3.108 \(\int (d+e x) (f+g x+h x^2+i x^3) (a+b \sin ^{-1}(c x)) \, dx\)

Optimal. Leaf size=308 \[ \frac {1}{2} x^2 (d g+e f) \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{3} x^3 (d h+e g) \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{4} x^4 (d i+e h) \left (a+b \sin ^{-1}(c x)\right )+d f x \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{5} e i x^5 \left (a+b \sin ^{-1}(c x)\right )+\frac {b x^3 \sqrt {1-c^2 x^2} (d i+e h)}{16 c}+\frac {b e i x^4 \sqrt {1-c^2 x^2}}{25 c}-\frac {b \sin ^{-1}(c x) \left (8 c^2 (d g+e f)+3 (d i+e h)\right )}{32 c^4}+\frac {b x^2 \sqrt {1-c^2 x^2} \left (25 c^2 (d h+e g)+12 e i\right )}{225 c^3}+\frac {b \sqrt {1-c^2 x^2} \left (225 c^2 x \left (8 c^2 (d g+e f)+3 (d i+e h)\right )+32 \left (225 c^4 d f+50 c^2 (d h+e g)+24 e i\right )\right )}{7200 c^5} \]

[Out]

-1/32*b*(8*c^2*(d*g+e*f)+3*d*i+3*e*h)*arcsin(c*x)/c^4+d*f*x*(a+b*arcsin(c*x))+1/2*(d*g+e*f)*x^2*(a+b*arcsin(c*
x))+1/3*(d*h+e*g)*x^3*(a+b*arcsin(c*x))+1/4*(d*i+e*h)*x^4*(a+b*arcsin(c*x))+1/5*e*i*x^5*(a+b*arcsin(c*x))+1/22
5*b*(25*c^2*(d*h+e*g)+12*e*i)*x^2*(-c^2*x^2+1)^(1/2)/c^3+1/16*b*(d*i+e*h)*x^3*(-c^2*x^2+1)^(1/2)/c+1/25*b*e*i*
x^4*(-c^2*x^2+1)^(1/2)/c+1/7200*b*(7200*c^4*d*f+1600*c^2*(d*h+e*g)+768*e*i+225*c^2*(8*c^2*(d*g+e*f)+3*d*i+3*e*
h)*x)*(-c^2*x^2+1)^(1/2)/c^5

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Rubi [A]  time = 0.95, antiderivative size = 308, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.172, Rules used = {4749, 12, 1809, 780, 216} \[ \frac {1}{2} x^2 (d g+e f) \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{3} x^3 (d h+e g) \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{4} x^4 (d i+e h) \left (a+b \sin ^{-1}(c x)\right )+d f x \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{5} e i x^5 \left (a+b \sin ^{-1}(c x)\right )+\frac {b \sqrt {1-c^2 x^2} \left (225 c^2 x \left (8 c^2 (d g+e f)+3 (d i+e h)\right )+32 \left (50 c^2 (d h+e g)+225 c^4 d f+24 e i\right )\right )}{7200 c^5}-\frac {b \sin ^{-1}(c x) \left (8 c^2 (d g+e f)+3 (d i+e h)\right )}{32 c^4}+\frac {b x^2 \sqrt {1-c^2 x^2} \left (25 c^2 (d h+e g)+12 e i\right )}{225 c^3}+\frac {b x^3 \sqrt {1-c^2 x^2} (d i+e h)}{16 c}+\frac {b e i x^4 \sqrt {1-c^2 x^2}}{25 c} \]

Antiderivative was successfully verified.

[In]

Int[(d + e*x)*(f + g*x + h*x^2 + i*x^3)*(a + b*ArcSin[c*x]),x]

[Out]

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

Rule 12

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

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]

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 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 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]

