3.5.87 \(\int \frac {x^2}{(d+e x) (a d e+(c d^2+a e^2) x+c d e x^2)^{5/2}} \, dx\) [487]

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

[Out]

2/5*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)^(3/2)-8/15*(a*d*e*(-a*e^2+c*d^2)*(3*a*e^2+c*d
^2)+(-2*a^3*e^6+a^2*c*d^2*e^4+c^3*d^6)*x)/e/(-a*e^2+c*d^2)^4/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)+8/15*(5*a
^2*e^4+10*a*c*d^2*e^2+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)^(1
/2)

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Rubi [A]
time = 0.15, antiderivative size = 259, normalized size of antiderivative = 1.00, 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 = {868, 791, 627} \begin {gather*} \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 (-2 a^3 e^6+a^2 c d^2 e^4+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}} \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)^(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 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 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 )^{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*}

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Mathematica [A]
time = 0.22, size = 235, normalized size = 0.91 \begin {gather*} \frac {2 \left (c^4 d^6 x^2 \left (15 d^2+20 d e x+8 e^2 x^2\right )+a^4 e^6 \left (8 d^2+20 d e x+15 e^2 x^2\right )+4 a^3 c d e^4 \left (20 d^3+53 d^2 e x+45 d e^2 x^2+15 e^3 x^3\right )+4 a c^3 d^4 e x \left (15 d^3+45 d^2 e x+53 d e^2 x^2+20 e^3 x^3\right )+2 a^2 c^2 d^2 e^2 \left (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\right )\right )}{15 \left (c d^2-a e^2\right )^5 (d+e x) ((a e+c d x) (d+e x))^{3/2}} \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)^(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 [B] Leaf count of result is larger than twice the leaf count of optimal. \(620\) vs. \(2(247)=494\).
time = 0.09, size = 621, normalized size = 2.40

