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

Optimal. Leaf size=60 \[ \frac{1}{3} \text{Unintegrable}\left (\frac{x}{\sqrt{\cos (a+b x)}},x\right )+\frac{4 \cos ^{\frac{3}{2}}(a+b x)}{9 b^2}+\frac{2 x \sin (a+b x) \sqrt{\cos (a+b x)}}{3 b} \]

[Out]

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

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Rubi [A]  time = 0.0377989, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int x \cos ^{\frac{3}{2}}(a+b x) \, dx \]

Verification is Not applicable to the result.

[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*} \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} \int \frac{x}{\sqrt{\cos (a+b x)}} \, dx\\ \end{align*}

Mathematica [A]  time = 2.02598, size = 0, normalized size = 0. \[ \int x \cos ^{\frac{3}{2}}(a+b x) \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[x*Cos[a + b*x]^(3/2),x]

[Out]

Integrate[x*Cos[a + b*x]^(3/2), x]

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Maple [A]  time = 0.122, size = 0, normalized size = 0. \begin{align*} \int x \left ( \cos \left ( bx+a \right ) \right ) ^{{\frac{3}{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x \cos \left (b x + a\right )^{\frac{3}{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(x*cos(b*x + a)^(3/2), x)

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Fricas [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: UnboundLocalError

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x \cos \left (b x + a\right )^{\frac{3}{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(x*cos(b*x + a)^(3/2), x)