3.5.88 \(\int \frac {x^2}{(d+e x) (a d e+(c d^2+a e^2) x+c d e x^2)^{7/2}} \, dx\) [488]

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

[Out]

2/7*x^2/(-a*e^2+c*d^2)/(e*x+d)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)-8/35*(2*a*d*e*(2*a*e^2+c*d^2)+(3*a^2*e^
4+a*c*d^2*e^2+2*c^2*d^4)*x)/e/(-a*e^2+c*d^2)^3/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)+16/105*(7*a^2*e^4+14*a*
c*d^2*e^2+3*c^2*d^4)*(2*c*d*e*x+a*e^2+c*d^2)/e/(-a*e^2+c*d^2)^5/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)-128/10
5*c*d*(7*a^2*e^4+14*a*c*d^2*e^2+3*c^2*d^4)*(2*c*d*e*x+a*e^2+c*d^2)/(-a*e^2+c*d^2)^7/(a*d*e+(a*e^2+c*d^2)*x+c*d
*e*x^2)^(1/2)

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Rubi [A]
time = 0.18, antiderivative size = 341, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 40, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {868, 791, 628, 627} \begin {gather*} -\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}} \end {gather*}

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 627

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]

Rule 628

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 791

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] &&
NeQ[b^2 - 4*a*c, 0] && LtQ[p, -1]

Rule 868

Int[(((f_.) + (g_.)*(x_))^(n_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_))/((d_) + (e_.)*(x_)), x_Symbol] :> Sim
p[(-(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]

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*}

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Mathematica [A]
time = 0.34, size = 440, normalized size = 1.29 \begin {gather*} -\frac {2 \sqrt {(a e+c d x) (d+e x)} \left (-15 d^2 e^4 (a e+c d x)^6+84 c d^3 e^3 (a e+c d x)^5 (d+e x)+42 a d e^5 (a e+c d x)^5 (d+e x)-210 c^2 d^4 e^2 (a e+c d x)^4 (d+e x)^2-280 a c d^2 e^4 (a e+c d x)^4 (d+e x)^2-35 a^2 e^6 (a e+c d x)^4 (d+e x)^2+420 c^3 d^5 e (a e+c d x)^3 (d+e x)^3+1260 a c^2 d^3 e^3 (a e+c d x)^3 (d+e x)^3+420 a^2 c d e^5 (a e+c d x)^3 (d+e x)^3+105 c^4 d^6 (a e+c d x)^2 (d+e x)^4+840 a c^3 d^4 e^2 (a e+c d x)^2 (d+e x)^4+630 a^2 c^2 d^2 e^4 (a e+c d x)^2 (d+e x)^4-70 a c^4 d^5 e (a e+c d x) (d+e x)^5-140 a^2 c^3 d^3 e^3 (a e+c d x) (d+e x)^5+21 a^2 c^4 d^4 e^2 (d+e x)^6\right )}{105 \left (c d^2-a e^2\right )^7 (a e+c d x)^3 (d+e x)^4} \end {gather*}

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)]*(-15*d^2*e^4*(a*e + c*d*x)^6 + 84*c*d^3*e^3*(a*e + c*d*x)^5*(d + e*x) + 42*a
*d*e^5*(a*e + c*d*x)^5*(d + e*x) - 210*c^2*d^4*e^2*(a*e + c*d*x)^4*(d + e*x)^2 - 280*a*c*d^2*e^4*(a*e + c*d*x)
^4*(d + e*x)^2 - 35*a^2*e^6*(a*e + c*d*x)^4*(d + e*x)^2 + 420*c^3*d^5*e*(a*e + c*d*x)^3*(d + e*x)^3 + 1260*a*c
^2*d^3*e^3*(a*e + c*d*x)^3*(d + e*x)^3 + 420*a^2*c*d*e^5*(a*e + c*d*x)^3*(d + e*x)^3 + 105*c^4*d^6*(a*e + c*d*
x)^2*(d + e*x)^4 + 840*a*c^3*d^4*e^2*(a*e + c*d*x)^2*(d + e*x)^4 + 630*a^2*c^2*d^2*e^4*(a*e + c*d*x)^2*(d + e*
x)^4 - 70*a*c^4*d^5*e*(a*e + c*d*x)*(d + e*x)^5 - 140*a^2*c^3*d^3*e^3*(a*e + c*d*x)*(d + e*x)^5 + 21*a^2*c^4*d
^4*e^2*(d + e*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] Leaf count of result is larger than twice the leaf count of optimal. \(932\) vs. \(2(325)=650\).
time = 0.09, size = 933, normalized size = 2.74 Too large to display

