3.91 \(\int \frac{(d-c^2 d x^2)^{5/2} (a+b \sin ^{-1}(c x))}{x^{10}} \, dx\)

Optimal. Leaf size=282 \[ -\frac{2 c^2 \left (d-c^2 d x^2\right )^{7/2} \left (a+b \sin ^{-1}(c x)\right )}{63 d x^7}-\frac{\left (d-c^2 d x^2\right )^{7/2} \left (a+b \sin ^{-1}(c x)\right )}{9 d x^9}-\frac{b c^7 d^2 \sqrt{d-c^2 d x^2}}{21 x^2 \sqrt{1-c^2 x^2}}+\frac{b c^5 d^2 \sqrt{d-c^2 d x^2}}{42 x^4 \sqrt{1-c^2 x^2}}-\frac{b c^3 d^2 \sqrt{d-c^2 d x^2}}{189 x^6 \sqrt{1-c^2 x^2}}-\frac{b c d^2 \left (1-c^2 x^2\right )^{7/2} \sqrt{d-c^2 d x^2}}{72 x^8}-\frac{2 b c^9 d^2 \log (x) \sqrt{d-c^2 d x^2}}{63 \sqrt{1-c^2 x^2}} \]

[Out]

-(b*c^3*d^2*Sqrt[d - c^2*d*x^2])/(189*x^6*Sqrt[1 - c^2*x^2]) + (b*c^5*d^2*Sqrt[d - c^2*d*x^2])/(42*x^4*Sqrt[1
- c^2*x^2]) - (b*c^7*d^2*Sqrt[d - c^2*d*x^2])/(21*x^2*Sqrt[1 - c^2*x^2]) - (b*c*d^2*(1 - c^2*x^2)^(7/2)*Sqrt[d
- c^2*d*x^2])/(72*x^8) - ((d - c^2*d*x^2)^(7/2)*(a + b*ArcSin[c*x]))/(9*d*x^9) - (2*c^2*(d - c^2*d*x^2)^(7/2)
*(a + b*ArcSin[c*x]))/(63*d*x^7) - (2*b*c^9*d^2*Sqrt[d - c^2*d*x^2]*Log[x])/(63*Sqrt[1 - c^2*x^2])

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Rubi [A]  time = 0.178928, antiderivative size = 282, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.259, Rules used = {271, 264, 4691, 12, 446, 78, 43} \[ -\frac{2 c^2 \left (d-c^2 d x^2\right )^{7/2} \left (a+b \sin ^{-1}(c x)\right )}{63 d x^7}-\frac{\left (d-c^2 d x^2\right )^{7/2} \left (a+b \sin ^{-1}(c x)\right )}{9 d x^9}-\frac{b c^7 d^2 \sqrt{d-c^2 d x^2}}{21 x^2 \sqrt{1-c^2 x^2}}+\frac{b c^5 d^2 \sqrt{d-c^2 d x^2}}{42 x^4 \sqrt{1-c^2 x^2}}-\frac{b c^3 d^2 \sqrt{d-c^2 d x^2}}{189 x^6 \sqrt{1-c^2 x^2}}-\frac{b c d^2 \left (1-c^2 x^2\right )^{7/2} \sqrt{d-c^2 d x^2}}{72 x^8}-\frac{2 b c^9 d^2 \log (x) \sqrt{d-c^2 d x^2}}{63 \sqrt{1-c^2 x^2}} \]

Antiderivative was successfully verified.

[In]

Int[((d - c^2*d*x^2)^(5/2)*(a + b*ArcSin[c*x]))/x^10,x]

[Out]

