3.1.23 \(\int (d-c^2 d x^2)^3 (a+b \text {ArcSin}(c x)) \, dx\) [23]

Optimal. Leaf size=175 \[ \frac {16 b d^3 \sqrt {1-c^2 x^2}}{35 c}+\frac {8 b d^3 \left (1-c^2 x^2\right )^{3/2}}{105 c}+\frac {6 b d^3 \left (1-c^2 x^2\right )^{5/2}}{175 c}+\frac {b d^3 \left (1-c^2 x^2\right )^{7/2}}{49 c}+d^3 x (a+b \text {ArcSin}(c x))-c^2 d^3 x^3 (a+b \text {ArcSin}(c x))+\frac {3}{5} c^4 d^3 x^5 (a+b \text {ArcSin}(c x))-\frac {1}{7} c^6 d^3 x^7 (a+b \text {ArcSin}(c x)) \]

[Out]

8/105*b*d^3*(-c^2*x^2+1)^(3/2)/c+6/175*b*d^3*(-c^2*x^2+1)^(5/2)/c+1/49*b*d^3*(-c^2*x^2+1)^(7/2)/c+d^3*x*(a+b*a
rcsin(c*x))-c^2*d^3*x^3*(a+b*arcsin(c*x))+3/5*c^4*d^3*x^5*(a+b*arcsin(c*x))-1/7*c^6*d^3*x^7*(a+b*arcsin(c*x))+
16/35*b*d^3*(-c^2*x^2+1)^(1/2)/c

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Rubi [A]
time = 0.12, antiderivative size = 175, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {200, 4739, 12, 1813, 1864} \begin {gather*} -\frac {1}{7} c^6 d^3 x^7 (a+b \text {ArcSin}(c x))+\frac {3}{5} c^4 d^3 x^5 (a+b \text {ArcSin}(c x))-c^2 d^3 x^3 (a+b \text {ArcSin}(c x))+d^3 x (a+b \text {ArcSin}(c x))+\frac {b d^3 \left (1-c^2 x^2\right )^{7/2}}{49 c}+\frac {6 b d^3 \left (1-c^2 x^2\right )^{5/2}}{175 c}+\frac {8 b d^3 \left (1-c^2 x^2\right )^{3/2}}{105 c}+\frac {16 b d^3 \sqrt {1-c^2 x^2}}{35 c} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(d - c^2*d*x^2)^3*(a + b*ArcSin[c*x]),x]

[Out]

(16*b*d^3*Sqrt[1 - c^2*x^2])/(35*c) + (8*b*d^3*(1 - c^2*x^2)^(3/2))/(105*c) + (6*b*d^3*(1 - c^2*x^2)^(5/2))/(1
75*c) + (b*d^3*(1 - c^2*x^2)^(7/2))/(49*c) + d^3*x*(a + b*ArcSin[c*x]) - c^2*d^3*x^3*(a + b*ArcSin[c*x]) + (3*
c^4*d^3*x^5*(a + b*ArcSin[c*x]))/5 - (c^6*d^3*x^7*(a + b*ArcSin[c*x]))/7

Rule 12

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

Rule 200

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Int[ExpandIntegrand[(a + b*x^n)^p, x], x] /; FreeQ[{a, b}, x]
&& IGtQ[n, 0] && IGtQ[p, 0]

Rule 1813

Int[(Pq_)*(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Dist[1/2, Subst[Int[x^((m - 1)/2)*SubstFor[x^2,
Pq, x]*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, p}, x] && PolyQ[Pq, x^2] && IntegerQ[(m - 1)/2]

Rule 1864

Int[(Pq_)*((a_) + (b_.)*(x_)^(n_.))^(p_.), x_Symbol] :> Int[ExpandIntegrand[Pq*(a + b*x^n)^p, x], x] /; FreeQ[
{a, b, n}, x] && PolyQ[Pq, x] && (IGtQ[p, 0] || EqQ[n, 1])

