3.5.80 \(\int \frac {x^2}{(d+e x) (a d e+(c d^2+a e^2) x+c d e x^2)^{3/2}} \, dx\) [480]

Optimal. Leaf size=126 \[ \frac {2 x^2}{3 \left (c d^2-a e^2\right ) (d+e x) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}-\frac {8 a e \left (2 a d e+\left (c d^2+a e^2\right ) x\right )}{3 \left (c d^2-a e^2\right )^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \]

[Out]

2/3*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)^(1/2)-8/3*a*e*(2*a*d*e+(a*e^2+c*d^2)*x)/(-a*e
^2+c*d^2)^3/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)

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Rubi [A]
time = 0.07, antiderivative size = 126, 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, 12, 650} \begin {gather*} \frac {2 x^2}{3 (d+e x) \left (c d^2-a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}-\frac {8 a e \left (x \left (a e^2+c d^2\right )+2 a d e\right )}{3 \left (c d^2-a e^2\right )^3 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^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)^(3/2)),x]

[Out]

(2*x^2)/(3*(c*d^2 - a*e^2)*(d + e*x)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]) - (8*a*e*(2*a*d*e + (c*d^2 +
a*e^2)*x))/(3*(c*d^2 - a*e^2)^3*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 650

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

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 )^{3/2}} \, dx &=\frac {2 x^2}{3 \left (c d^2-a e^2\right ) (d+e x) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {2 \int -\frac {2 a d e^2 \left (c d^2-a e^2\right ) x}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx}{3 d e \left (c d^2-a e^2\right )^2}\\ &=\frac {2 x^2}{3 \left (c d^2-a e^2\right ) (d+e x) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}-\frac {(4 a e) \int \frac {x}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx}{3 \left (c d^2-a e^2\right )}\\ &=\frac {2 x^2}{3 \left (c d^2-a e^2\right ) (d+e x) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}-\frac {8 a e \left (2 a d e+\left (c d^2+a e^2\right ) x\right )}{3 \left (c d^2-a e^2\right )^3 \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 = 0.15, size = 95, normalized size = 0.75 \begin {gather*} \frac {2 (a e+c d x)^3 \left (d^2-\frac {6 a d e (d+e x)}{a e+c d x}-\frac {3 a^2 e^2 (d+e x)^2}{(a e+c d x)^2}\right )}{3 \left (c d^2-a e^2\right )^3 ((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)^(3/2)),x]

[Out]

(2*(a*e + c*d*x)^3*(d^2 - (6*a*d*e*(d + e*x))/(a*e + c*d*x) - (3*a^2*e^2*(d + e*x)^2)/(a*e + c*d*x)^2))/(3*(c*
d^2 - a*e^2)^3*((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. \(364\) vs. \(2(118)=236\).
time = 0.10, size = 365, normalized size = 2.90

method	result	size
gosper	\(\frac {2 \left (c d x +a e \right ) \left (3 a^{2} e^{4} x^{2}+6 a c \,d^{2} e^{2} x^{2}-c^{2} d^{4} x^{2}+12 a^{2} d \,e^{3} x +4 a c \,d^{3} e x +8 a^{2} d^{2} e^{2}\right )}{3 \left (a^{3} e^{6}-3 a^{2} c \,d^{2} e^{4}+3 a \,c^{2} d^{4} e^{2}-c^{3} d^{6}\right ) \left (c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e \right )^{\frac {3}{2}}}\)	\(145\)
trager	\(\frac {2 \left (3 a^{2} e^{4} x^{2}+6 a c \,d^{2} e^{2} x^{2}-c^{2} d^{4} x^{2}+12 a^{2} d \,e^{3} x +4 a c \,d^{3} e x +8 a^{2} d^{2} e^{2}\right ) \sqrt {c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e}}{3 \left (a^{2} e^{4}-2 a c \,d^{2} e^{2}+c^{2} d^{4}\right ) \left (e x +d \right )^{2} \left (a \,e^{2}-c \,d^{2}\right ) \left (c d x +a e \right )}\)	\(153\)
default	\(\frac {-\frac {1}{c d e \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}}}-\frac {\left (a \,e^{2}+c \,d^{2}\right ) \left (2 c d e x +a \,e^{2}+c \,d^{2}\right )}{c d e \left (4 a c \,d^{2} e^{2}-\left (a \,e^{2}+c \,d^{2}\right )^{2}\right ) \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}}}}{e}-\frac {2 d \left (2 c d e x +a \,e^{2}+c \,d^{2}\right )}{e^{2} \left (4 a c \,d^{2} e^{2}-\left (a \,e^{2}+c \,d^{2}\right )^{2}\right ) \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}}}+\frac {d^{2} \left (-\frac {2}{3 \left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right ) \sqrt {c d e \left (x +\frac {d}{e}\right )^{2}+\left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right )}}+\frac {8 c d e \left (2 c d e \left (x +\frac {d}{e}\right )+a \,e^{2}-c \,d^{2}\right )}{3 \left (a \,e^{2}-c \,d^{2}\right )^{3} \sqrt {c d e \left (x +\frac {d}{e}\right )^{2}+\left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right )}}\right )}{e^{3}}\)	\(365\)