method	result	size
gosper	\(-\frac {2 \left (c d x +a e \right ) \left (40 a^{2} c^{2} d^{2} e^{6} x^{4}+80 a \,c^{3} d^{4} e^{4} x^{4}+8 c^{4} d^{6} e^{2} x^{4}+60 a^{3} c d \,e^{7} x^{3}+220 a^{2} c^{2} d^{3} e^{5} x^{3}+212 a \,c^{3} d^{5} e^{3} x^{3}+20 c^{4} d^{7} e \,x^{3}+15 a^{4} e^{8} x^{2}+180 a^{3} c \,d^{2} e^{6} x^{2}+378 a^{2} c^{2} d^{4} e^{4} x^{2}+180 a \,c^{3} d^{6} e^{2} x^{2}+15 c^{4} d^{8} x^{2}+20 a^{4} d \,e^{7} x +212 a^{3} c \,d^{3} e^{5} x +220 a^{2} c^{2} d^{5} e^{3} x +60 a \,c^{3} d^{7} e x +8 a^{4} d^{2} e^{6}+80 a^{3} c \,d^{4} e^{4}+40 a^{2} c^{2} d^{6} e^{2}\right )}{15 \left (a^{5} e^{10}-5 a^{4} c \,d^{2} e^{8}+10 a^{3} c^{2} d^{4} e^{6}-10 a^{2} c^{3} d^{6} e^{4}+5 a \,c^{4} d^{8} e^{2}-c^{5} d^{10}\right ) \left (c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e \right )^{\frac {5}{2}}}\)	\(366\)
trager	\(-\frac {2 \left (40 a^{2} c^{2} d^{2} e^{6} x^{4}+80 a \,c^{3} d^{4} e^{4} x^{4}+8 c^{4} d^{6} e^{2} x^{4}+60 a^{3} c d \,e^{7} x^{3}+220 a^{2} c^{2} d^{3} e^{5} x^{3}+212 a \,c^{3} d^{5} e^{3} x^{3}+20 c^{4} d^{7} e \,x^{3}+15 a^{4} e^{8} x^{2}+180 a^{3} c \,d^{2} e^{6} x^{2}+378 a^{2} c^{2} d^{4} e^{4} x^{2}+180 a \,c^{3} d^{6} e^{2} x^{2}+15 c^{4} d^{8} x^{2}+20 a^{4} d \,e^{7} x +212 a^{3} c \,d^{3} e^{5} x +220 a^{2} c^{2} d^{5} e^{3} x +60 a \,c^{3} d^{7} e x +8 a^{4} d^{2} e^{6}+80 a^{3} c \,d^{4} e^{4}+40 a^{2} c^{2} d^{6} e^{2}\right ) \sqrt {c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e}}{15 \left (a^{4} e^{8}-4 a^{3} c \,d^{2} e^{6}+6 a^{2} c^{2} d^{4} e^{4}-4 a \,c^{3} d^{6} e^{2}+c^{4} d^{8}\right ) \left (c d x +a e \right )^{2} \left (a \,e^{2}-c \,d^{2}\right ) \left (e x +d \right )^{3}}\)	\(374\)
default	\(\frac {-\frac {1}{3 c d e \left (a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}\right )^{\frac {3}{2}}}-\frac {\left (a \,e^{2}+c \,d^{2}\right ) \left (\frac {\frac {4}{3} c d e x +\frac {2}{3} a \,e^{2}+\frac {2}{3} c \,d^{2}}{\left (4 a c \,d^{2} e^{2}-\left (a \,e^{2}+c \,d^{2}\right )^{2}\right ) \left (a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}\right )^{\frac {3}{2}}}+\frac {16 c d e \left (2 c d e x +a \,e^{2}+c \,d^{2}\right )}{3 \left (4 a c \,d^{2} e^{2}-\left (a \,e^{2}+c \,d^{2}\right )^{2}\right )^{2} \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}}}\right )}{2 c d e}}{e}-\frac {d \left (\frac {\frac {4}{3} c d e x +\frac {2}{3} a \,e^{2}+\frac {2}{3} c \,d^{2}}{\left (4 a c \,d^{2} e^{2}-\left (a \,e^{2}+c \,d^{2}\right )^{2}\right ) \left (a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}\right )^{\frac {3}{2}}}+\frac {16 c d e \left (2 c d e x +a \,e^{2}+c \,d^{2}\right )}{3 \left (4 a c \,d^{2} e^{2}-\left (a \,e^{2}+c \,d^{2}\right )^{2}\right )^{2} \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}}}\right )}{e^{2}}+\frac {d^{2} \left (-\frac {2}{5 \left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right ) \left (c d e \left (x +\frac {d}{e}\right )^{2}+\left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right )\right )^{\frac {3}{2}}}-\frac {8 c d e \left (-\frac {2 \left (2 c d e \left (x +\frac {d}{e}\right )+a \,e^{2}-c \,d^{2}\right )}{3 \left (a \,e^{2}-c \,d^{2}\right )^{2} \left (c d e \left (x +\frac {d}{e}\right )^{2}+\left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right )\right )^{\frac {3}{2}}}+\frac {16 c d e \left (2 c d e \left (x +\frac {d}{e}\right )+a \,e^{2}-c \,d^{2}\right )}{3 \left (a \,e^{2}-c \,d^{2}\right )^{4} \sqrt {c d e \left (x +\frac {d}{e}\right )^{2}+\left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right )}}\right )}{5 \left (a \,e^{2}-c \,d^{2}\right )}\right )}{e^{3}}\)	\(621\)

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,method=_RETURNVERBOSE)

[Out]

