3.1.51 \(\int (e x)^m (a+b \sin (c+d x^2))^p \, dx\) [51]

Optimal. Leaf size=23 \[ \text {Int}\left ((e x)^m \left (a+b \sin \left (c+d x^2\right )\right )^p,x\right ) \]

[Out]

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

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Rubi [A]
time = 0.02, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int (e x)^m \left (a+b \sin \left (c+d x^2\right )\right )^p \, dx \end {gather*}

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*}

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Mathematica [A]
time = 0.59, size = 0, normalized size = 0.00 \begin {gather*} \int (e x)^m \left (a+b \sin \left (c+d x^2\right )\right )^p \, dx \end {gather*}

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 = 0.02, size = 0, normalized size = 0.00 \[\int \left (e x \right )^{m} \left (a +b \sin \left (d \,x^{2}+c \right )\right )^{p}\, dx\]

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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((x*e)^m*(b*sin(d*x^2 + c) + a)^p, x)

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Fricas [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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((x*e)^m*(b*sin(d*x^2 + c) + a)^p, x)

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Sympy [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (e x\right )^{m} \left (a + b \sin {\left (c + d x^{2} \right )}\right )^{p}\, dx \end {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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((x*e)^m*(b*sin(d*x^2 + c) + a)^p, x)

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Mupad [A]
time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int {\left (e\,x\right )}^m\,{\left (a+b\,\sin \left (d\,x^2+c\right )\right )}^p \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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