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)^(3/2),x,method=_RETURNVERBOSE)

[Out]

1/e*(-1/c/d/e/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)-(a*e^2+c*d^2)/c/d/e*(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)^(1/2))-2*d/e^2*(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)^(1/2)+1/e^3*d^2*(-2/3/(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))^(1/2)+8/3*c*d*e/(a*e^2-c*d^2)^3*(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)^(3/2),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. 313 vs. \(2 (120) = 240\).
time = 6.51, size = 313, normalized size = 2.48 \begin {gather*} \frac {2 \, {\left (c^{2} d^{4} x^{2} - 4 \, a c d^{3} x e - 3 \, a^{2} x^{2} e^{4} - 12 \, a^{2} d x e^{3} - 2 \, {\left (3 \, a c d^{2} x^{2} + 4 \, a^{2} d^{2}\right )} e^{2}\right )} \sqrt {c d^{2} x + a x e^{2} + {\left (c d x^{2} + a d\right )} e}}{3 \, {\left (c^{4} d^{9} x - a^{4} x^{2} e^{9} - {\left (a^{3} c d x^{3} + 2 \, a^{4} d x\right )} e^{8} + {\left (a^{3} c d^{2} x^{2} - a^{4} d^{2}\right )} e^{7} + {\left (3 \, a^{2} c^{2} d^{3} x^{3} + 5 \, a^{3} c d^{3} x\right )} e^{6} + 3 \, {\left (a^{2} c^{2} d^{4} x^{2} + a^{3} c d^{4}\right )} e^{5} - 3 \, {\left (a c^{3} d^{5} x^{3} + a^{2} c^{2} d^{5} x\right )} e^{4} - {\left (5 \, a c^{3} d^{6} x^{2} + 3 \, a^{2} c^{2} d^{6}\right )} e^{3} + {\left (c^{4} d^{7} x^{3} - a c^{3} d^{7} x\right )} e^{2} + {\left (2 \, c^{4} d^{8} x^{2} + a c^{3} d^{8}\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)^(3/2),x, algorithm="fricas")

[Out]

2/3*(c^2*d^4*x^2 - 4*a*c*d^3*x*e - 3*a^2*x^2*e^4 - 12*a^2*d*x*e^3 - 2*(3*a*c*d^2*x^2 + 4*a^2*d^2)*e^2)*sqrt(c*
d^2*x + a*x*e^2 + (c*d*x^2 + a*d)*e)/(c^4*d^9*x - a^4*x^2*e^9 - (a^3*c*d*x^3 + 2*a^4*d*x)*e^8 + (a^3*c*d^2*x^2
- a^4*d^2)*e^7 + (3*a^2*c^2*d^3*x^3 + 5*a^3*c*d^3*x)*e^6 + 3*(a^2*c^2*d^4*x^2 + a^3*c*d^4)*e^5 - 3*(a*c^3*d^5
*x^3 + a^2*c^2*d^5*x)*e^4 - (5*a*c^3*d^6*x^2 + 3*a^2*c^2*d^6)*e^3 + (c^4*d^7*x^3 - a*c^3*d^7*x)*e^2 + (2*c^4*d
^8*x^2 + a*c^3*d^8)*e)

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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 (a e + c d x\right )\right )^{\frac {3}{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)/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(3/2),x)

[Out]

Integral(x**2/(((d + e*x)*(a*e + c*d*x))**(3/2)*(d + e*x)), x)

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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)^(3/2),x, algorithm="giac")

[Out]

integrate(x^2/((c*d*x^2*e + a*d*e + (c*d^2 + a*e^2)*x)^(3/2)*(x*e + d)), x)

