\(\int \frac {a d e+(c d^2+a e^2) x+c d e x^2}{(d+e x)^{5/2}} \, dx\) [1980]

Optimal result

Integrand size = 35, antiderivative size = 39 \[ \int \frac {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}{(d+e x)^{5/2}} \, dx=-\frac {2 \left (a-\frac {c d^2}{e^2}\right )}{\sqrt {d+e x}}+\frac {2 c d \sqrt {d+e x}}{e^2} \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.057, Rules used = {24, 45} \[ \int \frac {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}{(d+e x)^{5/2}} \, dx=\frac {2 c d \sqrt {d+e x}}{e^2}-\frac {2 \left (a-\frac {c d^2}{e^2}\right )}{\sqrt {d+e x}} \]

[In]

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

[Out]

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

Rule 24

Int[(u_.)*((a_) + (b_.)*(v_))^(m_)*((A_.) + (B_.)*(v_) + (C_.)*(v_)^2), x_Symbol] :> Dist[1/b^2, Int[u*(a + b*
v)^(m + 1)*Simp[b*B - a*C + b*C*v, x], x], x] /; FreeQ[{a, b, A, B, C}, x] && EqQ[A*b^2 - a*b*B + a^2*C, 0] &&
LeQ[m, -1]

Rule 45

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

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

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.79 \[ \int \frac {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}{(d+e x)^{5/2}} \, dx=\frac {-2 a e^2+2 c d (2 d+e x)}{e^2 \sqrt {d+e x}} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 2.56 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.79

method	result	size
gosper	\(-\frac {2 \left (-x c d e +e^{2} a -2 c \,d^{2}\right )}{\sqrt {e x +d}\, e^{2}}\)	\(31\)
trager	\(-\frac {2 \left (-x c d e +e^{2} a -2 c \,d^{2}\right )}{\sqrt {e x +d}\, e^{2}}\)	\(31\)
pseudoelliptic	\(-\frac {2 \left (-x c d e +e^{2} a -2 c \,d^{2}\right )}{\sqrt {e x +d}\, e^{2}}\)	\(31\)
derivativedivides	\(\frac {2 c d \sqrt {e x +d}-\frac {2 \left (e^{2} a -c \,d^{2}\right )}{\sqrt {e x +d}}}{e^{2}}\)	\(38\)
default	\(\frac {2 c d \sqrt {e x +d}-\frac {2 \left (e^{2} a -c \,d^{2}\right )}{\sqrt {e x +d}}}{e^{2}}\)	\(38\)
risch	\(\frac {2 c d \sqrt {e x +d}}{e^{2}}-\frac {2 \left (e^{2} a -c \,d^{2}\right )}{e^{2} \sqrt {e x +d}}\)	\(40\)

[In]

int((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)/(e*x+d)^(5/2),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.38 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.03 \[ \int \frac {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}{(d+e x)^{5/2}} \, dx=\frac {2 \, {\left (c d e x + 2 \, c d^{2} - a e^{2}\right )} \sqrt {e x + d}}{e^{3} x + d e^{2}} \]

[In]

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

[Out]

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

Sympy [A] (verification not implemented)

Time = 0.27 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.49 \[ \int \frac {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}{(d+e x)^{5/2}} \, dx=\begin {cases} - \frac {2 a}{\sqrt {d + e x}} + \frac {4 c d^{2}}{e^{2} \sqrt {d + e x}} + \frac {2 c d x}{e \sqrt {d + e x}} & \text {for}\: e \neq 0 \\\frac {c x^{2}}{2 \sqrt {d}} & \text {otherwise} \end {cases} \]

[In]

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

[Out]

Piecewise((-2*a/sqrt(d + e*x) + 4*c*d**2/(e**2*sqrt(d + e*x)) + 2*c*d*x/(e*sqrt(d + e*x)), Ne(e, 0)), (c*x**2/
(2*sqrt(d)), True))

Maxima [A] (verification not implemented)

none

Time = 0.18 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.08 \[ \int \frac {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}{(d+e x)^{5/2}} \, dx=\frac {2 \, {\left (\frac {\sqrt {e x + d} c d}{e} + \frac {c d^{2} - a e^{2}}{\sqrt {e x + d} e}\right )}}{e} \]

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.00 \[ \int \frac {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}{(d+e x)^{5/2}} \, dx=\frac {2 \, \sqrt {e x + d} c d}{e^{2}} + \frac {2 \, {\left (c d^{2} - a e^{2}\right )}}{\sqrt {e x + d} e^{2}} \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 0.06 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.77 \[ \int \frac {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}{(d+e x)^{5/2}} \, dx=\frac {4\,c\,d^2+2\,c\,x\,d\,e-2\,a\,e^2}{e^2\,\sqrt {d+e\,x}} \]

[In]

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

[Out]

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