Integrand size = 21, antiderivative size = 261 \[ \int (d+e x)^2 (f+g x) (a+b \arcsin (c x)) \, dx=\frac {b \left (3 c^2 d^2 f+e (e f+2 d g)\right ) \sqrt {1-c^2 x^2}}{3 c^3}+\frac {b \left (3 e^2 g+8 c^2 d (2 e f+d g)\right ) x \sqrt {1-c^2 x^2}}{32 c^3}+\frac {b e^2 g x^3 \sqrt {1-c^2 x^2}}{16 c}-\frac {b e (e f+2 d g) \left (1-c^2 x^2\right )^{3/2}}{9 c^3}-\frac {b \left (3 e^2 g+8 c^2 d (2 e f+d g)\right ) \arcsin (c x)}{32 c^4}+d^2 f x (a+b \arcsin (c x))+\frac {1}{2} d (2 e f+d g) x^2 (a+b \arcsin (c x))+\frac {1}{3} e (e f+2 d g) x^3 (a+b \arcsin (c x))+\frac {1}{4} e^2 g x^4 (a+b \arcsin (c x)) \] Output:
1/3*b*(3*c^2*d^2*f+e*(2*d*g+e*f))*(-c^2*x^2+1)^(1/2)/c^3+1/32*b*(3*e^2*g+8 *c^2*d*(d*g+2*e*f))*x*(-c^2*x^2+1)^(1/2)/c^3+1/16*b*e^2*g*x^3*(-c^2*x^2+1) ^(1/2)/c-1/9*b*e*(2*d*g+e*f)*(-c^2*x^2+1)^(3/2)/c^3-1/32*b*(3*e^2*g+8*c^2* d*(d*g+2*e*f))*arcsin(c*x)/c^4+d^2*f*x*(a+b*arcsin(c*x))+1/2*d*(d*g+2*e*f) *x^2*(a+b*arcsin(c*x))+1/3*e*(2*d*g+e*f)*x^3*(a+b*arcsin(c*x))+1/4*e^2*g*x ^4*(a+b*arcsin(c*x))
Time = 0.19 (sec) , antiderivative size = 211, normalized size of antiderivative = 0.81 \[ \int (d+e x)^2 (f+g x) (a+b \arcsin (c x)) \, dx=\frac {24 a c^4 x \left (6 d^2 (2 f+g x)+4 d e x (3 f+2 g x)+e^2 x^2 (4 f+3 g x)\right )+b c \sqrt {1-c^2 x^2} \left (e (64 e f+128 d g+27 e g x)+2 c^2 \left (36 d^2 (4 f+g x)+8 d e x (9 f+4 g x)+e^2 x^2 (16 f+9 g x)\right )\right )+3 b \left (-9 e^2 g-24 c^2 d (2 e f+d g)+8 c^4 x \left (6 d^2 (2 f+g x)+4 d e x (3 f+2 g x)+e^2 x^2 (4 f+3 g x)\right )\right ) \arcsin (c x)}{288 c^4} \] Input:
Integrate[(d + e*x)^2*(f + g*x)*(a + b*ArcSin[c*x]),x]
Output:
(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)
Time = 0.89 (sec) , antiderivative size = 275, normalized size of antiderivative = 1.05, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.476, 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.
| \(\displaystyle \int (d+e x)^2 (f+g x) (a+b \arcsin (c x)) \, dx\) |
| \(\Big \downarrow \) 5248 |
| \(\displaystyle -b c \int \frac {x \left (3 e^2 g x^3+4 e (e f+2 d g) x^2+6 d (2 e f+d g) x+12 d^2 f\right )}{12 \sqrt {1-c^2 x^2}}dx+d^2 f x (a+b \arcsin (c x))+\frac {1}{3} e x^3 (2 d g+e f) (a+b \arcsin (c x))+\frac {1}{2} d x^2 (d g+2 e f) (a+b \arcsin (c x))+\frac {1}{4} e^2 g x^4 (a+b \arcsin (c x))\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {1}{12} b c \int \frac {x \left (3 e^2 g x^3+4 e (e f+2 d g) x^2+6 d (2 e f+d g) x+12 d^2 f\right )}{\sqrt {1-c^2 x^2}}dx+d^2 f x (a+b \arcsin (c x))+\frac {1}{3} e x^3 (2 d g+e f) (a+b \arcsin (c x))+\frac {1}{2} d x^2 (d g+2 e f) (a+b \arcsin (c x))+\frac {1}{4} e^2 g x^4 (a+b \arcsin (c x))\) |
| \(\Big \downarrow \) 2340 |
