3.214 \(\int \frac{x}{(d+e x)^4 (d^2-e^2 x^2)^{7/2}} \, dx\)

Optimal. Leaf size=211 \[ \frac{512 x}{6435 d^9 e \sqrt{d^2-e^2 x^2}}+\frac{256 x}{6435 d^7 e \left (d^2-e^2 x^2\right )^{3/2}}+\frac{64 x}{2145 d^5 e \left (d^2-e^2 x^2\right )^{5/2}}-\frac{32}{1287 d^3 e^2 (d+e x) \left (d^2-e^2 x^2\right )^{5/2}}-\frac{32}{1287 d^2 e^2 (d+e x)^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{4}{143 d e^2 (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}+\frac{1}{13 e^2 (d+e x)^4 \left (d^2-e^2 x^2\right )^{5/2}} \]

[Out]

(64*x)/(2145*d^5*e*(d^2 - e^2*x^2)^(5/2)) + 1/(13*e^2*(d + e*x)^4*(d^2 - e^2*x^2)^(5/2)) - 4/(143*d*e^2*(d + e
*x)^3*(d^2 - e^2*x^2)^(5/2)) - 32/(1287*d^2*e^2*(d + e*x)^2*(d^2 - e^2*x^2)^(5/2)) - 32/(1287*d^3*e^2*(d + e*x
)*(d^2 - e^2*x^2)^(5/2)) + (256*x)/(6435*d^7*e*(d^2 - e^2*x^2)^(3/2)) + (512*x)/(6435*d^9*e*Sqrt[d^2 - e^2*x^2
])

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Rubi [A]  time = 0.104949, antiderivative size = 211, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 4, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.16, Rules used = {793, 659, 192, 191} \[ \frac{512 x}{6435 d^9 e \sqrt{d^2-e^2 x^2}}+\frac{256 x}{6435 d^7 e \left (d^2-e^2 x^2\right )^{3/2}}+\frac{64 x}{2145 d^5 e \left (d^2-e^2 x^2\right )^{5/2}}-\frac{32}{1287 d^3 e^2 (d+e x) \left (d^2-e^2 x^2\right )^{5/2}}-\frac{32}{1287 d^2 e^2 (d+e x)^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{4}{143 d e^2 (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}+\frac{1}{13 e^2 (d+e x)^4 \left (d^2-e^2 x^2\right )^{5/2}} \]

Antiderivative was successfully verified.

[In]

Int[x/((d + e*x)^4*(d^2 - e^2*x^2)^(7/2)),x]

[Out]

(64*x)/(2145*d^5*e*(d^2 - e^2*x^2)^(5/2)) + 1/(13*e^2*(d + e*x)^4*(d^2 - e^2*x^2)^(5/2)) - 4/(143*d*e^2*(d + e
*x)^3*(d^2 - e^2*x^2)^(5/2)) - 32/(1287*d^2*e^2*(d + e*x)^2*(d^2 - e^2*x^2)^(5/2)) - 32/(1287*d^3*e^2*(d + e*x
)*(d^2 - e^2*x^2)^(5/2)) + (256*x)/(6435*d^7*e*(d^2 - e^2*x^2)^(3/2)) + (512*x)/(6435*d^9*e*Sqrt[d^2 - e^2*x^2
])

Rule 793

Int[((d_) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((d*g - e*f)*(
d + e*x)^m*(a + c*x^2)^(p + 1))/(2*c*d*(m + p + 1)), x] + Dist[(m*(g*c*d + c*e*f) + 2*e*c*f*(p + 1))/(e*(2*c*d
)*(m + p + 1)), Int[(d + e*x)^(m + 1)*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g, m, p}, x] && EqQ[c*d^2
+ a*e^2, 0] && ((LtQ[m, -1] &&  !IGtQ[m + p + 1, 0]) || (LtQ[m, 0] && LtQ[p, -1]) || EqQ[m + 2*p + 2, 0]) &&
NeQ[m + p + 1, 0]