Rubi steps

\begin {align*} \int (d+e x) \left (f+g x+h x^2+108 x^3\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} (108 d+e h) x^4 \left (a+b \sin ^{-1}(c x)\right )+\frac {108}{5} e x^5 \left (a+b \sin ^{-1}(c x)\right )-(b c) \int \frac {x \left (60 d f+30 (e f+d g) x+20 (e g+d h) x^2+15 (108 d+e h) x^3+1296 e x^4\right )}{60 \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} (108 d+e h) x^4 \left (a+b \sin ^{-1}(c x)\right )+\frac {108}{5} e x^5 \left (a+b \sin ^{-1}(c x)\right )-\frac {1}{60} (b c) \int \frac {x \left (60 d f+30 (e f+d g) x+20 (e g+d h) x^2+15 (108 d+e h) x^3+1296 e x^4\right )}{\sqrt {1-c^2 x^2}} \, dx\\ &=\frac {108 b e x^4 \sqrt {1-c^2 x^2}}{25 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} (108 d+e h) x^4 \left (a+b \sin ^{-1}(c x)\right )+\frac {108}{5} e x^5 \left (a+b \sin ^{-1}(c x)\right )+\frac {b \int \frac {x \left (-300 c^2 d f-150 c^2 (e f+d g) x-4 \left (e \left (1296+25 c^2 g\right )+25 c^2 d h\right ) x^2-75 c^2 (108 d+e h) x^3\right )}{\sqrt {1-c^2 x^2}} \, dx}{300 c}\\ &=\frac {b (108 d+e h) x^3 \sqrt {1-c^2 x^2}}{16 c}+\frac {108 b e x^4 \sqrt {1-c^2 x^2}}{25 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} (108 d+e h) x^4 \left (a+b \sin ^{-1}(c x)\right )+\frac {108}{5} e x^5 \left (a+b \sin ^{-1}(c x)\right )-\frac {b \int \frac {x \left (1200 c^4 d f+75 c^2 \left (324 d+8 c^2 e f+8 c^2 d g+3 e h\right ) x+16 c^2 \left (e \left (1296+25 c^2 g\right )+25 c^2 d h\right ) x^2\right )}{\sqrt {1-c^2 x^2}} \, dx}{1200 c^3}\\ &=\frac {b \left (e \left (1296+25 c^2 g\right )+25 c^2 d h\right ) x^2 \sqrt {1-c^2 x^2}}{225 c^3}+\frac {b (108 d+e h) x^3 \sqrt {1-c^2 x^2}}{16 c}+\frac {108 b e x^4 \sqrt {1-c^2 x^2}}{25 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} (108 d+e h) x^4 \left (a+b \sin ^{-1}(c x)\right )+\frac {108}{5} e x^5 \left (a+b \sin ^{-1}(c x)\right )+\frac {b \int \frac {x \left (-16 c^2 \left (2592 e+225 c^4 d f+50 c^2 (e g+d h)\right )-225 c^4 \left (324 d+8 c^2 e f+8 c^2 d g+3 e h\right ) x\right )}{\sqrt {1-c^2 x^2}} \, dx}{3600 c^5}\\ &=\frac {b \left (e \left (1296+25 c^2 g\right )+25 c^2 d h\right ) x^2 \sqrt {1-c^2 x^2}}{225 c^3}+\frac {b (108 d+e h) x^3 \sqrt {1-c^2 x^2}}{16 c}+\frac {108 b e x^4 \sqrt {1-c^2 x^2}}{25 c}+\frac {b \left (32 \left (2592 e+225 c^4 d f+50 c^2 (e g+d h)\right )+225 c^2 \left (324 d+8 c^2 e f+8 c^2 d g+3 e h\right ) x\right ) \sqrt {1-c^2 x^2}}{7200 c^5}+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} (108 d+e h) x^4 \left (a+b \sin ^{-1}(c x)\right )+\frac {108}{5} e x^5 \left (a+b \sin ^{-1}(c x)\right )-\frac {\left (b \left (324 d+8 c^2 e f+8 c^2 d g+3 e h\right )\right ) \int \frac {1}{\sqrt {1-c^2 x^2}} \, dx}{32 c^3}\\ &=\frac {b \left (e \left (1296+25 c^2 g\right )+25 c^2 d h\right ) x^2 \sqrt {1-c^2 x^2}}{225 c^3}+\frac {b (108 d+e h) x^3 \sqrt {1-c^2 x^2}}{16 c}+\frac {108 b e x^4 \sqrt {1-c^2 x^2}}{25 c}+\frac {b \left (32 \left (2592 e+225 c^4 d f+50 c^2 (e g+d h)\right )+225 c^2 \left (324 d+8 c^2 e f+8 c^2 d g+3 e h\right ) x\right ) \sqrt {1-c^2 x^2}}{7200 c^5}-\frac {b \left (324 d+8 c^2 e f+8 c^2 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} (108 d+e h) x^4 \left (a+b \sin ^{-1}(c x)\right )+\frac {108}{5} e x^5 \left (a+b \sin ^{-1}(c x)\right )\\ \end {align*}

