### 3.854 $$\int \frac{d+e x}{(d^2-e^2 x^2)^{7/2}} \, dx$$

Optimal. Leaf size=80 $\frac{8 x}{15 d^5 \sqrt{d^2-e^2 x^2}}+\frac{4 x}{15 d^3 \left (d^2-e^2 x^2\right )^{3/2}}+\frac{d+e x}{5 d e \left (d^2-e^2 x^2\right )^{5/2}}$

[Out]

(d + e*x)/(5*d*e*(d^2 - e^2*x^2)^(5/2)) + (4*x)/(15*d^3*(d^2 - e^2*x^2)^(3/2)) + (8*x)/(15*d^5*Sqrt[d^2 - e^2*
x^2])

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Rubi [A]  time = 0.0193956, antiderivative size = 80, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 22, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.136, Rules used = {639, 192, 191} $\frac{8 x}{15 d^5 \sqrt{d^2-e^2 x^2}}+\frac{4 x}{15 d^3 \left (d^2-e^2 x^2\right )^{3/2}}+\frac{d+e x}{5 d e \left (d^2-e^2 x^2\right )^{5/2}}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(d + e*x)/(5*d*e*(d^2 - e^2*x^2)^(5/2)) + (4*x)/(15*d^3*(d^2 - e^2*x^2)^(3/2)) + (8*x)/(15*d^5*Sqrt[d^2 - e^2*
x^2])

Rule 639

Int[((d_) + (e_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((a*e - c*d*x)*(a + c*x^2)^(p + 1))/(2*a
*c*(p + 1)), x] + Dist[(d*(2*p + 3))/(2*a*(p + 1)), Int[(a + c*x^2)^(p + 1), x], x] /; FreeQ[{a, c, d, e}, x]
&& LtQ[p, -1] && NeQ[p, -3/2]

Rule 192

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> -Simp[(x*(a + b*x^n)^(p + 1))/(a*n*(p + 1)), x] + Dist[(n*(p +
1) + 1)/(a*n*(p + 1)), Int[(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b, n, p}, x] && ILtQ[Simplify[1/n + p + 1
], 0] && NeQ[p, -1]

Rule 191

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(x*(a + b*x^n)^(p + 1))/a, x] /; FreeQ[{a, b, n, p}, x] &
& EqQ[1/n + p + 1, 0]

Rubi steps

\begin{align*} \int \frac{d+e x}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx &=\frac{d+e x}{5 d e \left (d^2-e^2 x^2\right )^{5/2}}+\frac{4 \int \frac{1}{\left (d^2-e^2 x^2\right )^{5/2}} \, dx}{5 d}\\ &=\frac{d+e x}{5 d e \left (d^2-e^2 x^2\right )^{5/2}}+\frac{4 x}{15 d^3 \left (d^2-e^2 x^2\right )^{3/2}}+\frac{8 \int \frac{1}{\left (d^2-e^2 x^2\right )^{3/2}} \, dx}{15 d^3}\\ &=\frac{d+e x}{5 d e \left (d^2-e^2 x^2\right )^{5/2}}+\frac{4 x}{15 d^3 \left (d^2-e^2 x^2\right )^{3/2}}+\frac{8 x}{15 d^5 \sqrt{d^2-e^2 x^2}}\\ \end{align*}

Mathematica [A]  time = 0.0317078, size = 82, normalized size = 1.02 $\frac{-12 d^2 e^2 x^2+12 d^3 e x+3 d^4-8 d e^3 x^3+8 e^4 x^4}{15 d^5 e (d-e x)^2 (d+e x) \sqrt{d^2-e^2 x^2}}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(3*d^4 + 12*d^3*e*x - 12*d^2*e^2*x^2 - 8*d*e^3*x^3 + 8*e^4*x^4)/(15*d^5*e*(d - e*x)^2*(d + e*x)*Sqrt[d^2 - e^2
*x^2])

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Maple [A]  time = 0.045, size = 77, normalized size = 1. \begin{align*}{\frac{ \left ( ex+d \right ) ^{2} \left ( -ex+d \right ) \left ( 8\,{e}^{4}{x}^{4}-8\,{e}^{3}{x}^{3}d-12\,{e}^{2}{x}^{2}{d}^{2}+12\,x{d}^{3}e+3\,{d}^{4} \right ) }{15\,{d}^{5}e} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{-{\frac{7}{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]

