3.487 \(\int \frac{x^2}{(d+e x) (a d e+(c d^2+a e^2) x+c d e x^2)^{5/2}} \, dx\)

Optimal. Leaf size=259 \[ \frac{8 \left (5 a^2 e^4+10 a c d^2 e^2+c^2 d^4\right ) \left (a e^2+c d^2+2 c d e x\right )}{15 e \left (c d^2-a e^2\right )^5 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}-\frac{8 \left (x \left (a^2 c d^2 e^4-2 a^3 e^6+c^3 d^6\right )+a d e \left (c d^2-a e^2\right ) \left (3 a e^2+c d^2\right )\right )}{15 e \left (c d^2-a e^2\right )^4 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}+\frac{2 x^2}{5 (d+e x) \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}} \]

[Out]

(2*x^2)/(5*(c*d^2 - a*e^2)*(d + e*x)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)) - (8*(a*d*e*(c*d^2 - a*e^2
)*(c*d^2 + 3*a*e^2) + (c^3*d^6 + a^2*c*d^2*e^4 - 2*a^3*e^6)*x))/(15*e*(c*d^2 - a*e^2)^4*(a*d*e + (c*d^2 + a*e^
2)*x + c*d*e*x^2)^(3/2)) + (8*(c^2*d^4 + 10*a*c*d^2*e^2 + 5*a^2*e^4)*(c*d^2 + a*e^2 + 2*c*d*e*x))/(15*e*(c*d^2
- a*e^2)^5*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])

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Rubi [A]  time = 0.235826, antiderivative size = 259, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 40, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.075, Rules used = {854, 777, 613} \[ \frac{8 \left (5 a^2 e^4+10 a c d^2 e^2+c^2 d^4\right ) \left (a e^2+c d^2+2 c d e x\right )}{15 e \left (c d^2-a e^2\right )^5 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}-\frac{8 \left (x \left (a^2 c d^2 e^4-2 a^3 e^6+c^3 d^6\right )+a d e \left (c d^2-a e^2\right ) \left (3 a e^2+c d^2\right )\right )}{15 e \left (c d^2-a e^2\right )^4 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}+\frac{2 x^2}{5 (d+e x) \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*x^2)/(5*(c*d^2 - a*e^2)*(d + e*x)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)) - (8*(a*d*e*(c*d^2 - a*e^2
)*(c*d^2 + 3*a*e^2) + (c^3*d^6 + a^2*c*d^2*e^4 - 2*a^3*e^6)*x))/(15*e*(c*d^2 - a*e^2)^4*(a*d*e + (c*d^2 + a*e^
2)*x + c*d*e*x^2)^(3/2)) + (8*(c^2*d^4 + 10*a*c*d^2*e^2 + 5*a^2*e^4)*(c*d^2 + a*e^2 + 2*c*d*e*x))/(15*e*(c*d^2
- a*e^2)^5*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])

Rule 854

Int[(((f_.) + (g_.)*(x_))^(n_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_))/((d_) + (e_.)*(x_)), x_Symbol] :> -Si
mp[((2*c*d - b*e)*(f + g*x)^n*(a + b*x + c*x^2)^(p + 1))/(e*p*(b^2 - 4*a*c)*(d + e*x)), x] - Dist[1/(d*e*p*(b^
2 - 4*a*c)), Int[(f + g*x)^(n - 1)*(a + b*x + c*x^2)^p*Simp[b*(a*e*g*n - c*d*f*(2*p + 1)) - 2*a*c*(d*g*n - e*f
*(2*p + 1)) - c*g*(b*d - 2*a*e)*(n + 2*p + 1)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[e*f - d*
g, 0] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a*e^2, 0] &&  !IntegerQ[p] && IGtQ[n, 0] && ILtQ[n + 2*p,
0]

