### 3.834 $$\int \frac{1}{(d+e x)^4 \sqrt{d^2-e^2 x^2}} \, dx$$

Optimal. Leaf size=133 $-\frac{2 \sqrt{d^2-e^2 x^2}}{35 d^4 e (d+e x)}-\frac{2 \sqrt{d^2-e^2 x^2}}{35 d^3 e (d+e x)^2}-\frac{3 \sqrt{d^2-e^2 x^2}}{35 d^2 e (d+e x)^3}-\frac{\sqrt{d^2-e^2 x^2}}{7 d e (d+e x)^4}$

[Out]

-Sqrt[d^2 - e^2*x^2]/(7*d*e*(d + e*x)^4) - (3*Sqrt[d^2 - e^2*x^2])/(35*d^2*e*(d + e*x)^3) - (2*Sqrt[d^2 - e^2*
x^2])/(35*d^3*e*(d + e*x)^2) - (2*Sqrt[d^2 - e^2*x^2])/(35*d^4*e*(d + e*x))

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Rubi [A]  time = 0.0552447, antiderivative size = 133, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 24, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.083, Rules used = {659, 651} $-\frac{2 \sqrt{d^2-e^2 x^2}}{35 d^4 e (d+e x)}-\frac{2 \sqrt{d^2-e^2 x^2}}{35 d^3 e (d+e x)^2}-\frac{3 \sqrt{d^2-e^2 x^2}}{35 d^2 e (d+e x)^3}-\frac{\sqrt{d^2-e^2 x^2}}{7 d e (d+e x)^4}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

-Sqrt[d^2 - e^2*x^2]/(7*d*e*(d + e*x)^4) - (3*Sqrt[d^2 - e^2*x^2])/(35*d^2*e*(d + e*x)^3) - (2*Sqrt[d^2 - e^2*
x^2])/(35*d^3*e*(d + e*x)^2) - (2*Sqrt[d^2 - e^2*x^2])/(35*d^4*e*(d + e*x))

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 651

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*(p + 1)), x] /; FreeQ[{a, c, d, e, m, p}, x] && EqQ[c*d^2 + a*e^2, 0] &&  !IntegerQ[p] && EqQ[m + 2*p
+ 2, 0]

Rubi steps

\begin{align*} \int \frac{1}{(d+e x)^4 \sqrt{d^2-e^2 x^2}} \, dx &=-\frac{\sqrt{d^2-e^2 x^2}}{7 d e (d+e x)^4}+\frac{3 \int \frac{1}{(d+e x)^3 \sqrt{d^2-e^2 x^2}} \, dx}{7 d}\\ &=-\frac{\sqrt{d^2-e^2 x^2}}{7 d e (d+e x)^4}-\frac{3 \sqrt{d^2-e^2 x^2}}{35 d^2 e (d+e x)^3}+\frac{6 \int \frac{1}{(d+e x)^2 \sqrt{d^2-e^2 x^2}} \, dx}{35 d^2}\\ &=-\frac{\sqrt{d^2-e^2 x^2}}{7 d e (d+e x)^4}-\frac{3 \sqrt{d^2-e^2 x^2}}{35 d^2 e (d+e x)^3}-\frac{2 \sqrt{d^2-e^2 x^2}}{35 d^3 e (d+e x)^2}+\frac{2 \int \frac{1}{(d+e x) \sqrt{d^2-e^2 x^2}} \, dx}{35 d^3}\\ &=-\frac{\sqrt{d^2-e^2 x^2}}{7 d e (d+e x)^4}-\frac{3 \sqrt{d^2-e^2 x^2}}{35 d^2 e (d+e x)^3}-\frac{2 \sqrt{d^2-e^2 x^2}}{35 d^3 e (d+e x)^2}-\frac{2 \sqrt{d^2-e^2 x^2}}{35 d^4 e (d+e x)}\\ \end{align*}

Mathematica [A]  time = 0.0423093, size = 63, normalized size = 0.47 $-\frac{\sqrt{d^2-e^2 x^2} \left (13 d^2 e x+12 d^3+8 d e^2 x^2+2 e^3 x^3\right )}{35 d^4 e (d+e x)^4}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

-(Sqrt[d^2 - e^2*x^2]*(12*d^3 + 13*d^2*e*x + 8*d*e^2*x^2 + 2*e^3*x^3))/(35*d^4*e*(d + e*x)^4)

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Maple [A]  time = 0.043, size = 66, normalized size = 0.5 \begin{align*} -{\frac{ \left ( -ex+d \right ) \left ( 2\,{e}^{3}{x}^{3}+8\,{e}^{2}{x}^{2}d+13\,x{d}^{2}e+12\,{d}^{3} \right ) }{35\,e{d}^{4} \left ( ex+d \right ) ^{3}}{\frac{1}{\sqrt{-{e}^{2}{x}^{2}+{d}^{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)^(1/2),x)

[Out]

-1/35*(-e*x+d)*(2*e^3*x^3+8*d*e^2*x^2+13*d^2*e*x+12*d^3)/(e*x+d)^3/d^4/e/(-e^2*x^2+d^2)^(1/2)

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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)^(1/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 2.28272, size = 288, normalized size = 2.17 \begin{align*} -\frac{12 \, e^{4} x^{4} + 48 \, d e^{3} x^{3} + 72 \, d^{2} e^{2} x^{2} + 48 \, d^{3} e x + 12 \, d^{4} +{\left (2 \, e^{3} x^{3} + 8 \, d e^{2} x^{2} + 13 \, d^{2} e x + 12 \, d^{3}\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{35 \,{\left (d^{4} e^{5} x^{4} + 4 \, d^{5} e^{4} x^{3} + 6 \, d^{6} e^{3} x^{2} + 4 \, d^{7} e^{2} x + d^{8} 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)^(1/2),x, algorithm="fricas")

[Out]

-1/35*(12*e^4*x^4 + 48*d*e^3*x^3 + 72*d^2*e^2*x^2 + 48*d^3*e*x + 12*d^4 + (2*e^3*x^3 + 8*d*e^2*x^2 + 13*d^2*e*
x + 12*d^3)*sqrt(-e^2*x^2 + d^2))/(d^4*e^5*x^4 + 4*d^5*e^4*x^3 + 6*d^6*e^3*x^2 + 4*d^7*e^2*x + d^8*e)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{- \left (- d + e x\right ) \left (d + e x\right )} \left (d + e x\right )^{4}}\, dx \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)**(1/2),x)

[Out]

Integral(1/(sqrt(-(-d + e*x)*(d + e*x))*(d + e*x)**4), x)

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Giac [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: NotImplementedError} \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)^(1/2),x, algorithm="giac")

[Out]

Exception raised: NotImplementedError