3.3.99 \(\int \frac {(a+b \text {ArcSin}(c+d x))^2}{(c e+d e x)^{9/2}} \, dx\) [299]
Optimal. Leaf size=130 \[ -\frac {2 (a+b \text {ArcSin}(c+d x))^2}{7 d e (e (c+d x))^{7/2}}-\frac {8 b (a+b \text {ArcSin}(c+d x)) \, _2F_1\left (-\frac {5}{4},\frac {1}{2};-\frac {1}{4};(c+d x)^2\right )}{35 d e^2 (e (c+d x))^{5/2}}-\frac {16 b^2 \, _3F_2\left (-\frac {3}{4},-\frac {3}{4},1;-\frac {1}{4},\frac {1}{4};(c+d x)^2\right )}{105 d e^3 (e (c+d x))^{3/2}} \]
[Out]
-2/7*(a+b*arcsin(d*x+c))^2/d/e/(e*(d*x+c))^(7/2)-8/35*b*(a+b*arcsin(d*x+c))*hypergeom([-5/4, 1/2],[-1/4],(d*x+
c)^2)/d/e^2/(e*(d*x+c))^(5/2)-16/105*b^2*hypergeom([-3/4, -3/4, 1],[-1/4, 1/4],(d*x+c)^2)/d/e^3/(e*(d*x+c))^(3
/2)
________________________________________________________________________________________
Rubi [A]
time = 0.15, antiderivative size = 130, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {4889, 4723,
4805} \begin {gather*} -\frac {16 b^2 \, _3F_2\left (-\frac {3}{4},-\frac {3}{4},1;-\frac {1}{4},\frac {1}{4};(c+d x)^2\right )}{105 d e^3 (e (c+d x))^{3/2}}-\frac {8 b \, _2F_1\left (-\frac {5}{4},\frac {1}{2};-\frac {1}{4};(c+d x)^2\right ) (a+b \text {ArcSin}(c+d x))}{35 d e^2 (e (c+d x))^{5/2}}-\frac {2 (a+b \text {ArcSin}(c+d x))^2}{7 d e (e (c+d x))^{7/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[(a + b*ArcSin[c + d*x])^2/(c*e + d*e*x)^(9/2),x]
[Out]
(-2*(a + b*ArcSin[c + d*x])^2)/(7*d*e*(e*(c + d*x))^(7/2)) - (8*b*(a + b*ArcSin[c + d*x])*Hypergeometric2F1[-5
/4, 1/2, -1/4, (c + d*x)^2])/(35*d*e^2*(e*(c + d*x))^(5/2)) - (16*b^2*HypergeometricPFQ[{-3/4, -3/4, 1}, {-1/4
, 1/4}, (c + d*x)^2])/(105*d*e^3*(e*(c + d*x))^(3/2))
Rule 4723
Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[(d*x)^(m + 1)*((a + b*ArcSi
n[c*x])^n/(d*(m + 1))), x] - Dist[b*c*(n/(d*(m + 1))), Int[(d*x)^(m + 1)*((a + b*ArcSin[c*x])^(n - 1)/Sqrt[1 -
c^2*x^2]), x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] && NeQ[m, -1]
Rule 4805
Int[(((a_.) + ArcSin[(c_.)*(x_)]*(b_.))*((f_.)*(x_))^(m_))/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[((f*x)
^(m + 1)/(f*(m + 1)))*Simp[Sqrt[1 - c^2*x^2]/Sqrt[d + e*x^2]]*(a + b*ArcSin[c*x])*Hypergeometric2F1[1/2, (1 +
m)/2, (3 + m)/2, c^2*x^2], x] - Simp[b*c*((f*x)^(m + 2)/(f^2*(m + 1)*(m + 2)))*Simp[Sqrt[1 - c^2*x^2]/Sqrt[d +
e*x^2]]*HypergeometricPFQ[{1, 1 + m/2, 1 + m/2}, {3/2 + m/2, 2 + m/2}, c^2*x^2], x] /; FreeQ[{a, b, c, d, e,
f, m}, x] && EqQ[c^2*d + e, 0] && !IntegerQ[m]
