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

Optimal. Leaf size=361 \[ \frac{1}{3} x^3 \left (a+b \sin ^{-1}(c x)\right ) \left (d^2 h+2 d e g+e^2 f\right )+d^2 f x \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{2} d x^2 (d g+2 e f) \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{4} e x^4 (2 d h+e g) \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{5} e^2 h x^5 \left (a+b \sin ^{-1}(c x)\right )+\frac{b x^2 \sqrt{1-c^2 x^2} \left (25 c^2 \left (d^2 h+2 d e g+e^2 f\right )+12 e^2 h\right )}{225 c^3}+\frac{b \sqrt{1-c^2 x^2} \left (32 \left (50 c^2 \left (d^2 h+2 d e g+e^2 f\right )+225 c^4 d^2 f+24 e^2 h\right )+225 c^2 x \left (8 c^2 d (d g+2 e f)+3 e (2 d h+e g)\right )\right )}{7200 c^5}-\frac{b \sin ^{-1}(c x) \left (8 c^2 d (d g+2 e f)+3 e (2 d h+e g)\right )}{32 c^4}+\frac{b e x^3 \sqrt{1-c^2 x^2} (2 d h+e g)}{16 c}+\frac{b e^2 h x^4 \sqrt{1-c^2 x^2}}{25 c} \]

[Out]

(b*(12*e^2*h + 25*c^2*(e^2*f + 2*d*e*g + d^2*h))*x^2*Sqrt[1 - c^2*x^2])/(225*c^3) + (b*e*(e*g + 2*d*h)*x^3*Sqr
t[1 - c^2*x^2])/(16*c) + (b*e^2*h*x^4*Sqrt[1 - c^2*x^2])/(25*c) + (b*(32*(225*c^4*d^2*f + 24*e^2*h + 50*c^2*(e
^2*f + 2*d*e*g + d^2*h)) + 225*c^2*(8*c^2*d*(2*e*f + d*g) + 3*e*(e*g + 2*d*h))*x)*Sqrt[1 - c^2*x^2])/(7200*c^5
) - (b*(8*c^2*d*(2*e*f + d*g) + 3*e*(e*g + 2*d*h))*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^2*f + 2*d*e*g + d^2*h)*x^3*(a + b*ArcSin[c*x]))/3 + (e*(e*g + 2*d
*h)*x^4*(a + b*ArcSin[c*x]))/4 + (e^2*h*x^5*(a + b*ArcSin[c*x]))/5

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Rubi [A]  time = 1.21528, antiderivative size = 361, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.192, Rules used = {4749, 12, 1809, 780, 216} \[ \frac{1}{3} x^3 \left (a+b \sin ^{-1}(c x)\right ) \left (d^2 h+2 d e g+e^2 f\right )+d^2 f x \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{2} d x^2 (d g+2 e f) \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{4} e x^4 (2 d h+e g) \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{5} e^2 h x^5 \left (a+b \sin ^{-1}(c x)\right )+\frac{b x^2 \sqrt{1-c^2 x^2} \left (25 c^2 \left (d^2 h+2 d e g+e^2 f\right )+12 e^2 h\right )}{225 c^3}+\frac{b \sqrt{1-c^2 x^2} \left (32 \left (50 c^2 \left (d^2 h+2 d e g+e^2 f\right )+225 c^4 d^2 f+24 e^2 h\right )+225 c^2 x \left (8 c^2 d (d g+2 e f)+3 e (2 d h+e g)\right )\right )}{7200 c^5}-\frac{b \sin ^{-1}(c x) \left (8 c^2 d (d g+2 e f)+3 e (2 d h+e g)\right )}{32 c^4}+\frac{b e x^3 \sqrt{1-c^2 x^2} (2 d h+e g)}{16 c}+\frac{b e^2 h x^4 \sqrt{1-c^2 x^2}}{25 c} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(b*(12*e^2*h + 25*c^2*(e^2*f + 2*d*e*g + d^2*h))*x^2*Sqrt[1 - c^2*x^2])/(225*c^3) + (b*e*(e*g + 2*d*h)*x^3*Sqr
