3.74 \(\int (e x)^m \sin ^{\frac{3}{2}}(d (a+b \log (c x^n))) \, dx\)

Optimal. Leaf size=150 \[ \frac{2 (e x)^{m+1} \sin ^{\frac{3}{2}}\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \text{Hypergeometric2F1}\left (-\frac{3}{2},-\frac{3 b d n+2 i m+2 i}{4 b d n},-\frac{-b d n+2 i m+2 i}{4 b d n},e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )}{e (-3 i b d n+2 m+2) \left (1-e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )^{3/2}} \]

[Out]

(2*(e*x)^(1 + m)*Hypergeometric2F1[-3/2, -(2*I + (2*I)*m + 3*b*d*n)/(4*b*d*n), -(2*I + (2*I)*m - b*d*n)/(4*b*d
*n), E^((2*I)*a*d)*(c*x^n)^((2*I)*b*d)]*Sin[d*(a + b*Log[c*x^n])]^(3/2))/(e*(2 + 2*m - (3*I)*b*d*n)*(1 - E^((2
*I)*a*d)*(c*x^n)^((2*I)*b*d))^(3/2))

________________________________________________________________________________________

Rubi [A]  time = 0.125528, antiderivative size = 145, normalized size of antiderivative = 0.97, number of steps used = 3, number of rules used = 3, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.13, Rules used = {4493, 4491, 364} \[ \frac{2 (e x)^{m+1} \, _2F_1\left (-\frac{3}{2},\frac{1}{4} \left (-\frac{2 i (m+1)}{b d n}-3\right );-\frac{2 i m-b d n+2 i}{4 b d n};e^{2 i a d} \left (c x^n\right )^{2 i b d}\right ) \sin ^{\frac{3}{2}}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{e (-3 i b d n+2 m+2) \left (1-e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(e*x)^(1 + m)*Hypergeometric2F1[-3/2, (-3 - ((2*I)*(1 + m))/(b*d*n))/4, -(2*I + (2*I)*m - b*d*n)/(4*b*d*n),
E^((2*I)*a*d)*(c*x^n)^((2*I)*b*d)]*Sin[d*(a + b*Log[c*x^n])]^(3/2))/(e*(2 + 2*m - (3*I)*b*d*n)*(1 - E^((2*I)*
a*d)*(c*x^n)^((2*I)*b*d))^(3/2))

Rule 4493

Int[((e_.)*(x_))^(m_.)*Sin[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*(d_.)]^(p_.), x_Symbol] :> Dist[(e*x)^(m + 1)
/(e*n*(c*x^n)^((m + 1)/n)), Subst[Int[x^((m + 1)/n - 1)*Sin[d*(a + b*Log[x])]^p, x], x, c*x^n], x] /; FreeQ[{a
, b, c, d, e, m, n, p}, x] && (NeQ[c, 1] || NeQ[n, 1])

Rule 4491

Int[((e_.)*(x_))^(m_.)*Sin[((a_.) + Log[x_]*(b_.))*(d_.)]^(p_), x_Symbol] :> Dist[(Sin[d*(a + b*Log[x])]^p*x^(
I*b*d*p))/(1 - E^(2*I*a*d)*x^(2*I*b*d))^p, Int[((e*x)^m*(1 - E^(2*I*a*d)*x^(2*I*b*d))^p)/x^(I*b*d*p), x], x] /
; FreeQ[{a, b, d, e, m, p}, x] &&  !IntegerQ[p]

Rule 364

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a^p*(c*x)^(m + 1)*Hypergeometric2F1[-
p, (m + 1)/n, (m + 1)/n + 1, -((b*x^n)/a)])/(c*(m + 1)), x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[p, 0] &&
(ILtQ[p, 0] || GtQ[a, 0])

