3.3.90 \(\int \frac {a+b \text {ArcSin}(c+d x)}{(c e+d e x)^{11/2}} \, dx\) [290]

Optimal. Leaf size=139 \[ -\frac {4 b \sqrt {1-(c+d x)^2}}{63 d e^2 (e (c+d x))^{7/2}}-\frac {20 b \sqrt {1-(c+d x)^2}}{189 d e^4 (e (c+d x))^{3/2}}-\frac {2 (a+b \text {ArcSin}(c+d x))}{9 d e (e (c+d x))^{9/2}}+\frac {20 b F\left (\left .\text {ArcSin}\left (\frac {\sqrt {e (c+d x)}}{\sqrt {e}}\right )\right |-1\right )}{189 d e^{11/2}} \]

[Out]

-2/9*(a+b*arcsin(d*x+c))/d/e/(e*(d*x+c))^(9/2)+20/189*b*EllipticF((e*(d*x+c))^(1/2)/e^(1/2),I)/d/e^(11/2)-4/63
*b*(1-(d*x+c)^2)^(1/2)/d/e^2/(e*(d*x+c))^(7/2)-20/189*b*(1-(d*x+c)^2)^(1/2)/d/e^4/(e*(d*x+c))^(3/2)

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Rubi [A]
time = 0.09, antiderivative size = 139, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {4889, 4723, 331, 335, 227} \begin {gather*} -\frac {2 (a+b \text {ArcSin}(c+d x))}{9 d e (e (c+d x))^{9/2}}+\frac {20 b F\left (\left .\text {ArcSin}\left (\frac {\sqrt {e (c+d x)}}{\sqrt {e}}\right )\right |-1\right )}{189 d e^{11/2}}-\frac {20 b \sqrt {1-(c+d x)^2}}{189 d e^4 (e (c+d x))^{3/2}}-\frac {4 b \sqrt {1-(c+d x)^2}}{63 d e^2 (e (c+d x))^{7/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*ArcSin[c + d*x])/(c*e + d*e*x)^(11/2),x]

[Out]

(-4*b*Sqrt[1 - (c + d*x)^2])/(63*d*e^2*(e*(c + d*x))^(7/2)) - (20*b*Sqrt[1 - (c + d*x)^2])/(189*d*e^4*(e*(c +
d*x))^(3/2)) - (2*(a + b*ArcSin[c + d*x]))/(9*d*e*(e*(c + d*x))^(9/2)) + (20*b*EllipticF[ArcSin[Sqrt[e*(c + d*
x)]/Sqrt[e]], -1])/(189*d*e^(11/2))

Rule 227

Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> Simp[EllipticF[ArcSin[Rt[-b, 4]*(x/Rt[a, 4])], -1]/(Rt[a, 4]*Rt[
-b, 4]), x] /; FreeQ[{a, b}, x] && NegQ[b/a] && GtQ[a, 0]

Rule 331

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] - Dist[b*((m + n*(p + 1) + 1)/(a*c^n*(m + 1))), Int[(c*x)^(m + n)*(a + b*x^n)^p, x], x] /; Free
Q[{a, b, c, p}, x] && IGtQ[n, 0] && LtQ[m, -1] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 335

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = Denominator[m]}, Dist[k/c, Subst[I
nt[x^(k*(m + 1) - 1)*(a + b*(x^(k*n)/c^n))^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0]
&& FractionQ[m] && IntBinomialQ[a, b, c, n, m, p, x]

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 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 {a+b \sin ^{-1}(c+d x)}{(c e+d e x)^{11/2}} \, dx &=\frac {\text {Subst}\left (\int \frac {a+b \sin ^{-1}(x)}{(e x)^{11/2}} \, dx,x,c+d x\right )}{d}\\ &=-\frac {2 \left (a+b \sin ^{-1}(c+d x)\right )}{9 d e (e (c+d x))^{9/2}}+\frac {(2 b) \text {Subst}\left (\int \frac {1}{(e x)^{9/2} \sqrt {1-x^2}} \, dx,x,c+d x\right )}{9 d e}\\ &=-\frac {4 b \sqrt {1-(c+d x)^2}}{63 d e^2 (e (c+d x))^{7/2}}-\frac {2 \left (a+b \sin ^{-1}(c+d x)\right )}{9 d e (e (c+d x))^{9/2}}+\frac {(10 b) \text {Subst}\left (\int \frac {1}{(e x)^{5/2} \sqrt {1-x^2}} \, dx,x,c+d x\right )}{63 d e^3}\\ &=-\frac {4 b \sqrt {1-(c+d x)^2}}{63 d e^2 (e (c+d x))^{7/2}}-\frac {20 b \sqrt {1-(c+d x)^2}}{189 d e^4 (e (c+d x))^{3/2}}-\frac {2 \left (a+b \sin ^{-1}(c+d x)\right )}{9 d e (e (c+d x))^{9/2}}+\frac {(10 b) \text {Subst}\left (\int \frac {1}{\sqrt {e x} \sqrt {1-x^2}} \, dx,x,c+d x\right )}{189 d e^5}\\ &=-\frac {4 b \sqrt {1-(c+d x)^2}}{63 d e^2 (e (c+d x))^{7/2}}-\frac {20 b \sqrt {1-(c+d x)^2}}{189 d e^4 (e (c+d x))^{3/2}}-\frac {2 \left (a+b \sin ^{-1}(c+d x)\right )}{9 d e (e (c+d x))^{9/2}}+\frac {(20 b) \text {Subst}\left (\int \frac {1}{\sqrt {1-\frac {x^4}{e^2}}} \, dx,x,\sqrt {e (c+d x)}\right )}{189 d e^6}\\ &=-\frac {4 b \sqrt {1-(c+d x)^2}}{63 d e^2 (e (c+d x))^{7/2}}-\frac {20 b \sqrt {1-(c+d x)^2}}{189 d e^4 (e (c+d x))^{3/2}}-\frac {2 \left (a+b \sin ^{-1}(c+d x)\right )}{9 d e (e (c+d x))^{9/2}}+\frac {20 b F\left (\left .\sin ^{-1}\left (\frac {\sqrt {e (c+d x)}}{\sqrt {e}}\right )\right |-1\right )}{189 d e^{11/2}}\\ \end {align*}

