3.92 \(\int \frac{(d-c^2 d x^2)^{5/2} (a+b \sin ^{-1}(c x))}{x^{12}} \, dx\)

Optimal. Leaf size=361 \[ -\frac{8 c^4 \left (d-c^2 d x^2\right )^{7/2} \left (a+b \sin ^{-1}(c x)\right )}{693 d x^7}-\frac{4 c^2 \left (d-c^2 d x^2\right )^{7/2} \left (a+b \sin ^{-1}(c x)\right )}{99 d x^9}-\frac{\left (d-c^2 d x^2\right )^{7/2} \left (a+b \sin ^{-1}(c x)\right )}{11 d x^{11}}+\frac{2 b c^9 d^2 \sqrt{d-c^2 d x^2}}{693 x^2 \sqrt{1-c^2 x^2}}+\frac{b c^7 d^2 \sqrt{d-c^2 d x^2}}{924 x^4 \sqrt{1-c^2 x^2}}-\frac{113 b c^5 d^2 \sqrt{d-c^2 d x^2}}{4158 x^6 \sqrt{1-c^2 x^2}}+\frac{23 b c^3 d^2 \sqrt{d-c^2 d x^2}}{792 x^8 \sqrt{1-c^2 x^2}}-\frac{b c d^2 \sqrt{d-c^2 d x^2}}{110 x^{10} \sqrt{1-c^2 x^2}}-\frac{8 b c^{11} d^2 \log (x) \sqrt{d-c^2 d x^2}}{693 \sqrt{1-c^2 x^2}} \]

[Out]

-(b*c*d^2*Sqrt[d - c^2*d*x^2])/(110*x^10*Sqrt[1 - c^2*x^2]) + (23*b*c^3*d^2*Sqrt[d - c^2*d*x^2])/(792*x^8*Sqrt
[1 - c^2*x^2]) - (113*b*c^5*d^2*Sqrt[d - c^2*d*x^2])/(4158*x^6*Sqrt[1 - c^2*x^2]) + (b*c^7*d^2*Sqrt[d - c^2*d*
x^2])/(924*x^4*Sqrt[1 - c^2*x^2]) + (2*b*c^9*d^2*Sqrt[d - c^2*d*x^2])/(693*x^2*Sqrt[1 - c^2*x^2]) - ((d - c^2*
d*x^2)^(7/2)*(a + b*ArcSin[c*x]))/(11*d*x^11) - (4*c^2*(d - c^2*d*x^2)^(7/2)*(a + b*ArcSin[c*x]))/(99*d*x^9) -
(8*c^4*(d - c^2*d*x^2)^(7/2)*(a + b*ArcSin[c*x]))/(693*d*x^7) - (8*b*c^11*d^2*Sqrt[d - c^2*d*x^2]*Log[x])/(69
3*Sqrt[1 - c^2*x^2])

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Rubi [A]  time = 0.222861, antiderivative size = 361, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {271, 264, 4691, 12, 1251, 893} \[ -\frac{8 c^4 \left (d-c^2 d x^2\right )^{7/2} \left (a+b \sin ^{-1}(c x)\right )}{693 d x^7}-\frac{4 c^2 \left (d-c^2 d x^2\right )^{7/2} \left (a+b \sin ^{-1}(c x)\right )}{99 d x^9}-\frac{\left (d-c^2 d x^2\right )^{7/2} \left (a+b \sin ^{-1}(c x)\right )}{11 d x^{11}}+\frac{2 b c^9 d^2 \sqrt{d-c^2 d x^2}}{693 x^2 \sqrt{1-c^2 x^2}}+\frac{b c^7 d^2 \sqrt{d-c^2 d x^2}}{924 x^4 \sqrt{1-c^2 x^2}}-\frac{113 b c^5 d^2 \sqrt{d-c^2 d x^2}}{4158 x^6 \sqrt{1-c^2 x^2}}+\frac{23 b c^3 d^2 \sqrt{d-c^2 d x^2}}{792 x^8 \sqrt{1-c^2 x^2}}-\frac{b c d^2 \sqrt{d-c^2 d x^2}}{110 x^{10} \sqrt{1-c^2 x^2}}-\frac{8 b c^{11} d^2 \log (x) \sqrt{d-c^2 d x^2}}{693 \sqrt{1-c^2 x^2}} \]

Antiderivative was successfully verified.