-(b*c^3*d^2*Sqrt[d - c^2*d*x^2])/(189*x^6*Sqrt[1 - c^2*x^2]) + (b*c^5*d^2*Sqrt[d - c^2*d*x^2])/(42*x^4*Sqrt[1
- c^2*x^2]) - (b*c^7*d^2*Sqrt[d - c^2*d*x^2])/(21*x^2*Sqrt[1 - c^2*x^2]) - (b*c*d^2*(1 - c^2*x^2)^(7/2)*Sqrt[d
- c^2*d*x^2])/(72*x^8) - ((d - c^2*d*x^2)^(7/2)*(a + b*ArcSin[c*x]))/(9*d*x^9) - (2*c^2*(d - c^2*d*x^2)^(7/2)
*(a + b*ArcSin[c*x]))/(63*d*x^7) - (2*b*c^9*d^2*Sqrt[d - c^2*d*x^2]*Log[x])/(63*Sqrt[1 - c^2*x^2])

Rule 271

Int[(x_)^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(x^(m + 1)*(a + b*x^n)^(p + 1))/(a*(m + 1)), x]
- Dist[(b*(m + n*(p + 1) + 1))/(a*(m + 1)), Int[x^(m + n)*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, m, n, p}, x]
&& ILtQ[Simplify[(m + 1)/n + p + 1], 0] && NeQ[m, -1]

Rule 264

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[((c*x)^(m + 1)*(a + b*x^n)^(p + 1))/(a
*c*(m + 1)), x] /; FreeQ[{a, b, c, m, n, p}, x] && EqQ[(m + 1)/n + p + 1, 0] && NeQ[m, -1]

Rule 4691

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))*(x_)^(m_)*((d_) + (e_.)*(x_)^2)^(p_), x_Symbol] :> With[{u = IntHide[x^
m*(1 - c^2*x^2)^p, x]}, Dist[a + b*ArcSin[c*x], Int[x^m*(d + e*x^2)^p, x], x] - Dist[(b*c*d^(p - 1/2)*Sqrt[d +
e*x^2])/Sqrt[1 - c^2*x^2], Int[SimplifyIntegrand[u/Sqrt[1 - c^2*x^2], x], x], x]] /; FreeQ[{a, b, c, d, e}, x
] && EqQ[c^2*d + e, 0] && IGtQ[p + 1/2, 0] && (IGtQ[(m + 1)/2, 0] || ILtQ[(m + 2*p + 3)/2, 0])

Rule 12

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

Rule 446

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Dist[1/n, Subst[Int
[x^(Simplify[(m + 1)/n] - 1)*(a + b*x)^p*(c + d*x)^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] &&
NeQ[b*c - a*d, 0] && IntegerQ[Simplify[(m + 1)/n]]

Rule 78

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> -Simp[((b*e - a*f
)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/(f*(p + 1)*(c*f - d*e)), x] - Dist[(a*d*f*(n + p + 2) - b*(d*e*(n + 1)
+ c*f*(p + 1)))/(f*(p + 1)*(c*f - d*e)), Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f,
n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || IntegerQ[p] ||  !(IntegerQ[n] ||  !(EqQ[e, 0] ||  !(EqQ[c, 0] || LtQ
[p, n]))))

