3.774 \(\int \frac{(a+b \sin (e+f x))^{3/2}}{\sqrt{c+d \sin (e+f x)}} \, dx\)

Optimal. Leaf size=644 \[ \frac{\sqrt{a+b} (b (c-d)-2 a d) \sec (e+f x) (c+d \sin (e+f x)) \sqrt{\frac{(b c-a d) (1-\sin (e+f x))}{(a+b) (c+d \sin (e+f x))}} \sqrt{-\frac{(b c-a d) (\sin (e+f x)+1)}{(a-b) (c+d \sin (e+f x))}} F\left (\sin ^{-1}\left (\frac{\sqrt{c+d} \sqrt{a+b \sin (e+f x)}}{\sqrt{a+b} \sqrt{c+d \sin (e+f x)}}\right )|\frac{(a+b) (c-d)}{(a-b) (c+d)}\right )}{d^2 f \sqrt{c+d}}-\frac{\sqrt{a+b} (b c-3 a d) \sec (e+f x) (c+d \sin (e+f x)) \sqrt{\frac{(b c-a d) (1-\sin (e+f x))}{(a+b) (c+d \sin (e+f x))}} \sqrt{-\frac{(b c-a d) (\sin (e+f x)+1)}{(a-b) (c+d \sin (e+f x))}} \Pi \left (\frac{(a+b) d}{b (c+d)};\sin ^{-1}\left (\frac{\sqrt{c+d} \sqrt{a+b \sin (e+f x)}}{\sqrt{a+b} \sqrt{c+d \sin (e+f x)}}\right )|\frac{(a+b) (c-d)}{(a-b) (c+d)}\right )}{d^2 f \sqrt{c+d}}-\frac{b \cos (e+f x) \sqrt{a+b \sin (e+f x)}}{f \sqrt{c+d \sin (e+f x)}}-\frac{b (a-b) \sqrt{a+b} \sqrt{c+d} \sec (e+f x) (c+d \sin (e+f x)) \sqrt{\frac{(b c-a d) (1-\sin (e+f x))}{(a+b) (c+d \sin (e+f x))}} \sqrt{-\frac{(b c-a d) (\sin (e+f x)+1)}{(a-b) (c+d \sin (e+f x))}} E\left (\sin ^{-1}\left (\frac{\sqrt{c+d} \sqrt{a+b \sin (e+f x)}}{\sqrt{a+b} \sqrt{c+d \sin (e+f x)}}\right )|\frac{(a+b) (c-d)}{(a-b) (c+d)}\right )}{d f (b c-a d)} \]

[Out]

-((b*Cos[e + f*x]*Sqrt[a + b*Sin[e + f*x]])/(f*Sqrt[c + d*Sin[e + f*x]])) - ((a - b)*b*Sqrt[a + b]*Sqrt[c + d]
*EllipticE[ArcSin[(Sqrt[c + d]*Sqrt[a + b*Sin[e + f*x]])/(Sqrt[a + b]*Sqrt[c + d*Sin[e + f*x]])], ((a + b)*(c
- d))/((a - b)*(c + d))]*Sec[e + f*x]*Sqrt[((b*c - a*d)*(1 - Sin[e + f*x]))/((a + b)*(c + d*Sin[e + f*x]))]*Sq
rt[-(((b*c - a*d)*(1 + Sin[e + f*x]))/((a - b)*(c + d*Sin[e + f*x])))]*(c + d*Sin[e + f*x]))/(d*(b*c - a*d)*f)
+ (Sqrt[a + b]*(b*(c - d) - 2*a*d)*EllipticF[ArcSin[(Sqrt[c + d]*Sqrt[a + b*Sin[e + f*x]])/(Sqrt[a + b]*Sqrt[
c + d*Sin[e + f*x]])], ((a + b)*(c - d))/((a - b)*(c + d))]*Sec[e + f*x]*Sqrt[((b*c - a*d)*(1 - Sin[e + f*x]))
/((a + b)*(c + d*Sin[e + f*x]))]*Sqrt[-(((b*c - a*d)*(1 + Sin[e + f*x]))/((a - b)*(c + d*Sin[e + f*x])))]*(c +
d*Sin[e + f*x]))/(d^2*Sqrt[c + d]*f) - (Sqrt[a + b]*(b*c - 3*a*d)*EllipticPi[((a + b)*d)/(b*(c + d)), ArcSin[
(Sqrt[c + d]*Sqrt[a + b*Sin[e + f*x]])/(Sqrt[a + b]*Sqrt[c + d*Sin[e + f*x]])], ((a + b)*(c - d))/((a - b)*(c
+ d))]*Sec[e + f*x]*Sqrt[((b*c - a*d)*(1 - Sin[e + f*x]))/((a + b)*(c + d*Sin[e + f*x]))]*Sqrt[-(((b*c - a*d)*
(1 + Sin[e + f*x]))/((a - b)*(c + d*Sin[e + f*x])))]*(c + d*Sin[e + f*x]))/(d^2*Sqrt[c + d]*f)

