3.844 \(\int \frac{1}{(d+e x)^3 (d^2-e^2 x^2)^{5/2}} \, dx\)

Optimal. Leaf size=148 \[ \frac{16 x}{63 d^7 \sqrt{d^2-e^2 x^2}}+\frac{8 x}{63 d^5 \left (d^2-e^2 x^2\right )^{3/2}}-\frac{2}{21 d^3 e (d+e x) \left (d^2-e^2 x^2\right )^{3/2}}-\frac{2}{21 d^2 e (d+e x)^2 \left (d^2-e^2 x^2\right )^{3/2}}-\frac{1}{9 d e (d+e x)^3 \left (d^2-e^2 x^2\right )^{3/2}} \]

[Out]

(8*x)/(63*d^5*(d^2 - e^2*x^2)^(3/2)) - 1/(9*d*e*(d + e*x)^3*(d^2 - e^2*x^2)^(3/2)) - 2/(21*d^2*e*(d + e*x)^2*(
d^2 - e^2*x^2)^(3/2)) - 2/(21*d^3*e*(d + e*x)*(d^2 - e^2*x^2)^(3/2)) + (16*x)/(63*d^7*Sqrt[d^2 - e^2*x^2])

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Rubi [A]  time = 0.053541, antiderivative size = 148, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {659, 192, 191} \[ \frac{16 x}{63 d^7 \sqrt{d^2-e^2 x^2}}+\frac{8 x}{63 d^5 \left (d^2-e^2 x^2\right )^{3/2}}-\frac{2}{21 d^3 e (d+e x) \left (d^2-e^2 x^2\right )^{3/2}}-\frac{2}{21 d^2 e (d+e x)^2 \left (d^2-e^2 x^2\right )^{3/2}}-\frac{1}{9 d e (d+e x)^3 \left (d^2-e^2 x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

Int[1/((d + e*x)^3*(d^2 - e^2*x^2)^(5/2)),x]

[Out]

(8*x)/(63*d^5*(d^2 - e^2*x^2)^(3/2)) - 1/(9*d*e*(d + e*x)^3*(d^2 - e^2*x^2)^(3/2)) - 2/(21*d^2*e*(d + e*x)^2*(
d^2 - e^2*x^2)^(3/2)) - 2/(21*d^3*e*(d + e*x)*(d^2 - e^2*x^2)^(3/2)) + (16*x)/(63*d^7*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 )^{5/2}} \, dx &=-\frac{1}{9 d e (d+e x)^3 \left (d^2-e^2 x^2\right )^{3/2}}+\frac{2 \int \frac{1}{(d+e x)^2 \left (d^2-e^2 x^2\right )^{5/2}} \, dx}{3 d}\\ &=-\frac{1}{9 d e (d+e x)^3 \left (d^2-e^2 x^2\right )^{3/2}}-\frac{2}{21 d^2 e (d+e x)^2 \left (d^2-e^2 x^2\right )^{3/2}}+\frac{10 \int \frac{1}{(d+e x) \left (d^2-e^2 x^2\right )^{5/2}} \, dx}{21 d^2}\\ &=-\frac{1}{9 d e (d+e x)^3 \left (d^2-e^2 x^2\right )^{3/2}}-\frac{2}{21 d^2 e (d+e x)^2 \left (d^2-e^2 x^2\right )^{3/2}}-\frac{2}{21 d^3 e (d+e x) \left (d^2-e^2 x^2\right )^{3/2}}+\frac{8 \int \frac{1}{\left (d^2-e^2 x^2\right )^{5/2}} \, dx}{21 d^3}\\ &=\frac{8 x}{63 d^5 \left (d^2-e^2 x^2\right )^{3/2}}-\frac{1}{9 d e (d+e x)^3 \left (d^2-e^2 x^2\right )^{3/2}}-\frac{2}{21 d^2 e (d+e x)^2 \left (d^2-e^2 x^2\right )^{3/2}}-\frac{2}{21 d^3 e (d+e x) \left (d^2-e^2 x^2\right )^{3/2}}+\frac{16 \int \frac{1}{\left (d^2-e^2 x^2\right )^{3/2}} \, dx}{63 d^5}\\ &=\frac{8 x}{63 d^5 \left (d^2-e^2 x^2\right )^{3/2}}-\frac{1}{9 d e (d+e x)^3 \left (d^2-e^2 x^2\right )^{3/2}}-\frac{2}{21 d^2 e (d+e x)^2 \left (d^2-e^2 x^2\right )^{3/2}}-\frac{2}{21 d^3 e (d+e x) \left (d^2-e^2 x^2\right )^{3/2}}+\frac{16 x}{63 d^7 \sqrt{d^2-e^2 x^2}}\\ \end{align*}

