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

Optimal. Leaf size=95 \[ -\frac {x^2}{5 d e (d+e x) \left (d^2-e^2 x^2\right )^{3/2}}+\frac {2 (d+e x)}{15 d e^3 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {2 x}{15 d^3 e^2 \sqrt {d^2-e^2 x^2}} \]

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Rubi [A]  time = 0.05, antiderivative size = 95, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {855, 778, 191} \begin {gather*} -\frac {x^2}{5 d e (d+e x) \left (d^2-e^2 x^2\right )^{3/2}}-\frac {2 x}{15 d^3 e^2 \sqrt {d^2-e^2 x^2}}+\frac {2 (d+e x)}{15 d e^3 \left (d^2-e^2 x^2\right )^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-x^2/(5*d*e*(d + e*x)*(d^2 - e^2*x^2)^(3/2)) + (2*(d + e*x))/(15*d*e^3*(d^2 - e^2*x^2)^(3/2)) - (2*x)/(15*d^3*
e^2*Sqrt[d^2 - e^2*x^2])

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]

Rule 778

Int[((d_.) + (e_.)*(x_))*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((a*(e*f + d*g) -
(c*d*f - a*e*g)*x)*(a + c*x^2)^(p + 1))/(2*a*c*(p + 1)), x] - Dist[(a*e*g - c*d*f*(2*p + 3))/(2*a*c*(p + 1)),
Int[(a + c*x^2)^(p + 1), x], x] /; FreeQ[{a, c, d, e, f, g}, x] && LtQ[p, -1]

Rule 855

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

Rubi steps

\begin {align*} \int \frac {x^2}{(d+e x) \left (d^2-e^2 x^2\right )^{5/2}} \, dx &=-\frac {x^2}{5 d e (d+e x) \left (d^2-e^2 x^2\right )^{3/2}}+\frac {\int \frac {x (2 d+2 e x)}{\left (d^2-e^2 x^2\right )^{5/2}} \, dx}{5 d e}\\ &=-\frac {x^2}{5 d e (d+e x) \left (d^2-e^2 x^2\right )^{3/2}}+\frac {2 (d+e x)}{15 d e^3 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {2 \int \frac {1}{\left (d^2-e^2 x^2\right )^{3/2}} \, dx}{15 d e^2}\\ &=-\frac {x^2}{5 d e (d+e x) \left (d^2-e^2 x^2\right )^{3/2}}+\frac {2 (d+e x)}{15 d e^3 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {2 x}{15 d^3 e^2 \sqrt {d^2-e^2 x^2}}\\ \end {align*}

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Mathematica [A]  time = 0.06, size = 82, normalized size = 0.86 \begin {gather*} \frac {\sqrt {d^2-e^2 x^2} \left (2 d^4+2 d^3 e x-3 d^2 e^2 x^2+2 d e^3 x^3+2 e^4 x^4\right )}{15 d^3 e^3 (d-e x)^2 (d+e x)^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[d^2 - e^2*x^2]*(2*d^4 + 2*d^3*e*x - 3*d^2*e^2*x^2 + 2*d*e^3*x^3 + 2*e^4*x^4))/(15*d^3*e^3*(d - e*x)^2*(d
+ e*x)^3)

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IntegrateAlgebraic [A]  time = 0.44, size = 82, normalized size = 0.86 \begin {gather*} \frac {\sqrt {d^2-e^2 x^2} \left (2 d^4+2 d^3 e x-3 d^2 e^2 x^2+2 d e^3 x^3+2 e^4 x^4\right )}{15 d^3 e^3 (d-e x)^2 (d+e x)^3} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[x^2/((d + e*x)*(d^2 - e^2*x^2)^(5/2)),x]

[Out]

(Sqrt[d^2 - e^2*x^2]*(2*d^4 + 2*d^3*e*x - 3*d^2*e^2*x^2 + 2*d*e^3*x^3 + 2*e^4*x^4))/(15*d^3*e^3*(d - e*x)^2*(d
+ e*x)^3)