t[1 - c^2*x^2])/(16*c) + (b*e^2*h*x^4*Sqrt[1 - c^2*x^2])/(25*c) + (b*(32*(225*c^4*d^2*f + 24*e^2*h + 50*c^2*(e
^2*f + 2*d*e*g + d^2*h)) + 225*c^2*(8*c^2*d*(2*e*f + d*g) + 3*e*(e*g + 2*d*h))*x)*Sqrt[1 - c^2*x^2])/(7200*c^5
) - (b*(8*c^2*d*(2*e*f + d*g) + 3*e*(e*g + 2*d*h))*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^2*f + 2*d*e*g + d^2*h)*x^3*(a + b*ArcSin[c*x]))/3 + (e*(e*g + 2*d
*h)*x^4*(a + b*ArcSin[c*x]))/4 + (e^2*h*x^5*(a + b*ArcSin[c*x]))/5

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 \left (f+g x+h x^2\right ) \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} \left (e^2 f+2 d e g+d^2 h\right ) x^3 \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{4} e (e g+2 d h) x^4 \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{5} e^2 h x^5 \left (a+b \sin ^{-1}(c x)\right )-(b c) \int \frac{x \left (60 d^2 f+30 d (2 e f+d g) x+20 \left (e^2 f+2 d e g+d^2 h\right ) x^2+15 e (e g+2 d h) x^3+12 e^2 h x^4\right )}{60 \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} \left (e^2 f+2 d e g+d^2 h\right ) x^3 \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{4} e (e g+2 d h) x^4 \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{5} e^2 h x^5 \left (a+b \sin ^{-1}(c x)\right )-\frac{1}{60} (b c) \int \frac{x \left (60 d^2 f+30 d (2 e f+d g) x+20 \left (e^2 f+2 d e g+d^2 h\right ) x^2+15 e (e g+2 d h) x^3+12 e^2 h x^4\right )}{\sqrt{1-c^2 x^2}} \, dx\\ &=\frac{b e^2 h x^4 \sqrt{1-c^2 x^2}}{25 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} \left (e^2 f+2 d e g+d^2 h\right ) x^3 \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{4} e (e g+2 d h) x^4 \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{5} e^2 h x^5 \left (a+b \sin ^{-1}(c x)\right )+\frac{b \int \frac{x \left (-300 c^2 d^2 f-150 c^2 d (2 e f+d g) x-4 \left (12 e^2 h+25 c^2 \left (e^2 f+2 d e g+d^2 h\right )\right ) x^2-75 c^2 e (e g+2 d h) x^3\right )}{\sqrt{1-c^2 x^2}} \, dx}{300 c}\\ &=\frac{b e (e g+2 d h) x^3 \sqrt{1-c^2 x^2}}{16 c}+\frac{b e^2 h x^4 \sqrt{1-c^2 x^2}}{25 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} \left (e^2 f+2 d e g+d^2 h\right ) x^3 \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{4} e (e g+2 d h) x^4 \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{5} e^2 h x^5 \left (a+b \sin ^{-1}(c x)\right )-\frac{b \int \frac{x \left (1200 c^4 d^2 f+75 c^2 \left (8 c^2 d (2 e f+d g)+3 e (e g+2 d h)\right ) x+16 c^2 \left (12 e^2 h+25 c^2 \left (e^2 f+2 d e g+d^2 h\right )\right ) x^2\right )}{\sqrt{1-c^2 x^2}} \, dx}{1200 c^3}\\ &=\frac{b \left (12 e^2 h+25 c^2 \left (e^2 f+2 d e g+d^2 h\right )\right ) x^2 \sqrt{1-c^2 x^2}}{225 c^3}+\frac{b e (e g+2 d h) x^3 \sqrt{1-c^2 x^2}}{16 c}+\frac{b e^2 h x^4 \sqrt{1-c^2 x^2}}{25 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} \left (e^2 f+2 d e g+d^2 h\right ) x^3 \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{4} e (e g+2 d h) x^4 \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{5} e^2 h x^5 \left (a+b \sin ^{-1}(c x)\right )+\frac{b \int \frac{x \left (-16 c^2 \left (225 c^4 d^2 f+24 e^2 h+50 c^2 \left (e^2 f+2 d e g+d^2 h\right )\right )-225 c^4 \left (8 c^2 d (2 e f+d g)+3 e (e g+2 d h)\right ) x\right )}{\sqrt{1-c^2 x^2}} \, dx}{3600 c^5}\\ &=\frac{b \left (12 e^2 h+25 c^2 \left (e^2 f+2 d e g+d^2 h\right )\right ) x^2 \sqrt{1-c^2 x^2}}{225 c^3}+\frac{b e (e g+2 d h) x^3 \sqrt{1-c^2 x^2}}{16 c}+\frac{b e^2 h x^4 \sqrt{1-c^2 x^2}}{25 c}+\frac{b \left (32 \left (225 c^4 d^2 f+24 e^2 h+50 c^2 \left (e^2 f+2 d e g+d^2 h\right )\right )+225 c^2 \left (8 c^2 d (2 e f+d g)+3 e (e g+2 d h)\right ) x\right ) \sqrt{1-c^2 x^2}}{7200 c^5}+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} \left (e^2 f+2 d e g+d^2 h\right ) x^3 \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{4} e (e g+2 d h) x^4 \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{5} e^2 h x^5 \left (a+b \sin ^{-1}(c x)\right )-\frac{\left (b \left (8 c^2 d (2 e f+d g)+3 e (e g+2 d h)\right )\right ) \int \frac{1}{\sqrt{1-c^2 x^2}} \, dx}{32 c^3}\\ &=\frac{b \left (12 e^2 h+25 c^2 \left (e^2 f+2 d e g+d^2 h\right )\right ) x^2 \sqrt{1-c^2 x^2}}{225 c^3}+\frac{b e (e g+2 d h) x^3 \sqrt{1-c^2 x^2}}{16 c}+\frac{b e^2 h x^4 \sqrt{1-c^2 x^2}}{25 c}+\frac{b \left (32 \left (225 c^4 d^2 f+24 e^2 h+50 c^2 \left (e^2 f+2 d e g+d^2 h\right )\right )+225 c^2 \left (8 c^2 d (2 e f+d g)+3 e (e g+2 d h)\right ) x\right ) \sqrt{1-c^2 x^2}}{7200 c^5}-\frac{b \left (8 c^2 d (2 e f+d g)+3 e (e g+2 d h)\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} \left (e^2 f+2 d e g+d^2 h\right ) x^3 \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{4} e (e g+2 d h) x^4 \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{5} e^2 h x^5 \left (a+b \sin ^{-1}(c x)\right )\\ \end{align*}

