3.51 \(\int (e x)^m (a+b \sin (c+d x^2))^p \, dx\)

Optimal. Leaf size=22 \[ \text{Unintegrable}\left ((e x)^m \left (a+b \sin \left (c+d x^2\right )\right )^p,x\right ) \]

[Out]

Unintegrable[(e*x)^m*(a + b*Sin[c + d*x^2])^p, x]

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Rubi [A]  time = 0.0267236, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int (e x)^m \left (a+b \sin \left (c+d x^2\right )\right )^p \, dx \]

Verification is Not applicable to the result.

[In]

Int[(e*x)^m*(a + b*Sin[c + d*x^2])^p,x]

[Out]

Defer[Int][(e*x)^m*(a + b*Sin[c + d*x^2])^p, x]

Rubi steps

\begin{align*} \int (e x)^m \left (a+b \sin \left (c+d x^2\right )\right )^p \, dx &=\int (e x)^m \left (a+b \sin \left (c+d x^2\right )\right )^p \, dx\\ \end{align*}

Mathematica [A]  time = 0.846408, size = 0, normalized size = 0. \[ \int (e x)^m \left (a+b \sin \left (c+d x^2\right )\right )^p \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[(e*x)^m*(a + b*Sin[c + d*x^2])^p,x]

[Out]

Integrate[(e*x)^m*(a + b*Sin[c + d*x^2])^p, x]

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Maple [A]  time = 0.723, size = 0, normalized size = 0. \begin{align*} \int \left ( ex \right ) ^{m} \left ( a+b\sin \left ( d{x}^{2}+c \right ) \right ) ^{p}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x)^m*(a+b*sin(d*x^2+c))^p,x)

[Out]

int((e*x)^m*(a+b*sin(d*x^2+c))^p,x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((e*x)^m*(b*sin(d*x^2 + c) + a)^p, x)

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Fricas [A]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\left (e x\right )^{m}{\left (b \sin \left (d x^{2} + c\right ) + a\right )}^{p}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((e*x)^m*(b*sin(d*x^2 + c) + a)^p, x)

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Sympy [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (e x\right )^{m} \left (a + b \sin{\left (c + d x^{2} \right )}\right )^{p}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x)**m*(a+b*sin(d*x**2+c))**p,x)

[Out]

Integral((e*x)**m*(a + b*sin(c + d*x**2))**p, x)

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Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (e x\right )^{m}{\left (b \sin \left (d x^{2} + c\right ) + a\right )}^{p}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((e*x)^m*(b*sin(d*x^2 + c) + a)^p, x)