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{x}{(d+e x)^4 \left (d^2-e^2 x^2\right )^{7/2}} \, dx &=\frac{1}{13 e^2 (d+e x)^4 \left (d^2-e^2 x^2\right )^{5/2}}+\frac{4 \int \frac{1}{(d+e x)^3 \left (d^2-e^2 x^2\right )^{7/2}} \, dx}{13 e}\\ &=\frac{1}{13 e^2 (d+e x)^4 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{4}{143 d e^2 (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}+\frac{32 \int \frac{1}{(d+e x)^2 \left (d^2-e^2 x^2\right )^{7/2}} \, dx}{143 d e}\\ &=\frac{1}{13 e^2 (d+e x)^4 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{4}{143 d e^2 (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{32}{1287 d^2 e^2 (d+e x)^2 \left (d^2-e^2 x^2\right )^{5/2}}+\frac{224 \int \frac{1}{(d+e x) \left (d^2-e^2 x^2\right )^{7/2}} \, dx}{1287 d^2 e}\\ &=\frac{1}{13 e^2 (d+e x)^4 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{4}{143 d e^2 (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{32}{1287 d^2 e^2 (d+e x)^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{32}{1287 d^3 e^2 (d+e x) \left (d^2-e^2 x^2\right )^{5/2}}+\frac{64 \int \frac{1}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx}{429 d^3 e}\\ &=\frac{64 x}{2145 d^5 e \left (d^2-e^2 x^2\right )^{5/2}}+\frac{1}{13 e^2 (d+e x)^4 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{4}{143 d e^2 (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{32}{1287 d^2 e^2 (d+e x)^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{32}{1287 d^3 e^2 (d+e x) \left (d^2-e^2 x^2\right )^{5/2}}+\frac{256 \int \frac{1}{\left (d^2-e^2 x^2\right )^{5/2}} \, dx}{2145 d^5 e}\\ &=\frac{64 x}{2145 d^5 e \left (d^2-e^2 x^2\right )^{5/2}}+\frac{1}{13 e^2 (d+e x)^4 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{4}{143 d e^2 (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{32}{1287 d^2 e^2 (d+e x)^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{32}{1287 d^3 e^2 (d+e x) \left (d^2-e^2 x^2\right )^{5/2}}+\frac{256 x}{6435 d^7 e \left (d^2-e^2 x^2\right )^{3/2}}+\frac{512 \int \frac{1}{\left (d^2-e^2 x^2\right )^{3/2}} \, dx}{6435 d^7 e}\\ &=\frac{64 x}{2145 d^5 e \left (d^2-e^2 x^2\right )^{5/2}}+\frac{1}{13 e^2 (d+e x)^4 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{4}{143 d e^2 (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{32}{1287 d^2 e^2 (d+e x)^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{32}{1287 d^3 e^2 (d+e x) \left (d^2-e^2 x^2\right )^{5/2}}+\frac{256 x}{6435 d^7 e \left (d^2-e^2 x^2\right )^{3/2}}+\frac{512 x}{6435 d^9 e \sqrt{d^2-e^2 x^2}}\\ \end{align*}

Mathematica [A]  time = 0.0939344, size = 137, normalized size = 0.65 \[ \frac{\sqrt{d^2-e^2 x^2} \left (3200 d^7 e^2 x^2+4320 d^6 e^3 x^3-1280 d^5 e^4 x^4-6208 d^4 e^5 x^5-3072 d^3 e^6 x^6+1792 d^2 e^7 x^7-20 d^8 e x-5 d^9+2048 d e^8 x^8+512 e^9 x^9\right )}{6435 d^9 e^2 (d-e x)^3 (d+e x)^7} \]

Antiderivative was successfully verified.

[In]

Integrate[x/((d + e*x)^4*(d^2 - e^2*x^2)^(7/2)),x]

[Out]

(Sqrt[d^2 - e^2*x^2]*(-5*d^9 - 20*d^8*e*x + 3200*d^7*e^2*x^2 + 4320*d^6*e^3*x^3 - 1280*d^5*e^4*x^4 - 6208*d^4*
e^5*x^5 - 3072*d^3*e^6*x^6 + 1792*d^2*e^7*x^7 + 2048*d*e^8*x^8 + 512*e^9*x^9))/(6435*d^9*e^2*(d - e*x)^3*(d +
e*x)^7)