| \(\displaystyle -\frac {1}{12} b c \left (-\frac {\int -\frac {x \left (48 c^2 f d^2+16 c^2 e (e f+2 d g) x^2+3 \left (8 d (2 e f+d g) c^2+3 e^2 g\right ) x\right )}{\sqrt {1-c^2 x^2}}dx}{4 c^2}-\frac {3 e^2 g x^3 \sqrt {1-c^2 x^2}}{4 c^2}\right )+d^2 f x (a+b \arcsin (c x))+\frac {1}{3} e x^3 (2 d g+e f) (a+b \arcsin (c x))+\frac {1}{2} d x^2 (d g+2 e f) (a+b \arcsin (c x))+\frac {1}{4} e^2 g x^4 (a+b \arcsin (c x))\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {1}{12} b c \left (\frac {\int \frac {x \left (48 c^2 f d^2+16 c^2 e (e f+2 d g) x^2+3 \left (8 d (2 e f+d g) c^2+3 e^2 g\right ) x\right )}{\sqrt {1-c^2 x^2}}dx}{4 c^2}-\frac {3 e^2 g x^3 \sqrt {1-c^2 x^2}}{4 c^2}\right )+d^2 f x (a+b \arcsin (c x))+\frac {1}{3} e x^3 (2 d g+e f) (a+b \arcsin (c x))+\frac {1}{2} d x^2 (d g+2 e f) (a+b \arcsin (c x))+\frac {1}{4} e^2 g x^4 (a+b \arcsin (c x))\) |
| \(\Big \downarrow \) 2340 |
| \(\displaystyle -\frac {1}{12} b c \left (\frac {-\frac {\int -\frac {c^2 x \left (16 \left (9 c^2 f d^2+2 e (e f+2 d g)\right )+9 \left (8 d (2 e f+d g) c^2+3 e^2 g\right ) x\right )}{\sqrt {1-c^2 x^2}}dx}{3 c^2}-\frac {16}{3} e x^2 \sqrt {1-c^2 x^2} (2 d g+e f)}{4 c^2}-\frac {3 e^2 g x^3 \sqrt {1-c^2 x^2}}{4 c^2}\right )+d^2 f x (a+b \arcsin (c x))+\frac {1}{3} e x^3 (2 d g+e f) (a+b \arcsin (c x))+\frac {1}{2} d x^2 (d g+2 e f) (a+b \arcsin (c x))+\frac {1}{4} e^2 g x^4 (a+b \arcsin (c x))\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {1}{12} b c \left (\frac {\frac {\int \frac {c^2 x \left (16 \left (9 c^2 f d^2+2 e (e f+2 d g)\right )+9 \left (8 d (2 e f+d g) c^2+3 e^2 g\right ) x\right )}{\sqrt {1-c^2 x^2}}dx}{3 c^2}-\frac {16}{3} e x^2 \sqrt {1-c^2 x^2} (2 d g+e f)}{4 c^2}-\frac {3 e^2 g x^3 \sqrt {1-c^2 x^2}}{4 c^2}\right )+d^2 f x (a+b \arcsin (c x))+\frac {1}{3} e x^3 (2 d g+e f) (a+b \arcsin (c x))+\frac {1}{2} d x^2 (d g+2 e f) (a+b \arcsin (c x))+\frac {1}{4} e^2 g x^4 (a+b \arcsin (c x))\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {1}{12} b c \left (\frac {\frac {1}{3} \int \frac {x \left (16 \left (9 c^2 f d^2+2 e (e f+2 d g)\right )+9 \left (8 d (2 e f+d g) c^2+3 e^2 g\right ) x\right )}{\sqrt {1-c^2 x^2}}dx-\frac {16}{3} e x^2 \sqrt {1-c^2 x^2} (2 d g+e f)}{4 c^2}-\frac {3 e^2 g x^3 \sqrt {1-c^2 x^2}}{4 c^2}\right )+d^2 f x (a+b \arcsin (c x))+\frac {1}{3} e x^3 (2 d g+e f) (a+b \arcsin (c x))+\frac {1}{2} d x^2 (d g+2 e f) (a+b \arcsin (c x))+\frac {1}{4} e^2 g x^4 (a+b \arcsin (c x))\) |
| \(\Big \downarrow \) 533 |
| \(\displaystyle -\frac {1}{12} b c \left (\frac {\frac {1}{3} \left (\frac {\int \frac {32 \left (9 c^2 f d^2+2 e (e f+2 d g)\right ) x c^2+9 \left (8 d (2 e f+d g) c^2+3 e^2 g\right )}{\sqrt {1-c^2 x^2}}dx}{2 c^2}-\frac {9}{2} x \sqrt {1-c^2 x^2} \left (\frac {3 e^2 g}{c^2}+8 d (d g+2 e f)\right )\right )-\frac {16}{3} e x^2 \sqrt {1-c^2 x^2} (2 d g+e f)}{4 c^2}-\frac {3 e^2 g x^3 \sqrt {1-c^2 x^2}}{4 c^2}\right )+d^2 f x (a+b \arcsin (c x))+\frac {1}{3} e x^3 (2 d g+e f) (a+b \arcsin (c x))+\frac {1}{2} d x^2 (d g+2 e f) (a+b \arcsin (c x))+\frac {1}{4} e^2 g x^4 (a+b \arcsin (c x))\) |
