3.488 \(\int \frac{x^2}{(d+e x) (a d e+(c d^2+a e^2) x+c d e x^2)^{7/2}} \, dx\)

Optimal. Leaf size=341 \[ -\frac{128 c d \left (7 a^2 e^4+14 a c d^2 e^2+3 c^2 d^4\right ) \left (a e^2+c d^2+2 c d e x\right )}{105 \left (c d^2-a e^2\right )^7 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}+\frac{16 \left (7 a^2 e^4+14 a c d^2 e^2+3 c^2 d^4\right ) \left (a e^2+c d^2+2 c d e x\right )}{105 e \left (c d^2-a e^2\right )^5 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}-\frac{8 \left (x \left (3 a^2 e^4+a c d^2 e^2+2 c^2 d^4\right )+2 a d e \left (2 a e^2+c d^2\right )\right )}{35 e \left (c d^2-a e^2\right )^3 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}+\frac{2 x^2}{7 (d+e x) \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}} \]

[Out]

(2*x^2)/(7*(c*d^2 - a*e^2)*(d + e*x)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)) - (8*(2*a*d*e*(c*d^2 + 2*a
*e^2) + (2*c^2*d^4 + a*c*d^2*e^2 + 3*a^2*e^4)*x))/(35*e*(c*d^2 - a*e^2)^3*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x
^2)^(5/2)) + (16*(3*c^2*d^4 + 14*a*c*d^2*e^2 + 7*a^2*e^4)*(c*d^2 + a*e^2 + 2*c*d*e*x))/(105*e*(c*d^2 - a*e^2)^
5*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)) - (128*c*d*(3*c^2*d^4 + 14*a*c*d^2*e^2 + 7*a^2*e^4)*(c*d^2 +
a*e^2 + 2*c*d*e*x))/(105*(c*d^2 - a*e^2)^7*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])

________________________________________________________________________________________

Rubi [A]  time = 0.288833, antiderivative size = 341, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 40, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {854, 777, 614, 613} \[ -\frac{128 c d \left (7 a^2 e^4+14 a c d^2 e^2+3 c^2 d^4\right ) \left (a e^2+c d^2+2 c d e x\right )}{105 \left (c d^2-a e^2\right )^7 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}+\frac{16 \left (7 a^2 e^4+14 a c d^2 e^2+3 c^2 d^4\right ) \left (a e^2+c d^2+2 c d e x\right )}{105 e \left (c d^2-a e^2\right )^5 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}-\frac{8 \left (x \left (3 a^2 e^4+a c d^2 e^2+2 c^2 d^4\right )+2 a d e \left (2 a e^2+c d^2\right )\right )}{35 e \left (c d^2-a e^2\right )^3 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}+\frac{2 x^2}{7 (d+e x) \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}} \]

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)^(7/2)),x]

[Out]

(2*x^2)/(7*(c*d^2 - a*e^2)*(d + e*x)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)) - (8*(2*a*d*e*(c*d^2 + 2*a
*e^2) + (2*c^2*d^4 + a*c*d^2*e^2 + 3*a^2*e^4)*x))/(35*e*(c*d^2 - a*e^2)^3*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x
^2)^(5/2)) + (16*(3*c^2*d^4 + 14*a*c*d^2*e^2 + 7*a^2*e^4)*(c*d^2 + a*e^2 + 2*c*d*e*x))/(105*e*(c*d^2 - a*e^2)^
5*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)) - (128*c*d*(3*c^2*d^4 + 14*a*c*d^2*e^2 + 7*a^2*e^4)*(c*d^2 +
a*e^2 + 2*c*d*e*x))/(105*(c*d^2 - a*e^2)^7*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 614

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((b + 2*c*x)*(a + b*x + c*x^2)^(p + 1))/((p +
1)*(b^2 - 4*a*c)), x] - Dist[(2*c*(2*p + 3))/((p + 1)*(b^2 - 4*a*c)), Int[(a + b*x + c*x^2)^(p + 1), x], x] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && LtQ[p, -1] && NeQ[p, -3/2] && IntegerQ[4*p]