[In]

Int[((d - c^2*d*x^2)^(5/2)*(a + b*ArcSin[c*x]))/x^12,x]

[Out]

-(b*c*d^2*Sqrt[d - c^2*d*x^2])/(110*x^10*Sqrt[1 - c^2*x^2]) + (23*b*c^3*d^2*Sqrt[d - c^2*d*x^2])/(792*x^8*Sqrt
[1 - c^2*x^2]) - (113*b*c^5*d^2*Sqrt[d - c^2*d*x^2])/(4158*x^6*Sqrt[1 - c^2*x^2]) + (b*c^7*d^2*Sqrt[d - c^2*d*
x^2])/(924*x^4*Sqrt[1 - c^2*x^2]) + (2*b*c^9*d^2*Sqrt[d - c^2*d*x^2])/(693*x^2*Sqrt[1 - c^2*x^2]) - ((d - c^2*
d*x^2)^(7/2)*(a + b*ArcSin[c*x]))/(11*d*x^11) - (4*c^2*(d - c^2*d*x^2)^(7/2)*(a + b*ArcSin[c*x]))/(99*d*x^9) -
(8*c^4*(d - c^2*d*x^2)^(7/2)*(a + b*ArcSin[c*x]))/(693*d*x^7) - (8*b*c^11*d^2*Sqrt[d - c^2*d*x^2]*Log[x])/(69
3*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 1251

Int[(x_)^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Dist[1/2,
Subst[Int[x^((m - 1)/2)*(d + e*x)^q*(a + b*x + c*x^2)^p, x], x, x^2], x] /; FreeQ[{a, b, c, d, e, p, q}, x] &&
IntegerQ[(m - 1)/2]

Rule 893

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :
> Int[ExpandIntegrand[(d + e*x)^m*(f + g*x)^n*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] &
& NeQ[e*f - d*g, 0] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IntegerQ[p] && ((EqQ[p, 1] && I
ntegersQ[m, n]) || (ILtQ[m, 0] && ILtQ[n, 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^{12}} \, dx &=-\frac{\left (b c d^2 \sqrt{d-c^2 d x^2}\right ) \int \frac{\left (1-c^2 x^2\right )^3 \left (-63-28 c^2 x^2-8 c^4 x^4\right )}{693 x^{11}} \, 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^{12}} \, dx\\ &=-\frac{\left (d-c^2 d x^2\right )^{7/2} \left (a+b \sin ^{-1}(c x)\right )}{11 d x^{11}}-\frac{\left (b c d^2 \sqrt{d-c^2 d x^2}\right ) \int \frac{\left (1-c^2 x^2\right )^3 \left (-63-28 c^2 x^2-8 c^4 x^4\right )}{x^{11}} \, dx}{693 \sqrt{1-c^2 x^2}}+\frac{1}{11} \left (4 c^2 \left (a+b \sin ^{-1}(c x)\right )\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 )}{11 d x^{11}}-\frac{4 c^2 \left (d-c^2 d x^2\right )^{7/2} \left (a+b \sin ^{-1}(c x)\right )}{99 d x^9}-\frac{\left (b c d^2 \sqrt{d-c^2 d x^2}\right ) \operatorname{Subst}\left (\int \frac{\left (1-c^2 x\right )^3 \left (-63-28 c^2 x-8 c^4 x^2\right )}{x^6} \, dx,x,x^2\right )}{1386 \sqrt{1-c^2 x^2}}+\frac{1}{99} \left (8 c^4 \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 )}{11 d x^{11}}-\frac{4 c^2 \left (d-c^2 d x^2\right )^{7/2} \left (a+b \sin ^{-1}(c x)\right )}{99 d x^9}-\frac{8 c^4 \left (d-c^2 d x^2\right )^{7/2} \left (a+b \sin ^{-1}(c x)\right )}{693 d x^7}-\frac{\left (b c d^2 \sqrt{d-c^2 d x^2}\right ) \operatorname{Subst}\left (\int \left (-\frac{63}{x^6}+\frac{161 c^2}{x^5}-\frac{113 c^4}{x^4}+\frac{3 c^6}{x^3}+\frac{4 c^8}{x^2}+\frac{8 c^{10}}{x}\right ) \, dx,x,x^2\right )}{1386 \sqrt{1-c^2 x^2}}\\ &=-\frac{b c d^2 \sqrt{d-c^2 d x^2}}{110 x^{10} \sqrt{1-c^2 x^2}}+\frac{23 b c^3 d^2 \sqrt{d-c^2 d x^2}}{792 x^8 \sqrt{1-c^2 x^2}}-\frac{113 b c^5 d^2 \sqrt{d-c^2 d x^2}}{4158 x^6 \sqrt{1-c^2 x^2}}+\frac{b c^7 d^2 \sqrt{d-c^2 d x^2}}{924 x^4 \sqrt{1-c^2 x^2}}+\frac{2 b c^9 d^2 \sqrt{d-c^2 d x^2}}{693 x^2 \sqrt{1-c^2 x^2}}-\frac{\left (d-c^2 d x^2\right )^{7/2} \left (a+b \sin ^{-1}(c x)\right )}{11 d x^{11}}-\frac{4 c^2 \left (d-c^2 d x^2\right )^{7/2} \left (a+b \sin ^{-1}(c x)\right )}{99 d x^9}-\frac{8 c^4 \left (d-c^2 d x^2\right )^{7/2} \left (a+b \sin ^{-1}(c x)\right )}{693 d x^7}-\frac{8 b c^{11} d^2 \sqrt{d-c^2 d x^2} \log (x)}{693 \sqrt{1-c^2 x^2}}\\ \end{align*}