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fricas [B]  time = 0.41, size = 170, normalized size = 1.79 \begin {gather*} \frac {2 \, e^{5} x^{5} + 2 \, d e^{4} x^{4} - 4 \, d^{2} e^{3} x^{3} - 4 \, d^{3} e^{2} x^{2} + 2 \, d^{4} e x + 2 \, d^{5} + {\left (2 \, e^{4} x^{4} + 2 \, d e^{3} x^{3} - 3 \, d^{2} e^{2} x^{2} + 2 \, d^{3} e x + 2 \, d^{4}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{15 \, {\left (d^{3} e^{8} x^{5} + d^{4} e^{7} x^{4} - 2 \, d^{5} e^{6} x^{3} - 2 \, d^{6} e^{5} x^{2} + d^{7} e^{4} x + d^{8} e^{3}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2/(e*x+d)/(-e^2*x^2+d^2)^(5/2),x, algorithm="fricas")

[Out]

1/15*(2*e^5*x^5 + 2*d*e^4*x^4 - 4*d^2*e^3*x^3 - 4*d^3*e^2*x^2 + 2*d^4*e*x + 2*d^5 + (2*e^4*x^4 + 2*d*e^3*x^3 -
3*d^2*e^2*x^2 + 2*d^3*e*x + 2*d^4)*sqrt(-e^2*x^2 + d^2))/(d^3*e^8*x^5 + d^4*e^7*x^4 - 2*d^5*e^6*x^3 - 2*d^6*e
^5*x^2 + d^7*e^4*x + d^8*e^3)

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giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,x):;OUTPUT:Unab
le to transpose Error: Bad Argument Value

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maple [A]  time = 0.01, size = 70, normalized size = 0.74 \begin {gather*} \frac {\left (-e x +d \right ) \left (2 x^{4} e^{4}+2 x^{3} d \,e^{3}-3 d^{2} x^{2} e^{2}+2 d^{3} x e +2 d^{4}\right )}{15 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}} d^{3} e^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2/(e*x+d)/(-e^2*x^2+d^2)^(5/2),x)

[Out]

1/15*(-e*x+d)*(2*e^4*x^4+2*d*e^3*x^3-3*d^2*e^2*x^2+2*d^3*e*x+2*d^4)/d^3/e^3/(-e^2*x^2+d^2)^(5/2)

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maxima [A]  time = 0.48, size = 110, normalized size = 1.16 \begin {gather*} -\frac {d}{5 \, {\left ({\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} e^{4} x + {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} d e^{3}\right )}} - \frac {x}{15 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} d e^{2}} + \frac {1}{3 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} e^{3}} - \frac {2 \, x}{15 \, \sqrt {-e^{2} x^{2} + d^{2}} d^{3} e^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2/(e*x+d)/(-e^2*x^2+d^2)^(5/2),x, algorithm="maxima")

[Out]

-1/5*d/((-e^2*x^2 + d^2)^(3/2)*e^4*x + (-e^2*x^2 + d^2)^(3/2)*d*e^3) - 1/15*x/((-e^2*x^2 + d^2)^(3/2)*d*e^2) +
1/3/((-e^2*x^2 + d^2)^(3/2)*e^3) - 2/15*x/(sqrt(-e^2*x^2 + d^2)*d^3*e^2)

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mupad [B]  time = 2.79, size = 78, normalized size = 0.82 \begin {gather*} \frac {\sqrt {d^2-e^2\,x^2}\,\left (2\,d^4+2\,d^3\,e\,x-3\,d^2\,e^2\,x^2+2\,d\,e^3\,x^3+2\,e^4\,x^4\right )}{15\,d^3\,e^3\,{\left (d+e\,x\right )}^3\,{\left (d-e\,x\right )}^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2/((d^2 - e^2*x^2)^(5/2)*(d + e*x)),x)

[Out]

((d^2 - e^2*x^2)^(1/2)*(2*d^4 + 2*e^4*x^4 + 2*d*e^3*x^3 - 3*d^2*e^2*x^2 + 2*d^3*e*x))/(15*d^3*e^3*(d + e*x)^3*
(d - e*x)^2)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{2}}{\left (- \left (- d + e x\right ) \left (d + e x\right )\right )^{\frac {5}{2}} \left (d + e x\right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**2/(e*x+d)/(-e**2*x**2+d**2)**(5/2),x)

[Out]

Integral(x**2/((-(-d + e*x)*(d + e*x))**(5/2)*(d + e*x)), x)

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