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

Optimal. Leaf size=351 \[ \frac{1}{2} d^2 x^2 (d g+3 e f) \left (a+b \sin ^{-1}(c x)\right )+d^3 f x \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{4} e^2 x^4 (3 d g+e f) \left (a+b \sin ^{-1}(c x)\right )+d e x^3 (d g+e f) \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{5} e^3 g x^5 \left (a+b \sin ^{-1}(c x)\right )+\frac{b \sqrt{1-c^2 x^2} \left (75 c^2 x \left (8 c^2 d^2 (d g+3 e f)+3 e^2 (3 d g+e f)\right )+32 \left (75 c^4 d^3 f+50 c^2 d e (d g+e f)+8 e^3 g\right )\right )}{2400 c^5}-\frac{b \sin ^{-1}(c x) \left (8 c^2 d^2 (d g+3 e f)+3 e^2 (3 d g+e f)\right )}{32 c^4}+\frac{b e^2 x^3 \sqrt{1-c^2 x^2} (3 d g+e f)}{16 c}+\frac{b e x^2 \sqrt{1-c^2 x^2} \left (25 c^2 d (d g+e f)+4 e^2 g\right )}{75 c^3}+\frac{b e^3 g x^4 \sqrt{1-c^2 x^2}}{25 c} \]

[Out]

(b*e*(4*e^2*g + 25*c^2*d*(e*f + d*g))*x^2*Sqrt[1 - c^2*x^2])/(75*c^3) + (b*e^2*(e*f + 3*d*g)*x^3*Sqrt[1 - c^2*
x^2])/(16*c) + (b*e^3*g*x^4*Sqrt[1 - c^2*x^2])/(25*c) + (b*(32*(75*c^4*d^3*f + 8*e^3*g + 50*c^2*d*e*(e*f + d*g
)) + 75*c^2*(8*c^2*d^2*(3*e*f + d*g) + 3*e^2*(e*f + 3*d*g))*x)*Sqrt[1 - c^2*x^2])/(2400*c^5) - (b*(8*c^2*d^2*(
3*e*f + d*g) + 3*e^2*(e*f + 3*d*g))*ArcSin[c*x])/(32*c^4) + d^3*f*x*(a + b*ArcSin[c*x]) + (d^2*(3*e*f + d*g)*x
^2*(a + b*ArcSin[c*x]))/2 + d*e*(e*f + d*g)*x^3*(a + b*ArcSin[c*x]) + (e^2*(e*f + 3*d*g)*x^4*(a + b*ArcSin[c*x
]))/4 + (e^3*g*x^5*(a + b*ArcSin[c*x]))/5

