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

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

[Out]

-1/32*b*(8*c^2*d^2*(d*g+3*e*f)+3*e^2*(3*d*g+e*f))*arcsin(c*x)/c^4+d^3*f*x*(a+b*arcsin(c*x))+1/2*d^2*(d*g+3*e*f
)*x^2*(a+b*arcsin(c*x))+d*e*(d*g+e*f)*x^3*(a+b*arcsin(c*x))+1/4*e^2*(3*d*g+e*f)*x^4*(a+b*arcsin(c*x))+1/5*e^3*
g*x^5*(a+b*arcsin(c*x))+1/75*b*e*(4*e^2*g+25*c^2*d*(d*g+e*f))*x^2*(-c^2*x^2+1)^(1/2)/c^3+1/16*b*e^2*(3*d*g+e*f
)*x^3*(-c^2*x^2+1)^(1/2)/c+1/25*b*e^3*g*x^4*(-c^2*x^2+1)^(1/2)/c+1/2400*b*(2400*c^4*d^3*f+256*e^3*g+1600*c^2*d
*e*(d*g+e*f)+75*c^2*(8*c^2*d^2*(d*g+3*e*f)+3*e^2*(3*d*g+e*f))*x)*(-c^2*x^2+1)^(1/2)/c^5

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Rubi [A]  time = 0.99, antiderivative size = 351, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, 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 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)^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*}

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Mathematica [A]  time = 0.43, size = 305, normalized size = 0.87 \[ \frac {120 a c^5 x \left (10 d^3 (2 f+g x)+10 d^2 e x (3 f+2 g x)+5 d e^2 x^2 (4 f+3 g x)+e^3 x^3 (5 f+4 g x)\right )+b \sqrt {1-c^2 x^2} \left (2 c^4 \left (300 d^3 (4 f+g x)+100 d^2 e x (9 f+4 g x)+25 d e^2 x^2 (16 f+9 g x)+3 e^3 x^3 (25 f+16 g x)\right )+c^2 e \left (1600 d^2 g+25 d e (64 f+27 g x)+e^2 x (225 f+128 g x)\right )+256 e^3 g\right )+15 b c \sin ^{-1}(c x) \left (8 c^4 x \left (10 d^3 (2 f+g x)+10 d^2 e x (3 f+2 g x)+5 d e^2 x^2 (4 f+3 g x)+e^3 x^3 (5 f+4 g x)\right )-40 c^2 d^2 (d g+3 e f)-15 e^2 (3 d g+e f)\right )}{2400 c^5} \]

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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fricas [A]  time = 0.76, size = 441, normalized size = 1.26 \[ \frac {480 \, a c^{5} e^{3} g x^{5} + 2400 \, a c^{5} d^{3} f x + 600 \, {\left (a c^{5} e^{3} f + 3 \, a c^{5} d e^{2} g\right )} x^{4} + 2400 \, {\left (a c^{5} d e^{2} f + a c^{5} d^{2} e g\right )} x^{3} + 1200 \, {\left (3 \, a c^{5} d^{2} e f + a c^{5} d^{3} g\right )} x^{2} + 15 \, {\left (32 \, b c^{5} e^{3} g x^{5} + 160 \, b c^{5} d^{3} f x + 40 \, {\left (b c^{5} e^{3} f + 3 \, b c^{5} d e^{2} g\right )} x^{4} + 160 \, {\left (b c^{5} d e^{2} f + b c^{5} d^{2} e g\right )} x^{3} + 80 \, {\left (3 \, b c^{5} d^{2} e f + b c^{5} d^{3} g\right )} x^{2} - 15 \, {\left (8 \, b c^{3} d^{2} e + b c e^{3}\right )} f - 5 \, {\left (8 \, b c^{3} d^{3} + 9 \, b c d e^{2}\right )} g\right )} \arcsin \left (c x\right ) + {\left (96 \, b c^{4} e^{3} g x^{4} + 150 \, {\left (b c^{4} e^{3} f + 3 \, b c^{4} d e^{2} g\right )} x^{3} + 32 \, {\left (25 \, b c^{4} d e^{2} f + {\left (25 \, b c^{4} d^{2} e + 4 \, b c^{2} e^{3}\right )} g\right )} x^{2} + 800 \, {\left (3 \, b c^{4} d^{3} + 2 \, b c^{2} d e^{2}\right )} f + 64 \, {\left (25 \, b c^{2} d^{2} e + 4 \, b e^{3}\right )} g + 75 \, {\left (3 \, {\left (8 \, b c^{4} d^{2} e + b c^{2} e^{3}\right )} f + {\left (8 \, b c^{4} d^{3} + 9 \, b c^{2} d e^{2}\right )} g\right )} x\right )} \sqrt {-c^{2} x^{2} + 1}}{2400 \, c^{5}} \]