Mathematica [A]  time = 0.500644, size = 307, normalized size = 0.85 \[ \frac{120 a c^5 x \left (10 d^2 (6 f+x (3 g+2 h x))+10 d e x (6 f+x (4 g+3 h x))+e^2 x^2 (20 f+3 x (5 g+4 h x))\right )+b \sqrt{1-c^2 x^2} \left (2 c^4 \left (100 d^2 (36 f+x (9 g+4 h x))+50 d e x (36 f+x (16 g+9 h x))+e^2 x^2 (400 f+9 x (25 g+16 h x))\right )+c^2 \left (1600 d^2 h+50 d e (64 g+27 h x)+e^2 \left (1600 f+675 g x+384 h x^2\right )\right )+768 e^2 h\right )+15 b c \sin ^{-1}(c x) \left (8 c^4 x \left (10 d^2 (6 f+x (3 g+2 h x))+10 d e x (6 f+x (4 g+3 h x))+e^2 x^2 (20 f+3 x (5 g+4 h x))\right )-120 c^2 d (d g+2 e f)-45 e (2 d h+e g)\right )}{7200 c^5} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(120*a*c^5*x*(10*d^2*(6*f + x*(3*g + 2*h*x)) + 10*d*e*x*(6*f + x*(4*g + 3*h*x)) + e^2*x^2*(20*f + 3*x*(5*g + 4
*h*x))) + b*Sqrt[1 - c^2*x^2]*(768*e^2*h + c^2*(1600*d^2*h + 50*d*e*(64*g + 27*h*x) + e^2*(1600*f + 675*g*x +
384*h*x^2)) + 2*c^4*(100*d^2*(36*f + x*(9*g + 4*h*x)) + 50*d*e*x*(36*f + x*(16*g + 9*h*x)) + e^2*x^2*(400*f +
9*x*(25*g + 16*h*x)))) + 15*b*c*(-120*c^2*d*(2*e*f + d*g) - 45*e*(e*g + 2*d*h) + 8*c^4*x*(10*d^2*(6*f + x*(3*g
+ 2*h*x)) + 10*d*e*x*(6*f + x*(4*g + 3*h*x)) + e^2*x^2*(20*f + 3*x*(5*g + 4*h*x))))*ArcSin[c*x])/(7200*c^5)

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Maple [A]  time = 0.005, size = 502, normalized size = 1.4 \begin{align*}{\frac{1}{c} \left ({\frac{a}{{c}^{4}} \left ({\frac{{e}^{2}h{c}^{5}{x}^{5}}{5}}+{\frac{ \left ( 2\,cdeh+{e}^{2}cg \right ){c}^{4}{x}^{4}}{4}}+{\frac{ \left ({c}^{2}{d}^{2}h+2\,{c}^{2}deg+{e}^{2}f{c}^{2} \right ){c}^{3}{x}^{3}}{3}}+{\frac{ \left ({c}^{3}{d}^{2}g+2\,{c}^{3}def \right ){c}^{2}{x}^{2}}{2}}+{c}^{5}{d}^{2}fx \right ) }+{\frac{b}{{c}^{4}} \left ({\frac{\arcsin \left ( cx \right ){e}^{2}h{c}^{5}{x}^{5}}{5}}+{\frac{\arcsin \left ( cx \right ){c}^{5}{x}^{4}deh}{2}}+{\frac{\arcsin \left ( cx \right ){c}^{5}{x}^{4}{e}^{2}g}{4}}+{\frac{\arcsin \left ( cx \right ){c}^{5}{x}^{3}{d}^{2}h}{3}}+{\frac{2\,\arcsin \left ( cx \right ){c}^{5}{x}^{3}deg}{3}}+{\frac{\arcsin \left ( cx \right ){c}^{5}{x}^{3}{e}^{2}f}{3}}+{\frac{\arcsin \left ( cx \right ){c}^{5}{x}^{2}{d}^{2}g}{2}}+\arcsin \left ( cx \right ){c}^{5}{x}^{2}def+\arcsin \left ( cx \right ){c}^{5}{d}^{2}fx-{\frac{{e}^{2}h}{5} \left ( -{\frac{{c}^{4}{x}^{4}}{5}\sqrt{-{c}^{2}{x}^{2}+1}}-{\frac{4\,{c}^{2}{x}^{2}}{15}\sqrt{-{c}^{2}{x}^{2}+1}}-{\frac{8}{15}\sqrt{-{c}^{2}{x}^{2}+1}} \right ) }-{\frac{30\,cdeh+15\,{e}^{2}cg}{60} \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{20\,{c}^{2}{d}^{2}h+40\,{c}^{2}deg+20\,{e}^{2}f{c}^{2}}{60} \left ( -{\frac{{c}^{2}{x}^{2}}{3}\sqrt{-{c}^{2}{x}^{2}+1}}-{\frac{2}{3}\sqrt{-{c}^{2}{x}^{2}+1}} \right ) }-{\frac{30\,{c}^{3}{d}^{2}g+60\,{c}^{3}def}{60} \left ( -{\frac{cx}{2}\sqrt{-{c}^{2}{x}^{2}+1}}+{\frac{\arcsin \left ( cx \right ) }{2}} \right ) }+{c}^{4}{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*(h*x^2+g*x+f)*(a+b*arcsin(c*x)),x)

[Out]

1/c*(a/c^4*(1/5*e^2*h*c^5*x^5+1/4*(2*c*d*e*h+c*e^2*g)*c^4*x^4+1/3*(c^2*d^2*h+2*c^2*d*e*g+c^2*e^2*f)*c^3*x^3+1/
2*(c^3*d^2*g+2*c^3*d*e*f)*c^2*x^2+c^5*d^2*f*x)+b/c^4*(1/5*arcsin(c*x)*e^2*h*c^5*x^5+1/2*arcsin(c*x)*c^5*x^4*d*
e*h+1/4*arcsin(c*x)*c^5*x^4*e^2*g+1/3*arcsin(c*x)*c^5*x^3*d^2*h+2/3*arcsin(c*x)*c^5*x^3*d*e*g+1/3*arcsin(c*x)*
c^5*x^3*e^2*f+1/2*arcsin(c*x)*c^5*x^2*d^2*g+arcsin(c*x)*c^5*x^2*d*e*f+arcsin(c*x)*c^5*d^2*f*x-1/5*e^2*h*(-1/5*
c^4*x^4*(-c^2*x^2+1)^(1/2)-4/15*c^2*x^2*(-c^2*x^2+1)^(1/2)-8/15*(-c^2*x^2+1)^(1/2))-1/60*(30*c*d*e*h+15*c*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))-1/60*(20*c^2*d^2*h+40*c^2*d*e*
g+20*c^2*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))-1/60*(30*c^3*d^2*g+60*c^3*d*e*f)*(-1/
2*c*x*(-c^2*x^2+1)^(1/2)+1/2*arcsin(c*x))+c^4*d^2*f*(-c^2*x^2+1)^(1/2)))

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Maxima [A]  time = 1.65624, size = 849, normalized size = 2.35 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/5*a*e^2*h*x^5 + 1/4*a*e^2*g*x^4 + 1/2*a*d*e*h*x^4 + 1/3*a*e^2*f*x^3 + 2/3*a*d*e*g*x^3 + 1/3*a*d^2*h*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*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^2*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^2*h + 1/16*(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*d*e*h + 1/75*(15*x^5*arcsin(c*x) + (3*sqrt(-c^2*x^2 + 1)*x^4/c^2 + 4*sqrt(-c
