### 3.801 $$\int (d+e x)^2 (d^2-e^2 x^2)^{7/2} \, dx$$

Optimal. Leaf size=179 $\frac{77}{256} d^8 x \sqrt{d^2-e^2 x^2}+\frac{77}{384} d^6 x \left (d^2-e^2 x^2\right )^{3/2}+\frac{77}{480} d^4 x \left (d^2-e^2 x^2\right )^{5/2}+\frac{11}{80} d^2 x \left (d^2-e^2 x^2\right )^{7/2}-\frac{11 d \left (d^2-e^2 x^2\right )^{9/2}}{90 e}-\frac{(d+e x) \left (d^2-e^2 x^2\right )^{9/2}}{10 e}+\frac{77 d^{10} \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{256 e}$

[Out]

(77*d^8*x*Sqrt[d^2 - e^2*x^2])/256 + (77*d^6*x*(d^2 - e^2*x^2)^(3/2))/384 + (77*d^4*x*(d^2 - e^2*x^2)^(5/2))/4
80 + (11*d^2*x*(d^2 - e^2*x^2)^(7/2))/80 - (11*d*(d^2 - e^2*x^2)^(9/2))/(90*e) - ((d + e*x)*(d^2 - e^2*x^2)^(9
/2))/(10*e) + (77*d^10*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]])/(256*e)

________________________________________________________________________________________

Rubi [A]  time = 0.0669763, antiderivative size = 179, 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} $\frac{77}{256} d^8 x \sqrt{d^2-e^2 x^2}+\frac{77}{384} d^6 x \left (d^2-e^2 x^2\right )^{3/2}+\frac{77}{480} d^4 x \left (d^2-e^2 x^2\right )^{5/2}+\frac{11}{80} d^2 x \left (d^2-e^2 x^2\right )^{7/2}-\frac{11 d \left (d^2-e^2 x^2\right )^{9/2}}{90 e}-\frac{(d+e x) \left (d^2-e^2 x^2\right )^{9/2}}{10 e}+\frac{77 d^{10} \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{256 e}$

Antiderivative was successfully veriﬁed.

[In]

Int[(d + e*x)^2*(d^2 - e^2*x^2)^(7/2),x]

[Out]

