3.1952 \(\int \frac{1}{(d+e x)^3 \sqrt{a d e+(c d^2+a e^2) x+c d e x^2}} \, dx\)

Optimal. Leaf size=171 \[ \frac{16 c^2 d^2 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{15 (d+e x) \left (c d^2-a e^2\right )^3}+\frac{8 c d \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{15 (d+e x)^2 \left (c d^2-a e^2\right )^2}+\frac{2 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{5 (d+e x)^3 \left (c d^2-a e^2\right )} \]

[Out]

(2*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(5*(c*d^2 - a*e^2)*(d + e*x)^3) + (8*c*d*Sqrt[a*d*e + (c*d^2 +
a*e^2)*x + c*d*e*x^2])/(15*(c*d^2 - a*e^2)^2*(d + e*x)^2) + (16*c^2*d^2*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*
e*x^2])/(15*(c*d^2 - a*e^2)^3*(d + e*x))

________________________________________________________________________________________

Rubi [A]  time = 0.078461, antiderivative size = 171, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 37, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.054, Rules used = {658, 650} \[ \frac{16 c^2 d^2 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{15 (d+e x) \left (c d^2-a e^2\right )^3}+\frac{8 c d \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{15 (d+e x)^2 \left (c d^2-a e^2\right )^2}+\frac{2 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{5 (d+e x)^3 \left (c d^2-a e^2\right )} \]

Antiderivative was successfully verified.

[In]

Int[1/((d + e*x)^3*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]),x]

[Out]

(2*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(5*(c*d^2 - a*e^2)*(d + e*x)^3) + (8*c*d*Sqrt[a*d*e + (c*d^2 +
a*e^2)*x + c*d*e*x^2])/(15*(c*d^2 - a*e^2)^2*(d + e*x)^2) + (16*c^2*d^2*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*
e*x^2])/(15*(c*d^2 - a*e^2)^3*(d + e*x))

Rule 658

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> -Simp[(e*(d + e*x)^m*(a +
b*x + c*x^2)^(p + 1))/((m + p + 1)*(2*c*d - b*e)), x] + Dist[(c*Simplify[m + 2*p + 2])/((m + p + 1)*(2*c*d -
b*e)), Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && NeQ[b^2 - 4*a*c
, 0] && EqQ[c*d^2 - b*d*e + a*e^2, 0] &&  !IntegerQ[p] && ILtQ[Simplify[m + 2*p + 2], 0]

Rule 650

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(e*(d + e*x)^m*(a +
b*x + c*x^2)^(p + 1))/((p + 1)*(2*c*d - b*e)), x] /; FreeQ[{a, b, c, d, e, m, p}, x] && NeQ[b^2 - 4*a*c, 0] &&
EqQ[c*d^2 - b*d*e + a*e^2, 0] &&  !IntegerQ[p] && EqQ[m + 2*p + 2, 0]

Rubi steps

\begin{align*} \int \frac{1}{(d+e x)^3 \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx &=\frac{2 \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{5 \left (c d^2-a e^2\right ) (d+e x)^3}+\frac{(4 c d) \int \frac{1}{(d+e x)^2 \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{5 \left (c d^2-a e^2\right )}\\ &=\frac{2 \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{5 \left (c d^2-a e^2\right ) (d+e x)^3}+\frac{8 c d \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{15 \left (c d^2-a e^2\right )^2 (d+e x)^2}+\frac{\left (8 c^2 d^2\right ) \int \frac{1}{(d+e x) \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{15 \left (c d^2-a e^2\right )^2}\\ &=\frac{2 \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{5 \left (c d^2-a e^2\right ) (d+e x)^3}+\frac{8 c d \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{15 \left (c d^2-a e^2\right )^2 (d+e x)^2}+\frac{16 c^2 d^2 \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{15 \left (c d^2-a e^2\right )^3 (d+e x)}\\ \end{align*}

Mathematica [A]  time = 0.0484204, size = 94, normalized size = 0.55 \[ \frac{2 \sqrt{(d+e x) (a e+c d x)} \left (3 a^2 e^4-2 a c d e^2 (5 d+2 e x)+c^2 d^2 \left (15 d^2+20 d e x+8 e^2 x^2\right )\right )}{15 (d+e x)^3 \left (c d^2-a e^2\right )^3} \]

Antiderivative was successfully verified.

