3.859 \(\int \frac{1}{(d+e x)^5 (d^2-e^2 x^2)^{7/2}} \, dx\)

Optimal. Leaf size=238 \[ \frac{256 x}{2145 d^{11} \sqrt{d^2-e^2 x^2}}+\frac{128 x}{2145 d^9 \left (d^2-e^2 x^2\right )^{3/2}}+\frac{32 x}{715 d^7 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{16}{429 d^5 e (d+e x) \left (d^2-e^2 x^2\right )^{5/2}}-\frac{16}{429 d^4 e (d+e x)^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{6}{143 d^3 e (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{2}{39 d^2 e (d+e x)^4 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{1}{15 d e (d+e x)^5 \left (d^2-e^2 x^2\right )^{5/2}} \]

[Out]

(32*x)/(715*d^7*(d^2 - e^2*x^2)^(5/2)) - 1/(15*d*e*(d + e*x)^5*(d^2 - e^2*x^2)^(5/2)) - 2/(39*d^2*e*(d + e*x)^
4*(d^2 - e^2*x^2)^(5/2)) - 6/(143*d^3*e*(d + e*x)^3*(d^2 - e^2*x^2)^(5/2)) - 16/(429*d^4*e*(d + e*x)^2*(d^2 -
e^2*x^2)^(5/2)) - 16/(429*d^5*e*(d + e*x)*(d^2 - e^2*x^2)^(5/2)) + (128*x)/(2145*d^9*(d^2 - e^2*x^2)^(3/2)) +
(256*x)/(2145*d^11*Sqrt[d^2 - e^2*x^2])

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Rubi [A]  time = 0.112964, antiderivative size = 238, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {659, 192, 191} \[ \frac{256 x}{2145 d^{11} \sqrt{d^2-e^2 x^2}}+\frac{128 x}{2145 d^9 \left (d^2-e^2 x^2\right )^{3/2}}+\frac{32 x}{715 d^7 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{16}{429 d^5 e (d+e x) \left (d^2-e^2 x^2\right )^{5/2}}-\frac{16}{429 d^4 e (d+e x)^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{6}{143 d^3 e (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{2}{39 d^2 e (d+e x)^4 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{1}{15 d e (d+e x)^5 \left (d^2-e^2 x^2\right )^{5/2}} \]

Antiderivative was successfully verified.

[In]

Int[1/((d + e*x)^5*(d^2 - e^2*x^2)^(7/2)),x]

[Out]