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Rubi [A]  time = 0.991871, antiderivative size = 351, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.19, Rules used = {4749, 1809, 780, 216} \[ \frac{1}{2} d^2 x^2 (d g+3 e f) \left (a+b \sin ^{-1}(c x)\right )+d^3 f x \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{4} e^2 x^4 (3 d g+e f) \left (a+b \sin ^{-1}(c x)\right )+d e x^3 (d g+e f) \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{5} e^3 g x^5 \left (a+b \sin ^{-1}(c x)\right )+\frac{b \sqrt{1-c^2 x^2} \left (75 c^2 x \left (8 c^2 d^2 (d g+3 e f)+3 e^2 (3 d g+e f)\right )+32 \left (75 c^4 d^3 f+50 c^2 d e (d g+e f)+8 e^3 g\right )\right )}{2400 c^5}-\frac{b \sin ^{-1}(c x) \left (8 c^2 d^2 (d g+3 e f)+3 e^2 (3 d g+e f)\right )}{32 c^4}+\frac{b e^2 x^3 \sqrt{1-c^2 x^2} (3 d g+e f)}{16 c}+\frac{b e x^2 \sqrt{1-c^2 x^2} \left (25 c^2 d (d g+e f)+4 e^2 g\right )}{75 c^3}+\frac{b e^3 g x^4 \sqrt{1-c^2 x^2}}{25 c} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(b*e*(4*e^2*g + 25*c^2*d*(e*f + d*g))*x^2*Sqrt[1 - c^2*x^2])/(75*c^3) + (b*e^2*(e*f + 3*d*g)*x^3*Sqrt[1 - c^2*
x^2])/(16*c) + (b*e^3*g*x^4*Sqrt[1 - c^2*x^2])/(25*c) + (b*(32*(75*c^4*d^3*f + 8*e^3*g + 50*c^2*d*e*(e*f + d*g
)) + 75*c^2*(8*c^2*d^2*(3*e*f + d*g) + 3*e^2*(e*f + 3*d*g))*x)*Sqrt[1 - c^2*x^2])/(2400*c^5) - (b*(8*c^2*d^2*(
3*e*f + d*g) + 3*e^2*(e*f + 3*d*g))*ArcSin[c*x])/(32*c^4) + d^3*f*x*(a + b*ArcSin[c*x]) + (d^2*(3*e*f + d*g)*x
^2*(a + b*ArcSin[c*x]))/2 + d*e*(e*f + d*g)*x^3*(a + b*ArcSin[c*x]) + (e^2*(e*f + 3*d*g)*x^4*(a + b*ArcSin[c*x
]))/4 + (e^3*g*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 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)^3 (f+g x) \left (a+b \sin ^{-1}(c x)\right ) \, dx &=d^3 f x \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{2} d^2 (3 e f+d g) x^2 \left (a+b \sin ^{-1}(c x)\right )+d e (e f+d g) x^3 \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{4} e^2 (e f+3 d g) x^4 \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{5} e^3 g x^5 \left (a+b \sin ^{-1}(c x)\right )-(b c) \int \frac{x \left (d^3 f+\frac{1}{2} d^2 (3 e f+d g) x+d e (e f+d g) x^2+\frac{1}{4} e^2 (e f+3 d g) x^3+\frac{1}{5} e^3 g x^4\right )}{\sqrt{1-c^2 x^2}} \, dx\\ &=\frac{b e^3 g x^4 \sqrt{1-c^2 x^2}}{25 c}+d^3 f x \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{2} d^2 (3 e f+d g) x^2 \left (a+b \sin ^{-1}(c x)\right )+d e (e f+d g) x^3 \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{4} e^2 (e f+3 d g) x^4 \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{5} e^3 g x^5 \left (a+b \sin ^{-1}(c x)\right )+\frac{b \int \frac{x \left (-5 c^2 d^3 f-\frac{5}{2} c^2 d^2 (3 e f+d g) x-\frac{1}{5} e \left (4 e^2 g+25 c^2 d (e f+d g)\right ) x^2-\frac{5}{4} c^2 e^2 (e f+3 d g) x^3\right )}{\sqrt{1-c^2 x^2}} \, dx}{5 c}\\ &=\frac{b e^2 (e f+3 d g) x^3 \sqrt{1-c^2 x^2}}{16 c}+\frac{b e^3 g x^4 \sqrt{1-c^2 x^2}}{25 c}+d^3 f x \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{2} d^2 (3 e f+d g) x^2 \left (a+b \sin ^{-1}(c x)\right )+d e (e f+d g) x^3 \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{4} e^2 (e f+3 d g) x^4 \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{5} e^3 g x^5 \left (a+b \sin ^{-1}(c x)\right )-\frac{b \int \frac{x \left (20 c^4 d^3 f+\frac{5}{4} c^2 \left (8 c^2 d^2 (3 e f+d g)+3 e^2 (e f+3 d g)\right ) x+\frac{4}{5} c^2 e \left (4 e^2 g+25 c^2 d (e f+d g)\right ) x^2\right )}{\sqrt{1-c^2 x^2}} \, dx}{20 c^3}\\ &=\frac{b e \left (4 e^2 g+25 c^2 d (e f+d g)\right ) x^2 \sqrt{1-c^2 x^2}}{75 c^3}+\frac{b e^2 (e f+3 d g) x^3 \sqrt{1-c^2 x^2}}{16 c}+\frac{b e^3 g x^4 \sqrt{1-c^2 x^2}}{25 c}+d^3 f x \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{2} d^2 (3 e f+d g) x^2 \left (a+b \sin ^{-1}(c x)\right )+d e (e f+d g) x^3 \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{4} e^2 (e f+3 d g) x^4 \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{5} e^3 g x^5 \left (a+b \sin ^{-1}(c x)\right )+\frac{b \int \frac{x \left (-\frac{4}{5} c^2 \left (75 c^4 d^3 f+8 e^3 g+50 c^2 d e (e f+d g)\right )-\frac{15}{4} c^4 \left (8 c^2 d^2 (3 e f+d g)+3 e^2 (e f+3 d g)\right ) x\right )}{\sqrt{1-c^2 x^2}} \, dx}{60 c^5}\\ &=\frac{b e \left (4 e^2 g+25 c^2 d (e f+d g)\right ) x^2 \sqrt{1-c^2 x^2}}{75 c^3}+\frac{b e^2 (e f+3 d g) x^3 \sqrt{1-c^2 x^2}}{16 c}+\frac{b e^3 g x^4 \sqrt{1-c^2 x^2}}{25 c}+\frac{b \left (32 \left (75 c^4 d^3 f+8 e^3 g+50 c^2 d e (e f+d g)\right )+75 c^2 \left (8 c^2 d^2 (3 e f+d g)+3 e^2 (e f+3 d g)\right ) x\right ) \sqrt{1-c^2 x^2}}{2400 c^5}+d^3 f x \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{2} d^2 (3 e f+d g) x^2 \left (a+b \sin ^{-1}(c x)\right )+d e (e f+d g) x^3 \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{4} e^2 (e f+3 d g) x^4 \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{5} e^3 g x^5 \left (a+b \sin ^{-1}(c x)\right )-\frac{\left (b \left (8 c^2 d^2 (3 e f+d g)+3 e^2 (e f+3 d g)\right )\right ) \int \frac{1}{\sqrt{1-c^2 x^2}} \, dx}{32 c^3}\\ &=\frac{b e \left (4 e^2 g+25 c^2 d (e f+d g)\right ) x^2 \sqrt{1-c^2 x^2}}{75 c^3}+\frac{b e^2 (e f+3 d g) x^3 \sqrt{1-c^2 x^2}}{16 c}+\frac{b e^3 g x^4 \sqrt{1-c^2 x^2}}{25 c}+\frac{b \left (32 \left (75 c^4 d^3 f+8 e^3 g+50 c^2 d e (e f+d g)\right )+75 c^2 \left (8 c^2 d^2 (3 e f+d g)+3 e^2 (e f+3 d g)\right ) x\right ) \sqrt{1-c^2 x^2}}{2400 c^5}-\frac{b \left (8 c^2 d^2 (3 e f+d g)+3 e^2 (e f+3 d g)\right ) \sin ^{-1}(c x)}{32 c^4}+d^3 f x \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{2} d^2 (3 e f+d g) x^2 \left (a+b \sin ^{-1}(c x)\right )+d e (e f+d g) x^3 \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{4} e^2 (e f+3 d g) x^4 \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{5} e^3 g x^5 \left (a+b \sin ^{-1}(c x)\right )\\ \end{align*}

