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

Optimal. Leaf size=248 \[ d^2 f x \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{3} e x^3 (2 d g+e f) \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{2} d x^2 (d g+2 e f) \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{4} e^2 g x^4 \left (a+b \sin ^{-1}(c x)\right )+\frac{b \sqrt{1-c^2 x^2} \left (32 \left (9 c^2 d^2 f+2 e (2 d g+e f)\right )+9 x \left (8 c^2 d (d g+2 e f)+3 e^2 g\right )\right )}{288 c^3}-\frac{b \sin ^{-1}(c x) \left (8 c^2 d (d g+2 e f)+3 e^2 g\right )}{32 c^4}+\frac{b e x^2 \sqrt{1-c^2 x^2} (2 d g+e f)}{9 c}+\frac{b e^2 g x^3 \sqrt{1-c^2 x^2}}{16 c} \]

[Out]

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

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Rubi [A]  time = 0.535084, antiderivative size = 248, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.238, Rules used = {4749, 12, 1809, 780, 216} \[ d^2 f x \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{3} e x^3 (2 d g+e f) \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{2} d x^2 (d g+2 e f) \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{4} e^2 g x^4 \left (a+b \sin ^{-1}(c x)\right )+\frac{b \sqrt{1-c^2 x^2} \left (32 \left (9 c^2 d^2 f+2 e (2 d g+e f)\right )+9 x \left (8 c^2 d (d g+2 e f)+3 e^2 g\right )\right )}{288 c^3}-\frac{b \sin ^{-1}(c x) \left (8 c^2 d (d g+2 e f)+3 e^2 g\right )}{32 c^4}+\frac{b e x^2 \sqrt{1-c^2 x^2} (2 d g+e f)}{9 c}+\frac{b e^2 g x^3 \sqrt{1-c^2 x^2}}{16 c} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Mathematica [A]  time = 0.298787, size = 211, normalized size = 0.85 \[ \frac{24 a c^4 x \left (6 d^2 (2 f+g x)+4 d e x (3 f+2 g x)+e^2 x^2 (4 f+3 g x)\right )+b c \sqrt{1-c^2 x^2} \left (2 c^2 \left (36 d^2 (4 f+g x)+8 d e x (9 f+4 g x)+e^2 x^2 (16 f+9 g x)\right )+e (128 d g+64 e f+27 e g x)\right )+3 b \sin ^{-1}(c x) \left (8 c^4 x \left (6 d^2 (2 f+g x)+4 d e x (3 f+2 g x)+e^2 x^2 (4 f+3 g x)\right )-24 c^2 d (d g+2 e f)-9 e^2 g\right )}{288 c^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(24*a*c^4*x*(6*d^2*(2*f + g*x) + 4*d*e*x*(3*f + 2*g*x) + e^2*x^2*(4*f + 3*g*x)) + b*c*Sqrt[1 - c^2*x^2]*(e*(64
*e*f + 128*d*g + 27*e*g*x) + 2*c^2*(36*d^2*(4*f + g*x) + 8*d*e*x*(9*f + 4*g*x) + e^2*x^2*(16*f + 9*g*x))) + 3*
b*(-9*e^2*g - 24*c^2*d*(2*e*f + d*g) + 8*c^4*x*(6*d^2*(2*f + g*x) + 4*d*e*x*(3*f + 2*g*x) + e^2*x^2*(4*f + 3*g
*x)))*ArcSin[c*x])/(288*c^4)

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Maple [A]  time = 0.006, size = 338, normalized size = 1.4 \begin{align*}{\frac{1}{c} \left ({\frac{a}{{c}^{3}} \left ({\frac{{e}^{2}g{c}^{4}{x}^{4}}{4}}+{\frac{ \left ( 2\,dceg+{e}^{2}cf \right ){c}^{3}{x}^{3}}{3}}+{\frac{ \left ({c}^{2}{d}^{2}g+2\,d{c}^{2}ef \right ){c}^{2}{x}^{2}}{2}}+{c}^{4}{d}^{2}fx \right ) }+{\frac{b}{{c}^{3}} \left ({\frac{\arcsin \left ( cx \right ){e}^{2}g{c}^{4}{x}^{4}}{4}}+{\frac{2\,\arcsin \left ( cx \right ){c}^{4}{x}^{3}deg}{3}}+{\frac{\arcsin \left ( cx \right ){c}^{4}{x}^{3}{e}^{2}f}{3}}+{\frac{\arcsin \left ( cx \right ){c}^{4}{x}^{2}{d}^{2}g}{2}}+\arcsin \left ( cx \right ){c}^{4}{x}^{2}def+\arcsin \left ( cx \right ){c}^{4}{d}^{2}fx-{\frac{{e}^{2}g}{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{8\,dceg+4\,{e}^{2}cf}{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\,{c}^{2}{d}^{2}g+12\,d{c}^{2}ef}{12} \left ( -{\frac{cx}{2}\sqrt{-{c}^{2}{x}^{2}+1}}+{\frac{\arcsin \left ( cx \right ) }{2}} \right ) }+{c}^{3}{d}^{2}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)^2*(g*x+f)*(a+b*arcsin(c*x)),x)

