3.430 \(\int (\sqrt{b^2+c^2}+b \cos (d+e x)+c \sin (d+e x))^{7/2} \, dx\)

Optimal. Leaf size=258 \[ -\frac{2 (c \cos (d+e x)-b \sin (d+e x)) \left (\sqrt{b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )^{5/2}}{7 e}-\frac{24 \sqrt{b^2+c^2} (c \cos (d+e x)-b \sin (d+e x)) \left (\sqrt{b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )^{3/2}}{35 e}-\frac{64 \left (b^2+c^2\right ) (c \cos (d+e x)-b \sin (d+e x)) \sqrt{\sqrt{b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)}}{35 e}-\frac{256 \left (b^2+c^2\right )^{3/2} (c \cos (d+e x)-b \sin (d+e x))}{35 e \sqrt{\sqrt{b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)}} \]

[Out]

(-256*(b^2 + c^2)^(3/2)*(c*Cos[d + e*x] - b*Sin[d + e*x]))/(35*e*Sqrt[Sqrt[b^2 + c^2] + b*Cos[d + e*x] + c*Sin
[d + e*x]]) - (64*(b^2 + c^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]])/(35*e) - (24*Sqrt[b^2 + c^2]*(c*Cos[d + e*x] - b*Sin[d + e*x])*(Sqrt[b^2 + c^2] + b*Cos[d + e*x] +
c*Sin[d + e*x])^(3/2))/(35*e) - (2*(c*Cos[d + e*x] - b*Sin[d + e*x])*(Sqrt[b^2 + c^2] + b*Cos[d + e*x] + c*Sin
[d + e*x])^(5/2))/(7*e)

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Rubi [A]  time = 0.178605, antiderivative size = 258, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.062, Rules used = {3113, 3112} \[ -\frac{2 (c \cos (d+e x)-b \sin (d+e x)) \left (\sqrt{b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )^{5/2}}{7 e}-\frac{24 \sqrt{b^2+c^2} (c \cos (d+e x)-b \sin (d+e x)) \left (\sqrt{b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )^{3/2}}{35 e}-\frac{64 \left (b^2+c^2\right ) (c \cos (d+e x)-b \sin (d+e x)) \sqrt{\sqrt{b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)}}{35 e}-\frac{256 \left (b^2+c^2\right )^{3/2} (c \cos (d+e x)-b \sin (d+e x))}{35 e \sqrt{\sqrt{b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)}} \]

Antiderivative was successfully verified.

[In]

Int[(Sqrt[b^2 + c^2] + b*Cos[d + e*x] + c*Sin[d + e*x])^(7/2),x]

[Out]

(-256*(b^2 + c^2)^(3/2)*(c*Cos[d + e*x] - b*Sin[d + e*x]))/(35*e*Sqrt[Sqrt[b^2 + c^2] + b*Cos[d + e*x] + c*Sin
[d + e*x]]) - (64*(b^2 + c^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]])/(35*e) - (24*Sqrt[b^2 + c^2]*(c*Cos[d + e*x] - b*Sin[d + e*x])*(Sqrt[b^2 + c^2] + b*Cos[d + e*x] +
c*Sin[d + e*x])^(3/2))/(35*e) - (2*(c*Cos[d + e*x] - b*Sin[d + e*x])*(Sqrt[b^2 + c^2] + b*Cos[d + e*x] + c*Sin
[d + e*x])^(5/2))/(7*e)

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]

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]