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Mupad [B]
time = 3.60, size = 1071, normalized size = 8.50 \begin {gather*} \frac {4\,c\,d^3\,\sqrt {c\,d^2\,x+c\,d\,e\,x^2+a\,d\,e+a\,e^2\,x}}{3\,\left (a^3\,d\,e^7+x\,a^3\,e^8-3\,a^2\,c\,d^3\,e^5-3\,x\,a^2\,c\,d^2\,e^6+3\,a\,c^2\,d^5\,e^3+3\,x\,a\,c^2\,d^4\,e^4-c^3\,d^7\,e-x\,c^3\,d^6\,e^2\right )}-\frac {2\,d^2\,\sqrt {c\,d^2\,x+c\,d\,e\,x^2+a\,d\,e+a\,e^2\,x}}{3\,a^2\,d^2\,e^5+6\,a^2\,d\,e^6\,x+3\,a^2\,e^7\,x^2-6\,a\,c\,d^4\,e^3-12\,a\,c\,d^3\,e^4\,x-6\,a\,c\,d^2\,e^5\,x^2+3\,c^2\,d^6\,e+6\,c^2\,d^5\,e^2\,x+3\,c^2\,d^4\,e^3\,x^2}-\frac {4\,a\,d\,e^2\,\sqrt {c\,d^2\,x+c\,d\,e\,x^2+a\,d\,e+a\,e^2\,x}}{3\,\left (a^3\,d\,e^7+x\,a^3\,e^8-3\,a^2\,c\,d^3\,e^5-3\,x\,a^2\,c\,d^2\,e^6+3\,a\,c^2\,d^5\,e^3+3\,x\,a\,c^2\,d^4\,e^4-c^3\,d^7\,e-x\,c^3\,d^6\,e^2\right )}+\frac {2\,c^4\,d^7\,x}{\sqrt {c\,d^2\,x+c\,d\,e\,x^2+a\,d\,e+a\,e^2\,x}\,\left (a^4\,c\,d\,e^9-4\,a^3\,c^2\,d^3\,e^7+6\,a^2\,c^3\,d^5\,e^5-4\,a\,c^4\,d^7\,e^3+c^5\,d^9\,e\right )}+\frac {22\,a^3\,c\,d^2\,e^5}{3\,\sqrt {c\,d^2\,x+c\,d\,e\,x^2+a\,d\,e+a\,e^2\,x}\,\left (a^4\,c\,d\,e^9-4\,a^3\,c^2\,d^3\,e^7+6\,a^2\,c^3\,d^5\,e^5-4\,a\,c^4\,d^7\,e^3+c^5\,d^9\,e\right )}-\frac {28\,a^2\,c^2\,d^4\,e^3}{3\,\sqrt {c\,d^2\,x+c\,d\,e\,x^2+a\,d\,e+a\,e^2\,x}\,\left (a^4\,c\,d\,e^9-4\,a^3\,c^2\,d^3\,e^7+6\,a^2\,c^3\,d^5\,e^5-4\,a\,c^4\,d^7\,e^3+c^5\,d^9\,e\right )}+\frac {2\,a\,c^3\,d^6\,e}{\sqrt {c\,d^2\,x+c\,d\,e\,x^2+a\,d\,e+a\,e^2\,x}\,\left (a^4\,c\,d\,e^9-4\,a^3\,c^2\,d^3\,e^7+6\,a^2\,c^3\,d^5\,e^5-4\,a\,c^4\,d^7\,e^3+c^5\,d^9\,e\right )}+\frac {10\,a^2\,c^2\,d^3\,e^4\,x}{3\,\sqrt {c\,d^2\,x+c\,d\,e\,x^2+a\,d\,e+a\,e^2\,x}\,\left (a^4\,c\,d\,e^9-4\,a^3\,c^2\,d^3\,e^7+6\,a^2\,c^3\,d^5\,e^5-4\,a\,c^4\,d^7\,e^3+c^5\,d^9\,e\right )}+\frac {2\,a^3\,c\,d\,e^6\,x}{\sqrt {c\,d^2\,x+c\,d\,e\,x^2+a\,d\,e+a\,e^2\,x}\,\left (a^4\,c\,d\,e^9-4\,a^3\,c^2\,d^3\,e^7+6\,a^2\,c^3\,d^5\,e^5-4\,a\,c^4\,d^7\,e^3+c^5\,d^9\,e\right )}-\frac {22\,a\,c^3\,d^5\,e^2\,x}{3\,\sqrt {c\,d^2\,x+c\,d\,e\,x^2+a\,d\,e+a\,e^2\,x}\,\left (a^4\,c\,d\,e^9-4\,a^3\,c^2\,d^3\,e^7+6\,a^2\,c^3\,d^5\,e^5-4\,a\,c^4\,d^7\,e^3+c^5\,d^9\,e\right )} \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)^(3/2)),x)