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Mathematica [A]  time = 0.49, size = 253, normalized size = 0.82 \[ \frac {120 a c^5 x (5 d (12 f+x (6 g+x (4 h+3 i x)))+e x (30 f+x (20 g+3 x (5 h+4 i x))))+b \sqrt {1-c^2 x^2} \left (2 c^4 (25 d (144 f+x (36 g+x (16 h+9 i x)))+e x (900 f+x (400 g+9 x (25 h+16 i x))))+c^2 \left (25 d (64 h+27 i x)+e \left (1600 g+675 h x+384 i x^2\right )\right )+768 e i\right )+15 b c \sin ^{-1}(c x) \left (8 c^4 x (5 d (12 f+x (6 g+x (4 h+3 i x)))+e x (30 f+x (20 g+3 x (5 h+4 i x))))-120 c^2 (d g+e f)-45 (d i+e h)\right )}{7200 c^5} \]

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x)*(f + g*x + h*x^2 + i*x^3)*(a + b*ArcSin[c*x]),x]

[Out]

(120*a*c^5*x*(5*d*(12*f + x*(6*g + x*(4*h + 3*i*x))) + e*x*(30*f + x*(20*g + 3*x*(5*h + 4*i*x)))) + b*Sqrt[1 -
c^2*x^2]*(768*e*i + c^2*(25*d*(64*h + 27*i*x) + e*(1600*g + 675*h*x + 384*i*x^2)) + 2*c^4*(25*d*(144*f + x*(3
6*g + x*(16*h + 9*i*x))) + e*x*(900*f + x*(400*g + 9*x*(25*h + 16*i*x))))) + 15*b*c*(-120*c^2*(e*f + d*g) - 45
*(e*h + d*i) + 8*c^4*x*(5*d*(12*f + x*(6*g + x*(4*h + 3*i*x))) + e*x*(30*f + x*(20*g + 3*x*(5*h + 4*i*x)))))*A
rcSin[c*x])/(7200*c^5)

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fricas [A]  time = 0.98, size = 341, normalized size = 1.11 \[ \frac {1440 \, a c^{5} e i x^{5} + 7200 \, a c^{5} d f x + 1800 \, {\left (a c^{5} e h + a c^{5} d i\right )} x^{4} + 2400 \, {\left (a c^{5} e g + a c^{5} d h\right )} x^{3} + 3600 \, {\left (a c^{5} e f + a c^{5} d g\right )} x^{2} + 15 \, {\left (96 \, b c^{5} e i x^{5} + 480 \, b c^{5} d f x - 120 \, b c^{3} e f - 120 \, b c^{3} d g + 120 \, {\left (b c^{5} e h + b c^{5} d i\right )} x^{4} - 45 \, b c e h - 45 \, b c d i + 160 \, {\left (b c^{5} e g + b c^{5} d h\right )} x^{3} + 240 \, {\left (b c^{5} e f + b c^{5} d g\right )} x^{2}\right )} \arcsin \left (c x\right ) + {\left (288 \, b c^{4} e i x^{4} + 7200 \, b c^{4} d f + 1600 \, b c^{2} e g + 1600 \, b c^{2} d h + 450 \, {\left (b c^{4} e h + b c^{4} d i\right )} x^{3} + 768 \, b e i + 32 \, {\left (25 \, b c^{4} e g + 25 \, b c^{4} d h + 12 \, b c^{2} e i\right )} x^{2} + 225 \, {\left (8 \, b c^{4} e f + 8 \, b c^{4} d g + 3 \, b c^{2} e h + 3 \, b c^{2} d i\right )} x\right )} \sqrt {-c^{2} x^{2} + 1}}{7200 \, c^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)*(i*x^3+h*x^2+g*x+f)*(a+b*arcsin(c*x)),x, algorithm="fricas")

[Out]

