3.4.3 \(\int \frac {(e+f x)^2}{a+b \sin (c+\frac {d}{x})} \, dx\) [303]

Optimal. Leaf size=25 \[ \text {Int}\left (\frac {(e+f x)^2}{a+b \sin \left (c+\frac {d}{x}\right )},x\right ) \]

[Out]

Unintegrable((f*x+e)^2/(a+b*sin(c+d/x)),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 \frac {(e+f x)^2}{a+b \sin \left (c+\frac {d}{x}\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]
time = 0.07, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(e+f x)^2}{a+b \sin \left (c+\frac {d}{x}\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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Maple [A]
time = 0.00, size = 0, normalized size = 0.00 \[\int \frac {\left (f x +e \right )^{2}}{a +b \sin \left (c +\frac {d}{x}\right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((f*x+e)^2/(a+b*sin(c+d/x)),x)

[Out]

int((f*x+e)^2/(a+b*sin(c+d/x)),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((f*x+e)^2/(a+b*sin(c+d/x)),x, algorithm="maxima")

[Out]

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

[Out]

integral((f^2*x^2 + 2*f*x*e + e^2)/(b*sin((c*x + d)/x) + a), x)

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x+e)**2/(a+b*sin(c+d/x)),x)

[Out]

Timed out

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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((f*x+e)^2/(a+b*sin(c+d/x)),x, algorithm="giac")

[Out]

integrate((f*x + e)^2/(b*sin(c + d/x) + a), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e + f*x)^2/(a + b*sin(c + d/x)),x)

[Out]

int((e + f*x)^2/(a + b*sin(c + d/x)), x)

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