\(\int \frac {a d e+(c d^2+a e^2) x+c d e x^2}{(d+e x)^{7/2}} \, dx\) [1981]

Optimal result

Integrand size = 35, antiderivative size = 41 \[ \int \frac {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}{(d+e x)^{7/2}} \, dx=-\frac {2 \left (a-\frac {c d^2}{e^2}\right )}{3 (d+e x)^{3/2}}-\frac {2 c d}{e^2 \sqrt {d+e x}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 41, 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} \[ \int \frac {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}{(d+e x)^{7/2}} \, dx=-\frac {2 \left (a-\frac {c d^2}{e^2}\right )}{3 (d+e x)^{3/2}}-\frac {2 c d}{e^2 \sqrt {d+e x}} \]

[In]

Int[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)/(d + e*x)^(7/2),x]

[Out]

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

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

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.80 \[ \int \frac {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}{(d+e x)^{7/2}} \, dx=-\frac {2 \left (a e^2+c d (2 d+3 e x)\right )}{3 e^2 (d+e x)^{3/2}} \]

[In]

Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)/(d + e*x)^(7/2),x]

[Out]

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

Maple [A] (verified)

Time = 2.47 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.76

method	result	size
gosper	\(-\frac {2 \left (3 x c d e +e^{2} a +2 c \,d^{2}\right )}{3 \left (e x +d \right )^{\frac {3}{2}} e^{2}}\)	\(31\)
trager	\(-\frac {2 \left (3 x c d e +e^{2} a +2 c \,d^{2}\right )}{3 \left (e x +d \right )^{\frac {3}{2}} e^{2}}\)	\(31\)
pseudoelliptic	\(-\frac {2 \left (3 x c d e +e^{2} a +2 c \,d^{2}\right )}{3 \left (e x +d \right )^{\frac {3}{2}} e^{2}}\)	\(31\)
derivativedivides	\(\frac {-\frac {2 c d}{\sqrt {e x +d}}-\frac {2 \left (e^{2} a -c \,d^{2}\right )}{3 \left (e x +d \right )^{\frac {3}{2}}}}{e^{2}}\)	\(39\)
default	\(\frac {-\frac {2 c d}{\sqrt {e x +d}}-\frac {2 \left (e^{2} a -c \,d^{2}\right )}{3 \left (e x +d \right )^{\frac {3}{2}}}}{e^{2}}\)	\(39\)

[In]

int((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)/(e*x+d)^(7/2),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.40 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.24 \[ \int \frac {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}{(d+e x)^{7/2}} \, dx=-\frac {2 \, {\left (3 \, c d e x + 2 \, c d^{2} + a e^{2}\right )} \sqrt {e x + d}}{3 \, {\left (e^{4} x^{2} + 2 \, d e^{3} x + d^{2} e^{2}\right )}} \]

[In]

integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)/(e*x+d)^(7/2),x, algorithm="fricas")

[Out]

-2/3*(3*c*d*e*x + 2*c*d^2 + a*e^2)*sqrt(e*x + d)/(e^4*x^2 + 2*d*e^3*x + d^2*e^2)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 126 vs. \(2 (41) = 82\).

Time = 0.46 (sec) , antiderivative size = 126, normalized size of antiderivative = 3.07 \[ \int \frac {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}{(d+e x)^{7/2}} \, dx=\begin {cases} - \frac {2 a e^{2}}{3 d e^{2} \sqrt {d + e x} + 3 e^{3} x \sqrt {d + e x}} - \frac {4 c d^{2}}{3 d e^{2} \sqrt {d + e x} + 3 e^{3} x \sqrt {d + e x}} - \frac {6 c d e x}{3 d e^{2} \sqrt {d + e x} + 3 e^{3} x \sqrt {d + e x}} & \text {for}\: e \neq 0 \\\frac {c x^{2}}{2 d^{\frac {3}{2}}} & \text {otherwise} \end {cases} \]

[In]

integrate((a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)/(e*x+d)**(7/2),x)

[Out]

Piecewise((-2*a*e**2/(3*d*e**2*sqrt(d + e*x) + 3*e**3*x*sqrt(d + e*x)) - 4*c*d**2/(3*d*e**2*sqrt(d + e*x) + 3*
e**3*x*sqrt(d + e*x)) - 6*c*d*e*x/(3*d*e**2*sqrt(d + e*x) + 3*e**3*x*sqrt(d + e*x)), Ne(e, 0)), (c*x**2/(2*d**
(3/2)), True))

Maxima [A] (verification not implemented)

none

Time = 0.18 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.80 \[ \int \frac {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}{(d+e x)^{7/2}} \, dx=-\frac {2 \, {\left (3 \, {\left (e x + d\right )} c d - c d^{2} + a e^{2}\right )}}{3 \, {\left (e x + d\right )}^{\frac {3}{2}} e^{2}} \]

[In]

integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)/(e*x+d)^(7/2),x, algorithm="maxima")

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.80 \[ \int \frac {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}{(d+e x)^{7/2}} \, dx=-\frac {2 \, {\left (3 \, {\left (e x + d\right )} c d - c d^{2} + a e^{2}\right )}}{3 \, {\left (e x + d\right )}^{\frac {3}{2}} e^{2}} \]

[In]

integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)/(e*x+d)^(7/2),x, algorithm="giac")

[Out]

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

Mupad [B] (verification not implemented)

Time = 9.68 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.83 \[ \int \frac {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}{(d+e x)^{7/2}} \, dx=-\frac {2\,a\,e^2-2\,c\,d^2+6\,c\,d\,\left (d+e\,x\right )}{3\,e^2\,{\left (d+e\,x\right )}^{3/2}} \]

[In]

int((x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)/(d + e*x)^(7/2),x)

[Out]

-(2*a*e^2 - 2*c*d^2 + 6*c*d*(d + e*x))/(3*e^2*(d + e*x)^(3/2))