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 )^{7/2}} \, dx &=\frac{2 x^2}{7 \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 )^{5/2}}+\frac{2 \int \frac{x \left (-2 a d e^2 \left (c d^2-a e^2\right )+4 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 )^{7/2}} \, dx}{7 d e \left (c d^2-a e^2\right )^2}\\ &=\frac{2 x^2}{7 \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 )^{5/2}}-\frac{8 \left (2 a d e \left (c d^2+2 a e^2\right )+\left (2 c^2 d^4+a c d^2 e^2+3 a^2 e^4\right ) x\right )}{35 e \left (c d^2-a e^2\right )^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}-\frac{\left (8 \left (3 c^2 d^4+14 a c d^2 e^2+7 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 )^{5/2}} \, dx}{35 e \left (c d^2-a e^2\right )^3}\\ &=\frac{2 x^2}{7 \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 )^{5/2}}-\frac{8 \left (2 a d e \left (c d^2+2 a e^2\right )+\left (2 c^2 d^4+a c d^2 e^2+3 a^2 e^4\right ) x\right )}{35 e \left (c d^2-a e^2\right )^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}+\frac{16 \left (3 c^2 d^4+14 a c d^2 e^2+7 a^2 e^4\right ) \left (c d^2+a e^2+2 c d e x\right )}{105 e \left (c d^2-a e^2\right )^5 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}+\frac{\left (64 c d \left (3 c^2 d^4+14 a c d^2 e^2+7 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}{105 \left (c d^2-a e^2\right )^5}\\ &=\frac{2 x^2}{7 \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 )^{5/2}}-\frac{8 \left (2 a d e \left (c d^2+2 a e^2\right )+\left (2 c^2 d^4+a c d^2 e^2+3 a^2 e^4\right ) x\right )}{35 e \left (c d^2-a e^2\right )^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}+\frac{16 \left (3 c^2 d^4+14 a c d^2 e^2+7 a^2 e^4\right ) \left (c d^2+a e^2+2 c d e x\right )}{105 e \left (c d^2-a e^2\right )^5 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}-\frac{128 c d \left (3 c^2 d^4+14 a c d^2 e^2+7 a^2 e^4\right ) \left (c d^2+a e^2+2 c d e x\right )}{105 \left (c d^2-a e^2\right )^7 \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\\ \end{align*}

Mathematica [A]  time = 0.213206, size = 433, normalized size = 1.27 \[ -\frac{2 \sqrt{(d+e x) (a e+c d x)} \left (5 a^4 c^2 d^2 e^6 \left (1859 d^2 e^2 x^2+1288 d^3 e x+336 d^4+1288 d e^3 x^3+336 e^4 x^4\right )+20 a^3 c^3 d^3 e^4 \left (1001 d^3 e^2 x^2+1084 d^2 e^3 x^3+406 d^4 e x+56 d^5+560 d e^4 x^4+112 e^5 x^5\right )+a^2 c^4 d^4 e^2 \left (13195 d^4 e^2 x^2+24080 d^3 e^3 x^3+20320 d^2 e^4 x^4+2996 d^5 e x+56 d^6+7616 d e^5 x^5+896 e^6 x^6\right )+2 a^5 c d e^8 \left (382 d^2 e x+112 d^3+455 d e^2 x^2+140 e^3 x^3\right )-a^6 e^{10} \left (8 d^2+28 d e x+35 e^2 x^2\right )+2 a c^5 d^6 e x \left (4060 d^3 e^2 x^2+5600 d^2 e^3 x^3+1295 d^4 e x+70 d^5+3616 d e^4 x^4+896 e^5 x^5\right )+3 c^6 d^8 x^2 \left (560 d^2 e^2 x^2+280 d^3 e x+35 d^4+448 d e^3 x^3+128 e^4 x^4\right )\right )}{105 (d+e x)^4 \left (c d^2-a e^2\right )^7 (a e+c d x)^3} \]

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)^(7/2)),x]

