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

Optimal. Leaf size=120 \[ -\frac{20 b d^{5/2} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{d x}}{\sqrt{d}}\right ),-1\right )}{147 c^{7/2}}+\frac{2 (d x)^{7/2} \left (a+b \sin ^{-1}(c x)\right )}{7 d}+\frac{20 b d^2 \sqrt{1-c^2 x^2} \sqrt{d x}}{147 c^3}+\frac{4 b \sqrt{1-c^2 x^2} (d x)^{5/2}}{49 c} \]

[Out]

(20*b*d^2*Sqrt[d*x]*Sqrt[1 - c^2*x^2])/(147*c^3) + (4*b*(d*x)^(5/2)*Sqrt[1 - c^2*x^2])/(49*c) + (2*(d*x)^(7/2)
*(a + b*ArcSin[c*x]))/(7*d) - (20*b*d^(5/2)*EllipticF[ArcSin[(Sqrt[c]*Sqrt[d*x])/Sqrt[d]], -1])/(147*c^(7/2))

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Rubi [A]  time = 0.075921, antiderivative size = 120, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {4627, 321, 329, 221} \[ \frac{2 (d x)^{7/2} \left (a+b \sin ^{-1}(c x)\right )}{7 d}+\frac{20 b d^2 \sqrt{1-c^2 x^2} \sqrt{d x}}{147 c^3}-\frac{20 b d^{5/2} F\left (\left .\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{d x}}{\sqrt{d}}\right )\right |-1\right )}{147 c^{7/2}}+\frac{4 b \sqrt{1-c^2 x^2} (d x)^{5/2}}{49 c} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(20*b*d^2*Sqrt[d*x]*Sqrt[1 - c^2*x^2])/(147*c^3) + (4*b*(d*x)^(5/2)*Sqrt[1 - c^2*x^2])/(49*c) + (2*(d*x)^(7/2)
*(a + b*ArcSin[c*x]))/(7*d) - (20*b*d^(5/2)*EllipticF[ArcSin[(Sqrt[c]*Sqrt[d*x])/Sqrt[d]], -1])/(147*c^(7/2))

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 329

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 221

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]

Rubi steps

\begin{align*} \int (d x)^{5/2} \left (a+b \sin ^{-1}(c x)\right ) \, dx &=\frac{2 (d x)^{7/2} \left (a+b \sin ^{-1}(c x)\right )}{7 d}-\frac{(2 b c) \int \frac{(d x)^{7/2}}{\sqrt{1-c^2 x^2}} \, dx}{7 d}\\ &=\frac{4 b (d x)^{5/2} \sqrt{1-c^2 x^2}}{49 c}+\frac{2 (d x)^{7/2} \left (a+b \sin ^{-1}(c x)\right )}{7 d}-\frac{(10 b d) \int \frac{(d x)^{3/2}}{\sqrt{1-c^2 x^2}} \, dx}{49 c}\\ &=\frac{20 b d^2 \sqrt{d x} \sqrt{1-c^2 x^2}}{147 c^3}+\frac{4 b (d x)^{5/2} \sqrt{1-c^2 x^2}}{49 c}+\frac{2 (d x)^{7/2} \left (a+b \sin ^{-1}(c x)\right )}{7 d}-\frac{\left (10 b d^3\right ) \int \frac{1}{\sqrt{d x} \sqrt{1-c^2 x^2}} \, dx}{147 c^3}\\ &=\frac{20 b d^2 \sqrt{d x} \sqrt{1-c^2 x^2}}{147 c^3}+\frac{4 b (d x)^{5/2} \sqrt{1-c^2 x^2}}{49 c}+\frac{2 (d x)^{7/2} \left (a+b \sin ^{-1}(c x)\right )}{7 d}-\frac{\left (20 b d^2\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-\frac{c^2 x^4}{d^2}}} \, dx,x,\sqrt{d x}\right )}{147 c^3}\\ &=\frac{20 b d^2 \sqrt{d x} \sqrt{1-c^2 x^2}}{147 c^3}+\frac{4 b (d x)^{5/2} \sqrt{1-c^2 x^2}}{49 c}+\frac{2 (d x)^{7/2} \left (a+b \sin ^{-1}(c x)\right )}{7 d}-\frac{20 b d^{5/2} F\left (\left .\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{d x}}{\sqrt{d}}\right )\right |-1\right )}{147 c^{7/2}}\\ \end{align*}