| \(\Big \downarrow \) 455 |
| \(\displaystyle -\frac {1}{12} b c \left (\frac {\frac {1}{3} \left (\frac {9 \left (8 c^2 d (d g+2 e f)+3 e^2 g\right ) \int \frac {1}{\sqrt {1-c^2 x^2}}dx-32 \sqrt {1-c^2 x^2} \left (9 c^2 d^2 f+2 e (2 d g+e f)\right )}{2 c^2}-\frac {9}{2} x \sqrt {1-c^2 x^2} \left (\frac {3 e^2 g}{c^2}+8 d (d g+2 e f)\right )\right )-\frac {16}{3} e x^2 \sqrt {1-c^2 x^2} (2 d g+e f)}{4 c^2}-\frac {3 e^2 g x^3 \sqrt {1-c^2 x^2}}{4 c^2}\right )+d^2 f x (a+b \arcsin (c x))+\frac {1}{3} e x^3 (2 d g+e f) (a+b \arcsin (c x))+\frac {1}{2} d x^2 (d g+2 e f) (a+b \arcsin (c x))+\frac {1}{4} e^2 g x^4 (a+b \arcsin (c x))\) |
| \(\Big \downarrow \) 223 |
| \(\displaystyle d^2 f x (a+b \arcsin (c x))+\frac {1}{3} e x^3 (2 d g+e f) (a+b \arcsin (c x))+\frac {1}{2} d x^2 (d g+2 e f) (a+b \arcsin (c x))+\frac {1}{4} e^2 g x^4 (a+b \arcsin (c x))-\frac {1}{12} b c \left (\frac {\frac {1}{3} \left (\frac {\frac {9 \arcsin (c x) \left (8 c^2 d (d g+2 e f)+3 e^2 g\right )}{c}-32 \sqrt {1-c^2 x^2} \left (9 c^2 d^2 f+2 e (2 d g+e f)\right )}{2 c^2}-\frac {9}{2} x \sqrt {1-c^2 x^2} \left (\frac {3 e^2 g}{c^2}+8 d (d g+2 e f)\right )\right )-\frac {16}{3} e x^2 \sqrt {1-c^2 x^2} (2 d g+e f)}{4 c^2}-\frac {3 e^2 g x^3 \sqrt {1-c^2 x^2}}{4 c^2}\right )\) |
Input:
Int[(d + e*x)^2*(f + g*x)*(a + b*ArcSin[c*x]),x]
Output:
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 - (b*c*((-3*e^2*g*x^3*Sqrt[1 - c^2*x^2])/(4*c^2) + ((-16*e*(e*f + 2*d*g)*x^2*Sqrt[1 - c^2*x^2])/3 + ((-9*((3*e^2*g)/c^2 + 8*d*(2*e*f + d*g)) *x*Sqrt[1 - c^2*x^2])/2 + (-32*(9*c^2*d^2*f + 2*e*(e*f + 2*d*g))*Sqrt[1 - c^2*x^2] + (9*(3*e^2*g + 8*c^2*d*(2*e*f + d*g))*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 = 0.25 (sec) , antiderivative size = 296, normalized size of antiderivative = 1.13
| method | result | size |
| parts | \(a \left (\frac {e^{2} g \,x^{4}}{4}+\frac {\left (2 d e g +e^{2} f \right ) x^{3}}{3}+\frac {\left (d^{2} g +2 d e f \right ) x^{2}}{2}+d^{2} f x \right )+\frac {b \left (\frac {c \arcsin \left (c x \right ) e^{2} g \,x^{4}}{4}+\frac {2 c \arcsin \left (c x \right ) x^{3} d e g}{3}+\frac {c \arcsin \left (c x \right ) x^{3} e^{2} f}{3}+\frac {c \arcsin \left (c x \right ) x^{2} d^{2} g}{2}+c \arcsin \left (c x \right ) x^{2} d e f +\arcsin \left (c x \right ) d^{2} f c x -\frac {4 e c \left (2 d g +e f \right ) \left (-\frac {c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{3}-\frac {2 \sqrt {-c^{2} x^{2}+1}}{3}\right )+6 d \,c^{2} \left (d g +2 e f \right ) \left (-\frac {c x \sqrt {-c^{2} x^{2}+1}}{2}+\frac {\arcsin \left (c x \right )}{2}\right )+3 e^{2} g \left (-\frac {c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}}{4}-\frac {3 c x \sqrt {-c^{2} x^{2}+1}}{8}+\frac {3 \arcsin \left (c x \right )}{8}\right )-12 c^{3} d^{2} f \sqrt {-c^{2} x^{2}+1}}{12 c^{3}}\right )}{c}\) | \(296\) |