[Out]

(-2*Sqrt[(a*e + c*d*x)*(d + e*x)]*(-(a^6*e^10*(8*d^2 + 28*d*e*x + 35*e^2*x^2)) + 2*a^5*c*d*e^8*(112*d^3 + 382*
d^2*e*x + 455*d*e^2*x^2 + 140*e^3*x^3) + 3*c^6*d^8*x^2*(35*d^4 + 280*d^3*e*x + 560*d^2*e^2*x^2 + 448*d*e^3*x^3
+ 128*e^4*x^4) + 5*a^4*c^2*d^2*e^6*(336*d^4 + 1288*d^3*e*x + 1859*d^2*e^2*x^2 + 1288*d*e^3*x^3 + 336*e^4*x^4)
+ 20*a^3*c^3*d^3*e^4*(56*d^5 + 406*d^4*e*x + 1001*d^3*e^2*x^2 + 1084*d^2*e^3*x^3 + 560*d*e^4*x^4 + 112*e^5*x^
5) + 2*a*c^5*d^6*e*x*(70*d^5 + 1295*d^4*e*x + 4060*d^3*e^2*x^2 + 5600*d^2*e^3*x^3 + 3616*d*e^4*x^4 + 896*e^5*x
^5) + a^2*c^4*d^4*e^2*(56*d^6 + 2996*d^5*e*x + 13195*d^4*e^2*x^2 + 24080*d^3*e^3*x^3 + 20320*d^2*e^4*x^4 + 761
6*d*e^5*x^5 + 896*e^6*x^6)))/(105*(c*d^2 - a*e^2)^7*(a*e + c*d*x)^3*(d + e*x)^4)

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Maple [B]  time = 0.06, size = 663, normalized size = 1.9 \begin{align*} -{\frac{ \left ( 2\,cdx+2\,ae \right ) \left ( -896\,{a}^{2}{c}^{4}{d}^{4}{e}^{8}{x}^{6}-1792\,a{c}^{5}{d}^{6}{e}^{6}{x}^{6}-384\,{c}^{6}{d}^{8}{e}^{4}{x}^{6}-2240\,{a}^{3}{c}^{3}{d}^{3}{e}^{9}{x}^{5}-7616\,{a}^{2}{c}^{4}{d}^{5}{e}^{7}{x}^{5}-7232\,a{c}^{5}{d}^{7}{e}^{5}{x}^{5}-1344\,{c}^{6}{d}^{9}{e}^{3}{x}^{5}-1680\,{a}^{4}{c}^{2}{d}^{2}{e}^{10}{x}^{4}-11200\,{a}^{3}{c}^{3}{d}^{4}{e}^{8}{x}^{4}-20320\,{a}^{2}{c}^{4}{d}^{6}{e}^{6}{x}^{4}-11200\,a{c}^{5}{d}^{8}{e}^{4}{x}^{4}-1680\,{c}^{6}{d}^{10}{e}^{2}{x}^{4}-280\,{a}^{5}cd{e}^{11}{x}^{3}-6440\,{a}^{4}{c}^{2}{d}^{3}{e}^{9}{x}^{3}-21680\,{a}^{3}{c}^{3}{d}^{5}{e}^{7}{x}^{3}-24080\,{a}^{2}{c}^{4}{d}^{7}{e}^{5}{x}^{3}-8120\,a{c}^{5}{d}^{9}{e}^{3}{x}^{3}-840\,{c}^{6}{d}^{11}e{x}^{3}+35\,{a}^{6}{e}^{12}{x}^{2}-910\,{a}^{5}c{d}^{2}{e}^{10}{x}^{2}-9295\,{a}^{4}{c}^{2}{d}^{4}{e}^{8}{x}^{2}-20020\,{a}^{3}{c}^{3}{d}^{6}{e}^{6}{x}^{2}-13195\,{a}^{2}{c}^{4}{d}^{8}{e}^{4}{x}^{2}-2590\,a{c}^{5}{d}^{10}{e}^{2}{x}^{2}-105\,{c}^{6}{d}^{12}{x}^{2}+28\,{a}^{6}d{e}^{11}x-764\,{a}^{5}c{d}^{3}{e}^{9}x-6440\,{a}^{4}{c}^{2}{d}^{5}{e}^{7}x-8120\,{a}^{3}{c}^{3}{d}^{7}{e}^{5}x-2996\,{a}^{2}{c}^{4}{d}^{9}{e}^{3}x-140\,a{c}^{5}{d}^{11}ex+8\,{a}^{6}{d}^{2}{e}^{10}-224\,{a}^{5}c{d}^{4}{e}^{8}-1680\,{a}^{4}{c}^{2}{d}^{6}{e}^{6}-1120\,{a}^{3}{c}^{3}{d}^{8}{e}^{4}-56\,{a}^{2}{c}^{4}{d}^{10}{e}^{2} \right ) }{105\,{a}^{7}{e}^{14}-735\,{a}^{6}c{d}^{2}{e}^{12}+2205\,{a}^{5}{c}^{2}{d}^{4}{e}^{10}-3675\,{a}^{4}{c}^{3}{d}^{6}{e}^{8}+3675\,{a}^{3}{c}^{4}{d}^{8}{e}^{6}-2205\,{a}^{2}{c}^{5}{d}^{10}{e}^{4}+735\,a{c}^{6}{d}^{12}{e}^{2}-105\,{c}^{7}{d}^{14}} \left ( cde{x}^{2}+a{e}^{2}x+c{d}^{2}x+ade \right ) ^{-{\frac{7}{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)^(7/2),x)