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,method=_RETURNVERBOSE)

[Out]

1/e*(-1/5/c/d/e/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)-1/2*(a*e^2+c*d^2)/c/d/e*(2/5*(2*c*d*e*x+a*e^2+c*d^2)/(
4*a*c*d^2*e^2-(a*e^2+c*d^2)^2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)+16/5*c*d*e/(4*a*c*d^2*e^2-(a*e^2+c*d^2)
^2)*(2/3*(2*c*d*e*x+a*e^2+c*d^2)/(4*a*c*d^2*e^2-(a*e^2+c*d^2)^2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)+16/3*
c*d*e/(4*a*c*d^2*e^2-(a*e^2+c*d^2)^2)^2*(2*c*d*e*x+a*e^2+c*d^2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2))))-d/e
^2*(2/5*(2*c*d*e*x+a*e^2+c*d^2)/(4*a*c*d^2*e^2-(a*e^2+c*d^2)^2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)+16/5*c
*d*e/(4*a*c*d^2*e^2-(a*e^2+c*d^2)^2)*(2/3*(2*c*d*e*x+a*e^2+c*d^2)/(4*a*c*d^2*e^2-(a*e^2+c*d^2)^2)/(a*d*e+(a*e^
2+c*d^2)*x+c*d*e*x^2)^(3/2)+16/3*c*d*e/(4*a*c*d^2*e^2-(a*e^2+c*d^2)^2)^2*(2*c*d*e*x+a*e^2+c*d^2)/(a*d*e+(a*e^2
+c*d^2)*x+c*d*e*x^2)^(1/2)))+1/e^3*d^2*(-2/7/(a*e^2-c*d^2)/(x+d/e)/(c*d*e*(x+d/e)^2+(a*e^2-c*d^2)*(x+d/e))^(5/
2)-12/7*c*d*e/(a*e^2-c*d^2)*(-2/5*(2*c*d*e*(x+d/e)+a*e^2-c*d^2)/(a*e^2-c*d^2)^2/(c*d*e*(x+d/e)^2+(a*e^2-c*d^2)
*(x+d/e))^(5/2)-16/5*c*d*e/(a*e^2-c*d^2)^2*(-2/3*(2*c*d*e*(x+d/e)+a*e^2-c*d^2)/(a*e^2-c*d^2)^2/(c*d*e*(x+d/e)^
2+(a*e^2-c*d^2)*(x+d/e))^(3/2)+16/3*c*d*e/(a*e^2-c*d^2)^4*(2*c*d*e*(x+d/e)+a*e^2-c*d^2)/(c*d*e*(x+d/e)^2+(a*e^
2-c*d^2)*(x+d/e))^(1/2))))

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

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 >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(c*d^2-%e^2*a>0)', see `assume?
` for more d