| derivativedivides | \(\frac {\frac {a \left (\frac {e^{2} g \,c^{4} x^{4}}{4}+\frac {\left (2 c d e g +c \,e^{2} f \right ) c^{3} x^{3}}{3}+\frac {\left (c^{2} d^{2} g +2 d e f \,c^{2}\right ) c^{2} x^{2}}{2}+c^{4} d^{2} f x \right )}{c^{3}}+\frac {b \left (\frac {\arcsin \left (c x \right ) e^{2} g \,c^{4} x^{4}}{4}+\frac {2 \arcsin \left (c x \right ) c^{4} d e g \,x^{3}}{3}+\frac {\arcsin \left (c x \right ) c^{4} e^{2} f \,x^{3}}{3}+\frac {\arcsin \left (c x \right ) c^{4} d^{2} g \,x^{2}}{2}+\arcsin \left (c x \right ) c^{4} d e f \,x^{2}+\arcsin \left (c x \right ) c^{4} d^{2} f x -\frac {e c \left (2 d g +e f \right ) \left (-\frac {c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{3}-\frac {2 \sqrt {-c^{2} x^{2}+1}}{3}\right )}{3}-\frac {d \,c^{2} \left (d g +2 e f \right ) \left (-\frac {c x \sqrt {-c^{2} x^{2}+1}}{2}+\frac {\arcsin \left (c x \right )}{2}\right )}{2}-\frac {e^{2} g \left (-\frac {c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}}{4}-\frac {3 c x \sqrt {-c^{2} x^{2}+1}}{8}+\frac {3 \arcsin \left (c x \right )}{8}\right )}{4}+c^{3} d^{2} f \sqrt {-c^{2} x^{2}+1}\right )}{c^{3}}}{c}\) | \(328\) |
| default | \(\frac {\frac {a \left (\frac {e^{2} g \,c^{4} x^{4}}{4}+\frac {\left (2 c d e g +c \,e^{2} f \right ) c^{3} x^{3}}{3}+\frac {\left (c^{2} d^{2} g +2 d e f \,c^{2}\right ) c^{2} x^{2}}{2}+c^{4} d^{2} f x \right )}{c^{3}}+\frac {b \left (\frac {\arcsin \left (c x \right ) e^{2} g \,c^{4} x^{4}}{4}+\frac {2 \arcsin \left (c x \right ) c^{4} d e g \,x^{3}}{3}+\frac {\arcsin \left (c x \right ) c^{4} e^{2} f \,x^{3}}{3}+\frac {\arcsin \left (c x \right ) c^{4} d^{2} g \,x^{2}}{2}+\arcsin \left (c x \right ) c^{4} d e f \,x^{2}+\arcsin \left (c x \right ) c^{4} d^{2} f x -\frac {e c \left (2 d g +e f \right ) \left (-\frac {c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{3}-\frac {2 \sqrt {-c^{2} x^{2}+1}}{3}\right )}{3}-\frac {d \,c^{2} \left (d g +2 e f \right ) \left (-\frac {c x \sqrt {-c^{2} x^{2}+1}}{2}+\frac {\arcsin \left (c x \right )}{2}\right )}{2}-\frac {e^{2} g \left (-\frac {c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}}{4}-\frac {3 c x \sqrt {-c^{2} x^{2}+1}}{8}+\frac {3 \arcsin \left (c x \right )}{8}\right )}{4}+c^{3} d^{2} f \sqrt {-c^{2} x^{2}+1}\right )}{c^{3}}}{c}\) | \(328\) |
| orering | \(\frac {\left (126 c^{4} e^{3} g^{2} x^{6}+474 c^{4} d \,e^{2} g^{2} x^{5}+300 c^{4} e^{3} f g \,x^{5}+616 c^{4} d^{2} e \,g^{2} x^{4}+1240 c^{4} d \,e^{2} f g \,x^{4}+160 c^{4} e^{3} f^{2} x^{4}+216 c^{4} d^{3} g^{2} x^{3}+2064 c^{4} d^{2} e f g \,x^{3}+672 c^{4} d \,e^{2} f^{2} x^{3}+720 c^{4} d^{3} f g \,x^{2}+1152 c^{4} d^{2} e \,f^{2} x^{2}+27 c^{2} e^{3} g^{2} x^{4}+288 c^{4} d^{3} f^{2} x +201 c^{2} d \,e^{2} g^{2} x^{3}+114 c^{2} e^{3} f g \,x^{3}-224 c^{2} d^{2} e \,g^{2} x^{2}-416 c^{2} d \,e^{2} f