3.206 \(\int \frac{\sin (a+\frac{b}{(c+d x)^{3/2}})}{(e+f x)^2} \, dx\)

Optimal. Leaf size=24 \[ \text{Unintegrable}\left (\frac{\sin \left (a+\frac{b}{(c+d x)^{3/2}}\right )}{(e+f x)^2},x\right ) \]

[Out]

Unintegrable[Sin[a + b/(c + d*x)^(3/2)]/(e + f*x)^2, x]

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Rubi [A]  time = 0.0150614, 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 = {} \[ \int \frac{\sin \left (a+\frac{b}{(c+d x)^{3/2}}\right )}{(e+f x)^2} \, dx \]

Verification is Not applicable to the result.

[In]

Int[Sin[a + b/(c + d*x)^(3/2)]/(e + f*x)^2,x]

[Out]

Defer[Int][Sin[a + b/(c + d*x)^(3/2)]/(e + f*x)^2, x]

Rubi steps

\begin{align*} \int \frac{\sin \left (a+\frac{b}{(c+d x)^{3/2}}\right )}{(e+f x)^2} \, dx &=\int \frac{\sin \left (a+\frac{b}{(c+d x)^{3/2}}\right )}{(e+f x)^2} \, dx\\ \end{align*}

Mathematica [A]  time = 15.474, size = 0, normalized size = 0. \[ \int \frac{\sin \left (a+\frac{b}{(c+d x)^{3/2}}\right )}{(e+f x)^2} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[Sin[a + b/(c + d*x)^(3/2)]/(e + f*x)^2,x]

[Out]

Integrate[Sin[a + b/(c + d*x)^(3/2)]/(e + f*x)^2, x]

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Maple [A]  time = 0.055, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{ \left ( fx+e \right ) ^{2}}\sin \left ( a+{b \left ( dx+c \right ) ^{-{\frac{3}{2}}}} \right ) }\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(a+b/(d*x+c)^(3/2))/(f*x+e)^2,x)

[Out]

int(sin(a+b/(d*x+c)^(3/2))/(f*x+e)^2,x)

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Maxima [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sin \left (a + \frac{b}{{\left (d x + c\right )}^{\frac{3}{2}}}\right )}{{\left (f x + e\right )}^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(a+b/(d*x+c)^(3/2))/(f*x+e)^2,x, algorithm="maxima")

[Out]

integrate(sin(a + b/(d*x + c)^(3/2))/(f*x + e)^2, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(a+b/(d*x+c)^(3/2))/(f*x+e)^2,x, algorithm="fricas")

[Out]

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

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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(sin(a+b/(d*x+c)**(3/2))/(f*x+e)**2,x)

[Out]

Timed out

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Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sin \left (a + \frac{b}{{\left (d x + c\right )}^{\frac{3}{2}}}\right )}{{\left (f x + e\right )}^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(a+b/(d*x+c)^(3/2))/(f*x+e)^2,x, algorithm="giac")

[Out]

integrate(sin(a + b/(d*x + c)^(3/2))/(f*x + e)^2, x)