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1540 vs. \(2 (325) = 650\).
time = 99.68, size = 1540, normalized size = 4.52 \begin {gather*} -\frac {2 \, {\left (56 \, a^{2} c^{4} d^{10} e^{2} + 1120 \, a^{3} c^{3} d^{8} e^{4} + 1680 \, a^{4} c^{2} d^{6} e^{6} + 224 \, a^{5} c d^{4} e^{8} - 8 \, a^{6} d^{2} e^{10} + 128 \, {\left (3 \, c^{6} d^{8} e^{4} + 14 \, a c^{5} d^{6} e^{6} + 7 \, a^{2} c^{4} d^{4} e^{8}\right )} x^{6} + 64 \, {\left (21 \, c^{6} d^{9} e^{3} + 113 \, a c^{5} d^{7} e^{5} + 119 \, a^{2} c^{4} d^{5} e^{7} + 35 \, a^{3} c^{3} d^{3} e^{9}\right )} x^{5} + 80 \, {\left (21 \, c^{6} d^{10} e^{2} + 140 \, a c^{5} d^{8} e^{4} + 254 \, a^{2} c^{4} d^{6} e^{6} + 140 \, a^{3} c^{3} d^{4} e^{8} + 21 \, a^{4} c^{2} d^{2} e^{10}\right )} x^{4} + 40 \, {\left (21 \, c^{6} d^{11} e + 203 \, a c^{5} d^{9} e^{3} + 602 \, a^{2} c^{4} d^{7} e^{5} + 542 \, a^{3} c^{3} d^{5} e^{7} + 161 \, a^{4} c^{2} d^{3} e^{9} + 7 \, a^{5} c d e^{11}\right )} x^{3} + 5 \, {\left (21 \, c^{6} d^{12} + 518 \, a c^{5} d^{10} e^{2} + 2639 \, a^{2} c^{4} d^{8} e^{4} + 4004 \, a^{3} c^{3} d^{6} e^{6} + 1859 \, a^{4} c^{2} d^{4} e^{8} + 182 \, a^{5} c d^{2} e^{10} - 7 \, a^{6} e^{12}\right )} x^{2} + 4 \, {\left (35 \, a c^{5} d^{11} e + 749 \, a^{2} c^{4} d^{9} e^{3} + 2030 \, a^{3} c^{3} d^{7} e^{5} + 1610 \, a^{4} c^{2} d^{5} e^{7} + 191 \, a^{5} c d^{3} e^{9} - 7 \, a^{6} d e^{11}\right )} x\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x}}{105 \, {\left (a^{3} c^{7} d^{18} e^{3} - 7 \, a^{4} c^{6} d^{16} e^{5} + 21 \, a^{5} c^{5} d^{14} e^{7} - 35 \, a^{6} c^{4} d^{12} e^{9} + 35 \, a^{7} c^{3} d^{10} e^{11} - 21 \, a^{8} c^{2} d^{8} e^{13} + 7 \, a^{9} c d^{6} e^{15} - a^{10} d^{4} e^{17} + {\left (c^{10} d^{17} e^{4} - 7 \, a c^{9} d^{15} e^{6} + 21 \, a^{2} c^{8} d^{13} e^{8} - 35 \, a^{3} c^{7} d^{11} e^{10} + 35 \, a^{4} c^{6} d^{9} e^{12} - 21 \, a^{5} c^{5} d^{7} e^{14} + 7 \, a^{6} c^{4} d^{5} e^{16} - a^{7} c^{3} d^{3} e^{18}\right )} x^{7} + {\left (4 \, c^{10} d^{18} e^{3} - 25 \, a c^{9} d^{16} e^{5} + 63 \, a^{2} c^{8} d^{14} e^{7} - 77 \, a^{3} c^{7} d^{12} e^{9} + 35 \, a^{4} c^{6} d^{10} e^{11} + 21 \, a^{5} c^{5} d^{8} e^{13} - 35 \, a^{6} c^{4} d^{6} e^{15} + 17 \, a^{7} c^{3} d^{4} e^{17} - 3 \, a^{8} c^{2} d^{2} e^{19}\right )} x^{6} + 3 \, {\left (2 \, c^{10} d^{19} e^{2} - 10 \, a c^{9} d^{17} e^{4} + 15 \, a^{2} c^{8} d^{15} e^{6} + 7 \, a^{3} c^{7} d^{13} e^{8} - 49 \, a^{4} c^{6} d^{11} e^{10} + 63 \, a^{5} c^{5} d^{9} e^{12} - 35 \, a^{6} c^{4} d^{7} e^{14} + 5 \, a^{7} c^{3} d^{5} e^{16} + 3 \, a^{8} c^{2} d^{3} e^{18} - a^{9} c d e^{20}\right )} x^{5} + {\left (4 \, c^{10} d^{20} e - 10 \, a c^{9} d^{18} e^{3} - 30 \, a^{2} c^{8} d^{16} e^{5} + 155 \, a^{3} c^{7} d^{14} e^{7} - 245 \, a^{4} c^{6} d^{12} e^{9} + 147 \, a^{5} c^{5} d^{10} e^{11} + 35 \, a^{6} c^{4} d^{8} e^{13} - 95 \, a^{7} c^{3} d^{6} e^{15} + 45 \, a^{8} c^{2} d^{4} e^{17} - 5 \, a^{9} c d^{2} e^{19} - a^{10} e^{21}\right )} x^{4} + {\left (c^{10} d^{21} + 5 \, a c^{9} d^{19} e^{2} - 45 \, a^{2} c^{8} d^{17} e^{4} + 95 \, a^{3} c^{7} d^{15} e^{6} - 35 \, a^{4} c^{6} d^{13} e^{8} - 147 \, a^{5} c^{5} d^{11} e^{10} + 245 \, a^{6} c^{4} d^{9} e^{12} - 155 \, a^{7} c^{3} d^{7} e^{14} + 30 \, a^{8} c^{2} d^{5} e^{16} + 10 \, a^{9} c d^{3} e^{18} - 4 \, a^{10} d e^{20}\right )} x^{3} + 3 \, {\left (a c^{9} d^{20} e - 3 \, a^{2} c^{8} d^{18} e^{3} - 5 \, a^{3} c^{7} d^{16} e^{5} + 35 \, a^{4} c^{6} d^{14} e^{7} - 63 \, a^{5} c^{5} d^{12} e^{9} + 49 \, a^{6} c^{4} d^{10} e^{11} - 7 \, a^{7} c^{3} d^{8} e^{13} - 15 \, a^{8} c^{2} d^{6} e^{15} + 10 \, a^{9} c d^{4} e^{17} - 2 \, a^{10} d^{2} e^{19}\right )} x^{2} + {\left (3 \, a^{2} c^{8} d^{19} e^{2} - 17 \, a^{3} c^{7} d^{17} e^{4} + 35 \, a^{4} c^{6} d^{15} e^{6} - 21 \, a^{5} c^{5} d^{13} e^{8} - 35 \, a^{6} c^{4} d^{11} e^{10} + 77 \, a^{7} c^{3} d^{9} e^{12} - 63 \, a^{8} c^{2} d^{7} e^{14} + 25 \, a^{9} c d^{5} e^{16} - 4 \, a^{10} d^{3} e^{18}\right )} x\right )}} \end {gather*}