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int \frac{\left (d-c^2 d x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )}{x^{10}} \, dx &=-\frac{\left (b c d^2 \sqrt{d-c^2 d x^2}\right ) \int \frac{\left (-7-2 c^2 x^2\right ) \left (1-c^2 x^2\right )^3}{63 x^9} \, dx}{\sqrt{1-c^2 x^2}}+\left (a+b \sin ^{-1}(c x)\right ) \int \frac{\left (d-c^2 d x^2\right )^{5/2}}{x^{10}} \, dx\\ &=-\frac{\left (d-c^2 d x^2\right )^{7/2} \left (a+b \sin ^{-1}(c x)\right )}{9 d x^9}-\frac{\left (b c d^2 \sqrt{d-c^2 d x^2}\right ) \int \frac{\left (-7-2 c^2 x^2\right ) \left (1-c^2 x^2\right )^3}{x^9} \, dx}{63 \sqrt{1-c^2 x^2}}+\frac{1}{9} \left (2 c^2 \left (a+b \sin ^{-1}(c x)\right )\right ) \int \frac{\left (d-c^2 d x^2\right )^{5/2}}{x^8} \, dx\\ &=-\frac{\left (d-c^2 d x^2\right )^{7/2} \left (a+b \sin ^{-1}(c x)\right )}{9 d x^9}-\frac{2 c^2 \left (d-c^2 d x^2\right )^{7/2} \left (a+b \sin ^{-1}(c x)\right )}{63 d x^7}-\frac{\left (b c d^2 \sqrt{d-c^2 d x^2}\right ) \operatorname{Subst}\left (\int \frac{\left (-7-2 c^2 x\right ) \left (1-c^2 x\right )^3}{x^5} \, dx,x,x^2\right )}{126 \sqrt{1-c^2 x^2}}\\ &=-\frac{b c d^2 \left (1-c^2 x^2\right )^{7/2} \sqrt{d-c^2 d x^2}}{72 x^8}-\frac{\left (d-c^2 d x^2\right )^{7/2} \left (a+b \sin ^{-1}(c x)\right )}{9 d x^9}-\frac{2 c^2 \left (d-c^2 d x^2\right )^{7/2} \left (a+b \sin ^{-1}(c x)\right )}{63 d x^7}+\frac{\left (b c^3 d^2 \sqrt{d-c^2 d x^2}\right ) \operatorname{Subst}\left (\int \frac{\left (1-c^2 x\right )^3}{x^4} \, dx,x,x^2\right )}{63 \sqrt{1-c^2 x^2}}\\ &=-\frac{b c d^2 \left (1-c^2 x^2\right )^{7/2} \sqrt{d-c^2 d x^2}}{72 x^8}-\frac{\left (d-c^2 d x^2\right )^{7/2} \left (a+b \sin ^{-1}(c x)\right )}{9 d x^9}-\frac{2 c^2 \left (d-c^2 d x^2\right )^{7/2} \left (a+b \sin ^{-1}(c x)\right )}{63 d x^7}+\frac{\left (b c^3 d^2 \sqrt{d-c^2 d x^2}\right ) \operatorname{Subst}\left (\int \left (\frac{1}{x^4}-\frac{3 c^2}{x^3}+\frac{3 c^4}{x^2}-\frac{c^6}{x}\right ) \, dx,x,x^2\right )}{63 \sqrt{1-c^2 x^2}}\\ &=-\frac{b c^3 d^2 \sqrt{d-c^2 d x^2}}{189 x^6 \sqrt{1-c^2 x^2}}+\frac{b c^5 d^2 \sqrt{d-c^2 d x^2}}{42 x^4 \sqrt{1-c^2 x^2}}-\frac{b c^7 d^2 \sqrt{d-c^2 d x^2}}{21 x^2 \sqrt{1-c^2 x^2}}-\frac{b c d^2 \left (1-c^2 x^2\right )^{7/2} \sqrt{d-c^2 d x^2}}{72 x^8}-\frac{\left (d-c^2 d x^2\right )^{7/2} \left (a+b \sin ^{-1}(c x)\right )}{9 d x^9}-\frac{2 c^2 \left (d-c^2 d x^2\right )^{7/2} \left (a+b \sin ^{-1}(c x)\right )}{63 d x^7}-\frac{2 b c^9 d^2 \sqrt{d-c^2 d x^2} \log (x)}{63 \sqrt{1-c^2 x^2}}\\ \end{align*}

Mathematica [A]  time = 0.217809, size = 184, normalized size = 0.65 \[ \frac{d^2 \sqrt{d-c^2 d x^2} \left (840 a \left (2 c^2 x^2+7\right ) \left (c^2 x^2-1\right )^4+b c x \sqrt{1-c^2 x^2} \left (-4566 c^8 x^8-420 c^6 x^6+3150 c^4 x^4-2660 c^2 x^2+735\right )+840 b \left (2 c^2 x^2+7\right ) \left (c^2 x^2-1\right )^4 \sin ^{-1}(c x)\right )}{52920 x^9 \left (c^2 x^2-1\right )}-\frac{2 b c^9 d^2 \log (x) \sqrt{d-c^2 d x^2}}{63 \sqrt{1-c^2 x^2}} \]

Antiderivative was successfully verified.

