### 3.857 $$\int \frac{1}{(d+e x)^3 (d^2-e^2 x^2)^{7/2}} \, dx$$

Optimal. Leaf size=172 $\frac{128 x}{495 d^9 \sqrt{d^2-e^2 x^2}}+\frac{64 x}{495 d^7 \left (d^2-e^2 x^2\right )^{3/2}}+\frac{16 x}{165 d^5 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{8}{99 d^3 e (d+e x) \left (d^2-e^2 x^2\right )^{5/2}}-\frac{8}{99 d^2 e (d+e x)^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{1}{11 d e (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}$

[Out]

(16*x)/(165*d^5*(d^2 - e^2*x^2)^(5/2)) - 1/(11*d*e*(d + e*x)^3*(d^2 - e^2*x^2)^(5/2)) - 8/(99*d^2*e*(d + e*x)^
2*(d^2 - e^2*x^2)^(5/2)) - 8/(99*d^3*e*(d + e*x)*(d^2 - e^2*x^2)^(5/2)) + (64*x)/(495*d^7*(d^2 - e^2*x^2)^(3/2
)) + (128*x)/(495*d^9*Sqrt[d^2 - e^2*x^2])

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Rubi [A]  time = 0.0639118, antiderivative size = 172, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 3, integrand size = 24, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.125, Rules used = {659, 192, 191} $\frac{128 x}{495 d^9 \sqrt{d^2-e^2 x^2}}+\frac{64 x}{495 d^7 \left (d^2-e^2 x^2\right )^{3/2}}+\frac{16 x}{165 d^5 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{8}{99 d^3 e (d+e x) \left (d^2-e^2 x^2\right )^{5/2}}-\frac{8}{99 d^2 e (d+e x)^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{1}{11 d e (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}$

Antiderivative was successfully veriﬁed.

[In]

Int[1/((d + e*x)^3*(d^2 - e^2*x^2)^(7/2)),x]

[Out]

(16*x)/(165*d^5*(d^2 - e^2*x^2)^(5/2)) - 1/(11*d*e*(d + e*x)^3*(d^2 - e^2*x^2)^(5/2)) - 8/(99*d^2*e*(d + e*x)^
2*(d^2 - e^2*x^2)^(5/2)) - 8/(99*d^3*e*(d + e*x)*(d^2 - e^2*x^2)^(5/2)) + (64*x)/(495*d^7*(d^2 - e^2*x^2)^(3/2
)) + (128*x)/(495*d^9*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)^3 \left (d^2-e^2 x^2\right )^{7/2}} \, dx &=-\frac{1}{11 d e (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}+\frac{8 \int \frac{1}{(d+e x)^2 \left (d^2-e^2 x^2\right )^{7/2}} \, dx}{11 d}\\ &=-\frac{1}{11 d e (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{8}{99 d^2 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}{99 d^2}\\ &=-\frac{1}{11 d e (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{8}{99 d^2 e (d+e x)^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{8}{99 d^3 e (d+e x) \left (d^2-e^2 x^2\right )^{5/2}}+\frac{16 \int \frac{1}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx}{33 d^3}\\ &=\frac{16 x}{165 d^5 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{1}{11 d e (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{8}{99 d^2 e (d+e x)^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{8}{99 d^3 e (d+e x) \left (d^2-e^2 x^2\right )^{5/2}}+\frac{64 \int \frac{1}{\left (d^2-e^2 x^2\right )^{5/2}} \, dx}{165 d^5}\\ &=\frac{16 x}{165 d^5 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{1}{11 d e (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{8}{99 d^2 e (d+e x)^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{8}{99 d^3 e (d+e x) \left (d^2-e^2 x^2\right )^{5/2}}+\frac{64 x}{495 d^7 \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}{495 d^7}\\ &=\frac{16 x}{165 d^5 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{1}{11 d e (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{8}{99 d^2 e (d+e x)^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{8}{99 d^3 e (d+e x) \left (d^2-e^2 x^2\right )^{5/2}}+\frac{64 x}{495 d^7 \left (d^2-e^2 x^2\right )^{3/2}}+\frac{128 x}{495 d^9 \sqrt{d^2-e^2 x^2}}\\ \end{align*}

Mathematica [A]  time = 0.0825094, size = 126, normalized size = 0.73 $\frac{\sqrt{d^2-e^2 x^2} \left (680 d^6 e^2 x^2+400 d^5 e^3 x^3-720 d^4 e^4 x^4-832 d^3 e^5 x^5+64 d^2 e^6 x^6+120 d^7 e x-125 d^8+384 d e^7 x^7+128 e^8 x^8\right )}{495 d^9 e (d-e x)^3 (d+e x)^6}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[1/((d + e*x)^3*(d^2 - e^2*x^2)^(7/2)),x]