Mathematica [C]  time = 0.0394848, size = 100, normalized size = 0.83 \[ \frac{2 d^2 \sqrt{d x} \left (-10 b \text{Hypergeometric2F1}\left (\frac{1}{4},\frac{1}{2},\frac{5}{4},c^2 x^2\right )+21 a c^3 x^3+6 b c^2 x^2 \sqrt{1-c^2 x^2}+10 b \sqrt{1-c^2 x^2}+21 b c^3 x^3 \sin ^{-1}(c x)\right )}{147 c^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*d^2*Sqrt[d*x]*(21*a*c^3*x^3 + 10*b*Sqrt[1 - c^2*x^2] + 6*b*c^2*x^2*Sqrt[1 - c^2*x^2] + 21*b*c^3*x^3*ArcSin[
c*x] - 10*b*Hypergeometric2F1[1/4, 1/2, 5/4, c^2*x^2]))/(147*c^3)

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Maple [A]  time = 0.036, size = 144, normalized size = 1.2 \begin{align*} 2\,{\frac{1}{d} \left ( 1/7\, \left ( dx \right ) ^{7/2}a+b \left ( 1/7\, \left ( dx \right ) ^{7/2}\arcsin \left ( cx \right ) -2/7\,{\frac{c}{d} \left ( -1/7\,{\frac{{d}^{2} \left ( dx \right ) ^{5/2}\sqrt{-{c}^{2}{x}^{2}+1}}{{c}^{2}}}-{\frac{5\,{d}^{4}\sqrt{dx}\sqrt{-{c}^{2}{x}^{2}+1}}{21\,{c}^{4}}}+{\frac{5\,{d}^{4}\sqrt{-cx+1}\sqrt{cx+1}}{21\,{c}^{4}\sqrt{-{c}^{2}{x}^{2}+1}}{\it EllipticF} \left ( \sqrt{dx}\sqrt{{\frac{c}{d}}},i \right ){\frac{1}{\sqrt{{\frac{c}{d}}}}}} \right ) } \right ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x)^(5/2)*(a+b*arcsin(c*x)),x)

[Out]

2/d*(1/7*(d*x)^(7/2)*a+b*(1/7*(d*x)^(7/2)*arcsin(c*x)-2/7*c/d*(-1/7/c^2*d^2*(d*x)^(5/2)*(-c^2*x^2+1)^(1/2)-5/2
1/c^4*d^4*(d*x)^(1/2)*(-c^2*x^2+1)^(1/2)+5/21/c^4*d^4/(c/d)^(1/2)*(-c*x+1)^(1/2)*(c*x+1)^(1/2)/(-c^2*x^2+1)^(1
/2)*EllipticF((d*x)^(1/2)*(c/d)^(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*x)^(5/2)*(a+b*arcsin(c*x)),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 (b d^{2} x^{2} \arcsin \left (c x\right ) + a d^{2} x^{2}\right )} \sqrt{d x}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x)^(5/2)*(a+b*arcsin(c*x)),x, algorithm="fricas")

[Out]

integral((b*d^2*x^2*arcsin(c*x) + a*d^2*x^2)*sqrt(d*x), 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*x)**(5/2)*(a+b*asin(c*x)),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (d x\right )^{\frac{5}{2}}{\left (b \arcsin \left (c x\right ) + a\right )}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x)^(5/2)*(a+b*arcsin(c*x)),x, algorithm="giac")

[Out]

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