### 3.1970 $$\int \frac{1}{(a d e+(c d^2+a e^2) x+c d e x^2)^{5/2}} \, dx$$

Optimal. Leaf size=132 $\frac{16 c d e \left (a e^2+c d^2+2 c d e x\right )}{3 \left (c d^2-a e^2\right )^4 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}-\frac{2 \left (a e^2+c d^2+2 c d e x\right )}{3 \left (c d^2-a e^2\right )^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}$

[Out]

(-2*(c*d^2 + a*e^2 + 2*c*d*e*x))/(3*(c*d^2 - a*e^2)^2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)) + (16*c*d
*e*(c*d^2 + a*e^2 + 2*c*d*e*x))/(3*(c*d^2 - a*e^2)^4*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])

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Rubi [A]  time = 0.024465, antiderivative size = 132, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 29, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.069, Rules used = {614, 613} $\frac{16 c d e \left (a e^2+c d^2+2 c d e x\right )}{3 \left (c d^2-a e^2\right )^4 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}-\frac{2 \left (a e^2+c d^2+2 c d e x\right )}{3 \left (c d^2-a e^2\right )^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(-2*(c*d^2 + a*e^2 + 2*c*d*e*x))/(3*(c*d^2 - a*e^2)^2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)) + (16*c*d
*e*(c*d^2 + a*e^2 + 2*c*d*e*x))/(3*(c*d^2 - a*e^2)^4*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])

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{1}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx &=-\frac{2 \left (c d^2+a e^2+2 c d e x\right )}{3 \left (c d^2-a e^2\right )^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}-\frac{(8 c d e) \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}{3 \left (c d^2-a e^2\right )^2}\\ &=-\frac{2 \left (c d^2+a e^2+2 c d e x\right )}{3 \left (c d^2-a e^2\right )^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}+\frac{16 c d e \left (c d^2+a e^2+2 c d e x\right )}{3 \left (c d^2-a e^2\right )^4 \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\\ \end{align*}

Mathematica [A]  time = 0.0657726, size = 132, normalized size = 1. $\frac{6 a^2 c d e^4 (3 d+2 e x)-2 a^3 e^6+6 a c^2 d^2 e^2 \left (3 d^2+12 d e x+8 e^2 x^2\right )+2 c^3 d^3 \left (6 d^2 e x-d^3+24 d e^2 x^2+16 e^3 x^3\right )}{3 \left (c d^2-a e^2\right )^4 ((d+e x) (a e+c d x))^{3/2}}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(-2*a^3*e^6 + 6*a^2*c*d*e^4*(3*d + 2*e*x) + 6*a*c^2*d^2*e^2*(3*d^2 + 12*d*e*x + 8*e^2*x^2) + 2*c^3*d^3*(-d^3 +
6*d^2*e*x + 24*d*e^2*x^2 + 16*e^3*x^3))/(3*(c*d^2 - a*e^2)^4*((a*e + c*d*x)*(d + e*x))^(3/2))

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Maple [A]  time = 0.045, size = 213, normalized size = 1.6 \begin{align*} -{\frac{ \left ( 2\,cdx+2\,ae \right ) \left ( ex+d \right ) \left ( -16\,{c}^{3}{d}^{3}{e}^{3}{x}^{3}-24\,a{c}^{2}{d}^{2}{e}^{4}{x}^{2}-24\,{c}^{3}{d}^{4}{e}^{2}{x}^{2}-6\,{a}^{2}cd{e}^{5}x-36\,a{c}^{2}{d}^{3}{e}^{3}x-6\,{c}^{3}{d}^{5}ex+{a}^{3}{e}^{6}-9\,{a}^{2}c{d}^{2}{e}^{4}-9\,a{c}^{2}{d}^{4}{e}^{2}+{c}^{3}{d}^{6} \right ) }{3\,{a}^{4}{e}^{8}-12\,{a}^{3}c{d}^{2}{e}^{6}+18\,{a}^{2}{c}^{2}{d}^{4}{e}^{4}-12\,a{c}^{3}{d}^{6}{e}^{2}+3\,{c}^{4}{d}^{8}} \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(1/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2),x)

[Out]