1/7200*(1440*a*c^5*e*i*x^5 + 7200*a*c^5*d*f*x + 1800*(a*c^5*e*h + a*c^5*d*i)*x^4 + 2400*(a*c^5*e*g + a*c^5*d*h
)*x^3 + 3600*(a*c^5*e*f + a*c^5*d*g)*x^2 + 15*(96*b*c^5*e*i*x^5 + 480*b*c^5*d*f*x - 120*b*c^3*e*f - 120*b*c^3*
d*g + 120*(b*c^5*e*h + b*c^5*d*i)*x^4 - 45*b*c*e*h - 45*b*c*d*i + 160*(b*c^5*e*g + b*c^5*d*h)*x^3 + 240*(b*c^5
*e*f + b*c^5*d*g)*x^2)*arcsin(c*x) + (288*b*c^4*e*i*x^4 + 7200*b*c^4*d*f + 1600*b*c^2*e*g + 1600*b*c^2*d*h + 4
50*(b*c^4*e*h + b*c^4*d*i)*x^3 + 768*b*e*i + 32*(25*b*c^4*e*g + 25*b*c^4*d*h + 12*b*c^2*e*i)*x^2 + 225*(8*b*c^
4*e*f + 8*b*c^4*d*g + 3*b*c^2*e*h + 3*b*c^2*d*i)*x)*sqrt(-c^2*x^2 + 1))/c^5

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giac [B]  time = 0.48, size = 714, normalized size = 2.32 \[ \frac {1}{5} \, a i x^{5} e + \frac {1}{4} \, a d i x^{4} + \frac {1}{4} \, a h x^{4} e + \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 {{\left (c^{2} x^{2} - 1\right )}^{2} b i x \arcsin \left (c x\right ) e}{5 \, c^{4}} + \frac {\sqrt {-c^{2} x^{2} + 1} b d f}{c} - \frac {{\left (-c^{2} x^{2} + 1\right )}^{\frac {3}{2}} b d i x}{16 \, c^{3}} - \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 )}^{2} b d i \arcsin \left (c x\right )}{4 \, c^{4}} + \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 {2 \, {\left (c^{2} x^{2} - 1\right )} b i x \arcsin \left (c x\right ) e}{5 \, c^{4}} - \frac {{\left (-c^{2} x^{2} + 1\right )}^{\frac {3}{2}} b d h}{9 \, c^{3}} + \frac {5 \, \sqrt {-c^{2} x^{2} + 1} b d i x}{32 \, 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 )} b d i \arcsin \left (c x\right )}{2 \, c^{4}} + \frac {{\left (c^{2} x^{2} - 1\right )} b h \arcsin \left (c x\right ) e}{2 \, c^{4}} + \frac {b i x \arcsin \left (c x\right ) e}{5 \, 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 )}^{2} \sqrt {-c^{2} x^{2} + 1} b i e}{25 \, c^{5}} + \frac {5 \, b d i \arcsin \left (c x\right )}{32 \, c^{4}} + \frac {5 \, b h \arcsin \left (c x\right ) e}{32 \, c^{4}} - \frac {2 \, {\left (-c^{2} x^{2} + 1\right )}^{\frac {3}{2}} b i e}{15 \, c^{5}} + \frac {\sqrt {-c^{2} x^{2} + 1} b i e}{5 \, c^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)*(i*x^3+h*x^2+g*x+f)*(a+b*arcsin(c*x)),x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.01, size = 428, normalized size = 1.39 \[ \frac {\frac {a \left (\frac {e i \,c^{5} x^{5}}{5}+\frac {\left (d c i +e c h \right ) c^{4} x^{4}}{4}+\frac {\left (d \,c^{2} h +e \,c^{2} g \right ) c^{3} x^{3}}{3}+\frac {\left (d \,c^{3} g +e f \,c^{3}\right ) c^{2} x^{2}}{2}+c^{5} f d x \right )}{c^{4}}+\frac {b \left (\frac {\arcsin \left (c x \right ) e i \,c^{5} x^{5}}{5}+\frac {\arcsin \left (c x \right ) c^{5} x^{4} d i}{4}+\frac {\arcsin \left (c x \right ) c^{5} x^{4} e h}{4}+\frac {\arcsin \left (c x \right ) c^{5} x^{3} d h}{3}+\frac {\arcsin \left (c x \right ) c^{5} x^{3} e g}{3}+\frac {\arcsin \left (c x \right ) c^{5} x^{2} d g}{2}+\frac {\arcsin \left (c x \right ) c^{5} x^{2} e f}{2}+\arcsin \left (c x \right ) c^{5} f d x -\frac {e i \left (-\frac {c^{4} x^{4} \sqrt {-c^{2} x^{2}+1}}{5}-\frac {4 c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{15}-\frac {8 \sqrt {-c^{2} x^{2}+1}}{15}\right )}{5}-\frac {\left (15 d c i +15 e c h \right ) \left (-\frac {c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}}{4}-\frac {3 c x \sqrt {-c^{2} x^{2}+1}}{8}+\frac {3 \arcsin \left (c x \right )}{8}\right )}{60}-\frac {\left (20 d \,c^{2} h +20 e \,c^{2} g \right ) \left (-\frac {c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{3}-\frac {2 \sqrt {-c^{2} x^{2}+1}}{3}\right )}{60}-\frac {\left (30 d \,c^{3} g +30 e f \,c^{3}\right ) \left (-\frac {c x \sqrt {-c^{2} x^{2}+1}}{2}+\frac {\arcsin \left (c x \right )}{2}\right )}{60}+c^{4} d f \sqrt {-c^{2} x^{2}+1}\right )}{c^{4}}}{c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)*(i*x^3+h*x^2+g*x+f)*(a+b*arcsin(c*x)),x)