[Out]

-2/105*(c*d*x+a*e)*(-896*a^2*c^4*d^4*e^8*x^6-1792*a*c^5*d^6*e^6*x^6-384*c^6*d^8*e^4*x^6-2240*a^3*c^3*d^3*e^9*x
^5-7616*a^2*c^4*d^5*e^7*x^5-7232*a*c^5*d^7*e^5*x^5-1344*c^6*d^9*e^3*x^5-1680*a^4*c^2*d^2*e^10*x^4-11200*a^3*c^
3*d^4*e^8*x^4-20320*a^2*c^4*d^6*e^6*x^4-11200*a*c^5*d^8*e^4*x^4-1680*c^6*d^10*e^2*x^4-280*a^5*c*d*e^11*x^3-644
0*a^4*c^2*d^3*e^9*x^3-21680*a^3*c^3*d^5*e^7*x^3-24080*a^2*c^4*d^7*e^5*x^3-8120*a*c^5*d^9*e^3*x^3-840*c^6*d^11*
e*x^3+35*a^6*e^12*x^2-910*a^5*c*d^2*e^10*x^2-9295*a^4*c^2*d^4*e^8*x^2-20020*a^3*c^3*d^6*e^6*x^2-13195*a^2*c^4*
d^8*e^4*x^2-2590*a*c^5*d^10*e^2*x^2-105*c^6*d^12*x^2+28*a^6*d*e^11*x-764*a^5*c*d^3*e^9*x-6440*a^4*c^2*d^5*e^7*
x-8120*a^3*c^3*d^7*e^5*x-2996*a^2*c^4*d^9*e^3*x-140*a*c^5*d^11*e*x+8*a^6*d^2*e^10-224*a^5*c*d^4*e^8-1680*a^4*c
^2*d^6*e^6-1120*a^3*c^3*d^8*e^4-56*a^2*c^4*d^10*e^2)/(a^7*e^14-7*a^6*c*d^2*e^12+21*a^5*c^2*d^4*e^10-35*a^4*c^3
*d^6*e^8+35*a^3*c^4*d^8*e^6-21*a^2*c^5*d^10*e^4+7*a*c^6*d^12*e^2-c^7*d^14)/(c*d*e*x^2+a*e^2*x+c*d^2*x+a*d*e)^(
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)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(7/2),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)^(7/2),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)**(7/2),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}, \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)^(7/2),x, algorithm="giac")

[Out]

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