3.432 \(\int (\sqrt {b^2+c^2}+b \cos (d+e x)+c \sin (d+e x))^{3/2} \, dx\)

Optimal. Leaf size=126 \[ -\frac {2 \sqrt {\sqrt {b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)} (c \cos (d+e x)-b \sin (d+e x))}{3 e}-\frac {8 \sqrt {b^2+c^2} (c \cos (d+e x)-b \sin (d+e x))}{3 e \sqrt {\sqrt {b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)}} \]

[Out]

-8/3*(c*cos(e*x+d)-b*sin(e*x+d))*(b^2+c^2)^(1/2)/e/(b*cos(e*x+d)+c*sin(e*x+d)+(b^2+c^2)^(1/2))^(1/2)-2/3*(c*co
s(e*x+d)-b*sin(e*x+d))*(b*cos(e*x+d)+c*sin(e*x+d)+(b^2+c^2)^(1/2))^(1/2)/e

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Rubi [A]  time = 0.07, antiderivative size = 126, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {3113, 3112} \[ -\frac {2 \sqrt {\sqrt {b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)} (c \cos (d+e x)-b \sin (d+e x))}{3 e}-\frac {8 \sqrt {b^2+c^2} (c \cos (d+e x)-b \sin (d+e x))}{3 e \sqrt {\sqrt {b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)}} \]

Antiderivative was successfully verified.

[In]

Int[(Sqrt[b^2 + c^2] + b*Cos[d + e*x] + c*Sin[d + e*x])^(3/2),x]

[Out]

(-8*Sqrt[b^2 + c^2]*(c*Cos[d + e*x] - b*Sin[d + e*x]))/(3*e*Sqrt[Sqrt[b^2 + c^2] + b*Cos[d + e*x] + c*Sin[d +
e*x]]) - (2*(c*Cos[d + e*x] - b*Sin[d + e*x])*Sqrt[Sqrt[b^2 + c^2] + b*Cos[d + e*x] + c*Sin[d + e*x]])/(3*e)

Rule 3112

Int[Sqrt[cos[(d_.) + (e_.)*(x_)]*(b_.) + (a_) + (c_.)*sin[(d_.) + (e_.)*(x_)]], x_Symbol] :> Simp[(-2*(c*Cos[d
+ e*x] - b*Sin[d + e*x]))/(e*Sqrt[a + b*Cos[d + e*x] + c*Sin[d + e*x]]), x] /; FreeQ[{a, b, c, d, e}, x] && E
qQ[a^2 - b^2 - c^2, 0]

Rule 3113

Int[(cos[(d_.) + (e_.)*(x_)]*(b_.) + (a_) + (c_.)*sin[(d_.) + (e_.)*(x_)])^(n_), x_Symbol] :> -Simp[((c*Cos[d
+ e*x] - b*Sin[d + e*x])*(a + b*Cos[d + e*x] + c*Sin[d + e*x])^(n - 1))/(e*n), x] + Dist[(a*(2*n - 1))/n, Int[
(a + b*Cos[d + e*x] + c*Sin[d + e*x])^(n - 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[a^2 - b^2 - c^2, 0]
&& GtQ[n, 0]

Rubi steps

\begin {align*} \int \left (\sqrt {b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )^{3/2} \, dx &=-\frac {2 (c \cos (d+e x)-b \sin (d+e x)) \sqrt {\sqrt {b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)}}{3 e}+\frac {1}{3} \left (4 \sqrt {b^2+c^2}\right ) \int \sqrt {\sqrt {b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)} \, dx\\ &=-\frac {8 \sqrt {b^2+c^2} (c \cos (d+e x)-b \sin (d+e x))}{3 e \sqrt {\sqrt {b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)}}-\frac {2 (c \cos (d+e x)-b \sin (d+e x)) \sqrt {\sqrt {b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)}}{3 e}\\ \end {align*}

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Mathematica [C]  time = 33.01, size = 11679, normalized size = 92.69 \[ \text {Result too large to show} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[(Sqrt[b^2 + c^2] + b*Cos[d + e*x] + c*Sin[d + e*x])^(3/2),x]

[Out]