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]

-2/105*(56*a^2*c^4*d^10*e^2 + 1120*a^3*c^3*d^8*e^4 + 1680*a^4*c^2*d^6*e^6 + 224*a^5*c*d^4*e^8 - 8*a^6*d^2*e^10
+ 128*(3*c^6*d^8*e^4 + 14*a*c^5*d^6*e^6 + 7*a^2*c^4*d^4*e^8)*x^6 + 64*(21*c^6*d^9*e^3 + 113*a*c^5*d^7*e^5 + 1
19*a^2*c^4*d^5*e^7 + 35*a^3*c^3*d^3*e^9)*x^5 + 80*(21*c^6*d^10*e^2 + 140*a*c^5*d^8*e^4 + 254*a^2*c^4*d^6*e^6 +
140*a^3*c^3*d^4*e^8 + 21*a^4*c^2*d^2*e^10)*x^4 + 40*(21*c^6*d^11*e + 203*a*c^5*d^9*e^3 + 602*a^2*c^4*d^7*e^5
+ 542*a^3*c^3*d^5*e^7 + 161*a^4*c^2*d^3*e^9 + 7*a^5*c*d*e^11)*x^3 + 5*(21*c^6*d^12 + 518*a*c^5*d^10*e^2 + 2639
*a^2*c^4*d^8*e^4 + 4004*a^3*c^3*d^6*e^6 + 1859*a^4*c^2*d^4*e^8 + 182*a^5*c*d^2*e^10 - 7*a^6*e^12)*x^2 + 4*(35*
a*c^5*d^11*e + 749*a^2*c^4*d^9*e^3 + 2030*a^3*c^3*d^7*e^5 + 1610*a^4*c^2*d^5*e^7 + 191*a^5*c*d^3*e^9 - 7*a^6*d
*e^11)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)/(a^3*c^7*d^18*e^3 - 7*a^4*c^6*d^16*e^5 + 21*a^5*c^5*d^14
*e^7 - 35*a^6*c^4*d^12*e^9 + 35*a^7*c^3*d^10*e^11 - 21*a^8*c^2*d^8*e^13 + 7*a^9*c*d^6*e^15 - a^10*d^4*e^17 + (
c^10*d^17*e^4 - 7*a*c^9*d^15*e^6 + 21*a^2*c^8*d^13*e^8 - 35*a^3*c^7*d^11*e^10 + 35*a^4*c^6*d^9*e^12 - 21*a^5*c
^5*d^7*e^14 + 7*a^6*c^4*d^5*e^16 - a^7*c^3*d^3*e^18)*x^7 + (4*c^10*d^18*e^3 - 25*a*c^9*d^16*e^5 + 63*a^2*c^8*d
^14*e^7 - 77*a^3*c^7*d^12*e^9 + 35*a^4*c^6*d^10*e^11 + 21*a^5*c^5*d^8*e^13 - 35*a^6*c^4*d^6*e^15 + 17*a^7*c^3*
d^4*e^17 - 3*a^8*c^2*d^2*e^19)*x^6 + 3*(2*c^10*d^19*e^2 - 10*a*c^9*d^17*e^4 + 15*a^2*c^8*d^15*e^6 + 7*a^3*c^7*
d^13*e^8 - 49*a^4*c^6*d^11*e^10 + 63*a^5*c^5*d^9*e^12 - 35*a^6*c^4*d^7*e^14 + 5*a^7*c^3*d^5*e^16 + 3*a^8*c^2*d
^3*e^18 - a^9*c*d*e^20)*x^5 + (4*c^10*d^20*e - 10*a*c^9*d^18*e^3 - 30*a^2*c^8*d^16*e^5 + 155*a^3*c^7*d^14*e^7
- 245*a^4*c^6*d^12*e^9 + 147*a^5*c^5*d^10*e^11 + 35*a^6*c^4*d^8*e^13 - 95*a^7*c^3*d^6*e^15 + 45*a^8*c^2*d^4*e^
17 - 5*a^9*c*d^2*e^19 - a^10*e^21)*x^4 + (c^10*d^21 + 5*a*c^9*d^19*e^2 - 45*a^2*c^8*d^17*e^4 + 95*a^3*c^7*d^15
*e^6 - 35*a^4*c^6*d^13*e^8 - 147*a^5*c^5*d^11*e^10 + 245*a^6*c^4*d^9*e^12 - 155*a^7*c^3*d^7*e^14 + 30*a^8*c^2*
d^5*e^16 + 10*a^9*c*d^3*e^18 - 4*a^10*d*e^20)*x^3 + 3*(a*c^9*d^20*e - 3*a^2*c^8*d^18*e^3 - 5*a^3*c^7*d^16*e^5
+ 35*a^4*c^6*d^14*e^7 - 63*a^5*c^5*d^12*e^9 + 49*a^6*c^4*d^10*e^11 - 7*a^7*c^3*d^8*e^13 - 15*a^8*c^2*d^6*e^15
+ 10*a^9*c*d^4*e^17 - 2*a^10*d^2*e^19)*x^2 + (3*a^2*c^8*d^19*e^2 - 17*a^3*c^7*d^17*e^4 + 35*a^4*c^6*d^15*e^6 -
21*a^5*c^5*d^13*e^8 - 35*a^6*c^4*d^11*e^10 + 77*a^7*c^3*d^9*e^12 - 63*a^8*c^2*d^7*e^14 + 25*a^9*c*d^5*e^16 -
4*a^10*d^3*e^18)*x)

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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]

integrate(x^2/((c*d*x^2*e + a*d*e + (c*d^2 + a*e^2)*x)^(7/2)*(x*e + d)), x)

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Mupad [B]
time = 7.72, size = 2500, normalized size = 7.33 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2/((d + e*x)*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(7/2)),x)

[Out]