Rule 4739

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> With[{u = IntHide[(d + e*x^2)
^p, x]}, Dist[a + b*ArcSin[c*x], u, x] - Dist[b*c, Int[SimplifyIntegrand[u/Sqrt[1 - c^2*x^2], x], x], x]] /; F
reeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IGtQ[p, 0]

Rubi steps

\begin {align*} \int \left (d-c^2 d x^2\right )^3 \left (a+b \sin ^{-1}(c x)\right ) \, dx &=d^3 x \left (a+b \sin ^{-1}(c x)\right )-c^2 d^3 x^3 \left (a+b \sin ^{-1}(c x)\right )+\frac {3}{5} c^4 d^3 x^5 \left (a+b \sin ^{-1}(c x)\right )-\frac {1}{7} c^6 d^3 x^7 \left (a+b \sin ^{-1}(c x)\right )-(b c) \int \frac {d^3 x \left (35-35 c^2 x^2+21 c^4 x^4-5 c^6 x^6\right )}{35 \sqrt {1-c^2 x^2}} \, dx\\ &=d^3 x \left (a+b \sin ^{-1}(c x)\right )-c^2 d^3 x^3 \left (a+b \sin ^{-1}(c x)\right )+\frac {3}{5} c^4 d^3 x^5 \left (a+b \sin ^{-1}(c x)\right )-\frac {1}{7} c^6 d^3 x^7 \left (a+b \sin ^{-1}(c x)\right )-\frac {1}{35} \left (b c d^3\right ) \int \frac {x \left (35-35 c^2 x^2+21 c^4 x^4-5 c^6 x^6\right )}{\sqrt {1-c^2 x^2}} \, dx\\ &=d^3 x \left (a+b \sin ^{-1}(c x)\right )-c^2 d^3 x^3 \left (a+b \sin ^{-1}(c x)\right )+\frac {3}{5} c^4 d^3 x^5 \left (a+b \sin ^{-1}(c x)\right )-\frac {1}{7} c^6 d^3 x^7 \left (a+b \sin ^{-1}(c x)\right )-\frac {1}{70} \left (b c d^3\right ) \text {Subst}\left (\int \frac {35-35 c^2 x+21 c^4 x^2-5 c^6 x^3}{\sqrt {1-c^2 x}} \, dx,x,x^2\right )\\ &=d^3 x \left (a+b \sin ^{-1}(c x)\right )-c^2 d^3 x^3 \left (a+b \sin ^{-1}(c x)\right )+\frac {3}{5} c^4 d^3 x^5 \left (a+b \sin ^{-1}(c x)\right )-\frac {1}{7} c^6 d^3 x^7 \left (a+b \sin ^{-1}(c x)\right )-\frac {1}{70} \left (b c d^3\right ) \text {Subst}\left (\int \left (\frac {16}{\sqrt {1-c^2 x}}+8 \sqrt {1-c^2 x}+6 \left (1-c^2 x\right )^{3/2}+5 \left (1-c^2 x\right )^{5/2}\right ) \, dx,x,x^2\right )\\ &=\frac {16 b d^3 \sqrt {1-c^2 x^2}}{35 c}+\frac {8 b d^3 \left (1-c^2 x^2\right )^{3/2}}{105 c}+\frac {6 b d^3 \left (1-c^2 x^2\right )^{5/2}}{175 c}+\frac {b d^3 \left (1-c^2 x^2\right )^{7/2}}{49 c}+d^3 x \left (a+b \sin ^{-1}(c x)\right )-c^2 d^3 x^3 \left (a+b \sin ^{-1}(c x)\right )+\frac {3}{5} c^4 d^3 x^5 \left (a+b \sin ^{-1}(c x)\right )-\frac {1}{7} c^6 d^3 x^7 \left (a+b \sin ^{-1}(c x)\right )\\ \end {align*}