Mathematica [A]  time = 0.0679166, size = 104, normalized size = 0.7 \[ -\frac{\sqrt{d^2-e^2 x^2} \left (-66 d^4 e^2 x^2-56 d^3 e^3 x^3+24 d^2 e^4 x^4-6 d^5 e x+19 d^6+48 d e^5 x^5+16 e^6 x^6\right )}{63 d^7 e (d-e x)^2 (d+e x)^5} \]

Antiderivative was successfully verified.

[In]

Integrate[1/((d + e*x)^3*(d^2 - e^2*x^2)^(5/2)),x]

[Out]

-(Sqrt[d^2 - e^2*x^2]*(19*d^6 - 6*d^5*e*x - 66*d^4*e^2*x^2 - 56*d^3*e^3*x^3 + 24*d^2*e^4*x^4 + 48*d*e^5*x^5 +
16*e^6*x^6))/(63*d^7*e*(d - e*x)^2*(d + e*x)^5)

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Maple [A]  time = 0.045, size = 99, normalized size = 0.7 \begin{align*} -{\frac{ \left ( -ex+d \right ) \left ( 16\,{e}^{6}{x}^{6}+48\,{e}^{5}{x}^{5}d+24\,{e}^{4}{x}^{4}{d}^{2}-56\,{e}^{3}{x}^{3}{d}^{3}-66\,{e}^{2}{x}^{2}{d}^{4}-6\,x{d}^{5}e+19\,{d}^{6} \right ) }{63\,e{d}^{7} \left ( ex+d \right ) ^{2}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{-{\frac{5}{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)^(5/2),x)

[Out]

-1/63*(-e*x+d)*(16*e^6*x^6+48*d*e^5*x^5+24*d^2*e^4*x^4-56*d^3*e^3*x^3-66*d^4*e^2*x^2-6*d^5*e*x+19*d^6)/(e*x+d)
^2/d^7/e/(-e^2*x^2+d^2)^(5/2)

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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)^(5/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 2.9897, size = 493, normalized size = 3.33 \begin{align*} -\frac{19 \, e^{7} x^{7} + 57 \, d e^{6} x^{6} + 19 \, d^{2} e^{5} x^{5} - 95 \, d^{3} e^{4} x^{4} - 95 \, d^{4} e^{3} x^{3} + 19 \, d^{5} e^{2} x^{2} + 57 \, d^{6} e x + 19 \, d^{7} +{\left (16 \, e^{6} x^{6} + 48 \, d e^{5} x^{5} + 24 \, d^{2} e^{4} x^{4} - 56 \, d^{3} e^{3} x^{3} - 66 \, d^{4} e^{2} x^{2} - 6 \, d^{5} e x + 19 \, d^{6}\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{63 \,{\left (d^{7} e^{8} x^{7} + 3 \, d^{8} e^{7} x^{6} + d^{9} e^{6} x^{5} - 5 \, d^{10} e^{5} x^{4} - 5 \, d^{11} e^{4} x^{3} + d^{12} e^{3} x^{2} + 3 \, d^{13} e^{2} x + d^{14} 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)^(5/2),x, algorithm="fricas")

[Out]

-1/63*(19*e^7*x^7 + 57*d*e^6*x^6 + 19*d^2*e^5*x^5 - 95*d^3*e^4*x^4 - 95*d^4*e^3*x^3 + 19*d^5*e^2*x^2 + 57*d^6*
e*x + 19*d^7 + (16*e^6*x^6 + 48*d*e^5*x^5 + 24*d^2*e^4*x^4 - 56*d^3*e^3*x^3 - 66*d^4*e^2*x^2 - 6*d^5*e*x + 19*
d^6)*sqrt(-e^2*x^2 + d^2))/(d^7*e^8*x^7 + 3*d^8*e^7*x^6 + d^9*e^6*x^5 - 5*d^10*e^5*x^4 - 5*d^11*e^4*x^3 + d^12
*e^3*x^2 + 3*d^13*e^2*x + d^14*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{5}{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)**(5/2),x)

[Out]

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

[Out]

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