Integrand size = 24, antiderivative size = 235 \[ \int (d+e x) \left (f+g x+h x^2\right ) (a+b \arcsin (c x)) \, dx=\frac {b \left (3 c^2 d f+e g+d h\right ) \sqrt {1-c^2 x^2}}{3 c^3}+\frac {b \left (8 c^2 (e f+d g)+3 e h\right ) x \sqrt {1-c^2 x^2}}{32 c^3}+\frac {b e h x^3 \sqrt {1-c^2 x^2}}{16 c}-\frac {b (e g+d h) \left (1-c^2 x^2\right )^{3/2}}{9 c^3}-\frac {b \left (8 c^2 (e f+d g)+3 e h\right ) \arcsin (c x)}{32 c^4}+d f x (a+b \arcsin (c x))+\frac {1}{2} (e f+d g) x^2 (a+b \arcsin (c x))+\frac {1}{3} (e g+d h) x^3 (a+b \arcsin (c x))+\frac {1}{4} e h x^4 (a+b \arcsin (c x)) \] Output:
1/3*b*(3*c^2*d*f+d*h+e*g)*(-c^2*x^2+1)^(1/2)/c^3+1/32*b*(8*c^2*(d*g+e*f)+3 *e*h)*x*(-c^2*x^2+1)^(1/2)/c^3+1/16*b*e*h*x^3*(-c^2*x^2+1)^(1/2)/c-1/9*b*( d*h+e*g)*(-c^2*x^2+1)^(3/2)/c^3-1/32*b*(8*c^2*(d*g+e*f)+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*e*h*x^4*(a+b*arcsin(c*x))
Time = 0.18 (sec) , antiderivative size = 186, normalized size of antiderivative = 0.79 \[ \int (d+e x) \left (f+g x+h x^2\right ) (a+b \arcsin (c x)) \, dx=\frac {24 a c^4 x (2 d (6 f+x (3 g+2 h x))+e x (6 f+x (4 g+3 h x)))+b c \sqrt {1-c^2 x^2} \left (64 e g+64 d h+27 e h x+2 c^2 \left (4 d \left (36 f+9 g x+4 h x^2\right )+e x \left (36 f+16 g x+9 h x^2\right )\right )\right )+3 b \left (-24 c^2 (e f+d g)-9 e h+8 c^4 x \left (2 d \left (6 f+3 g x+2 h x^2\right )+e x \left (6 f+4 g x+3 h x^2\right )\right )\right ) \arcsin (c x)}{288 c^4} \] Input:
Integrate[(d + e*x)*(f + g*x + h*x^2)*(a + b*ArcSin[c*x]),x]
Output:
(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)
Time = 0.80 (sec) , antiderivative size = 249, normalized size of antiderivative = 1.06, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {5248, 27, 2340, 25, 2340, 25, 27, 533, 455, 223}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int (d+e x) \left (f+g x+h x^2\right ) (a+b \arcsin (c x)) \, dx\) |
| \(\Big \downarrow \) 5248 |
| \(\displaystyle -b c \int \frac {x \left (3 e h x^3+4 (e g+d h) x^2+6 (e f+d g) x+12 d f\right )}{12 \sqrt {1-c^2 x^2}}dx+\frac {1}{2} x^2 (d g+e f) (a+b \arcsin (c x))+\frac {1}{3} x^3 (d h+e g) (a+b \arcsin (c x))+d f x (a+b \arcsin (c x))+\frac {1}{4} e h x^4 (a+b \arcsin (c x))\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {1}{12} b c \int \frac {x \left (3 e h x^3+4 (e g+d h) x^2+6 (e f+d g) x+12 d f\right )}{\sqrt {1-c^2 x^2}}dx+\frac {1}{2} x^2 (d g+e f) (a+b \arcsin (c x))+\frac {1}{3} x^3 (d h+e g) (a+b \arcsin (c x))+d f x (a+b \arcsin (c x))+\frac {1}{4} e h x^4 (a+b \arcsin (c x))\) |
| \(\Big \downarrow \) 2340 |
| \(\displaystyle -\frac {1}{12} b c \left (-\frac {\int -\frac {x \left (16 (e g+d h) x^2 c^2+48 d f c^2+3 \left (8 (e f+d g) c^2+3 e h\right ) x\right )}{\sqrt {1-c^2 x^2}}dx}{4 c^2}-\frac {3 e h x^3 \sqrt {1-c^2 x^2}}{4 c^2}\right )+\frac {1}{2} x^2 (d g+e f) (a+b \arcsin (c x))+\frac {1}{3} x^3 (d h+e g) (a+b \arcsin (c x))+d f x (a+b \arcsin (c x))+\frac {1}{4} e h x^4 (a+b \arcsin (c x))\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {1}{12} b c \left (\frac {\int \frac {x \left (16 (e g+d h) x^2 c^2+48 d f c^2+3 \left (8 (e f+d g) c^2+3 e h\right ) x\right )}{\sqrt {1-c^2 x^2}}dx}{4 c^2}-\frac {3 e h x^3 \sqrt {1-c^2 x^2}}{4 c^2}\right )+\frac {1}{2} x^2 (d g+e f) (a+b \arcsin (c x))+\frac {1}{3} x^3 (d h+e g) (a+b \arcsin (c x))+d f x (a+b \arcsin (c x))+\frac {1}{4} e h x^4 (a+b \arcsin (c x))\) |
