3.2.96 \(\int \frac {\sin (a+b (c+d x)^{3/2})}{(e+f x)^2} \, dx\) [196]

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

[Out]

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

________________________________________________________________________________________

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

________________________________________________________________________________________

Mathematica [A]
time = 8.43, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sin \left (a+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]

________________________________________________________________________________________

Maple [A]
time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {\sin \left (a +b \left (d x +c \right )^{\frac {3}{2}}\right )}{\left (f x +e \right )^{2}}\, dx\]

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)

________________________________________________________________________________________

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

________________________________________________________________________________________

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

________________________________________________________________________________________

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

________________________________________________________________________________________

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

________________________________________________________________________________________

Mupad [A]
time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int \frac {\sin \left (a+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)

________________________________________________________________________________________