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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
time = 0.03, size = 66, normalized size = 0.47 \begin {gather*} -\frac {2 \sqrt {e (c+d x)} \left (7 (a+b \text {ArcSin}(c+d x))+2 b (c+d x) \, _2F_1\left (-\frac {7}{4},\frac {1}{2};-\frac {3}{4};(c+d x)^2\right )\right )}{63 d e^6 (c+d x)^5} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*ArcSin[c + d*x])/(c*e + d*e*x)^(11/2),x]

[Out]

(-2*Sqrt[e*(c + d*x)]*(7*(a + b*ArcSin[c + d*x]) + 2*b*(c + d*x)*Hypergeometric2F1[-7/4, 1/2, -3/4, (c + d*x)^
2]))/(63*d*e^6*(c + d*x)^5)

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Maple [A]
time = 0.14, size = 203, normalized size = 1.46

method	result	size
derivativedivides	\(\frac {-\frac {2 a}{9 \left (d e x +c e \right )^{\frac {9}{2}}}+2 b \left (-\frac {\arcsin \left (\frac {d e x +c e}{e}\right )}{9 \left (d e x +c e \right )^{\frac {9}{2}}}+\frac {-\frac {2 \sqrt {-\frac {\left (d e x +c e \right )^{2}}{e^{2}}+1}}{63 \left (d e x +c e \right )^{\frac {7}{2}}}-\frac {10 \sqrt {-\frac {\left (d e x +c e \right )^{2}}{e^{2}}+1}}{189 e^{2} \left (d e x +c e \right )^{\frac {3}{2}}}+\frac {10 \sqrt {1-\frac {d e x +c e}{e}}\, \sqrt {1+\frac {d e x +c e}{e}}\, \EllipticF \left (\sqrt {d e x +c e}\, \sqrt {\frac {1}{e}}, i\right )}{189 e^{4} \sqrt {\frac {1}{e}}\, \sqrt {-\frac {\left (d e x +c e \right )^{2}}{e^{2}}+1}}}{e}\right )}{d e}\)	\(203\)
default	\(\frac {-\frac {2 a}{9 \left (d e x +c e \right )^{\frac {9}{2}}}+2 b \left (-\frac {\arcsin \left (\frac {d e x +c e}{e}\right )}{9 \left (d e x +c e \right )^{\frac {9}{2}}}+\frac {-\frac {2 \sqrt {-\frac {\left (d e x +c e \right )^{2}}{e^{2}}+1}}{63 \left (d e x +c e \right )^{\frac {7}{2}}}-\frac {10 \sqrt {-\frac {\left (d e x +c e \right )^{2}}{e^{2}}+1}}{189 e^{2} \left (d e x +c e \right )^{\frac {3}{2}}}+\frac {10 \sqrt {1-\frac {d e x +c e}{e}}\, \sqrt {1+\frac {d e x +c e}{e}}\, \EllipticF \left (\sqrt {d e x +c e}\, \sqrt {\frac {1}{e}}, i\right )}{189 e^{4} \sqrt {\frac {1}{e}}\, \sqrt {-\frac {\left (d e x +c e \right )^{2}}{e^{2}}+1}}}{e}\right )}{d e}\)	\(203\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*arcsin(d*x+c))/(d*e*x+c*e)^(11/2),x,method=_RETURNVERBOSE)

[Out]

2/d/e*(-1/9*a/(d*e*x+c*e)^(9/2)+b*(-1/9/(d*e*x+c*e)^(9/2)*arcsin((d*e*x+c*e)/e)+2/9/e*(-1/7*(-(d*e*x+c*e)^2/e^
2+1)^(1/2)/(d*e*x+c*e)^(7/2)-5/21/e^2*(-(d*e*x+c*e)^2/e^2+1)^(1/2)/(d*e*x+c*e)^(3/2)+5/21/e^4/(1/e)^(1/2)*(1-(
d*e*x+c*e)/e)^(1/2)*(1+(d*e*x+c*e)/e)^(1/2)/(-(d*e*x+c*e)^2/e^2+1)^(1/2)*EllipticF((d*e*x+c*e)^(1/2)*(1/e)^(1/
2),I))))

