3.257 \(\int (d \cos (a+b x))^{9/2} \sqrt{c \sin (a+b x)} \, dx\)

Optimal. Leaf size=132 \[ \frac{7 d^3 (c \sin (a+b x))^{3/2} (d \cos (a+b x))^{3/2}}{30 b c}+\frac{7 d^4 E\left (\left .a+b x-\frac{\pi }{4}\right |2\right ) \sqrt{c \sin (a+b x)} \sqrt{d \cos (a+b x)}}{20 b \sqrt{\sin (2 a+2 b x)}}+\frac{d (c \sin (a+b x))^{3/2} (d \cos (a+b x))^{7/2}}{5 b c} \]

[Out]

(7*d^3*(d*Cos[a + b*x])^(3/2)*(c*Sin[a + b*x])^(3/2))/(30*b*c) + (d*(d*Cos[a + b*x])^(7/2)*(c*Sin[a + b*x])^(3
/2))/(5*b*c) + (7*d^4*Sqrt[d*Cos[a + b*x]]*EllipticE[a - Pi/4 + b*x, 2]*Sqrt[c*Sin[a + b*x]])/(20*b*Sqrt[Sin[2
*a + 2*b*x]])

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Rubi [A]  time = 0.159601, antiderivative size = 132, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.12, Rules used = {2569, 2572, 2639} \[ \frac{7 d^3 (c \sin (a+b x))^{3/2} (d \cos (a+b x))^{3/2}}{30 b c}+\frac{7 d^4 E\left (\left .a+b x-\frac{\pi }{4}\right |2\right ) \sqrt{c \sin (a+b x)} \sqrt{d \cos (a+b x)}}{20 b \sqrt{\sin (2 a+2 b x)}}+\frac{d (c \sin (a+b x))^{3/2} (d \cos (a+b x))^{7/2}}{5 b c} \]

Antiderivative was successfully verified.

[In]

Int[(d*Cos[a + b*x])^(9/2)*Sqrt[c*Sin[a + b*x]],x]

[Out]

(7*d^3*(d*Cos[a + b*x])^(3/2)*(c*Sin[a + b*x])^(3/2))/(30*b*c) + (d*(d*Cos[a + b*x])^(7/2)*(c*Sin[a + b*x])^(3
/2))/(5*b*c) + (7*d^4*Sqrt[d*Cos[a + b*x]]*EllipticE[a - Pi/4 + b*x, 2]*Sqrt[c*Sin[a + b*x]])/(20*b*Sqrt[Sin[2
*a + 2*b*x]])

Rule 2569

Int[(cos[(e_.) + (f_.)*(x_)]*(a_.))^(m_)*((b_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(a*(b*Sin[e +
f*x])^(n + 1)*(a*Cos[e + f*x])^(m - 1))/(b*f*(m + n)), x] + Dist[(a^2*(m - 1))/(m + n), Int[(b*Sin[e + f*x])^
n*(a*Cos[e + f*x])^(m - 2), x], x] /; FreeQ[{a, b, e, f, n}, x] && GtQ[m, 1] && NeQ[m + n, 0] && IntegersQ[2*m
, 2*n]

Rule 2572

Int[Sqrt[cos[(e_.) + (f_.)*(x_)]*(b_.)]*Sqrt[(a_.)*sin[(e_.) + (f_.)*(x_)]], x_Symbol] :> Dist[(Sqrt[a*Sin[e +
f*x]]*Sqrt[b*Cos[e + f*x]])/Sqrt[Sin[2*e + 2*f*x]], Int[Sqrt[Sin[2*e + 2*f*x]], x], x] /; FreeQ[{a, b, e, f},
x]

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 (d \cos (a+b x))^{9/2} \sqrt{c \sin (a+b x)} \, dx &=\frac{d (d \cos (a+b x))^{7/2} (c \sin (a+b x))^{3/2}}{5 b c}+\frac{1}{10} \left (7 d^2\right ) \int (d \cos (a+b x))^{5/2} \sqrt{c \sin (a+b x)} \, dx\\ &=\frac{7 d^3 (d \cos (a+b x))^{3/2} (c \sin (a+b x))^{3/2}}{30 b c}+\frac{d (d \cos (a+b x))^{7/2} (c \sin (a+b x))^{3/2}}{5 b c}+\frac{1}{20} \left (7 d^4\right ) \int \sqrt{d \cos (a+b x)} \sqrt{c \sin (a+b x)} \, dx\\ &=\frac{7 d^3 (d \cos (a+b x))^{3/2} (c \sin (a+b x))^{3/2}}{30 b c}+\frac{d (d \cos (a+b x))^{7/2} (c \sin (a+b x))^{3/2}}{5 b c}+\frac{\left (7 d^4 \sqrt{d \cos (a+b x)} \sqrt{c \sin (a+b x)}\right ) \int \sqrt{\sin (2 a+2 b x)} \, dx}{20 \sqrt{\sin (2 a+2 b x)}}\\ &=\frac{7 d^3 (d \cos (a+b x))^{3/2} (c \sin (a+b x))^{3/2}}{30 b c}+\frac{d (d \cos (a+b x))^{7/2} (c \sin (a+b x))^{3/2}}{5 b c}+\frac{7 d^4 \sqrt{d \cos (a+b x)} E\left (\left .a-\frac{\pi }{4}+b x\right |2\right ) \sqrt{c \sin (a+b x)}}{20 b \sqrt{\sin (2 a+2 b x)}}\\ \end{align*}