g \,x^{2}+64 c^{2} e^{3} f^{2} x^{2}-144 c^{2} d^{3} g^{2} x -1368 c^{2} d^{2} e f g x -432 c^{2} d \,e^{2} f^{2} x -360 c^{2} d^{3} f g -720 c^{2} d^{2} e \,f^{2}-108 e^{3} g^{2} x^{2}-438 d \,e^{2} g^{2} x -273 e^{3} f g x -128 d^{2} e \,g^{2}-347 d \,e^{2} f g -128 e^{3} f^{2}\right ) \left (a +b \arcsin \left (c x \right )\right )}{288 \left (e x +d \right ) \left (g x +f \right ) c^{4}}-\frac {\left (18 c^{2} e^{2} g \,x^{3}+64 c^{2} d e g \,x^{2}+32 c^{2} e^{2} f \,x^{2}+72 c^{2} d^{2} g x +144 c^{2} d e f x +288 c^{2} d^{2} f +27 e^{2} g x +128 d e g +64 e^{2} f \right ) \left (c x -1\right ) \left (c x +1\right ) \left (2 \left (e x +d \right ) \left (g x +f \right ) \left (a +b \arcsin \left (c x \right )\right ) e +\left (e x +d \right )^{2} g \left (a +b \arcsin \left (c x \right )\right )+\frac {\left (e x +d \right )^{2} \left (g x +f \right ) b c}{\sqrt {-c^{2} x^{2}+1}}\right )}{288 c^{4} \left (g x +f \right ) \left (e x +d \right )^{2}}\) | \(573\) |
Input:
int((e*x+d)^2*(g*x+f)*(a+b*arcsin(c*x)),x,method=_RETURNVERBOSE)
Output:
a*(1/4*e^2*g*x^4+1/3*(2*d*e*g+e^2*f)*x^3+1/2*(d^2*g+2*d*e*f)*x^2+d^2*f*x)+ b/c*(1/4*c*arcsin(c*x)*e^2*g*x^4+2/3*c*arcsin(c*x)*x^3*d*e*g+1/3*c*arcsin( c*x)*x^3*e^2*f+1/2*c*arcsin(c*x)*x^2*d^2*g+c*arcsin(c*x)*x^2*d*e*f+arcsin( c*x)*d^2*f*c*x-1/12/c^3*(4*e*c*(2*d*g+e*f)*(-1/3*c^2*x^2*(-c^2*x^2+1)^(1/2 )-2/3*(-c^2*x^2+1)^(1/2))+6*d*c^2*(d*g+2*e*f)*(-1/2*c*x*(-c^2*x^2+1)^(1/2) +1/2*arcsin(c*x))+3*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))-12*c^3*d^2*f*(-c^2*x^2+1)^(1/2)))
Time = 0.12 (sec) , antiderivative size = 295, normalized size of antiderivative = 1.13 \[ \int (d+e x)^2 (f+g x) (a+b \arcsin (c x)) \, dx=\frac {72 \, a c^{4} e^{2} g x^{4} + 288 \, a c^{4} d^{2} f x + 96 \, {\left (a c^{4} e^{2} f + 2 \, a c^{4} d e g\right )} x^{3} + 144 \, {\left (2 \, a c^{4} d e f + a c^{4} d^{2} g\right )} x^{2} + 3 \, {\left (24 \, b c^{4} e^{2} g x^{4} + 96 \, b c^{4} d^{2} f x - 48 \, b c^{2} d e f + 32 \, {\left (b c^{4} e^{2} f + 2 \, b c^{4} d e g\right )} x^{3} + 48 \, {\left (2 \, b c^{4} d e f + b c^{4} d^{2} g\right )} x^{2} - 3 \, {\left (8 \, b c^{2} d^{2} + 3 \, b e^{2}\right )} g\right )} \arcsin \left (c x\right ) + {\left (18 \, b c^{3} e^{2} g x^{3} + 128 \, b c d e g + 32 \, {\left (b c^{3} e^{2} f + 2 \, b c^{3} d e g\right )} x^{2} + 32 \, {\left (9 \, b c^{3} d^{2} + 2 \, b c e^{2}\right )} f + 9 \, {\left (16 \, b c^{3} d e f + {\left (8 \, b c^{3} d^{2} + 3 \, b c e^{2}\right )} g\right )} x\right )} \sqrt {-c^{2} x^{2} + 1}}{288 \, c^{4}} \] Input:
integrate((e*x+d)^2*(g*x+f)*(a+b*arcsin(c*x)),x, algorithm="fricas")
Output:
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)*ar csin(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
Leaf count of result is larger than twice the leaf count of optimal. 502 vs. \(2 (250) = 500\).