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Mathematica [A]
time = 0.10, size = 119, normalized size = 0.68 \begin {gather*} -\frac {d^3 \left (105 a c x \left (-35+35 c^2 x^2-21 c^4 x^4+5 c^6 x^6\right )+b \sqrt {1-c^2 x^2} \left (-2161+757 c^2 x^2-351 c^4 x^4+75 c^6 x^6\right )+105 b c x \left (-35+35 c^2 x^2-21 c^4 x^4+5 c^6 x^6\right ) \text {ArcSin}(c x)\right )}{3675 c} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(d - c^2*d*x^2)^3*(a + b*ArcSin[c*x]),x]

[Out]

-1/3675*(d^3*(105*a*c*x*(-35 + 35*c^2*x^2 - 21*c^4*x^4 + 5*c^6*x^6) + b*Sqrt[1 - c^2*x^2]*(-2161 + 757*c^2*x^2
- 351*c^4*x^4 + 75*c^6*x^6) + 105*b*c*x*(-35 + 35*c^2*x^2 - 21*c^4*x^4 + 5*c^6*x^6)*ArcSin[c*x]))/c

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Maple [A]
time = 0.07, size = 164, normalized size = 0.94

method	result	size
derivativedivides	\(\frac {-d^{3} a \left (\frac {1}{7} c^{7} x^{7}-\frac {3}{5} c^{5} x^{5}+c^{3} x^{3}-c x \right )-d^{3} b \left (\frac {\arcsin \left (c x \right ) c^{7} x^{7}}{7}-\frac {3 \arcsin \left (c x \right ) c^{5} x^{5}}{5}+c^{3} x^{3} \arcsin \left (c x \right )-c x \arcsin \left (c x \right )+\frac {c^{6} x^{6} \sqrt {-c^{2} x^{2}+1}}{49}-\frac {117 c^{4} x^{4} \sqrt {-c^{2} x^{2}+1}}{1225}+\frac {757 c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{3675}-\frac {2161 \sqrt {-c^{2} x^{2}+1}}{3675}\right )}{c}\)	\(164\)
default	\(\frac {-d^{3} a \left (\frac {1}{7} c^{7} x^{7}-\frac {3}{5} c^{5} x^{5}+c^{3} x^{3}-c x \right )-d^{3} b \left (\frac {\arcsin \left (c x \right ) c^{7} x^{7}}{7}-\frac {3 \arcsin \left (c x \right ) c^{5} x^{5}}{5}+c^{3} x^{3} \arcsin \left (c x \right )-c x \arcsin \left (c x \right )+\frac {c^{6} x^{6} \sqrt {-c^{2} x^{2}+1}}{49}-\frac {117 c^{4} x^{4} \sqrt {-c^{2} x^{2}+1}}{1225}+\frac {757 c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{3675}-\frac {2161 \sqrt {-c^{2} x^{2}+1}}{3675}\right )}{c}\)	\(164\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-c^2*d*x^2+d)^3*(a+b*arcsin(c*x)),x,method=_RETURNVERBOSE)

[Out]

1/c*(-d^3*a*(1/7*c^7*x^7-3/5*c^5*x^5+c^3*x^3-c*x)-d^3*b*(1/7*arcsin(c*x)*c^7*x^7-3/5*arcsin(c*x)*c^5*x^5+c^3*x
^3*arcsin(c*x)-c*x*arcsin(c*x)+1/49*c^6*x^6*(-c^2*x^2+1)^(1/2)-117/1225*c^4*x^4*(-c^2*x^2+1)^(1/2)+757/3675*c^
2*x^2*(-c^2*x^2+1)^(1/2)-2161/3675*(-c^2*x^2+1)^(1/2)))

