\(\int (e x)^m \sqrt {\sin (d (a+b \log (c x^n)))} \, dx\) [75]

Optimal result

Integrand size = 23, antiderivative size = 149 \[ \int (e x)^m \sqrt {\sin \left (d \left (a+b \log \left (c x^n\right )\right )\right )} \, dx=\frac {2 (e x)^{1+m} \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},-\frac {2 i+2 i m+b d n}{4 b d n},-\frac {2 i+2 i m-3 b d n}{4 b d n},e^{2 i a d} \left (c x^n\right )^{2 i b d}\right ) \sqrt {\sin \left (d \left (a+b \log \left (c x^n\right )\right )\right )}}{e (2+2 m-i b d n) \sqrt {1-e^{2 i a d} \left (c x^n\right )^{2 i b d}}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.14 (sec) , antiderivative size = 145, normalized size of antiderivative = 0.97, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {4581, 4579, 371} \[ \int (e x)^m \sqrt {\sin \left (d \left (a+b \log \left (c x^n\right )\right )\right )} \, dx=\frac {2 (e x)^{m+1} \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},\frac {1}{4} \left (-\frac {2 i (m+1)}{b d n}-1\right ),-\frac {2 i m-3 b d n+2 i}{4 b d n},e^{2 i a d} \left (c x^n\right )^{2 i b d}\right ) \sqrt {\sin \left (d \left (a+b \log \left (c x^n\right )\right )\right )}}{e (-i b d n+2 m+2) \sqrt {1-e^{2 i a d} \left (c x^n\right )^{2 i b d}}} \]

[In]

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

[Out]

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

Rule 371

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^p*((c*x)^(m + 1)/(c*(m + 1)))*Hyperg
eometric2F1[-p, (m + 1)/n, (m + 1)/n + 1, (-b)*(x^n/a)], x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[p, 0] &&
(ILtQ[p, 0] || GtQ[a, 0])

Rule 4579

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 4581

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])

Rubi steps \begin{align*} \text {integral}& = \frac {\left ((e x)^{1+m} \left (c x^n\right )^{-\frac {1+m}{n}}\right ) \text {Subst}\left (\int x^{-1+\frac {1+m}{n}} \sqrt {\sin (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 {i b d}{2}-\frac {1+m}{n}} \sqrt {\sin \left (d \left (a+b \log \left (c x^n\right )\right )\right )}\right ) \text {Subst}\left (\int x^{-1-\frac {i b d}{2}+\frac {1+m}{n}} \sqrt {1-e^{2 i a d} x^{2 i b d}} \, dx,x,c x^n\right )}{e n \sqrt {1-e^{2 i a d} \left (c x^n\right )^{2 i b d}}} \\ & = \frac {2 (e x)^{1+m} \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},\frac {1}{4} \left (-1-\frac {2 i (1+m)}{b d n}\right ),-\frac {2 i+2 i m-3 b d n}{4 b d n},e^{2 i a d} \left (c x^n\right )^{2 i b d}\right ) \sqrt {\sin \left (d \left (a+b \log \left (c x^n\right )\right )\right )}}{e (2+2 m-i b d n) \sqrt {1-e^{2 i a d} \left (c x^n\right )^{2 i b d}}} \\ \end{align*}

Mathematica [B] (verified)

Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(488\) vs. \(2(149)=298\).