((6*c^3*d^5 + 36*a*c^2*d^3*e^2 - 10*a^2*c*d*e^4)/(105*(a*e^2 - c*d^2)^6) - x*((16*c^2*d^2*e)/(105*(a*e^2 - c*d
^2)^5) - (8*c^2*d^2*e*(a*e^2 + c*d^2))/(105*(a*e^2 - c*d^2)^6)) + (8*a*c^2*d^3*e^2)/(105*(a*e^2 - c*d^2)^6))/(
x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2) + (x*((a*((64*c^5*d^5*e^4*(a*e^2 + c*d^2))/(105*(a*e^2 - c*d^2)^6
*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5)) - (64*c^5*d^5*e^4*(5*a*e^2 - 3*c*d^2))/(105*(a*e^2 - c*d^2)^6*(c
^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5))))/c - ((a*e^2 + c*d^2)*(((a*e^2 + c*d^2)*((64*c^5*d^5*e^4*(a*e^2 +
c*d^2))/(105*(a*e^2 - c*d^2)^6*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5)) - (64*c^5*d^5*e^4*(5*a*e^2 - 3*c*d
^2))/(105*(a*e^2 - c*d^2)^6*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5))))/(c*d*e) - (32*c^4*d^4*e^3*(7*c^2*d^
4 - 9*a^2*e^4 + 18*a*c*d^2*e^2))/(105*(a*e^2 - c*d^2)^6*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5)) - (128*a*
c^5*d^6*e^5)/(105*(a*e^2 - c*d^2)^6*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5)) + (32*c^4*d^4*e^3*(a*e^2 + c*
d^2)*(5*a*e^2 - 3*c*d^2))/(105*(a*e^2 - c*d^2)^6*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5))))/(c*d*e) + (2*c
^2*d^2*e^2*(60*c^4*d^7 - 204*a*c^3*d^5*e^2 - 156*a^2*c^2*d^3*e^4 + 44*a^3*c*d*e^6))/(105*(a*e^2 - c*d^2)^6*(c^
3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5)) - (16*c^3*d^3*e^2*(a*e^2 + c*d^2)*(7*c^2*d^4 - 9*a^2*e^4 + 18*a*c*d^
2*e^2))/(105*(a*e^2 - c*d^2)^6*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5))) - (a*(((a*e^2 + c*d^2)*((64*c^5*d
^5*e^4*(a*e^2 + c*d^2))/(105*(a*e^2 - c*d^2)^6*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5)) - (64*c^5*d^5*e^4*
(5*a*e^2 - 3*c*d^2))/(105*(a*e^2 - c*d^2)^6*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5))))/(c*d*e) - (32*c^4*d
^4*e^3*(7*c^2*d^4 - 9*a^2*e^4 + 18*a*c*d^2*e^2))/(105*(a*e^2 - c*d^2)^6*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d
*e^5)) - (128*a*c^5*d^6*e^5)/(105*(a*e^2 - c*d^2)^6*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5)) + (32*c^4*d^4
*e^3*(a*e^2 + c*d^2)*(5*a*e^2 - 3*c*d^2))/(105*(a*e^2 - c*d^2)^6*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5)))
)/c + (c*d*e*(a*e^2 + c*d^2)*(60*c^4*d^7 - 204*a*c^3*d^5*e^2 - 156*a^2*c^2*d^3*e^4 + 44*a^3*c*d*e^6))/(105*(a*
e^2 - c*d^2)^6*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5)))/(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2) + (
x*((a*((8*c^3*d^3*e^2*(a*e^2 + c*d^2))/(105*(a*e^2 - c*d^2)^6) - (8*c^3*d^3*e^2*(3*a*e^2 - c*d^2))/(105*(a*e^2
- c*d^2)^6)))/c + (36*c^4*d^7*e - 76*a*c^3*d^5*e^3 - 36*a^2*c^2*d^3*e^5 + 12*a^3*c*d*e^7)/(105*e*(a*e^2 - c*d
^2)^6) + ((a*e^2 + c*d^2)*((8*a*c^3*d^4*e^3)/(105*(a*e^2 - c*d^2)^6) - (((8*c^3*d^3*e^2*(a*e^2 + c*d^2))/(105*
