*x^2+1)^(1/2)-3/8*c*x*(-c^2*x^2+1)^(1/2
)+3/8*arcsin(c*x))-1/12*(4*c*d*h+4*c*e*g)*(-1/3*c^2*x^2*(-c^2*x^2+1)^(1/2)-2/3*(-c^2*x^2+1)^(1/2))-1/12*(6*c^2
*d*g+6*c^2*e*f)*(-1/2*c*x*(-c^2*x^2+1)^(1/2)+1/2*arcsin(c*x))+d*c^3*f*(-c^2*x^2+1)^(1/2)))

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Maxima [A]  time = 1.67339, size = 500, normalized size = 2.24 \begin{align*} \frac{1}{4} \, a e h x^{4} + \frac{1}{3} \, a e g x^{3} + \frac{1}{3} \, a d h x^{3} + \frac{1}{2} \, a e f x^{2} + \frac{1}{2} \, a d g x^{2} + \frac{1}{4} \,{\left (2 \, x^{2} \arcsin \left (c x\right ) + c{\left (\frac{\sqrt{-c^{2} x^{2} + 1} x}{c^{2}} - \frac{\arcsin \left (\frac{c^{2} x}{\sqrt{c^{2}}}\right )}{\sqrt{c^{2}} c^{2}}\right )}\right )} b e f + \frac{1}{4} \,{\left (2 \, x^{2} \arcsin \left (c x\right ) + c{\left (\frac{\sqrt{-c^{2} x^{2} + 1} x}{c^{2}} - \frac{\arcsin \left (\frac{c^{2} x}{\sqrt{c^{2}}}\right )}{\sqrt{c^{2}} c^{2}}\right )}\right )} b d g + \frac{1}{9} \,{\left (3 \, x^{3} \arcsin \left (c x\right ) + c{\left (\frac{\sqrt{-c^{2} x^{2} + 1} x^{2}}{c^{2}} + \frac{2 \, \sqrt{-c^{2} x^{2} + 1}}{c^{4}}\right )}\right )} b e g + \frac{1}{9} \,{\left (3 \, x^{3} \arcsin \left (c x\right ) + c{\left (\frac{\sqrt{-c^{2} x^{2} + 1} x^{2}}{c^{2}} + \frac{2 \, \sqrt{-c^{2} x^{2} + 1}}{c^{4}}\right )}\right )} b d h + \frac{1}{32} \,{\left (8 \, x^{4} \arcsin \left (c x\right ) +{\left (\frac{2 \, \sqrt{-c^{2} x^{2} + 1} x^{3}}{c^{2}} + \frac{3 \, \sqrt{-c^{2} x^{2} + 1} x}{c^{4}} - \frac{3 \, \arcsin \left (\frac{c^{2} x}{\sqrt{c^{2}}}\right )}{\sqrt{c^{2}} c^{4}}\right )} c\right )} b e h + a d f x + \frac{{\left (c x \arcsin \left (c x\right ) + \sqrt{-c^{2} x^{2} + 1}\right )} b d f}{c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*a*e*h*x^4 + 1/3*a*e*g*x^3 + 1/3*a*d*h*x^3 + 1/2*a*e*f*x^2 + 1/2*a*d*g*x^2 + 1/4*(2*x^2*arcsin(c*x) + c*(sq
rt(-c^2*x^2 + 1)*x/c^2 - arcsin(c^2*x/sqrt(c^2))/(sqrt(c^2)*c^2)))*b*e*f + 1/4*(2*x^2*arcsin(c*x) + c*(sqrt(-c
^2*x^2 + 1)*x/c^2 - arcsin(c^2*x/sqrt(c^2))/(sqrt(c^2)*c^2)))*b*d*g + 1/9*(3*x^3*arcsin(c*x) + c*(sqrt(-c^2*x^
2 + 1)*x^2/c^2 + 2*sqrt(-c^2*x^2 + 1)/c^4))*b*e*g + 1/9*(3*x^3*arcsin(c*x) + c*(sqrt(-c^2*x^2 + 1)*x^2/c^2 + 2