Rule 777

Int[((d_.) + (e_.)*(x_))*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> -Simp[((2
*a*c*(e*f + d*g) - b*(c*d*f + a*e*g) - (b^2*e*g - b*c*(e*f + d*g) + 2*c*(c*d*f - a*e*g))*x)*(a + b*x + c*x^2)^
(p + 1))/(c*(p + 1)*(b^2 - 4*a*c)), x] - Dist[(b^2*e*g*(p + 2) - 2*a*c*e*g + c*(2*c*d*f - b*(e*f + d*g))*(2*p
+ 3))/(c*(p + 1)*(b^2 - 4*a*c)), Int[(a + b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && N
eQ[b^2 - 4*a*c, 0] && LtQ[p, -1]

Rule 613

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-3/2), x_Symbol] :> Simp[(-2*(b + 2*c*x))/((b^2 - 4*a*c)*Sqrt[a + b*x
+ c*x^2]), x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rubi steps

\begin{align*} \int \frac{x^2}{(d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx &=\frac{2 x^2}{5 \left (c d^2-a e^2\right ) (d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}+\frac{2 \int \frac{x \left (-2 a d e^2 \left (c d^2-a e^2\right )+2 c d^2 e \left (c d^2-a e^2\right ) x\right )}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx}{5 d e \left (c d^2-a e^2\right )^2}\\ &=\frac{2 x^2}{5 \left (c d^2-a e^2\right ) (d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}-\frac{8 \left (a d e \left (c d^2-a e^2\right ) \left (c d^2+3 a e^2\right )+\left (c^3 d^6+a^2 c d^2 e^4-2 a^3 e^6\right ) x\right )}{15 e \left (c d^2-a e^2\right )^4 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}-\frac{\left (4 \left (c^2 d^4+10 a c d^2 e^2+5 a^2 e^4\right )\right ) \int \frac{1}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx}{15 e \left (c d^2-a e^2\right )^3}\\ &=\frac{2 x^2}{5 \left (c d^2-a e^2\right ) (d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}-\frac{8 \left (a d e \left (c d^2-a e^2\right ) \left (c d^2+3 a e^2\right )+\left (c^3 d^6+a^2 c d^2 e^4-2 a^3 e^6\right ) x\right )}{15 e \left (c d^2-a e^2\right )^4 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}+\frac{8 \left (c^2 d^4+10 a c d^2 e^2+5 a^2 e^4\right ) \left (c d^2+a e^2+2 c d e x\right )}{15 e \left (c d^2-a e^2\right )^5 \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\\ \end{align*}

Mathematica [A]  time = 0.142492, size = 235, normalized size = 0.91 \[ \frac{2 \left (2 a^2 c^2 d^2 e^2 \left (189 d^2 e^2 x^2+110 d^3 e x+20 d^4+110 d e^3 x^3+20 e^4 x^4\right )+4 a^3 c d e^4 \left (53 d^2 e x+20 d^3+45 d e^2 x^2+15 e^3 x^3\right )+a^4 e^6 \left (8 d^2+20 d e x+15 e^2 x^2\right )+4 a c^3 d^4 e x \left (45 d^2 e x+15 d^3+53 d e^2 x^2+20 e^3 x^3\right )+c^4 d^6 x^2 \left (15 d^2+20 d e x+8 e^2 x^2\right )\right )}{15 (d+e x) \left (c d^2-a e^2\right )^5 ((d+e x) (a e+c d x))^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(c^4*d^6*x^2*(15*d^2 + 20*d*e*x + 8*e^2*x^2) + a^4*e^6*(8*d^2 + 20*d*e*x + 15*e^2*x^2) + 4*a^3*c*d*e^4*(20*
d^3 + 53*d^2*e*x + 45*d*e^2*x^2 + 15*e^3*x^3) + 4*a*c^3*d^4*e*x*(15*d^3 + 45*d^2*e*x + 53*d*e^2*x^2 + 20*e^3*x
^3) + 2*a^2*c^2*d^2*e^2*(20*d^4 + 110*d^3*e*x + 189*d^2*e^2*x^2 + 110*d*e^3*x^3 + 20*e^4*x^4)))/(15*(c*d^2 - a
*e^2)^5*(d + e*x)*((a*e + c*d*x)*(d + e*x))^(3/2))

