### 3.802 $$\int (d+e x) (d^2-e^2 x^2)^{7/2} \, dx$$

Optimal. Leaf size=148 $\frac{35}{128} d^7 x \sqrt{d^2-e^2 x^2}+\frac{35}{192} d^5 x \left (d^2-e^2 x^2\right )^{3/2}+\frac{7}{48} d^3 x \left (d^2-e^2 x^2\right )^{5/2}+\frac{1}{8} d x \left (d^2-e^2 x^2\right )^{7/2}-\frac{\left (d^2-e^2 x^2\right )^{9/2}}{9 e}+\frac{35 d^9 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{128 e}$

[Out]

(35*d^7*x*Sqrt[d^2 - e^2*x^2])/128 + (35*d^5*x*(d^2 - e^2*x^2)^(3/2))/192 + (7*d^3*x*(d^2 - e^2*x^2)^(5/2))/48
+ (d*x*(d^2 - e^2*x^2)^(7/2))/8 - (d^2 - e^2*x^2)^(9/2)/(9*e) + (35*d^9*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]])/(1
28*e)

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Rubi [A]  time = 0.0466568, antiderivative size = 148, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 4, integrand size = 22, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.182, Rules used = {641, 195, 217, 203} $\frac{35}{128} d^7 x \sqrt{d^2-e^2 x^2}+\frac{35}{192} d^5 x \left (d^2-e^2 x^2\right )^{3/2}+\frac{7}{48} d^3 x \left (d^2-e^2 x^2\right )^{5/2}+\frac{1}{8} d x \left (d^2-e^2 x^2\right )^{7/2}-\frac{\left (d^2-e^2 x^2\right )^{9/2}}{9 e}+\frac{35 d^9 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{128 e}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(35*d^7*x*Sqrt[d^2 - e^2*x^2])/128 + (35*d^5*x*(d^2 - e^2*x^2)^(3/2))/192 + (7*d^3*x*(d^2 - e^2*x^2)^(5/2))/48
+ (d*x*(d^2 - e^2*x^2)^(7/2))/8 - (d^2 - e^2*x^2)^(9/2)/(9*e) + (35*d^9*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]])/(1
28*e)

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) \left (d^2-e^2 x^2\right )^{7/2} \, dx &=-\frac{\left (d^2-e^2 x^2\right )^{9/2}}{9 e}+d \int \left (d^2-e^2 x^2\right )^{7/2} \, dx\\ &=\frac{1}{8} d x \left (d^2-e^2 x^2\right )^{7/2}-\frac{\left (d^2-e^2 x^2\right )^{9/2}}{9 e}+\frac{1}{8} \left (7 d^3\right ) \int \left (d^2-e^2 x^2\right )^{5/2} \, dx\\ &=\frac{7}{48} d^3 x \left (d^2-e^2 x^2\right )^{5/2}+\frac{1}{8} d x \left (d^2-e^2 x^2\right )^{7/2}-\frac{\left (d^2-e^2 x^2\right )^{9/2}}{9 e}+\frac{1}{48} \left (35 d^5\right ) \int \left (d^2-e^2 x^2\right )^{3/2} \, dx\\ &=\frac{35}{192} d^5 x \left (d^2-e^2 x^2\right )^{3/2}+\frac{7}{48} d^3 x \left (d^2-e^2 x^2\right )^{5/2}+\frac{1}{8} d x \left (d^2-e^2 x^2\right )^{7/2}-\frac{\left (d^2-e^2 x^2\right )^{9/2}}{9 e}+\frac{1}{64} \left (35 d^7\right ) \int \sqrt{d^2-e^2 x^2} \, dx\\ &=\frac{35}{128} d^7 x \sqrt{d^2-e^2 x^2}+\frac{35}{192} d^5 x \left (d^2-e^2 x^2\right )^{3/2}+\frac{7}{48} d^3 x \left (d^2-e^2 x^2\right )^{5/2}+\frac{1}{8} d x \left (d^2-e^2 x^2\right )^{7/2}-\frac{\left (d^2-e^2 x^2\right )^{9/2}}{9 e}+\frac{1}{128} \left (35 d^9\right ) \int \frac{1}{\sqrt{d^2-e^2 x^2}} \, dx\\ &=\frac{35}{128} d^7 x \sqrt{d^2-e^2 x^2}+\frac{35}{192} d^5 x \left (d^2-e^2 x^2\right )^{3/2}+\frac{7}{48} d^3 x \left (d^2-e^2 x^2\right )^{5/2}+\frac{1}{8} d x \left (d^2-e^2 x^2\right )^{7/2}-\frac{\left (d^2-e^2 x^2\right )^{9/2}}{9 e}+\frac{1}{128} \left (35 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{35}{128} d^7 x \sqrt{d^2-e^2 x^2}+\frac{35}{192} d^5 x \left (d^2-e^2 x^2\right )^{3/2}+\frac{7}{48} d^3 x \left (d^2-e^2 x^2\right )^{5/2}+\frac{1}{8} d x \left (d^2-e^2 x^2\right )^{7/2}-\frac{\left (d^2-e^2 x^2\right )^{9/2}}{9 e}+\frac{35 d^9 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{128 e}\\ \end{align*}

Mathematica [A]  time = 0.263427, size = 114, normalized size = 0.77 $\frac{1}{384} d \sqrt{d^2-e^2 x^2} \left (-326 d^4 e^2 x^3+200 d^2 e^4 x^5+\frac{105 d^7 \sin ^{-1}\left (\frac{e x}{d}\right )}{e \sqrt{1-\frac{e^2 x^2}{d^2}}}+279 d^6 x-48 e^6 x^7\right )-\frac{\left (d^2-e^2 x^2\right )^{9/2}}{9 e}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

