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

Optimal. Leaf size=61 \[ \frac {4 \cos ^{\frac {3}{2}}(a+b x)}{9 b^2}+\frac {2 x \sqrt {\cos (a+b x)} \sin (a+b x)}{3 b}+\frac {1}{3} \text {Int}\left (\frac {x}{\sqrt {\cos (a+b x)}},x\right ) \]

[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)

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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 = {} \begin {gather*} \int x \cos ^{\frac {3}{2}}(a+b x) \, dx \end {gather*}

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*}

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Mathematica [A]
time = 1.20, size = 0, normalized size = 0.00 \begin {gather*} \int x \cos ^{\frac {3}{2}}(a+b x) \, dx \end {gather*}

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 = 0.02, size = 0, normalized size = 0.00 \[\int x \left (\cos ^{\frac {3}{2}}\left (b x +a \right )\right )\, dx\]

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 = 0.00, 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(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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

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

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Sympy [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int x \cos ^{\frac {3}{2}}{\left (a + b x \right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(x*cos(a + b*x)**(3/2), x)

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

[Out]

integrate(x*cos(b*x + a)^(3/2), x)

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Mupad [A]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int x\,{\cos \left (a+b\,x\right )}^{3/2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*cos(a + b*x)^(3/2),x)

[Out]

int(x*cos(a + b*x)^(3/2), x)

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