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

Optimal. Leaf size=209 \[ -\frac{d}{13 e^3 (d+e x)^4 \left (d^2-e^2 x^2\right )^{5/2}}+\frac{17}{143 e^3 (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{7}{1287 d e^3 (d+e x)^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{7}{1287 d^2 e^3 (d+e x) \left (d^2-e^2 x^2\right )^{5/2}}+\frac{14 x}{2145 d^4 e^2 \left (d^2-e^2 x^2\right )^{5/2}}+\frac{56 x}{6435 d^6 e^2 \left (d^2-e^2 x^2\right )^{3/2}}+\frac{112 x}{6435 d^8 e^2 \sqrt{d^2-e^2 x^2}} \]

[Out]

(14*x)/(2145*d^4*e^2*(d^2 - e^2*x^2)^(5/2)) - d/(13*e^3*(d + e*x)^4*(d^2 - e^2*x^2)^(5/2)) + 17/(143*e^3*(d +
e*x)^3*(d^2 - e^2*x^2)^(5/2)) - 7/(1287*d*e^3*(d + e*x)^2*(d^2 - e^2*x^2)^(5/2)) - 7/(1287*d^2*e^3*(d + e*x)*(
d^2 - e^2*x^2)^(5/2)) + (56*x)/(6435*d^6*e^2*(d^2 - e^2*x^2)^(3/2)) + (112*x)/(6435*d^8*e^2*Sqrt[d^2 - e^2*x^2
])

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Rubi [A]  time = 0.207456, antiderivative size = 209, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.185, Rules used = {1639, 793, 659, 192, 191} \[ -\frac{d}{13 e^3 (d+e x)^4 \left (d^2-e^2 x^2\right )^{5/2}}+\frac{17}{143 e^3 (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{7}{1287 d e^3 (d+e x)^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{7}{1287 d^2 e^3 (d+e x) \left (d^2-e^2 x^2\right )^{5/2}}+\frac{14 x}{2145 d^4 e^2 \left (d^2-e^2 x^2\right )^{5/2}}+\frac{56 x}{6435 d^6 e^2 \left (d^2-e^2 x^2\right )^{3/2}}+\frac{112 x}{6435 d^8 e^2 \sqrt{d^2-e^2 x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(14*x)/(2145*d^4*e^2*(d^2 - e^2*x^2)^(5/2)) - d/(13*e^3*(d + e*x)^4*(d^2 - e^2*x^2)^(5/2)) + 17/(143*e^3*(d +
e*x)^3*(d^2 - e^2*x^2)^(5/2)) - 7/(1287*d*e^3*(d + e*x)^2*(d^2 - e^2*x^2)^(5/2)) - 7/(1287*d^2*e^3*(d + e*x)*(
d^2 - e^2*x^2)^(5/2)) + (56*x)/(6435*d^6*e^2*(d^2 - e^2*x^2)^(3/2)) + (112*x)/(6435*d^8*e^2*Sqrt[d^2 - e^2*x^2
])

Rule 1639

Int[(Pq_)*((d_) + (e_.)*(x_))^(m_.)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{q = Expon[Pq, x], f = Coeff
[Pq, x, Expon[Pq, x]]}, Simp[(f*(d + e*x)^(m + q - 1)*(a + c*x^2)^(p + 1))/(c*e^(q - 1)*(m + q + 2*p + 1)), x]
+ Dist[1/(c*e^q*(m + q + 2*p + 1)), Int[(d + e*x)^m*(a + c*x^2)^p*ExpandToSum[c*e^q*(m + q + 2*p + 1)*Pq - c*
f*(m + q + 2*p + 1)*(d + e*x)^q - 2*e*f*(m + p + q)*(d + e*x)^(q - 2)*(a*e - c*d*x), x], x], x] /; NeQ[m + q +
2*p + 1, 0]] /; FreeQ[{a, c, d, e, m, p}, x] && PolyQ[Pq, x] && EqQ[c*d^2 + a*e^2, 0] &&  !IGtQ[m, 0]