[In]

Integrate[((d - c^2*d*x^2)^(5/2)*(a + b*ArcSin[c*x]))/x^10,x]

[Out]

(d^2*Sqrt[d - c^2*d*x^2]*(840*a*(-1 + c^2*x^2)^4*(7 + 2*c^2*x^2) + b*c*x*Sqrt[1 - c^2*x^2]*(735 - 2660*c^2*x^2
+ 3150*c^4*x^4 - 420*c^6*x^6 - 4566*c^8*x^8) + 840*b*(-1 + c^2*x^2)^4*(7 + 2*c^2*x^2)*ArcSin[c*x]))/(52920*x^
9*(-1 + c^2*x^2)) - (2*b*c^9*d^2*Sqrt[d - c^2*d*x^2]*Log[x])/(63*Sqrt[1 - c^2*x^2])

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Maple [C]  time = 0.477, size = 5323, normalized size = 18.9 \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-c^2*d*x^2+d)^(5/2)*(a+b*arcsin(c*x))/x^10,x)

[Out]

result too large to display

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-c^2*d*x^2+d)^(5/2)*(a+b*arcsin(c*x))/x^10,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 3.12368, size = 1609, normalized size = 5.71 \begin{align*} \left [\frac{24 \,{\left (b c^{11} d^{2} x^{11} - b c^{9} d^{2} x^{9}\right )} \sqrt{d} \log \left (\frac{c^{2} d x^{6} + c^{2} d x^{2} - d x^{4} + \sqrt{-c^{2} d x^{2} + d} \sqrt{-c^{2} x^{2} + 1}{\left (x^{4} - 1\right )} \sqrt{d} - d}{c^{2} x^{4} - x^{2}}\right ) -{\left (12 \, b c^{7} d^{2} x^{7} - 90 \, b c^{5} d^{2} x^{5} -{\left (12 \, b c^{7} - 90 \, b c^{5} + 76 \, b c^{3} - 21 \, b c\right )} d^{2} x^{9} + 76 \, b c^{3} d^{2} x^{3} - 21 \, b c d^{2} x\right )} \sqrt{-c^{2} d x^{2} + d} \sqrt{-c^{2} x^{2} + 1} + 24 \,{\left (2 \, a c^{10} d^{2} x^{10} - a c^{8} d^{2} x^{8} - 16 \, a c^{6} d^{2} x^{6} + 34 \, a c^{4} d^{2} x^{4} - 26 \, a c^{2} d^{2} x^{2} + 7 \, a d^{2} +{\left (2 \, b c^{10} d^{2} x^{10} - b c^{8} d^{2} x^{8} - 16 \, b c^{6} d^{2} x^{6} + 34 \, b c^{4} d^{2} x^{4} - 26 \, b c^{2} d^{2} x^{2} + 7 \, b d^{2}\right )} \arcsin \left (c x\right )\right )} \sqrt{-c^{2} d x^{2} + d}}{1512 \,{\left (c^{2} x^{11} - x^{9}\right )}}, -\frac{48 \,{\left (b c^{11} d^{2} x^{11} - b c^{9} d^{2} x^{9}\right )} \sqrt{-d} \arctan \left (\frac{\sqrt{-c^{2} d x^{2} + d} \sqrt{-c^{2} x^{2} + 1}{\left (x^{2} + 1\right )} \sqrt{-d}}{c^{2} d x^{4} -{\left (c^{2} + 1\right )} d x^{2} + d}\right ) +{\left (12 \, b c^{7} d^{2} x^{7} - 90 \, b c^{5} d^{2} x^{5} -{\left (12 \, b c^{7} - 90 \, b c^{5} + 76 \, b c^{3} - 21 \, b c\right )} d^{2} x^{9} + 76 \, b c^{3} d^{2} x^{3} - 21 \, b c d^{2} x\right )} \sqrt{-c^{2} d x^{2} + d} \sqrt{-c^{2} x^{2} + 1} - 24 \,{\left (2 \, a c^{10} d^{2} x^{10} - a c^{8} d^{2} x^{8} - 16 \, a c^{6} d^{2} x^{6} + 34 \, a c^{4} d^{2} x^{4} - 26 \, a c^{2} d^{2} x^{2} + 7 \, a d^{2} +{\left (2 \, b c^{10} d^{2} x^{10} - b c^{8} d^{2} x^{8} - 16 \, b c^{6} d^{2} x^{6} + 34 \, b c^{4} d^{2} x^{4} - 26 \, b c^{2} d^{2} x^{2} + 7 \, b d^{2}\right )} \arcsin \left (c x\right )\right )} \sqrt{-c^{2} d x^{2} + d}}{1512 \,{\left (c^{2} x^{11} - x^{9}\right )}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-c^2*d*x^2+d)^(5/2)*(a+b*arcsin(c*x))/x^10,x, algorithm="fricas")

