3.6.73 \(\int (a+a \sin (e+f x))^{3/2} \sqrt {c+d \sin (e+f x)} \, dx\) [573]

Optimal. Leaf size=171 \[ \frac {a^{3/2} (c-7 d) (c+d) \tan ^{-1}\left (\frac {\sqrt {a} \sqrt {d} \cos (e+f x)}{\sqrt {a+a \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}\right )}{4 d^{3/2} f}+\frac {a^2 (c-7 d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{4 d f \sqrt {a+a \sin (e+f x)}}-\frac {a^2 \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{2 d f \sqrt {a+a \sin (e+f x)}} \]

[Out]

1/4*a^(3/2)*(c-7*d)*(c+d)*arctan(cos(f*x+e)*a^(1/2)*d^(1/2)/(a+a*sin(f*x+e))^(1/2)/(c+d*sin(f*x+e))^(1/2))/d^(
3/2)/f-1/2*a^2*cos(f*x+e)*(c+d*sin(f*x+e))^(3/2)/d/f/(a+a*sin(f*x+e))^(1/2)+1/4*a^2*(c-7*d)*cos(f*x+e)*(c+d*si
n(f*x+e))^(1/2)/d/f/(a+a*sin(f*x+e))^(1/2)

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Rubi [A]
time = 0.21, antiderivative size = 171, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.172, Rules used = {2842, 21, 2849, 2854, 211} \begin {gather*} \frac {a^{3/2} (c-7 d) (c+d) \text {ArcTan}\left (\frac {\sqrt {a} \sqrt {d} \cos (e+f x)}{\sqrt {a \sin (e+f x)+a} \sqrt {c+d \sin (e+f x)}}\right )}{4 d^{3/2} f}-\frac {a^2 \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{2 d f \sqrt {a \sin (e+f x)+a}}+\frac {a^2 (c-7 d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{4 d f \sqrt {a \sin (e+f x)+a}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(a^(3/2)*(c - 7*d)*(c + d)*ArcTan[(Sqrt[a]*Sqrt[d]*Cos[e + f*x])/(Sqrt[a + a*Sin[e + f*x]]*Sqrt[c + d*Sin[e +
f*x]])])/(4*d^(3/2)*f) + (a^2*(c - 7*d)*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(4*d*f*Sqrt[a + a*Sin[e + f*x]]
) - (a^2*Cos[e + f*x]*(c + d*Sin[e + f*x])^(3/2))/(2*d*f*Sqrt[a + a*Sin[e + f*x]])

Rule 21

Int[(u_.)*((a_) + (b_.)*(v_))^(m_.)*((c_) + (d_.)*(v_))^(n_.), x_Symbol] :> Dist[(b/d)^m, Int[u*(c + d*v)^(m +
n), x], x] /; FreeQ[{a, b, c, d, n}, x] && EqQ[b*c - a*d, 0] && IntegerQ[m] && ( !IntegerQ[n] || SimplerQ[c +
d*x, a + b*x])

Rule 211

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/Rt[a/b, 2]], x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rule 2842

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Sim
p[(-b^2)*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m - 2)*((c + d*Sin[e + f*x])^(n + 1)/(d*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*Simp[a*b*c*(m - 2) + b^2*d*(n + 1) + a^2*d
*(m + n) - b*(b*c*(m - 1) - a*d*(3*m + 2*n - 2))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] &
& NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[m, 1] &&  !LtQ[n, -1] && (IntegersQ[2*m,
2*n] || IntegerQ[m + 1/2] || (IntegerQ[m] && EqQ[c, 0]))