1/e*(-1/3/c/d/e/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)-1/2*(a*e^2+c*d^2)/c/d/e*(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*(2/3*(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*(-2/5/(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))^(3/2)-8/5*c*d*e/(a*e^2-c*d^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)^(5/2),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. 841 vs. \(2 (241) = 482\).
time = 53.61, size = 841, normalized size = 3.25 \begin {gather*} \frac {2 \, {\left (15 \, c^{4} d^{8} x^{2} + 15 \, a^{4} x^{2} e^{8} + 20 \, {\left (3 \, a^{3} c d x^{3} + a^{4} d x\right )} e^{7} + 4 \, {\left (10 \, a^{2} c^{2} d^{2} x^{4} + 45 \, a^{3} c d^{2} x^{2} + 2 \, a^{4} d^{2}\right )} e^{6} + 4 \, {\left (55 \, a^{2} c^{2} d^{3} x^{3} + 53 \, a^{3} c d^{3} x\right )} e^{5} + 2 \, {\left (40 \, a c^{3} d^{4} x^{4} + 189 \, a^{2} c^{2} d^{4} x^{2} + 40 \, a^{3} c d^{4}\right )} e^{4} + 4 \, {\left (53 \, a c^{3} d^{5} x^{3} + 55 \, a^{2} c^{2} d^{5} x\right )} e^{3} + 4 \, {\left (2 \, c^{4} d^{6} x^{4} + 45 \, a c^{3} d^{6} x^{2} + 10 \, a^{2} c^{2} d^{6}\right )} e^{2} + 20 \, {\left (c^{4} d^{7} x^{3} + 3 \, a c^{3} d^{7} x\right )} e\right )} \sqrt {c d^{2} x + a x e^{2} + {\left (c d x^{2} + a d\right )} e}}{15 \, {\left (c^{7} d^{15} x^{2} - a^{7} x^{3} e^{15} - {\left (2 \, a^{6} c d x^{4} + 3 \, a^{7} d x^{2}\right )} e^{14} - {\left (a^{5} c^{2} d^{2} x^{5} + a^{6} c d^{2} x^{3} + 3 \, a^{7} d^{2} x\right )} e^{13} + {\left (7 \, a^{5} c^{2} d^{3} x^{4} + 9 \, a^{6} c d^{3} x^{2} - a^{7} d^{3}\right )} e^{12} + {\left (5 \, a^{4} c^{3} d^{4} x^{5} + 17 \, a^{5} c^{2} d^{4} x^{3} + 13 \, a^{6} c d^{4} x\right )} e^{11} - {\left (5 \, a^{4} c^{3} d^{5} x^{4} + a^{5} c^{2} d^{5} x^{2} - 5 \, a^{6} c d^{5}\right )} e^{10} - 5 \, {\left (2 \, a^{3} c^{4} d^{6} x^{5} + 7 \, a^{4} c^{3} d^{6} x^{3} + 4 \, a^{5} c^{2} d^{6} x\right )} e^{9} - 5 \, {\left (2 \, a^{3} c^{4} d^{7} x^{4} + 5 \, a^{4} c^{3} d^{7} x^{2} + 2 \, a^{5} c^{2} d^{7}\right )} e^{8} + 5 \, {\left (2 \, a^{2} c^{5} d^{8} x^{5} + 5 \, a^{3} c^{4} d^{8} x^{3} + 2 \, a^{4} c^{3} d^{8} x\right )} e^{7} + 5 \, {\left (4 \, a^{2} c^{5} d^{9} x^{4} + 7 \, a^{3} c^{4} d^{9} x^{2} + 2 \, a^{4} c^{3} d^{9}\right )} e^{6} - {\left (5 \, a c^{6} d^{10} x^{5} - a^{2} c^{5} d^{10} x^{3} - 5 \, a^{3} c^{4} d^{10} x\right )} e^{5} - {\left (13 \, a c^{6} d^{11} x^{4} + 17 \, a^{2} c^{5} d^{11} x^{2} + 5 \, a^{3} c^{4} d^{11}\right )} e^{4} + {\left (c^{7} d^{12} x^{5} - 9 \, a c^{6} d^{12} x^{3} - 7 \, a^{2} c^{5} d^{12} x\right )} e^{3} + {\left (3 \, c^{7} d^{13} x^{4} + a c^{6} d^{13} x^{2} + a^{2} c^{5} d^{13}\right )} e^{2} + {\left (3 \, c^{7} d^{14} x^{3} + 2 \, a c^{6} d^{14} x\right )} e\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)^(5/2),x, algorithm="fricas")

[Out]