[Out]

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

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maxima [A]  time = 0.51, size = 490, normalized size = 1.59 \[ \frac {1}{5} \, a e i x^{5} + \frac {1}{4} \, a e h x^{4} + \frac {1}{4} \, a d i x^{4} + \frac {1}{3} \, a e g x^{3} + \frac {1}{3} \, a d h x^{3} + \frac {1}{2} \, a e f x^{2} + \frac {1}{2} \, a d g x^{2} + \frac {1}{4} \, {\left (2 \, x^{2} \arcsin \left (c x\right ) + c {\left (\frac {\sqrt {-c^{2} x^{2} + 1} x}{c^{2}} - \frac {\arcsin \left (c x\right )}{c^{3}}\right )}\right )} b e f + \frac {1}{4} \, {\left (2 \, x^{2} \arcsin \left (c x\right ) + c {\left (\frac {\sqrt {-c^{2} x^{2} + 1} x}{c^{2}} - \frac {\arcsin \left (c x\right )}{c^{3}}\right )}\right )} b d g + \frac {1}{9} \, {\left (3 \, x^{3} \arcsin \left (c x\right ) + c {\left (\frac {\sqrt {-c^{2} x^{2} + 1} x^{2}}{c^{2}} + \frac {2 \, \sqrt {-c^{2} x^{2} + 1}}{c^{4}}\right )}\right )} b e g + \frac {1}{9} \, {\left (3 \, x^{3} \arcsin \left (c x\right ) + c {\left (\frac {\sqrt {-c^{2} x^{2} + 1} x^{2}}{c^{2}} + \frac {2 \, \sqrt {-c^{2} x^{2} + 1}}{c^{4}}\right )}\right )} b d h + \frac {1}{32} \, {\left (8 \, x^{4} \arcsin \left (c x\right ) + {\left (\frac {2 \, \sqrt {-c^{2} x^{2} + 1} x^{3}}{c^{2}} + \frac {3 \, \sqrt {-c^{2} x^{2} + 1} x}{c^{4}} - \frac {3 \, \arcsin \left (c x\right )}{c^{5}}\right )} c\right )} b e h + \frac {1}{32} \, {\left (8 \, x^{4} \arcsin \left (c x\right ) + {\left (\frac {2 \, \sqrt {-c^{2} x^{2} + 1} x^{3}}{c^{2}} + \frac {3 \, \sqrt {-c^{2} x^{2} + 1} x}{c^{4}} - \frac {3 \, \arcsin \left (c x\right )}{c^{5}}\right )} c\right )} b d i + \frac {1}{75} \, {\left (15 \, x^{5} \arcsin \left (c x\right ) + {\left (\frac {3 \, \sqrt {-c^{2} x^{2} + 1} x^{4}}{c^{2}} + \frac {4 \, \sqrt {-c^{2} x^{2} + 1} x^{2}}{c^{4}} + \frac {8 \, \sqrt {-c^{2} x^{2} + 1}}{c^{6}}\right )} c\right )} b e i + a d f x + \frac {{\left (c x \arcsin \left (c x\right ) + \sqrt {-c^{2} x^{2} + 1}\right )} b d f}{c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)*(i*x^3+h*x^2+g*x+f)*(a+b*arcsin(c*x)),x, algorithm="maxima")

