3.845 \(\int \frac{1}{(d+e x)^4 (d^2-e^2 x^2)^{5/2}} \, dx\)

Optimal. Leaf size=181 \[ \frac{16 x}{99 d^8 \sqrt{d^2-e^2 x^2}}+\frac{8 x}{99 d^6 \left (d^2-e^2 x^2\right )^{3/2}}-\frac{2}{33 d^4 e (d+e x) \left (d^2-e^2 x^2\right )^{3/2}}-\frac{2}{33 d^3 e (d+e x)^2 \left (d^2-e^2 x^2\right )^{3/2}}-\frac{7}{99 d^2 e (d+e x)^3 \left (d^2-e^2 x^2\right )^{3/2}}-\frac{1}{11 d e (d+e x)^4 \left (d^2-e^2 x^2\right )^{3/2}} \]

[Out]

(8*x)/(99*d^6*(d^2 - e^2*x^2)^(3/2)) - 1/(11*d*e*(d + e*x)^4*(d^2 - e^2*x^2)^(3/2)) - 7/(99*d^2*e*(d + e*x)^3*
(d^2 - e^2*x^2)^(3/2)) - 2/(33*d^3*e*(d + e*x)^2*(d^2 - e^2*x^2)^(3/2)) - 2/(33*d^4*e*(d + e*x)*(d^2 - e^2*x^2
)^(3/2)) + (16*x)/(99*d^8*Sqrt[d^2 - e^2*x^2])

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Rubi [A]  time = 0.0747313, antiderivative size = 181, 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} \[ \frac{16 x}{99 d^8 \sqrt{d^2-e^2 x^2}}+\frac{8 x}{99 d^6 \left (d^2-e^2 x^2\right )^{3/2}}-\frac{2}{33 d^4 e (d+e x) \left (d^2-e^2 x^2\right )^{3/2}}-\frac{2}{33 d^3 e (d+e x)^2 \left (d^2-e^2 x^2\right )^{3/2}}-\frac{7}{99 d^2 e (d+e x)^3 \left (d^2-e^2 x^2\right )^{3/2}}-\frac{1}{11 d e (d+e x)^4 \left (d^2-e^2 x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

Int[1/((d + e*x)^4*(d^2 - e^2*x^2)^(5/2)),x]

[Out]

(8*x)/(99*d^6*(d^2 - e^2*x^2)^(3/2)) - 1/(11*d*e*(d + e*x)^4*(d^2 - e^2*x^2)^(3/2)) - 7/(99*d^2*e*(d + e*x)^3*
(d^2 - e^2*x^2)^(3/2)) - 2/(33*d^3*e*(d + e*x)^2*(d^2 - e^2*x^2)^(3/2)) - 2/(33*d^4*e*(d + e*x)*(d^2 - e^2*x^2
)^(3/2)) + (16*x)/(99*d^8*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)^4 \left (d^2-e^2 x^2\right )^{5/2}} \, dx &=-\frac{1}{11 d e (d+e x)^4 \left (d^2-e^2 x^2\right )^{3/2}}+\frac{7 \int \frac{1}{(d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}} \, dx}{11 d}\\ &=-\frac{1}{11 d e (d+e x)^4 \left (d^2-e^2 x^2\right )^{3/2}}-\frac{7}{99 d^2 e (d+e x)^3 \left (d^2-e^2 x^2\right )^{3/2}}+\frac{14 \int \frac{1}{(d+e x)^2 \left (d^2-e^2 x^2\right )^{5/2}} \, dx}{33 d^2}\\ &=-\frac{1}{11 d e (d+e x)^4 \left (d^2-e^2 x^2\right )^{3/2}}-\frac{7}{99 d^2 e (d+e x)^3 \left (d^2-e^2 x^2\right )^{3/2}}-\frac{2}{33 d^3 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}{33 d^3}\\ &=-\frac{1}{11 d e (d+e x)^4 \left (d^2-e^2 x^2\right )^{3/2}}-\frac{7}{99 d^2 e (d+e x)^3 \left (d^2-e^2 x^2\right )^{3/2}}-\frac{2}{33 d^3 e (d+e x)^2 \left (d^2-e^2 x^2\right )^{3/2}}-\frac{2}{33 d^4 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}{33 d^4}\\ &=\frac{8 x}{99 d^6 \left (d^2-e^2 x^2\right )^{3/2}}-\frac{1}{11 d e (d+e x)^4 \left (d^2-e^2 x^2\right )^{3/2}}-\frac{7}{99 d^2 e (d+e x)^3 \left (d^2-e^2 x^2\right )^{3/2}}-\frac{2}{33 d^3 e (d+e x)^2 \left (d^2-e^2 x^2\right )^{3/2}}-\frac{2}{33 d^4 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}{99 d^6}\\ &=\frac{8 x}{99 d^6 \left (d^2-e^2 x^2\right )^{3/2}}-\frac{1}{11 d e (d+e x)^4 \left (d^2-e^2 x^2\right )^{3/2}}-\frac{7}{99 d^2 e (d+e x)^3 \left (d^2-e^2 x^2\right )^{3/2}}-\frac{2}{33 d^3 e (d+e x)^2 \left (d^2-e^2 x^2\right )^{3/2}}-\frac{2}{33 d^4 e (d+e x) \left (d^2-e^2 x^2\right )^{3/2}}+\frac{16 x}{99 d^8 \sqrt{d^2-e^2 x^2}}\\ \end{align*}