Mathematica [A]  time = 0.395729, size = 305, normalized size = 0.87 \[ \frac{120 a c^5 x \left (10 d^2 e x (3 f+2 g x)+10 d^3 (2 f+g x)+5 d e^2 x^2 (4 f+3 g x)+e^3 x^3 (5 f+4 g x)\right )+b \sqrt{1-c^2 x^2} \left (2 c^4 \left (100 d^2 e x (9 f+4 g x)+300 d^3 (4 f+g x)+25 d e^2 x^2 (16 f+9 g x)+3 e^3 x^3 (25 f+16 g x)\right )+c^2 e \left (1600 d^2 g+25 d e (64 f+27 g x)+e^2 x (225 f+128 g x)\right )+256 e^3 g\right )+15 b c \sin ^{-1}(c x) \left (8 c^4 x \left (10 d^2 e x (3 f+2 g x)+10 d^3 (2 f+g x)+5 d e^2 x^2 (4 f+3 g x)+e^3 x^3 (5 f+4 g x)\right )-40 c^2 d^2 (d g+3 e f)-15 e^2 (3 d g+e f)\right )}{2400 c^5} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(120*a*c^5*x*(10*d^3*(2*f + g*x) + 10*d^2*e*x*(3*f + 2*g*x) + 5*d*e^2*x^2*(4*f + 3*g*x) + e^3*x^3*(5*f + 4*g*x
)) + b*Sqrt[1 - c^2*x^2]*(256*e^3*g + 2*c^4*(300*d^3*(4*f + g*x) + 100*d^2*e*x*(9*f + 4*g*x) + 25*d*e^2*x^2*(1
6*f + 9*g*x) + 3*e^3*x^3*(25*f + 16*g*x)) + c^2*e*(1600*d^2*g + 25*d*e*(64*f + 27*g*x) + e^2*x*(225*f + 128*g*
x))) + 15*b*c*(-40*c^2*d^2*(3*e*f + d*g) - 15*e^2*(e*f + 3*d*g) + 8*c^4*x*(10*d^3*(2*f + g*x) + 10*d^2*e*x*(3*
f + 2*g*x) + 5*d*e^2*x^2*(4*f + 3*g*x) + e^3*x^3*(5*f + 4*g*x)))*ArcSin[c*x])/(2400*c^5)