| \(\Big \downarrow \) 2340 |
| \(\displaystyle -\frac {1}{12} b c \left (\frac {-\frac {\int -\frac {c^2 x \left (16 \left (9 d f c^2+2 e g+2 d h\right )+9 \left (8 (e f+d g) c^2+3 e h\right ) x\right )}{\sqrt {1-c^2 x^2}}dx}{3 c^2}-\frac {16}{3} x^2 \sqrt {1-c^2 x^2} (d h+e g)}{4 c^2}-\frac {3 e h x^3 \sqrt {1-c^2 x^2}}{4 c^2}\right )+\frac {1}{2} x^2 (d g+e f) (a+b \arcsin (c x))+\frac {1}{3} x^3 (d h+e g) (a+b \arcsin (c x))+d f x (a+b \arcsin (c x))+\frac {1}{4} e h x^4 (a+b \arcsin (c x))\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {1}{12} b c \left (\frac {\frac {\int \frac {c^2 x \left (16 \left (9 d f c^2+2 e g+2 d h\right )+9 \left (8 (e f+d g) c^2+3 e h\right ) x\right )}{\sqrt {1-c^2 x^2}}dx}{3 c^2}-\frac {16}{3} x^2 \sqrt {1-c^2 x^2} (d h+e g)}{4 c^2}-\frac {3 e h x^3 \sqrt {1-c^2 x^2}}{4 c^2}\right )+\frac {1}{2} x^2 (d g+e f) (a+b \arcsin (c x))+\frac {1}{3} x^3 (d h+e g) (a+b \arcsin (c x))+d f x (a+b \arcsin (c x))+\frac {1}{4} e h x^4 (a+b \arcsin (c x))\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {1}{12} b c \left (\frac {\frac {1}{3} \int \frac {x \left (16 \left (9 d f c^2+2 e g+2 d h\right )+9 \left (8 (e f+d g) c^2+3 e h\right ) x\right )}{\sqrt {1-c^2 x^2}}dx-\frac {16}{3} x^2 \sqrt {1-c^2 x^2} (d h+e g)}{4 c^2}-\frac {3 e h x^3 \sqrt {1-c^2 x^2}}{4 c^2}\right )+\frac {1}{2} x^2 (d g+e f) (a+b \arcsin (c x))+\frac {1}{3} x^3 (d h+e g) (a+b \arcsin (c x))+d f x (a+b \arcsin (c x))+\frac {1}{4} e h x^4 (a+b \arcsin (c x))\) |
| \(\Big \downarrow \) 533 |
| \(\displaystyle -\frac {1}{12} b c \left (\frac {\frac {1}{3} \left (\frac {\int \frac {32 \left (9 d f c^2+2 e g+2 d h\right ) x c^2+9 \left (8 (e f+d g) c^2+3 e h\right )}{\sqrt {1-c^2 x^2}}dx}{2 c^2}-\frac {9}{2} x \sqrt {1-c^2 x^2} \left (\frac {3 e h}{c^2}+8 d g+8 e f\right )\right )-\frac {16}{3} x^2 \sqrt {1-c^2 x^2} (d h+e g)}{4 c^2}-\frac {3 e h x^3 \sqrt {1-c^2 x^2}}{4 c^2}\right )+\frac {1}{2} x^2 (d g+e f) (a+b \arcsin (c x))+\frac {1}{3} x^3 (d h+e g) (a+b \arcsin (c x))+d f x (a+b \arcsin (c x))+\frac {1}{4} e h x^4 (a+b \arcsin (c x))\) |
| \(\Big \downarrow \) 455 |
| \(\displaystyle -\frac {1}{12} b c \left (\frac {\frac {1}{3} \left (\frac {9 \left (8 c^2 (d g+e f)+3 e h\right ) \int \frac {1}{\sqrt {1-c^2 x^2}}dx-32 \sqrt {1-c^2 x^2} \left (9 c^2 d f+2 d h+2 e g\right )}{2 c^2}-\frac {9}{2} x \sqrt {1-c^2 x^2} \left (\frac {3 e h}{c^2}+8 d g+8 e f\right )\right )-\frac {16}{3} x^2 \sqrt {1-c^2 x^2} (d h+e g)}{4 c^2}-\frac {3 e h x^3 \sqrt {1-c^2 x^2}}{4 c^2}\right )+\frac {1}{2} x^2 (d g+e f) (a+b \arcsin (c x))+\frac {1}{3} x^3 (d h+e g) (a+b \arcsin (c x))+d f x (a+b \arcsin (c x))+\frac {1}{4} e h x^4 (a+b \arcsin (c x))\) |
