3.281 \(\int (c e+d e x)^{7/2} (a+b \sin ^{-1}(c+d x)) \, dx\)

Optimal. Leaf size=156 \[ \frac{2 (e (c+d x))^{9/2} \left (a+b \sin ^{-1}(c+d x)\right )}{9 d e}+\frac{28 b e^2 \sqrt{1-(c+d x)^2} (e (c+d x))^{3/2}}{405 d}+\frac{28 b e^3 \sqrt{e (c+d x)} E\left (\left .\sin ^{-1}\left (\frac{\sqrt{-c-d x+1}}{\sqrt{2}}\right )\right |2\right )}{135 d \sqrt{c+d x}}+\frac{4 b \sqrt{1-(c+d x)^2} (e (c+d x))^{7/2}}{81 d} \]

[Out]

(28*b*e^2*(e*(c + d*x))^(3/2)*Sqrt[1 - (c + d*x)^2])/(405*d) + (4*b*(e*(c + d*x))^(7/2)*Sqrt[1 - (c + d*x)^2])
/(81*d) + (2*(e*(c + d*x))^(9/2)*(a + b*ArcSin[c + d*x]))/(9*d*e) + (28*b*e^3*Sqrt[e*(c + d*x)]*EllipticE[ArcS
in[Sqrt[1 - c - d*x]/Sqrt[2]], 2])/(135*d*Sqrt[c + d*x])

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Rubi [A]  time = 0.124094, antiderivative size = 156, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.261, Rules used = {4805, 4627, 321, 320, 318, 424} \[ \frac{2 (e (c+d x))^{9/2} \left (a+b \sin ^{-1}(c+d x)\right )}{9 d e}+\frac{28 b e^2 \sqrt{1-(c+d x)^2} (e (c+d x))^{3/2}}{405 d}+\frac{28 b e^3 \sqrt{e (c+d x)} E\left (\left .\sin ^{-1}\left (\frac{\sqrt{-c-d x+1}}{\sqrt{2}}\right )\right |2\right )}{135 d \sqrt{c+d x}}+\frac{4 b \sqrt{1-(c+d x)^2} (e (c+d x))^{7/2}}{81 d} \]

Antiderivative was successfully verified.

[In]

Int[(c*e + d*e*x)^(7/2)*(a + b*ArcSin[c + d*x]),x]

[Out]

(28*b*e^2*(e*(c + d*x))^(3/2)*Sqrt[1 - (c + d*x)^2])/(405*d) + (4*b*(e*(c + d*x))^(7/2)*Sqrt[1 - (c + d*x)^2])
/(81*d) + (2*(e*(c + d*x))^(9/2)*(a + b*ArcSin[c + d*x]))/(9*d*e) + (28*b*e^3*Sqrt[e*(c + d*x)]*EllipticE[ArcS
in[Sqrt[1 - c - d*x]/Sqrt[2]], 2])/(135*d*Sqrt[c + d*x])

Rule 4805

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]

Rule 4627

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 321

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

Rule 320

Int[Sqrt[(c_)*(x_)]/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Dist[Sqrt[c*x]/Sqrt[x], Int[Sqrt[x]/Sqrt[a + b*x^2
], x], x] /; FreeQ[{a, b, c}, x] && GtQ[-(b/a), 0]

Rule 318

Int[Sqrt[x_]/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Dist[-2/(Sqrt[a]*(-(b/a))^(3/4)), Subst[Int[Sqrt[1 - 2*x^
2]/Sqrt[1 - x^2], x], x, Sqrt[1 - Sqrt[-(b/a)]*x]/Sqrt[2]], x] /; FreeQ[{a, b}, x] && GtQ[-(b/a), 0] && GtQ[a,
0]

Rule 424

Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[(Sqrt[a]*EllipticE[ArcSin[Rt[-(d/c)
, 2]*x], (b*c)/(a*d)])/(Sqrt[c]*Rt[-(d/c), 2]), x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[
a, 0]

