3.70 \(\int (d+e x)^3 (d^2-e^2 x^2)^{5/2} \, dx\)

Optimal. Leaf size=188 \[ \frac{55}{128} d^7 x \sqrt{d^2-e^2 x^2}+\frac{55}{192} d^5 x \left (d^2-e^2 x^2\right )^{3/2}+\frac{11}{48} d^3 x \left (d^2-e^2 x^2\right )^{5/2}-\frac{11 d^2 \left (d^2-e^2 x^2\right )^{7/2}}{56 e}-\frac{11 d (d+e x) \left (d^2-e^2 x^2\right )^{7/2}}{72 e}-\frac{(d+e x)^2 \left (d^2-e^2 x^2\right )^{7/2}}{9 e}+\frac{55 d^9 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{128 e} \]

[Out]

(55*d^7*x*Sqrt[d^2 - e^2*x^2])/128 + (55*d^5*x*(d^2 - e^2*x^2)^(3/2))/192 + (11*d^3*x*(d^2 - e^2*x^2)^(5/2))/4
8 - (11*d^2*(d^2 - e^2*x^2)^(7/2))/(56*e) - (11*d*(d + e*x)*(d^2 - e^2*x^2)^(7/2))/(72*e) - ((d + e*x)^2*(d^2
- e^2*x^2)^(7/2))/(9*e) + (55*d^9*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]])/(128*e)

________________________________________________________________________________________

Rubi [A]  time = 0.082442, antiderivative size = 188, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208, Rules used = {671, 641, 195, 217, 203} \[ \frac{55}{128} d^7 x \sqrt{d^2-e^2 x^2}+\frac{55}{192} d^5 x \left (d^2-e^2 x^2\right )^{3/2}+\frac{11}{48} d^3 x \left (d^2-e^2 x^2\right )^{5/2}-\frac{11 d^2 \left (d^2-e^2 x^2\right )^{7/2}}{56 e}-\frac{11 d (d+e x) \left (d^2-e^2 x^2\right )^{7/2}}{72 e}-\frac{(d+e x)^2 \left (d^2-e^2 x^2\right )^{7/2}}{9 e}+\frac{55 d^9 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{128 e} \]

Antiderivative was successfully verified.

[In]

Int[(d + e*x)^3*(d^2 - e^2*x^2)^(5/2),x]

[Out]

(55*d^7*x*Sqrt[d^2 - e^2*x^2])/128 + (55*d^5*x*(d^2 - e^2*x^2)^(3/2))/192 + (11*d^3*x*(d^2 - e^2*x^2)^(5/2))/4
8 - (11*d^2*(d^2 - e^2*x^2)^(7/2))/(56*e) - (11*d*(d + e*x)*(d^2 - e^2*x^2)^(7/2))/(72*e) - ((d + e*x)^2*(d^2
- e^2*x^2)^(7/2))/(9*e) + (55*d^9*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]])/(128*e)

Rule 671

Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(e*(d + e*x)^(m - 1)*(a + c*x^2)^(p
+ 1))/(c*(m + 2*p + 1)), x] + Dist[(2*c*d*(m + p))/(c*(m + 2*p + 1)), Int[(d + e*x)^(m - 1)*(a + c*x^2)^p, x]
, x] /; FreeQ[{a, c, d, e, p}, x] && EqQ[c*d^2 + a*e^2, 0] && GtQ[m, 1] && NeQ[m + 2*p + 1, 0] && IntegerQ[2*p
]

Rule 641

Int[((d_) + (e_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(e*(a + c*x^2)^(p + 1))/(2*c*(p + 1)),
x] + Dist[d, Int[(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, p}, x] && NeQ[p, -1]

Rule 195

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(x*(a + b*x^n)^p)/(n*p + 1), x] + Dist[(a*n*p)/(n*p + 1),
Int[(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && GtQ[p, 0] && (IntegerQ[2*p] || (EqQ[n, 2
] && IntegerQ[4*p]) || (EqQ[n, 2] && IntegerQ[3*p]) || LtQ[Denominator[p + 1/n], Denominator[p]])

Rule 217

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] &&  !GtQ[a, 0]

