3.1.7 \(\int (d \sin (e+f x))^n (a+a \sin (e+f x))^{5/2} (A+B \sin (e+f x)) \, dx\) [7]

Optimal. Leaf size=336 \[ -\frac {2 a^3 \left (2 B \left (115+203 n+104 n^2+16 n^3\right )+A \left (301+478 n+224 n^2+32 n^3\right )\right ) \cos (e+f x) \, _2F_1\left (\frac {1}{2},-n;\frac {3}{2};1-\sin (e+f x)\right ) \sin ^{-n}(e+f x) (d \sin (e+f x))^n}{f (3+2 n) (5+2 n) (7+2 n) \sqrt {a+a \sin (e+f x)}}-\frac {2 a^3 \left (2 B \left (35+23 n+4 n^2\right )+A \left (77+50 n+8 n^2\right )\right ) \cos (e+f x) (d \sin (e+f x))^{1+n}}{d f (3+2 n) (5+2 n) (7+2 n) \sqrt {a+a \sin (e+f x)}}-\frac {2 a^2 (2 B (5+n)+A (7+2 n)) \cos (e+f x) (d \sin (e+f x))^{1+n} \sqrt {a+a \sin (e+f x)}}{d f (5+2 n) (7+2 n)}-\frac {2 a B \cos (e+f x) (d \sin (e+f x))^{1+n} (a+a \sin (e+f x))^{3/2}}{d f (7+2 n)} \]

[Out]

-2*a*B*cos(f*x+e)*(d*sin(f*x+e))^(1+n)*(a+a*sin(f*x+e))^(3/2)/d/f/(7+2*n)-2*a^3*(2*B*(16*n^3+104*n^2+203*n+115
)+A*(32*n^3+224*n^2+478*n+301))*cos(f*x+e)*hypergeom([1/2, -n],[3/2],1-sin(f*x+e))*(d*sin(f*x+e))^n/f/(3+2*n)/
(5+2*n)/(7+2*n)/(sin(f*x+e)^n)/(a+a*sin(f*x+e))^(1/2)-2*a^3*(2*B*(4*n^2+23*n+35)+A*(8*n^2+50*n+77))*cos(f*x+e)
*(d*sin(f*x+e))^(1+n)/d/f/(3+2*n)/(5+2*n)/(7+2*n)/(a+a*sin(f*x+e))^(1/2)-2*a^2*(2*B*(5+n)+A*(7+2*n))*cos(f*x+e
)*(d*sin(f*x+e))^(1+n)*(a+a*sin(f*x+e))^(1/2)/d/f/(5+2*n)/(7+2*n)

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Rubi [A]
time = 0.61, antiderivative size = 336, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {3055, 3060, 2855, 69, 67} \begin {gather*} -\frac {2 a^3 \left (A \left (32 n^3+224 n^2+478 n+301\right )+2 B \left (16 n^3+104 n^2+203 n+115\right )\right ) \cos (e+f x) \sin ^{-n}(e+f x) (d \sin (e+f x))^n \, _2F_1\left (\frac {1}{2},-n;\frac {3}{2};1-\sin (e+f x)\right )}{f (2 n+3) (2 n+5) (2 n+7) \sqrt {a \sin (e+f x)+a}}-\frac {2 a^3 \left (A \left (8 n^2+50 n+77\right )+2 B \left (4 n^2+23 n+35\right )\right ) \cos (e+f x) (d \sin (e+f x))^{n+1}}{d f (2 n+3) (2 n+5) (2 n+7) \sqrt {a \sin (e+f x)+a}}-\frac {2 a^2 (A (2 n+7)+2 B (n+5)) \cos (e+f x) \sqrt {a \sin (e+f x)+a} (d \sin (e+f x))^{n+1}}{d f (2 n+5) (2 n+7)}-\frac {2 a B \cos (e+f x) (a \sin (e+f x)+a)^{3/2} (d \sin (e+f x))^{n+1}}{d f (2 n+7)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(d*Sin[e + f*x])^n*(a + a*Sin[e + f*x])^(5/2)*(A + B*Sin[e + f*x]),x]

[Out]