| \(\Big \downarrow \) 223 |
| \(\displaystyle \frac {1}{2} x^2 (d g+e f) (a+b \arcsin (c x))+\frac {1}{3} x^3 (d h+e g) (a+b \arcsin (c x))+d f x (a+b \arcsin (c x))+\frac {1}{4} e h x^4 (a+b \arcsin (c x))-\frac {1}{12} b c \left (\frac {\frac {1}{3} \left (\frac {\frac {9 \arcsin (c x) \left (8 c^2 (d g+e f)+3 e h\right )}{c}-32 \sqrt {1-c^2 x^2} \left (9 c^2 d f+2 d h+2 e g\right )}{2 c^2}-\frac {9}{2} x \sqrt {1-c^2 x^2} \left (\frac {3 e h}{c^2}+8 d g+8 e f\right )\right )-\frac {16}{3} x^2 \sqrt {1-c^2 x^2} (d h+e g)}{4 c^2}-\frac {3 e h x^3 \sqrt {1-c^2 x^2}}{4 c^2}\right )\) |
Input:
Int[(d + e*x)*(f + g*x + h*x^2)*(a + b*ArcSin[c*x]),x]
Output:
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 - (b *c*((-3*e*h*x^3*Sqrt[1 - c^2*x^2])/(4*c^2) + ((-16*(e*g + d*h)*x^2*Sqrt[1 - c^2*x^2])/3 + ((-9*(8*e*f + 8*d*g + (3*e*h)/c^2)*x*Sqrt[1 - c^2*x^2])/2 + (-32*(9*c^2*d*f + 2*e*g + 2*d*h)*Sqrt[1 - c^2*x^2] + (9*(8*c^2*(e*f + d* g) + 3*e*h)*ArcSin[c*x])/c)/(2*c^2))/3)/(4*c^2)))/12
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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]
Int[((c_) + (d_.)*(x_))*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[d*(( a + b*x^2)^(p + 1)/(2*b*(p + 1))), x] + Simp[c Int[(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, p}, x] && !LeQ[p, -1]
Int[(x_)^(m_.)*((c_) + (d_.)*(x_))*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[d*x^m*((a + b*x^2)^(p + 1)/(b*(m + 2*p + 2))), x] - Simp[1/(b*(m + 2* p + 2)) Int[x^(m - 1)*(a + b*x^2)^p*Simp[a*d*m - b*c*(m + 2*p + 2)*x, x], x], x] /; FreeQ[{a, b, c, d, p}, x] && IGtQ[m, 0] && GtQ[p, -1] && Integer Q[2*p]
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] + Simp[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])
Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))*(Px_), x_Symbol] :> With[{u = IntHid e[ExpandExpression[Px, x], x]}, Simp[(a + b*ArcSin[c*x]) u, x] - Simp[b*c Int[SimplifyIntegrand[u/Sqrt[1 - c^2*x^2], x], x], x]] /; FreeQ[{a, b, c }, x] && PolynomialQ[Px, x]
Time = 0.22 (sec) , antiderivative size = 267, normalized size of antiderivative = 1.14
| method | result | size |
| parts | \(a \left (\frac {e h \,x^{4}}{4}+\frac {\left (d h +e g \right ) x^{3}}{3}+\frac {\left (d g +e f \right ) x^{2}}{2}+d f x \right )+\frac {b \left (\frac {c \arcsin \left (c x \right ) e h \,x^{4}}{4}+\frac {c \arcsin \left (c x \right ) x^{3} d h}{3}+\frac {c \arcsin \left (c x \right ) x^{3} e g}{3}+\frac {c \arcsin \left (c x \right ) x^{2} d g}{2}+\frac {c \arcsin \left (c x \right ) x^{2} e f}{2}+\arcsin \left (c x \right ) f d c x -\frac {6 c^{2} \left (d g +e f \right ) \left (-\frac {c x \sqrt {-c^{2} x^{2}+1}}{2}+\frac {\arcsin \left (c x \right )}{2}\right )+4 c \left (d h +e g \right ) \left (-\frac {c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{3}-\frac {2 \sqrt {-c^{2} x^{2}+1}}{3}\right )+3 e h \left (-\frac {c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}}{4}-\frac {3 c x \sqrt {-c^{2} x^{2}+1}}{8}+\frac {3 \arcsin \left (c x \right )}{8}\right )-12 d \,c^{3} f \sqrt {-c^{2} x^{2}+1}}{12 c^{3}}\right )}{c}\) | \(267\) |