[Out]

(Sqrt[d^2 - e^2*x^2]*(-125*d^8 + 120*d^7*e*x + 680*d^6*e^2*x^2 + 400*d^5*e^3*x^3 - 720*d^4*e^4*x^4 - 832*d^3*e
^5*x^5 + 64*d^2*e^6*x^6 + 384*d*e^7*x^7 + 128*e^8*x^8))/(495*d^9*e*(d - e*x)^3*(d + e*x)^6)

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Maple [A]  time = 0.046, size = 121, normalized size = 0.7 \begin{align*} -{\frac{ \left ( -ex+d \right ) \left ( -128\,{e}^{8}{x}^{8}-384\,{e}^{7}{x}^{7}d-64\,{e}^{6}{x}^{6}{d}^{2}+832\,{e}^{5}{x}^{5}{d}^{3}+720\,{e}^{4}{x}^{4}{d}^{4}-400\,{e}^{3}{x}^{3}{d}^{5}-680\,{e}^{2}{x}^{2}{d}^{6}-120\,x{d}^{7}e+125\,{d}^{8} \right ) }{495\,e{d}^{9} \left ( ex+d \right ) ^{2}} \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)^3/(-e^2*x^2+d^2)^(7/2),x)

[Out]

-1/495*(-e*x+d)*(-128*e^8*x^8-384*d*e^7*x^7-64*d^2*e^6*x^6+832*d^3*e^5*x^5+720*d^4*e^4*x^4-400*d^5*e^3*x^3-680
*d^6*e^2*x^2-120*d^7*e*x+125*d^8)/(e*x+d)^2/d^9/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)^3/(-e^2*x^2+d^2)^(7/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 5.7211, size = 575, normalized size = 3.34 \begin{align*} -\frac{125 \, e^{9} x^{9} + 375 \, d e^{8} x^{8} - 1000 \, d^{3} e^{6} x^{6} - 750 \, d^{4} e^{5} x^{5} + 750 \, d^{5} e^{4} x^{4} + 1000 \, d^{6} e^{3} x^{3} - 375 \, d^{8} e x - 125 \, d^{9} +{\left (128 \, e^{8} x^{8} + 384 \, d e^{7} x^{7} + 64 \, d^{2} e^{6} x^{6} - 832 \, d^{3} e^{5} x^{5} - 720 \, d^{4} e^{4} x^{4} + 400 \, d^{5} e^{3} x^{3} + 680 \, d^{6} e^{2} x^{2} + 120 \, d^{7} e x - 125 \, d^{8}\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{495 \,{\left (d^{9} e^{10} x^{9} + 3 \, d^{10} e^{9} x^{8} - 8 \, d^{12} e^{7} x^{6} - 6 \, d^{13} e^{6} x^{5} + 6 \, d^{14} e^{5} x^{4} + 8 \, d^{15} e^{4} x^{3} - 3 \, d^{17} e^{2} x - d^{18} e\right )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)^3/(-e^2*x^2+d^2)^(7/2),x, algorithm="fricas")

[Out]

-1/495*(125*e^9*x^9 + 375*d*e^8*x^8 - 1000*d^3*e^6*x^6 - 750*d^4*e^5*x^5 + 750*d^5*e^4*x^4 + 1000*d^6*e^3*x^3
- 375*d^8*e*x - 125*d^9 + (128*e^8*x^8 + 384*d*e^7*x^7 + 64*d^2*e^6*x^6 - 832*d^3*e^5*x^5 - 720*d^4*e^4*x^4 +
400*d^5*e^3*x^3 + 680*d^6*e^2*x^2 + 120*d^7*e*x - 125*d^8)*sqrt(-e^2*x^2 + d^2))/(d^9*e^10*x^9 + 3*d^10*e^9*x^
8 - 8*d^12*e^7*x^6 - 6*d^13*e^6*x^5 + 6*d^14*e^5*x^4 + 8*d^15*e^4*x^3 - 3*d^17*e^2*x - d^18*e)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (- \left (- d + e x\right ) \left (d + e x\right )\right )^{\frac{7}{2}} \left (d + e x\right )^{3}}\, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)**3/(-e**2*x**2+d**2)**(7/2),x)

[Out]

Integral(1/((-(-d + e*x)*(d + e*x))**(7/2)*(d + e*x)**3), x)

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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)^3/(-e^2*x^2+d^2)^(7/2),x, algorithm="giac")

[Out]

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