-2/3*(c*d*x+a*e)*(e*x+d)*(-16*c^3*d^3*e^3*x^3-24*a*c^2*d^2*e^4*x^2-24*c^3*d^4*e^2*x^2-6*a^2*c*d*e^5*x-36*a*c^2
*d^3*e^3*x-6*c^3*d^5*e*x+a^3*e^6-9*a^2*c*d^2*e^4-9*a*c^2*d^4*e^2+c^3*d^6)/(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)/(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(1/(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 [B]  time = 58.0201, size = 972, normalized size = 7.36 \begin{align*} \frac{2 \,{\left (16 \, c^{3} d^{3} e^{3} x^{3} - c^{3} d^{6} + 9 \, a c^{2} d^{4} e^{2} + 9 \, a^{2} c d^{2} e^{4} - a^{3} e^{6} + 24 \,{\left (c^{3} d^{4} e^{2} + a c^{2} d^{2} e^{4}\right )} x^{2} + 6 \,{\left (c^{3} d^{5} e + 6 \, a c^{2} d^{3} e^{3} + a^{2} c d e^{5}\right )} x\right )} \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x}}{3 \,{\left (a^{2} c^{4} d^{10} e^{2} - 4 \, a^{3} c^{3} d^{8} e^{4} + 6 \, a^{4} c^{2} d^{6} e^{6} - 4 \, a^{5} c d^{4} e^{8} + a^{6} d^{2} e^{10} +{\left (c^{6} d^{10} e^{2} - 4 \, a c^{5} d^{8} e^{4} + 6 \, a^{2} c^{4} d^{6} e^{6} - 4 \, a^{3} c^{3} d^{4} e^{8} + a^{4} c^{2} d^{2} e^{10}\right )} x^{4} + 2 \,{\left (c^{6} d^{11} e - 3 \, a c^{5} d^{9} e^{3} + 2 \, a^{2} c^{4} d^{7} e^{5} + 2 \, a^{3} c^{3} d^{5} e^{7} - 3 \, a^{4} c^{2} d^{3} e^{9} + a^{5} c d e^{11}\right )} x^{3} +{\left (c^{6} d^{12} - 9 \, a^{2} c^{4} d^{8} e^{4} + 16 \, a^{3} c^{3} d^{6} e^{6} - 9 \, a^{4} c^{2} d^{4} e^{8} + a^{6} e^{12}\right )} x^{2} + 2 \,{\left (a c^{5} d^{11} e - 3 \, a^{2} c^{4} d^{9} e^{3} + 2 \, a^{3} c^{3} d^{7} e^{5} + 2 \, a^{4} c^{2} d^{5} e^{7} - 3 \, a^{5} c d^{3} e^{9} + a^{6} d e^{11}\right )} x\right )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2),x, algorithm="fricas")

[Out]

2/3*(16*c^3*d^3*e^3*x^3 - c^3*d^6 + 9*a*c^2*d^4*e^2 + 9*a^2*c*d^2*e^4 - a^3*e^6 + 24*(c^3*d^4*e^2 + a*c^2*d^2*
e^4)*x^2 + 6*(c^3*d^5*e + 6*a*c^2*d^3*e^3 + a^2*c*d*e^5)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)/(a^2*c
^4*d^10*e^2 - 4*a^3*c^3*d^8*e^4 + 6*a^4*c^2*d^6*e^6 - 4*a^5*c*d^4*e^8 + a^6*d^2*e^10 + (c^6*d^10*e^2 - 4*a*c^5
*d^8*e^4 + 6*a^2*c^4*d^6*e^6 - 4*a^3*c^3*d^4*e^8 + a^4*c^2*d^2*e^10)*x^4 + 2*(c^6*d^11*e - 3*a*c^5*d^9*e^3 + 2
*a^2*c^4*d^7*e^5 + 2*a^3*c^3*d^5*e^7 - 3*a^4*c^2*d^3*e^9 + a^5*c*d*e^11)*x^3 + (c^6*d^12 - 9*a^2*c^4*d^8*e^4 +
16*a^3*c^3*d^6*e^6 - 9*a^4*c^2*d^4*e^8 + a^6*e^12)*x^2 + 2*(a*c^5*d^11*e - 3*a^2*c^4*d^9*e^3 + 2*a^3*c^3*d^7*
e^5 + 2*a^4*c^2*d^5*e^7 - 3*a^5*c*d^3*e^9 + a^6*d*e^11)*x)

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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(1/(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 [B]  time = 1.3058, size = 466, normalized size = 3.53 \begin{align*} \frac{2 \,{\left (2 \,{\left (4 \,{\left (\frac{2 \, c^{3} d^{3} x e^{3}}{c^{4} d^{8} - 4 \, a c^{3} d^{6} e^{2} + 6 \, a^{2} c^{2} d^{4} e^{4} - 4 \, a^{3} c d^{2} e^{6} + a^{4} e^{8}} + \frac{3 \,{\left (c^{3} d^{4} e^{2} + a c^{2} d^{2} e^{4}\right )}}{c^{4} d^{8} - 4 \, a c^{3} d^{6} e^{2} + 6 \, a^{2} c^{2} d^{4} e^{4} - 4 \, a^{3} c d^{2} e^{6} + a^{4} e^{8}}\right )} x + \frac{3 \,{\left (c^{3} d^{5} e + 6 \, a c^{2} d^{3} e^{3} + a^{2} c d e^{5}\right )}}{c^{4} d^{8} - 4 \, a c^{3} d^{6} e^{2} + 6 \, a^{2} c^{2} d^{4} e^{4} - 4 \, a^{3} c d^{2} e^{6} + a^{4} e^{8}}\right )} x - \frac{c^{3} d^{6} - 9 \, a c^{2} d^{4} e^{2} - 9 \, a^{2} c d^{2} e^{4} + a^{3} e^{6}}{c^{4} d^{8} - 4 \, a c^{3} d^{6} e^{2} + 6 \, a^{2} c^{2} d^{4} e^{4} - 4 \, a^{3} c d^{2} e^{6} + a^{4} e^{8}}\right )}}{3 \,{\left (c d x^{2} e + a d e +{\left (c d^{2} + a e^{2}\right )} x\right )}^{\frac{3}{2}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2),x, algorithm="giac")

[Out]

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