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

Optimal. Leaf size=347 \[ -\frac{2 \left (c^2-d^2\right ) \left (35 a^2 d^2+42 a b c d+b^2 \left (-\left (6 c^2-25 d^2\right )\right )\right ) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left (\frac{1}{2} \left (e+f x-\frac{\pi }{2}\right )|\frac{2 d}{c+d}\right )}{105 d^2 f \sqrt{c+d \sin (e+f x)}}+\frac{4 \left (70 a^2 c d^2+21 a b d \left (c^2+3 d^2\right )+b^2 \left (-\left (3 c^3-41 c d^2\right )\right )\right ) \sqrt{c+d \sin (e+f x)} E\left (\frac{1}{2} \left (e+f x-\frac{\pi }{2}\right )|\frac{2 d}{c+d}\right )}{105 d^2 f \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}-\frac{2 \left (5 d^2 \left (7 a^2+5 b^2\right )-6 b c (b c-7 a d)\right ) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{105 d f}+\frac{4 b (b c-7 a d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{35 d f}-\frac{2 b^2 \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{7 d f} \]

[Out]

(-2*(5*(7*a^2 + 5*b^2)*d^2 - 6*b*c*(b*c - 7*a*d))*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(105*d*f) + (4*b*(b*c
- 7*a*d)*Cos[e + f*x]*(c + d*Sin[e + f*x])^(3/2))/(35*d*f) - (2*b^2*Cos[e + f*x]*(c + d*Sin[e + f*x])^(5/2))/
(7*d*f) + (4*(70*a^2*c*d^2 + 21*a*b*d*(c^2 + 3*d^2) - b^2*(3*c^3 - 41*c*d^2))*EllipticE[(e - Pi/2 + f*x)/2, (2
*d)/(c + d)]*Sqrt[c + d*Sin[e + f*x]])/(105*d^2*f*Sqrt[(c + d*Sin[e + f*x])/(c + d)]) - (2*(c^2 - d^2)*(42*a*b
*c*d + 35*a^2*d^2 - b^2*(6*c^2 - 25*d^2))*EllipticF[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x
])/(c + d)])/(105*d^2*f*Sqrt[c + d*Sin[e + f*x]])

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Rubi [A]  time = 0.633734, antiderivative size = 347, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.259, Rules used = {2791, 2753, 2752, 2663, 2661, 2655, 2653} \[ -\frac{2 \left (c^2-d^2\right ) \left (35 a^2 d^2+42 a b c d+b^2 \left (-\left (6 c^2-25 d^2\right )\right )\right ) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left (\frac{1}{2} \left (e+f x-\frac{\pi }{2}\right )|\frac{2 d}{c+d}\right )}{105 d^2 f \sqrt{c+d \sin (e+f x)}}+\frac{4 \left (70 a^2 c d^2+21 a b d \left (c^2+3 d^2\right )+b^2 \left (-\left (3 c^3-41 c d^2\right )\right )\right ) \sqrt{c+d \sin (e+f x)} E\left (\frac{1}{2} \left (e+f x-\frac{\pi }{2}\right )|\frac{2 d}{c+d}\right )}{105 d^2 f \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}-\frac{2 \left (5 d^2 \left (7 a^2+5 b^2\right )-6 b c (b c-7 a d)\right ) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{105 d f}+\frac{4 b (b c-7 a d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{35 d f}-\frac{2 b^2 \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{7 d f} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*(5*(7*a^2 + 5*b^2)*d^2 - 6*b*c*(b*c - 7*a*d))*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(105*d*f) + (4*b*(b*c
- 7*a*d)*Cos[e + f*x]*(c + d*Sin[e + f*x])^(3/2))/(35*d*f) - (2*b^2*Cos[e + f*x]*(c + d*Sin[e + f*x])^(5/2))/
(7*d*f) + (4*(70*a^2*c*d^2 + 21*a*b*d*(c^2 + 3*d^2) - b^2*(3*c^3 - 41*c*d^2))*EllipticE[(e - Pi/2 + f*x)/2, (2
*d)/(c + d)]*Sqrt[c + d*Sin[e + f*x]])/(105*d^2*f*Sqrt[(c + d*Sin[e + f*x])/(c + d)]) - (2*(c^2 - d^2)*(42*a*b
*c*d + 35*a^2*d^2 - b^2*(6*c^2 - 25*d^2))*EllipticF[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x
])/(c + d)])/(105*d^2*f*Sqrt[c + d*Sin[e + f*x]])