| derivativedivides | \(\frac {\frac {a \left (\frac {e h \,c^{4} x^{4}}{4}+\frac {\left (h c d +c g e \right ) c^{3} x^{3}}{3}+\frac {\left (c^{2} g d +f \,c^{2} e \right ) c^{2} x^{2}}{2}+d \,c^{4} f x \right )}{c^{3}}+\frac {b \left (\frac {\arcsin \left (c x \right ) e h \,c^{4} x^{4}}{4}+\frac {\arcsin \left (c x \right ) c^{4} d h \,x^{3}}{3}+\frac {\arcsin \left (c x \right ) c^{4} e g \,x^{3}}{3}+\frac {\arcsin \left (c x \right ) c^{4} d g \,x^{2}}{2}+\frac {\arcsin \left (c x \right ) c^{4} e f \,x^{2}}{2}+\arcsin \left (c x \right ) d \,c^{4} f x -\frac {c^{2} \left (d g +e f \right ) \left (-\frac {c x \sqrt {-c^{2} x^{2}+1}}{2}+\frac {\arcsin \left (c x \right )}{2}\right )}{2}-\frac {c \left (d h +e g \right ) \left (-\frac {c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{3}-\frac {2 \sqrt {-c^{2} x^{2}+1}}{3}\right )}{3}-\frac {e h \left (-\frac {c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}}{4}-\frac {3 c x \sqrt {-c^{2} x^{2}+1}}{8}+\frac {3 \arcsin \left (c x \right )}{8}\right )}{4}+d \,c^{3} f \sqrt {-c^{2} x^{2}+1}\right )}{c^{3}}}{c}\) | \(299\) |
| default | \(\frac {\frac {a \left (\frac {e h \,c^{4} x^{4}}{4}+\frac {\left (h c d +c g e \right ) c^{3} x^{3}}{3}+\frac {\left (c^{2} g d +f \,c^{2} e \right ) c^{2} x^{2}}{2}+d \,c^{4} f x \right )}{c^{3}}+\frac {b \left (\frac {\arcsin \left (c x \right ) e h \,c^{4} x^{4}}{4}+\frac {\arcsin \left (c x \right ) c^{4} d h \,x^{3}}{3}+\frac {\arcsin \left (c x \right ) c^{4} e g \,x^{3}}{3}+\frac {\arcsin \left (c x \right ) c^{4} d g \,x^{2}}{2}+\frac {\arcsin \left (c x \right ) c^{4} e f \,x^{2}}{2}+\arcsin \left (c x \right ) d \,c^{4} f x -\frac {c^{2} \left (d g +e f \right ) \left (-\frac {c x \sqrt {-c^{2} x^{2}+1}}{2}+\frac {\arcsin \left (c x \right )}{2}\right )}{2}-\frac {c \left (d h +e g \right ) \left (-\frac {c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{3}-\frac {2 \sqrt {-c^{2} x^{2}+1}}{3}\right )}{3}-\frac {e h \left (-\frac {c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}}{4}-\frac {3 c x \sqrt {-c^{2} x^{2}+1}}{8}+\frac {3 \arcsin \left (c x \right )}{8}\right )}{4}+d \,c^{3} f \sqrt {-c^{2} x^{2}+1}\right )}{c^{3}}}{c}\) | \(299\) |
| orering | \(\frac {\left (126 c^{4} e^{2} h^{2} x^{7}+300 c^{4} d e \,h^{2} x^{6}+300 c^{4} e^{2} g h \,x^{6}+160 c^{4} d^{2} h^{2} x^{5}+770 c^{4} d e g h \,x^{5}+450 c^{4} e^{2} f h \,x^{5}+160 c^{4} e^{2} g^{2} x^{5}+416 c^{4} d^{2} g h \,x^{4}+1640 c^{4} d e f h \,x^{4}+416 c^{4} d e \,g^{2} x^{4}+416 c^{4} e^{2} f g \,x^{4}+960 c^{4} d^{2} f h \,x^{3}+216 c^{4} d^{2} g^{2} x^{3}+1392 c^{4} d e f g \,x^{3}+216 c^{4} e^{2} f^{2} x^{3}+27 c^{2} e^{2} h^{2} x^{5}+720 c^{4} d^{2} f g \,x^{2}+720 c^{4} d e \,f^{2} x^{2}+114 c^{2} d e \,h^{2} x^{4}+114 c^{2} e^{2} g h \,x^{4}+288 c^{4} d^{2} f^{2} x +64 c^{2} d^{2} h^{2} x^{3}-151 c^{2} d e g h \,x^{3}-279 c^{2} e^{2} f h \,x^{3}+64 c^{2} e^{2} g^{2} x^{3}-184 c^{2} d^{2} g h \,x^{2}-1048 c^{2} d e f h \,x^{2}-184 c^{2} d e \,g^{2} x^{2}-184 c^{2} e^{2} f g \,x^{2}-576 c^{2} d^{2} f h x -144 c^{2} d^{2} g^{2} x -864 c^{2} d e f g x -144 c^{2} e^{2} f^{2} x -108 e^{2} h^{2} x^{3}-360 c^{2} d^{2} f g -360 c^{2} d e \,f^{2}-273 d e \,h^{2} x^{2}-273 e^{2} g h \,x^{2}-128 d^{2} h^{2} x -310 d e g h x -54 e^{2} f h x -128 e^{2} g^{2} x -64 d^{2} g h -91 d e f h -64 d e \,g^{2}-64 e^{2} f g \right ) \left (a +b \arcsin \left (c x \right )\right )}{288 \left (e x +d \right ) \left (h \,x^{2}+g x +f \right ) c^{4}}-\frac {\left (18 c^{2} e h \,x^{3}+32 c^{2} d h \,x^{2}+32 c^{2} e g \,x^{2}+72 c^{2} d g x +72 c^{2} e f x +288 c^{2} d f +27 e h x +64 d h +64 e g \right ) \left (c x -1\right ) \left (c x +1\right ) \left (e \left (h \,x^{2}+g x +f \right ) \left (a +b \arcsin \left (c x \right )\right )+\left (e x +d \right ) \left (2 h x +g \right ) \left (a +b \arcsin \left (c x \right )\right )+\frac {\left (e x +d \right ) \left (h \,x^{2}+g x +f \right ) b c}{\sqrt {-c^{2} x^{2}+1}}\right )}{288 c^{4} \left (h \,x^{2}+g x +f \right ) \left (e x +d \right )}\) | \(742\) |
Input:
int((e*x+d)*(h*x^2+g*x+f)*(a+b*arcsin(c*x)),x,method=_RETURNVERBOSE)
Output:
a*(1/4*e*h*x^4+1/3*(d*h+e*g)*x^3+1/2*(d*g+e*f)*x^2+d*f*x)+b/c*(1/4*c*arcsi n(c*x)*e*h*x^4+1/3*c*arcsin(c*x)*x^3*d*h+1/3*c*arcsin(c*x)*x^3*e*g+1/2*c*a rcsin(c*x)*x^2*d*g+1/2*c*arcsin(c*x)*x^2*e*f+arcsin(c*x)*f*d*c*x-1/12/c^3* (6*c^2*(d*g+e*f)*(-1/2*c*x*(-c^2*x^2+1)^(1/2)+1/2*arcsin(c*x))+4*c*(d*h+e* g)*(-1/3*c^2*x^2*(-c^2*x^2+1)^(1/2)-2/3*(-c^2*x^2+1)^(1/2))+3*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))-12*d* c^3*f*(-c^2*x^2+1)^(1/2)))
Time = 0.12 (sec) , antiderivative size = 245, normalized size of antiderivative = 1.04 \[ \int (d+e x) \left (f+g x+h x^2\right ) (a+b \arcsin (c x)) \, dx=\frac {72 \, a c^{4} e h x^{4} + 288 \, a c^{4} d f x + 96 \, {\left (a c^{4} e g + a c^{4} d h\right )} x^{3} + 144 \, {\left (a c^{4} e f + a c^{4} d g\right )} x^{2} + 3 \, {\left (24 \, b c^{4} e h x^{4} + 96 \, b c^{4} d f x - 24 \, b c^{2} e f - 24 \, b c^{2} d g + 32 \, {\left (b c^{4} e g + b c^{4} d h\right )} x^{3} - 9 \, b e h + 48 \, {\left (b c^{4} e f + b c^{4} d g\right )} x^{2}\right )} \arcsin \left (c x\right ) + {\left (18 \, b c^{3} e h x^{3} + 288 \, b c^{3} d f + 64 \, b c e g + 64 \, b c d h + 32 \, {\left (b c^{3} e g + b c^{3} d h\right )} x^{2} + 9 \, {\left (8 \, b c^{3} e f + 8 \, b c^{3} d g + 3 \, b c e h\right )} x\right )} \sqrt {-c^{2} x^{2} + 1}}{288 \, c^{4}} \] Input:
integrate((e*x+d)*(h*x^2+g*x+f)*(a+b*arcsin(c*x)),x, algorithm="fricas")
Output:
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
Leaf count of result is larger than twice the leaf count of optimal. 449 vs. \(2 (223) = 446\).
