3.3.95 \(\int \frac {(a+b \text {ArcSin}(c+d x))^2}{\sqrt {c e+d e x}} \, dx\) [295]

Optimal. Leaf size=128 \[ \frac {2 \sqrt {e (c+d x)} (a+b \text {ArcSin}(c+d x))^2}{d e}-\frac {8 b (e (c+d x))^{3/2} (a+b \text {ArcSin}(c+d x)) \, _2F_1\left (\frac {1}{2},\frac {3}{4};\frac {7}{4};(c+d x)^2\right )}{3 d e^2}+\frac {16 b^2 (e (c+d x))^{5/2} \, _3F_2\left (1,\frac {5}{4},\frac {5}{4};\frac {7}{4},\frac {9}{4};(c+d x)^2\right )}{15 d e^3} \]

[Out]

-8/3*b*(e*(d*x+c))^(3/2)*(a+b*arcsin(d*x+c))*hypergeom([1/2, 3/4],[7/4],(d*x+c)^2)/d/e^2+16/15*b^2*(e*(d*x+c))
^(5/2)*hypergeom([1, 5/4, 5/4],[7/4, 9/4],(d*x+c)^2)/d/e^3+2*(a+b*arcsin(d*x+c))^2*(e*(d*x+c))^(1/2)/d/e

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Rubi [A]
time = 0.13, antiderivative size = 128, 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 (e (c+d x))^{5/2} \, _3F_2\left (1,\frac {5}{4},\frac {5}{4};\frac {7}{4},\frac {9}{4};(c+d x)^2\right )}{15 d e^3}-\frac {8 b (e (c+d x))^{3/2} \, _2F_1\left (\frac {1}{2},\frac {3}{4};\frac {7}{4};(c+d x)^2\right ) (a+b \text {ArcSin}(c+d x))}{3 d e^2}+\frac {2 \sqrt {e (c+d x)} (a+b \text {ArcSin}(c+d x))^2}{d e} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[e*(c + d*x)]*(a + b*ArcSin[c + d*x])^2)/(d*e) - (8*b*(e*(c + d*x))^(3/2)*(a + b*ArcSin[c + d*x])*Hyper
geometric2F1[1/2, 3/4, 7/4, (c + d*x)^2])/(3*d*e^2) + (16*b^2*(e*(c + d*x))^(5/2)*HypergeometricPFQ[{1, 5/4, 5
/4}, {7/4, 9/4}, (c + d*x)^2])/(15*d*e^3)

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}{\sqrt {c e+d e x}} \, dx &=\frac {\text {Subst}\left (\int \frac {\left (a+b \sin ^{-1}(x)\right )^2}{\sqrt {e x}} \, dx,x,c+d x\right )}{d}\\ &=\frac {2 \sqrt {e (c+d x)} \left (a+b \sin ^{-1}(c+d x)\right )^2}{d e}-\frac {(4 b) \text {Subst}\left (\int \frac {\sqrt {e x} \left (a+b \sin ^{-1}(x)\right )}{\sqrt {1-x^2}} \, dx,x,c+d x\right )}{d e}\\ &=\frac {2 \sqrt {e (c+d x)} \left (a+b \sin ^{-1}(c+d x)\right )^2}{d e}-\frac {8 b (e (c+d x))^{3/2} \left (a+b \sin ^{-1}(c+d x)\right ) \, _2F_1\left (\frac {1}{2},\frac {3}{4};\frac {7}{4};(c+d x)^2\right )}{3 d e^2}+\frac {16 b^2 (e (c+d x))^{5/2} \, _3F_2\left (1,\frac {5}{4},\frac {5}{4};\frac {7}{4},\frac {9}{4};(c+d x)^2\right )}{15 d e^3}\\ \end {align*}

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Mathematica [A]
time = 0.06, size = 107, normalized size = 0.84 \begin {gather*} \frac {2 \sqrt {e (c+d x)} \left (5 (a+b \text {ArcSin}(c+d x)) \left (3 (a+b \text {ArcSin}(c+d x))-4 b (c+d x) \, _2F_1\left (\frac {1}{2},\frac {3}{4};\frac {7}{4};(c+d x)^2\right )\right )+8 b^2 (c+d x)^2 \, _3F_2\left (1,\frac {5}{4},\frac {5}{4};\frac {7}{4},\frac {9}{4};(c+d x)^2\right )\right )}{15 d e} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[e*(c + d*x)]*(5*(a + b*ArcSin[c + d*x])*(3*(a + b*ArcSin[c + d*x]) - 4*b*(c + d*x)*Hypergeometric2F1[1
/2, 3/4, 7/4, (c + d*x)^2]) + 8*b^2*(c + d*x)^2*HypergeometricPFQ[{1, 5/4, 5/4}, {7/4, 9/4}, (c + d*x)^2]))/(1
5*d*e)