Rule 203

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTan[(Rt[b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rubi steps

\begin{align*} \int (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2} \, dx &=-\frac{(d+e x)^2 \left (d^2-e^2 x^2\right )^{7/2}}{9 e}+\frac{1}{9} (11 d) \int (d+e x)^2 \left (d^2-e^2 x^2\right )^{5/2} \, dx\\ &=-\frac{11 d (d+e x) \left (d^2-e^2 x^2\right )^{7/2}}{72 e}-\frac{(d+e x)^2 \left (d^2-e^2 x^2\right )^{7/2}}{9 e}+\frac{1}{8} \left (11 d^2\right ) \int (d+e x) \left (d^2-e^2 x^2\right )^{5/2} \, dx\\ &=-\frac{11 d^2 \left (d^2-e^2 x^2\right )^{7/2}}{56 e}-\frac{11 d (d+e x) \left (d^2-e^2 x^2\right )^{7/2}}{72 e}-\frac{(d+e x)^2 \left (d^2-e^2 x^2\right )^{7/2}}{9 e}+\frac{1}{8} \left (11 d^3\right ) \int \left (d^2-e^2 x^2\right )^{5/2} \, dx\\ &=\frac{11}{48} d^3 x \left (d^2-e^2 x^2\right )^{5/2}-\frac{11 d^2 \left (d^2-e^2 x^2\right )^{7/2}}{56 e}-\frac{11 d (d+e x) \left (d^2-e^2 x^2\right )^{7/2}}{72 e}-\frac{(d+e x)^2 \left (d^2-e^2 x^2\right )^{7/2}}{9 e}+\frac{1}{48} \left (55 d^5\right ) \int \left (d^2-e^2 x^2\right )^{3/2} \, dx\\ &=\frac{55}{192} d^5 x \left (d^2-e^2 x^2\right )^{3/2}+\frac{11}{48} d^3 x \left (d^2-e^2 x^2\right )^{5/2}-\frac{11 d^2 \left (d^2-e^2 x^2\right )^{7/2}}{56 e}-\frac{11 d (d+e x) \left (d^2-e^2 x^2\right )^{7/2}}{72 e}-\frac{(d+e x)^2 \left (d^2-e^2 x^2\right )^{7/2}}{9 e}+\frac{1}{64} \left (55 d^7\right ) \int \sqrt{d^2-e^2 x^2} \, dx\\ &=\frac{55}{128} d^7 x \sqrt{d^2-e^2 x^2}+\frac{55}{192} d^5 x \left (d^2-e^2 x^2\right )^{3/2}+\frac{11}{48} d^3 x \left (d^2-e^2 x^2\right )^{5/2}-\frac{11 d^2 \left (d^2-e^2 x^2\right )^{7/2}}{56 e}-\frac{11 d (d+e x) \left (d^2-e^2 x^2\right )^{7/2}}{72 e}-\frac{(d+e x)^2 \left (d^2-e^2 x^2\right )^{7/2}}{9 e}+\frac{1}{128} \left (55 d^9\right ) \int \frac{1}{\sqrt{d^2-e^2 x^2}} \, dx\\ &=\frac{55}{128} d^7 x \sqrt{d^2-e^2 x^2}+\frac{55}{192} d^5 x \left (d^2-e^2 x^2\right )^{3/2}+\frac{11}{48} d^3 x \left (d^2-e^2 x^2\right )^{5/2}-\frac{11 d^2 \left (d^2-e^2 x^2\right )^{7/2}}{56 e}-\frac{11 d (d+e x) \left (d^2-e^2 x^2\right )^{7/2}}{72 e}-\frac{(d+e x)^2 \left (d^2-e^2 x^2\right )^{7/2}}{9 e}+\frac{1}{128} \left (55 d^9\right ) \operatorname{Subst}\left (\int \frac{1}{1+e^2 x^2} \, dx,x,\frac{x}{\sqrt{d^2-e^2 x^2}}\right )\\ &=\frac{55}{128} d^7 x \sqrt{d^2-e^2 x^2}+\frac{55}{192} d^5 x \left (d^2-e^2 x^2\right )^{3/2}+\frac{11}{48} d^3 x \left (d^2-e^2 x^2\right )^{5/2}-\frac{11 d^2 \left (d^2-e^2 x^2\right )^{7/2}}{56 e}-\frac{11 d (d+e x) \left (d^2-e^2 x^2\right )^{7/2}}{72 e}-\frac{(d+e x)^2 \left (d^2-e^2 x^2\right )^{7/2}}{9 e}+\frac{55 d^9 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{128 e}\\ \end{align*}