Rule 2849

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

Rule 2854

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

Rubi steps

\begin {align*} \int (a+a \sin (e+f x))^{3/2} \sqrt {c+d \sin (e+f x)} \, dx &=-\frac {a^2 \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{2 d f \sqrt {a+a \sin (e+f x)}}+\frac {\int \frac {\left (-\frac {1}{2} a^2 (c-7 d)-\frac {1}{2} a^2 (c-7 d) \sin (e+f x)\right ) \sqrt {c+d \sin (e+f x)}}{\sqrt {a+a \sin (e+f x)}} \, dx}{2 d}\\ &=-\frac {a^2 \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{2 d f \sqrt {a+a \sin (e+f x)}}-\frac {(a (c-7 d)) \int \sqrt {a+a \sin (e+f x)} \sqrt {c+d \sin (e+f x)} \, dx}{4 d}\\ &=\frac {a^2 (c-7 d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{4 d f \sqrt {a+a \sin (e+f x)}}-\frac {a^2 \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{2 d f \sqrt {a+a \sin (e+f x)}}-\frac {(a (c-7 d) (c+d)) \int \frac {\sqrt {a+a \sin (e+f x)}}{\sqrt {c+d \sin (e+f x)}} \, dx}{8 d}\\ &=\frac {a^2 (c-7 d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{4 d f \sqrt {a+a \sin (e+f x)}}-\frac {a^2 \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{2 d f \sqrt {a+a \sin (e+f x)}}+\frac {\left (a^2 (c-7 d) (c+d)\right ) \text {Subst}\left (\int \frac {1}{a+d x^2} \, dx,x,\frac {a \cos (e+f x)}{\sqrt {a+a \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}\right )}{4 d f}\\ &=\frac {a^{3/2} (c-7 d) (c+d) \tan ^{-1}\left (\frac {\sqrt {a} \sqrt {d} \cos (e+f x)}{\sqrt {a+a \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}\right )}{4 d^{3/2} f}+\frac {a^2 (c-7 d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{4 d f \sqrt {a+a \sin (e+f x)}}-\frac {a^2 \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{2 d f \sqrt {a+a \sin (e+f x)}}\\ \end {align*}

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Mathematica [A]
time = 0.42, size = 247, normalized size = 1.44 \begin {gather*} \frac {(a (1+\sin (e+f x)))^{3/2} \left (\frac {(c-7 d) (c+d) \left (-2 \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {d} \sin \left (\frac {1}{4} (2 e-\pi +2 f x)\right )}{\sqrt {c+d \sin (e+f x)}}\right )-\tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {d} \cos \left (\frac {1}{4} (2 e-\pi +2 f x)\right )}{\sqrt {c+d \sin (e+f x)}}\right )+\log \left (\sqrt {2} \sqrt {d} \cos \left (\frac {1}{4} (2 e-\pi +2 f x)\right )+\sqrt {c+d \sin (e+f x)}\right )\right )}{d^{3/2}}-\frac {2 \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right ) \sqrt {c+d \sin (e+f x)} (c+7 d+2 d \sin (e+f x))}{d}\right )}{8 f \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((a*(1 + Sin[e + f*x]))^(3/2)*(((c - 7*d)*(c + d)*(-2*ArcTan[(Sqrt[2]*Sqrt[d]*Sin[(2*e - Pi + 2*f*x)/4])/Sqrt[
c + d*Sin[e + f*x]]] - ArcTanh[(Sqrt[2]*Sqrt[d]*Cos[(2*e - Pi + 2*f*x)/4])/Sqrt[c + d*Sin[e + f*x]]] + Log[Sqr
t[2]*Sqrt[d]*Cos[(2*e - Pi + 2*f*x)/4] + Sqrt[c + d*Sin[e + f*x]]]))/d^(3/2) - (2*(Cos[(e + f*x)/2] - Sin[(e +
f*x)/2])*Sqrt[c + d*Sin[e + f*x]]*(c + 7*d + 2*d*Sin[e + f*x]))/d))/(8*f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2]
)^3)