(32*x)/(715*d^7*(d^2 - e^2*x^2)^(5/2)) - 1/(15*d*e*(d + e*x)^5*(d^2 - e^2*x^2)^(5/2)) - 2/(39*d^2*e*(d + e*x)^
4*(d^2 - e^2*x^2)^(5/2)) - 6/(143*d^3*e*(d + e*x)^3*(d^2 - e^2*x^2)^(5/2)) - 16/(429*d^4*e*(d + e*x)^2*(d^2 -
e^2*x^2)^(5/2)) - 16/(429*d^5*e*(d + e*x)*(d^2 - e^2*x^2)^(5/2)) + (128*x)/(2145*d^9*(d^2 - e^2*x^2)^(3/2)) +
(256*x)/(2145*d^11*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)^5 \left (d^2-e^2 x^2\right )^{7/2}} \, dx &=-\frac{1}{15 d e (d+e x)^5 \left (d^2-e^2 x^2\right )^{5/2}}+\frac{2 \int \frac{1}{(d+e x)^4 \left (d^2-e^2 x^2\right )^{7/2}} \, dx}{3 d}\\ &=-\frac{1}{15 d e (d+e x)^5 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{2}{39 d^2 e (d+e x)^4 \left (d^2-e^2 x^2\right )^{5/2}}+\frac{6 \int \frac{1}{(d+e x)^3 \left (d^2-e^2 x^2\right )^{7/2}} \, dx}{13 d^2}\\ &=-\frac{1}{15 d e (d+e x)^5 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{2}{39 d^2 e (d+e x)^4 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{6}{143 d^3 e (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}+\frac{48 \int \frac{1}{(d+e x)^2 \left (d^2-e^2 x^2\right )^{7/2}} \, dx}{143 d^3}\\ &=-\frac{1}{15 d e (d+e x)^5 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{2}{39 d^2 e (d+e x)^4 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{6}{143 d^3 e (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{16}{429 d^4 e (d+e x)^2 \left (d^2-e^2 x^2\right )^{5/2}}+\frac{112 \int \frac{1}{(d+e x) \left (d^2-e^2 x^2\right )^{7/2}} \, dx}{429 d^4}\\ &=-\frac{1}{15 d e (d+e x)^5 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{2}{39 d^2 e (d+e x)^4 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{6}{143 d^3 e (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{16}{429 d^4 e (d+e x)^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{16}{429 d^5 e (d+e x) \left (d^2-e^2 x^2\right )^{5/2}}+\frac{32 \int \frac{1}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx}{143 d^5}\\ &=\frac{32 x}{715 d^7 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{1}{15 d e (d+e x)^5 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{2}{39 d^2 e (d+e x)^4 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{6}{143 d^3 e (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{16}{429 d^4 e (d+e x)^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{16}{429 d^5 e (d+e x) \left (d^2-e^2 x^2\right )^{5/2}}+\frac{128 \int \frac{1}{\left (d^2-e^2 x^2\right )^{5/2}} \, dx}{715 d^7}\\ &=\frac{32 x}{715 d^7 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{1}{15 d e (d+e x)^5 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{2}{39 d^2 e (d+e x)^4 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{6}{143 d^3 e (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{16}{429 d^4 e (d+e x)^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{16}{429 d^5 e (d+e x) \left (d^2-e^2 x^2\right )^{5/2}}+\frac{128 x}{2145 d^9 \left (d^2-e^2 x^2\right )^{3/2}}+\frac{256 \int \frac{1}{\left (d^2-e^2 x^2\right )^{3/2}} \, dx}{2145 d^9}\\ &=\frac{32 x}{715 d^7 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{1}{15 d e (d+e x)^5 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{2}{39 d^2 e (d+e x)^4 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{6}{143 d^3 e (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{16}{429 d^4 e (d+e x)^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{16}{429 d^5 e (d+e x) \left (d^2-e^2 x^2\right )^{5/2}}+\frac{128 x}{2145 d^9 \left (d^2-e^2 x^2\right )^{3/2}}+\frac{256 x}{2145 d^{11} \sqrt{d^2-e^2 x^2}}\\ \end{align*}

Mathematica [A]  time = 0.0965126, size = 148, normalized size = 0.62 \[ \frac{\sqrt{d^2-e^2 x^2} \left (1590 d^8 e^2 x^2+3760 d^7 e^3 x^3+1520 d^6 e^4 x^4-3744 d^5 e^5 x^5-4640 d^4 e^6 x^6-640 d^3 e^7 x^7+1920 d^2 e^8 x^8-370 d^9 e x-503 d^{10}+1280 d e^9 x^9+256 e^{10} x^{10}\right )}{2145 d^{11} e (d-e x)^3 (d+e x)^8} \]

Antiderivative was successfully verified.

[In]

Integrate[1/((d + e*x)^5*(d^2 - e^2*x^2)^(7/2)),x]

[Out]

(Sqrt[d^2 - e^2*x^2]*(-503*d^10 - 370*d^9*e*x + 1590*d^8*e^2*x^2 + 3760*d^7*e^3*x^3 + 1520*d^6*e^4*x^4 - 3744*
d^5*e^5*x^5 - 4640*d^4*e^6*x^6 - 640*d^3*e^7*x^7 + 1920*d^2*e^8*x^8 + 1280*d*e^9*x^9 + 256*e^10*x^10))/(2145*d
^11*e*(d - e*x)^3*(d + e*x)^8)

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Maple [A]  time = 0.048, size = 143, normalized size = 0.6 \begin{align*} -{\frac{ \left ( -ex+d \right ) \left ( -256\,{e}^{10}{x}^{10}-1280\,{e}^{9}{x}^{9}d-1920\,{e}^{8}{x}^{8}{d}^{2}+640\,{e}^{7}{x}^{7}{d}^{3}+4640\,{e}^{6}{x}^{6}{d}^{4}+3744\,{e}^{5}{x}^{5}{d}^{5}-1520\,{e}^{4}{x}^{4}{d}^{6}-3760\,{e}^{3}{x}^{3}{d}^{7}-1590\,{e}^{2}{x}^{2}{d}^{8}+370\,x{d}^{9}e+503\,{d}^{10} \right ) }{2145\,e{d}^{11} \left ( ex+d \right ) ^{4}} \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)^5/(-e^2*x^2+d^2)^(7/2),x)