Rubi steps

\begin{align*} \int \left (\sqrt{b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )^{7/2} \, dx &=-\frac{2 (c \cos (d+e x)-b \sin (d+e x)) \left (\sqrt{b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )^{5/2}}{7 e}+\frac{1}{7} \left (12 \sqrt{b^2+c^2}\right ) \int \left (\sqrt{b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )^{5/2} \, dx\\ &=-\frac{24 \sqrt{b^2+c^2} (c \cos (d+e x)-b \sin (d+e x)) \left (\sqrt{b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )^{3/2}}{35 e}-\frac{2 (c \cos (d+e x)-b \sin (d+e x)) \left (\sqrt{b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )^{5/2}}{7 e}+\frac{1}{35} \left (96 \left (b^2+c^2\right )\right ) \int \left (\sqrt{b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )^{3/2} \, dx\\ &=-\frac{64 \left (b^2+c^2\right ) (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)}}{35 e}-\frac{24 \sqrt{b^2+c^2} (c \cos (d+e x)-b \sin (d+e x)) \left (\sqrt{b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )^{3/2}}{35 e}-\frac{2 (c \cos (d+e x)-b \sin (d+e x)) \left (\sqrt{b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )^{5/2}}{7 e}+\frac{1}{35} \left (128 \left (b^2+c^2\right )^{3/2}\right ) \int \sqrt{\sqrt{b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)} \, dx\\ &=-\frac{256 \left (b^2+c^2\right )^{3/2} (c \cos (d+e x)-b \sin (d+e x))}{35 e \sqrt{\sqrt{b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)}}-\frac{64 \left (b^2+c^2\right ) (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)}}{35 e}-\frac{24 \sqrt{b^2+c^2} (c \cos (d+e x)-b \sin (d+e x)) \left (\sqrt{b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )^{3/2}}{35 e}-\frac{2 (c \cos (d+e x)-b \sin (d+e x)) \left (\sqrt{b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )^{5/2}}{7 e}\\ \end{align*}

Mathematica [C]  time = 32.745, size = 11888, normalized size = 46.08 \[ \text{Result too large to show} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[(Sqrt[b^2 + c^2] + b*Cos[d + e*x] + c*Sin[d + e*x])^(7/2),x]

[Out]

Result too large to show

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Maple [A]  time = 1.937, size = 306, normalized size = 1.2 \begin{align*}{\frac{ \left ( 2+2\,\sin \left ( ex+d-\arctan \left ( -b,c \right ) \right ) \right ) \left ( \sin \left ( ex+d-\arctan \left ( -b,c \right ) \right ) -1 \right ) \left ( 5\,{b}^{4} \left ( \sin \left ( ex+d-\arctan \left ( -b,c \right ) \right ) \right ) ^{3}+10\,{b}^{2}{c}^{2} \left ( \sin \left ( ex+d-\arctan \left ( -b,c \right ) \right ) \right ) ^{3}+5\,{c}^{4} \left ( \sin \left ( ex+d-\arctan \left ( -b,c \right ) \right ) \right ) ^{3}+27\,{b}^{4} \left ( \sin \left ( ex+d-\arctan \left ( -b,c \right ) \right ) \right ) ^{2}+54\,{b}^{2}{c}^{2} \left ( \sin \left ( ex+d-\arctan \left ( -b,c \right ) \right ) \right ) ^{2}+27\,{c}^{4} \left ( \sin \left ( ex+d-\arctan \left ( -b,c \right ) \right ) \right ) ^{2}+71\,{b}^{4}\sin \left ( ex+d-\arctan \left ( -b,c \right ) \right ) +142\,{b}^{2}{c}^{2}\sin \left ( ex+d-\arctan \left ( -b,c \right ) \right ) +71\,{c}^{4}\sin \left ( ex+d-\arctan \left ( -b,c \right ) \right ) +177\,{b}^{4}+354\,{b}^{2}{c}^{2}+177\,{c}^{4} \right ) }{35\,\cos \left ( ex+d-\arctan \left ( -b,c \right ) \right ) e}{\frac{1}{\sqrt{{({b}^{2}\sin \left ( ex+d-\arctan \left ( -b,c \right ) \right ) +{c}^{2}\sin \left ( ex+d-\arctan \left ( -b,c \right ) \right ) +{b}^{2}+{c}^{2}){\frac{1}{\sqrt{{b}^{2}+{c}^{2}}}}}}}}} \end{align*}