(-2*a^3*(2*B*(115 + 203*n + 104*n^2 + 16*n^3) + A*(301 + 478*n + 224*n^2 + 32*n^3))*Cos[e + f*x]*Hypergeometri
c2F1[1/2, -n, 3/2, 1 - Sin[e + f*x]]*(d*Sin[e + f*x])^n)/(f*(3 + 2*n)*(5 + 2*n)*(7 + 2*n)*Sin[e + f*x]^n*Sqrt[
a + a*Sin[e + f*x]]) - (2*a^3*(2*B*(35 + 23*n + 4*n^2) + A*(77 + 50*n + 8*n^2))*Cos[e + f*x]*(d*Sin[e + f*x])^
(1 + n))/(d*f*(3 + 2*n)*(5 + 2*n)*(7 + 2*n)*Sqrt[a + a*Sin[e + f*x]]) - (2*a^2*(2*B*(5 + n) + A*(7 + 2*n))*Cos
[e + f*x]*(d*Sin[e + f*x])^(1 + n)*Sqrt[a + a*Sin[e + f*x]])/(d*f*(5 + 2*n)*(7 + 2*n)) - (2*a*B*Cos[e + f*x]*(
d*Sin[e + f*x])^(1 + n)*(a + a*Sin[e + f*x])^(3/2))/(d*f*(7 + 2*n))

Rule 67

Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((c + d*x)^(n + 1)/(d*(n + 1)*(-d/(b*c))^m))
*Hypergeometric2F1[-m, n + 1, n + 2, 1 + d*(x/c)], x] /; FreeQ[{b, c, d, m, n}, x] &&  !IntegerQ[n] && (Intege
rQ[m] || GtQ[-d/(b*c), 0])

Rule 69

Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Dist[((-b)*(c/d))^IntPart[m]*((b*x)^FracPart[m]/(
(-d)*(x/c))^FracPart[m]), Int[((-d)*(x/c))^m*(c + d*x)^n, x], x] /; FreeQ[{b, c, d, m, n}, x] &&  !IntegerQ[m]
&&  !IntegerQ[n] &&  !GtQ[c, 0] &&  !GtQ[-d/(b*c), 0]

Rule 2855

Int[Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Dist
[a^2*(Cos[e + f*x]/(f*Sqrt[a + b*Sin[e + f*x]]*Sqrt[a - b*Sin[e + f*x]])), Subst[Int[(c + d*x)^n/Sqrt[a - b*x]
, x], x, Sin[e + f*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] &&  !IntegerQ[2*n]