Time = 0.35 (sec) , antiderivative size = 449, normalized size of antiderivative = 1.91 \[ \int (d+e x) \left (f+g x+h x^2\right ) (a+b \arcsin (c x)) \, dx=\begin {cases} a d f x + \frac {a d g x^{2}}{2} + \frac {a d h x^{3}}{3} + \frac {a e f x^{2}}{2} + \frac {a e g x^{3}}{3} + \frac {a e h x^{4}}{4} + b d f x \operatorname {asin}{\left (c x \right )} + \frac {b d g x^{2} \operatorname {asin}{\left (c x \right )}}{2} + \frac {b d h x^{3} \operatorname {asin}{\left (c x \right )}}{3} + \frac {b e f x^{2} \operatorname {asin}{\left (c x \right )}}{2} + \frac {b e g x^{3} \operatorname {asin}{\left (c x \right )}}{3} + \frac {b e h x^{4} \operatorname {asin}{\left (c x \right )}}{4} + \frac {b d f \sqrt {- c^{2} x^{2} + 1}}{c} + \frac {b d g x \sqrt {- c^{2} x^{2} + 1}}{4 c} + \frac {b d h x^{2} \sqrt {- c^{2} x^{2} + 1}}{9 c} + \frac {b e f x \sqrt {- c^{2} x^{2} + 1}}{4 c} + \frac {b e g x^{2} \sqrt {- c^{2} x^{2} + 1}}{9 c} + \frac {b e h x^{3} \sqrt {- c^{2} x^{2} + 1}}{16 c} - \frac {b d g \operatorname {asin}{\left (c x \right )}}{4 c^{2}} - \frac {b e f \operatorname {asin}{\left (c x \right )}}{4 c^{2}} + \frac {2 b d h \sqrt {- c^{2} x^{2} + 1}}{9 c^{3}} + \frac {2 b e g \sqrt {- c^{2} x^{2} + 1}}{9 c^{3}} + \frac {3 b e h x \sqrt {- c^{2} x^{2} + 1}}{32 c^{3}} - \frac {3 b e h \operatorname {asin}{\left (c x \right )}}{32 c^{4}} & \text {for}\: c \neq 0 \\a \left (d f x + \frac {d g x^{2}}{2} + \frac {d h x^{3}}{3} + \frac {e f x^{2}}{2} + \frac {e g x^{3}}{3} + \frac {e h x^{4}}{4}\right ) & \text {otherwise} \end {cases} \] Input:
integrate((e*x+d)*(h*x**2+g*x+f)*(a+b*asin(c*x)),x)
Output:
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*sqrt(-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))
Time = 0.13 (sec) , antiderivative size = 334, normalized size of antiderivative = 1.42 \[ \int (d+e x) \left (f+g x+h x^2\right ) (a+b \arcsin (c x)) \, dx=\frac {1}{4} \, a e h x^{4} + \frac {1}{3} \, a e g x^{3} + \frac {1}{3} \, a d h x^{3} + \frac {1}{2} \, a e f x^{2} + \frac {1}{2} \, a d g x^{2} + \frac {1}{4} \, {\left (2 \, x^{2} \arcsin \left (c x\right ) + c {\left (\frac {\sqrt {-c^{2} x^{2} + 1} x}{c^{2}} - \frac {\arcsin \left (c x\right )}{c^{3}}\right )}\right )} b e f + \frac {1}{4} \, {\left (2 \, x^{2} \arcsin \left (c x\right ) + c {\left (\frac {\sqrt {-c^{2} x^{2} + 1} x}{c^{2}} - \frac {\arcsin \left (c x\right )}{c^{3}}\right )}\right )} b d g + \frac {1}{9} \, {\left (3 \, x^{3} \arcsin \left (c x\right ) + c {\left (\frac {\sqrt {-c^{2} x^{2} + 1} x^{2}}{c^{2}} + \frac {2 \, \sqrt {-c^{2} x^{2} + 1}}{c^{4}}\right )}\right )} b e g + \frac {1}{9} \, {\left (3 \, x^{3} \arcsin \left (c x\right ) + c {\left (\frac {\sqrt {-c^{2} x^{2} + 1} x^{2}}{c^{2}} + \frac {2 \, \sqrt {-c^{2} x^{2} + 1}}{c^{4}}\right )}\right )} b d h + \frac {1}{32} \, {\left (8 \, x^{4} \arcsin \left (c x\right ) + {\left (\frac {2 \, \sqrt {-c^{2} x^{2} + 1} x^{3}}{c^{2}} + \frac {3 \, \sqrt {-c^{2} x^{2} + 1} x}{c^{4}} - \frac {3 \, \arcsin \left (c x\right )}{c^{5}}\right )} c\right )} b e h + a d f x + \frac {{\left (c x \arcsin \left (c x\right ) + \sqrt {-c^{2} x^{2} + 1}\right )} b d f}{c} \] Input:
integrate((e*x+d)*(h*x^2+g*x+f)*(a+b*arcsin(c*x)),x, algorithm="maxima")
Output:
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*(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 + a*d*f*x + (c*x*arcsin(c*x) + sqrt(-c^2*x^2 + 1))* b*d*f/c
Leaf count of result is larger than twice the leaf count of optimal. 448 vs. \(2 (211) = 422\).