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Maple [A]  time = 0.055, size = 132, normalized size = 0.6 \begin{align*} -{\frac{ \left ( -ex+d \right ) \left ( -512\,{e}^{9}{x}^{9}-2048\,{e}^{8}{x}^{8}d-1792\,{e}^{7}{x}^{7}{d}^{2}+3072\,{e}^{6}{x}^{6}{d}^{3}+6208\,{e}^{5}{x}^{5}{d}^{4}+1280\,{e}^{4}{x}^{4}{d}^{5}-4320\,{e}^{3}{x}^{3}{d}^{6}-3200\,{x}^{2}{d}^{7}{e}^{2}+20\,x{d}^{8}e+5\,{d}^{9} \right ) }{6435\,{e}^{2}{d}^{9} \left ( ex+d \right ) ^{3}} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{-{\frac{7}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x/(e*x+d)^4/(-e^2*x^2+d^2)^(7/2),x)

[Out]

-1/6435*(-e*x+d)*(-512*e^9*x^9-2048*d*e^8*x^8-1792*d^2*e^7*x^7+3072*d^3*e^6*x^6+6208*d^4*e^5*x^5+1280*d^5*e^4*
x^4-4320*d^6*e^3*x^3-3200*d^7*e^2*x^2+20*d^8*e*x+5*d^9)/(e*x+d)^3/d^9/e^2/(-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(x/(e*x+d)^4/(-e^2*x^2+d^2)^(7/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 5.40865, size = 697, normalized size = 3.3 \begin{align*} -\frac{5 \, e^{10} x^{10} + 20 \, d e^{9} x^{9} + 15 \, d^{2} e^{8} x^{8} - 40 \, d^{3} e^{7} x^{7} - 70 \, d^{4} e^{6} x^{6} + 70 \, d^{6} e^{4} x^{4} + 40 \, d^{7} e^{3} x^{3} - 15 \, d^{8} e^{2} x^{2} - 20 \, d^{9} e x - 5 \, d^{10} +{\left (512 \, e^{9} x^{9} + 2048 \, d e^{8} x^{8} + 1792 \, d^{2} e^{7} x^{7} - 3072 \, d^{3} e^{6} x^{6} - 6208 \, d^{4} e^{5} x^{5} - 1280 \, d^{5} e^{4} x^{4} + 4320 \, d^{6} e^{3} x^{3} + 3200 \, d^{7} e^{2} x^{2} - 20 \, d^{8} e x - 5 \, d^{9}\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{6435 \,{\left (d^{9} e^{12} x^{10} + 4 \, d^{10} e^{11} x^{9} + 3 \, d^{11} e^{10} x^{8} - 8 \, d^{12} e^{9} x^{7} - 14 \, d^{13} e^{8} x^{6} + 14 \, d^{15} e^{6} x^{4} + 8 \, d^{16} e^{5} x^{3} - 3 \, d^{17} e^{4} x^{2} - 4 \, d^{18} e^{3} x - d^{19} e^{2}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(e*x+d)^4/(-e^2*x^2+d^2)^(7/2),x, algorithm="fricas")

[Out]

-1/6435*(5*e^10*x^10 + 20*d*e^9*x^9 + 15*d^2*e^8*x^8 - 40*d^3*e^7*x^7 - 70*d^4*e^6*x^6 + 70*d^6*e^4*x^4 + 40*d
^7*e^3*x^3 - 15*d^8*e^2*x^2 - 20*d^9*e*x - 5*d^10 + (512*e^9*x^9 + 2048*d*e^8*x^8 + 1792*d^2*e^7*x^7 - 3072*d^
3*e^6*x^6 - 6208*d^4*e^5*x^5 - 1280*d^5*e^4*x^4 + 4320*d^6*e^3*x^3 + 3200*d^7*e^2*x^2 - 20*d^8*e*x - 5*d^9)*sq
rt(-e^2*x^2 + d^2))/(d^9*e^12*x^10 + 4*d^10*e^11*x^9 + 3*d^11*e^10*x^8 - 8*d^12*e^9*x^7 - 14*d^13*e^8*x^6 + 14
*d^15*e^6*x^4 + 8*d^16*e^5*x^3 - 3*d^17*e^4*x^2 - 4*d^18*e^3*x - d^19*e^2)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x}{\left (- \left (- d + e x\right ) \left (d + e x\right )\right )^{\frac{7}{2}} \left (d + e x\right )^{4}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(e*x+d)**4/(-e**2*x**2+d**2)**(7/2),x)

[Out]

Integral(x/((-(-d + e*x)*(d + e*x))**(7/2)*(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}, \mathit{undef}, \mathit{undef}, 1\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(e*x+d)^4/(-e^2*x^2+d^2)^(7/2),x, algorithm="giac")

[Out]

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