*sqrt(-c^2*x^2 + 1)/c^4))*b*d*h + 1/32*(8*x^4*arcsin(c*x) + (2*sqrt(-c^2*x^2 + 1)*x^3/c^2 + 3*sqrt(-c^2*x^2 +
1)*x/c^4 - 3*arcsin(c^2*x/sqrt(c^2))/(sqrt(c^2)*c^4))*c)*b*e*h + a*d*f*x + (c*x*arcsin(c*x) + sqrt(-c^2*x^2 +
1))*b*d*f/c

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Fricas [A]  time = 3.00993, size = 586, normalized size = 2.63 \begin{align*} \frac{72 \, a c^{4} e h x^{4} + 288 \, a c^{4} d f x + 96 \,{\left (a c^{4} e g + a c^{4} d h\right )} x^{3} + 144 \,{\left (a c^{4} e f + a c^{4} d g\right )} x^{2} + 3 \,{\left (24 \, b c^{4} e h x^{4} + 96 \, b c^{4} d f x - 24 \, b c^{2} e f - 24 \, b c^{2} d g + 32 \,{\left (b c^{4} e g + b c^{4} d h\right )} x^{3} - 9 \, b e h + 48 \,{\left (b c^{4} e f + b c^{4} d g\right )} x^{2}\right )} \arcsin \left (c x\right ) +{\left (18 \, b c^{3} e h x^{3} + 288 \, b c^{3} d f + 64 \, b c e g + 64 \, b c d h + 32 \,{\left (b c^{3} e g + b c^{3} d h\right )} x^{2} + 9 \,{\left (8 \, b c^{3} e f + 8 \, b c^{3} d g + 3 \, b c e h\right )} x\right )} \sqrt{-c^{2} x^{2} + 1}}{288 \, c^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/288*(72*a*c^4*e*h*x^4 + 288*a*c^4*d*f*x + 96*(a*c^4*e*g + a*c^4*d*h)*x^3 + 144*(a*c^4*e*f + a*c^4*d*g)*x^2 +
3*(24*b*c^4*e*h*x^4 + 96*b*c^4*d*f*x - 24*b*c^2*e*f - 24*b*c^2*d*g + 32*(b*c^4*e*g + b*c^4*d*h)*x^3 - 9*b*e*h
+ 48*(b*c^4*e*f + b*c^4*d*g)*x^2)*arcsin(c*x) + (18*b*c^3*e*h*x^3 + 288*b*c^3*d*f + 64*b*c*e*g + 64*b*c*d*h +
32*(b*c^3*e*g + b*c^3*d*h)*x^2 + 9*(8*b*c^3*e*f + 8*b*c^3*d*g + 3*b*c*e*h)*x)*sqrt(-c^2*x^2 + 1))/c^4

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Sympy [A]  time = 2.68895, size = 449, normalized size = 2.01 \begin{align*} \begin{cases} a d f x + \frac{a d g x^{2}}{2} + \frac{a d h x^{3}}{3} + \frac{a e f x^{2}}{2} + \frac{a e g x^{3}}{3} + \frac{a e h x^{4}}{4} + b d f x \operatorname{asin}{\left (c x \right )} + \frac{b d g x^{2} \operatorname{asin}{\left (c x \right )}}{2} + \frac{b d h x^{3} \operatorname{asin}{\left (c x \right )}}{3} + \frac{b e f x^{2} \operatorname{asin}{\left (c x \right )}}{2} + \frac{b e g x^{3} \operatorname{asin}{\left (c x \right )}}{3} + \frac{b e h x^{4} \operatorname{asin}{\left (c x \right )}}{4} + \frac{b d f \sqrt{- c^{2} x^{2} + 