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Maple [A]  time = 0.019, size = 490, normalized size = 1.4 \begin{align*}{\frac{1}{c} \left ({\frac{a}{{c}^{4}} \left ({\frac{{e}^{3}g{c}^{5}{x}^{5}}{5}}+{\frac{ \left ( 3\,dc{e}^{2}g+{e}^{3}cf \right ){c}^{4}{x}^{4}}{4}}+{\frac{ \left ( 3\,{c}^{2}{d}^{2}eg+3\,d{c}^{2}{e}^{2}f \right ){c}^{3}{x}^{3}}{3}}+{\frac{ \left ({c}^{3}{d}^{3}g+3\,{c}^{3}{d}^{2}ef \right ){c}^{2}{x}^{2}}{2}}+{c}^{5}{d}^{3}fx \right ) }+{\frac{b}{{c}^{4}} \left ({\frac{\arcsin \left ( cx \right ){e}^{3}g{c}^{5}{x}^{5}}{5}}+{\frac{3\,\arcsin \left ( cx \right ){c}^{5}{x}^{4}d{e}^{2}g}{4}}+{\frac{\arcsin \left ( cx \right ){c}^{5}{x}^{4}{e}^{3}f}{4}}+\arcsin \left ( cx \right ){c}^{5}{x}^{3}{d}^{2}eg+\arcsin \left ( cx \right ){c}^{5}{x}^{3}d{e}^{2}f+{\frac{\arcsin \left ( cx \right ){c}^{5}{x}^{2}{d}^{3}g}{2}}+{\frac{3\,\arcsin \left ( cx \right ){c}^{5}{x}^{2}{d}^{2}ef}{2}}+\arcsin \left ( cx \right ){c}^{5}{d}^{3}fx-{\frac{{e}^{3}g}{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{15\,dc{e}^{2}g+5\,{e}^{3}cf}{20} \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}eg+20\,d{c}^{2}{e}^{2}f}{20} \left ( -{\frac{{c}^{2}{x}^{2}}{3}\sqrt{-{c}^{2}{x}^{2}+1}}-{\frac{2}{3}\sqrt{-{c}^{2}{x}^{2}+1}} \right ) }-{\frac{10\,{c}^{3}{d}^{3}g+30\,{c}^{3}{d}^{2}ef}{20} \left ( -{\frac{cx}{2}\sqrt{-{c}^{2}{x}^{2}+1}}+{\frac{\arcsin \left ( cx \right ) }{2}} \right ) }+{c}^{4}{d}^{3}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)^3*(g*x+f)*(a+b*arcsin(c*x)),x)

[Out]