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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giac [B]  time = 0.42, size = 769, normalized size = 2.19 \[ \frac {1}{5} \, a g x^{5} e^{3} + \frac {3}{4} \, a d g x^{4} e^{2} + a d^{2} g x^{3} e + b d^{3} f x \arcsin \left (c x\right ) + \frac {1}{4} \, a f x^{4} e^{3} + a d f x^{3} e^{2} + a d^{3} f x + \frac {{\left (c^{2} x^{2} - 1\right )} b d^{2} g x \arcsin \left (c x\right ) e}{c^{2}} + \frac {\sqrt {-c^{2} x^{2} + 1} b d^{3} g x}{4 \, c} + \frac {3 \, \sqrt {-c^{2} x^{2} + 1} b d^{2} f x e}{4 \, c} + \frac {{\left (c^{2} x^{2} - 1\right )} b d^{3} g \arcsin \left (c x\right )}{2 \, c^{2}} + \frac {{\left (c^{2} x^{2} - 1\right )} b d f x \arcsin \left (c x\right ) e^{2}}{c^{2}} + \frac {3 \, {\left (c^{2} x^{2} - 1\right )} b d^{2} f \arcsin \left (c x\right ) e}{2 \, c^{2}} + \frac {b d^{2} g x \arcsin \left (c x\right ) e}{c^{2}} + \frac {\sqrt {-c^{2} x^{2} + 1} b d^{3} f}{c} + \frac {{\left (c^{2} x^{2} - 1\right )} a d^{3} g}{2 \, c^{2}} + \frac {b d^{3} g \arcsin \left (c x\right )}{4 \, c^{2}} + \frac {b d f x \arcsin \left (c x\right ) e^{2}}{c^{2}} + \frac {3 \, {\left (c^{2} x^{2} - 1\right )} a d^{2} f e}{2 \, c^{2}} + \frac {3 \, b d^{2} f \arcsin \left (c x\right ) e}{4 \, c^{2}} - \frac {3 \, {\left (-c^{2} x^{2} + 1\right )}^{\frac {3}{2}} b d g x e^{2}}{16 \, c^{3}} - \frac {{\left (-c^{2} x^{2} + 1\right )}^{\frac {3}{2}} b d^{2} g e}{3 \, c^{3}} + \frac {{\left (c^{2} x^{2} - 1\right )}^{2} b g x \arcsin \left (c x\right ) e^{3}}{5 \, c^{4}} + \frac {3 \, {\left (c^{2} x^{2} - 1\right )}^{2} b d g \arcsin \left (c x\right ) e^{2}}{4 \, c^{4}} - \frac {{\left (-c^{2} x^{2} + 1\right )}^{\frac {3}{2}} b f x e^{3}}{16 \, c^{3}} - \frac {{\left (-c^{2} x^{2} + 1\right )}^{\frac {3}{2}} b d f e^{2}}{3 \, c^{3}} + \frac {15 \, \sqrt {-c^{2} x^{2} + 1} b d g x e^{2}}{32 \, c^{3}} + \frac {\sqrt {-c^{2} x^{2} + 1} b d^{2} g e}{c^{3}} + \frac {{\left (c^{2} x^{2} - 1\right )}^{2} b f \arcsin \left (c x\right ) e^{3}}{4 \, c^{4}} + \frac {2 \, {\left (c^{2} x^{2} - 1\right )} b g x \arcsin \left (c x\right ) e^{3}}{5 \, c^{4}} + \frac {3 \, {\left (c^{2} x^{2} - 1\right )} b d g \arcsin \left (c x\right ) e^{2}}{2 \, c^{4}} + \frac {5 \, \sqrt {-c^{2} x^{2} + 1} b f x e^{3}}{32 \, c^{3}} + \frac {\sqrt {-c^{2} x^{2} + 1} b d f e^{2}}{c^{3}} + \frac {{\left (c^{2} x^{2} - 1\right )} b f \arcsin \left (c x\right ) e^{3}}{2 \, c^{4}} + \frac {b g x \arcsin \left (c x\right ) e^{3}}{5 \, c^{4}} + \frac {15 \, b d g \arcsin \left (c x\right ) e^{2}}{32 \, c^{4}} + \frac {{\left (c^{2} x^{2} - 1\right )}^{2} \sqrt {-c^{2} x^{2} + 1} b g e^{3}}{25 \, c^{5}} + \frac {5 \, b f \arcsin \left (c x\right ) e^{3}}{32 \, c^{4}} - \frac {2 \, {\left (-c^{2} x^{2} + 1\right )}^{\frac {3}{2}} b g e^{3}}{15 \, c^{5}} + \frac {\sqrt {-c^{2} x^{2} + 1} b g e^{3}}{5 \, c^{5}} \]