Time = 0.14 (sec) , antiderivative size = 448, normalized size of antiderivative = 1.91 \[ \int (d+e x) \left (f+g x+h x^2\right ) (a+b \arcsin (c x)) \, dx=\frac {1}{4} \, a e h x^{4} + \frac {1}{3} \, a e g x^{3} + \frac {1}{3} \, a d h x^{3} + b d f x \arcsin \left (c x\right ) + a d f x + \frac {{\left (c^{2} x^{2} - 1\right )} b e g x \arcsin \left (c x\right )}{3 \, c^{2}} + \frac {{\left (c^{2} x^{2} - 1\right )} b d h x \arcsin \left (c x\right )}{3 \, c^{2}} + \frac {\sqrt {-c^{2} x^{2} + 1} b e f x}{4 \, c} + \frac {\sqrt {-c^{2} x^{2} + 1} b d g x}{4 \, c} + \frac {{\left (c^{2} x^{2} - 1\right )} b e f \arcsin \left (c x\right )}{2 \, c^{2}} + \frac {{\left (c^{2} x^{2} - 1\right )} b d g \arcsin \left (c x\right )}{2 \, c^{2}} + \frac {b e g x \arcsin \left (c x\right )}{3 \, c^{2}} + \frac {b d h x \arcsin \left (c x\right )}{3 \, c^{2}} + \frac {\sqrt {-c^{2} x^{2} + 1} b d f}{c} - \frac {{\left (-c^{2} x^{2} + 1\right )}^{\frac {3}{2}} b e h x}{16 \, c^{3}} + \frac {{\left (c^{2} x^{2} - 1\right )} a e f}{2 \, c^{2}} + \frac {{\left (c^{2} x^{2} - 1\right )} a d g}{2 \, c^{2}} + \frac {b e f \arcsin \left (c x\right )}{4 \, c^{2}} + \frac {b d g \arcsin \left (c x\right )}{4 \, c^{2}} + \frac {{\left (c^{2} x^{2} - 1\right )}^{2} b e h \arcsin \left (c x\right )}{4 \, c^{4}} - \frac {{\left (-c^{2} x^{2} + 1\right )}^{\frac {3}{2}} b e g}{9 \, c^{3}} - \frac {{\left (-c^{2} x^{2} + 1\right )}^{\frac {3}{2}} b d h}{9 \, c^{3}} + \frac {5 \, \sqrt {-c^{2} x^{2} + 1} b e h x}{32 \, c^{3}} + \frac {{\left (c^{2} x^{2} - 1\right )} b e h \arcsin \left (c x\right )}{2 \, c^{4}} + \frac {\sqrt {-c^{2} x^{2} + 1} b e g}{3 \, c^{3}} + \frac {\sqrt {-c^{2} x^{2} + 1} b d h}{3 \, c^{3}} + \frac {5 \, b e h \arcsin \left (c x\right )}{32 \, c^{4}} \] Input:
integrate((e*x+d)*(h*x^2+g*x+f)*(a+b*arcsin(c*x)),x, algorithm="giac")
Output:
1/4*a*e*h*x^4 + 1/3*a*e*g*x^3 + 1/3*a*d*h*x^3 + b*d*f*x*arcsin(c*x) + a*d* f*x + 1/3*(c^2*x^2 - 1)*b*e*g*x*arcsin(c*x)/c^2 + 1/3*(c^2*x^2 - 1)*b*d*h* x*arcsin(c*x)/c^2 + 1/4*sqrt(-c^2*x^2 + 1)*b*e*f*x/c + 1/4*sqrt(-c^2*x^2 + 1)*b*d*g*x/c + 1/2*(c^2*x^2 - 1)*b*e*f*arcsin(c*x)/c^2 + 1/2*(c^2*x^2 - 1 )*b*d*g*arcsin(c*x)/c^2 + 1/3*b*e*g*x*arcsin(c*x)/c^2 + 1/3*b*d*h*x*arcsin (c*x)/c^2 + sqrt(-c^2*x^2 + 1)*b*d*f/c - 1/16*(-c^2*x^2 + 1)^(3/2)*b*e*h*x /c^3 + 1/2*(c^2*x^2 - 1)*a*e*f/c^2 + 1/2*(c^2*x^2 - 1)*a*d*g/c^2 + 1/4*b*e *f*arcsin(c*x)/c^2 + 1/4*b*d*g*arcsin(c*x)/c^2 + 1/4*(c^2*x^2 - 1)^2*b*e*h *arcsin(c*x)/c^4 - 1/9*(-c^2*x^2 + 1)^(3/2)*b*e*g/c^3 - 1/9*(-c^2*x^2 + 1) ^(3/2)*b*d*h/c^3 + 5/32*sqrt(-c^2*x^2 + 1)*b*e*h*x/c^3 + 1/2*(c^2*x^2 - 1) *b*e*h*arcsin(c*x)/c^4 + 1/3*sqrt(-c^2*x^2 + 1)*b*e*g/c^3 + 1/3*sqrt(-c^2* x^2 + 1)*b*d*h/c^3 + 5/32*b*e*h*arcsin(c*x)/c^4
