### 3.858 $$\int \frac{1}{(d+e x)^4 (d^2-e^2 x^2)^{7/2}} \, dx$$

Optimal. Leaf size=205 $\frac{128 x}{715 d^{10} \sqrt{d^2-e^2 x^2}}+\frac{64 x}{715 d^8 \left (d^2-e^2 x^2\right )^{3/2}}+\frac{48 x}{715 d^6 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{8}{143 d^4 e (d+e x) \left (d^2-e^2 x^2\right )^{5/2}}-\frac{8}{143 d^3 e (d+e x)^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{9}{143 d^2 e (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{1}{13 d e (d+e x)^4 \left (d^2-e^2 x^2\right )^{5/2}}$

[Out]

(48*x)/(715*d^6*(d^2 - e^2*x^2)^(5/2)) - 1/(13*d*e*(d + e*x)^4*(d^2 - e^2*x^2)^(5/2)) - 9/(143*d^2*e*(d + e*x)
^3*(d^2 - e^2*x^2)^(5/2)) - 8/(143*d^3*e*(d + e*x)^2*(d^2 - e^2*x^2)^(5/2)) - 8/(143*d^4*e*(d + e*x)*(d^2 - e^
2*x^2)^(5/2)) + (64*x)/(715*d^8*(d^2 - e^2*x^2)^(3/2)) + (128*x)/(715*d^10*Sqrt[d^2 - e^2*x^2])

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Rubi [A]  time = 0.0889444, antiderivative size = 205, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 3, integrand size = 24, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.125, Rules used = {659, 192, 191} $\frac{128 x}{715 d^{10} \sqrt{d^2-e^2 x^2}}+\frac{64 x}{715 d^8 \left (d^2-e^2 x^2\right )^{3/2}}+\frac{48 x}{715 d^6 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{8}{143 d^4 e (d+e x) \left (d^2-e^2 x^2\right )^{5/2}}-\frac{8}{143 d^3 e (d+e x)^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{9}{143 d^2 e (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{1}{13 d e (d+e x)^4 \left (d^2-e^2 x^2\right )^{5/2}}$

Antiderivative was successfully veriﬁed.

[In]

Int[1/((d + e*x)^4*(d^2 - e^2*x^2)^(7/2)),x]

[Out]

(48*x)/(715*d^6*(d^2 - e^2*x^2)^(5/2)) - 1/(13*d*e*(d + e*x)^4*(d^2 - e^2*x^2)^(5/2)) - 9/(143*d^2*e*(d + e*x)
^3*(d^2 - e^2*x^2)^(5/2)) - 8/(143*d^3*e*(d + e*x)^2*(d^2 - e^2*x^2)^(5/2)) - 8/(143*d^4*e*(d + e*x)*(d^2 - e^
2*x^2)^(5/2)) + (64*x)/(715*d^8*(d^2 - e^2*x^2)^(3/2)) + (128*x)/(715*d^10*Sqrt[d^2 - e^2*x^2])

Rule 659

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

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{1}{(d+e x)^4 \left (d^2-e^2 x^2\right )^{7/2}} \, dx &=-\frac{1}{13 d e (d+e x)^4 \left (d^2-e^2 x^2\right )^{5/2}}+\frac{9 \int \frac{1}{(d+e x)^3 \left (d^2-e^2 x^2\right )^{7/2}} \, dx}{13 d}\\ &=-\frac{1}{13 d e (d+e x)^4 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{9}{143 d^2 e (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}+\frac{72 \int \frac{1}{(d+e x)^2 \left (d^2-e^2 x^2\right )^{7/2}} \, dx}{143 d^2}\\ &=-\frac{1}{13 d e (d+e x)^4 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{9}{143 d^2 e (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{8}{143 d^3 e (d+e x)^2 \left (d^2-e^2 x^2\right )^{5/2}}+\frac{56 \int \frac{1}{(d+e x) \left (d^2-e^2 x^2\right )^{7/2}} \, dx}{143 d^3}\\ &=-\frac{1}{13 d e (d+e x)^4 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{9}{143 d^2 e (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{8}{143 d^3 e (d+e x)^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{8}{143 d^4 e (d+e x) \left (d^2-e^2 x^2\right )^{5/2}}+\frac{48 \int \frac{1}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx}{143 d^4}\\ &=\frac{48 x}{715 d^6 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{1}{13 d e (d+e x)^4 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{9}{143 d^2 e (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{8}{143 d^3 e (d+e x)^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{8}{143 d^4 e (d+e x) \left (d^2-e^2 x^2\right )^{5/2}}+\frac{192 \int \frac{1}{\left (d^2-e^2 x^2\right )^{5/2}} \, dx}{715 d^6}\\ &=\frac{48 x}{715 d^6 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{1}{13 d e (d+e x)^4 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{9}{143 d^2 e (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{8}{143 d^3 e (d+e x)^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{8}{143 d^4 e (d+e x) \left (d^2-e^2 x^2\right )^{5/2}}+\frac{64 x}{715 d^8 \left (d^2-e^2 x^2\right )^{3/2}}+\frac{128 \int \frac{1}{\left (d^2-e^2 x^2\right )^{3/2}} \, dx}{715 d^8}\\ &=\frac{48 x}{715 d^6 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{1}{13 d e (d+e x)^4 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{9}{143 d^2 e (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{8}{143 d^3 e (d+e x)^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{8}{143 d^4 e (d+e x) \left (d^2-e^2 x^2\right )^{5/2}}+\frac{64 x}{715 d^8 \left (d^2-e^2 x^2\right )^{3/2}}+\frac{128 x}{715 d^{10} \sqrt{d^2-e^2 x^2}}\\ \end{align*}