Rule 2791

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

Rule 2753

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

Rule 2752

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

Rule 2663

Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Dist[Sqrt[(a + b*Sin[c + d*x])/(a + b)]/Sqrt[a
+ b*Sin[c + d*x]], Int[1/Sqrt[a/(a + b) + (b*Sin[c + d*x])/(a + b)], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a
^2 - b^2, 0] &&  !GtQ[a + b, 0]

Rule 2661

Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2*EllipticF[(1*(c - Pi/2 + d*x))/2, (2*b)
/(a + b)])/(d*Sqrt[a + b]), x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && GtQ[a + b, 0]

Rule 2655

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

Rule 2653

Int[Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2*Sqrt[a + b]*EllipticE[(1*(c - Pi/2 + d*x)
)/2, (2*b)/(a + b)])/d, x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && GtQ[a + b, 0]

Rubi steps

\begin{align*} \int (a+b \sin (e+f x))^2 (c+d \sin (e+f x))^{3/2} \, dx &=-\frac{2 b^2 \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{7 d f}+\frac{2 \int (c+d \sin (e+f x))^{3/2} \left (\frac{1}{2} \left (7 a^2+5 b^2\right ) d-b (b c-7 a d) \sin (e+f x)\right ) \, dx}{7 d}\\ &=\frac{4 b (b c-7 a d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{35 d f}-\frac{2 b^2 \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{7 d f}+\frac{4 \int \sqrt{c+d \sin (e+f x)} \left (\frac{1}{4} d \left (35 a^2 c+19 b^2 c+42 a b d\right )+\frac{1}{4} \left (5 \left (7 a^2+5 b^2\right ) d^2-6 b c (b c-7 a d)\right ) \sin (e+f x)\right ) \, dx}{35 d}\\ &=-\frac{2 \left (5 \left (7 a^2+5 b^2\right ) d^2-6 b c (b c-7 a d)\right ) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{105 d f}+\frac{4 b (b c-7 a d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{35 d f}-\frac{2 b^2 \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{7 d f}+\frac{8 \int \frac{\frac{1}{8} d \left (168 a b c d+35 a^2 \left (3 c^2+d^2\right )+b^2 \left (51 c^2+25 d^2\right )\right )+\frac{1}{4} \left (70 a^2 c d^2+21 a b d \left (c^2+3 d^2\right )-b^2 \left (3 c^3-41 c d^2\right )\right ) \sin (e+f x)}{\sqrt{c+d \sin (e+f x)}} \, dx}{105 d}\\ &=-\frac{2 \left (5 \left (7 a^2+5 b^2\right ) d^2-6 b c (b c-7 a d)\right ) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{105 d f}+\frac{4 b (b c-7 a d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{35 d f}-\frac{2 b^2 \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{7 d f}-\frac{\left (\left (c^2-d^2\right ) \left (42 a b c d+35 a^2 d^2-b^2 \left (6 c^2-25 d^2\right )\right )\right ) \int \frac{1}{\sqrt{c+d \sin (e+f x)}} \, dx}{105 d^2}+\frac{\left (2 \left (70 a^2 c d^2+21 a b d \left (c^2+3 d^2\right )-b^2 \left (3 c^3-41 c d^2\right )\right )\right ) \int \sqrt{c+d \sin (e+f x)} \, dx}{105 d^2}\\ &=-\frac{2 \left (5 \left (7 a^2+5 b^2\right ) d^2-6 b c (b c-7 a d)\right ) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{105 d f}+\frac{4 b (b c-7 a d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{35 d f}-\frac{2 b^2 \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{7 d f}+\frac{\left (2 \left (70 a^2 c d^2+21 a b d \left (c^2+3 d^2\right )-b^2 \left (3 c^3-41 c d^2\right )\right ) \sqrt{c+d \sin (e+f x)}\right ) \int \sqrt{\frac{c}{c+d}+\frac{d \sin (e+f x)}{c+d}} \, dx}{105 d^2 \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}-\frac{\left (\left (c^2-d^2\right ) \left (42 a b c d+35 a^2 d^2-b^2 \left (6 c^2-25 d^2\right )\right ) \sqrt{\frac{c+d \sin (e+f x)}{c+d}}\right ) \int \frac{1}{\sqrt{\frac{c}{c+d}+\frac{d \sin (e+f x)}{c+d}}} \, dx}{105 d^2 \sqrt{c+d \sin (e+f x)}}\\ &=-\frac{2 \left (5 \left (7 a^2+5 b^2\right ) d^2-6 b c (b c-7 a d)\right ) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{105 d f}+\frac{4 b (b c-7 a d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{35 d f}-\frac{2 b^2 \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{7 d f}+\frac{4 \left (70 a^2 c d^2+21 a b d \left (c^2+3 d^2\right )-b^2 \left (3 c^3-41 c d^2\right )\right ) E\left (\frac{1}{2} \left (e-\frac{\pi }{2}+f x\right )|\frac{2 d}{c+d}\right ) \sqrt{c+d \sin (e+f x)}}{105 d^2 f \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}-\frac{2 \left (c^2-d^2\right ) \left (42 a b c d+35 a^2 d^2-b^2 \left (6 c^2-25 d^2\right )\right ) F\left (\frac{1}{2} \left (e-\frac{\pi }{2}+f x\right )|\frac{2 d}{c+d}\right ) \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}{105 d^2 f \sqrt{c+d \sin (e+f x)}}\\ \end{align*}