Timed out. \[ \int (d+e x) \left (f+g x+h x^2\right ) (a+b \arcsin (c x)) \, dx=\int \left (a+b\,\mathrm {asin}\left (c\,x\right )\right )\,\left (d+e\,x\right )\,\left (h\,x^2+g\,x+f\right ) \,d x \] Input:
int((a + b*asin(c*x))*(d + e*x)*(f + g*x + h*x^2),x)
Output:
int((a + b*asin(c*x))*(d + e*x)*(f + g*x + h*x^2), x)
Time = 0.30 (sec) , antiderivative size = 368, normalized size of antiderivative = 1.57 \[ \int (d+e x) \left (f+g x+h x^2\right ) (a+b \arcsin (c x)) \, dx=\frac {288 \mathit {asin} \left (c x \right ) b \,c^{4} d f x +144 \mathit {asin} \left (c x \right ) b \,c^{4} d g \,x^{2}+96 \mathit {asin} \left (c x \right ) b \,c^{4} d h \,x^{3}+144 \mathit {asin} \left (c x \right ) b \,c^{4} e f \,x^{2}+96 \mathit {asin} \left (c x \right ) b \,c^{4} e g \,x^{3}+72 \mathit {asin} \left (c x \right ) b \,c^{4} e h \,x^{4}-72 \mathit {asin} \left (c x \right ) b \,c^{2} d g -72 \mathit {asin} \left (c x \right ) b \,c^{2} e f -27 \mathit {asin} \left (c x \right ) b e h +288 \sqrt {-c^{2} x^{2}+1}\, b \,c^{3} d f +72 \sqrt {-c^{2} x^{2}+1}\, b \,c^{3} d g x +32 \sqrt {-c^{2} x^{2}+1}\, b \,c^{3} d h \,x^{2}+72 \sqrt {-c^{2} x^{2}+1}\, b \,c^{3} e f x +32 \sqrt {-c^{2} x^{2}+1}\, b \,c^{3} e g \,x^{2}+18 \sqrt {-c^{2} x^{2}+1}\, b \,c^{3} e h \,x^{3}+64 \sqrt {-c^{2} x^{2}+1}\, b c d h +64 \sqrt {-c^{2} x^{2}+1}\, b c e g +27 \sqrt {-c^{2} x^{2}+1}\, b c e h x +288 a \,c^{4} d f x +144 a \,c^{4} d g \,x^{2}+96 a \,c^{4} d h \,x^{3}+144 a \,c^{4} e f \,x^{2}+96 a \,c^{4} e g \,x^{3}+72 a \,c^{4} e h \,x^{4}}{288 c^{4}} \] Input:
int((e*x+d)*(h*x^2+g*x+f)*(a+b*asin(c*x)),x)
Output:
(288*asin(c*x)*b*c**4*d*f*x + 144*asin(c*x)*b*c**4*d*g*x**2 + 96*asin(c*x) *b*c**4*d*h*x**3 + 144*asin(c*x)*b*c**4*e*f*x**2 + 96*asin(c*x)*b*c**4*e*g *x**3 + 72*asin(c*x)*b*c**4*e*h*x**4 - 72*asin(c*x)*b*c**2*d*g - 72*asin(c *x)*b*c**2*e*f - 27*asin(c*x)*b*e*h + 288*sqrt( - c**2*x**2 + 1)*b*c**3*d* f + 72*sqrt( - c**2*x**2 + 1)*b*c**3*d*g*x + 32*sqrt( - c**2*x**2 + 1)*b*c **3*d*h*x**2 + 72*sqrt( - c**2*x**2 + 1)*b*c**3*e*f*x + 32*sqrt( - c**2*x* *2 + 1)*b*c**3*e*g*x**2 + 18*sqrt( - c**2*x**2 + 1)*b*c**3*e*h*x**3 + 64*s qrt( - c**2*x**2 + 1)*b*c*d*h + 64*sqrt( - c**2*x**2 + 1)*b*c*e*g + 27*sqr t( - c**2*x**2 + 1)*b*c*e*h*x + 288*a*c**4*d*f*x + 144*a*c**4*d*g*x**2 + 9 6*a*c**4*d*h*x**3 + 144*a*c**4*e*f*x**2 + 96*a*c**4*e*g*x**3 + 72*a*c**4*e *h*x**4)/(288*c**4)