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

Optimal. Leaf size=149 \[ \frac{2 (e x)^{m+1} \sqrt{\sin \left (d \left (a+b \log \left (c x^n\right )\right )\right )} \text{Hypergeometric2F1}\left (-\frac{1}{2},-\frac{b d n+2 i m+2 i}{4 b d n},-\frac{-3 b d n+2 i m+2 i}{4 b d n},e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )}{e (-i b d n+2 m+2) \sqrt{1-e^{2 i a d} \left (c x^n\right )^{2 i b d}}} \]

[Out]

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

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Rubi [A]  time = 0.11187, 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} \[ \frac{2 (e x)^{m+1} \, _2F_1\left (-\frac{1}{2},\frac{1}{4} \left (-\frac{2 i (m+1)}{b d n}-1\right );-\frac{2 i m-3 b d n+2 i}{4 b d n};e^{2 i a d} \left (c x^n\right )^{2 i b d}\right ) \sqrt{\sin \left (d \left (a+b \log \left (c x^n\right )\right )\right )}}{e (-i b d n+2 m+2) \sqrt{1-e^{2 i a d} \left (c x^n\right )^{2 i b d}}} \]

Antiderivative was successfully verified.

[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, -(2*I + (2*I)*m - 3*b*d*n)/(4*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 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 \sqrt{\sin \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}} \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 ) \operatorname{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} \, _2F_1\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]  time = 5.46119, size = 488, normalized size = 3.28 \[ 2 x (e x)^m \left (\frac{\sqrt{\sin \left (d \left (a+b \log \left (c x^n\right )\right )\right )} \sin \left (d \left (a+b \log \left (c x^n\right )-b n \log (x)\right )\right )}{2 (m+1) \sin \left (d \left (a+b \log \left (c x^n\right )-b n \log (x)\right )\right )+b d n \cos \left (d \left (a+b \log \left (c x^n\right )-b n \log (x)\right )\right )}-\frac{b d n x^{-i b d n} \sqrt{2-2 e^{2 i a d} \left (c x^n\right )^{2 i b d}} e^{i d \left (a+b \log \left (c x^n\right )-b n \log (x)\right )} \left ((b d n+2 i m+2 i) x^{2 i b d n} \text{Hypergeometric2F1}\left (\frac{1}{2},-\frac{i \left (\frac{3}{2} i b d n+m+1\right )}{2 b d n},-\frac{-7 b d n+2 i m+2 i}{4 b d n},e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )+(3 b d n-2 i m-2 i) \text{Hypergeometric2F1}\left (\frac{1}{2},-\frac{b d n+2 i m+2 i}{4 b d n},-\frac{-3 b d n+2 i m+2 i}{4 b d n},e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )\right )}{(-i b d n+2 m+2) (3 i b d n+2 m+2) \sqrt{-i e^{-i a d} \left (c x^n\right )^{-i b d} \left (-1+e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )} \left ((b d n-2 i m-2 i) e^{2 i d \left (a+b \log \left (c x^n\right )-b n \log (x)\right )}+b d n+2 i m+2 i\right )}\right ) \]

Warning: Unable to verify antiderivative.

[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, ((-I/2)*(1 + m + ((3*I)/2)*b*d*n))/(b*d*n), -(2*
I + (2*I)*m - 7*b*d*n)/(4*b*d*n), E^((2*I)*a*d)*(c*x^n)^((2*I)*b*d)] + (-2*I - (2*I)*m + 3*b*d*n)*Hypergeometr
ic2F1[1/2, -(2*I + (2*I)*m + b*d*n)/(4*b*d*n), -(2*I + (2*I)*m - 3*b*d*n)/(4*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])]))

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Maple [F]  time = 0.269, size = 0, normalized size = 0. \begin{align*} \int \left ( ex \right ) ^{m}\sqrt{\sin \left ( d \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) \right ) }\, 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)))^(1/2),x)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (e x\right )^{m} \sqrt{\sin \left ({\left (b \log \left (c x^{n}\right ) + a\right )} d\right )}\,{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)))^(1/2),x, algorithm="maxima")

[Out]

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

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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)))^(1/2),x, algorithm="fricas")

[Out]

Exception raised: UnboundLocalError

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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)))**(1/2),x)

[Out]

Timed out

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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)))^(1/2),x, algorithm="giac")

[Out]

Timed out