### 3.1057 $$\int \frac{(c d^2+2 c d e x+c e^2 x^2)^{5/2}}{(d+e x)^4} \, dx$$

Optimal. Leaf size=39 $\frac{c^2 (d+e x) \sqrt{c d^2+2 c d e x+c e^2 x^2}}{2 e}$

[Out]

(c^2*(d + e*x)*Sqrt[c*d^2 + 2*c*d*e*x + c*e^2*x^2])/(2*e)

________________________________________________________________________________________

Rubi [A]  time = 0.0208637, antiderivative size = 39, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 32, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.062, Rules used = {642, 609} $\frac{c^2 (d+e x) \sqrt{c d^2+2 c d e x+c e^2 x^2}}{2 e}$

Antiderivative was successfully veriﬁed.

[In]

Int[(c*d^2 + 2*c*d*e*x + c*e^2*x^2)^(5/2)/(d + e*x)^4,x]

[Out]

(c^2*(d + e*x)*Sqrt[c*d^2 + 2*c*d*e*x + c*e^2*x^2])/(2*e)

Rule 642

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

Rule 609

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

Rubi steps

\begin{align*} \int \frac{\left (c d^2+2 c d e x+c e^2 x^2\right )^{5/2}}{(d+e x)^4} \, dx &=c^2 \int \sqrt{c d^2+2 c d e x+c e^2 x^2} \, dx\\ &=\frac{c^2 (d+e x) \sqrt{c d^2+2 c d e x+c e^2 x^2}}{2 e}\\ \end{align*}

Mathematica [A]  time = 0.0022836, size = 33, normalized size = 0.85 $\frac{c^3 x (d+e x) (2 d+e x)}{2 \sqrt{c (d+e x)^2}}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(c*d^2 + 2*c*d*e*x + c*e^2*x^2)^(5/2)/(d + e*x)^4,x]

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 0.039, size = 40, normalized size = 1. \begin{align*}{\frac{x \left ( ex+2\,d \right ) }{2\, \left ( ex+d \right ) ^{5}} \left ( c{e}^{2}{x}^{2}+2\,cdex+c{d}^{2} \right ) ^{{\frac{5}{2}}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int((c*e^2*x^2+2*c*d*e*x+c*d^2)^(5/2)/(e*x+d)^4,x)

[Out]

1/2*x*(e*x+2*d)*(c*e^2*x^2+2*c*d*e*x+c*d^2)^(5/2)/(e*x+d)^5

________________________________________________________________________________________

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*e^2*x^2+2*c*d*e*x+c*d^2)^(5/2)/(e*x+d)^4,x, algorithm="maxima")

[Out]

Exception raised: RuntimeError

________________________________________________________________________________________

Fricas [A]  time = 2.21686, size = 101, normalized size = 2.59 \begin{align*} \frac{{\left (c^{2} e x^{2} + 2 \, c^{2} d x\right )} \sqrt{c e^{2} x^{2} + 2 \, c d e x + c d^{2}}}{2 \,{\left (e x + d\right )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*e^2*x^2+2*c*d*e*x+c*d^2)^(5/2)/(e*x+d)^4,x, algorithm="fricas")

[Out]

1/2*(c^2*e*x^2 + 2*c^2*d*x)*sqrt(c*e^2*x^2 + 2*c*d*e*x + c*d^2)/(e*x + d)

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (c \left (d + e x\right )^{2}\right )^{\frac{5}{2}}}{\left (d + e x\right )^{4}}\, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*e**2*x**2+2*c*d*e*x+c*d**2)**(5/2)/(e*x+d)**4,x)

[Out]

Integral((c*(d + e*x)**2)**(5/2)/(d + e*x)**4, x)

________________________________________________________________________________________

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((c*e^2*x^2+2*c*d*e*x+c*d^2)^(5/2)/(e*x+d)^4,x, algorithm="giac")

[Out]

Exception raised: NotImplementedError