[Out]

1/5*a*e*i*x^5 + 1/4*a*e*h*x^4 + 1/4*a*d*i*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*(sqrt(-c^2*x^2 + 1)*x/c^2 - arcsin(c*x)/c^3))*b*e*f + 1/4*(2*x^2*arcsin(c*x) + c*
(sqrt(-c^2*x^2 + 1)*x/c^2 - arcsin(c*x)/c^3))*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*arc
sin(c*x)/c^5)*c)*b*e*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*x)/c^5)*c)*b*d*i + 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*i + a*d*f*x + (c*x*arcsin(c*x) + sqrt(-c^2*x^2 + 1))*b*d*f/c

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \left (a+b\,\mathrm {asin}\left (c\,x\right )\right )\,\left (d+e\,x\right )\,\left (i\,x^3+h\,x^2+g\,x+f\right ) \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*asin(c*x))*(d + e*x)*(f + g*x + h*x^2 + i*x^3),x)

[Out]

int((a + b*asin(c*x))*(d + e*x)*(f + g*x + h*x^2 + i*x^3), x)

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sympy [A]  time = 3.36, size = 658, normalized size = 2.14 \[ \begin {cases} a d f x + \frac {a d g x^{2}}{2} + \frac {a d h x^{3}}{3} + \frac {a d i x^{4}}{4} + \frac {a e f x^{2}}{2} + \frac {a e g x^{3}}{3} + \frac {a e h x^{4}}{4} + \frac {a e i x^{5}}{5} + b d f x \operatorname {asin}{\left (c x \right )} + \frac {b d g x^{2} \operatorname {asin}{\left (c x \right )}}{2} + \frac {b d h x^{3} \operatorname {asin}{\left (c x \right )}}{3} + \frac {b d i x^{4} \operatorname {asin}{\left (c x \right )}}{4} + \frac {b e f x^{2} \operatorname {asin}{\left (c x \right )}}{2} + \frac {b e g x^{3} \operatorname {asin}{\left (c x \right )}}{3} + \frac {b e h x^{4} \operatorname {asin}{\left (c x \right )}}{4} + \frac {b e i x^{5} \operatorname {asin}{\left (c x \right )}}{5} + \frac {b d f \sqrt {- c^{2} x^{2} + 1}}{c} + \frac {b d g x \sqrt {- c^{2} x^{2} + 1}}{4 c} + \frac {b d h x^{2} \sqrt {- c^{2} x^{2} + 1}}{9 c} + \frac {b d i x^{3} \sqrt {- c^{2} x^{2} + 1}}{16 c} + \frac {b e f x \sqrt {- c^{2} x^{2} + 1}}{4 c} + \frac {b e g x^{2} \sqrt {- c^{2} x^{2} + 1}}{9 c} + \frac {b e h x^{3} \sqrt {- c^{2} x^{2} + 1}}{16 c} + \frac {b e i x^{4} \sqrt {- c^{2} x^{2} + 1}}{25 c} - \frac {b d g \operatorname {asin}{\left (c x \right )}}{4 c^{2}} - \frac {b e f \operatorname {asin}{\left (c x \right )}}{4 c^{2}} + \frac {2 b d h \sqrt {- c^{2} x^{2} + 1}}{9 c^{3}} + \frac {3 b d i x \sqrt {- c^{2} x^{2} + 1}}{32 c^{3}} + \frac {2 b e g \sqrt {- c^{2} x^{2} + 1}}{9 c^{3}} + \frac {3 b e h x \sqrt {- c^{2} x^{2} + 1}}{32 c^{3}} + \frac {4 b e i x^{2} \sqrt {- c^{2} x^{2} + 1}}{75 c^{3}} - \frac {3 b d i \operatorname {asin}{\left (c x \right )}}{32 c^{4}} - \frac {3 b e h \operatorname {asin}{\left (c x \right )}}{32 c^{4}} + \frac {8 b e i \sqrt {- c^{2} x^{2} + 1}}{75 c^{5}} & \text {for}\: c \neq 0 \\a \left (d f x + \frac {d g x^{2}}{2} + \frac {d h x^{3}}{3} + \frac {d i x^{4}}{4} + \frac {e f x^{2}}{2} + \frac {e g x^{3}}{3} + \frac {e h x^{4}}{4} + \frac {e i x^{5}}{5}\right ) & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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