1/15*(e*x+d)^2*(-e*x+d)*(8*e^4*x^4-8*d*e^3*x^3-12*d^2*e^2*x^2+12*d^3*e*x+3*d^4)/d^5/e/(-e^2*x^2+d^2)^(7/2)

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Maxima [A]  time = 1.27569, size = 108, normalized size = 1.35 \begin{align*} \frac{x}{5 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}} d} + \frac{1}{5 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}} e} + \frac{4 \, x}{15 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{3}{2}} d^{3}} + \frac{8 \, x}{15 \, \sqrt{-e^{2} x^{2} + d^{2}} d^{5}} \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]

1/5*x/((-e^2*x^2 + d^2)^(5/2)*d) + 1/5/((-e^2*x^2 + d^2)^(5/2)*e) + 4/15*x/((-e^2*x^2 + d^2)^(3/2)*d^3) + 8/15
*x/(sqrt(-e^2*x^2 + d^2)*d^5)

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Fricas [B]  time = 2.28307, size = 340, normalized size = 4.25 \begin{align*} \frac{3 \, e^{5} x^{5} - 3 \, d e^{4} x^{4} - 6 \, d^{2} e^{3} x^{3} + 6 \, d^{3} e^{2} x^{2} + 3 \, d^{4} e x - 3 \, d^{5} -{\left (8 \, e^{4} x^{4} - 8 \, d e^{3} x^{3} - 12 \, d^{2} e^{2} x^{2} + 12 \, d^{3} e x + 3 \, d^{4}\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{15 \,{\left (d^{5} e^{6} x^{5} - d^{6} e^{5} x^{4} - 2 \, d^{7} e^{4} x^{3} + 2 \, d^{8} e^{3} x^{2} + d^{9} e^{2} x - d^{10} e\right )}} \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]

1/15*(3*e^5*x^5 - 3*d*e^4*x^4 - 6*d^2*e^3*x^3 + 6*d^3*e^2*x^2 + 3*d^4*e*x - 3*d^5 - (8*e^4*x^4 - 8*d*e^3*x^3 -
12*d^2*e^2*x^2 + 12*d^3*e*x + 3*d^4)*sqrt(-e^2*x^2 + d^2))/(d^5*e^6*x^5 - d^6*e^5*x^4 - 2*d^7*e^4*x^3 + 2*d^8
*e^3*x^2 + d^9*e^2*x - d^10*e)