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Maple [F]
time = 180.00, size = 0, normalized size = 0.00 \[\int \left (a +a \sin \left (f x +e \right )\right )^{\frac {3}{2}} \sqrt {c +d \sin \left (f x +e \right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 352 vs. \(2 (154) = 308\).
time = 0.68, size = 1177, normalized size = 6.88 \begin {gather*} \left [-\frac {{\left (a c^{2} - 6 \, a c d - 7 \, a d^{2} + {\left (a c^{2} - 6 \, a c d - 7 \, a d^{2}\right )} \cos \left (f x + e\right ) + {\left (a c^{2} - 6 \, a c d - 7 \, a d^{2}\right )} \sin \left (f x + e\right )\right )} \sqrt {-\frac {a}{d}} \log \left (\frac {128 \, a d^{4} \cos \left (f x + e\right )^{5} + a c^{4} + 4 \, a c^{3} d + 6 \, a c^{2} d^{2} + 4 \, a c d^{3} + a d^{4} + 128 \, {\left (2 \, a c d^{3} - a d^{4}\right )} \cos \left (f x + e\right )^{4} - 32 \, {\left (5 \, a c^{2} d^{2} - 14 \, a c d^{3} + 13 \, a d^{4}\right )} \cos \left (f x + e\right )^{3} - 32 \, {\left (a c^{3} d - 2 \, a c^{2} d^{2} + 9 \, a c d^{3} - 4 \, a d^{4}\right )} \cos \left (f x + e\right )^{2} - 8 \, {\left (16 \, d^{4} \cos \left (f x + e\right )^{4} - c^{3} d + 17 \, c^{2} d^{2} - 59 \, c d^{3} + 51 \, d^{4} + 24 \, {\left (c d^{3} - d^{4}\right )} \cos \left (f x + e\right )^{3} - 2 \, {\left (5 \, c^{2} d^{2} - 26 \, c d^{3} + 33 \, d^{4}\right )} \cos \left (f x + e\right )^{2} - {\left (c^{3} d - 7 \, c^{2} d^{2} + 31 \, c d^{3} - 25 \, d^{4}\right )} \cos \left (f x + e\right ) + {\left (16 \, d^{4} \cos \left (f x + e\right )^{3} + c^{3} d - 17 \, c^{2} d^{2} + 59 \, c d^{3} - 51 \, d^{4} - 8 \, {\left (3 \, c d^{3} - 5 \, d^{4}\right )} \cos \left (f x + e\right )^{2} - 2 \, {\left (5 \, c^{2} d^{2} - 14 \, c d^{3} + 13 \, d^{4}\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )\right )} \sqrt {a \sin \left (f x + e\right ) + a} \sqrt {d \sin \left (f x + e\right ) + c} \sqrt {-\frac {a}{d}} + {\left (a c^{4} - 28 \, a c^{3} d + 230 \, a c^{2} d^{2} - 476 \, a c d^{3} + 289 \, a d^{4}\right )} \cos \left (f x + e\right ) + {\left (128 \, a d^{4} \cos \left (f x + e\right )^{4} + a c^{4} + 4 \, a c^{3} d + 6 \, a c^{2} d^{2} + 4 \, a c d^{3} + a d^{4} - 256 \, {\left (a c d^{3} - a d^{4}\right )} \cos \left (f x + e\right )^{3} - 32 \, {\left (5 \, a c^{2} d^{2} - 6 \, a c d^{3} + 5 \, a d^{4}\right )} \cos \left (f x + e\right )^{2} + 32 \, {\left (a c^{3} d - 7 \, a c^{2} d^{2} + 15 \, a c d^{3} - 9 \, a d^{4}\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + \sin \left (f x + e\right ) + 1}\right ) + 8 \, {\left (2 \, a d \cos \left (f x + e\right )^{2} + a c + 5 \, a d + {\left (a c + 7 \, a d\right )} \cos \left (f x + e\right ) + {\left (2 \, a d \cos \left (f x + e\right ) - a c - 5 \, a d\right )} \sin \left (f x + e\right )\right )} \sqrt {a \sin \left (f x + e\right ) + a} \sqrt {d \sin \left (f x + e\right ) + c}}{32 \, {\left (d f \cos \left (f x + e\right ) + d f \sin \left (f x + e\right ) + d f\right )}}, -\frac {{\left (a c^{2} - 6 \, a c d - 7 \, a d^{2} + {\left (a c^{2} - 6 \, a c d - 7 \, a d^{2}\right )} \cos \left (f x + e\right ) + {\left (a c^{2} - 6 \, a c d - 7 \, a d^{2}\right )} \sin \left (f x + e\right )\right )} \sqrt {\frac {a}{d}} \arctan \left (\frac {{\left (8 \, d^{2} \cos \left (f x + e\right )^{2} - c^{2} + 6 \, c d - 9 \, d^{2} - 8 \, {\left (c d - d^{2}\right )} \sin \left (f x + e\right )\right )} \sqrt {a \sin \left (f x + e\right ) + a} \sqrt {d \sin \left (f x + e\right ) + c} \sqrt {\frac {a}{d}}}{4 \, {\left (2 \, a d^{2} \cos \left (f x + e\right )^{3} - {\left (3 \, a c d - a d^{2}\right )} \cos \left (f x + e\right ) \sin \left (f x + e\right ) - {\left (a c^{2} - a c d + 2 \, a d^{2}\right )} \cos \left (f x + e\right )\right )}}\right ) + 4 \, {\left (2 \, a d \cos \left (f x + e\right )^{2} + a c + 5 \, a d + {\left (a c + 7 \, a d\right )} \cos \left (f x + e\right ) + {\left (2 \, a d \cos \left (f x + e\right ) - a c - 5 \, a d\right )} \sin \left (f x + e\right )\right )} \sqrt {a \sin \left (f x + e\right ) + a} \sqrt {d \sin \left (f x + e\right ) + c}}{16 \, {\left (d f \cos \left (f x + e\right ) + d f \sin \left (f x + e\right ) + d f\right )}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/32*((a*c^2 - 6*a*c*d - 7*a*d^2 + (a*c^2 - 6*a*c*d - 7*a*d^2)*cos(f*x + e) + (a*c^2 - 6*a*c*d - 7*a*d^2)*si
n(f*x + e))*sqrt(-a/d)*log((128*a*d^4*cos(f*x + e)^5 + a*c^4 + 4*a*c^3*d + 6*a*c^2*d^2 + 4*a*c*d^3 + a*d^4 + 1
28*(2*a*c*d^3 - a*d^4)*cos(f*x + e)^4 - 32*(5*a*c^2*d^2 - 14*a*c*d^3 + 13*a*d^4)*cos(f*x + e)^3 - 32*(a*c^3*d
- 2*a*c^2*d^2 + 9*a*c*d^3 - 4*a*d^4)*cos(f*x + e)^2 - 8*(16*d^4*cos(f*x + e)^4 - c^3*d + 17*c^2*d^2 - 59*c*d^3
+ 51*d^4 + 24*(c*d^3 - d^4)*cos(f*x + e)^3 - 2*(5*c^2*d^2 - 26*c*d^3 + 33*d^4)*cos(f*x + e)^2 - (c^3*d - 7*c^
2*d^2 + 31*c*d^3 - 25*d^4)*cos(f*x + e) + (16*d^4*cos(f*x + e)^3 + c^3*d - 17*c^2*d^2 + 59*c*d^3 - 51*d^4 - 8*
(3*c*d^3 - 5*d^4)*cos(f*x + e)^2 - 2*(5*c^2*d^2 - 14*c*d^3 + 13*d^4)*cos(f*x + e))*sin(f*x + e))*sqrt(a*sin(f*
x + e) + a)*sqrt(d*sin(f*x + e) + c)*sqrt(-a/d) + (a*c^4 - 28*a*c^3*d + 230*a*c^2*d^2 - 476*a*c*d^3 + 289*a*d^
4)*cos(f*x + e) + (128*a*d^4*cos(f*x + e)^4 + a*c^4 + 4*a*c^3*d + 6*a*c^2*d^2 + 4*a*c*d^3 + a*d^4 - 256*(a*c*d
^3 - a*d^4)*cos(f*x + e)^3 - 32*(5*a*c^2*d^2 - 6*a*c*d^3 + 5*a*d^4)*cos(f*x + e)^2 + 32*(a*c^3*d - 7*a*c^2*d^2
+ 15*a*c*d^3 - 9*a*d^4)*cos(f*x + e))*sin(f*x + e))/(cos(f*x + e) + sin(f*x + e) + 1)) + 8*(2*a*d*cos(f*x + e
)^2 + a*c + 5*a*d + (a*c + 7*a*d)*cos(f*x + e) + (2*a*d*cos(f*x + e) - a*c - 5*a*d)*sin(f*x + e))*sqrt(a*sin(f
*x + e) + a)*sqrt(d*sin(f*x + e) + c))/(d*f*cos(f*x + e) + d*f*sin(f*x + e) + d*f), -1/16*((a*c^2 - 6*a*c*d -
7*a*d^2 + (a*c^2 - 6*a*c*d - 7*a*d^2)*cos(f*x + e) + (a*c^2 - 6*a*c*d - 7*a*d^2)*sin(f*x + e))*sqrt(a/d)*arcta
n(1/4*(8*d^2*cos(f*x + e)^2 - c^2 + 6*c*d - 9*d^2 - 8*(c*d - d^2)*sin(f*x + e))*sqrt(a*sin(f*x + e) + a)*sqrt(
d*sin(f*x + e) + c)*sqrt(a/d)/(2*a*d^2*cos(f*x + e)^3 - (3*a*c*d - a*d^2)*cos(f*x + e)*sin(f*x + e) - (a*c^2 -
a*c*d + 2*a*d^2)*cos(f*x + e))) + 4*(2*a*d*cos(f*x + e)^2 + a*c + 5*a*d + (a*c + 7*a*d)*cos(f*x + e) + (2*a*d
*cos(f*x + e) - a*c - 5*a*d)*sin(f*x + e))*sqrt(a*sin(f*x + e) + a)*sqrt(d*sin(f*x + e) + c))/(d*f*cos(f*x + e
) + d*f*sin(f*x + e) + d*f)]

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (a \left (\sin {\left (e + f x \right )} + 1\right )\right )^{\frac {3}{2}} \sqrt {c + d \sin {\left (e + f x \right )}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int {\left (a+a\,\sin \left (e+f\,x\right )\right )}^{3/2}\,\sqrt {c+d\,\sin \left (e+f\,x\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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