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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))/(d*e*x+c*e)^(11/2),x, algorithm="maxima")

[Out]

-2/9*((d*x + c)^(9/2)*b*arctan2(d*x + c, sqrt(d*x + c + 1)*sqrt(-d*x - c + 1)) + (a*d^4*x^4 + 4*a*c*d^3*x^3 +
6*a*c^2*d^2*x^2 + 9*(b*d^5*x^4*e^6 + 4*b*c*d^4*x^3*e^6 + 6*b*c^2*d^3*x^2*e^6 + 4*b*c^3*d^2*x*e^6 + b*c^4*d*e^6
)*(d*x + c)^(9/2)*integrate(1/9*sqrt(d*x + c + 1)*sqrt(d*x + c)*sqrt(-d*x - c + 1)/(d^7*x^7*e^6 + 7*c*d^6*x^6*
e^6 + (21*c^2*e^6 - e^6)*d^5*x^5 + 5*(7*c^3*e^6 - c*e^6)*d^4*x^4 + c^7*e^6 + 5*(7*c^4*e^6 - 2*c^2*e^6)*d^3*x^3
- c^5*e^6 + (21*c^5*e^6 - 10*c^3*e^6)*d^2*x^2 + (7*c^6*e^6 - 5*c^4*e^6)*d*x), x) + 4*a*c^3*d*x + a*c^4)*sqrt(
d*x + c))*e^(-9/2*log(d*x + c) - 1/2)/((d^5*x^4*e^5 + 4*c*d^4*x^3*e^5 + 6*c^2*d^3*x^2*e^5 + 4*c^3*d^2*x*e^5 +
c^4*d*e^5)*sqrt(d*x + c))

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Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.71, size = 247, normalized size = 1.78 \begin {gather*} -\frac {2 \, {\left ({\left (21 \, b d^{2} \arcsin \left (d x + c\right ) + 21 \, a d^{2} + 2 \, {\left (5 \, b d^{5} x^{3} + 15 \, b c d^{4} x^{2} + 3 \, {\left (5 \, b c^{2} + b\right )} d^{3} x + {\left (5 \, b c^{3} + 3 \, b c\right )} d^{2}\right )} \sqrt {-d^{2} x^{2} - 2 \, c d x - c^{2} + 1}\right )} \sqrt {d x + c} e^{\frac {1}{2}} + 10 \, {\left (b d^{5} x^{5} + 5 \, b c d^{4} x^{4} + 10 \, b c^{2} d^{3} x^{3} + 10 \, b c^{3} d^{2} x^{2} + 5 \, b c^{4} d x + b c^{5}\right )} \sqrt {-d^{3} e} {\rm weierstrassPInverse}\left (\frac {4}{d^{2}}, 0, \frac {d x + c}{d}\right )\right )} e^{\left (-6\right )}}{189 \, {\left (d^{8} x^{5} + 5 \, c d^{7} x^{4} + 10 \, c^{2} d^{6} x^{3} + 10 \, c^{3} d^{5} x^{2} + 5 \, c^{4} d^{4} x + c^{5} d^{3}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arcsin(d*x+c))/(d*e*x+c*e)^(11/2),x, algorithm="fricas")

[Out]

-2/189*((21*b*d^2*arcsin(d*x + c) + 21*a*d^2 + 2*(5*b*d^5*x^3 + 15*b*c*d^4*x^2 + 3*(5*b*c^2 + b)*d^3*x + (5*b*
c^3 + 3*b*c)*d^2)*sqrt(-d^2*x^2 - 2*c*d*x - c^2 + 1))*sqrt(d*x + c)*e^(1/2) + 10*(b*d^5*x^5 + 5*b*c*d^4*x^4 +
10*b*c^2*d^3*x^3 + 10*b*c^3*d^2*x^2 + 5*b*c^4*d*x + b*c^5)*sqrt(-d^3*e)*weierstrassPInverse(4/d^2, 0, (d*x + c
)/d))*e^(-6)/(d^8*x^5 + 5*c*d^7*x^4 + 10*c^2*d^6*x^3 + 10*c^3*d^5*x^2 + 5*c^4*d^4*x + c^5*d^3)

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*asin(d*x+c))/(d*e*x+c*e)**(11/2),x)

[Out]

Timed out

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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))/(d*e*x+c*e)^(11/2),x, algorithm="giac")

[Out]

integrate((b*arcsin(d*x + c) + a)/(d*e*x + c*e)^(11/2), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {a+b\,\mathrm {asin}\left (c+d\,x\right )}{{\left (c\,e+d\,e\,x\right )}^{11/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*asin(c + d*x))/(c*e + d*e*x)^(11/2),x)

[Out]

int((a + b*asin(c + d*x))/(c*e + d*e*x)^(11/2), x)

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