2/15*(15*c^4*d^8*x^2 + 15*a^4*x^2*e^8 + 20*(3*a^3*c*d*x^3 + a^4*d*x)*e^7 + 4*(10*a^2*c^2*d^2*x^4 + 45*a^3*c*d^
2*x^2 + 2*a^4*d^2)*e^6 + 4*(55*a^2*c^2*d^3*x^3 + 53*a^3*c*d^3*x)*e^5 + 2*(40*a*c^3*d^4*x^4 + 189*a^2*c^2*d^4*x
^2 + 40*a^3*c*d^4)*e^4 + 4*(53*a*c^3*d^5*x^3 + 55*a^2*c^2*d^5*x)*e^3 + 4*(2*c^4*d^6*x^4 + 45*a*c^3*d^6*x^2 + 1
0*a^2*c^2*d^6)*e^2 + 20*(c^4*d^7*x^3 + 3*a*c^3*d^7*x)*e)*sqrt(c*d^2*x + a*x*e^2 + (c*d*x^2 + a*d)*e)/(c^7*d^15
*x^2 - a^7*x^3*e^15 - (2*a^6*c*d*x^4 + 3*a^7*d*x^2)*e^14 - (a^5*c^2*d^2*x^5 + a^6*c*d^2*x^3 + 3*a^7*d^2*x)*e^1
3 + (7*a^5*c^2*d^3*x^4 + 9*a^6*c*d^3*x^2 - a^7*d^3)*e^12 + (5*a^4*c^3*d^4*x^5 + 17*a^5*c^2*d^4*x^3 + 13*a^6*c*
d^4*x)*e^11 - (5*a^4*c^3*d^5*x^4 + a^5*c^2*d^5*x^2 - 5*a^6*c*d^5)*e^10 - 5*(2*a^3*c^4*d^6*x^5 + 7*a^4*c^3*d^6*
x^3 + 4*a^5*c^2*d^6*x)*e^9 - 5*(2*a^3*c^4*d^7*x^4 + 5*a^4*c^3*d^7*x^2 + 2*a^5*c^2*d^7)*e^8 + 5*(2*a^2*c^5*d^8*
x^5 + 5*a^3*c^4*d^8*x^3 + 2*a^4*c^3*d^8*x)*e^7 + 5*(4*a^2*c^5*d^9*x^4 + 7*a^3*c^4*d^9*x^2 + 2*a^4*c^3*d^9)*e^6
- (5*a*c^6*d^10*x^5 - a^2*c^5*d^10*x^3 - 5*a^3*c^4*d^10*x)*e^5 - (13*a*c^6*d^11*x^4 + 17*a^2*c^5*d^11*x^2 + 5
*a^3*c^4*d^11)*e^4 + (c^7*d^12*x^5 - 9*a*c^6*d^12*x^3 - 7*a^2*c^5*d^12*x)*e^3 + (3*c^7*d^13*x^4 + a*c^6*d^13*x
^2 + a^2*c^5*d^13)*e^2 + (3*c^7*d^14*x^3 + 2*a*c^6*d^14*x)*e)

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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)**(5/2),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)^(5/2),x, algorithm="giac")

[Out]

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

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Mupad [B]
time = 4.33, size = 3099, normalized size = 11.97 \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)^(5/2)),x)

[Out]