[Out]

(4*c*d^3*(a*d*e + a*e^2*x + c*d^2*x + c*d*e*x^2)^(1/2))/(3*(a^3*d*e^7 - c^3*d^7*e + a^3*e^8*x + 3*a*c^2*d^5*e^
3 - 3*a^2*c*d^3*e^5 - c^3*d^6*e^2*x + 3*a*c^2*d^4*e^4*x - 3*a^2*c*d^2*e^6*x)) - (2*d^2*(a*d*e + a*e^2*x + c*d^
2*x + c*d*e*x^2)^(1/2))/(3*c^2*d^6*e + 3*a^2*d^2*e^5 + 3*a^2*e^7*x^2 + 6*c^2*d^5*e^2*x + 3*c^2*d^4*e^3*x^2 - 6
*a*c*d^4*e^3 + 6*a^2*d*e^6*x - 12*a*c*d^3*e^4*x - 6*a*c*d^2*e^5*x^2) - (4*a*d*e^2*(a*d*e + a*e^2*x + c*d^2*x +
c*d*e*x^2)^(1/2))/(3*(a^3*d*e^7 - c^3*d^7*e + a^3*e^8*x + 3*a*c^2*d^5*e^3 - 3*a^2*c*d^3*e^5 - c^3*d^6*e^2*x +
3*a*c^2*d^4*e^4*x - 3*a^2*c*d^2*e^6*x)) + (2*c^4*d^7*x)/((a*d*e + a*e^2*x + c*d^2*x + c*d*e*x^2)^(1/2)*(c^5*d
^9*e - 4*a*c^4*d^7*e^3 + 6*a^2*c^3*d^5*e^5 - 4*a^3*c^2*d^3*e^7 + a^4*c*d*e^9)) + (22*a^3*c*d^2*e^5)/(3*(a*d*e
+ a*e^2*x + c*d^2*x + c*d*e*x^2)^(1/2)*(c^5*d^9*e - 4*a*c^4*d^7*e^3 + 6*a^2*c^3*d^5*e^5 - 4*a^3*c^2*d^3*e^7 +
a^4*c*d*e^9)) - (28*a^2*c^2*d^4*e^3)/(3*(a*d*e + a*e^2*x + c*d^2*x + c*d*e*x^2)^(1/2)*(c^5*d^9*e - 4*a*c^4*d^7
*e^3 + 6*a^2*c^3*d^5*e^5 - 4*a^3*c^2*d^3*e^7 + a^4*c*d*e^9)) + (2*a*c^3*d^6*e)/((a*d*e + a*e^2*x + c*d^2*x + c
*d*e*x^2)^(1/2)*(c^5*d^9*e - 4*a*c^4*d^7*e^3 + 6*a^2*c^3*d^5*e^5 - 4*a^3*c^2*d^3*e^7 + a^4*c*d*e^9)) + (10*a^2
*c^2*d^3*e^4*x)/(3*(a*d*e + a*e^2*x + c*d^2*x + c*d*e*x^2)^(1/2)*(c^5*d^9*e - 4*a*c^4*d^7*e^3 + 6*a^2*c^3*d^5*
e^5 - 4*a^3*c^2*d^3*e^7 + a^4*c*d*e^9)) + (2*a^3*c*d*e^6*x)/((a*d*e + a*e^2*x + c*d^2*x + c*d*e*x^2)^(1/2)*(c^
5*d^9*e - 4*a*c^4*d^7*e^3 + 6*a^2*c^3*d^5*e^5 - 4*a^3*c^2*d^3*e^7 + a^4*c*d*e^9)) - (22*a*c^3*d^5*e^2*x)/(3*(a
*d*e + a*e^2*x + c*d^2*x + c*d*e*x^2)^(1/2)*(c^5*d^9*e - 4*a*c^4*d^7*e^3 + 6*a^2*c^3*d^5*e^5 - 4*a^3*c^2*d^3*e
^7 + a^4*c*d*e^9))

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