(a*e^2 - c*d^2)^6) - (8*c^3*d^3*e^2*(3*a*e^2 - c*d^2))/(105*(a*e^2 - c*d^2)^6))*(a*e^2 + c*d^2))/(c*d*e) + (2*
c^2*d^2*e*(11*c^2*d^4 - 13*a^2*e^4 + 14*a*c*d^2*e^2))/(105*(a*e^2 - c*d^2)^6)))/(c*d*e)) + (30*a^4*e^8 - 22*c^
4*d^8 + 20*a*c^3*d^6*e^2 - 132*a^3*c*d^2*e^6 + 72*a^2*c^2*d^4*e^4)/(105*e*(a*e^2 - c*d^2)^6) + (a*((8*a*c^3*d^
4*e^3)/(105*(a*e^2 - c*d^2)^6) - (((8*c^3*d^3*e^2*(a*e^2 + c*d^2))/(105*(a*e^2 - c*d^2)^6) - (8*c^3*d^3*e^2*(3
*a*e^2 - c*d^2))/(105*(a*e^2 - c*d^2)^6))*(a*e^2 + c*d^2))/(c*d*e) + (2*c^2*d^2*e*(11*c^2*d^4 - 13*a^2*e^4 + 1
4*a*c*d^2*e^2))/(105*(a*e^2 - c*d^2)^6)))/c)/(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(3/2) - (((d*((e*(2*a*e^4
- 2*c*d^2*e^2))/(7*(a*e^2 - c*d^2)^4*(5*a*e^3 - 5*c*d^2*e)) - (4*c*d^2*e^3)/(7*(a*e^2 - c*d^2)^4*(5*a*e^3 - 5
*c*d^2*e))))/e + (e*(2*a*d*e^3 + 2*c*d^3*e))/(7*(a*e^2 - c*d^2)^4*(5*a*e^3 - 5*c*d^2*e)))*(x*(a*e^2 + c*d^2) +
a*d*e + c*d*e*x^2)^(1/2))/(d + e*x)^3 + (((e*(10*a*e^3 - 14*c*d^2*e))/(35*(a*e^2 - c*d^2)^4*(3*a*e^3 - 3*c*d^
2*e)) - (4*c*d^2*e^2)/(7*(a*e^2 - c*d^2)^4*(3*a*e^3 - 3*c*d^2*e)))*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/
2))/(d + e*x)^2 + ((x*((a*((a*((4*c^5*d^5*e^4*(a*e^2 + c*d^2))/(35*(a*e^2 - c*d^2)^4*(c^3*d^5*e - 2*a*c^2*d^3*
e^3 + a^2*c*d*e^5)) - (4*c^5*d^5*e^4*(7*a*e^2 - c*d^2))/(35*(a*e^2 - c*d^2)^4*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a
^2*c*d*e^5))))/c - ((a*e^2 + c*d^2)*(((a*e^2 + c*d^2)*((4*c^5*d^5*e^4*(a*e^2 + c*d^2))/(35*(a*e^2 - c*d^2)^4*(
c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5)) - (4*c^5*d^5*e^4*(7*a*e^2 - c*d^2))/(35*(a*e^2 - c*d^2)^4*(c^3*d^5
*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5))))/(c*d*e) - (4*c^4*d^4*e^3*(7*c^2*d^4 - 9*a^2*e^4 + 4*a*c*d^2*e^2))/(35*(
a*e^2 - c*d^2)^4*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5)) - (8*a*c^5*d^6*e^5)/(35*(a*e^2 - c*d^2)^4*(c^3*d
^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5)) + (2*c^4*d^4*e^3*(a*e^2 + c*d^2)*(7*a*e^2 - c*d^2))/(35*(a*e^2 - c*d^2)
^4*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5))))/(c*d*e) + (2*c^2*d^2*e^2*(14*c^4*d^7 - 56*a*c^3*d^5*e^2 - 12
*a^2*c^2*d^3*e^4 + 10*a^3*c*d*e^6))/(35*(a*e^2 - c*d^2)^4*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5)) - (2*c^
3*d^3*e^2*(a*e^2 + c*d^2)*(7*c^2*d^4 - 9*a^2*e^4 + 4*a*c*d^2*e^2))/(35*(a*e^2 - c*d^2)^4*(c^3*d^5*e - 2*a*c^2*
d^3*e^3 + a^2*c*d*e^5))))/c - ((a*e^2 + c*d^2)*((a*(((a*e^2 + c*d^2)*((4*c^5*d^5*e^4*(a*e^2 + c*d^2))/(35*(a*e
^2 - c*d^2)^4*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5)) - (4*c^5*d^5*e^4*(7*a*e^2 - c*d^2))/(35*(a*e^2 - c*
d^2)^4*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e...

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