Rubi steps

\begin{align*} \int (c e+d e x)^{7/2} \left (a+b \sin ^{-1}(c+d x)\right ) \, dx &=\frac{\operatorname{Subst}\left (\int (e x)^{7/2} \left (a+b \sin ^{-1}(x)\right ) \, dx,x,c+d x\right )}{d}\\ &=\frac{2 (e (c+d x))^{9/2} \left (a+b \sin ^{-1}(c+d x)\right )}{9 d e}-\frac{(2 b) \operatorname{Subst}\left (\int \frac{(e x)^{9/2}}{\sqrt{1-x^2}} \, dx,x,c+d x\right )}{9 d e}\\ &=\frac{4 b (e (c+d x))^{7/2} \sqrt{1-(c+d x)^2}}{81 d}+\frac{2 (e (c+d x))^{9/2} \left (a+b \sin ^{-1}(c+d x)\right )}{9 d e}-\frac{(14 b e) \operatorname{Subst}\left (\int \frac{(e x)^{5/2}}{\sqrt{1-x^2}} \, dx,x,c+d x\right )}{81 d}\\ &=\frac{28 b e^2 (e (c+d x))^{3/2} \sqrt{1-(c+d x)^2}}{405 d}+\frac{4 b (e (c+d x))^{7/2} \sqrt{1-(c+d x)^2}}{81 d}+\frac{2 (e (c+d x))^{9/2} \left (a+b \sin ^{-1}(c+d x)\right )}{9 d e}-\frac{\left (14 b e^3\right ) \operatorname{Subst}\left (\int \frac{\sqrt{e x}}{\sqrt{1-x^2}} \, dx,x,c+d x\right )}{135 d}\\ &=\frac{28 b e^2 (e (c+d x))^{3/2} \sqrt{1-(c+d x)^2}}{405 d}+\frac{4 b (e (c+d x))^{7/2} \sqrt{1-(c+d x)^2}}{81 d}+\frac{2 (e (c+d x))^{9/2} \left (a+b \sin ^{-1}(c+d x)\right )}{9 d e}-\frac{\left (14 b e^3 \sqrt{e (c+d x)}\right ) \operatorname{Subst}\left (\int \frac{\sqrt{x}}{\sqrt{1-x^2}} \, dx,x,c+d x\right )}{135 d \sqrt{c+d x}}\\ &=\frac{28 b e^2 (e (c+d x))^{3/2} \sqrt{1-(c+d x)^2}}{405 d}+\frac{4 b (e (c+d x))^{7/2} \sqrt{1-(c+d x)^2}}{81 d}+\frac{2 (e (c+d x))^{9/2} \left (a+b \sin ^{-1}(c+d x)\right )}{9 d e}+\frac{\left (28 b e^3 \sqrt{e (c+d x)}\right ) \operatorname{Subst}\left (\int \frac{\sqrt{1-2 x^2}}{\sqrt{1-x^2}} \, dx,x,\frac{\sqrt{1-c-d x}}{\sqrt{2}}\right )}{135 d \sqrt{c+d x}}\\ &=\frac{28 b e^2 (e (c+d x))^{3/2} \sqrt{1-(c+d x)^2}}{405 d}+\frac{4 b (e (c+d x))^{7/2} \sqrt{1-(c+d x)^2}}{81 d}+\frac{2 (e (c+d x))^{9/2} \left (a+b \sin ^{-1}(c+d x)\right )}{9 d e}+\frac{28 b e^3 \sqrt{e (c+d x)} E\left (\left .\sin ^{-1}\left (\frac{\sqrt{1-c-d x}}{\sqrt{2}}\right )\right |2\right )}{135 d \sqrt{c+d x}}\\ \end{align*}

Mathematica [C]  time = 0.221958, size = 115, normalized size = 0.74 \[ \frac{2 (e (c+d x))^{7/2} \left (-14 b \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{3}{4},\frac{7}{4},(c+d x)^2\right )+45 a (c+d x)^3+10 b \sqrt{1-(c+d x)^2} (c+d x)^2+14 b \sqrt{1-(c+d x)^2}+45 b (c+d x)^3 \sin ^{-1}(c+d x)\right )}{405 d (c+d x)^2} \]

Antiderivative was successfully verified.

