### 3.835 $$\int \frac{1}{(d+e x)^5 \sqrt{d^2-e^2 x^2}} \, dx$$

Optimal. Leaf size=166 $-\frac{8 \sqrt{d^2-e^2 x^2}}{315 d^5 e (d+e x)}-\frac{8 \sqrt{d^2-e^2 x^2}}{315 d^4 e (d+e x)^2}-\frac{4 \sqrt{d^2-e^2 x^2}}{105 d^3 e (d+e x)^3}-\frac{4 \sqrt{d^2-e^2 x^2}}{63 d^2 e (d+e x)^4}-\frac{\sqrt{d^2-e^2 x^2}}{9 d e (d+e x)^5}$

[Out]

-Sqrt[d^2 - e^2*x^2]/(9*d*e*(d + e*x)^5) - (4*Sqrt[d^2 - e^2*x^2])/(63*d^2*e*(d + e*x)^4) - (4*Sqrt[d^2 - e^2*
x^2])/(105*d^3*e*(d + e*x)^3) - (8*Sqrt[d^2 - e^2*x^2])/(315*d^4*e*(d + e*x)^2) - (8*Sqrt[d^2 - e^2*x^2])/(315
*d^5*e*(d + e*x))

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Rubi [A]  time = 0.0721729, antiderivative size = 166, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 2, integrand size = 24, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.083, Rules used = {659, 651} $-\frac{8 \sqrt{d^2-e^2 x^2}}{315 d^5 e (d+e x)}-\frac{8 \sqrt{d^2-e^2 x^2}}{315 d^4 e (d+e x)^2}-\frac{4 \sqrt{d^2-e^2 x^2}}{105 d^3 e (d+e x)^3}-\frac{4 \sqrt{d^2-e^2 x^2}}{63 d^2 e (d+e x)^4}-\frac{\sqrt{d^2-e^2 x^2}}{9 d e (d+e x)^5}$

Antiderivative was successfully veriﬁed.

[In]

Int[1/((d + e*x)^5*Sqrt[d^2 - e^2*x^2]),x]

[Out]

-Sqrt[d^2 - e^2*x^2]/(9*d*e*(d + e*x)^5) - (4*Sqrt[d^2 - e^2*x^2])/(63*d^2*e*(d + e*x)^4) - (4*Sqrt[d^2 - e^2*
x^2])/(105*d^3*e*(d + e*x)^3) - (8*Sqrt[d^2 - e^2*x^2])/(315*d^4*e*(d + e*x)^2) - (8*Sqrt[d^2 - e^2*x^2])/(315
*d^5*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)^5 \sqrt{d^2-e^2 x^2}} \, dx &=-\frac{\sqrt{d^2-e^2 x^2}}{9 d e (d+e x)^5}+\frac{4 \int \frac{1}{(d+e x)^4 \sqrt{d^2-e^2 x^2}} \, dx}{9 d}\\ &=-\frac{\sqrt{d^2-e^2 x^2}}{9 d e (d+e x)^5}-\frac{4 \sqrt{d^2-e^2 x^2}}{63 d^2 e (d+e x)^4}+\frac{4 \int \frac{1}{(d+e x)^3 \sqrt{d^2-e^2 x^2}} \, dx}{21 d^2}\\ &=-\frac{\sqrt{d^2-e^2 x^2}}{9 d e (d+e x)^5}-\frac{4 \sqrt{d^2-e^2 x^2}}{63 d^2 e (d+e x)^4}-\frac{4 \sqrt{d^2-e^2 x^2}}{105 d^3 e (d+e x)^3}+\frac{8 \int \frac{1}{(d+e x)^2 \sqrt{d^2-e^2 x^2}} \, dx}{105 d^3}\\ &=-\frac{\sqrt{d^2-e^2 x^2}}{9 d e (d+e x)^5}-\frac{4 \sqrt{d^2-e^2 x^2}}{63 d^2 e (d+e x)^4}-\frac{4 \sqrt{d^2-e^2 x^2}}{105 d^3 e (d+e x)^3}-\frac{8 \sqrt{d^2-e^2 x^2}}{315 d^4 e (d+e x)^2}+\frac{8 \int \frac{1}{(d+e x) \sqrt{d^2-e^2 x^2}} \, dx}{315 d^4}\\ &=-\frac{\sqrt{d^2-e^2 x^2}}{9 d e (d+e x)^5}-\frac{4 \sqrt{d^2-e^2 x^2}}{63 d^2 e (d+e x)^4}-\frac{4 \sqrt{d^2-e^2 x^2}}{105 d^3 e (d+e x)^3}-\frac{8 \sqrt{d^2-e^2 x^2}}{315 d^4 e (d+e x)^2}-\frac{8 \sqrt{d^2-e^2 x^2}}{315 d^5 e (d+e x)}\\ \end{align*}