Mathematica [A]  time = 1.17567, size = 292, normalized size = 0.84 \[ \frac{4 \sqrt{\frac{c+d \sin (e+f x)}{c+d}} \left (d^2 \left (-\left (35 a^2 \left (3 c^2+d^2\right )+168 a b c d+b^2 \left (51 c^2+25 d^2\right )\right )\right ) F\left (\frac{1}{4} (-2 e-2 f x+\pi )|\frac{2 d}{c+d}\right )-2 \left (70 a^2 c d^2+21 a b d \left (c^2+3 d^2\right )+b^2 \left (41 c d^2-3 c^3\right )\right ) \left ((c+d) E\left (\frac{1}{4} (-2 e-2 f x+\pi )|\frac{2 d}{c+d}\right )-c F\left (\frac{1}{4} (-2 e-2 f x+\pi )|\frac{2 d}{c+d}\right )\right )\right )-d (c+d \sin (e+f x)) \left (\left (140 a^2 d^2+336 a b c d+b^2 \left (12 c^2+115 d^2\right )\right ) \cos (e+f x)+3 b d (4 (7 a d+4 b c) \sin (2 (e+f x))-5 b d \cos (3 (e+f x)))\right )}{210 d^2 f \sqrt{c+d \sin (e+f x)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(4*(-(d^2*(168*a*b*c*d + 35*a^2*(3*c^2 + d^2) + b^2*(51*c^2 + 25*d^2))*EllipticF[(-2*e + Pi - 2*f*x)/4, (2*d)/
(c + d)]) - 2*(70*a^2*c*d^2 + 21*a*b*d*(c^2 + 3*d^2) + b^2*(-3*c^3 + 41*c*d^2))*((c + d)*EllipticE[(-2*e + Pi
- 2*f*x)/4, (2*d)/(c + d)] - c*EllipticF[(-2*e + Pi - 2*f*x)/4, (2*d)/(c + d)]))*Sqrt[(c + d*Sin[e + f*x])/(c
+ d)] - d*(c + d*Sin[e + f*x])*((336*a*b*c*d + 140*a^2*d^2 + b^2*(12*c^2 + 115*d^2))*Cos[e + f*x] + 3*b*d*(-5*
b*d*Cos[3*(e + f*x)] + 4*(4*b*c + 7*a*d)*Sin[2*(e + f*x)])))/(210*d^2*f*Sqrt[c + d*Sin[e + f*x]])