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))^(7/2),x)

[Out]

2/35*(1+sin(e*x+d-arctan(-b,c)))*(sin(e*x+d-arctan(-b,c))-1)*(5*b^4*sin(e*x+d-arctan(-b,c))^3+10*b^2*c^2*sin(e
*x+d-arctan(-b,c))^3+5*c^4*sin(e*x+d-arctan(-b,c))^3+27*b^4*sin(e*x+d-arctan(-b,c))^2+54*b^2*c^2*sin(e*x+d-arc
tan(-b,c))^2+27*c^4*sin(e*x+d-arctan(-b,c))^2+71*b^4*sin(e*x+d-arctan(-b,c))+142*b^2*c^2*sin(e*x+d-arctan(-b,c
))+71*c^4*sin(e*x+d-arctan(-b,c))+177*b^4+354*b^2*c^2+177*c^4)/cos(e*x+d-arctan(-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., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: RuntimeError} \end{align*}

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))^(7/2),x, algorithm="maxima")

[Out]

Exception raised: RuntimeError

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Fricas [A]  time = 2.00733, size = 645, normalized size = 2.5 \begin{align*} \frac{2 \,{\left (5 \,{\left (b^{4} - 6 \, b^{2} c^{2} + c^{4}\right )} \cos \left (e x + d\right )^{4} - 177 \, b^{4} - 310 \, b^{2} c^{2} - 128 \, c^{4} + 2 \,{\left (22 \, b^{4} + 15 \, b^{2} c^{2} - 27 \, c^{4}\right )} \cos \left (e x + d\right )^{2} + 4 \,{\left (5 \,{\left (b^{3} c - b c^{3}\right )} \cos \left (e x + d\right )^{3} +{\left (22 \, b^{3} c + 27 \, b c^{3}\right )} \cos \left (e x + d\right )\right )} \sin \left (e x + d\right ) + 2 \,{\left (11 \,{\left (b^{3} - 3 \, b c^{2}\right )} \cos \left (e x + d\right )^{3} +{\left (53 \, b^{3} + 86 \, b c^{2}\right )} \cos \left (e x + d\right ) +{\left (53 \, b^{2} c + 64 \, c^{3} + 11 \,{\left (3 \, b^{2} c - c^{3}\right )} \cos \left (e x + d\right )^{2}\right )} \sin \left (e x + d\right )\right )} \sqrt{b^{2} + c^{2}}\right )} \sqrt{b \cos \left (e x + d\right ) + c \sin \left (e x + d\right ) + \sqrt{b^{2} + c^{2}}}}{35 \,{\left (c e \cos \left (e x + d\right ) - b e \sin \left (e x + d\right )\right )}} \end{align*}

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))^(7/2),x, algorithm="fricas")

[Out]

2/35*(5*(b^4 - 6*b^2*c^2 + c^4)*cos(e*x + d)^4 - 177*b^4 - 310*b^2*c^2 - 128*c^4 + 2*(22*b^4 + 15*b^2*c^2 - 27
*c^4)*cos(e*x + d)^2 + 4*(5*(b^3*c - b*c^3)*cos(e*x + d)^3 + (22*b^3*c + 27*b*c^3)*cos(e*x + d))*sin(e*x + d)
+ 2*(11*(b^3 - 3*b*c^2)*cos(e*x + d)^3 + (53*b^3 + 86*b*c^2)*cos(e*x + d) + (53*b^2*c + 64*c^3 + 11*(3*b^2*c -
c^3)*cos(e*x + d)^2)*sin(e*x + d))*sqrt(b^2 + c^2))*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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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((b*cos(e*x+d)+c*sin(e*x+d)+(b**2+c**2)**(1/2))**(7/2),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((b*cos(e*x+d)+c*sin(e*x+d)+(b^2+c^2)^(1/2))^(7/2),x, algorithm="giac")

[Out]

Exception raised: TypeError