Rule 3055

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

Rule 3060

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

Rubi steps

\begin {align*} \int (d \sin (e+f x))^n (a+a \sin (e+f x))^{5/2} (A+B \sin (e+f x)) \, dx &=-\frac {2 a B \cos (e+f x) (d \sin (e+f x))^{1+n} (a+a \sin (e+f x))^{3/2}}{d f (7+2 n)}+\frac {2 \int (d \sin (e+f x))^n (a+a \sin (e+f x))^{3/2} \left (\frac {1}{2} a d \left (2 B (1+n)+2 A \left (\frac {7}{2}+n\right )\right )+\frac {1}{2} a d (2 B (5+n)+A (7+2 n)) \sin (e+f x)\right ) \, dx}{d (7+2 n)}\\ &=-\frac {2 a^2 (2 B (5+n)+A (7+2 n)) \cos (e+f x) (d \sin (e+f x))^{1+n} \sqrt {a+a \sin (e+f x)}}{d f (5+2 n) (7+2 n)}-\frac {2 a B \cos (e+f x) (d \sin (e+f x))^{1+n} (a+a \sin (e+f x))^{3/2}}{d f (7+2 n)}+\frac {4 \int (d \sin (e+f x))^n \sqrt {a+a \sin (e+f x)} \left (\frac {1}{4} a^2 d^2 \left (2 B \left (15+19 n+4 n^2\right )+A \left (49+42 n+8 n^2\right )\right )+\frac {1}{4} a^2 d^2 \left (2 B \left (35+23 n+4 n^2\right )+A \left (77+50 n+8 n^2\right )\right ) \sin (e+f x)\right ) \, dx}{d^2 (5+2 n) (7+2 n)}\\ &=-\frac {2 a^3 \left (2 B \left (35+23 n+4 n^2\right )+A \left (77+50 n+8 n^2\right )\right ) \cos (e+f x) (d \sin (e+f x))^{1+n}}{d f (3+2 n) (5+2 n) (7+2 n) \sqrt {a+a \sin (e+f x)}}-\frac {2 a^2 (2 B (5+n)+A (7+2 n)) \cos (e+f x) (d \sin (e+f x))^{1+n} \sqrt {a+a \sin (e+f x)}}{d f (5+2 n) (7+2 n)}-\frac {2 a B \cos (e+f x) (d \sin (e+f x))^{1+n} (a+a \sin (e+f x))^{3/2}}{d f (7+2 n)}+\frac {\left (a^2 \left (2 B \left (115+203 n+104 n^2+16 n^3\right )+A \left (301+478 n+224 n^2+32 n^3\right )\right )\right ) \int (d \sin (e+f x))^n \sqrt {a+a \sin (e+f x)} \, dx}{(3+2 n) (5+2 n) (7+2 n)}\\ &=-\frac {2 a^3 \left (2 B \left (35+23 n+4 n^2\right )+A \left (77+50 n+8 n^2\right )\right ) \cos (e+f x) (d \sin (e+f x))^{1+n}}{d f (3+2 n) (5+2 n) (7+2 n) \sqrt {a+a \sin (e+f x)}}-\frac {2 a^2 (2 B (5+n)+A (7+2 n)) \cos (e+f x) (d \sin (e+f x))^{1+n} \sqrt {a+a \sin (e+f x)}}{d f (5+2 n) (7+2 n)}-\frac {2 a B \cos (e+f x) (d \sin (e+f x))^{1+n} (a+a \sin (e+f x))^{3/2}}{d f (7+2 n)}+\frac {\left (a^4 \left (2 B \left (115+203 n+104 n^2+16 n^3\right )+A \left (301+478 n+224 n^2+32 n^3\right )\right ) \cos (e+f x)\right ) \text {Subst}\left (\int \frac {(d x)^n}{\sqrt {a-a x}} \, dx,x,\sin (e+f x)\right )}{f (3+2 n) (5+2 n) (7+2 n) \sqrt {a-a \sin (e+f x)} \sqrt {a+a \sin (e+f x)}}\\ &=-\frac {2 a^3 \left (2 B \left (35+23 n+4 n^2\right )+A \left (77+50 n+8 n^2\right )\right ) \cos (e+f x) (d \sin (e+f x))^{1+n}}{d f (3+2 n) (5+2 n) (7+2 n) \sqrt {a+a \sin (e+f x)}}-\frac {2 a^2 (2 B (5+n)+A (7+2 n)) \cos (e+f x) (d \sin (e+f x))^{1+n} \sqrt {a+a \sin (e+f x)}}{d f (5+2 n) (7+2 n)}-\frac {2 a B \cos (e+f x) (d \sin (e+f x))^{1+n} (a+a \sin (e+f x))^{3/2}}{d f (7+2 n)}+\frac {\left (a^4 \left (2 B \left (115+203 n+104 n^2+16 n^3\right )+A \left (301+478 n+224 n^2+32 n^3\right )\right ) \cos (e+f x) \sin ^{-n}(e+f x) (d \sin (e+f x))^n\right ) \text {Subst}\left (\int \frac {x^n}{\sqrt {a-a x}} \, dx,x,\sin (e+f x)\right )}{f (3+2 n) (5+2 n) (7+2 n) \sqrt {a-a \sin (e+f x)} \sqrt {a+a \sin (e+f x)}}\\ &=-\frac {2 a^3 \left (2 B \left (115+203 n+104 n^2+16 n^3\right )+A \left (301+478 n+224 n^2+32 n^3\right )\right ) \cos (e+f x) \, _2F_1\left (\frac {1}{2},-n;\frac {3}{2};1-\sin (e+f x)\right ) \sin ^{-n}(e+f x) (d \sin (e+f x))^n}{f (3+2 n) (5+2 n) (7+2 n) \sqrt {a+a \sin (e+f x)}}-\frac {2 a^3 \left (2 B \left (35+23 n+4 n^2\right )+A \left (77+50 n+8 n^2\right )\right ) \cos (e+f x) (d \sin (e+f x))^{1+n}}{d f (3+2 n) (5+2 n) (7+2 n) \sqrt {a+a \sin (e+f x)}}-\frac {2 a^2 (2 B (5+n)+A (7+2 n)) \cos (e+f x) (d \sin (e+f x))^{1+n} \sqrt {a+a \sin (e+f x)}}{d f (5+2 n) (7+2 n)}-\frac {2 a B \cos (e+f x) (d \sin (e+f x))^{1+n} (a+a \sin (e+f x))^{3/2}}{d f (7+2 n)}\\ \end {align*}