Mathematica [A]  time = 0.350777, size = 156, normalized size = 0.83 \[ \frac{\sqrt{d^2-e^2 x^2} \left (\sqrt{1-\frac{e^2 x^2}{d^2}} \left (10240 d^6 e^2 x^2+3066 d^5 e^3 x^3-8448 d^4 e^4 x^4-7224 d^3 e^5 x^5+1024 d^2 e^6 x^6+4599 d^7 e x-3712 d^8+3024 d e^7 x^7+896 e^8 x^8\right )+3465 d^8 \sin ^{-1}\left (\frac{e x}{d}\right )\right )}{8064 e \sqrt{1-\frac{e^2 x^2}{d^2}}} \]

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x)^3*(d^2 - e^2*x^2)^(5/2),x]

[Out]

(Sqrt[d^2 - e^2*x^2]*(Sqrt[1 - (e^2*x^2)/d^2]*(-3712*d^8 + 4599*d^7*e*x + 10240*d^6*e^2*x^2 + 3066*d^5*e^3*x^3
- 8448*d^4*e^4*x^4 - 7224*d^3*e^5*x^5 + 1024*d^2*e^6*x^6 + 3024*d*e^7*x^7 + 896*e^8*x^8) + 3465*d^8*ArcSin[(e
*x)/d]))/(8064*e*Sqrt[1 - (e^2*x^2)/d^2])

________________________________________________________________________________________

Maple [A]  time = 0.06, size = 154, normalized size = 0.8 \begin{align*} -{\frac{e{x}^{2}}{9} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{{\frac{7}{2}}}}-{\frac{29\,{d}^{2}}{63\,e} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{{\frac{7}{2}}}}-{\frac{3\,dx}{8} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{{\frac{7}{2}}}}+{\frac{11\,{d}^{3}x}{48} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{{\frac{5}{2}}}}+{\frac{55\,{d}^{5}x}{192} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{{\frac{3}{2}}}}+{\frac{55\,{d}^{7}x}{128}\sqrt{-{x}^{2}{e}^{2}+{d}^{2}}}+{\frac{55\,{d}^{9}}{128}\arctan \left ({x\sqrt{{e}^{2}}{\frac{1}{\sqrt{-{x}^{2}{e}^{2}+{d}^{2}}}}} \right ){\frac{1}{\sqrt{{e}^{2}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^3*(-e^2*x^2+d^2)^(5/2),x)

[Out]

-1/9*e*x^2*(-e^2*x^2+d^2)^(7/2)-29/63*d^2*(-e^2*x^2+d^2)^(7/2)/e-3/8*d*x*(-e^2*x^2+d^2)^(7/2)+11/48*d^3*x*(-e^
2*x^2+d^2)^(5/2)+55/192*d^5*x*(-e^2*x^2+d^2)^(3/2)+55/128*d^7*x*(-e^2*x^2+d^2)^(1/2)+55/128*d^9/(e^2)^(1/2)*ar
ctan((e^2)^(1/2)*x/(-e^2*x^2+d^2)^(1/2))

________________________________________________________________________________________

Maxima [A]  time = 1.4854, size = 197, normalized size = 1.05 \begin{align*} \frac{55 \, d^{9} \arcsin \left (\frac{e^{2} x}{\sqrt{d^{2} e^{2}}}\right )}{128 \, \sqrt{e^{2}}} + \frac{55}{128} \, \sqrt{-e^{2} x^{2} + d^{2}} d^{7} x + \frac{55}{192} \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{3}{2}} d^{5} x + \frac{11}{48} \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}} d^{3} x - \frac{1}{9} \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{7}{2}} e x^{2} - \frac{3}{8} \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{7}{2}} d x - \frac{29 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{7}{2}} d^{2}}{63 \, e} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^3*(-e^2*x^2+d^2)^(5/2),x, algorithm="maxima")

[Out]