^2*x^2 + 1)*x^2/c^4 + 8*sqrt(-c^2*x^2 + 1)/c^6)*c)*b*e^2*h + 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 = 3.91403, size = 1042, normalized size = 2.89 \begin{align*} \frac{1440 \, a c^{5} e^{2} h x^{5} + 7200 \, a c^{5} d^{2} f x + 1800 \,{\left (a c^{5} e^{2} g + 2 \, a c^{5} d e h\right )} x^{4} + 2400 \,{\left (a c^{5} e^{2} f + 2 \, a c^{5} d e g + a c^{5} d^{2} h\right )} x^{3} + 3600 \,{\left (2 \, a c^{5} d e f + a c^{5} d^{2} g\right )} x^{2} + 15 \,{\left (96 \, b c^{5} e^{2} h x^{5} + 480 \, b c^{5} d^{2} f x - 240 \, b c^{3} d e f - 90 \, b c d e h + 120 \,{\left (b c^{5} e^{2} g + 2 \, b c^{5} d e h\right )} x^{4} + 160 \,{\left (b c^{5} e^{2} f + 2 \, b c^{5} d e g + b c^{5} d^{2} h\right )} x^{3} + 240 \,{\left (2 \, b c^{5} d e f + b c^{5} d^{2} g\right )} x^{2} - 15 \,{\left (8 \, b c^{3} d^{2} + 3 \, b c e^{2}\right )} g\right )} \arcsin \left (c x\right ) +{\left (288 \, b c^{4} e^{2} h x^{4} + 3200 \, b c^{2} d e g + 450 \,{\left (b c^{4} e^{2} g + 2 \, b c^{4} d e h\right )} x^{3} + 32 \,{\left (25 \, b c^{4} e^{2} f + 50 \, b c^{4} d e g +{\left (25 \, b c^{4} d^{2} + 12 \, b c^{2} e^{2}\right )} h\right )} x^{2} + 800 \,{\left (9 \, b c^{4} d^{2} + 2 \, b c^{2} e^{2}\right )} f + 64 \,{\left (25 \, b c^{2} d^{2} + 12 \, b e^{2}\right )} h + 225 \,{\left (16 \, b c^{4} d e f + 6 \, b c^{2} d e h +{\left (8 \, b c^{4} d^{2} + 3 \, b c^{2} e^{2}\right )} g\right )} x\right )} \sqrt{-c^{2} x^{2} + 1}}{7200 \, c^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/7200*(1440*a*c^5*e^2*h*x^5 + 7200*a*c^5*d^2*f*x + 1800*(a*c^5*e^2*g + 2*a*c^5*d*e*h)*x^4 + 2400*(a*c^5*e^2*f
+ 2*a*c^5*d*e*g + a*c^5*d^2*h)*x^3 + 3600*(2*a*c^5*d*e*f + a*c^5*d^2*g)*x^2 + 15*(96*b*c^5*e^2*h*x^5 + 480*b*
c^5*d^2*f*x - 240*b*c^3*d*e*f - 90*b*c*d*e*h + 120*(b*c^5*e^2*g + 2*b*c^5*d*e*h)*x^4 + 160*(b*c^5*e^2*f + 2*b*
c^5*d*e*g + b*c^5*d^2*h)*x^3 + 240*(2*b*c^5*d*e*f + b*c^5*d^2*g)*x^2 - 15*(8*b*c^3*d^2 + 3*b*c*e^2)*g)*arcsin(
c*x) + (288*b*c^4*e^2*h*x^4 + 3200*b*c^2*d*e*g + 450*(b*c^4*e^2*g + 2*b*c^4*d*e*h)*x^3 + 32*(25*b*c^4*e^2*f +
50*b*c^4*d*e*g + (25*b*c^4*d^2 + 12*b*c^2*e^2)*h)*x^2 + 800*(9*b*c^4*d^2 + 2*b*c^2*e^2)*f + 64*(25*b*c^2*d^2 +
12*b*e^2)*h + 225*(16*b*c^4*d*e*f + 6*b*c^2*d*e*h + (8*b*c^4*d^2 + 3*b*c^2*e^2)*g)*x)*sqrt(-c^2*x^2 + 1))/c^5

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Sympy [A]  time = 5.68296, size = 821, normalized size = 2.27 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)**2*(h*x**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**2*h*x**3/3 + a*d*e*f*x**2 + 2*a*d*e*g*x**3/3 + a*d*e*h*x**4/2 +
a*e**2*f*x**3/3 + a*e**2*g*x**4/4 + a*e**2*h*x**5/5 + b*d**2*f*x*asin(c*x) + b*d**2*g*x**2*asin(c*x)/2 + b*d*