[In]

Integrate[1/((d + e*x)^3*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]),x]

[Out]

(2*Sqrt[(a*e + c*d*x)*(d + e*x)]*(3*a^2*e^4 - 2*a*c*d*e^2*(5*d + 2*e*x) + c^2*d^2*(15*d^2 + 20*d*e*x + 8*e^2*x
^2)))/(15*(c*d^2 - a*e^2)^3*(d + e*x)^3)

________________________________________________________________________________________

Maple [A]  time = 0.046, size = 146, normalized size = 0.9 \begin{align*} -{\frac{ \left ( 2\,cdx+2\,ae \right ) \left ( 8\,{c}^{2}{d}^{2}{e}^{2}{x}^{2}-4\,acd{e}^{3}x+20\,{c}^{2}{d}^{3}ex+3\,{a}^{2}{e}^{4}-10\,ac{d}^{2}{e}^{2}+15\,{c}^{2}{d}^{4} \right ) }{15\, \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 ( ex+d \right ) ^{2}}{\frac{1}{\sqrt{cde{x}^{2}+a{e}^{2}x+c{d}^{2}x+ade}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(e*x+d)^3/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2),x)

[Out]

-2/15*(c*d*x+a*e)*(8*c^2*d^2*e^2*x^2-4*a*c*d*e^3*x+20*c^2*d^3*e*x+3*a^2*e^4-10*a*c*d^2*e^2+15*c^2*d^4)/(e*x+d)
^2/(a^3*e^6-3*a^2*c*d^2*e^4+3*a*c^2*d^4*e^2-c^3*d^6)/(c*d*e*x^2+a*e^2*x+c*d^2*x+a*d*e)^(1/2)

________________________________________________________________________________________

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/(e*x+d)^3/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [A]  time = 9.18276, size = 551, normalized size = 3.22 \begin{align*} \frac{2 \,{\left (8 \, c^{2} d^{2} e^{2} x^{2} + 15 \, c^{2} d^{4} - 10 \, a c d^{2} e^{2} + 3 \, a^{2} e^{4} + 4 \,{\left (5 \, c^{2} d^{3} e - a c d e^{3}\right )} x\right )} \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x}}{15 \,{\left (c^{3} d^{9} - 3 \, a c^{2} d^{7} e^{2} + 3 \, a^{2} c d^{5} e^{4} - a^{3} d^{3} e^{6} +{\left (c^{3} d^{6} e^{3} - 3 \, a c^{2} d^{4} e^{5} + 3 \, a^{2} c d^{2} e^{7} - a^{3} e^{9}\right )} x^{3} + 3 \,{\left (c^{3} d^{7} e^{2} - 3 \, a c^{2} d^{5} e^{4} + 3 \, a^{2} c d^{3} e^{6} - a^{3} d e^{8}\right )} x^{2} + 3 \,{\left (c^{3} d^{8} e - 3 \, a c^{2} d^{6} e^{3} + 3 \, a^{2} c d^{4} e^{5} - a^{3} d^{2} e^{7}\right )} x\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)^3/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2),x, algorithm="fricas")

[Out]

2/15*(8*c^2*d^2*e^2*x^2 + 15*c^2*d^4 - 10*a*c*d^2*e^2 + 3*a^2*e^4 + 4*(5*c^2*d^3*e - a*c*d*e^3)*x)*sqrt(c*d*e*
x^2 + a*d*e + (c*d^2 + a*e^2)*x)/(c^3*d^9 - 3*a*c^2*d^7*e^2 + 3*a^2*c*d^5*e^4 - a^3*d^3*e^6 + (c^3*d^6*e^3 - 3
*a*c^2*d^4*e^5 + 3*a^2*c*d^2*e^7 - a^3*e^9)*x^3 + 3*(c^3*d^7*e^2 - 3*a*c^2*d^5*e^4 + 3*a^2*c*d^3*e^6 - a^3*d*e
^8)*x^2 + 3*(c^3*d^8*e - 3*a*c^2*d^6*e^3 + 3*a^2*c*d^4*e^5 - a^3*d^2*e^7)*x)

________________________________________________________________________________________

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/(e*x+d)**3/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(1/2),x)

[Out]

Timed out

________________________________________________________________________________________

Giac [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: NotImplementedError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)^3/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2),x, algorithm="giac")

[Out]

Exception raised: NotImplementedError