Time = 0.34 (sec) , antiderivative size = 502, normalized size of antiderivative = 1.92 \[ \int (d+e x)^2 (f+g x) (a+b \arcsin (c x)) \, dx=\begin {cases} a d^{2} f x + \frac {a d^{2} g x^{2}}{2} + a d e f x^{2} + \frac {2 a d e g x^{3}}{3} + \frac {a e^{2} f x^{3}}{3} + \frac {a e^{2} g x^{4}}{4} + b d^{2} f x \operatorname {asin}{\left (c x \right )} + \frac {b d^{2} g x^{2} \operatorname {asin}{\left (c x \right )}}{2} + b d e f x^{2} \operatorname {asin}{\left (c x \right )} + \frac {2 b d e g x^{3} \operatorname {asin}{\left (c x \right )}}{3} + \frac {b e^{2} f x^{3} \operatorname {asin}{\left (c x \right )}}{3} + \frac {b e^{2} g x^{4} \operatorname {asin}{\left (c x \right )}}{4} + \frac {b d^{2} f \sqrt {- c^{2} x^{2} + 1}}{c} + \frac {b d^{2} g x \sqrt {- c^{2} x^{2} + 1}}{4 c} + \frac {b d e f x \sqrt {- c^{2} x^{2} + 1}}{2 c} + \frac {2 b d e g x^{2} \sqrt {- c^{2} x^{2} + 1}}{9 c} + \frac {b e^{2} f x^{2} \sqrt {- c^{2} x^{2} + 1}}{9 c} + \frac {b e^{2} g x^{3} \sqrt {- c^{2} x^{2} + 1}}{16 c} - \frac {b d^{2} g \operatorname {asin}{\left (c x \right )}}{4 c^{2}} - \frac {b d e f \operatorname {asin}{\left (c x \right )}}{2 c^{2}} + \frac {4 b d e g \sqrt {- c^{2} x^{2} + 1}}{9 c^{3}} + \frac {2 b e^{2} f \sqrt {- c^{2} x^{2} + 1}}{9 c^{3}} + \frac {3 b e^{2} g x \sqrt {- c^{2} x^{2} + 1}}{32 c^{3}} - \frac {3 b e^{2} g \operatorname {asin}{\left (c x \right )}}{32 c^{4}} & \text {for}\: c \neq 0 \\a \left (d^{2} f x + \frac {d^{2} g x^{2}}{2} + d e f x^{2} + \frac {2 d e g x^{3}}{3} + \frac {e^{2} f x^{3}}{3} + \frac {e^{2} g x^{4}}{4}\right ) & \text {otherwise} \end {cases} \] Input:
integrate((e*x+d)**2*(g*x+f)*(a+b*asin(c*x)),x)
Output:
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), N e(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))
Time = 0.13 (sec) , antiderivative size = 353, normalized size of antiderivative = 1.35 \[ \int (d+e x)^2 (f+g x) (a+b \arcsin (c x)) \, dx=\frac {1}{4} \, a e^{2} g x^{4} + \frac {1}{3} \, a e^{2} f x^{3} + \frac {2}{3} \, a d e g x^{3} + a d e f x^{2} + \frac {1}{2} \, a d^{2} g x^{2} + \frac {1}{2} \, {\left (2 \, x^{2} \arcsin \left (c x\right ) + c {\left (\frac {\sqrt {-c^{2} x^{2} + 1} x}{c^{2}} - \frac {\arcsin \left (c x\right )}{c^{3}}\right )}\right )} b d e f + \frac {1}{9} \, {\left (3 \, x^{3} \arcsin \left (c x\right ) + c {\left (\frac {\sqrt {-c^{2} x^{2} + 1} x^{2}}{c^{2}} + \frac {2 \, \sqrt {-c^{2} x^{2} + 1}}{c^{4}}\right )}\right )} b e^{2} f + \frac {1}{4} \, {\left (2 \, x^{2} \arcsin \left (c x\right ) + c {\left (\frac {\sqrt {-c^{2} x^{2} + 1} x}{c^{2}} - \frac {\arcsin \left (c x\right )}{c^{3}}\right )}\right )} b d^{2} g + \frac {2}{9} \, {\left (3 \, x^{3} \arcsin \left (c x\right ) + c {\left (\frac {\sqrt {-c^{2} x^{2} + 1} x^{2}}{c^{2}} + \frac {2 \, \sqrt {-c^{2} x^{2} + 1}}{c^{4}}\right )}\right )} b d e g + \frac {1}{32} \, {\left (8 \, x^{4} \arcsin \left (c x\right ) + {\left (\frac {2 \, \sqrt {-c^{2} x^{2} + 1} x^{3}}{c^{2}} + \frac {3 \, \sqrt {-c^{2} x^{2} + 1} x}{c^{4}} - \frac {3 \, \arcsin \left (c x\right )}{c^{5}}\right )} c\right )} b e^{2} g + a d^{2} f x + \frac {{\left (c x \arcsin \left (c x\right ) + \sqrt {-c^{2} x^{2} + 1}\right )} b d^{2} f}{c} \] Input:
integrate((e*x+d)^2*(g*x+f)*(a+b*arcsin(c*x)),x, algorithm="maxima")
Output:
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*x)/c^3))*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*x)/c^3))*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*x)/c^5)*c)*b*e^2*g + a*d^2*f*x + (c*x*arcsin(c*x) + sq rt(-c^2*x^2 + 1))*b*d^2*f/c
Leaf count of result is larger than twice the leaf count of optimal. 491 vs. \(2 (237) = 474\).