*2*h*x**3*asin(c*x)/3 + b*d*e*f*x**2*asin(c*x) + 2*b*d*e*g*x**3*asin(c*x)/3 + b*d*e*h*x**4*asin(c*x)/2 + b*e**
2*f*x**3*asin(c*x)/3 + b*e**2*g*x**4*asin(c*x)/4 + b*e**2*h*x**5*asin(c*x)/5 + 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**2*h*x**2*sqrt(-c**2*x**2 + 1)/(9*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*d*e*h*x**3*sqrt(-c**2*x**2 + 1)/(8*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*e**2*h*x**4*sqrt(-c**2*x**2 +
1)/(25*c) - b*d**2*g*asin(c*x)/(4*c**2) - b*d*e*f*asin(c*x)/(2*c**2) + 2*b*d**2*h*sqrt(-c**2*x**2 + 1)/(9*c**3
) + 4*b*d*e*g*sqrt(-c**2*x**2 + 1)/(9*c**3) + 3*b*d*e*h*x*sqrt(-c**2*x**2 + 1)/(16*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) + 4*b*e**2*h*x**2*sqrt(-c**2*x**2 + 1)/(75
*c**3) - 3*b*d*e*h*asin(c*x)/(16*c**4) - 3*b*e**2*g*asin(c*x)/(32*c**4) + 8*b*e**2*h*sqrt(-c**2*x**2 + 1)/(75*
c**5), Ne(c, 0)), (a*(d**2*f*x + d**2*g*x**2/2 + d**2*h*x**3/3 + d*e*f*x**2 + 2*d*e*g*x**3/3 + d*e*h*x**4/2 +
e**2*f*x**3/3 + e**2*g*x**4/4 + e**2*h*x**5/5), True))

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Giac [B]  time = 1.30556, size = 1218, normalized size = 3.37 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/5*a*h*x^5*e^2 + 1/3*a*d^2*h*x^3 + 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 + 1/
3*(c^2*x^2 - 1)*b*d^2*h*x*arcsin(c*x)/c^2 + 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*arcsin(c*x)/c^2 + 1/3*b*d^2*h*
x*arcsin(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 - 1/8*(-c^2*x^2 + 1)^(3/2)*b*d*h*x*e/c^3 + 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 + 1/5*(c^2*x^2 - 1)^2*b*h*x*
arcsin(c*x)*e^2/c^4 + (c^2*x^2 - 1)*a*d*f*e/c^2 + 1/2*b*d*f*arcsin(c*x)*e/c^2 + 1/2*(c^2*x^2 - 1)^2*b*d*h*arcs
in(c*x)*e/c^4 - 1/9*(-c^2*x^2 + 1)^(3/2)*b*d^2*h/c^3 - 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 + 5/16*sqrt(-c^2*x^2 + 1)*b*d*h*x*e/c^3 + 1/4*(c^2*x^2 - 1)^2*b*g*arcsin(c*x)*e^2/c^4
+ 2/5*(c^2*x^2 - 1)*b*h*x*arcsin(c*x)*e^2/c^4 + 1/2*(c^2*x^2 - 1)^2*a*d*h*e/c^4 + (c^2*x^2 - 1)*b*d*h*arcsin(c
*x)*e/c^4 + 1/3*sqrt(-c^2*x^2 + 1)*b*d^2*h/c^3 - 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 + 1/5*b*h*x*arcsin(c*x)*e^2/c^4 + (c^2*x^2 - 1)*a*d*h*e/c^4 + 5/16*b*d*h*arcsin(c*x)*e/c^
4 + 1/3*sqrt(-c^2*x^2 + 1)*b*f*e^2/c^3 + 1/25*(c^2*x^2 - 1)^2*sqrt(-c^2*x^2 + 1)*b*h*e^2/c^5 + 1/2*(c^2*x^2 -
1)*a*g*e^2/c^4 + 5/32*b*g*arcsin(c*x)*e^2/c^4 - 2/15*(-c^2*x^2 + 1)^(3/2)*b*h*e^2/c^5 + 1/5*sqrt(-c^2*x^2 + 1)
*b*h*e^2/c^5