[Out]

1/c*(a/c^3*(1/4*e^2*g*c^4*x^4+1/3*(2*c*d*e*g+c*e^2*f)*c^3*x^3+1/2*(c^2*d^2*g+2*c^2*d*e*f)*c^2*x^2+c^4*d^2*f*x)
+b/c^3*(1/4*arcsin(c*x)*e^2*g*c^4*x^4+2/3*arcsin(c*x)*c^4*x^3*d*e*g+1/3*arcsin(c*x)*c^4*x^3*e^2*f+1/2*arcsin(c
*x)*c^4*x^2*d^2*g+arcsin(c*x)*c^4*x^2*d*e*f+arcsin(c*x)*c^4*d^2*f*x-1/4*e^2*g*(-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*(8*c*d*e*g+4*c*e^2*f)*(-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^2*g+12*c^2*d*e*f)*(-1/2*c*x*(-c^2*x^2+1)^(1/2)+1/2*arcsin(c*x))+c^3*d^2*f*(-c
^2*x^2+1)^(1/2)))

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Maxima [A]  time = 1.74651, size = 525, normalized size = 2.12 \begin{align*} \frac{1}{4} \, a e^{2} g x^{4} + \frac{1}{3} \, a e^{2} f x^{3} + \frac{2}{3} \, a d e g x^{3} + a d e f x^{2} + \frac{1}{2} \, a d^{2} g x^{2} + \frac{1}{2} \,{\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 e f + \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^{2} 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^{2} g + \frac{2}{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 e g + \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^{2} g + a d^{2} f x + \frac{{\left (c x \arcsin \left (c x\right ) + \sqrt{-c^{2} x^{2} + 1}\right )} b d^{2} f}{c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.76757, size = 668, normalized size = 2.69 \begin{align*} \frac{72 \, a c^{4} e^{2} g x^{4} + 288 \, a c^{4} d^{2} f x + 96 \,{\left (a c^{4} e^{2} f + 2 \, a c^{4} d e g\right )} x^{3} + 144 \,{\left (2 \, a c^{4} d e f + a c^{4} d^{2} g\right )} x^{2} + 3 \,{\left (24 \, b c^{4} e^{2} g x^{4} + 96 \, b c^{4} d^{2} f x - 48 \, b c^{2} d e f + 32 \,{\left (b c^{4} e^{2} f + 2 \, b c^{4} d e g\right )} x^{3} + 48 \,{\left (2 \, b c^{4} d e f + b c^{4} d^{2} g\right )} x^{2} - 3 \,{\left (8 \, b c^{2} d^{2} + 3 \, b e^{2}\right )} g\right )} \arcsin \left (c x\right ) +{\left (18 \, b c^{3} e^{2} g x^{3} + 128 \, b c d e g + 32 \,{\left (b c^{3} e^{2} f + 2 \, b c^{3} d e g\right )} x^{2} + 32 \,{\left (9 \, b c^{3} d^{2} + 2 \, b c e^{2}\right )} f + 9 \,{\left (16 \, b c^{3} d e f +{\left (8 \, b c^{3} d^{2} + 3 \, b c e^{2}\right )} g\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)^2*(g*x+f)*(a+b*arcsin(c*x)),x, algorithm="fricas")

[Out]

1/288*(72*a*c^4*e^2*g*x^4 + 288*a*c^4*d^2*f*x + 96*(a*c^4*e^2*f + 2*a*c^4*d*e*g)*x^3 + 144*(2*a*c^4*d*e*f + a*
c^4*d^2*g)*x^2 + 3*(24*b*c^4*e^2*g*x^4 + 96*b*c^4*d^2*f*x - 48*b*c^2*d*e*f + 32*(b*c^4*e^2*f + 2*b*c^4*d*e*g)*
x^3 + 48*(2*b*c^4*d*e*f + b*c^4*d^2*g)*x^2 - 3*(8*b*c^2*d^2 + 3*b*e^2)*g)*arcsin(c*x) + (18*b*c^3*e^2*g*x^3 +
128*b*c*d*e*g + 32*(b*c^3*e^2*f + 2*b*c^3*d*e*g)*x^2 + 32*(9*b*c^3*d^2 + 2*b*c*e^2)*f + 9*(16*b*c^3*d*e*f + (8
*b*c^3*d^2 + 3*b*c*e^2)*g)*x)*sqrt(-c^2*x^2 + 1))/c^4