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Maxima [A]
time = 0.49, size = 307, normalized size = 1.75 \begin {gather*} -\frac {1}{7} \, a c^{6} d^{3} x^{7} + \frac {3}{5} \, a c^{4} d^{3} x^{5} - \frac {1}{245} \, {\left (35 \, x^{7} \arcsin \left (c x\right ) + {\left (\frac {5 \, \sqrt {-c^{2} x^{2} + 1} x^{6}}{c^{2}} + \frac {6 \, \sqrt {-c^{2} x^{2} + 1} x^{4}}{c^{4}} + \frac {8 \, \sqrt {-c^{2} x^{2} + 1} x^{2}}{c^{6}} + \frac {16 \, \sqrt {-c^{2} x^{2} + 1}}{c^{8}}\right )} c\right )} b c^{6} d^{3} + \frac {1}{25} \, {\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 c^{4} d^{3} - a c^{2} d^{3} x^{3} - \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 c^{2} d^{3} + a d^{3} x + \frac {{\left (c x \arcsin \left (c x\right ) + \sqrt {-c^{2} x^{2} + 1}\right )} b d^{3}}{c} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-c^2*d*x^2+d)^3*(a+b*arcsin(c*x)),x, algorithm="maxima")

[Out]

-1/7*a*c^6*d^3*x^7 + 3/5*a*c^4*d^3*x^5 - 1/245*(35*x^7*arcsin(c*x) + (5*sqrt(-c^2*x^2 + 1)*x^6/c^2 + 6*sqrt(-c
^2*x^2 + 1)*x^4/c^4 + 8*sqrt(-c^2*x^2 + 1)*x^2/c^6 + 16*sqrt(-c^2*x^2 + 1)/c^8)*c)*b*c^6*d^3 + 1/25*(15*x^5*ar
csin(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*c^4*
d^3 - a*c^2*d^3*x^3 - 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*c^
2*d^3 + a*d^3*x + (c*x*arcsin(c*x) + sqrt(-c^2*x^2 + 1))*b*d^3/c

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Fricas [A]
time = 2.83, size = 157, normalized size = 0.90 \begin {gather*} -\frac {525 \, a c^{7} d^{3} x^{7} - 2205 \, a c^{5} d^{3} x^{5} + 3675 \, a c^{3} d^{3} x^{3} - 3675 \, a c d^{3} x + 105 \, {\left (5 \, b c^{7} d^{3} x^{7} - 21 \, b c^{5} d^{3} x^{5} + 35 \, b c^{3} d^{3} x^{3} - 35 \, b c d^{3} x\right )} \arcsin \left (c x\right ) + {\left (75 \, b c^{6} d^{3} x^{6} - 351 \, b c^{4} d^{3} x^{4} + 757 \, b c^{2} d^{3} x^{2} - 2161 \, b d^{3}\right )} \sqrt {-c^{2} x^{2} + 1}}{3675 \, c} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-c^2*d*x^2+d)^3*(a+b*arcsin(c*x)),x, algorithm="fricas")

[Out]

-1/3675*(525*a*c^7*d^3*x^7 - 2205*a*c^5*d^3*x^5 + 3675*a*c^3*d^3*x^3 - 3675*a*c*d^3*x + 105*(5*b*c^7*d^3*x^7 -
21*b*c^5*d^3*x^5 + 35*b*c^3*d^3*x^3 - 35*b*c*d^3*x)*arcsin(c*x) + (75*b*c^6*d^3*x^6 - 351*b*c^4*d^3*x^4 + 757
*b*c^2*d^3*x^2 - 2161*b*d^3)*sqrt(-c^2*x^2 + 1))/c

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Sympy [A]
time = 0.77, size = 221, normalized size = 1.26 \begin {gather*} \begin {cases} - \frac {a c^{6} d^{3} x^{7}}{7} + \frac {3 a c^{4} d^{3} x^{5}}{5} - a c^{2} d^{3} x^{3} + a d^{3} x - \frac {b c^{6} d^{3} x^{7} \operatorname {asin}{\left (c x \right )}}{7} - \frac {b c^{5} d^{3} x^{6} \sqrt {- c^{2} x^{2} + 1}}{49} + \frac {3 b c^{4} d^{3} x^{5} \operatorname {asin}{\left (c x \right )}}{5} + \frac {117 b c^{3} d^{3} x^{4} \sqrt {- c^{2} x^{2} + 1}}{1225} - b c^{2} d^{3} x^{3} \operatorname {asin}{\left (c x \right )} - \frac {757 b c d^{3} x^{2} \sqrt {- c^{2} x^{2} + 1}}{3675} + b d^{3} x \operatorname {asin}{\left (c x \right )} + \frac {2161 b d^{3} \sqrt {- c^{2} x^{2} + 1}}{3675 c} & \text {for}\: c \neq 0 \\a d^{3} x & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-c**2*d*x**2+d)**3*(a+b*asin(c*x)),x)