Mathematica [A]  time = 0.237405, size = 209, normalized size = 0.58 \[ \frac{d^2 \sqrt{d-c^2 d x^2} \left (2520 a \left (8 c^4 x^4+28 c^2 x^2+63\right ) \left (c^2 x^2-1\right )^4-b c x \sqrt{1-c^2 x^2} \left (59048 c^{10} x^{10}+5040 c^8 x^8+1890 c^6 x^6-47460 c^4 x^4+50715 c^2 x^2-15876\right )+2520 b \left (8 c^4 x^4+28 c^2 x^2+63\right ) \left (c^2 x^2-1\right )^4 \sin ^{-1}(c x)\right )}{1746360 x^{11} \left (c^2 x^2-1\right )}-\frac{8 b c^{11} d^2 \log (x) \sqrt{d-c^2 d x^2}}{693 \sqrt{1-c^2 x^2}} \]

Antiderivative was successfully verified.

[In]

Integrate[((d - c^2*d*x^2)^(5/2)*(a + b*ArcSin[c*x]))/x^12,x]

[Out]

(d^2*Sqrt[d - c^2*d*x^2]*(2520*a*(-1 + c^2*x^2)^4*(63 + 28*c^2*x^2 + 8*c^4*x^4) - b*c*x*Sqrt[1 - c^2*x^2]*(-15
876 + 50715*c^2*x^2 - 47460*c^4*x^4 + 1890*c^6*x^6 + 5040*c^8*x^8 + 59048*c^10*x^10) + 2520*b*(-1 + c^2*x^2)^4
*(63 + 28*c^2*x^2 + 8*c^4*x^4)*ArcSin[c*x]))/(1746360*x^11*(-1 + c^2*x^2)) - (8*b*c^11*d^2*Sqrt[d - c^2*d*x^2]
*Log[x])/(693*Sqrt[1 - c^2*x^2])

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Maple [C]  time = 0.648, size = 6758, normalized size = 18.7 \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^12,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^12,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 3.6102, size = 1871, normalized size = 5.18 \begin{align*} \text{result too large to display} \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^12,x, algorithm="fricas")

[Out]

[1/83160*(480*(b*c^13*d^2*x^13 - b*c^11*d^2*x^11)*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)) - (240*b*c^9*d^2*x^9 + 90*b*c^7*d^2*x^7 - (24
0*b*c^9 + 90*b*c^7 - 2260*b*c^5 + 2415*b*c^3 - 756*b*c)*d^2*x^11 - 2260*b*c^5*d^2*x^5 + 2415*b*c^3*d^2*x^3 - 7
56*b*c*d^2*x)*sqrt(-c^2*d*x^2 + d)*sqrt(-c^2*x^2 + 1) + 120*(8*a*c^12*d^2*x^12 - 4*a*c^10*d^2*x^10 - a*c^8*d^2
*x^8 - 116*a*c^6*d^2*x^6 + 274*a*c^4*d^2*x^4 - 224*a*c^2*d^2*x^2 + 63*a*d^2 + (8*b*c^12*d^2*x^12 - 4*b*c^10*d^
2*x^10 - b*c^8*d^2*x^8 - 116*b*c^6*d^2*x^6 + 274*b*c^4*d^2*x^4 - 224*b*c^2*d^2*x^2 + 63*b*d^2)*arcsin(c*x))*sq
rt(-c^2*d*x^2 + d))/(c^2*x^13 - x^11), -1/83160*(960*(b*c^13*d^2*x^13 - b*c^11*d^2*x^11)*sqrt(-d)*arctan(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)) + (240*b*c^9*d^2*x^9
+ 90*b*c^7*d^2*x^7 - (240*b*c^9 + 90*b*c^7 - 2260*b*c^5 + 2415*b*c^3 - 756*b*c)*d^2*x^11 - 2260*b*c^5*d^2*x^5
+ 2415*b*c^3*d^2*x^3 - 756*b*c*d^2*x)*sqrt(-c^2*d*x^2 + d)*sqrt(-c^2*x^2 + 1) - 120*(8*a*c^12*d^2*x^12 - 4*a*c
^10*d^2*x^10 - a*c^8*d^2*x^8 - 116*a*c^6*d^2*x^6 + 274*a*c^4*d^2*x^4 - 224*a*c^2*d^2*x^2 + 63*a*d^2 + (8*b*c^1
2*d^2*x^12 - 4*b*c^10*d^2*x^10 - b*c^8*d^2*x^8 - 116*b*c^6*d^2*x^6 + 274*b*c^4*d^2*x^4 - 224*b*c^2*d^2*x^2 + 6
3*b*d^2)*arcsin(c*x))*sqrt(-c^2*d*x^2 + d))/(c^2*x^13 - x^11)]

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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**12,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^{12}}\,{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^12,x, algorithm="giac")

[Out]

integrate((-c^2*d*x^2 + d)^(5/2)*(b*arcsin(c*x) + a)/x^12, x)