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Maple [A]  time = 0.054, size = 366, normalized size = 1.4 \begin{align*} -{\frac{ \left ( 2\,cdx+2\,ae \right ) \left ( 40\,{a}^{2}{c}^{2}{d}^{2}{e}^{6}{x}^{4}+80\,a{c}^{3}{d}^{4}{e}^{4}{x}^{4}+8\,{c}^{4}{d}^{6}{e}^{2}{x}^{4}+60\,{a}^{3}cd{e}^{7}{x}^{3}+220\,{a}^{2}{c}^{2}{d}^{3}{e}^{5}{x}^{3}+212\,a{c}^{3}{d}^{5}{e}^{3}{x}^{3}+20\,{c}^{4}{d}^{7}e{x}^{3}+15\,{a}^{4}{e}^{8}{x}^{2}+180\,{a}^{3}c{d}^{2}{e}^{6}{x}^{2}+378\,{a}^{2}{c}^{2}{d}^{4}{e}^{4}{x}^{2}+180\,a{c}^{3}{d}^{6}{e}^{2}{x}^{2}+15\,{c}^{4}{d}^{8}{x}^{2}+20\,{a}^{4}d{e}^{7}x+212\,{a}^{3}c{d}^{3}{e}^{5}x+220\,{a}^{2}{c}^{2}{d}^{5}{e}^{3}x+60\,a{c}^{3}{d}^{7}ex+8\,{a}^{4}{d}^{2}{e}^{6}+80\,{a}^{3}c{d}^{4}{e}^{4}+40\,{a}^{2}{c}^{2}{d}^{6}{e}^{2} \right ) }{15\,{a}^{5}{e}^{10}-75\,{a}^{4}c{d}^{2}{e}^{8}+150\,{a}^{3}{c}^{2}{d}^{4}{e}^{6}-150\,{a}^{2}{c}^{3}{d}^{6}{e}^{4}+75\,a{c}^{4}{d}^{8}{e}^{2}-15\,{c}^{5}{d}^{10}} \left ( cde{x}^{2}+a{e}^{2}x+c{d}^{2}x+ade \right ) ^{-{\frac{5}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/15*(c*d*x+a*e)*(40*a^2*c^2*d^2*e^6*x^4+80*a*c^3*d^4*e^4*x^4+8*c^4*d^6*e^2*x^4+60*a^3*c*d*e^7*x^3+220*a^2*c^
2*d^3*e^5*x^3+212*a*c^3*d^5*e^3*x^3+20*c^4*d^7*e*x^3+15*a^4*e^8*x^2+180*a^3*c*d^2*e^6*x^2+378*a^2*c^2*d^4*e^4*
x^2+180*a*c^3*d^6*e^2*x^2+15*c^4*d^8*x^2+20*a^4*d*e^7*x+212*a^3*c*d^3*e^5*x+220*a^2*c^2*d^5*e^3*x+60*a*c^3*d^7
*e*x+8*a^4*d^2*e^6+80*a^3*c*d^4*e^4+40*a^2*c^2*d^6*e^2)/(a^5*e^10-5*a^4*c*d^2*e^8+10*a^3*c^2*d^4*e^6-10*a^2*c^
3*d^6*e^4+5*a*c^4*d^8*e^2-c^5*d^10)/(c*d*e*x^2+a*e^2*x+c*d^2*x+a*d*e)^(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(x^2/(e*x+d)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [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(x^2/(e*x+d)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2),x, algorithm="fricas")

[Out]

Timed out

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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(x**2/(e*x+d)/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(5/2),x)

[Out]

Timed out

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

[Out]

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