Time = 0.14 (sec) , antiderivative size = 491, normalized size of antiderivative = 1.88 \[ \int (d+e x)^2 (f+g x) (a+b \arcsin (c x)) \, dx=\frac {1}{4} \, a e^{2} g x^{4} + \frac {1}{3} \, a e^{2} f x^{3} + \frac {2}{3} \, a d e g x^{3} + b d^{2} f x \arcsin \left (c x\right ) + a d^{2} f x + \frac {{\left (c^{2} x^{2} - 1\right )} b e^{2} f x \arcsin \left (c x\right )}{3 \, c^{2}} + \frac {2 \, {\left (c^{2} x^{2} - 1\right )} b d e g x \arcsin \left (c x\right )}{3 \, c^{2}} + \frac {\sqrt {-c^{2} x^{2} + 1} b d e f x}{2 \, c} + \frac {\sqrt {-c^{2} x^{2} + 1} b d^{2} g x}{4 \, c} + \frac {{\left (c^{2} x^{2} - 1\right )} b d e f \arcsin \left (c x\right )}{c^{2}} + \frac {{\left (c^{2} x^{2} - 1\right )} b d^{2} g \arcsin \left (c x\right )}{2 \, c^{2}} + \frac {b e^{2} f x \arcsin \left (c x\right )}{3 \, c^{2}} + \frac {2 \, b d e g x \arcsin \left (c x\right )}{3 \, c^{2}} + \frac {\sqrt {-c^{2} x^{2} + 1} b d^{2} f}{c} - \frac {{\left (-c^{2} x^{2} + 1\right )}^{\frac {3}{2}} b e^{2} g x}{16 \, c^{3}} + \frac {{\left (c^{2} x^{2} - 1\right )} a d e f}{c^{2}} + \frac {{\left (c^{2} x^{2} - 1\right )} a d^{2} g}{2 \, c^{2}} + \frac {b d e f \arcsin \left (c x\right )}{2 \, c^{2}} + \frac {b d^{2} g \arcsin \left (c x\right )}{4 \, c^{2}} + \frac {{\left (c^{2} x^{2} - 1\right )}^{2} b e^{2} g \arcsin \left (c x\right )}{4 \, c^{4}} - \frac {{\left (-c^{2} x^{2} + 1\right )}^{\frac {3}{2}} b e^{2} f}{9 \, c^{3}} - \frac {2 \, {\left (-c^{2} x^{2} + 1\right )}^{\frac {3}{2}} b d e g}{9 \, c^{3}} + \frac {5 \, \sqrt {-c^{2} x^{2} + 1} b e^{2} g x}{32 \, c^{3}} + \frac {{\left (c^{2} x^{2} - 1\right )} b e^{2} g \arcsin \left (c x\right )}{2 \, c^{4}} + \frac {\sqrt {-c^{2} x^{2} + 1} b e^{2} f}{3 \, c^{3}} + \frac {2 \, \sqrt {-c^{2} x^{2} + 1} b d e g}{3 \, c^{3}} + \frac {5 \, b e^{2} g \arcsin \left (c x\right )}{32 \, c^{4}} \] Input:
integrate((e*x+d)^2*(g*x+f)*(a+b*arcsin(c*x)),x, algorithm="giac")
Output:
1/4*a*e^2*g*x^4 + 1/3*a*e^2*f*x^3 + 2/3*a*d*e*g*x^3 + b*d^2*f*x*arcsin(c*x ) + a*d^2*f*x + 1/3*(c^2*x^2 - 1)*b*e^2*f*x*arcsin(c*x)/c^2 + 2/3*(c^2*x^2 - 1)*b*d*e*g*x*arcsin(c*x)/c^2 + 1/2*sqrt(-c^2*x^2 + 1)*b*d*e*f*x/c + 1/4 *sqrt(-c^2*x^2 + 1)*b*d^2*g*x/c + (c^2*x^2 - 1)*b*d*e*f*arcsin(c*x)/c^2 + 1/2*(c^2*x^2 - 1)*b*d^2*g*arcsin(c*x)/c^2 + 1/3*b*e^2*f*x*arcsin(c*x)/c^2 + 2/3*b*d*e*g*x*arcsin(c*x)/c^2 + sqrt(-c^2*x^2 + 1)*b*d^2*f/c - 1/16*(-c^ 2*x^2 + 1)^(3/2)*b*e^2*g*x/c^3 + (c^2*x^2 - 1)*a*d*e*f/c^2 + 1/2*(c^2*x^2 - 1)*a*d^2*g/c^2 + 1/2*b*d*e*f*arcsin(c*x)/c^2 + 1/4*b*d^2*g*arcsin(c*x)/c ^2 + 1/4*(c^2*x^2 - 1)^2*b*e^2*g*arcsin(c*x)/c^4 - 1/9*(-c^2*x^2 + 1)^(3/2 )*b*e^2*f/c^3 - 2/9*(-c^2*x^2 + 1)^(3/2)*b*d*e*g/c^3 + 5/32*sqrt(-c^2*x^2 + 1)*b*e^2*g*x/c^3 + 1/2*(c^2*x^2 - 1)*b*e^2*g*arcsin(c*x)/c^4 + 1/3*sqrt( -c^2*x^2 + 1)*b*e^2*f/c^3 + 2/3*sqrt(-c^2*x^2 + 1)*b*d*e*g/c^3 + 5/32*b*e^ 2*g*arcsin(c*x)/c^4
