\(\int x \cos ^{\frac {3}{2}}(a+b x) \, dx\) [78]

Optimal result

Integrand size = 12, antiderivative size = 12 \[ \int x \cos ^{\frac {3}{2}}(a+b x) \, dx=\frac {4 \cos ^{\frac {3}{2}}(a+b x)}{9 b^2}+\frac {2 x \sqrt {\cos (a+b x)} \sin (a+b x)}{3 b}+\frac {1}{3} \text {Int}\left (\frac {x}{\sqrt {\cos (a+b x)}},x\right ) \]

[Out]

4/9*cos(b*x+a)^(3/2)/b^2+2/3*x*sin(b*x+a)*cos(b*x+a)^(1/2)/b+1/3*Unintegrable(x/cos(b*x+a)^(1/2),x)

Rubi [N/A]

Not integrable

Time = 0.05 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.00, number of steps used = 0, number of rules used = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int x \cos ^{\frac {3}{2}}(a+b x) \, dx=\int x \cos ^{\frac {3}{2}}(a+b x) \, dx \]

[In]

Int[x*Cos[a + b*x]^(3/2),x]

[Out]

(4*Cos[a + b*x]^(3/2))/(9*b^2) + (2*x*Sqrt[Cos[a + b*x]]*Sin[a + b*x])/(3*b) + Defer[Int][x/Sqrt[Cos[a + b*x]]
, x]/3

Rubi steps \begin{align*} \text {integral}& = \frac {4 \cos ^{\frac {3}{2}}(a+b x)}{9 b^2}+\frac {2 x \sqrt {\cos (a+b x)} \sin (a+b x)}{3 b}+\frac {1}{3} \int \frac {x}{\sqrt {\cos (a+b x)}} \, dx \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 2.29 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.17 \[ \int x \cos ^{\frac {3}{2}}(a+b x) \, dx=\int x \cos ^{\frac {3}{2}}(a+b x) \, dx \]

[In]

Integrate[x*Cos[a + b*x]^(3/2),x]

[Out]

Integrate[x*Cos[a + b*x]^(3/2), x]

Maple [N/A] (verified)

Not integrable

Time = 0.13 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.83

\[\int x \left (\cos ^{\frac {3}{2}}\left (b x +a \right )\right )d x\]

[In]

int(x*cos(b*x+a)^(3/2),x)

[Out]

int(x*cos(b*x+a)^(3/2),x)

Fricas [F(-2)]

Exception generated. \[ \int x \cos ^{\frac {3}{2}}(a+b x) \, dx=\text {Exception raised: TypeError} \]

[In]

integrate(x*cos(b*x+a)^(3/2),x, algorithm="fricas")

[Out]

Exception raised: TypeError >>  Error detected within library code:   integrate: implementation incomplete (ha
s polynomial part)

Sympy [N/A]

Not integrable

Time = 87.66 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.00 \[ \int x \cos ^{\frac {3}{2}}(a+b x) \, dx=\int x \cos ^{\frac {3}{2}}{\left (a + b x \right )}\, dx \]

[In]

integrate(x*cos(b*x+a)**(3/2),x)

[Out]

Integral(x*cos(a + b*x)**(3/2), x)

Maxima [N/A]

Not integrable

Time = 0.70 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.00 \[ \int x \cos ^{\frac {3}{2}}(a+b x) \, dx=\int { x \cos \left (b x + a\right )^{\frac {3}{2}} \,d x } \]

[In]

integrate(x*cos(b*x+a)^(3/2),x, algorithm="maxima")

[Out]

integrate(x*cos(b*x + a)^(3/2), x)

Giac [N/A]

Not integrable

Time = 0.37 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.00 \[ \int x \cos ^{\frac {3}{2}}(a+b x) \, dx=\int { x \cos \left (b x + a\right )^{\frac {3}{2}} \,d x } \]

[In]

integrate(x*cos(b*x+a)^(3/2),x, algorithm="giac")

[Out]

integrate(x*cos(b*x + a)^(3/2), x)

Mupad [N/A]

Not integrable

Time = 13.73 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.00 \[ \int x \cos ^{\frac {3}{2}}(a+b x) \, dx=\int x\,{\cos \left (a+b\,x\right )}^{3/2} \,d x \]

[In]

int(x*cos(a + b*x)^(3/2),x)

[Out]

int(x*cos(a + b*x)^(3/2), x)