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Sympy [A]  time = 2.95428, size = 502, normalized size = 2.02 \begin{align*} \begin{cases} a d^{2} f x + \frac{a d^{2} g x^{2}}{2} + a d e f x^{2} + \frac{2 a d e g x^{3}}{3} + \frac{a e^{2} f x^{3}}{3} + \frac{a e^{2} g x^{4}}{4} + b d^{2} f x \operatorname{asin}{\left (c x \right )} + \frac{b d^{2} g x^{2} \operatorname{asin}{\left (c x \right )}}{2} + b d e f x^{2} \operatorname{asin}{\left (c x \right )} + \frac{2 b d e g x^{3} \operatorname{asin}{\left (c x \right )}}{3} + \frac{b e^{2} f x^{3} \operatorname{asin}{\left (c x \right )}}{3} + \frac{b e^{2} g x^{4} \operatorname{asin}{\left (c x \right )}}{4} + \frac{b d^{2} f \sqrt{- c^{2} x^{2} + 1}}{c} + \frac{b d^{2} g x \sqrt{- c^{2} x^{2} + 1}}{4 c} + \frac{b d e f x \sqrt{- c^{2} x^{2} + 1}}{2 c} + \frac{2 b d e g x^{2} \sqrt{- c^{2} x^{2} + 1}}{9 c} + \frac{b e^{2} f x^{2} \sqrt{- c^{2} x^{2} + 1}}{9 c} + \frac{b e^{2} g x^{3} \sqrt{- c^{2} x^{2} + 1}}{16 c} - \frac{b d^{2} g \operatorname{asin}{\left (c x \right )}}{4 c^{2}} - \frac{b d e f \operatorname{asin}{\left (c x \right )}}{2 c^{2}} + \frac{4 b d e g \sqrt{- c^{2} x^{2} + 1}}{9 c^{3}} + \frac{2 b e^{2} f \sqrt{- c^{2} x^{2} + 1}}{9 c^{3}} + \frac{3 b e^{2} g x \sqrt{- c^{2} x^{2} + 1}}{32 c^{3}} - \frac{3 b e^{2} g \operatorname{asin}{\left (c x \right )}}{32 c^{4}} & \text{for}\: c \neq 0 \\a \left (d^{2} f x + \frac{d^{2} g x^{2}}{2} + d e f x^{2} + \frac{2 d e g x^{3}}{3} + \frac{e^{2} f x^{3}}{3} + \frac{e^{2} g 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)**2*(g*x+f)*(a+b*asin(c*x)),x)

[Out]

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

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Giac [B]  time = 1.3287, size = 699, normalized size = 2.82 \begin{align*} \frac{2}{3} \, a d g x^{3} e + b d^{2} f x \arcsin \left (c x\right ) + \frac{1}{3} \, a f x^{3} e^{2} + a d^{2} f x + \frac{2 \,{\left (c^{2} x^{2} - 1\right )} b d g x \arcsin \left (c x\right ) e}{3 \, c^{2}} + \frac{\sqrt{-c^{2} x^{2} + 1} b d^{2} g x}{4 \, c} + \frac{\sqrt{-c^{2} x^{2} + 1} b d f x e}{2 \, c} + \frac{{\left (c^{2} x^{2} - 1\right )} b d^{2} g \arcsin \left (c x\right )}{2 \, c^{2}} + \frac{{\left (c^{2} x^{2} - 1\right )} b f x \arcsin \left (c x\right ) e^{2}}{3 \, c^{2}} + \frac{{\left (c^{2} x^{2} - 1\right )} b d f \arcsin \left (c x\right ) e}{c^{2}} + \frac{2 \, b d g x \arcsin \left (c x\right ) e}{3 \, c^{2}} + \frac{\sqrt{-c^{2} x^{2} + 1} b d^{2} f}{c} + \frac{{\left (c^{2} x^{2} - 1\right )} a d^{2} g}{2 \, c^{2}} + \frac{b d^{2} g \arcsin \left (c x\right )}{4 \, c^{2}} + \frac{b f x \arcsin \left (c x\right ) e^{2}}{3 \, c^{2}} + \frac{{\left (c^{2} x^{2} - 1\right )} a d f e}{c^{2}} + \frac{b d f \arcsin \left (c x\right ) e}{2 \, c^{2}} - \frac{{\left (-c^{2} x^{2} + 1\right )}^{\frac{3}{2}} b g x e^{2}}{16 \, c^{3}} - \frac{2 \,{\left (-c^{2} x^{2} + 1\right )}^{\frac{3}{2}} b d g e}{9 \, c^{3}} + \frac{{\left (c^{2} x^{2} - 1\right )}^{2} b g \arcsin \left (c x\right ) e^{2}}{4 \, c^{4}} - \frac{{\left (-c^{2} x^{2} + 1\right )}^{\frac{3}{2}} b f e^{2}}{9 \, c^{3}} + \frac{5 \, \sqrt{-c^{2} x^{2} + 1} b g x e^{2}}{32 \, c^{3}} + \frac{2 \, \sqrt{-c^{2} x^{2} + 1} b d g e}{3 \, c^{3}} + \frac{{\left (c^{2} x^{2} - 1\right )}^{2} a g e^{2}}{4 \, c^{4}} + \frac{{\left (c^{2} x^{2} - 1\right )} b g \arcsin \left (c x\right ) e^{2}}{2 \, c^{4}} + \frac{\sqrt{-c^{2} x^{2} + 1} b f e^{2}}{3 \, c^{3}} + \frac{{\left (c^{2} x^{2} - 1\right )} a g e^{2}}{2 \, c^{4}} + \frac{5 \, b g \arcsin \left (c x\right ) e^{2}}{32 \, c^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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