(77*d^8*x*Sqrt[d^2 - e^2*x^2])/256 + (77*d^6*x*(d^2 - e^2*x^2)^(3/2))/384 + (77*d^4*x*(d^2 - e^2*x^2)^(5/2))/4
80 + (11*d^2*x*(d^2 - e^2*x^2)^(7/2))/80 - (11*d*(d^2 - e^2*x^2)^(9/2))/(90*e) - ((d + e*x)*(d^2 - e^2*x^2)^(9
/2))/(10*e) + (77*d^10*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]])/(256*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)^2 \left (d^2-e^2 x^2\right )^{7/2} \, dx &=-\frac{(d+e x) \left (d^2-e^2 x^2\right )^{9/2}}{10 e}+\frac{1}{10} (11 d) \int (d+e x) \left (d^2-e^2 x^2\right )^{7/2} \, dx\\ &=-\frac{11 d \left (d^2-e^2 x^2\right )^{9/2}}{90 e}-\frac{(d+e x) \left (d^2-e^2 x^2\right )^{9/2}}{10 e}+\frac{1}{10} \left (11 d^2\right ) \int \left (d^2-e^2 x^2\right )^{7/2} \, dx\\ &=\frac{11}{80} d^2 x \left (d^2-e^2 x^2\right )^{7/2}-\frac{11 d \left (d^2-e^2 x^2\right )^{9/2}}{90 e}-\frac{(d+e x) \left (d^2-e^2 x^2\right )^{9/2}}{10 e}+\frac{1}{80} \left (77 d^4\right ) \int \left (d^2-e^2 x^2\right )^{5/2} \, dx\\ &=\frac{77}{480} d^4 x \left (d^2-e^2 x^2\right )^{5/2}+\frac{11}{80} d^2 x \left (d^2-e^2 x^2\right )^{7/2}-\frac{11 d \left (d^2-e^2 x^2\right )^{9/2}}{90 e}-\frac{(d+e x) \left (d^2-e^2 x^2\right )^{9/2}}{10 e}+\frac{1}{96} \left (77 d^6\right ) \int \left (d^2-e^2 x^2\right )^{3/2} \, dx\\ &=\frac{77}{384} d^6 x \left (d^2-e^2 x^2\right )^{3/2}+\frac{77}{480} d^4 x \left (d^2-e^2 x^2\right )^{5/2}+\frac{11}{80} d^2 x \left (d^2-e^2 x^2\right )^{7/2}-\frac{11 d \left (d^2-e^2 x^2\right )^{9/2}}{90 e}-\frac{(d+e x) \left (d^2-e^2 x^2\right )^{9/2}}{10 e}+\frac{1}{128} \left (77 d^8\right ) \int \sqrt{d^2-e^2 x^2} \, dx\\ &=\frac{77}{256} d^8 x \sqrt{d^2-e^2 x^2}+\frac{77}{384} d^6 x \left (d^2-e^2 x^2\right )^{3/2}+\frac{77}{480} d^4 x \left (d^2-e^2 x^2\right )^{5/2}+\frac{11}{80} d^2 x \left (d^2-e^2 x^2\right )^{7/2}-\frac{11 d \left (d^2-e^2 x^2\right )^{9/2}}{90 e}-\frac{(d+e x) \left (d^2-e^2 x^2\right )^{9/2}}{10 e}+\frac{1}{256} \left (77 d^{10}\right ) \int \frac{1}{\sqrt{d^2-e^2 x^2}} \, dx\\ &=\frac{77}{256} d^8 x \sqrt{d^2-e^2 x^2}+\frac{77}{384} d^6 x \left (d^2-e^2 x^2\right )^{3/2}+\frac{77}{480} d^4 x \left (d^2-e^2 x^2\right )^{5/2}+\frac{11}{80} d^2 x \left (d^2-e^2 x^2\right )^{7/2}-\frac{11 d \left (d^2-e^2 x^2\right )^{9/2}}{90 e}-\frac{(d+e x) \left (d^2-e^2 x^2\right )^{9/2}}{10 e}+\frac{1}{256} \left (77 d^{10}\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{77}{256} d^8 x \sqrt{d^2-e^2 x^2}+\frac{77}{384} d^6 x \left (d^2-e^2 x^2\right )^{3/2}+\frac{77}{480} d^4 x \left (d^2-e^2 x^2\right )^{5/2}+\frac{11}{80} d^2 x \left (d^2-e^2 x^2\right )^{7/2}-\frac{11 d \left (d^2-e^2 x^2\right )^{9/2}}{90 e}-\frac{(d+e x) \left (d^2-e^2 x^2\right )^{9/2}}{10 e}+\frac{77 d^{10} \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{256 e}\\ \end{align*}

Mathematica [A]  time = 0.355226, size = 118, normalized size = 0.66 $\frac{\left (d^2-e^2 x^2\right )^{9/2} \left (\frac{33 d^2 \left (-326 d^4 e^2 x^3+200 d^2 e^4 x^5+\frac{105 d^7 \sin ^{-1}\left (\frac{e x}{d}\right )}{e \sqrt{1-\frac{e^2 x^2}{d^2}}}+279 d^6 x-48 e^6 x^7\right )}{\left (d^2-e^2 x^2\right )^4}-\frac{2560 d}{e}-1152 x\right )}{11520}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(d + e*x)^2*(d^2 - e^2*x^2)^(7/2),x]

[Out]

((d^2 - e^2*x^2)^(9/2)*((-2560*d)/e - 1152*x + (33*d^2*(279*d^6*x - 326*d^4*e^2*x^3 + 200*d^2*e^4*x^5 - 48*e^6
*x^7 + (105*d^7*ArcSin[(e*x)/d])/(e*Sqrt[1 - (e^2*x^2)/d^2])))/(d^2 - e^2*x^2)^4))/11520