Mathematica [A]  time = 0.0886175, size = 137, normalized size = 0.67 $\frac{\sqrt{d^2-e^2 x^2} \left (800 d^7 e^2 x^2+1080 d^6 e^3 x^3-320 d^5 e^4 x^4-1552 d^4 e^5 x^5-768 d^3 e^6 x^6+448 d^2 e^7 x^7-5 d^8 e x-180 d^9+512 d e^8 x^8+128 e^9 x^9\right )}{715 d^{10} e (d-e x)^3 (d+e x)^7}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[1/((d + e*x)^4*(d^2 - e^2*x^2)^(7/2)),x]

[Out]

(Sqrt[d^2 - e^2*x^2]*(-180*d^9 - 5*d^8*e*x + 800*d^7*e^2*x^2 + 1080*d^6*e^3*x^3 - 320*d^5*e^4*x^4 - 1552*d^4*e
^5*x^5 - 768*d^3*e^6*x^6 + 448*d^2*e^7*x^7 + 512*d*e^8*x^8 + 128*e^9*x^9))/(715*d^10*e*(d - e*x)^3*(d + e*x)^7
)

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Maple [A]  time = 0.047, size = 132, normalized size = 0.6 \begin{align*} -{\frac{ \left ( -ex+d \right ) \left ( -128\,{e}^{9}{x}^{9}-512\,{e}^{8}{x}^{8}d-448\,{e}^{7}{x}^{7}{d}^{2}+768\,{e}^{6}{x}^{6}{d}^{3}+1552\,{e}^{5}{x}^{5}{d}^{4}+320\,{e}^{4}{x}^{4}{d}^{5}-1080\,{e}^{3}{x}^{3}{d}^{6}-800\,{e}^{2}{x}^{2}{d}^{7}+5\,x{d}^{8}e+180\,{d}^{9} \right ) }{715\,e{d}^{10} \left ( ex+d \right ) ^{3}} \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(1/(e*x+d)^4/(-e^2*x^2+d^2)^(7/2),x)

[Out]

-1/715*(-e*x+d)*(-128*e^9*x^9-512*d*e^8*x^8-448*d^2*e^7*x^7+768*d^3*e^6*x^6+1552*d^4*e^5*x^5+320*d^5*e^4*x^4-1
080*d^6*e^3*x^3-800*d^7*e^2*x^2+5*d^8*e*x+180*d^9)/(e*x+d)^3/d^10/e/(-e^2*x^2+d^2)^(7/2)

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)^4/(-e^2*x^2+d^2)^(7/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 7.73555, size = 709, normalized size = 3.46 \begin{align*} -\frac{180 \, e^{10} x^{10} + 720 \, d e^{9} x^{9} + 540 \, d^{2} e^{8} x^{8} - 1440 \, d^{3} e^{7} x^{7} - 2520 \, d^{4} e^{6} x^{6} + 2520 \, d^{6} e^{4} x^{4} + 1440 \, d^{7} e^{3} x^{3} - 540 \, d^{8} e^{2} x^{2} - 720 \, d^{9} e x - 180 \, d^{10} +{\left (128 \, e^{9} x^{9} + 512 \, d e^{8} x^{8} + 448 \, d^{2} e^{7} x^{7} - 768 \, d^{3} e^{6} x^{6} - 1552 \, d^{4} e^{5} x^{5} - 320 \, d^{5} e^{4} x^{4} + 1080 \, d^{6} e^{3} x^{3} + 800 \, d^{7} e^{2} x^{2} - 5 \, d^{8} e x - 180 \, d^{9}\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{715 \,{\left (d^{10} e^{11} x^{10} + 4 \, d^{11} e^{10} x^{9} + 3 \, d^{12} e^{9} x^{8} - 8 \, d^{13} e^{8} x^{7} - 14 \, d^{14} e^{7} x^{6} + 14 \, d^{16} e^{5} x^{4} + 8 \, d^{17} e^{4} x^{3} - 3 \, d^{18} e^{3} x^{2} - 4 \, d^{19} e^{2} x - d^{20} e\right )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)^4/(-e^2*x^2+d^2)^(7/2),x, algorithm="fricas")

[Out]

-1/715*(180*e^10*x^10 + 720*d*e^9*x^9 + 540*d^2*e^8*x^8 - 1440*d^3*e^7*x^7 - 2520*d^4*e^6*x^6 + 2520*d^6*e^4*x
^4 + 1440*d^7*e^3*x^3 - 540*d^8*e^2*x^2 - 720*d^9*e*x - 180*d^10 + (128*e^9*x^9 + 512*d*e^8*x^8 + 448*d^2*e^7*
x^7 - 768*d^3*e^6*x^6 - 1552*d^4*e^5*x^5 - 320*d^5*e^4*x^4 + 1080*d^6*e^3*x^3 + 800*d^7*e^2*x^2 - 5*d^8*e*x -
180*d^9)*sqrt(-e^2*x^2 + d^2))/(d^10*e^11*x^10 + 4*d^11*e^10*x^9 + 3*d^12*e^9*x^8 - 8*d^13*e^8*x^7 - 14*d^14*e
^7*x^6 + 14*d^16*e^5*x^4 + 8*d^17*e^4*x^3 - 3*d^18*e^3*x^2 - 4*d^19*e^2*x - d^20*e)

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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(1/(e*x+d)**4/(-e**2*x**2+d**2)**(7/2),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \left [\mathit{undef}, \mathit{undef}, \mathit{undef}, \mathit{undef}, \mathit{undef}, \mathit{undef}, \mathit{undef}, 1\right ] \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)^4/(-e^2*x^2+d^2)^(7/2),x, algorithm="giac")

[Out]

[undef, undef, undef, undef, undef, undef, undef, 1]