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Sympy [C]  time = 10.657, size = 605, normalized size = 7.56 \begin{align*} d \left (\begin{cases} - \frac{15 i d^{4} x}{15 d^{11} \sqrt{-1 + \frac{e^{2} x^{2}}{d^{2}}} - 30 d^{9} e^{2} x^{2} \sqrt{-1 + \frac{e^{2} x^{2}}{d^{2}}} + 15 d^{7} e^{4} x^{4} \sqrt{-1 + \frac{e^{2} x^{2}}{d^{2}}}} + \frac{20 i d^{2} e^{2} x^{3}}{15 d^{11} \sqrt{-1 + \frac{e^{2} x^{2}}{d^{2}}} - 30 d^{9} e^{2} x^{2} \sqrt{-1 + \frac{e^{2} x^{2}}{d^{2}}} + 15 d^{7} e^{4} x^{4} \sqrt{-1 + \frac{e^{2} x^{2}}{d^{2}}}} - \frac{8 i e^{4} x^{5}}{15 d^{11} \sqrt{-1 + \frac{e^{2} x^{2}}{d^{2}}} - 30 d^{9} e^{2} x^{2} \sqrt{-1 + \frac{e^{2} x^{2}}{d^{2}}} + 15 d^{7} e^{4} x^{4} \sqrt{-1 + \frac{e^{2} x^{2}}{d^{2}}}} & \text{for}\: \frac{\left |{e^{2} x^{2}}\right |}{\left |{d^{2}}\right |} > 1 \\\frac{15 d^{4} x}{15 d^{11} \sqrt{1 - \frac{e^{2} x^{2}}{d^{2}}} - 30 d^{9} e^{2} x^{2} \sqrt{1 - \frac{e^{2} x^{2}}{d^{2}}} + 15 d^{7} e^{4} x^{4} \sqrt{1 - \frac{e^{2} x^{2}}{d^{2}}}} - \frac{20 d^{2} e^{2} x^{3}}{15 d^{11} \sqrt{1 - \frac{e^{2} x^{2}}{d^{2}}} - 30 d^{9} e^{2} x^{2} \sqrt{1 - \frac{e^{2} x^{2}}{d^{2}}} + 15 d^{7} e^{4} x^{4} \sqrt{1 - \frac{e^{2} x^{2}}{d^{2}}}} + \frac{8 e^{4} x^{5}}{15 d^{11} \sqrt{1 - \frac{e^{2} x^{2}}{d^{2}}} - 30 d^{9} e^{2} x^{2} \sqrt{1 - \frac{e^{2} x^{2}}{d^{2}}} + 15 d^{7} e^{4} x^{4} \sqrt{1 - \frac{e^{2} x^{2}}{d^{2}}}} & \text{otherwise} \end{cases}\right ) + e \left (\begin{cases} \frac{1}{5 d^{4} e^{2} \sqrt{d^{2} - e^{2} x^{2}} - 10 d^{2} e^{4} x^{2} \sqrt{d^{2} - e^{2} x^{2}} + 5 e^{6} x^{4} \sqrt{d^{2} - e^{2} x^{2}}} & \text{for}\: e \neq 0 \\\frac{x^{2}}{2 \left (d^{2}\right )^{\frac{7}{2}}} & \text{otherwise} \end{cases}\right ) \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*Piecewise((-15*I*d**4*x/(15*d**11*sqrt(-1 + e**2*x**2/d**2) - 30*d**9*e**2*x**2*sqrt(-1 + e**2*x**2/d**2) +
15*d**7*e**4*x**4*sqrt(-1 + e**2*x**2/d**2)) + 20*I*d**2*e**2*x**3/(15*d**11*sqrt(-1 + e**2*x**2/d**2) - 30*d*
*9*e**2*x**2*sqrt(-1 + e**2*x**2/d**2) + 15*d**7*e**4*x**4*sqrt(-1 + e**2*x**2/d**2)) - 8*I*e**4*x**5/(15*d**1
1*sqrt(-1 + e**2*x**2/d**2) - 30*d**9*e**2*x**2*sqrt(-1 + e**2*x**2/d**2) + 15*d**7*e**4*x**4*sqrt(-1 + e**2*x
**2/d**2)), Abs(e**2*x**2)/Abs(d**2) > 1), (15*d**4*x/(15*d**11*sqrt(1 - e**2*x**2/d**2) - 30*d**9*e**2*x**2*s
qrt(1 - e**2*x**2/d**2) + 15*d**7*e**4*x**4*sqrt(1 - e**2*x**2/d**2)) - 20*d**2*e**2*x**3/(15*d**11*sqrt(1 - e
**2*x**2/d**2) - 30*d**9*e**2*x**2*sqrt(1 - e**2*x**2/d**2) + 15*d**7*e**4*x**4*sqrt(1 - e**2*x**2/d**2)) + 8*
e**4*x**5/(15*d**11*sqrt(1 - e**2*x**2/d**2) - 30*d**9*e**2*x**2*sqrt(1 - e**2*x**2/d**2) + 15*d**7*e**4*x**4*
sqrt(1 - e**2*x**2/d**2)), True)) + e*Piecewise((1/(5*d**4*e**2*sqrt(d**2 - e**2*x**2) - 10*d**2*e**4*x**2*sqr
t(d**2 - e**2*x**2) + 5*e**6*x**4*sqrt(d**2 - e**2*x**2)), Ne(e, 0)), (x**2/(2*(d**2)**(7/2)), True))

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Giac [A]  time = 1.33083, size = 88, normalized size = 1.1 \begin{align*} -\frac{\sqrt{-x^{2} e^{2} + d^{2}}{\left ({\left (4 \, x^{2}{\left (\frac{2 \, x^{2} e^{4}}{d^{5}} - \frac{5 \, e^{2}}{d^{3}}\right )} + \frac{15}{d}\right )} x + 3 \, e^{\left (-1\right )}\right )}}{15 \,{\left (x^{2} e^{2} - d^{2}\right )}^{3}} \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]

-1/15*sqrt(-x^2*e^2 + d^2)*((4*x^2*(2*x^2*e^4/d^5 - 5*e^2/d^3) + 15/d)*x + 3*e^(-1))/(x^2*e^2 - d^2)^3