[Out]

Piecewise((-a*c**6*d**3*x**7/7 + 3*a*c**4*d**3*x**5/5 - a*c**2*d**3*x**3 + a*d**3*x - b*c**6*d**3*x**7*asin(c*
x)/7 - b*c**5*d**3*x**6*sqrt(-c**2*x**2 + 1)/49 + 3*b*c**4*d**3*x**5*asin(c*x)/5 + 117*b*c**3*d**3*x**4*sqrt(-
c**2*x**2 + 1)/1225 - b*c**2*d**3*x**3*asin(c*x) - 757*b*c*d**3*x**2*sqrt(-c**2*x**2 + 1)/3675 + b*d**3*x*asin
(c*x) + 2161*b*d**3*sqrt(-c**2*x**2 + 1)/(3675*c), Ne(c, 0)), (a*d**3*x, True))

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Giac [A]
time = 0.42, size = 224, normalized size = 1.28 \begin {gather*} -\frac {1}{7} \, a c^{6} d^{3} x^{7} + \frac {3}{5} \, a c^{4} d^{3} x^{5} - a c^{2} d^{3} x^{3} - \frac {1}{7} \, {\left (c^{2} x^{2} - 1\right )}^{3} b d^{3} x \arcsin \left (c x\right ) + \frac {6}{35} \, {\left (c^{2} x^{2} - 1\right )}^{2} b d^{3} x \arcsin \left (c x\right ) - \frac {8}{35} \, {\left (c^{2} x^{2} - 1\right )} b d^{3} x \arcsin \left (c x\right ) - \frac {{\left (c^{2} x^{2} - 1\right )}^{3} \sqrt {-c^{2} x^{2} + 1} b d^{3}}{49 \, c} + \frac {16}{35} \, b d^{3} x \arcsin \left (c x\right ) + \frac {6 \, {\left (c^{2} x^{2} - 1\right )}^{2} \sqrt {-c^{2} x^{2} + 1} b d^{3}}{175 \, c} + a d^{3} x + \frac {8 \, {\left (-c^{2} x^{2} + 1\right )}^{\frac {3}{2}} b d^{3}}{105 \, c} + \frac {16 \, \sqrt {-c^{2} x^{2} + 1} b d^{3}}{35 \, c} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-c^2*d*x^2+d)^3*(a+b*arcsin(c*x)),x, algorithm="giac")

[Out]

-1/7*a*c^6*d^3*x^7 + 3/5*a*c^4*d^3*x^5 - a*c^2*d^3*x^3 - 1/7*(c^2*x^2 - 1)^3*b*d^3*x*arcsin(c*x) + 6/35*(c^2*x
^2 - 1)^2*b*d^3*x*arcsin(c*x) - 8/35*(c^2*x^2 - 1)*b*d^3*x*arcsin(c*x) - 1/49*(c^2*x^2 - 1)^3*sqrt(-c^2*x^2 +
1)*b*d^3/c + 16/35*b*d^3*x*arcsin(c*x) + 6/175*(c^2*x^2 - 1)^2*sqrt(-c^2*x^2 + 1)*b*d^3/c + a*d^3*x + 8/105*(-
c^2*x^2 + 1)^(3/2)*b*d^3/c + 16/35*sqrt(-c^2*x^2 + 1)*b*d^3/c

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \left (a+b\,\mathrm {asin}\left (c\,x\right )\right )\,{\left (d-c^2\,d\,x^2\right )}^3 \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*asin(c*x))*(d - c^2*d*x^2)^3,x)

[Out]

int((a + b*asin(c*x))*(d - c^2*d*x^2)^3, x)

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