________________________________________________________________________________________

Maple [A]  time = 0.05, size = 151, normalized size = 0.8 \begin{align*} -{\frac{x}{10} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{9}{2}}}}+{\frac{11\,{d}^{2}x}{80} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{7}{2}}}}+{\frac{77\,{d}^{4}x}{480} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{5}{2}}}}+{\frac{77\,{d}^{6}x}{384} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{3}{2}}}}+{\frac{77\,{d}^{8}x}{256}\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}+{\frac{77\,{d}^{10}}{256}\arctan \left ({x\sqrt{{e}^{2}}{\frac{1}{\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}}} \right ){\frac{1}{\sqrt{{e}^{2}}}}}-{\frac{2\,d}{9\,e} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{9}{2}}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^2*(-e^2*x^2+d^2)^(7/2),x)

[Out]

-1/10*x*(-e^2*x^2+d^2)^(9/2)+11/80*d^2*x*(-e^2*x^2+d^2)^(7/2)+77/480*d^4*x*(-e^2*x^2+d^2)^(5/2)+77/384*d^6*x*(
-e^2*x^2+d^2)^(3/2)+77/256*d^8*x*(-e^2*x^2+d^2)^(1/2)+77/256*d^10/(e^2)^(1/2)*arctan((e^2)^(1/2)*x/(-e^2*x^2+d
^2)^(1/2))-2/9*d*(-e^2*x^2+d^2)^(9/2)/e

________________________________________________________________________________________

Maxima [A]  time = 1.71957, size = 193, normalized size = 1.08 \begin{align*} \frac{77 \, d^{10} \arcsin \left (\frac{e^{2} x}{\sqrt{d^{2} e^{2}}}\right )}{256 \, \sqrt{e^{2}}} + \frac{77}{256} \, \sqrt{-e^{2} x^{2} + d^{2}} d^{8} x + \frac{77}{384} \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{3}{2}} d^{6} x + \frac{77}{480} \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}} d^{4} x + \frac{11}{80} \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{7}{2}} d^{2} x - \frac{1}{10} \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{9}{2}} x - \frac{2 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{9}{2}} d}{9 \, e} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^2*(-e^2*x^2+d^2)^(7/2),x, algorithm="maxima")

[Out]

77/256*d^10*arcsin(e^2*x/sqrt(d^2*e^2))/sqrt(e^2) + 77/256*sqrt(-e^2*x^2 + d^2)*d^8*x + 77/384*(-e^2*x^2 + d^2
)^(3/2)*d^6*x + 77/480*(-e^2*x^2 + d^2)^(5/2)*d^4*x + 11/80*(-e^2*x^2 + d^2)^(7/2)*d^2*x - 1/10*(-e^2*x^2 + d^
2)^(9/2)*x - 2/9*(-e^2*x^2 + d^2)^(9/2)*d/e

________________________________________________________________________________________

Fricas [A]  time = 2.1594, size = 355, normalized size = 1.98 \begin{align*} -\frac{6930 \, d^{10} \arctan \left (-\frac{d - \sqrt{-e^{2} x^{2} + d^{2}}}{e x}\right ) +{\left (1152 \, e^{9} x^{9} + 2560 \, d e^{8} x^{8} - 3024 \, d^{2} e^{7} x^{7} - 10240 \, d^{3} e^{6} x^{6} + 312 \, d^{4} e^{5} x^{5} + 15360 \, d^{5} e^{4} x^{4} + 6150 \, d^{6} e^{3} x^{3} - 10240 \, d^{7} e^{2} x^{2} - 8055 \, d^{8} e x + 2560 \, d^{9}\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{11520 \, e} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^2*(-e^2*x^2+d^2)^(7/2),x, algorithm="fricas")

[Out]

-1/11520*(6930*d^10*arctan(-(d - sqrt(-e^2*x^2 + d^2))/(e*x)) + (1152*e^9*x^9 + 2560*d*e^8*x^8 - 3024*d^2*e^7*
x^7 - 10240*d^3*e^6*x^6 + 312*d^4*e^5*x^5 + 15360*d^5*e^4*x^4 + 6150*d^6*e^3*x^3 - 10240*d^7*e^2*x^2 - 8055*d^
8*e*x + 2560*d^9)*sqrt(-e^2*x^2 + d^2))/e