[Out]

-1/2145*(-e*x+d)*(-256*e^10*x^10-1280*d*e^9*x^9-1920*d^2*e^8*x^8+640*d^3*e^7*x^7+4640*d^4*e^6*x^6+3744*d^5*e^5
*x^5-1520*d^6*e^4*x^4-3760*d^7*e^3*x^3-1590*d^8*e^2*x^2+370*d^9*e*x+503*d^10)/(e*x+d)^4/d^11/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)^5/(-e^2*x^2+d^2)^(7/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 11.6702, size = 863, normalized size = 3.63 \begin{align*} -\frac{503 \, e^{11} x^{11} + 2515 \, d e^{10} x^{10} + 3521 \, d^{2} e^{9} x^{9} - 2515 \, d^{3} e^{8} x^{8} - 11066 \, d^{4} e^{7} x^{7} - 7042 \, d^{5} e^{6} x^{6} + 7042 \, d^{6} e^{5} x^{5} + 11066 \, d^{7} e^{4} x^{4} + 2515 \, d^{8} e^{3} x^{3} - 3521 \, d^{9} e^{2} x^{2} - 2515 \, d^{10} e x - 503 \, d^{11} +{\left (256 \, e^{10} x^{10} + 1280 \, d e^{9} x^{9} + 1920 \, d^{2} e^{8} x^{8} - 640 \, d^{3} e^{7} x^{7} - 4640 \, d^{4} e^{6} x^{6} - 3744 \, d^{5} e^{5} x^{5} + 1520 \, d^{6} e^{4} x^{4} + 3760 \, d^{7} e^{3} x^{3} + 1590 \, d^{8} e^{2} x^{2} - 370 \, d^{9} e x - 503 \, d^{10}\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{2145 \,{\left (d^{11} e^{12} x^{11} + 5 \, d^{12} e^{11} x^{10} + 7 \, d^{13} e^{10} x^{9} - 5 \, d^{14} e^{9} x^{8} - 22 \, d^{15} e^{8} x^{7} - 14 \, d^{16} e^{7} x^{6} + 14 \, d^{17} e^{6} x^{5} + 22 \, d^{18} e^{5} x^{4} + 5 \, d^{19} e^{4} x^{3} - 7 \, d^{20} e^{3} x^{2} - 5 \, d^{21} e^{2} x - d^{22} 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)^(7/2),x, algorithm="fricas")

[Out]

-1/2145*(503*e^11*x^11 + 2515*d*e^10*x^10 + 3521*d^2*e^9*x^9 - 2515*d^3*e^8*x^8 - 11066*d^4*e^7*x^7 - 7042*d^5
*e^6*x^6 + 7042*d^6*e^5*x^5 + 11066*d^7*e^4*x^4 + 2515*d^8*e^3*x^3 - 3521*d^9*e^2*x^2 - 2515*d^10*e*x - 503*d^
11 + (256*e^10*x^10 + 1280*d*e^9*x^9 + 1920*d^2*e^8*x^8 - 640*d^3*e^7*x^7 - 4640*d^4*e^6*x^6 - 3744*d^5*e^5*x^
5 + 1520*d^6*e^4*x^4 + 3760*d^7*e^3*x^3 + 1590*d^8*e^2*x^2 - 370*d^9*e*x - 503*d^10)*sqrt(-e^2*x^2 + d^2))/(d^
11*e^12*x^11 + 5*d^12*e^11*x^10 + 7*d^13*e^10*x^9 - 5*d^14*e^9*x^8 - 22*d^15*e^8*x^7 - 14*d^16*e^7*x^6 + 14*d^
17*e^6*x^5 + 22*d^18*e^5*x^4 + 5*d^19*e^4*x^3 - 7*d^20*e^3*x^2 - 5*d^21*e^2*x - d^22*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)**5/(-e**2*x**2+d**2)**(7/2),x)

[Out]

Timed out

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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)^(7/2),x, algorithm="giac")

[Out]

Exception raised: RuntimeError