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

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maple [A]  time = 0.02, size = 490, normalized size = 1.40 \[ \frac {\frac {a \left (\frac {e^{3} g \,c^{5} x^{5}}{5}+\frac {\left (3 d c \,e^{2} g +e^{3} c f \right ) c^{4} x^{4}}{4}+\frac {\left (3 c^{2} d^{2} e g +3 d \,c^{2} e^{2} f \right ) c^{3} x^{3}}{3}+\frac {\left (c^{3} d^{3} g +3 c^{3} d^{2} e f \right ) c^{2} x^{2}}{2}+c^{5} d^{3} f x \right )}{c^{4}}+\frac {b \left (\frac {\arcsin \left (c x \right ) e^{3} g \,c^{5} x^{5}}{5}+\frac {3 \arcsin \left (c x \right ) c^{5} x^{4} d \,e^{2} g}{4}+\frac {\arcsin \left (c x \right ) c^{5} x^{4} e^{3} f}{4}+\arcsin \left (c x \right ) c^{5} x^{3} d^{2} e g +\arcsin \left (c x \right ) c^{5} x^{3} d \,e^{2} f +\frac {\arcsin \left (c x \right ) c^{5} x^{2} d^{3} g}{2}+\frac {3 \arcsin \left (c x \right ) c^{5} x^{2} d^{2} e f}{2}+\arcsin \left (c x \right ) c^{5} d^{3} f x -\frac {e^{3} g \left (-\frac {c^{4} x^{4} \sqrt {-c^{2} x^{2}+1}}{5}-\frac {4 c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{15}-\frac {8 \sqrt {-c^{2} x^{2}+1}}{15}\right )}{5}-\frac {\left (15 d c \,e^{2} g +5 e^{3} c f \right ) \left (-\frac {c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}}{4}-\frac {3 c x \sqrt {-c^{2} x^{2}+1}}{8}+\frac {3 \arcsin \left (c x \right )}{8}\right )}{20}-\frac {\left (20 c^{2} d^{2} e g +20 d \,c^{2} e^{2} f \right ) \left (-\frac {c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{3}-\frac {2 \sqrt {-c^{2} x^{2}+1}}{3}\right )}{20}-\frac {\left (10 c^{3} d^{3} g +30 c^{3} d^{2} e f \right ) \left (-\frac {c x \sqrt {-c^{2} x^{2}+1}}{2}+\frac {\arcsin \left (c x \right )}{2}\right )}{20}+c^{4} d^{3} f \sqrt {-c^{2} x^{2}+1}\right )}{c^{4}}}{c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^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 = 1.25, size = 528, normalized