(((6*a*e^2 - 10*c*d^2)/(15*(a*e^2 - c*d^2)^4) - (4*c*d^2)/(5*(a*e^2 - c*d^2)^4))*(x*(a*e^2 + c*d^2) + a*d*e +
c*d*e*x^2)^(1/2))/(d + e*x) - (((d*((e*(2*a*e^3 - 2*c*d^2*e))/(5*(a*e^2 - c*d^2)^3*(3*a*e^3 - 3*c*d^2*e)) - (4
*c*d^2*e^2)/(5*(a*e^2 - c*d^2)^3*(3*a*e^3 - 3*c*d^2*e))))/e + (e*(2*c*d^3 + 2*a*d*e^2))/(5*(a*e^2 - c*d^2)^3*(
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*e^2 + c*d^2) + a*d*
e + c*d*e*x^2)^(1/2)*(x*((((12*c^3*d^3*e^2)/(5*(a*e^2 - c*d^2)^2*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5))
- (4*c^3*d^3*e^2*(a*e^2 + c*d^2))/(5*(a*e^2 - c*d^2)^3*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5)))*(a*e^2 +
c*d^2))/(c*d*e) - (6*c^2*d^2*e*(a*e^2 + c*d^2))/(5*(a*e^2 - c*d^2)^2*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^
5)) + (8*a*c^3*d^4*e^3)/(5*(a*e^2 - c*d^2)^3*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5)) - (2*c^2*d^2*e*(46*a
^2*e^4 + 4*c^2*d^4 + 66*a*c*d^2*e^2))/(15*(a*e^2 - c*d^2)^3*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5))) + (a
*((12*c^3*d^3*e^2)/(5*(a*e^2 - c*d^2)^2*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5)) - (4*c^3*d^3*e^2*(a*e^2 +
c*d^2))/(5*(a*e^2 - c*d^2)^3*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5))))/c - (c*d*(a*e^2 + c*d^2)*(46*a^2*
e^4 + 4*c^2*d^4 + 66*a*c*d^2*e^2))/(15*(a*e^2 - c*d^2)^3*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5))))/((a*e
+ c*d*x)*(d + e*x)) + ((x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2)*(x*((a*(((a*e^2 + c*d^2)*((4*c^4*d^4*e^3*
(a*e^2 + c*d^2))/(15*(a*e^2 - c*d^2)^3*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5)) - (4*c^4*d^4*e^3*(5*a*e^2
- c*d^2))/(15*(a*e^2 - c*d^2)^3*(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*(10*c^
3*d^5 + 6*a*c^2*d^3*e^2 - 8*a^2*c*d*e^4))/(15*(a*e^2 - c*d^2)^3*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5)) -
(8*a*c^4*d^5*e^4)/(15*(a*e^2 - c*d^2)^3*(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)*(5*a*e^2 - c*d^2))/(15*(a*e^2 - c*d^2)^3*(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*((4*c^4*d^4*e^3*(a*e^2 + c*d^2))/(15*(a*e^2 - c*d^2)^3*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5)
) - (4*c^4*d^4*e^3*(5*a*e^2 - c*d^2))/(15*(a*e^2 - c*d^2)^3*(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^4*d^4*e^3*(a*e^2 + c*d^2))/(15*(a*e^2 - c*d^2)^3*(c^3*d^5*e - 2*a*c^
2*d^3*e^3 + a^2*c*d*e^5)) - (4*c^4*d^4*e^3*(5*a*e^2 - c*d^2))/(15*(a*e^2 - c*d^2)^3*(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*(10*c^3*d^5 + 6*a*c^2*d^3*e^2 - 8*a^2*c*d*e^4))/(15*(a*e^2 - c*d
^2)^3*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5)) - (8*a*c^4*d^5*e^4)/(15*(a*e^2 - c*d^2)^3*(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)*(5*a*e^2 - c*d^2))/(15*(a*e^2 - c*d^2)^3*(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*(12*a^3*e^5 - 36*a^2*c*d^2*e^3))/(15*(a*e^2 -
c*d^2)^3*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5)) - (c*d*e*(a*e^2 + c*d^2)*(10*c^3*d^5 + 6*a*c^2*d^3*e^2 -
8*a^2*c*d*e^4))/(15*(a*e^2 - c*d^2)^3*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5))))/(c*d*e) + (8*a^3*c^2*d^3
*e^6)/(5*(a*e^2 - c*d^2)^3*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5)) - (c*d*e*(12*a^3*e^5 - 36*a^2*c*d^2*e^
3)*(a*e^2 + c*d^2))/(15*(a*e^2 - c*d^2)^3*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5))) + (a*((a*((4*c^4*d^4*e
^3*(a*e^2 + c*d^2))/(15*(a*e^2 - c*d^2)^3*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5)) - (4*c^4*d^4*e^3*(5*a*e
^2 - c*d^2))/(15*(a*e^2 - c*d^2)^3*(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^4*d^4*e^3*(a*e^2 + c*d^2))/(15*(a*e^2 - c*d^2)^3*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5))
- (4*c^4*d^4*e^3*(5*a*e^2 - c*d^2))/(15*(a*e^2 - c*d^2)^3*(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*(10*c^3*d^5 + 6*a*c^2*d^3*e^2 - 8*a^2*c*d*e^4))/(15*(a*e^2 - c*d^2)^3*(c^3*d^5*e - 2*a*c^
2*d^3*e^3 + a^2*c*d*e^5)) - (8*a*c^4*d^5*e^4)/(15*(a*e^2 - c*d^2)^3*(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)*(5*a*e^2 - c*d^2))/(15*(a*e^2 - c*d^2)^3*(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*(12*a^3*e^5 - 36*a^2*c*d^2*e^3))/(15*(a*e^2 - c*d^2)^3*(c^3*d^5*e - 2*a
*c^2*d^3*e^3 + a^2*c*d*e^5)) - (c*d*e*(a*e^2 + c*d^2)*(10*c^3*d^5 + 6*a*c^2*d^3*e^2 - 8*a^2*c*d*e^4))/(15*(a*e
^2 - c*d^2)^3*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5))))/c + (4*a^3*c*d^2*e^5*(a*e^2 + c*d^2))/(5*(a*e^2 -
c*d^2)^3*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5))))/((a*e + c*d*x)^2*(d + e*x)^2) - (2*d^2*e*(x*(a*e^2 +
c*d^2) + a*d*e + c*d*e*x^2)^(1/2))/((d + e*x)^3*(5*a^3*e^7 - 5*c^3*d^6*e + 15*a*c^2*d^4*e^3 - 15*a^2*c*d^2*e^5
))

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