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Maple [F]
time = 180.00, size = 0, normalized size = 0.00 \[\int \frac {\left (a +b \arcsin \left (d x +c \right )\right )^{2}}{\sqrt {d e x +c e}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*arcsin(d*x+c))^2/(d*e*x+c*e)^(1/2),x)

[Out]

int((a+b*arcsin(d*x+c))^2/(d*e*x+c*e)^(1/2),x)

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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))^2/(d*e*x+c*e)^(1/2),x, algorithm="maxima")

[Out]

1/2*(4*sqrt(d*x + c)*b^2*arctan2(d*x + c, sqrt(d*x + c + 1)*sqrt(-d*x - c + 1))^2*e^(1/2) + (4*a*b*d^2*e^(1/2)
*integrate(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^3*x^3*e + 3*c*d^2*x^2*e + 3*c^2*d*x*e + c^3*e - d*x*e - c*e), x) + 8*a*b*c*d*e^(1/2)*integrate
(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^3*x^3*e + 3*c*d^2*x^2*e + 3*c^2*d*x*e + c^3*e - d*x*e - c*e), x) + 4*a*b*c^2*e^(1/2)*integrate(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^3*x^3*e +
3*c*d^2*x^2*e + 3*c^2*d*x*e + c^3*e - d*x*e - c*e), x) - (2*arctan(sqrt(d*x + c))*e^(-1) + e^(-1)*log(sqrt(d*x
+ c) + 1) - e^(-1)*log(sqrt(d*x + c) - 1))*a^2*c^2*e^(1/2)/d + 8*b^2*d*e^(1/2)*integrate(sqrt(d*x + c + 1)*sq
rt(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^3*x^3*e + 3*c*d^2*x^2*e + 3*c^2*d*x*e + c^3*e - d*x*e - c*e), x) + 8*b^2*c*e^(1/2)*integra
te(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/(s
qrt(d*x + c + 1)*sqrt(-d*x - c + 1)))/(d^3*x^3*e + 3*c*d^2*x^2*e + 3*c^2*d*x*e + c^3*e - d*x*e - c*e), x) + 2*
(2*(c + 1)*arctan(sqrt(d*x + c))*e^(-1) + (c - 1)*e^(-1)*log(sqrt(d*x + c) + 1) - (c - 1)*e^(-1)*log(sqrt(d*x
+ c) - 1))*a^2*c*e^(1/2)/d - 4*a*b*e^(1/2)*integrate(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^3*x^3*e + 3*c*d^2*x^2*e + 3*c^2*d*x*e + c^3*e - d*x*e -
c*e), x) - (2*(c^2 + 2*c + 1)*arctan(sqrt(d*x + c))*e^(-1) + (c^2 - 2*c + 1)*e^(-1)*log(sqrt(d*x + c) + 1) - (
c^2 - 2*c + 1)*e^(-1)*log(sqrt(d*x + c) - 1) - 4*sqrt(d*x + c)*e^(-1))*a^2*e^(1/2)/d + (2*arctan(sqrt(d*x + c)
)*e^(-1) + e^(-1)*log(sqrt(d*x + c) + 1) - e^(-1)*log(sqrt(d*x + c) - 1))*a^2*e^(1/2)/d)*d*e)*e^(-1)/d

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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)^(1/2),x, algorithm="fricas")

[Out]

integral((b^2*arcsin(d*x + c)^2 + 2*a*b*arcsin(d*x + c) + a^2)*e^(-1/2)/sqrt(d*x + c), x)

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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \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)**(1/2),x)

[Out]

Exception raised: TypeError >> Invalid comparison of non-real zoo

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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))^2/(d*e*x+c*e)^(1/2),x, algorithm="giac")

[Out]

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

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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}{\sqrt {c\,e+d\,e\,x}} \,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)^(1/2),x)

[Out]

int((a + b*asin(c + d*x))^2/(c*e + d*e*x)^(1/2), x)

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