Timed out. \[ \int (d+e x)^2 (f+g x) (a+b \arcsin (c x)) \, dx=\int \left (f+g\,x\right )\,\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )\,{\left (d+e\,x\right )}^2 \,d x \] Input:
int((f + g*x)*(a + b*asin(c*x))*(d + e*x)^2,x)
Output:
int((f + g*x)*(a + b*asin(c*x))*(d + e*x)^2, x)
Time = 0.25 (sec) , antiderivative size = 408, normalized size of antiderivative = 1.56 \[ \int (d+e x)^2 (f+g x) (a+b \arcsin (c x)) \, dx=\frac {288 \mathit {asin} \left (c x \right ) b \,c^{4} d^{2} f x +144 \mathit {asin} \left (c x \right ) b \,c^{4} d^{2} g \,x^{2}+288 \mathit {asin} \left (c x \right ) b \,c^{4} d e f \,x^{2}+192 \mathit {asin} \left (c x \right ) b \,c^{4} d e g \,x^{3}+96 \mathit {asin} \left (c x \right ) b \,c^{4} e^{2} f \,x^{3}+72 \mathit {asin} \left (c x \right ) b \,c^{4} e^{2} g \,x^{4}-72 \mathit {asin} \left (c x \right ) b \,c^{2} d^{2} g -144 \mathit {asin} \left (c x \right ) b \,c^{2} d e f -27 \mathit {asin} \left (c x \right ) b \,e^{2} g +288 \sqrt {-c^{2} x^{2}+1}\, b \,c^{3} d^{2} f +72 \sqrt {-c^{2} x^{2}+1}\, b \,c^{3} d^{2} g x +144 \sqrt {-c^{2} x^{2}+1}\, b \,c^{3} d e f x +64 \sqrt {-c^{2} x^{2}+1}\, b \,c^{3} d e g \,x^{2}+32 \sqrt {-c^{2} x^{2}+1}\, b \,c^{3} e^{2} f \,x^{2}+18 \sqrt {-c^{2} x^{2}+1}\, b \,c^{3} e^{2} g \,x^{3}+128 \sqrt {-c^{2} x^{2}+1}\, b c d e g +64 \sqrt {-c^{2} x^{2}+1}\, b c \,e^{2} f +27 \sqrt {-c^{2} x^{2}+1}\, b c \,e^{2} g x +288 a \,c^{4} d^{2} f x +144 a \,c^{4} d^{2} g \,x^{2}+288 a \,c^{4} d e f \,x^{2}+192 a \,c^{4} d e g \,x^{3}+96 a \,c^{4} e^{2} f \,x^{3}+72 a \,c^{4} e^{2} g \,x^{4}}{288 c^{4}} \] Input:
int((e*x+d)^2*(g*x+f)*(a+b*asin(c*x)),x)
Output:
(288*asin(c*x)*b*c**4*d**2*f*x + 144*asin(c*x)*b*c**4*d**2*g*x**2 + 288*as in(c*x)*b*c**4*d*e*f*x**2 + 192*asin(c*x)*b*c**4*d*e*g*x**3 + 96*asin(c*x) *b*c**4*e**2*f*x**3 + 72*asin(c*x)*b*c**4*e**2*g*x**4 - 72*asin(c*x)*b*c** 2*d**2*g - 144*asin(c*x)*b*c**2*d*e*f - 27*asin(c*x)*b*e**2*g + 288*sqrt( - c**2*x**2 + 1)*b*c**3*d**2*f + 72*sqrt( - c**2*x**2 + 1)*b*c**3*d**2*g*x + 144*sqrt( - c**2*x**2 + 1)*b*c**3*d*e*f*x + 64*sqrt( - c**2*x**2 + 1)*b *c**3*d*e*g*x**2 + 32*sqrt( - c**2*x**2 + 1)*b*c**3*e**2*f*x**2 + 18*sqrt( - c**2*x**2 + 1)*b*c**3*e**2*g*x**3 + 128*sqrt( - c**2*x**2 + 1)*b*c*d*e* g + 64*sqrt( - c**2*x**2 + 1)*b*c*e**2*f + 27*sqrt( - c**2*x**2 + 1)*b*c*e **2*g*x + 288*a*c**4*d**2*f*x + 144*a*c**4*d**2*g*x**2 + 288*a*c**4*d*e*f* x**2 + 192*a*c**4*d*e*g*x**3 + 96*a*c**4*e**2*f*x**3 + 72*a*c**4*e**2*g*x* *4)/(288*c**4)