Rule 4889
Int[((a_.) + ArcSin[(c_) + (d_.)*(x_)]*(b_.))^(n_.)*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> Dist[1/d, Subst[I
nt[((d*e - c*f)/d + f*(x/d))^m*(a + b*ArcSin[x])^n, x], x, c + d*x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x]
Rubi steps
\begin {align*} \int \frac {\left (a+b \sin ^{-1}(c+d x)\right )^2}{(c e+d e x)^{9/2}} \, dx &=\frac {\text {Subst}\left (\int \frac {\left (a+b \sin ^{-1}(x)\right )^2}{(e x)^{9/2}} \, dx,x,c+d x\right )}{d}\\ &=-\frac {2 \left (a+b \sin ^{-1}(c+d x)\right )^2}{7 d e (e (c+d x))^{7/2}}+\frac {(4 b) \text {Subst}\left (\int \frac {a+b \sin ^{-1}(x)}{(e x)^{7/2} \sqrt {1-x^2}} \, dx,x,c+d x\right )}{7 d e}\\ &=-\frac {2 \left (a+b \sin ^{-1}(c+d x)\right )^2}{7 d e (e (c+d x))^{7/2}}-\frac {8 b \left (a+b \sin ^{-1}(c+d x)\right ) \, _2F_1\left (-\frac {5}{4},\frac {1}{2};-\frac {1}{4};(c+d x)^2\right )}{35 d e^2 (e (c+d x))^{5/2}}-\frac {16 b^2 \, _3F_2\left (-\frac {3}{4},-\frac {3}{4},1;-\frac {1}{4},\frac {1}{4};(c+d x)^2\right )}{105 d e^3 (e (c+d x))^{3/2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.06, size = 114, normalized size = 0.88 \begin {gather*} -\frac {2 \sqrt {e (c+d x)} \left (3 (a+b \text {ArcSin}(c+d x)) \left (5 (a+b \text {ArcSin}(c+d x))+4 b (c+d x) \, _2F_1\left (-\frac {5}{4},\frac {1}{2};-\frac {1}{4};(c+d x)^2\right )\right )+8 b^2 (c+d x)^2 \, _3F_2\left (-\frac {3}{4},-\frac {3}{4},1;-\frac {1}{4},\frac {1}{4};(c+d x)^2\right )\right )}{105 d e^5 (c+d x)^4} \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[(a + b*ArcSin[c + d*x])^2/(c*e + d*e*x)^(9/2),x]
[Out]
(-2*Sqrt[e*(c + d*x)]*(3*(a + b*ArcSin[c + d*x])*(5*(a + b*ArcSin[c + d*x]) + 4*b*(c + d*x)*Hypergeometric2F1[
-5/4, 1/2, -1/4, (c + d*x)^2]) + 8*b^2*(c + d*x)^2*HypergeometricPFQ[{-3/4, -3/4, 1}, {-1/4, 1/4}, (c + d*x)^2
]))/(105*d*e^5*(c + d*x)^4)
________________________________________________________________________________________
Maple [F]
time = 0.22, size = 0, normalized size = 0.00 \[\int \frac {\left (a +b \arcsin \left (d x +c \right )\right )^{2}}{\left (d e x +c e \right )^{\frac {9}{2}}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((a+b*arcsin(d*x+c))^2/(d*e*x+c*e)^(9/2),x)
[Out]
int((a+b*arcsin(d*x+c))^2/(d*e*x+c*e)^(9/2),x)
________________________________________________________________________________________
Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((a+b*arcsin(d*x+c))^2/(d*e*x+c*e)^(9/2),x, algorithm="maxima")