________________________________________________________________________________________

Sympy [C]  time = 31.3627, size = 1420, normalized size = 7.93 \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)**2*(-e**2*x**2+d**2)**(7/2),x)

[Out]

d**8*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)) + 2*d**7*e*Piecewise((x**2*sqrt(d**2)/2, Eq(e**2, 0)), (-(d**2 - e**2*x**2)**(3/2)/(3*e**2), True)) - 2*d*
*6*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
*sqrt(-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)) - 6*d**5*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)) + 6*d**3*e**5*Piecewise((-8*d**6*sqrt(d**2 - e**2*x**2)/(105*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*sqrt(d**2 - e**2*x**2)/7, Ne(e, 0)), (x**6*s
qrt(d**2)/6, True)) + 2*d**2*e**6*Piecewise((-5*I*d**8*acosh(e*x/d)/(128*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)) - 2*d*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*s
qrt(d**2)/8, True)) - e**8*Piecewise((-7*I*d**10*acosh(e*x/d)/(256*e**9) + 7*I*d**9*x/(256*e**8*sqrt(-1 + e**2
*x**2/d**2)) - 7*I*d**7*x**3/(768*e**6*sqrt(-1 + e**2*x**2/d**2)) - 7*I*d**5*x**5/(1920*e**4*sqrt(-1 + e**2*x*
*2/d**2)) - I*d**3*x**7/(480*e**2*sqrt(-1 + e**2*x**2/d**2)) - 9*I*d*x**9/(80*sqrt(-1 + e**2*x**2/d**2)) + I*e
**2*x**11/(10*d*sqrt(-1 + e**2*x**2/d**2)), Abs(e**2*x**2)/Abs(d**2) > 1), (7*d**10*asin(e*x/d)/(256*e**9) - 7
*d**9*x/(256*e**8*sqrt(1 - e**2*x**2/d**2)) + 7*d**7*x**3/(768*e**6*sqrt(1 - e**2*x**2/d**2)) + 7*d**5*x**5/(1
920*e**4*sqrt(1 - e**2*x**2/d**2)) + d**3*x**7/(480*e**2*sqrt(1 - e**2*x**2/d**2)) + 9*d*x**9/(80*sqrt(1 - e**
2*x**2/d**2)) - e**2*x**11/(10*d*sqrt(1 - e**2*x**2/d**2)), True))

________________________________________________________________________________________

Giac [A]  time = 1.26565, size = 173, normalized size = 0.97 \begin{align*} \frac{77}{256} \, d^{10} \arcsin \left (\frac{x e}{d}\right ) e^{\left (-1\right )} \mathrm{sgn}\left (d\right ) - \frac{1}{11520} \,{\left (2560 \, d^{9} e^{\left (-1\right )} -{\left (8055 \, d^{8} + 2 \,{\left (5120 \, d^{7} e -{\left (3075 \, d^{6} e^{2} + 4 \,{\left (1920 \, d^{5} e^{3} +{\left (39 \, d^{4} e^{4} - 2 \,{\left (640 \, d^{3} e^{5} +{\left (189 \, d^{2} e^{6} - 8 \,{\left (9 \, x e^{8} + 20 \, d e^{7}\right )} x\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)^2*(-e^2*x^2+d^2)^(7/2),x, algorithm="giac")

[Out]

77/256*d^10*arcsin(x*e/d)*e^(-1)*sgn(d) - 1/11520*(2560*d^9*e^(-1) - (8055*d^8 + 2*(5120*d^7*e - (3075*d^6*e^2
+ 4*(1920*d^5*e^3 + (39*d^4*e^4 - 2*(640*d^3*e^5 + (189*d^2*e^6 - 8*(9*x*e^8 + 20*d*e^7)*x)*x)*x)*x)*x)*x)*x)
*x)*sqrt(-x^2*e^2 + d^2)