3.3.6 \(\int \frac {\sin (a+\frac {b}{(c+d x)^{3/2}})}{(e+f x)^2} \, dx\) [206]

Optimal. Leaf size=25 \[ \text {Int}\left (\frac {\sin \left (a+\frac {b}{(c+d x)^{3/2}}\right )}{(e+f x)^2},x\right ) \]

[Out]

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

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Rubi [A]
time = 0.01, 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 {\sin \left (a+\frac {b}{(c+d x)^{3/2}}\right )}{(e+f x)^2} \, dx \end {gather*}

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*}

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Mathematica [A]
time = 11.73, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sin \left (a+\frac {b}{(c+d x)^{3/2}}\right )}{(e+f x)^2} \, dx \end {gather*}

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 = 0.03, size = 0, normalized size = 0.00 \[\int \frac {\sin \left (a +\frac {b}{\left (d x +c \right )^{\frac {3}{2}}}\right )}{\left (f x +e \right )^{2}}\, dx\]

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 = 0.00, 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(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 = 0.00, 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(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*f*x*e +
e^2), x)

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

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]

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

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Mupad [A]
time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int \frac {\sin \left (a+\frac {b}{{\left (c+d\,x\right )}^{3/2}}\right )}{{\left (e+f\,x\right )}^2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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