[Out]

[1/1512*(24*(b*c^11*d^2*x^11 - b*c^9*d^2*x^9)*sqrt(d)*log((c^2*d*x^6 + c^2*d*x^2 - d*x^4 + sqrt(-c^2*d*x^2 + d
)*sqrt(-c^2*x^2 + 1)*(x^4 - 1)*sqrt(d) - d)/(c^2*x^4 - x^2)) - (12*b*c^7*d^2*x^7 - 90*b*c^5*d^2*x^5 - (12*b*c^
7 - 90*b*c^5 + 76*b*c^3 - 21*b*c)*d^2*x^9 + 76*b*c^3*d^2*x^3 - 21*b*c*d^2*x)*sqrt(-c^2*d*x^2 + d)*sqrt(-c^2*x^
2 + 1) + 24*(2*a*c^10*d^2*x^10 - a*c^8*d^2*x^8 - 16*a*c^6*d^2*x^6 + 34*a*c^4*d^2*x^4 - 26*a*c^2*d^2*x^2 + 7*a*
d^2 + (2*b*c^10*d^2*x^10 - b*c^8*d^2*x^8 - 16*b*c^6*d^2*x^6 + 34*b*c^4*d^2*x^4 - 26*b*c^2*d^2*x^2 + 7*b*d^2)*a
rcsin(c*x))*sqrt(-c^2*d*x^2 + d))/(c^2*x^11 - x^9), -1/1512*(48*(b*c^11*d^2*x^11 - b*c^9*d^2*x^9)*sqrt(-d)*arc
tan(sqrt(-c^2*d*x^2 + d)*sqrt(-c^2*x^2 + 1)*(x^2 + 1)*sqrt(-d)/(c^2*d*x^4 - (c^2 + 1)*d*x^2 + d)) + (12*b*c^7*
d^2*x^7 - 90*b*c^5*d^2*x^5 - (12*b*c^7 - 90*b*c^5 + 76*b*c^3 - 21*b*c)*d^2*x^9 + 76*b*c^3*d^2*x^3 - 21*b*c*d^2
*x)*sqrt(-c^2*d*x^2 + d)*sqrt(-c^2*x^2 + 1) - 24*(2*a*c^10*d^2*x^10 - a*c^8*d^2*x^8 - 16*a*c^6*d^2*x^6 + 34*a*
c^4*d^2*x^4 - 26*a*c^2*d^2*x^2 + 7*a*d^2 + (2*b*c^10*d^2*x^10 - b*c^8*d^2*x^8 - 16*b*c^6*d^2*x^6 + 34*b*c^4*d^
2*x^4 - 26*b*c^2*d^2*x^2 + 7*b*d^2)*arcsin(c*x))*sqrt(-c^2*d*x^2 + d))/(c^2*x^11 - x^9)]

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-c**2*d*x**2+d)**(5/2)*(a+b*asin(c*x))/x**10,x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (-c^{2} d x^{2} + d\right )}^{\frac{5}{2}}{\left (b \arcsin \left (c x\right ) + a\right )}}{x^{10}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-c^2*d*x^2+d)^(5/2)*(a+b*arcsin(c*x))/x^10,x, algorithm="giac")

[Out]

integrate((-c^2*d*x^2 + d)^(5/2)*(b*arcsin(c*x) + a)/x^10, x)