[In]

Integrate[(c*e + d*e*x)^(7/2)*(a + b*ArcSin[c + d*x]),x]

[Out]

(2*(e*(c + d*x))^(7/2)*(45*a*(c + d*x)^3 + 14*b*Sqrt[1 - (c + d*x)^2] + 10*b*(c + d*x)^2*Sqrt[1 - (c + d*x)^2]
+ 45*b*(c + d*x)^3*ArcSin[c + d*x] - 14*b*Hypergeometric2F1[1/2, 3/4, 7/4, (c + d*x)^2]))/(405*d*(c + d*x)^2)

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Maple [C]  time = 0.027, size = 228, normalized size = 1.5 \begin{align*} 2\,{\frac{1}{de} \left ( 1/9\, \left ( dex+ce \right ) ^{9/2}a+b \left ( 1/9\, \left ( dex+ce \right ) ^{9/2}\arcsin \left ({\frac{dex+ce}{e}} \right ) -2/9\,{\frac{1}{e} \left ( -1/9\,{e}^{2} \left ( dex+ce \right ) ^{7/2}\sqrt{-{\frac{ \left ( dex+ce \right ) ^{2}}{{e}^{2}}}+1}-{\frac{7\,{e}^{4} \left ( dex+ce \right ) ^{3/2}}{45}\sqrt{-{\frac{ \left ( dex+ce \right ) ^{2}}{{e}^{2}}}+1}}-{\frac{7\,{e}^{5} \left ({\it EllipticF} \left ( \sqrt{dex+ce}\sqrt{{e}^{-1}},i \right ) -{\it EllipticE} \left ( \sqrt{dex+ce}\sqrt{{e}^{-1}},i \right ) \right ) }{15\,\sqrt{{e}^{-1}}}\sqrt{1-{\frac{dex+ce}{e}}}\sqrt{{\frac{dex+ce}{e}}+1}{\frac{1}{\sqrt{-{\frac{ \left ( dex+ce \right ) ^{2}}{{e}^{2}}}+1}}}} \right ) } \right ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*e*x+c*e)^(7/2)*(a+b*arcsin(d*x+c)),x)

[Out]

2/d/e*(1/9*(d*e*x+c*e)^(9/2)*a+b*(1/9*(d*e*x+c*e)^(9/2)*arcsin((d*e*x+c*e)/e)-2/9/e*(-1/9*e^2*(d*e*x+c*e)^(7/2
)*(-(d*e*x+c*e)^2/e^2+1)^(1/2)-7/45*e^4*(d*e*x+c*e)^(3/2)*(-(d*e*x+c*e)^2/e^2+1)^(1/2)-7/15*e^5/(1/e)^(1/2)*(1
-(d*e*x+c*e)/e)^(1/2)*((d*e*x+c*e)/e+1)^(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)-EllipticE((d*e*x+c*e)^(1/2)*(1/e)^(1/2),I)))))

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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((d*e*x+c*e)^(7/2)*(a+b*arcsin(d*x+c)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (a d^{3} e^{3} x^{3} + 3 \, a c d^{2} e^{3} x^{2} + 3 \, a c^{2} d e^{3} x + a c^{3} e^{3} +{\left (b d^{3} e^{3} x^{3} + 3 \, b c d^{2} e^{3} x^{2} + 3 \, b c^{2} d e^{3} x + b c^{3} e^{3}\right )} \arcsin \left (d x + c\right )\right )} \sqrt{d e x + c e}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*e*x+c*e)^(7/2)*(a+b*arcsin(d*x+c)),x, algorithm="fricas")

[Out]

integral((a*d^3*e^3*x^3 + 3*a*c*d^2*e^3*x^2 + 3*a*c^2*d*e^3*x + a*c^3*e^3 + (b*d^3*e^3*x^3 + 3*b*c*d^2*e^3*x^2
+ 3*b*c^2*d*e^3*x + b*c^3*e^3)*arcsin(d*x + c))*sqrt(d*e*x + c*e), x)

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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((d*e*x+c*e)**(7/2)*(a+b*asin(d*x+c)),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (d e x + c e\right )}^{\frac{7}{2}}{\left (b \arcsin \left (d x + c\right ) + a\right )}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*e*x+c*e)^(7/2)*(a+b*arcsin(d*x+c)),x, algorithm="giac")

[Out]

integrate((d*e*x + c*e)^(7/2)*(b*arcsin(d*x + c) + a), x)