-(d^2 - e^2*x^2)^(9/2)/(9*e) + (d*Sqrt[d^2 - e^2*x^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])))/384

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Maple [A]  time = 0.046, size = 131, normalized size = 0.9 \begin{align*} -{\frac{1}{9\,e} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{9}{2}}}}+{\frac{dx}{8} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{7}{2}}}}+{\frac{7\,{d}^{3}x}{48} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{5}{2}}}}+{\frac{35\,{d}^{5}x}{192} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{3}{2}}}}+{\frac{35\,{d}^{7}x}{128}\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}+{\frac{35\,{d}^{9}}{128}\arctan \left ({x\sqrt{{e}^{2}}{\frac{1}{\sqrt{-{e}^{2}{x}^{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)*(-e^2*x^2+d^2)^(7/2),x)

[Out]

-1/9*(-e^2*x^2+d^2)^(9/2)/e+1/8*d*x*(-e^2*x^2+d^2)^(7/2)+7/48*d^3*x*(-e^2*x^2+d^2)^(5/2)+35/192*d^5*x*(-e^2*x^
2+d^2)^(3/2)+35/128*d^7*x*(-e^2*x^2+d^2)^(1/2)+35/128*d^9/(e^2)^(1/2)*arctan((e^2)^(1/2)*x/(-e^2*x^2+d^2)^(1/2
))

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Maxima [A]  time = 1.52541, size = 166, normalized size = 1.12 \begin{align*} \frac{35 \, d^{9} \arcsin \left (\frac{e^{2} x}{\sqrt{d^{2} e^{2}}}\right )}{128 \, \sqrt{e^{2}}} + \frac{35}{128} \, \sqrt{-e^{2} x^{2} + d^{2}} d^{7} x + \frac{35}{192} \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{3}{2}} d^{5} x + \frac{7}{48} \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}} d^{3} x + \frac{1}{8} \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{7}{2}} d x - \frac{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{9}{2}}}{9 \, e} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

35/128*d^9*arcsin(e^2*x/sqrt(d^2*e^2))/sqrt(e^2) + 35/128*sqrt(-e^2*x^2 + d^2)*d^7*x + 35/192*(-e^2*x^2 + d^2)
^(3/2)*d^5*x + 7/48*(-e^2*x^2 + d^2)^(5/2)*d^3*x + 1/8*(-e^2*x^2 + d^2)^(7/2)*d*x - 1/9*(-e^2*x^2 + d^2)^(9/2)
/e

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Fricas [A]  time = 2.18489, size = 311, normalized size = 2.1 \begin{align*} -\frac{630 \, d^{9} \arctan \left (-\frac{d - \sqrt{-e^{2} x^{2} + d^{2}}}{e x}\right ) +{\left (128 \, e^{8} x^{8} + 144 \, d e^{7} x^{7} - 512 \, d^{2} e^{6} x^{6} - 600 \, d^{3} e^{5} x^{5} + 768 \, d^{4} e^{4} x^{4} + 978 \, d^{5} e^{3} x^{3} - 512 \, d^{6} e^{2} x^{2} - 837 \, d^{7} e x + 128 \, d^{8}\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{1152 \, e} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/1152*(630*d^9*arctan(-(d - sqrt(-e^2*x^2 + d^2))/(e*x)) + (128*e^8*x^8 + 144*d*e^7*x^7 - 512*d^2*e^6*x^6 -
600*d^3*e^5*x^5 + 768*d^4*e^4*x^4 + 978*d^5*e^3*x^3 - 512*d^6*e^2*x^2 - 837*d^7*e*x + 128*d^8)*sqrt(-e^2*x^2 +
d^2))/e

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Sympy [C]  time = 24.1894, size = 1290, normalized size = 8.72 \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)*(-e**2*x**2+d**2)**(7/2),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)) + 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)) - 3*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)) - 3*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)) + 3*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)) + 3*d**2*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*sqrt(d**2)/6, True)) - 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))

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Giac [A]  time = 1.22357, size = 161, normalized size = 1.09 \begin{align*} \frac{35}{128} \, d^{9} \arcsin \left (\frac{x e}{d}\right ) e^{\left (-1\right )} \mathrm{sgn}\left (d\right ) - \frac{1}{1152} \,{\left (128 \, d^{8} e^{\left (-1\right )} -{\left (837 \, d^{7} + 2 \,{\left (256 \, d^{6} e -{\left (489 \, d^{5} e^{2} + 4 \,{\left (96 \, d^{4} e^{3} -{\left (75 \, d^{3} e^{4} + 2 \,{\left (32 \, d^{2} e^{5} -{\left (8 \, x e^{7} + 9 \, 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)*(-e^2*x^2+d^2)^(7/2),x, algorithm="giac")

[Out]

35/128*d^9*arcsin(x*e/d)*e^(-1)*sgn(d) - 1/1152*(128*d^8*e^(-1) - (837*d^7 + 2*(256*d^6*e - (489*d^5*e^2 + 4*(
96*d^4*e^3 - (75*d^3*e^4 + 2*(32*d^2*e^5 - (8*x*e^7 + 9*d*e^6)*x)*x)*x)*x)*x)*x)*x)*sqrt(-x^2*e^2 + d^2)