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^2}{(d+e x)^4 \left (d^2-e^2 x^2\right )^{7/2}} \, dx &=\frac{1}{8 e^3 (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}+\frac{\int \frac{3 d^2 e^2-5 d e^3 x}{(d+e x)^4 \left (d^2-e^2 x^2\right )^{7/2}} \, dx}{8 e^4}\\ &=-\frac{d}{13 e^3 (d+e x)^4 \left (d^2-e^2 x^2\right )^{5/2}}+\frac{1}{8 e^3 (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}+\frac{(7 d) \int \frac{1}{(d+e x)^3 \left (d^2-e^2 x^2\right )^{7/2}} \, dx}{104 e^2}\\ &=-\frac{d}{13 e^3 (d+e x)^4 \left (d^2-e^2 x^2\right )^{5/2}}+\frac{17}{143 e^3 (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}+\frac{7 \int \frac{1}{(d+e x)^2 \left (d^2-e^2 x^2\right )^{7/2}} \, dx}{143 e^2}\\ &=-\frac{d}{13 e^3 (d+e x)^4 \left (d^2-e^2 x^2\right )^{5/2}}+\frac{17}{143 e^3 (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{7}{1287 d e^3 (d+e x)^2 \left (d^2-e^2 x^2\right )^{5/2}}+\frac{49 \int \frac{1}{(d+e x) \left (d^2-e^2 x^2\right )^{7/2}} \, dx}{1287 d e^2}\\ &=-\frac{d}{13 e^3 (d+e x)^4 \left (d^2-e^2 x^2\right )^{5/2}}+\frac{17}{143 e^3 (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{7}{1287 d e^3 (d+e x)^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{7}{1287 d^2 e^3 (d+e x) \left (d^2-e^2 x^2\right )^{5/2}}+\frac{14 \int \frac{1}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx}{429 d^2 e^2}\\ &=\frac{14 x}{2145 d^4 e^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{d}{13 e^3 (d+e x)^4 \left (d^2-e^2 x^2\right )^{5/2}}+\frac{17}{143 e^3 (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{7}{1287 d e^3 (d+e x)^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{7}{1287 d^2 e^3 (d+e x) \left (d^2-e^2 x^2\right )^{5/2}}+\frac{56 \int \frac{1}{\left (d^2-e^2 x^2\right )^{5/2}} \, dx}{2145 d^4 e^2}\\ &=\frac{14 x}{2145 d^4 e^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{d}{13 e^3 (d+e x)^4 \left (d^2-e^2 x^2\right )^{5/2}}+\frac{17}{143 e^3 (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{7}{1287 d e^3 (d+e x)^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{7}{1287 d^2 e^3 (d+e x) \left (d^2-e^2 x^2\right )^{5/2}}+\frac{56 x}{6435 d^6 e^2 \left (d^2-e^2 x^2\right )^{3/2}}+\frac{112 \int \frac{1}{\left (d^2-e^2 x^2\right )^{3/2}} \, dx}{6435 d^6 e^2}\\ &=\frac{14 x}{2145 d^4 e^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{d}{13 e^3 (d+e x)^4 \left (d^2-e^2 x^2\right )^{5/2}}+\frac{17}{143 e^3 (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{7}{1287 d e^3 (d+e x)^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{7}{1287 d^2 e^3 (d+e x) \left (d^2-e^2 x^2\right )^{5/2}}+\frac{56 x}{6435 d^6 e^2 \left (d^2-e^2 x^2\right )^{3/2}}+\frac{112 x}{6435 d^8 e^2 \sqrt{d^2-e^2 x^2}}\\ \end{align*}

Mathematica [A]  time = 0.111982, size = 137, normalized size = 0.66 \[ \frac{\sqrt{d^2-e^2 x^2} \left (700 d^7 e^2 x^2+945 d^6 e^3 x^3-280 d^5 e^4 x^4-1358 d^4 e^5 x^5-672 d^3 e^6 x^6+392 d^2 e^7 x^7+800 d^8 e x+200 d^9+448 d e^8 x^8+112 e^9 x^9\right )}{6435 d^8 e^3 (d-e x)^3 (d+e x)^7} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[d^2 - e^2*x^2]*(200*d^9 + 800*d^8*e*x + 700*d^7*e^2*x^2 + 945*d^6*e^3*x^3 - 280*d^5*e^4*x^4 - 1358*d^4*e
^5*x^5 - 672*d^3*e^6*x^6 + 392*d^2*e^7*x^7 + 448*d*e^8*x^8 + 112*e^9*x^9))/(6435*d^8*e^3*(d - e*x)^3*(d + e*x)
^7)