1/c*(a/c^4*(1/5*e^3*g*c^5*x^5+1/4*(3*c*d*e^2*g+c*e^3*f)*c^4*x^4+1/3*(3*c^2*d^2*e*g+3*c^2*d*e^2*f)*c^3*x^3+1/2*
(c^3*d^3*g+3*c^3*d^2*e*f)*c^2*x^2+c^5*d^3*f*x)+b/c^4*(1/5*arcsin(c*x)*e^3*g*c^5*x^5+3/4*arcsin(c*x)*c^5*x^4*d*
e^2*g+1/4*arcsin(c*x)*c^5*x^4*e^3*f+arcsin(c*x)*c^5*x^3*d^2*e*g+arcsin(c*x)*c^5*x^3*d*e^2*f+1/2*arcsin(c*x)*c^
5*x^2*d^3*g+3/2*arcsin(c*x)*c^5*x^2*d^2*e*f+arcsin(c*x)*c^5*d^3*f*x-1/5*e^3*g*(-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))-1/20*(15*c*d*e^2*g+5*c*e^3*f)*(-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))-1/20*(20*c^2*d^2*e*g+20*c^2*d*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))-1/20*(10*c^3*d^3*g+30*c^3*d^2*e*f)*(-1/2*c*x*(-c^2*x^2+1)^(1/2)+1/2*arc
sin(c*x))+c^4*d^3*f*(-c^2*x^2+1)^(1/2)))