[Out]
-1/210*(60*b^2*arctan2(d*x + c, sqrt(d*x + c + 1)*sqrt(-d*x - c + 1))^2*e^(1/2) - (d^4*x^3*e^5 + 3*c*d^3*x^2*e
^5 + 3*c^2*d^2*x*e^5 + c^3*d*e^5)*(2940*a*b*d^2*e^(1/2)*integrate(1/7*sqrt(d*x + c)*x^2*arctan(d*x/(sqrt(d*x +
c + 1)*sqrt(-d*x - c + 1)) + c/(sqrt(d*x + c + 1)*sqrt(-d*x - c + 1)))/(d^7*x^7*e^5 + 7*c*d^6*x^6*e^5 + 21*c^
2*d^5*x^5*e^5 + 35*c^3*d^4*x^4*e^5 + 35*c^4*d^3*x^3*e^5 - d^5*x^5*e^5 + 21*c^5*d^2*x^2*e^5 - 5*c*d^4*x^4*e^5 +
7*c^6*d*x*e^5 - 10*c^2*d^3*x^3*e^5 + c^7*e^5 - 10*c^3*d^2*x^2*e^5 - 5*c^4*d*x*e^5 - c^5*e^5), x) + 5880*a*b*c
*d*e^(1/2)*integrate(1/7*sqrt(d*x + c)*x*arctan(d*x/(sqrt(d*x + c + 1)*sqrt(-d*x - c + 1)) + c/(sqrt(d*x + c +
1)*sqrt(-d*x - c + 1)))/(d^7*x^7*e^5 + 7*c*d^6*x^6*e^5 + 21*c^2*d^5*x^5*e^5 + 35*c^3*d^4*x^4*e^5 + 35*c^4*d^3
*x^3*e^5 - d^5*x^5*e^5 + 21*c^5*d^2*x^2*e^5 - 5*c*d^4*x^4*e^5 + 7*c^6*d*x*e^5 - 10*c^2*d^3*x^3*e^5 + c^7*e^5 -
10*c^3*d^2*x^2*e^5 - 5*c^4*d*x*e^5 - c^5*e^5), x) + 2940*a*b*c^2*e^(1/2)*integrate(1/7*sqrt(d*x + c)*arctan(d
*x/(sqrt(d*x + c + 1)*sqrt(-d*x - c + 1)) + c/(sqrt(d*x + c + 1)*sqrt(-d*x - c + 1)))/(d^7*x^7*e^5 + 7*c*d^6*x
^6*e^5 + 21*c^2*d^5*x^5*e^5 + 35*c^3*d^4*x^4*e^5 + 35*c^4*d^3*x^3*e^5 - d^5*x^5*e^5 + 21*c^5*d^2*x^2*e^5 - 5*c
*d^4*x^4*e^5 + 7*c^6*d*x*e^5 - 10*c^2*d^3*x^3*e^5 + c^7*e^5 - 10*c^3*d^2*x^2*e^5 - 5*c^4*d*x*e^5 - c^5*e^5), x
) - 5*(42*arctan(sqrt(d*x + c))*e^(-5) + 21*e^(-5)*log(sqrt(d*x + c) + 1) - 21*e^(-5)*log(sqrt(d*x + c) - 1) -
4*(7*(d*x + c)^2 + 3)*e^(-5)/(d*x + c)^(7/2))*a^2*c^2*e^(1/2)/d - 840*b^2*d*e^(1/2)*integrate(1/7*sqrt(d*x +
c + 1)*sqrt(d*x + c)*sqrt(-d*x - c + 1)*x*arctan(d*x/(sqrt(d*x + c + 1)*sqrt(-d*x - c + 1)) + c/(sqrt(d*x + c
+ 1)*sqrt(-d*x - c + 1)))/(d^7*x^7*e^5 + 7*c*d^6*x^6*e^5 + 21*c^2*d^5*x^5*e^5 + 35*c^3*d^4*x^4*e^5 + 35*c^4*d^
3*x^3*e^5 - d^5*x^5*e^5 + 21*c^5*d^2*x^2*e^5 - 5*c*d^4*x^4*e^5 + 7*c^6*d*x*e^5 - 10*c^2*d^3*x^3*e^5 + c^7*e^5
- 10*c^3*d^2*x^2*e^5 - 5*c^4*d*x*e^5 - c^5*e^5), x) - 840*b^2*c*e^(1/2)*integrate(1/7*sqrt(d*x + c + 1)*sqrt(d
*x + c)*sqrt(-d*x - c + 1)*arctan(d*x/(sqrt(d*x + c + 1)*sqrt(-d*x - c + 1)) + c/(sqrt(d*x + c + 1)*sqrt(-d*x
- c + 1)))/(d^7*x^7*e^5 + 7*c*d^6*x^6*e^5 + 21*c^2*d^5*x^5*e^5 + 35*c^3*d^4*x^4*e^5 + 35*c^4*d^3*x^3*e^5 - d^5
*x^5*e^5 + 21*c^5*d^2*x^2*e^5 - 5*c*d^4*x^4*e^5 + 7*c^6*d*x*e^5 - 10*c^2*d^3*x^3*e^5 + c^7*e^5 - 10*c^3*d^2*x^
2*e^5 - 5*c^4*d*x*e^5 - c^5*e^5), x) + 2*(210*(c + 1)*arctan(sqrt(d*x + c))*e^(-5) + 105*(c - 1)*e^(-5)*log(sq