Rubi steps

\begin{align*} \int (e x)^m \sin ^{\frac{3}{2}}\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx &=\frac{\left ((e x)^{1+m} \left (c x^n\right )^{-\frac{1+m}{n}}\right ) \operatorname{Subst}\left (\int x^{-1+\frac{1+m}{n}} \sin ^{\frac{3}{2}}(d (a+b \log (x))) \, dx,x,c x^n\right )}{e n}\\ &=\frac{\left ((e x)^{1+m} \left (c x^n\right )^{\frac{3 i b d}{2}-\frac{1+m}{n}} \sin ^{\frac{3}{2}}\left (d \left (a+b \log \left (c x^n\right )\right )\right )\right ) \operatorname{Subst}\left (\int x^{-1-\frac{3 i b d}{2}+\frac{1+m}{n}} \left (1-e^{2 i a d} x^{2 i b d}\right )^{3/2} \, dx,x,c x^n\right )}{e n \left (1-e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )^{3/2}}\\ &=\frac{2 (e x)^{1+m} \, _2F_1\left (-\frac{3}{2},\frac{1}{4} \left (-3-\frac{2 i (1+m)}{b d n}\right );-\frac{2 i+2 i m-b d n}{4 b d n};e^{2 i a d} \left (c x^n\right )^{2 i b d}\right ) \sin ^{\frac{3}{2}}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{e (2+2 m-3 i b d n) \left (1-e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )^{3/2}}\\ \end{align*}

Mathematica [A]  time = 1.97296, size = 235, normalized size = 1.57 \[ \frac{2 (e x)^m \left (x (i b d n+2 m+2) \sin \left (d \left (a+b \log \left (c x^n\right )\right )\right ) \left (2 (m+1) \sin \left (d \left (a+b \log \left (c x^n\right )\right )\right )-3 b d n \cos \left (d \left (a+b \log \left (c x^n\right )\right )\right )\right )-3 b^2 d^2 n^2 x \left (-1+e^{2 i d \left (a+b \log \left (c x^n\right )\right )}\right ) \text{Hypergeometric2F1}\left (1,-\frac{-3 b d n+2 i m+2 i}{4 b d n},-\frac{-5 b d n+2 i m+2 i}{4 b d n},e^{2 i d \left (a+b \log \left (c x^n\right )\right )}\right )\right )}{(i b d n+2 m+2) (-3 i b d n+2 m+2) (3 i b d n+2 m+2) \sqrt{\sin \left (d \left (a+b \log \left (c x^n\right )\right )\right )}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(2*(e*x)^m*(-3*b^2*d^2*(-1 + E^((2*I)*d*(a + b*Log[c*x^n])))*n^2*x*Hypergeometric2F1[1, -(2*I + (2*I)*m - 3*b*
d*n)/(4*b*d*n), -(2*I + (2*I)*m - 5*b*d*n)/(4*b*d*n), E^((2*I)*d*(a + b*Log[c*x^n]))] + (2 + 2*m + I*b*d*n)*x*
Sin[d*(a + b*Log[c*x^n])]*(-3*b*d*n*Cos[d*(a + b*Log[c*x^n])] + 2*(1 + m)*Sin[d*(a + b*Log[c*x^n])])))/((2 + 2
*m + I*b*d*n)*(2 + 2*m - (3*I)*b*d*n)*(2 + 2*m + (3*I)*b*d*n)*Sqrt[Sin[d*(a + b*Log[c*x^n])]])

________________________________________________________________________________________

Maple [F]  time = 0.431, size = 0, normalized size = 0. \begin{align*} \int \left ( ex \right ) ^{m} \left ( \sin \left ( d \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) \right ) \right ) ^{{\frac{3}{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: UnboundLocalError

________________________________________________________________________________________

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((e*x)**m*sin(d*(a+b*ln(c*x**n)))**(3/2),x)

[Out]

Timed out

________________________________________________________________________________________

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((e*x)^m*sin(d*(a+b*log(c*x^n)))^(3/2),x, algorithm="giac")

[Out]

Timed out