55/128*d^9*arcsin(e^2*x/sqrt(d^2*e^2))/sqrt(e^2) + 55/128*sqrt(-e^2*x^2 + d^2)*d^7*x + 55/192*(-e^2*x^2 + d^2)
^(3/2)*d^5*x + 11/48*(-e^2*x^2 + d^2)^(5/2)*d^3*x - 1/9*(-e^2*x^2 + d^2)^(7/2)*e*x^2 - 3/8*(-e^2*x^2 + d^2)^(7
/2)*d*x - 29/63*(-e^2*x^2 + d^2)^(7/2)*d^2/e

________________________________________________________________________________________

Fricas [A]  time = 2.01678, size = 324, normalized size = 1.72 \begin{align*} -\frac{6930 \, d^{9} \arctan \left (-\frac{d - \sqrt{-e^{2} x^{2} + d^{2}}}{e x}\right ) -{\left (896 \, e^{8} x^{8} + 3024 \, d e^{7} x^{7} + 1024 \, d^{2} e^{6} x^{6} - 7224 \, d^{3} e^{5} x^{5} - 8448 \, d^{4} e^{4} x^{4} + 3066 \, d^{5} e^{3} x^{3} + 10240 \, d^{6} e^{2} x^{2} + 4599 \, d^{7} e x - 3712 \, d^{8}\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{8064 \, e} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^3*(-e^2*x^2+d^2)^(5/2),x, algorithm="fricas")

[Out]

-1/8064*(6930*d^9*arctan(-(d - sqrt(-e^2*x^2 + d^2))/(e*x)) - (896*e^8*x^8 + 3024*d*e^7*x^7 + 1024*d^2*e^6*x^6
- 7224*d^3*e^5*x^5 - 8448*d^4*e^4*x^4 + 3066*d^5*e^3*x^3 + 10240*d^6*e^2*x^2 + 4599*d^7*e*x - 3712*d^8)*sqrt(
-e^2*x^2 + d^2))/e

________________________________________________________________________________________

Sympy [C]  time = 40.9155, size = 1290, normalized size = 6.86 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)**3*(-e**2*x**2+d**2)**(5/2),x)

[Out]