Mathematica [A]  time = 0.0482122, size = 74, normalized size = 0.45 $-\frac{\sqrt{d^2-e^2 x^2} \left (84 d^2 e^2 x^2+100 d^3 e x+83 d^4+40 d e^3 x^3+8 e^4 x^4\right )}{315 d^5 e (d+e x)^5}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[1/((d + e*x)^5*Sqrt[d^2 - e^2*x^2]),x]

[Out]

-(Sqrt[d^2 - e^2*x^2]*(83*d^4 + 100*d^3*e*x + 84*d^2*e^2*x^2 + 40*d*e^3*x^3 + 8*e^4*x^4))/(315*d^5*e*(d + e*x)
^5)

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Maple [A]  time = 0.045, size = 77, normalized size = 0.5 \begin{align*} -{\frac{ \left ( -ex+d \right ) \left ( 8\,{e}^{4}{x}^{4}+40\,{e}^{3}{x}^{3}d+84\,{e}^{2}{x}^{2}{d}^{2}+100\,x{d}^{3}e+83\,{d}^{4} \right ) }{315\,e{d}^{5} \left ( ex+d \right ) ^{4}}{\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)^5/(-e^2*x^2+d^2)^(1/2),x)

[Out]

-1/315*(-e*x+d)*(8*e^4*x^4+40*d*e^3*x^3+84*d^2*e^2*x^2+100*d^3*e*x+83*d^4)/(e*x+d)^4/d^5/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)^5/(-e^2*x^2+d^2)^(1/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 2.40017, size = 369, normalized size = 2.22 \begin{align*} -\frac{83 \, e^{5} x^{5} + 415 \, d e^{4} x^{4} + 830 \, d^{2} e^{3} x^{3} + 830 \, d^{3} e^{2} x^{2} + 415 \, d^{4} e x + 83 \, d^{5} +{\left (8 \, e^{4} x^{4} + 40 \, d e^{3} x^{3} + 84 \, d^{2} e^{2} x^{2} + 100 \, d^{3} e x + 83 \, d^{4}\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{315 \,{\left (d^{5} e^{6} x^{5} + 5 \, d^{6} e^{5} x^{4} + 10 \, d^{7} e^{4} x^{3} + 10 \, d^{8} e^{3} x^{2} + 5 \, d^{9} e^{2} x + d^{10} e\right )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)^5/(-e^2*x^2+d^2)^(1/2),x, algorithm="fricas")

[Out]

-1/315*(83*e^5*x^5 + 415*d*e^4*x^4 + 830*d^2*e^3*x^3 + 830*d^3*e^2*x^2 + 415*d^4*e*x + 83*d^5 + (8*e^4*x^4 + 4
0*d*e^3*x^3 + 84*d^2*e^2*x^2 + 100*d^3*e*x + 83*d^4)*sqrt(-e^2*x^2 + d^2))/(d^5*e^6*x^5 + 5*d^6*e^5*x^4 + 10*d
^7*e^4*x^3 + 10*d^8*e^3*x^2 + 5*d^9*e^2*x + d^10*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 )^{5}}\, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)**5/(-e**2*x**2+d**2)**(1/2),x)

[Out]

Integral(1/(sqrt(-(-d + e*x)*(d + e*x))*(d + e*x)**5), x)

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)^5/(-e^2*x^2+d^2)^(1/2),x, algorithm="giac")

[Out]

Exception raised: RuntimeError