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Maple [A]  time = 0.056, size = 132, normalized size = 0.6 \begin{align*}{\frac{ \left ( -ex+d \right ) \left ( 112\,{e}^{9}{x}^{9}+448\,{e}^{8}{x}^{8}d+392\,{e}^{7}{x}^{7}{d}^{2}-672\,{e}^{6}{x}^{6}{d}^{3}-1358\,{e}^{5}{x}^{5}{d}^{4}-280\,{e}^{4}{x}^{4}{d}^{5}+945\,{x}^{3}{d}^{6}{e}^{3}+700\,{x}^{2}{d}^{7}{e}^{2}+800\,x{d}^{8}e+200\,{d}^{9} \right ) }{6435\,{e}^{3}{d}^{8} \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^2/(e*x+d)^4/(-e^2*x^2+d^2)^(7/2),x)

[Out]

1/6435*(-e*x+d)*(112*e^9*x^9+448*d*e^8*x^8+392*d^2*e^7*x^7-672*d^3*e^6*x^6-1358*d^4*e^5*x^5-280*d^5*e^4*x^4+94
5*d^6*e^3*x^3+700*d^7*e^2*x^2+800*d^8*e*x+200*d^9)/(e*x+d)^3/d^8/e^3/(-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^2/(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 = 4.99928, size = 713, normalized size = 3.41 \begin{align*} \frac{200 \, e^{10} x^{10} + 800 \, d e^{9} x^{9} + 600 \, d^{2} e^{8} x^{8} - 1600 \, d^{3} e^{7} x^{7} - 2800 \, d^{4} e^{6} x^{6} + 2800 \, d^{6} e^{4} x^{4} + 1600 \, d^{7} e^{3} x^{3} - 600 \, d^{8} e^{2} x^{2} - 800 \, d^{9} e x - 200 \, d^{10} -{\left (112 \, e^{9} x^{9} + 448 \, d e^{8} x^{8} + 392 \, d^{2} e^{7} x^{7} - 672 \, d^{3} e^{6} x^{6} - 1358 \, d^{4} e^{5} x^{5} - 280 \, d^{5} e^{4} x^{4} + 945 \, d^{6} e^{3} x^{3} + 700 \, d^{7} e^{2} x^{2} + 800 \, d^{8} e x + 200 \, d^{9}\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{6435 \,{\left (d^{8} e^{13} x^{10} + 4 \, d^{9} e^{12} x^{9} + 3 \, d^{10} e^{11} x^{8} - 8 \, d^{11} e^{10} x^{7} - 14 \, d^{12} e^{9} x^{6} + 14 \, d^{14} e^{7} x^{4} + 8 \, d^{15} e^{6} x^{3} - 3 \, d^{16} e^{5} x^{2} - 4 \, d^{17} e^{4} x - d^{18} e^{3}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6435*(200*e^10*x^10 + 800*d*e^9*x^9 + 600*d^2*e^8*x^8 - 1600*d^3*e^7*x^7 - 2800*d^4*e^6*x^6 + 2800*d^6*e^4*x
^4 + 1600*d^7*e^3*x^3 - 600*d^8*e^2*x^2 - 800*d^9*e*x - 200*d^10 - (112*e^9*x^9 + 448*d*e^8*x^8 + 392*d^2*e^7*
x^7 - 672*d^3*e^6*x^6 - 1358*d^4*e^5*x^5 - 280*d^5*e^4*x^4 + 945*d^6*e^3*x^3 + 700*d^7*e^2*x^2 + 800*d^8*e*x +
200*d^9)*sqrt(-e^2*x^2 + d^2))/(d^8*e^13*x^10 + 4*d^9*e^12*x^9 + 3*d^10*e^11*x^8 - 8*d^11*e^10*x^7 - 14*d^12*
e^9*x^6 + 14*d^14*e^7*x^4 + 8*d^15*e^6*x^3 - 3*d^16*e^5*x^2 - 4*d^17*e^4*x - d^18*e^3)

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

[Out]

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

[Out]

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