Result too large to show

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fricas [A]  time = 0.68, size = 125, normalized size = 0.99 \[ \frac {2 \, {\left (2 \, b c \cos \left (e x + d\right ) \sin \left (e x + d\right ) + {\left (b^{2} - c^{2}\right )} \cos \left (e x + d\right )^{2} - 5 \, b^{2} - 4 \, c^{2} + 4 \, \sqrt {b^{2} + c^{2}} {\left (b \cos \left (e x + d\right ) + c \sin \left (e x + d\right )\right )}\right )} \sqrt {b \cos \left (e x + d\right ) + c \sin \left (e x + d\right ) + \sqrt {b^{2} + c^{2}}}}{3 \, {\left (c e \cos \left (e x + d\right ) - b e \sin \left (e x + d\right )\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*cos(e*x+d)+c*sin(e*x+d)+(b^2+c^2)^(1/2))^(3/2),x, algorithm="fricas")

[Out]

2/3*(2*b*c*cos(e*x + d)*sin(e*x + d) + (b^2 - c^2)*cos(e*x + d)^2 - 5*b^2 - 4*c^2 + 4*sqrt(b^2 + c^2)*(b*cos(e
*x + d) + c*sin(e*x + d)))*sqrt(b*cos(e*x + d) + c*sin(e*x + d) + sqrt(b^2 + c^2))/(c*e*cos(e*x + d) - b*e*sin
(e*x + d))

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giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*cos(e*x+d)+c*sin(e*x+d)+(b^2+c^2)^(1/2))^(3/2),x, algorithm="giac")

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,x):;OUTPUT:Simp
lification assuming b near 0sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argum
ent Value

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maple [A]  time = 0.37, size = 126, normalized size = 1.00 \[ \frac {2 \left (1+\sin \left (e x +d -\arctan \left (-b , c\right )\right )\right ) \left (b^{2}+c^{2}\right ) \left (\sin \left (e x +d -\arctan \left (-b , c\right )\right )-1\right ) \left (\sin \left (e x +d -\arctan \left (-b , c\right )\right )+5\right )}{3 \cos \left (e x +d -\arctan \left (-b , c\right )\right ) \sqrt {\frac {b^{2} \sin \left (e x +d -\arctan \left (-b , c\right )\right )+c^{2} \sin \left (e x +d -\arctan \left (-b , c\right )\right )+b^{2}+c^{2}}{\sqrt {b^{2}+c^{2}}}}\, e} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*cos(e*x+d)+c*sin(e*x+d)+(b^2+c^2)^(1/2))^(3/2),x)

[Out]

2/3*(1+sin(e*x+d-arctan(-b,c)))*(b^2+c^2)*(sin(e*x+d-arctan(-b,c))-1)*(sin(e*x+d-arctan(-b,c))+5)/cos(e*x+d-ar
ctan(-b,c))/((b^2*sin(e*x+d-arctan(-b,c))+c^2*sin(e*x+d-arctan(-b,c))+b^2+c^2)/(b^2+c^2)^(1/2))^(1/2)/e

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maxima [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: RuntimeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*cos(e*x+d)+c*sin(e*x+d)+(b^2+c^2)^(1/2))^(3/2),x, algorithm="maxima")

[Out]

Exception raised: RuntimeError >> ECL says: THROW: The catch RAT-ERR is undefined.

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int {\left (b\,\cos \left (d+e\,x\right )+c\,\sin \left (d+e\,x\right )+\sqrt {b^2+c^2}\right )}^{3/2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*cos(d + e*x) + c*sin(d + e*x) + (b^2 + c^2)^(1/2))^(3/2),x)

[Out]

int((b*cos(d + e*x) + c*sin(d + e*x) + (b^2 + c^2)^(1/2))^(3/2), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (b \cos {\left (d + e x \right )} + c \sin {\left (d + e x \right )} + \sqrt {b^{2} + c^{2}}\right )^{\frac {3}{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*cos(e*x+d)+c*sin(e*x+d)+(b**2+c**2)**(1/2))**(3/2),x)

[Out]

Integral((b*cos(d + e*x) + c*sin(d + e*x) + sqrt(b**2 + c**2))**(3/2), x)

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