Optimal. Leaf size=25 \[ \text {Int}\left (\frac {\sin \left (a+\frac {b}{(c+d x)^{3/2}}\right )}{(e+f x)^2},x\right ) \]
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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 = {} \[ \int \frac {\sin \left (a+\frac {b}{(c+d x)^{3/2}}\right )}{(e+f x)^2} \, dx \]
Verification is Not applicable to the result.
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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 = 15.90, size = 0, normalized size = 0.00 \[ \int \frac {\sin \left (a+\frac {b}{(c+d x)^{3/2}}\right )}{(e+f x)^2} \, dx \]
Verification is Not applicable to the result.
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fricas [A] time = 0.81, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sin \left (\frac {a d^{2} x^{2} + 2 \, a c d x + a c^{2} + \sqrt {d x + c} b}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right )}{f^{2} x^{2} + 2 \, e f x + e^{2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sin \left (a + \frac {b}{{\left (d x + c\right )}^{\frac {3}{2}}}\right )}{{\left (f x + e\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.08, size = 0, normalized size = 0.00 \[ \int \frac {\sin \left (a +\frac {b}{\left (d x +c \right )^{\frac {3}{2}}}\right )}{\left (f x +e \right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sin \left (a + \frac {b}{{\left (d x + c\right )}^{\frac {3}{2}}}\right )}{{\left (f x + e\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [A] time = 0.00, size = -1, normalized size = -0.04 \[ \int \frac {\sin \left (a+\frac {b}{{\left (c+d\,x\right )}^{3/2}}\right )}{{\left (e+f\,x\right )}^2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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