d**7*Piecewise((-I*d**2*acosh(e*x/d)/(2*e) - I*d*x/(2*sqrt(-1 + e**2*x**2/d**2)) + I*e**2*x**3/(2*d*sqrt(-1 +
e**2*x**2/d**2)), Abs(e**2*x**2)/Abs(d**2) > 1), (d**2*asin(e*x/d)/(2*e) + d*x*sqrt(1 - e**2*x**2/d**2)/2, Tru
e)) + 3*d**6*e*Piecewise((x**2*sqrt(d**2)/2, Eq(e**2, 0)), (-(d**2 - e**2*x**2)**(3/2)/(3*e**2), True)) + d**5
*e**2*Piecewise((-I*d**4*acosh(e*x/d)/(8*e**3) + I*d**3*x/(8*e**2*sqrt(-1 + e**2*x**2/d**2)) - 3*I*d*x**3/(8*s
qrt(-1 + e**2*x**2/d**2)) + I*e**2*x**5/(4*d*sqrt(-1 + e**2*x**2/d**2)), Abs(e**2*x**2)/Abs(d**2) > 1), (d**4*
asin(e*x/d)/(8*e**3) - d**3*x/(8*e**2*sqrt(1 - e**2*x**2/d**2)) + 3*d*x**3/(8*sqrt(1 - e**2*x**2/d**2)) - e**2
*x**5/(4*d*sqrt(1 - e**2*x**2/d**2)), True)) - 5*d**4*e**3*Piecewise((-2*d**4*sqrt(d**2 - e**2*x**2)/(15*e**4)
- d**2*x**2*sqrt(d**2 - e**2*x**2)/(15*e**2) + x**4*sqrt(d**2 - e**2*x**2)/5, Ne(e, 0)), (x**4*sqrt(d**2)/4,
True)) - 5*d**3*e**4*Piecewise((-I*d**6*acosh(e*x/d)/(16*e**5) + I*d**5*x/(16*e**4*sqrt(-1 + e**2*x**2/d**2))
- I*d**3*x**3/(48*e**2*sqrt(-1 + e**2*x**2/d**2)) - 5*I*d*x**5/(24*sqrt(-1 + e**2*x**2/d**2)) + I*e**2*x**7/(6
*d*sqrt(-1 + e**2*x**2/d**2)), Abs(e**2*x**2)/Abs(d**2) > 1), (d**6*asin(e*x/d)/(16*e**5) - d**5*x/(16*e**4*sq
rt(1 - e**2*x**2/d**2)) + d**3*x**3/(48*e**2*sqrt(1 - e**2*x**2/d**2)) + 5*d*x**5/(24*sqrt(1 - e**2*x**2/d**2)
) - e**2*x**7/(6*d*sqrt(1 - e**2*x**2/d**2)), True)) + d**2*e**5*Piecewise((-8*d**6*sqrt(d**2 - e**2*x**2)/(10
5*e**6) - 4*d**4*x**2*sqrt(d**2 - e**2*x**2)/(105*e**4) - d**2*x**4*sqrt(d**2 - e**2*x**2)/(35*e**2) + x**6*sq
rt(d**2 - e**2*x**2)/7, Ne(e, 0)), (x**6*sqrt(d**2)/6, True)) + 3*d*e**6*Piecewise((-5*I*d**8*acosh(e*x/d)/(12
8*e**7) + 5*I*d**7*x/(128*e**6*sqrt(-1 + e**2*x**2/d**2)) - 5*I*d**5*x**3/(384*e**4*sqrt(-1 + e**2*x**2/d**2))
- I*d**3*x**5/(192*e**2*sqrt(-1 + e**2*x**2/d**2)) - 7*I*d*x**7/(48*sqrt(-1 + e**2*x**2/d**2)) + I*e**2*x**9/
(8*d*sqrt(-1 + e**2*x**2/d**2)), Abs(e**2*x**2)/Abs(d**2) > 1), (5*d**8*asin(e*x/d)/(128*e**7) - 5*d**7*x/(128
*e**6*sqrt(1 - e**2*x**2/d**2)) + 5*d**5*x**3/(384*e**4*sqrt(1 - e**2*x**2/d**2)) + d**3*x**5/(192*e**2*sqrt(1
- e**2*x**2/d**2)) + 7*d*x**7/(48*sqrt(1 - e**2*x**2/d**2)) - e**2*x**9/(8*d*sqrt(1 - e**2*x**2/d**2)), True)
) + e**7*Piecewise((-16*d**8*sqrt(d**2 - e**2*x**2)/(315*e**8) - 8*d**6*x**2*sqrt(d**2 - e**2*x**2)/(315*e**6)
- 2*d**4*x**4*sqrt(d**2 - e**2*x**2)/(105*e**4) - d**2*x**6*sqrt(d**2 - e**2*x**2)/(63*e**2) + x**8*sqrt(d**2
- e**2*x**2)/9, Ne(e, 0)), (x**8*sqrt(d**2)/8, True))

________________________________________________________________________________________

Giac [A]  time = 1.17023, size = 158, normalized size = 0.84 \begin{align*} \frac{55}{128} \, d^{9} \arcsin \left (\frac{x e}{d}\right ) e^{\left (-1\right )} \mathrm{sgn}\left (d\right ) - \frac{1}{8064} \,{\left (3712 \, d^{8} e^{\left (-1\right )} -{\left (4599 \, d^{7} + 2 \,{\left (5120 \, d^{6} e +{\left (1533 \, d^{5} e^{2} - 4 \,{\left (1056 \, d^{4} e^{3} +{\left (903 \, d^{3} e^{4} - 2 \,{\left (64 \, d^{2} e^{5} + 7 \,{\left (8 \, x e^{7} + 27 \, d e^{6}\right )} x\right )} x\right )} x\right )} x\right )} x\right )} x\right )} x\right )} \sqrt{-x^{2} e^{2} + d^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^3*(-e^2*x^2+d^2)^(5/2),x, algorithm="giac")

[Out]

55/128*d^9*arcsin(x*e/d)*e^(-1)*sgn(d) - 1/8064*(3712*d^8*e^(-1) - (4599*d^7 + 2*(5120*d^6*e + (1533*d^5*e^2 -
4*(1056*d^4*e^3 + (903*d^3*e^4 - 2*(64*d^2*e^5 + 7*(8*x*e^7 + 27*d*e^6)*x)*x)*x)*x)*x)*x)*x)*sqrt(-x^2*e^2 +
d^2)