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Rubi [A]  time = 1.53792, antiderivative size = 644, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.207, Rules used = {2821, 3053, 2811, 2998, 2818, 2996} \[ \frac{\sqrt{a+b} (b (c-d)-2 a d) \sec (e+f x) (c+d \sin (e+f x)) \sqrt{\frac{(b c-a d) (1-\sin (e+f x))}{(a+b) (c+d \sin (e+f x))}} \sqrt{-\frac{(b c-a d) (\sin (e+f x)+1)}{(a-b) (c+d \sin (e+f x))}} F\left (\sin ^{-1}\left (\frac{\sqrt{c+d} \sqrt{a+b \sin (e+f x)}}{\sqrt{a+b} \sqrt{c+d \sin (e+f x)}}\right )|\frac{(a+b) (c-d)}{(a-b) (c+d)}\right )}{d^2 f \sqrt{c+d}}-\frac{\sqrt{a+b} (b c-3 a d) \sec (e+f x) (c+d \sin (e+f x)) \sqrt{\frac{(b c-a d) (1-\sin (e+f x))}{(a+b) (c+d \sin (e+f x))}} \sqrt{-\frac{(b c-a d) (\sin (e+f x)+1)}{(a-b) (c+d \sin (e+f x))}} \Pi \left (\frac{(a+b) d}{b (c+d)};\sin ^{-1}\left (\frac{\sqrt{c+d} \sqrt{a+b \sin (e+f x)}}{\sqrt{a+b} \sqrt{c+d \sin (e+f x)}}\right )|\frac{(a+b) (c-d)}{(a-b) (c+d)}\right )}{d^2 f \sqrt{c+d}}-\frac{b \cos (e+f x) \sqrt{a+b \sin (e+f x)}}{f \sqrt{c+d \sin (e+f x)}}-\frac{b (a-b) \sqrt{a+b} \sqrt{c+d} \sec (e+f x) (c+d \sin (e+f x)) \sqrt{\frac{(b c-a d) (1-\sin (e+f x))}{(a+b) (c+d \sin (e+f x))}} \sqrt{-\frac{(b c-a d) (\sin (e+f x)+1)}{(a-b) (c+d \sin (e+f x))}} E\left (\sin ^{-1}\left (\frac{\sqrt{c+d} \sqrt{a+b \sin (e+f x)}}{\sqrt{a+b} \sqrt{c+d \sin (e+f x)}}\right )|\frac{(a+b) (c-d)}{(a-b) (c+d)}\right )}{d f (b c-a d)} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*Sin[e + f*x])^(3/2)/Sqrt[c + d*Sin[e + f*x]],x]

[Out]

-((b*Cos[e + f*x]*Sqrt[a + b*Sin[e + f*x]])/(f*Sqrt[c + d*Sin[e + f*x]])) - ((a - b)*b*Sqrt[a + b]*Sqrt[c + d]
*EllipticE[ArcSin[(Sqrt[c + d]*Sqrt[a + b*Sin[e + f*x]])/(Sqrt[a + b]*Sqrt[c + d*Sin[e + f*x]])], ((a + b)*(c
- d))/((a - b)*(c + d))]*Sec[e + f*x]*Sqrt[((b*c - a*d)*(1 - Sin[e + f*x]))/((a + b)*(c + d*Sin[e + f*x]))]*Sq
rt[-(((b*c - a*d)*(1 + Sin[e + f*x]))/((a - b)*(c + d*Sin[e + f*x])))]*(c + d*Sin[e + f*x]))/(d*(b*c - a*d)*f)
+ (Sqrt[a + b]*(b*(c - d) - 2*a*d)*EllipticF[ArcSin[(Sqrt[c + d]*Sqrt[a + b*Sin[e + f*x]])/(Sqrt[a + b]*Sqrt[
c + d*Sin[e + f*x]])], ((a + b)*(c - d))/((a - b)*(c + d))]*Sec[e + f*x]*Sqrt[((b*c - a*d)*(1 - Sin[e + f*x]))
/((a + b)*(c + d*Sin[e + f*x]))]*Sqrt[-(((b*c - a*d)*(1 + Sin[e + f*x]))/((a - b)*(c + d*Sin[e + f*x])))]*(c +
d*Sin[e + f*x]))/(d^2*Sqrt[c + d]*f) - (Sqrt[a + b]*(b*c - 3*a*d)*EllipticPi[((a + b)*d)/(b*(c + d)), ArcSin[
(Sqrt[c + d]*Sqrt[a + b*Sin[e + f*x]])/(Sqrt[a + b]*Sqrt[c + d*Sin[e + f*x]])], ((a + b)*(c - d))/((a - b)*(c
+ d))]*Sec[e + f*x]*Sqrt[((b*c - a*d)*(1 - Sin[e + f*x]))/((a + b)*(c + d*Sin[e + f*x]))]*Sqrt[-(((b*c - a*d)*
(1 + Sin[e + f*x]))/((a - b)*(c + d*Sin[e + f*x])))]*(c + d*Sin[e + f*x]))/(d^2*Sqrt[c + d]*f)