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Maple [B]  time = 4.46, size = 1575, normalized size = 4.5 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(-(-d*sin(f*x+e)-c)*cos(f*x+e)^2)^(1/2)*(b^2*d^2*(-2/7/d*sin(f*x+e)^2*(-(-d*sin(f*x+e)-c)*cos(f*x+e)^2)^(1/2)+
12/35*c/d^2*sin(f*x+e)*(-(-d*sin(f*x+e)-c)*cos(f*x+e)^2)^(1/2)-2/3*(5/7+24/35*c^2/d^2)/d*(-(-d*sin(f*x+e)-c)*c
os(f*x+e)^2)^(1/2)+2*(-4/35*c^2/d^2+5/21)*(c/d-1)*((c+d*sin(f*x+e))/(c-d))^(1/2)*(d*(1-sin(f*x+e))/(c+d))^(1/2
)*((-sin(f*x+e)-1)*d/(c-d))^(1/2)/(-(-d*sin(f*x+e)-c)*cos(f*x+e)^2)^(1/2)*EllipticF(((c+d*sin(f*x+e))/(c-d))^(
1/2),((c-d)/(c+d))^(1/2))+2/105*(-48*c^3-44*c*d^2)/d^3*(c/d-1)*((c+d*sin(f*x+e))/(c-d))^(1/2)*(d*(1-sin(f*x+e)
)/(c+d))^(1/2)*((-sin(f*x+e)-1)*d/(c-d))^(1/2)/(-(-d*sin(f*x+e)-c)*cos(f*x+e)^2)^(1/2)*((-c/d-1)*EllipticE(((c
+d*sin(f*x+e))/(c-d))^(1/2),((c-d)/(c+d))^(1/2))+EllipticF(((c+d*sin(f*x+e))/(c-d))^(1/2),((c-d)/(c+d))^(1/2))
))+(2*a*b*d^2+2*b^2*c*d)*(-2/5/d*sin(f*x+e)*(-(-d*sin(f*x+e)-c)*cos(f*x+e)^2)^(1/2)+8/15*c/d^2*(-(-d*sin(f*x+e
)-c)*cos(f*x+e)^2)^(1/2)+4/15*c/d*(c/d-1)*((c+d*sin(f*x+e))/(c-d))^(1/2)*(d*(1-sin(f*x+e))/(c+d))^(1/2)*((-sin
(f*x+e)-1)*d/(c-d))^(1/2)/(-(-d*sin(f*x+e)-c)*cos(f*x+e)^2)^(1/2)*EllipticF(((c+d*sin(f*x+e))/(c-d))^(1/2),((c
-d)/(c+d))^(1/2))+2*(3/5+8/15*c^2/d^2)*(c/d-1)*((c+d*sin(f*x+e))/(c-d))^(1/2)*(d*(1-sin(f*x+e))/(c+d))^(1/2)*(
(-sin(f*x+e)-1)*d/(c-d))^(1/2)/(-(-d*sin(f*x+e)-c)*cos(f*x+e)^2)^(1/2)*((-c/d-1)*EllipticE(((c+d*sin(f*x+e))/(
c-d))^(1/2),((c-d)/(c+d))^(1/2))+EllipticF(((c+d*sin(f*x+e))/(c-d))^(1/2),((c-d)/(c+d))^(1/2))))+(a^2*d^2+4*a*
b*c*d+b^2*c^2)*(-2/3/d*(-(-d*sin(f*x+e)-c)*cos(f*x+e)^2)^(1/2)+2/3*(c/d-1)*((c+d*sin(f*x+e))/(c-d))^(1/2)*(d*(
1-sin(f*x+e))/(c+d))^(1/2)*((-sin(f*x+e)-1)*d/(c-d))^(1/2)/(-(-d*sin(f*x+e)-c)*cos(f*x+e)^2)^(1/2)*EllipticF((
(c+d*sin(f*x+e))/(c-d))^(1/2),((c-d)/(c+d))^(1/2))-4/3*c/d*(c/d-1)*((c+d*sin(f*x+e))/(c-d))^(1/2)*(d*(1-sin(f*
x+e))/(c+d))^(1/2)*((-sin(f*x+e)-1)*d/(c-d))^(1/2)/(-(-d*sin(f*x+e)-c)*cos(f*x+e)^2)^(1/2)*((-c/d-1)*EllipticE
(((c+d*sin(f*x+e))/(c-d))^(1/2),((c-d)/(c+d))^(1/2))+EllipticF(((c+d*sin(f*x+e))/(c-d))^(1/2),((c-d)/(c+d))^(1
/2))))+2*(2*a^2*c*d+2*a*b*c^2)*(c/d-1)*((c+d*sin(f*x+e))/(c-d))^(1/2)*(d*(1-sin(f*x+e))/(c+d))^(1/2)*((-sin(f*
x+e)-1)*d/(c-d))^(1/2)/(-(-d*sin(f*x+e)-c)*cos(f*x+e)^2)^(1/2)*((-c/d-1)*EllipticE(((c+d*sin(f*x+e))/(c-d))^(1
/2),((c-d)/(c+d))^(1/2))+EllipticF(((c+d*sin(f*x+e))/(c-d))^(1/2),((c-d)/(c+d))^(1/2)))+2*c^2*a^2*(c/d-1)*((c+
d*sin(f*x+e))/(c-d))^(1/2)*(d*(1-sin(f*x+e))/(c+d))^(1/2)*((-sin(f*x+e)-1)*d/(c-d))^(1/2)/(-(-d*sin(f*x+e)-c)*
cos(f*x+e)^2)^(1/2)*EllipticF(((c+d*sin(f*x+e))/(c-d))^(1/2),((c-d)/(c+d))^(1/2)))/cos(f*x+e)/(c+d*sin(f*x+e))
^(1/2)/f

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (2 \, a b d -{\left (b^{2} c + 2 \, a b d\right )} \cos \left (f x + e\right )^{2} +{\left (a^{2} + b^{2}\right )} c -{\left (b^{2} d \cos \left (f x + e\right )^{2} - 2 \, a b c -{\left (a^{2} + b^{2}\right )} d\right )} \sin \left (f x + e\right )\right )} \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))^2*(c+d*sin(f*x+e))^(3/2),x, algorithm="fricas")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

Timed out