Mathematica [A]  time = 0.0675843, size = 115, normalized size = 0.64 \[ -\frac{\sqrt{d^2-e^2 x^2} \left (-72 d^5 e^2 x^2-122 d^4 e^3 x^3-32 d^3 e^4 x^4+72 d^2 e^5 x^5+13 d^6 e x+28 d^7+64 d e^6 x^6+16 e^7 x^7\right )}{99 d^8 e (d-e x)^2 (d+e x)^6} \]

Antiderivative was successfully verified.

[In]

Integrate[1/((d + e*x)^4*(d^2 - e^2*x^2)^(5/2)),x]

[Out]

-(Sqrt[d^2 - e^2*x^2]*(28*d^7 + 13*d^6*e*x - 72*d^5*e^2*x^2 - 122*d^4*e^3*x^3 - 32*d^3*e^4*x^4 + 72*d^2*e^5*x^
5 + 64*d*e^6*x^6 + 16*e^7*x^7))/(99*d^8*e*(d - e*x)^2*(d + e*x)^6)

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Maple [A]  time = 0.047, size = 110, normalized size = 0.6 \begin{align*} -{\frac{ \left ( -ex+d \right ) \left ( 16\,{e}^{7}{x}^{7}+64\,{e}^{6}{x}^{6}d+72\,{e}^{5}{x}^{5}{d}^{2}-32\,{e}^{4}{x}^{4}{d}^{3}-122\,{e}^{3}{x}^{3}{d}^{4}-72\,{e}^{2}{x}^{2}{d}^{5}+13\,x{d}^{6}e+28\,{d}^{7} \right ) }{99\,e{d}^{8} \left ( ex+d \right ) ^{3}} \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)^4/(-e^2*x^2+d^2)^(5/2),x)

[Out]

-1/99*(-e*x+d)*(16*e^7*x^7+64*d*e^6*x^6+72*d^2*e^5*x^5-32*d^3*e^4*x^4-122*d^4*e^3*x^3-72*d^5*e^2*x^2+13*d^6*e*
x+28*d^7)/(e*x+d)^3/d^8/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)^4/(-e^2*x^2+d^2)^(5/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 3.69455, size = 582, normalized size = 3.22 \begin{align*} -\frac{28 \, e^{8} x^{8} + 112 \, d e^{7} x^{7} + 112 \, d^{2} e^{6} x^{6} - 112 \, d^{3} e^{5} x^{5} - 280 \, d^{4} e^{4} x^{4} - 112 \, d^{5} e^{3} x^{3} + 112 \, d^{6} e^{2} x^{2} + 112 \, d^{7} e x + 28 \, d^{8} +{\left (16 \, e^{7} x^{7} + 64 \, d e^{6} x^{6} + 72 \, d^{2} e^{5} x^{5} - 32 \, d^{3} e^{4} x^{4} - 122 \, d^{4} e^{3} x^{3} - 72 \, d^{5} e^{2} x^{2} + 13 \, d^{6} e x + 28 \, d^{7}\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{99 \,{\left (d^{8} e^{9} x^{8} + 4 \, d^{9} e^{8} x^{7} + 4 \, d^{10} e^{7} x^{6} - 4 \, d^{11} e^{6} x^{5} - 10 \, d^{12} e^{5} x^{4} - 4 \, d^{13} e^{4} x^{3} + 4 \, d^{14} e^{3} x^{2} + 4 \, d^{15} e^{2} x + d^{16} 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)^(5/2),x, algorithm="fricas")

[Out]

-1/99*(28*e^8*x^8 + 112*d*e^7*x^7 + 112*d^2*e^6*x^6 - 112*d^3*e^5*x^5 - 280*d^4*e^4*x^4 - 112*d^5*e^3*x^3 + 11
2*d^6*e^2*x^2 + 112*d^7*e*x + 28*d^8 + (16*e^7*x^7 + 64*d*e^6*x^6 + 72*d^2*e^5*x^5 - 32*d^3*e^4*x^4 - 122*d^4*
e^3*x^3 - 72*d^5*e^2*x^2 + 13*d^6*e*x + 28*d^7)*sqrt(-e^2*x^2 + d^2))/(d^8*e^9*x^8 + 4*d^9*e^8*x^7 + 4*d^10*e^
7*x^6 - 4*d^11*e^6*x^5 - 10*d^12*e^5*x^4 - 4*d^13*e^4*x^3 + 4*d^14*e^3*x^2 + 4*d^15*e^2*x + d^16*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 )^{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)**(5/2),x)

[Out]

Integral(1/((-(-d + e*x)*(d + e*x))**(5/2)*(d + e*x)**4), 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)^4/(-e^2*x^2+d^2)^(5/2),x, algorithm="giac")

[Out]

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