\(\int (e x)^m \sin ^{\frac {3}{2}}(d (a+b \log (c x^n))) \, dx\) [74]

Optimal result

Integrand size = 23, antiderivative size = 150 \[ \int (e x)^m \sin ^{\frac {3}{2}}\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\frac {2 (e x)^{1+m} \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},-\frac {2 i+2 i m+3 b d n}{4 b d n},-\frac {2 i+2 i m-b d n}{4 b d n},e^{2 i a d} \left (c x^n\right )^{2 i b d}\right ) \sin ^{\frac {3}{2}}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{e (2+2 m-3 i b d n) \left (1-e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )^{3/2}} \] Output:

2*(e*x)^(1+m)*hypergeom([-3/2, -1/4*(2*I+2*I*m+3*b*d*n)/b/d/n],[-1/4*(2*I+ 
2*I*m-b*d*n)/b/d/n],exp(2*I*a*d)*(c*x^n)^(2*I*b*d))*sin(d*(a+b*ln(c*x^n))) 
^(3/2)/e/(2+2*m-3*I*b*d*n)/(1-exp(2*I*a*d)*(c*x^n)^(2*I*b*d))^(3/2)
 

Mathematica [A] (verified)

Time = 1.73 (sec) , antiderivative size = 235, normalized size of antiderivative = 1.57 \[ \int (e x)^m \sin ^{\frac {3}{2}}\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\frac {2 (e x)^m \left (-3 b^2 d^2 \left (-1+e^{2 i d \left (a+b \log \left (c x^n\right )\right )}\right ) n^2 x \operatorname {Hypergeometric2F1}\left (1,-\frac {2 i+2 i m-3 b d n}{4 b d n},-\frac {2 i+2 i m-5 b d n}{4 b d n},e^{2 i d \left (a+b \log \left (c x^n\right )\right )}\right )+(2+2 m+i b d n) x \sin \left (d \left (a+b \log \left (c x^n\right )\right )\right ) \left (-3 b d n \cos \left (d \left (a+b \log \left (c x^n\right )\right )\right )+2 (1+m) \sin \left (d \left (a+b \log \left (c x^n\right )\right )\right )\right )\right )}{(2+2 m+i b d n) (2+2 m-3 i b d n) (2+2 m+3 i b d n) \sqrt {\sin \left (d \left (a+b \log \left (c x^n\right )\right )\right )}} \] Input:

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

Output:

(2*(e*x)^m*(-3*b^2*d^2*(-1 + E^((2*I)*d*(a + b*Log[c*x^n])))*n^2*x*Hyperge 
ometric2F1[1, -1/4*(2*I + (2*I)*m - 3*b*d*n)/(b*d*n), -1/4*(2*I + (2*I)*m 
- 5*b*d*n)/(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])]])
 

Rubi [A] (verified)

Time = 0.37 (sec) , antiderivative size = 145, normalized size of antiderivative = 0.97, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {4996, 4994, 888}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

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

\(\Big \downarrow \) 4996

\(\displaystyle \frac {(e x)^{m+1} \left (c x^n\right )^{-\frac {m+1}{n}} \int \left (c x^n\right )^{\frac {m+1}{n}-1} \sin ^{\frac {3}{2}}\left (d \left (a+b \log \left (c x^n\right )\right )\right )d\left (c x^n\right )}{e n}\)

\(\Big \downarrow \) 4994

\(\displaystyle \frac {(e x)^{m+1} \left (c x^n\right )^{-\frac {m+1}{n}+\frac {3 i b d}{2}} \sin ^{\frac {3}{2}}\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \int \left (c x^n\right )^{-\frac {3}{2} i b d+\frac {m+1}{n}-1} \left (1-e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )^{3/2}d\left (c x^n\right )}{e n \left (1-e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )^{3/2}}\)

\(\Big \downarrow \) 888

\(\displaystyle \frac {2 (e x)^{m+1} \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},\frac {1}{4} \left (-\frac {2 i (m+1)}{b d n}-3\right ),-\frac {2 i m-b d n+2 i}{4 b d n},e^{2 i a d} \left (c x^n\right )^{2 i b d}\right ) \sin ^{\frac {3}{2}}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{e (-3 i b d n+2 m+2) \left (1-e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )^{3/2}}\)

Input:

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

Output:

(2*(e*x)^(1 + m)*Hypergeometric2F1[-3/2, (-3 - ((2*I)*(1 + m))/(b*d*n))/4, 
 -1/4*(2*I + (2*I)*m - b*d*n)/(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))
 

Defintions of rubi rules used

rule 888

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

rule 4994

Int[((e_.)*(x_))^(m_.)*Sin[((a_.) + Log[x_]*(b_.))*(d_.)]^(p_), x_Symbol] : 
> Simp[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] /; Fr 
eeQ[{a, b, d, e, m, p}, x] &&  !IntegerQ[p]
 

rule 4996

Int[((e_.)*(x_))^(m_.)*Sin[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*(d_.)]^(p_ 
.), x_Symbol] :> Simp[(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])
 

Maple [F]

\[\int \left (e x \right )^{m} {\sin \left (d \left (a +b \ln \left (c \,x^{n}\right )\right )\right )}^{\frac {3}{2}}d x\]

Input:

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

Output:

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

Fricas [F(-2)]

Exception generated. \[ \int (e x)^m \sin ^{\frac {3}{2}}\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\text {Exception raised: TypeError} \] Input:

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

Output:

Exception raised: TypeError >>  Error detected within library code:   inte 
grate: implementation incomplete (has polynomial part)
 

Sympy [F(-1)]

Timed out. \[ \int (e x)^m \sin ^{\frac {3}{2}}\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\text {Timed out} \] Input:

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

Output:

Timed out
 

Maxima [F]

\[ \int (e x)^m \sin ^{\frac {3}{2}}\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\int { \left (e x\right )^{m} \sin \left ({\left (b \log \left (c x^{n}\right ) + a\right )} d\right )^{\frac {3}{2}} \,d x } \] Input:

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

Output:

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

Giac [F]

\[ \int (e x)^m \sin ^{\frac {3}{2}}\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\int { \left (e x\right )^{m} \sin \left ({\left (b \log \left (c x^{n}\right ) + a\right )} d\right )^{\frac {3}{2}} \,d x } \] Input:

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

Output:

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

Mupad [F(-1)]

Timed out. \[ \int (e x)^m \sin ^{\frac {3}{2}}\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\int {\sin \left (d\,\left (a+b\,\ln \left (c\,x^n\right )\right )\right )}^{3/2}\,{\left (e\,x\right )}^m \,d x \] Input:

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

Output:

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

Reduce [F]

\[ \int (e x)^m \sin ^{\frac {3}{2}}\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\frac {e^{m} \left (2 x^{m} \sqrt {\sin \left (\mathrm {log}\left (x^{n} c \right ) b d +a d \right )}\, \sin \left (\mathrm {log}\left (x^{n} c \right ) b d +a d \right ) x -3 \left (\int x^{m} \sqrt {\sin \left (\mathrm {log}\left (x^{n} c \right ) b d +a d \right )}\, \cos \left (\mathrm {log}\left (x^{n} c \right ) b d +a d \right )d x \right ) b d n \right )}{2 m +2} \] Input:

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

Output:

(e**m*(2*x**m*sqrt(sin(log(x**n*c)*b*d + a*d))*sin(log(x**n*c)*b*d + a*d)* 
x - 3*int(x**m*sqrt(sin(log(x**n*c)*b*d + a*d))*cos(log(x**n*c)*b*d + a*d) 
,x)*b*d*n))/(2*(m + 1))