rt(d*x + c) + 1) - 105*(c - 1)*e^(-5)*log(sqrt(d*x + c) - 1) + 4*(105*(d*x + c)^3 - 35*(d*x + c)^2*c + 21*d*x
+ 6*c)*e^(-5)/(d*x + c)^(7/2))*a^2*c*e^(1/2)/d - 2940*a*b*e^(1/2)*integrate(1/7*sqrt(d*x + c)*arctan(d*x/(sqrt
(d*x + c + 1)*sqrt(-d*x - c + 1)) + c/(sqrt(d*x + c + 1)*sqrt(-d*x - c + 1)))/(d^7*x^7*e^5 + 7*c*d^6*x^6*e^5 +
21*c^2*d^5*x^5*e^5 + 35*c^3*d^4*x^4*e^5 + 35*c^4*d^3*x^3*e^5 - d^5*x^5*e^5 + 21*c^5*d^2*x^2*e^5 - 5*c*d^4*x^4
*e^5 + 7*c^6*d*x*e^5 - 10*c^2*d^3*x^3*e^5 + c^7*e^5 - 10*c^3*d^2*x^2*e^5 - 5*c^4*d*x*e^5 - c^5*e^5), x) - (210
*(c^2 + 2*c + 1)*arctan(sqrt(d*x + c))*e^(-5) + 105*(c^2 - 2*c + 1)*e^(-5)*log(sqrt(d*x + c) + 1) - 105*(c^2 -
2*c + 1)*e^(-5)*log(sqrt(d*x + c) - 1) + 4*(210*(d*x + c)^3*c - 35*(c^2 + 1)*(d*x + c)^2 + 42*(d*x + c)*c - 1
5*c^2)*e^(-5)/(d*x + c)^(7/2))*a^2*e^(1/2)/d + 5*(42*arctan(sqrt(d*x + c))*e^(-5) + 21*e^(-5)*log(sqrt(d*x + c
) + 1) - 21*e^(-5)*log(sqrt(d*x + c) - 1) - 4*(7*(d*x + c)^2 + 3)*e^(-5)/(d*x + c)^(7/2))*a^2*e^(1/2)/d)*sqrt(
d*x + c))/((d^4*x^3*e^5 + 3*c*d^3*x^2*e^5 + 3*c^2*d^2*x*e^5 + c^3*d*e^5)*sqrt(d*x + c))
________________________________________________________________________________________
Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((a+b*arcsin(d*x+c))^2/(d*e*x+c*e)^(9/2),x, algorithm="fricas")
[Out]
integral((b^2*arcsin(d*x + c)^2 + 2*a*b*arcsin(d*x + c) + a^2)*sqrt(d*x + c)*e^(-9/2)/(d^5*x^5 + 5*c*d^4*x^4 +
10*c^2*d^3*x^3 + 10*c^3*d^2*x^2 + 5*c^4*d*x + c^5), x)
________________________________________________________________________________________
Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((a+b*asin(d*x+c))**2/(d*e*x+c*e)**(9/2),x)
[Out]
Exception raised: SystemError >> excessive stack use: stack is 3877 deep
________________________________________________________________________________________
Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((a+b*arcsin(d*x+c))^2/(d*e*x+c*e)^(9/2),x, algorithm="giac")
[Out]
integrate((b*arcsin(d*x + c) + a)^2/(d*e*x + c*e)^(9/2), x)
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (a+b\,\mathrm {asin}\left (c+d\,x\right )\right )}^2}{{\left (c\,e+d\,e\,x\right )}^{9/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((a + b*asin(c + d*x))^2/(c*e + d*e*x)^(9/2),x)
[Out]
int((a + b*asin(c + d*x))^2/(c*e + d*e*x)^(9/2), x)
________________________________________________________________________________________