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

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

[Out]

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

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]  time = 13.38, size = 0, normalized size = 0.00 \[ \int \frac {\sin \left (a+\frac {b}{(c+d x)^{3/2}}\right )}{e+f x} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.70, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sin \left (\frac {a d^{2} x^{2} + 2 \, a c d x + a c^{2} + \sqrt {d x + c} b}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right )}{f x + e}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.08, size = 0, normalized size = 0.00 \[ \int \frac {\sin \left (a +\frac {b}{\left (d x +c \right )^{\frac {3}{2}}}\right )}{f x +e}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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