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Maxima [A]  time = 1.74747, size = 778, normalized size = 2.22 \begin{align*} \frac{1}{5} \, a e^{3} g x^{5} + \frac{1}{4} \, a e^{3} f x^{4} + \frac{3}{4} \, a d e^{2} g x^{4} + a d e^{2} f x^{3} + a d^{2} e g x^{3} + \frac{3}{2} \, a d^{2} e f x^{2} + \frac{1}{2} \, a d^{3} g x^{2} + \frac{3}{4} \,{\left (2 \, x^{2} \arcsin \left (c x\right ) + c{\left (\frac{\sqrt{-c^{2} x^{2} + 1} x}{c^{2}} - \frac{\arcsin \left (\frac{c^{2} x}{\sqrt{c^{2}}}\right )}{\sqrt{c^{2}} c^{2}}\right )}\right )} b d^{2} e f + \frac{1}{3} \,{\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^{2} f + \frac{1}{32} \,{\left (8 \, x^{4} \arcsin \left (c x\right ) +{\left (\frac{2 \, \sqrt{-c^{2} x^{2} + 1} x^{3}}{c^{2}} + \frac{3 \, \sqrt{-c^{2} x^{2} + 1} x}{c^{4}} - \frac{3 \, \arcsin \left (\frac{c^{2} x}{\sqrt{c^{2}}}\right )}{\sqrt{c^{2}} c^{4}}\right )} c\right )} b e^{3} f + \frac{1}{4} \,{\left (2 \, x^{2} \arcsin \left (c x\right ) + c{\left (\frac{\sqrt{-c^{2} x^{2} + 1} x}{c^{2}} - \frac{\arcsin \left (\frac{c^{2} x}{\sqrt{c^{2}}}\right )}{\sqrt{c^{2}} c^{2}}\right )}\right )} b d^{3} g + \frac{1}{3} \,{\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^{2} e g + \frac{3}{32} \,{\left (8 \, x^{4} \arcsin \left (c x\right ) +{\left (\frac{2 \, \sqrt{-c^{2} x^{2} + 1} x^{3}}{c^{2}} + \frac{3 \, \sqrt{-c^{2} x^{2} + 1} x}{c^{4}} - \frac{3 \, \arcsin \left (\frac{c^{2} x}{\sqrt{c^{2}}}\right )}{\sqrt{c^{2}} c^{4}}\right )} c\right )} b d e^{2} g + \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^{3} g + a d^{3} f x + \frac{{\left (c x \arcsin \left (c x\right ) + \sqrt{-c^{2} x^{2} + 1}\right )} b d^{3} f}{c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.85592, size = 980, normalized size = 2.79 \begin{align*} \frac{480 \, a c^{5} e^{3} g x^{5} + 2400 \, a c^{5} d^{3} f x + 600 \,{\left (a c^{5} e^{3} f + 3 \, a c^{5} d e^{2} g\right )} x^{4} + 2400 \,{\left (a c^{5} d e^{2} f + a c^{5} d^{2} e g\right )} x^{3} + 1200 \,{\left (3 \, a c^{5} d^{2} e f + a c^{5} d^{3} g\right )} x^{2} + 15 \,{\left (32 \, b c^{5} e^{3} g x^{5} + 160 \, b c^{5} d^{3} f x + 40 \,{\left (b c^{5} e^{3} f + 3 \, b c^{5} d e^{2} g\right )} x^{4} + 160 \,{\left (b c^{5} d e^{2} f + b c^{5} d^{2} e g\right )} x^{3} + 80 \,{\left (3 \, b c^{5} d^{2} e f + b c^{5} d^{3} g\right )} x^{2} - 15 \,{\left (8 \, b c^{3} d^{2} e + b c e^{3}\right )} f - 5 \,{\left (8 \, b c^{3} d^{3} + 9 \, b c d e^{2}\right )} g\right )} \arcsin \left (c x\right ) +{\left (96 \, b c^{4} e^{3} g x^{4} + 150 \,{\left (b c^{4} e^{3} f + 3 \, b c^{4} d e^{2} g\right )} x^{3} + 32 \,{\left (25 \, b c^{4} d e^{2} f +{\left (25 \, b c^{4} d^{2} e + 4 \, b c^{2} e^{3}\right )} g\right )} x^{2} + 800 \,{\left (3 \, b c^{4} d^{3} + 2 \, b c^{2} d e^{2}\right )} f + 64 \,{\left (25 \, b c^{2} d^{2} e + 4 \, b e^{3}\right )} g + 75 \,{\left (3 \,{\left (8 \, b c^{4} d^{2} e + b c^{2} e^{3}\right )} f +{\left (8 \, b c^{4} d^{3} + 9 \, b c^{2} d e^{2}\right )} g\right )} x\right )} \sqrt{-c^{2} x^{2} + 1}}{2400 \, c^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2400*(480*a*c^5*e^3*g*x^5 + 2400*a*c^5*d^3*f*x + 600*(a*c^5*e^3*f + 3*a*c^5*d*e^2*g)*x^4 + 2400*(a*c^5*d*e^2
*f + a*c^5*d^2*e*g)*x^3 + 1200*(3*a*c^5*d^2*e*f + a*c^5*d^3*g)*x^2 + 15*(32*b*c^5*e^3*g*x^5 + 160*b*c^5*d^3*f*
x + 40*(b*c^5*e^3*f + 3*b*c^5*d*e^2*g)*x^4 + 160*(b*c^5*d*e^2*f + b*c^5*d^2*e*g)*x^3 + 80*(3*b*c^5*d^2*e*f + b
*c^5*d^3*g)*x^2 - 15*(8*b*c^3*d^2*e + b*c*e^3)*f - 5*(8*b*c^3*d^3 + 9*b*c*d*e^2)*g)*arcsin(c*x) + (96*b*c^4*e^
3*g*x^4 + 150*(b*c^4*e^3*f + 3*b*c^4*d*e^2*g)*x^3 + 32*(25*b*c^4*d*e^2*f + (25*b*c^4*d^2*e + 4*b*c^2*e^3)*g)*x
^2 + 800*(3*b*c^4*d^3 + 2*b*c^2*d*e^2)*f + 64*(25*b*c^2*d^2*e + 4*b*e^3)*g + 75*(3*(8*b*c^4*d^2*e + b*c^2*e^3)
*f + (8*b*c^4*d^3 + 9*b*c^2*d*e^2)*g)*x)*sqrt(-c^2*x^2 + 1))/c^5

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Sympy [A]  time = 5.15809, size = 770, normalized size = 2.19 \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)**3*(g*x+f)*(a+b*asin(c*x)),x)

[Out]

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

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Giac [B]  time = 1.29463, size = 1118, normalized size = 3.19 \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)^3*(g*x+f)*(a+b*arcsin(c*x)),x, algorithm="giac")

[Out]

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