### 3.1914 $$\int \frac{\sqrt{a d e+(c d^2+a e^2) x+c d e x^2}}{(d+e x)^3} \, dx$$

Optimal. Leaf size=54 $\frac{2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{3 (d+e x)^3 \left (c d^2-a e^2\right )}$

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.020352, antiderivative size = 54, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 37, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.027, Rules used = {650} $\frac{2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{3 (d+e x)^3 \left (c d^2-a e^2\right )}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

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

Mathematica [A]  time = 0.0207341, size = 43, normalized size = 0.8 $\frac{2 ((d+e x) (a e+c d x))^{3/2}}{3 (d+e x)^3 \left (c d^2-a e^2\right )}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 0.044, size = 58, normalized size = 1.1 \begin{align*} -{\frac{2\,cdx+2\,ae}{3\, \left ( ex+d \right ) ^{2} \left ( a{e}^{2}-c{d}^{2} \right ) }\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((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/(e*x+d)^3,x)

[Out]

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

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [A]  time = 2.57038, size = 182, normalized size = 3.37 \begin{align*} \frac{2 \, \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x}{\left (c d x + a e\right )}}{3 \,{\left (c d^{4} - a d^{2} e^{2} +{\left (c d^{2} e^{2} - a e^{4}\right )} x^{2} + 2 \,{\left (c d^{3} e - a d e^{3}\right )} x\right )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*(c*d*x + a*e)/(c*d^4 - a*d^2*e^2 + (c*d^2*e^2 - a*e^4)*x^2 + 2
*(c*d^3*e - a*d*e^3)*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((a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(1/2)/(e*x+d)**3,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((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/(e*x+d)^3,x, algorithm="giac")

[Out]

Exception raised: NotImplementedError