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Mathematica [A]
time = 29.71, size = 596, normalized size = 1.77 \begin {gather*} \frac {2^{1+n} \sec \left (\frac {1}{2} (e+f x)\right ) \sin ^{-n}(e+f x) (d \sin (e+f x))^n (a (1+\sin (e+f x)))^{5/2} \tan \left (\frac {1}{2} (e+f x)\right ) \left (\frac {\tan \left (\frac {1}{2} (e+f x)\right )}{1+\tan ^2\left (\frac {1}{2} (e+f x)\right )}\right )^n \left (1+\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )^n \left (\frac {A \, _2F_1\left (\frac {1+n}{2},\frac {9}{2}+n;\frac {3+n}{2};-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )}{1+n}+\frac {A \, _2F_1\left (4+\frac {n}{2},\frac {9}{2}+n;5+\frac {n}{2};-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right ) \tan ^7\left (\frac {1}{2} (e+f x)\right )}{8+n}+\tan \left (\frac {1}{2} (e+f x)\right ) \left (\frac {(5 A+2 B) \, _2F_1\left (\frac {2+n}{2},\frac {9}{2}+n;\frac {4+n}{2};-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )}{2+n}+\tan \left (\frac {1}{2} (e+f x)\right ) \left (\frac {(11 A+10 B) \, _2F_1\left (\frac {3+n}{2},\frac {9}{2}+n;\frac {5+n}{2};-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )}{3+n}+\tan \left (\frac {1}{2} (e+f x)\right ) \left (\frac {5 (3 A+4 B) \, _2F_1\left (\frac {4+n}{2},\frac {9}{2}+n;\frac {6+n}{2};-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )}{4+n}+\tan \left (\frac {1}{2} (e+f x)\right ) \left (\frac {5 (3 A+4 B) \, _2F_1\left (\frac {9}{2}+n,\frac {5+n}{2};\frac {7+n}{2};-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )}{5+n}+\tan \left (\frac {1}{2} (e+f x)\right ) \left (\frac {(11 A+10 B) \, _2F_1\left (\frac {9}{2}+n,\frac {6+n}{2};\frac {8+n}{2};-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )}{6+n}+\frac {(5 A+2 B) \, _2F_1\left (\frac {9}{2}+n,\frac {7+n}{2};\frac {9+n}{2};-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right ) \tan \left (\frac {1}{2} (e+f x)\right )}{7+n}\right )\right )\right )\right )\right )\right )}{f \sqrt {\sec ^2\left (\frac {1}{2} (e+f x)\right )} \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^5} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Integrate[(d*Sin[e + f*x])^n*(a + a*Sin[e + f*x])^(5/2)*(A + B*Sin[e + f*x]),x]

[Out]