size = 1.50 \[ \frac {1}{5} \, a e^{3} g x^{5} + \frac {1}{4} \, a e^{3} f x^{4} + \frac {3}{4} \, a d e^{2} g x^{4} + a d e^{2} f x^{3} + a d^{2} e g x^{3} + \frac {3}{2} \, a d^{2} e f x^{2} + \frac {1}{2} \, a d^{3} g x^{2} + \frac {3}{4} \, {\left (2 \, x^{2} \arcsin \left (c x\right ) + c {\left (\frac {\sqrt {-c^{2} x^{2} + 1} x}{c^{2}} - \frac {\arcsin \left (c x\right )}{c^{3}}\right )}\right )} b d^{2} e f + \frac {1}{3} \, {\left (3 \, x^{3} \arcsin \left (c x\right ) + c {\left (\frac {\sqrt {-c^{2} x^{2} + 1} x^{2}}{c^{2}} + \frac {2 \, \sqrt {-c^{2} x^{2} + 1}}{c^{4}}\right )}\right )} b d e^{2} f + \frac {1}{32} \, {\left (8 \, x^{4} \arcsin \left (c x\right ) + {\left (\frac {2 \, \sqrt {-c^{2} x^{2} + 1} x^{3}}{c^{2}} + \frac {3 \, \sqrt {-c^{2} x^{2} + 1} x}{c^{4}} - \frac {3 \, \arcsin \left (c x\right )}{c^{5}}\right )} c\right )} b e^{3} f + \frac {1}{4} \, {\left (2 \, x^{2} \arcsin \left (c x\right ) + c {\left (\frac {\sqrt {-c^{2} x^{2} + 1} x}{c^{2}} - \frac {\arcsin \left (c x\right )}{c^{3}}\right )}\right )} b d^{3} g + \frac {1}{3} \, {\left (3 \, x^{3} \arcsin \left (c x\right ) + c {\left (\frac {\sqrt {-c^{2} x^{2} + 1} x^{2}}{c^{2}} + \frac {2 \, \sqrt {-c^{2} x^{2} + 1}}{c^{4}}\right )}\right )} b d^{2} e g + \frac {3}{32} \, {\left (8 \, x^{4} \arcsin \left (c x\right ) + {\left (\frac {2 \, \sqrt {-c^{2} x^{2} + 1} x^{3}}{c^{2}} + \frac {3 \, \sqrt {-c^{2} x^{2} + 1} x}{c^{4}} - \frac {3 \, \arcsin \left (c x\right )}{c^{5}}\right )} c\right )} b d e^{2} g + \frac {1}{75} \, {\left (15 \, x^{5} \arcsin \left (c x\right ) + {\left (\frac {3 \, \sqrt {-c^{2} x^{2} + 1} x^{4}}{c^{2}} + \frac {4 \, \sqrt {-c^{2} x^{2} + 1} x^{2}}{c^{4}} + \frac {8 \, \sqrt {-c^{2} x^{2} + 1}}{c^{6}}\right )} c\right )} b e^{3} g + a d^{3} f x + \frac {{\left (c x \arcsin \left (c x\right ) + \sqrt {-c^{2} x^{2} + 1}\right )} b d^{3} f}{c} \]