Time = 6.54 (sec) , antiderivative size = 488, normalized size of antiderivative = 3.28 \[ \int (e x)^m \sqrt {\sin \left (d \left (a+b \log \left (c x^n\right )\right )\right )} \, dx=2 x (e x)^m \left (-\frac {b d e^{i d \left (a-b n \log (x)+b \log \left (c x^n\right )\right )} n x^{-i b d n} \sqrt {2-2 e^{2 i a d} \left (c x^n\right )^{2 i b d}} \left ((2 i+2 i m+b d n) x^{2 i b d n} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-\frac {i \left (1+m+\frac {3}{2} i b d n\right )}{2 b d n},-\frac {2 i+2 i m-7 b d n}{4 b d n},e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )+(-2 i-2 i m+3 b d n) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-\frac {2 i+2 i m+b d n}{4 b d n},-\frac {2 i+2 i m-3 b d n}{4 b d n},e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )\right )}{(2+2 m-i b d n) (2+2 m+3 i b d n) \left (2 i+2 i m+b d n+e^{2 i d \left (a-b n \log (x)+b \log \left (c x^n\right )\right )} (-2 i-2 i m+b d n)\right ) \sqrt {-i e^{-i a d} \left (c x^n\right )^{-i b d} \left (-1+e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )}}+\frac {\sqrt {\sin \left (d \left (a+b \log \left (c x^n\right )\right )\right )} \sin \left (d \left (a-b n \log (x)+b \log \left (c x^n\right )\right )\right )}{b d n \cos \left (d \left (a-b n \log (x)+b \log \left (c x^n\right )\right )\right )+2 (1+m) \sin \left (d \left (a-b n \log (x)+b \log \left (c x^n\right )\right )\right )}\right ) \]

[In]

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

[Out]

2*x*(e*x)^m*(-((b*d*E^(I*d*(a - b*n*Log[x] + b*Log[c*x^n]))*n*Sqrt[2 - 2*E^((2*I)*a*d)*(c*x^n)^((2*I)*b*d)]*((
2*I + (2*I)*m + b*d*n)*x^((2*I)*b*d*n)*Hypergeometric2F1[1/2, ((-1/2*I)*(1 + m + ((3*I)/2)*b*d*n))/(b*d*n), -1
/4*(2*I + (2*I)*m - 7*b*d*n)/(b*d*n), E^((2*I)*a*d)*(c*x^n)^((2*I)*b*d)] + (-2*I - (2*I)*m + 3*b*d*n)*Hypergeo
metric2F1[1/2, -1/4*(2*I + (2*I)*m + b*d*n)/(b*d*n), -1/4*(2*I + (2*I)*m - 3*b*d*n)/(b*d*n), E^((2*I)*a*d)*(c*
x^n)^((2*I)*b*d)]))/((2 + 2*m - I*b*d*n)*(2 + 2*m + (3*I)*b*d*n)*(2*I + (2*I)*m + b*d*n + E^((2*I)*d*(a - b*n*
Log[x] + b*Log[c*x^n]))*(-2*I - (2*I)*m + b*d*n))*x^(I*b*d*n)*Sqrt[((-I)*(-1 + E^((2*I)*a*d)*(c*x^n)^((2*I)*b*
d)))/(E^(I*a*d)*(c*x^n)^(I*b*d))])) + (Sqrt[Sin[d*(a + b*Log[c*x^n])]]*Sin[d*(a - b*n*Log[x] + b*Log[c*x^n])])
/(b*d*n*Cos[d*(a - b*n*Log[x] + b*Log[c*x^n])] + 2*(1 + m)*Sin[d*(a - b*n*Log[x] + b*Log[c*x^n])]))

Maple [F]

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

[In]

int((e*x)^m*sin(d*(a+b*ln(c*x^n)))^(1/2),x)

[Out]

int((e*x)^m*sin(d*(a+b*ln(c*x^n)))^(1/2),x)

Fricas [F(-2)]

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

[In]

integrate((e*x)^m*sin(d*(a+b*log(c*x^n)))^(1/2),x, algorithm="fricas")

[Out]

Exception raised: TypeError >>  Error detected within library code:   integrate: implementation incomplete (ha
s polynomial part)

Sympy [F]

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

[In]

integrate((e*x)**m*sin(d*(a+b*ln(c*x**n)))**(1/2),x)

[Out]

Integral((e*x)**m*sqrt(sin(a*d + b*d*log(c*x**n))), x)

Maxima [F]

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

[In]

integrate((e*x)^m*sin(d*(a+b*log(c*x^n)))^(1/2),x, algorithm="maxima")

[Out]

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

Giac [F]

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

[In]

integrate((e*x)^m*sin(d*(a+b*log(c*x^n)))^(1/2),x, algorithm="giac")

[Out]

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

Mupad [F(-1)]

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

[In]

int(sin(d*(a + b*log(c*x^n)))^(1/2)*(e*x)^m,x)

[Out]

int(sin(d*(a + b*log(c*x^n)))^(1/2)*(e*x)^m, x)