Mathematica [C]  time = 0.102623, size = 70, normalized size = 0.53 \[ \frac{2 d^4 \sqrt [4]{\cos ^2(a+b x)} \tan (a+b x) \sqrt{c \sin (a+b x)} \sqrt{d \cos (a+b x)} \, _2F_1\left (-\frac{7}{4},\frac{3}{4};\frac{7}{4};\sin ^2(a+b x)\right )}{3 b} \]

Antiderivative was successfully verified.

[In]

Integrate[(d*Cos[a + b*x])^(9/2)*Sqrt[c*Sin[a + b*x]],x]

[Out]

(2*d^4*Sqrt[d*Cos[a + b*x]]*(Cos[a + b*x]^2)^(1/4)*Hypergeometric2F1[-7/4, 3/4, 7/4, Sin[a + b*x]^2]*Sqrt[c*Si
n[a + b*x]]*Tan[a + b*x])/(3*b)

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Maple [B]  time = 0.224, size = 540, normalized size = 4.1 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*cos(b*x+a))^(9/2)*(c*sin(b*x+a))^(1/2),x)

[Out]

-1/120/b*2^(1/2)*(c*sin(b*x+a))^(1/2)*(d*cos(b*x+a))^(9/2)*(12*cos(b*x+a)^6*2^(1/2)+2*cos(b*x+a)^4*2^(1/2)+42*
cos(b*x+a)*(-(-1+cos(b*x+a)-sin(b*x+a))/sin(b*x+a))^(1/2)*((-1+cos(b*x+a)+sin(b*x+a))/sin(b*x+a))^(1/2)*((-1+c
os(b*x+a))/sin(b*x+a))^(1/2)*EllipticE((-(-1+cos(b*x+a)-sin(b*x+a))/sin(b*x+a))^(1/2),1/2*2^(1/2))-21*cos(b*x+
a)*(-(-1+cos(b*x+a)-sin(b*x+a))/sin(b*x+a))^(1/2)*((-1+cos(b*x+a)+sin(b*x+a))/sin(b*x+a))^(1/2)*((-1+cos(b*x+a
))/sin(b*x+a))^(1/2)*EllipticF((-(-1+cos(b*x+a)-sin(b*x+a))/sin(b*x+a))^(1/2),1/2*2^(1/2))+42*(-(-1+cos(b*x+a)
-sin(b*x+a))/sin(b*x+a))^(1/2)*((-1+cos(b*x+a)+sin(b*x+a))/sin(b*x+a))^(1/2)*((-1+cos(b*x+a))/sin(b*x+a))^(1/2
)*EllipticE((-(-1+cos(b*x+a)-sin(b*x+a))/sin(b*x+a))^(1/2),1/2*2^(1/2))-21*(-(-1+cos(b*x+a)-sin(b*x+a))/sin(b*
x+a))^(1/2)*((-1+cos(b*x+a)+sin(b*x+a))/sin(b*x+a))^(1/2)*((-1+cos(b*x+a))/sin(b*x+a))^(1/2)*EllipticF((-(-1+c
os(b*x+a)-sin(b*x+a))/sin(b*x+a))^(1/2),1/2*2^(1/2))+7*cos(b*x+a)^2*2^(1/2)-21*cos(b*x+a)*2^(1/2))/cos(b*x+a)^
5/sin(b*x+a)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (d \cos \left (b x + a\right )\right )^{\frac{9}{2}} \sqrt{c \sin \left (b x + a\right )}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*cos(b*x+a))^(9/2)*(c*sin(b*x+a))^(1/2),x, algorithm="maxima")

[Out]

integrate((d*cos(b*x + a))^(9/2)*sqrt(c*sin(b*x + a)), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\sqrt{d \cos \left (b x + a\right )} \sqrt{c \sin \left (b x + a\right )} d^{4} \cos \left (b x + a\right )^{4}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*cos(b*x+a))^(9/2)*(c*sin(b*x+a))^(1/2),x, algorithm="fricas")

[Out]

integral(sqrt(d*cos(b*x + a))*sqrt(c*sin(b*x + a))*d^4*cos(b*x + a)^4, x)

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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((d*cos(b*x+a))**(9/2)*(c*sin(b*x+a))**(1/2),x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*cos(b*x+a))^(9/2)*(c*sin(b*x+a))^(1/2),x, algorithm="giac")

[Out]

Exception raised: TypeError