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*x)/c^3))*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*x)/c^5)*c)*b*e^3*f + 1/4*(2*x^2*arc
sin(c*x) + c*(sqrt(-c^2*x^2 + 1)*x/c^2 - arcsin(c*x)/c^3))*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*arcsin(c*x)/c^5)*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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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \left (f+g\,x\right )\,\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )\,{\left (d+e\,x\right )}^3 \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 3.65, size = 770, normalized size = 2.19 \[ \begin {cases} a d^{3} f x + \frac {a d^{3} g x^{2}}{2} + \frac {3 a d^{2} e f x^{2}}{2} + a d^{2} e g x^{3} + a d e^{2} f x^{3} + \frac {3 a d e^{2} g x^{4}}{4} + \frac {a e^{3} f x^{4}}{4} + \frac {a e^{3} g x^{5}}{5} + b d^{3} f x \operatorname {asin}{\left (c x \right )} + \frac {b d^{3} g x^{2} \operatorname {asin}{\left (c x \right )}}{2} + \frac {3 b d^{2} e f x^{2} \operatorname {asin}{\left (c x \right )}}{2} + b d^{2} e g x^{3} \operatorname {asin}{\left (c x \right )} + b d e^{2} f x^{3} \operatorname {asin}{\left (c x \right )} + \frac {3 b d e^{2} g x^{4} \operatorname {asin}{\left (c x \right )}}{4} + \frac {b e^{3} f x^{4} \operatorname {asin}{\left (c x \right )}}{4} + \frac {b e^{3} g x^{5} \operatorname {asin}{\left (c x \right )}}{5} + \frac {b d^{3} f \sqrt {- c^{2} x^{2} + 1}}{c} + \frac {b d^{3} g x \sqrt {- c^{2} x^{2} + 1}}{4 c} + \frac {3 b d^{2} e f x \sqrt {- c^{2} x^{2} + 1}}{4 c} + \frac {b d^{2} e g x^{2} \sqrt {- c^{2} x^{2} + 1}}{3 c} + \frac {b d e^{2} f x^{2} \sqrt {- c^{2} x^{2} + 1}}{3 c} + \frac {3 b d e^{2} g x^{3} \sqrt {- c^{2} x^{2} + 1}}{16 c} + \frac {b e^{3} f x^{3} \sqrt {- c^{2} x^{2} + 1}}{16 c} + \frac {b e^{3} g x^{4} \sqrt {- c^{2} x^{2} + 1}}{25 c} - \frac {b d^{3} g \operatorname {asin}{\left (c x \right )}}{4 c^{2}} - \frac {3 b d^{2} e f \operatorname {asin}{\left (c x \right )}}{4 c^{2}} + \frac {2 b d^{2} e g \sqrt {- c^{2} x^{2} + 1}}{3 c^{3}} + \frac {2 b d e^{2} f \sqrt {- c^{2} x^{2} + 1}}{3 c^{3}} + \frac {9 b d e^{2} g x \sqrt {- c^{2} x^{2} + 1}}{32 c^{3}} + \frac {3 b e^{3} f x \sqrt {- c^{2} x^{2} + 1}}{32 c^{3}} + \frac {4 b e^{3} g x^{2} \sqrt {- c^{2} x^{2} + 1}}{75 c^{3}} - \frac {9 b d e^{2} g \operatorname {asin}{\left (c x \right )}}{32 c^{4}} - \frac {3 b e^{3} f \operatorname {asin}{\left (c x \right )}}{32 c^{4}} + \frac {8 b e^{3} g \sqrt {- c^{2} x^{2} + 1}}{75 c^{5}} & \text {for}\: c \neq 0 \\a \left (d^{3} f x + \frac {d^{3} g x^{2}}{2} + \frac {3 d^{2} e f x^{2}}{2} + d^{2} e g x^{3} + d e^{2} f x^{3} + \frac {3 d e^{2} g x^{4}}{4} + \frac {e^{3} f x^{4}}{4} + \frac {e^{3} g x^{5}}{5}\right ) & \text {otherwise} \end {cases} \]

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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