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

Optimal. Leaf size=60 \[ \frac{12 E\left (\left .\frac{1}{2} (a+b x)\right |2\right )}{25 b^2}+\frac{4 \sin (a+b x) \cos ^{\frac{3}{2}}(a+b x)}{25 b^2}-\frac{2 x \cos ^{\frac{5}{2}}(a+b x)}{5 b} \]

[Out]

(-2*x*Cos[a + b*x]^(5/2))/(5*b) + (12*EllipticE[(a + b*x)/2, 2])/(25*b^2) + (4*Cos[a + b*x]^(3/2)*Sin[a + b*x]
)/(25*b^2)

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Rubi [A]  time = 0.0397231, antiderivative size = 60, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {3444, 2635, 2639} \[ \frac{12 E\left (\left .\frac{1}{2} (a+b x)\right |2\right )}{25 b^2}+\frac{4 \sin (a+b x) \cos ^{\frac{3}{2}}(a+b x)}{25 b^2}-\frac{2 x \cos ^{\frac{5}{2}}(a+b x)}{5 b} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*x*Cos[a + b*x]^(5/2))/(5*b) + (12*EllipticE[(a + b*x)/2, 2])/(25*b^2) + (4*Cos[a + b*x]^(3/2)*Sin[a + b*x]
)/(25*b^2)

Rule 3444

Int[Cos[(a_.) + (b_.)*(x_)^(n_.)]^(p_.)*(x_)^(m_.)*Sin[(a_.) + (b_.)*(x_)^(n_.)], x_Symbol] :> -Simp[(x^(m - n
+ 1)*Cos[a + b*x^n]^(p + 1))/(b*n*(p + 1)), x] + Dist[(m - n + 1)/(b*n*(p + 1)), Int[x^(m - n)*Cos[a + b*x^n]
^(p + 1), x], x] /; FreeQ[{a, b, p}, x] && LtQ[0, n, m + 1] && NeQ[p, -1]

Rule 2635

Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> -Simp[(b*Cos[c + d*x]*(b*Sin[c + d*x])^(n - 1))/(d*n),
x] + Dist[(b^2*(n - 1))/n, Int[(b*Sin[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && Integer
Q[2*n]

Rule 2639

Int[Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2*EllipticE[(1*(c - Pi/2 + d*x))/2, 2])/d, x] /; FreeQ[{
c, d}, x]

Rubi steps

\begin{align*} \int x \cos ^{\frac{3}{2}}(a+b x) \sin (a+b x) \, dx &=-\frac{2 x \cos ^{\frac{5}{2}}(a+b x)}{5 b}+\frac{2 \int \cos ^{\frac{5}{2}}(a+b x) \, dx}{5 b}\\ &=-\frac{2 x \cos ^{\frac{5}{2}}(a+b x)}{5 b}+\frac{4 \cos ^{\frac{3}{2}}(a+b x) \sin (a+b x)}{25 b^2}+\frac{6 \int \sqrt{\cos (a+b x)} \, dx}{25 b}\\ &=-\frac{2 x \cos ^{\frac{5}{2}}(a+b x)}{5 b}+\frac{12 E\left (\left .\frac{1}{2} (a+b x)\right |2\right )}{25 b^2}+\frac{4 \cos ^{\frac{3}{2}}(a+b x) \sin (a+b x)}{25 b^2}\\ \end{align*}

Mathematica [A]  time = 0.390156, size = 51, normalized size = 0.85 \[ -\frac{2 \left (\cos ^{\frac{3}{2}}(a+b x) (5 b x \cos (a+b x)-2 \sin (a+b x))-6 E\left (\left .\frac{1}{2} (a+b x)\right |2\right )\right )}{25 b^2} \]

Antiderivative was successfully verified.

[In]

Integrate[x*Cos[a + b*x]^(3/2)*Sin[a + b*x],x]

[Out]

(-2*(-6*EllipticE[(a + b*x)/2, 2] + Cos[a + b*x]^(3/2)*(5*b*x*Cos[a + b*x] - 2*Sin[a + b*x])))/(25*b^2)

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Maple [F]  time = 0.087, size = 0, normalized size = 0. \begin{align*} \int x \left ( \cos \left ( bx+a \right ) \right ) ^{{\frac{3}{2}}}\sin \left ( bx+a \right ) \, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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