Rule 2821

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> -S
imp[(b*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x])^n)/(f*(m + n)), x] + Dist[1/(d*(m + n)),
Int[(a + b*Sin[e + f*x])^(m - 2)*(c + d*Sin[e + f*x])^(n - 1)*Simp[a^2*c*d*(m + n) + b*d*(b*c*(m - 1) + a*d*n
) + (a*d*(2*b*c + a*d)*(m + n) - b*d*(a*c - b*d*(m + n - 1)))*Sin[e + f*x] + b*d*(b*c*n + a*d*(2*m + n - 1))*S
in[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2
- d^2, 0] && LtQ[0, m, 2] && LtQ[-1, n, 2] && NeQ[m + n, 0] && (IntegerQ[m] || IntegersQ[2*m, 2*n])

Rule 3053

Int[((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^2)/(((a_.) + (b_.)*sin[(e_.) + (f_.
)*(x_)])^(3/2)*Sqrt[(c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]]), x_Symbol] :> Dist[C/b^2, Int[Sqrt[a + b*Sin[e + f
*x]]/Sqrt[c + d*Sin[e + f*x]], x], x] + Dist[1/b^2, Int[(A*b^2 - a^2*C + b*(b*B - 2*a*C)*Sin[e + f*x])/((a + b
*Sin[e + f*x])^(3/2)*Sqrt[c + d*Sin[e + f*x]]), x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C}, x] && NeQ[b*c - a
*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0]

Rule 2811

Int[Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]/Sqrt[(c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]], x_Symbol] :> Simp[
(2*(a + b*Sin[e + f*x])*Sqrt[((b*c - a*d)*(1 + Sin[e + f*x]))/((c - d)*(a + b*Sin[e + f*x]))]*Sqrt[-(((b*c - a
*d)*(1 - Sin[e + f*x]))/((c + d)*(a + b*Sin[e + f*x])))]*EllipticPi[(b*(c + d))/(d*(a + b)), ArcSin[(Rt[(a + b
)/(c + d), 2]*Sqrt[c + d*Sin[e + f*x]])/Sqrt[a + b*Sin[e + f*x]]], ((a - b)*(c + d))/((a + b)*(c - d))])/(d*f*
Rt[(a + b)/(c + d), 2]*Cos[e + f*x]), x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2
, 0] && NeQ[c^2 - d^2, 0] && PosQ[(a + b)/(c + d)]

Rule 2998

Int[((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])/(((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(3/2)*Sqrt[(c_) + (d_.)*s
in[(e_.) + (f_.)*(x_)]]), x_Symbol] :> Dist[(A - B)/(a - b), Int[1/(Sqrt[a + b*Sin[e + f*x]]*Sqrt[c + d*Sin[e
+ f*x]]), x], x] - Dist[(A*b - a*B)/(a - b), Int[(1 + Sin[e + f*x])/((a + b*Sin[e + f*x])^(3/2)*Sqrt[c + d*Sin
[e + f*x]]), x], x] /; FreeQ[{a, b, c, d, e, f, A, B}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2
- d^2, 0] && NeQ[A, B]