1}}{c} + \frac{b d g x \sqrt{- c^{2} x^{2} + 1}}{4 c} + \frac{b d h x^{2} \sqrt{- c^{2} x^{2} + 1}}{9 c} + \frac{b e f x \sqrt{- c^{2} x^{2} + 1}}{4 c} + \frac{b e g x^{2} \sqrt{- c^{2} x^{2} + 1}}{9 c} + \frac{b e h x^{3} \sqrt{- c^{2} x^{2} + 1}}{16 c} - \frac{b d g \operatorname{asin}{\left (c x \right )}}{4 c^{2}} - \frac{b e f \operatorname{asin}{\left (c x \right )}}{4 c^{2}} + \frac{2 b d h \sqrt{- c^{2} x^{2} + 1}}{9 c^{3}} + \frac{2 b e g \sqrt{- c^{2} x^{2} + 1}}{9 c^{3}} + \frac{3 b e h x \sqrt{- c^{2} x^{2} + 1}}{32 c^{3}} - \frac{3 b e h \operatorname{asin}{\left (c x \right )}}{32 c^{4}} & \text{for}\: c \neq 0 \\a \left (d f x + \frac{d g x^{2}}{2} + \frac{d h x^{3}}{3} + \frac{e f x^{2}}{2} + \frac{e g x^{3}}{3} + \frac{e h x^{4}}{4}\right ) & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)*(h*x**2+g*x+f)*(a+b*asin(c*x)),x)

[Out]

Piecewise((a*d*f*x + a*d*g*x**2/2 + a*d*h*x**3/3 + a*e*f*x**2/2 + a*e*g*x**3/3 + a*e*h*x**4/4 + b*d*f*x*asin(c
*x) + b*d*g*x**2*asin(c*x)/2 + b*d*h*x**3*asin(c*x)/3 + b*e*f*x**2*asin(c*x)/2 + b*e*g*x**3*asin(c*x)/3 + b*e*
h*x**4*asin(c*x)/4 + b*d*f*sqrt(-c**2*x**2 + 1)/c + b*d*g*x*sqrt(-c**2*x**2 + 1)/(4*c) + b*d*h*x**2*sqrt(-c**2
*x**2 + 1)/(9*c) + b*e*f*x*sqrt(-c**2*x**2 + 1)/(4*c) + b*e*g*x**2*sqrt(-c**2*x**2 + 1)/(9*c) + b*e*h*x**3*sqr
t(-c**2*x**2 + 1)/(16*c) - b*d*g*asin(c*x)/(4*c**2) - b*e*f*asin(c*x)/(4*c**2) + 2*b*d*h*sqrt(-c**2*x**2 + 1)/
(9*c**3) + 2*b*e*g*sqrt(-c**2*x**2 + 1)/(9*c**3) + 3*b*e*h*x*sqrt(-c**2*x**2 + 1)/(32*c**3) - 3*b*e*h*asin(c*x
)/(32*c**4), Ne(c, 0)), (a*(d*f*x + d*g*x**2/2 + d*h*x**3/3 + e*f*x**2/2 + e*g*x**3/3 + e*h*x**4/4), True))

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Giac [B]  time = 1.26511, size = 664, normalized size = 2.98 \begin{align*} \frac{1}{3} \, a d h x^{3} + \frac{1}{3} \, a g x^{3} e + b d f x \arcsin \left (c x\right ) + a d f x + \frac{{\left (c^{2} x^{2} - 1\right )} b d h x \arcsin \left (c x\right )}{3 \, c^{2}} + \frac{{\left (c^{2} x^{2} - 1\right )} b g x \arcsin \left (c x\right ) e}{3 \, c^{2}} + \frac{\sqrt{-c^{2} x^{2} + 1} b d g x}{4 \, c} + \frac{\sqrt{-c^{2} x^{2} + 1} b f x e}{4 \, c} + \frac{{\left (c^{2} x^{2} - 1\right )} b d g \arcsin \left (c x\right )}{2 \, c^{2}} + \frac{b d h x \arcsin \left (c x\right )}{3 \, c^{2}} + \frac{{\left (c^{2} x^{2} - 1\right )} b f \arcsin \left (c x\right ) e}{2 \, c^{2}} + \frac{b g x \arcsin \left (c x\right ) e}{3 \, c^{2}} + \frac{\sqrt{-c^{2} x^{2} + 1} b d f}{c} - \frac{{\left (-c^{2} x^{2} + 1\right )}^{\frac{3}{2}} b h x e}{16 \, c^{3}} + \frac{{\left (c^{2} x^{2} - 1\right )} a d g}{2 \, c^{2}} + \frac{b d g \arcsin \left (c x\right )}{4 \, c^{2}} + \frac{{\left (c^{2} x^{2} - 1\right )} a f e}{2 \, c^{2}} + \frac{b f \arcsin \left (c x\right ) e}{4 \, c^{2}} + \frac{{\left (c^{2} x^{2} - 1\right )}^{2} b h \arcsin \left (c x\right ) e}{4 \, c^{4}} - \frac{{\left (-c^{2} x^{2} + 1\right )}^{\frac{3}{2}} b d h}{9 \, c^{3}} - \frac{{\left (-c^{2} x^{2} + 1\right )}^{\frac{3}{2}} b g e}{9 \, c^{3}} + \frac{5 \, \sqrt{-c^{2} x^{2} + 1} b h x e}{32 \, c^{3}} + \frac{{\left (c^{2} x^{2} - 1\right )}^{2} a h e}{4 \, c^{4}} + \frac{{\left (c^{2} x^{2} - 1\right )} b h \arcsin \left (c x\right ) e}{2 \, c^{4}} + \frac{\sqrt{-c^{2} x^{2} + 1} b d h}{3 \, c^{3}} + \frac{\sqrt{-c^{2} x^{2} + 1} b g e}{3 \, c^{3}} + \frac{{\left (c^{2} x^{2} - 1\right )} a h e}{2 \, c^{4}} + \frac{5 \, b h \arcsin \left (c x\right ) e}{32 \, c^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*a*d*h*x^3 + 1/3*a*g*x^3*e + b*d*f*x*arcsin(c*x) + a*d*f*x + 1/3*(c^2*x^2 - 1)*b*d*h*x*arcsin(c*x)/c^2 + 1/
3*(c^2*x^2 - 1)*b*g*x*arcsin(c*x)*e/c^2 + 1/4*sqrt(-c^2*x^2 + 1)*b*d*g*x/c + 1/4*sqrt(-c^2*x^2 + 1)*b*f*x*e/c
+ 1/2*(c^2*x^2 - 1)*b*d*g*arcsin(c*x)/c^2 + 1/3*b*d*h*x*arcsin(c*x)/c^2 + 1/2*(c^2*x^2 - 1)*b*f*arcsin(c*x)*e/
c^2 + 1/3*b*g*x*arcsin(c*x)*e/c^2 + sqrt(-c^2*x^2 + 1)*b*d*f/c - 1/16*(-c^2*x^2 + 1)^(3/2)*b*h*x*e/c^3 + 1/2*(
c^2*x^2 - 1)*a*d*g/c^2 + 1/4*b*d*g*arcsin(c*x)/c^2 + 1/2*(c^2*x^2 - 1)*a*f*e/c^2 + 1/4*b*f*arcsin(c*x)*e/c^2 +
1/4*(c^2*x^2 - 1)^2*b*h*arcsin(c*x)*e/c^4 - 1/9*(-c^2*x^2 + 1)^(3/2)*b*d*h/c^3 - 1/9*(-c^2*x^2 + 1)^(3/2)*b*g
*e/c^3 + 5/32*sqrt(-c^2*x^2 + 1)*b*h*x*e/c^3 + 1/4*(c^2*x^2 - 1)^2*a*h*e/c^4 + 1/2*(c^2*x^2 - 1)*b*h*arcsin(c*
x)*e/c^4 + 1/3*sqrt(-c^2*x^2 + 1)*b*d*h/c^3 + 1/3*sqrt(-c^2*x^2 + 1)*b*g*e/c^3 + 1/2*(c^2*x^2 - 1)*a*h*e/c^4 +
5/32*b*h*arcsin(c*x)*e/c^4