(2^(1 + n)*Sec[(e + f*x)/2]*(d*Sin[e + f*x])^n*(a*(1 + Sin[e + f*x]))^(5/2)*Tan[(e + f*x)/2]*(Tan[(e + f*x)/2]
/(1 + Tan[(e + f*x)/2]^2))^n*(1 + Tan[(e + f*x)/2]^2)^n*((A*Hypergeometric2F1[(1 + n)/2, 9/2 + n, (3 + n)/2, -
Tan[(e + f*x)/2]^2])/(1 + n) + (A*Hypergeometric2F1[4 + n/2, 9/2 + n, 5 + n/2, -Tan[(e + f*x)/2]^2]*Tan[(e + f
*x)/2]^7)/(8 + n) + Tan[(e + f*x)/2]*(((5*A + 2*B)*Hypergeometric2F1[(2 + n)/2, 9/2 + n, (4 + n)/2, -Tan[(e +
f*x)/2]^2])/(2 + n) + Tan[(e + f*x)/2]*(((11*A + 10*B)*Hypergeometric2F1[(3 + n)/2, 9/2 + n, (5 + n)/2, -Tan[(
e + f*x)/2]^2])/(3 + n) + Tan[(e + f*x)/2]*((5*(3*A + 4*B)*Hypergeometric2F1[(4 + n)/2, 9/2 + n, (6 + n)/2, -T
an[(e + f*x)/2]^2])/(4 + n) + Tan[(e + f*x)/2]*((5*(3*A + 4*B)*Hypergeometric2F1[9/2 + n, (5 + n)/2, (7 + n)/2
, -Tan[(e + f*x)/2]^2])/(5 + n) + Tan[(e + f*x)/2]*(((11*A + 10*B)*Hypergeometric2F1[9/2 + n, (6 + n)/2, (8 +
n)/2, -Tan[(e + f*x)/2]^2])/(6 + n) + ((5*A + 2*B)*Hypergeometric2F1[9/2 + n, (7 + n)/2, (9 + n)/2, -Tan[(e +
f*x)/2]^2]*Tan[(e + f*x)/2])/(7 + n))))))))/(f*Sqrt[Sec[(e + f*x)/2]^2]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^
5*Sin[e + f*x]^n)

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Maple [F]
time = 0.23, size = 0, normalized size = 0.00 \[\int \left (d \sin \left (f x +e \right )\right )^{n} \left (a +a \sin \left (f x +e \right )\right )^{\frac {5}{2}} \left (A +B \sin \left (f x +e \right )\right )\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*sin(f*x+e))^n*(a+a*sin(f*x+e))^(5/2)*(A+B*sin(f*x+e)),x)

[Out]

int((d*sin(f*x+e))^n*(a+a*sin(f*x+e))^(5/2)*(A+B*sin(f*x+e)),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((d*sin(f*x+e))^n*(a+a*sin(f*x+e))^(5/2)*(A+B*sin(f*x+e)),x, algorithm="maxima")

[Out]

integrate((B*sin(f*x + e) + A)*(a*sin(f*x + e) + a)^(5/2)*(d*sin(f*x + e))^n, x)

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Fricas [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((d*sin(f*x+e))^n*(a+a*sin(f*x+e))^(5/2)*(A+B*sin(f*x+e)),x, algorithm="fricas")

[Out]

integral(-((A + 2*B)*a^2*cos(f*x + e)^2 - 2*(A + B)*a^2 + (B*a^2*cos(f*x + e)^2 - 2*(A + B)*a^2)*sin(f*x + e))
*sqrt(a*sin(f*x + e) + a)*(d*sin(f*x + e))^n, x)

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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*sin(f*x+e))**n*(a+a*sin(f*x+e))**(5/2)*(A+B*sin(f*x+e)),x)

[Out]

Exception raised: SystemError >> excessive stack use: stack is 8569 deep

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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((d*sin(f*x+e))^n*(a+a*sin(f*x+e))^(5/2)*(A+B*sin(f*x+e)),x, algorithm="giac")

[Out]

integrate((B*sin(f*x + e) + A)*(a*sin(f*x + e) + a)^(5/2)*(d*sin(f*x + e))^n, x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int {\left (d\,\sin \left (e+f\,x\right )\right )}^n\,\left (A+B\,\sin \left (e+f\,x\right )\right )\,{\left (a+a\,\sin \left (e+f\,x\right )\right )}^{5/2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*sin(e + f*x))^n*(A + B*sin(e + f*x))*(a + a*sin(e + f*x))^(5/2),x)

[Out]

int((d*sin(e + f*x))^n*(A + B*sin(e + f*x))*(a + a*sin(e + f*x))^(5/2), x)

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