Rule 2818

Int[1/(Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]*Sqrt[(c_) + (d_.)*sin[(e_.) + (f_.)*(x_)]]), x_Symbol] :> Si
mp[(2*(c + d*Sin[e + f*x])*Sqrt[((b*c - a*d)*(1 - Sin[e + f*x]))/((a + b)*(c + d*Sin[e + f*x]))]*Sqrt[-(((b*c
- a*d)*(1 + Sin[e + f*x]))/((a - b)*(c + d*Sin[e + f*x])))]*EllipticF[ArcSin[Rt[(c + d)/(a + b), 2]*(Sqrt[a +
b*Sin[e + f*x]]/Sqrt[c + d*Sin[e + f*x]])], ((a + b)*(c - d))/((a - b)*(c + d))])/(f*(b*c - a*d)*Rt[(c + d)/(a
+ b), 2]*Cos[e + f*x]), x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c
^2 - d^2, 0] && PosQ[(c + d)/(a + b)]

Rule 2996

Int[((A_) + (B_.)*sin[(e_.) + (f_.)*(x_)])/(((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(3/2)*Sqrt[(c_) + (d_.)*sin
[(e_.) + (f_.)*(x_)]]), x_Symbol] :> Simp[(-2*A*(c - d)*(a + b*Sin[e + f*x])*Sqrt[((b*c - a*d)*(1 + Sin[e + f*
x]))/((c - d)*(a + b*Sin[e + f*x]))]*Sqrt[-(((b*c - a*d)*(1 - Sin[e + f*x]))/((c + d)*(a + b*Sin[e + f*x])))]*
EllipticE[ArcSin[(Rt[(a + b)/(c + d), 2]*Sqrt[c + d*Sin[e + f*x]])/Sqrt[a + b*Sin[e + f*x]]], ((a - b)*(c + d)
)/((a + b)*(c - d))])/(f*(b*c - a*d)^2*Rt[(a + b)/(c + d), 2]*Cos[e + f*x]), x] /; FreeQ[{a, b, c, d, e, f, A,
B}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && EqQ[A, B] && PosQ[(a + b)/(c + d)]

Rubi steps

\begin{align*} \int \frac{(a+b \sin (e+f x))^{3/2}}{\sqrt{c+d \sin (e+f x)}} \, dx &=-\frac{b \cos (e+f x) \sqrt{a+b \sin (e+f x)}}{f \sqrt{c+d \sin (e+f x)}}+\frac{\int \frac{\frac{1}{2} d \left (2 a^2 c+b^2 c-a b d\right )+a d (b c+a d) \sin (e+f x)-\frac{1}{2} b d (b c-3 a d) \sin ^2(e+f x)}{\sqrt{a+b \sin (e+f x)} (c+d \sin (e+f x))^{3/2}} \, dx}{d}\\ &=-\frac{b \cos (e+f x) \sqrt{a+b \sin (e+f x)}}{f \sqrt{c+d \sin (e+f x)}}+\frac{\int \frac{\frac{1}{2} b c^2 d (b c-3 a d)+\frac{1}{2} d^3 \left (2 a^2 c+b^2 c-a b d\right )+d \left (b c d (b c-3 a d)+a d^2 (b c+a d)\right ) \sin (e+f x)}{\sqrt{a+b \sin (e+f x)} (c+d \sin (e+f x))^{3/2}} \, dx}{d^3}-\frac{(b (b c-3 a d)) \int \frac{\sqrt{c+d \sin (e+f x)}}{\sqrt{a+b \sin (e+f x)}} \, dx}{2 d^2}\\ &=-\frac{b \cos (e+f x) \sqrt{a+b \sin (e+f x)}}{f \sqrt{c+d \sin (e+f x)}}-\frac{\sqrt{a+b} (b c-3 a d) \Pi \left (\frac{(a+b) d}{b (c+d)};\sin ^{-1}\left (\frac{\sqrt{c+d} \sqrt{a+b \sin (e+f x)}}{\sqrt{a+b} \sqrt{c+d \sin (e+f x)}}\right )|\frac{(a+b) (c-d)}{(a-b) (c+d)}\right ) \sec (e+f x) \sqrt{\frac{(b c-a d) (1-\sin (e+f x))}{(a+b) (c+d \sin (e+f x))}} \sqrt{-\frac{(b c-a d) (1+\sin (e+f x))}{(a-b) (c+d \sin (e+f x))}} (c+d \sin (e+f x))}{d^2 \sqrt{c+d} f}+\frac{(b (c+d) (b c-a d)) \int \frac{1+\sin (e+f x)}{\sqrt{a+b \sin (e+f x)} (c+d \sin (e+f x))^{3/2}} \, dx}{2 d}+\frac{\left (\frac{1}{2} b c^2 d (b c-3 a d)+\frac{1}{2} d^3 \left (2 a^2 c+b^2 c-a b d\right )-d \left (b c d (b c-3 a d)+a d^2 (b c+a d)\right )\right ) \int \frac{1}{\sqrt{a+b \sin (e+f x)} \sqrt{c+d \sin (e+f x)}} \, dx}{(c-d) d^3}\\ &=-\frac{b \cos (e+f x) \sqrt{a+b \sin (e+f x)}}{f \sqrt{c+d \sin (e+f x)}}-\frac{(a-b) b \sqrt{a+b} \sqrt{c+d} E\left (\sin ^{-1}\left (\frac{\sqrt{c+d} \sqrt{a+b \sin (e+f x)}}{\sqrt{a+b} \sqrt{c+d \sin (e+f x)}}\right )|\frac{(a+b) (c-d)}{(a-b) (c+d)}\right ) \sec (e+f x) \sqrt{\frac{(b c-a d) (1-\sin (e+f x))}{(a+b) (c+d \sin (e+f x))}} \sqrt{-\frac{(b c-a d) (1+\sin (e+f x))}{(a-b) (c+d \sin (e+f x))}} (c+d \sin (e+f x))}{d (b c-a d) f}+\frac{\sqrt{a+b} (b (c-d)-2 a d) F\left (\sin ^{-1}\left (\frac{\sqrt{c+d} \sqrt{a+b \sin (e+f x)}}{\sqrt{a+b} \sqrt{c+d \sin (e+f x)}}\right )|\frac{(a+b) (c-d)}{(a-b) (c+d)}\right ) \sec (e+f x) \sqrt{\frac{(b c-a d) (1-\sin (e+f x))}{(a+b) (c+d \sin (e+f x))}} \sqrt{-\frac{(b c-a d) (1+\sin (e+f x))}{(a-b) (c+d \sin (e+f x))}} (c+d \sin (e+f x))}{d^2 \sqrt{c+d} f}-\frac{\sqrt{a+b} (b c-3 a d) \Pi \left (\frac{(a+b) d}{b (c+d)};\sin ^{-1}\left (\frac{\sqrt{c+d} \sqrt{a+b \sin (e+f x)}}{\sqrt{a+b} \sqrt{c+d \sin (e+f x)}}\right )|\frac{(a+b) (c-d)}{(a-b) (c+d)}\right ) \sec (e+f x) \sqrt{\frac{(b c-a d) (1-\sin (e+f x))}{(a+b) (c+d \sin (e+f x))}} \sqrt{-\frac{(b c-a d) (1+\sin (e+f x))}{(a-b) (c+d \sin (e+f x))}} (c+d \sin (e+f x))}{d^2 \sqrt{c+d} f}\\ \end{align*}

Mathematica [C]  time = 32.6955, size = 222963, normalized size = 346.22 \[ \text{Result too large to show} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[(a + b*Sin[e + f*x])^(3/2)/Sqrt[c + d*Sin[e + f*x]],x]

[Out]

Result too large to show

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Maple [B]  time = 8.722, size = 529273, normalized size = 821.9 \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*sin(f*x+e))^(3/2)/(c+d*sin(f*x+e))^(1/2),x)

[Out]

result too large to display

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b \sin \left (f x + e\right ) + a\right )}^{\frac{3}{2}}}{\sqrt{d \sin \left (f x + e\right ) + c}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*sin(f*x+e))^(3/2)/(c+d*sin(f*x+e))^(1/2),x, algorithm="maxima")

[Out]

integrate((b*sin(f*x + e) + a)^(3/2)/sqrt(d*sin(f*x + e) + c), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*sin(f*x+e))^(3/2)/(c+d*sin(f*x+e))^(1/2),x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (a + b \sin{\left (e + f x \right )}\right )^{\frac{3}{2}}}{\sqrt{c + d \sin{\left (e + f x \right )}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*sin(f*x+e))**(3/2)/(c+d*sin(f*x+e))**(1/2),x)

[Out]

Integral((a + b*sin(e + f*x))**(3/2)/sqrt(c + d*sin(e + f*x)), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b \sin \left (f x + e\right ) + a\right )}^{\frac{3}{2}}}{\sqrt{d \sin \left (f x + e\right ) + c}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*sin(f*x+e))^(3/2)/(c+d*sin(f*x+e))^(1/2),x, algorithm="giac")

[Out]

integrate((b*sin(f*x + e) + a)^(3/2)/sqrt(d*sin(f*x + e) + c), x)