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

Optimal. Leaf size=39 \[ -\frac {2 \left (a-\frac {c d^2}{e^2}\right )}{\sqrt {d+e x}}+\frac {2 c d \sqrt {d+e x}}{e^2} \]

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 39, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.057, Rules used = {24, 45} \begin {gather*} \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}} \end {gather*}

Antiderivative was successfully verified.

[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*} \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 {\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*}

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Mathematica [A]
time = 0.03, size = 31, normalized size = 0.79 \begin {gather*} \frac {-2 a e^2+2 c d (2 d+e x)}{e^2 \sqrt {d+e x}} \end {gather*}

Antiderivative was successfully verified.

[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])

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Maple [A]
time = 0.47, size = 38, normalized size = 0.97

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

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

[Out]

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

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Maxima [A]
time = 0.28, size = 40, normalized size = 1.03 \begin {gather*} 2 \, {\left (\sqrt {x e + d} c d e^{\left (-1\right )} + \frac {{\left (c d^{2} - a e^{2}\right )} e^{\left (-1\right )}}{\sqrt {x e + d}}\right )} e^{\left (-1\right )} \end {gather*}

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

[Out]

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

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Fricas [A]
time = 2.78, size = 39, normalized size = 1.00 \begin {gather*} \frac {2 \, {\left (c d x e + 2 \, c d^{2} - a e^{2}\right )} \sqrt {x e + d}}{x e^{3} + d e^{2}} \end {gather*}

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

[Out]

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

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Sympy [A]
time = 0.42, size = 58, normalized size = 1.49 \begin {gather*} \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} \end {gather*}

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)/(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))

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Giac [A]
time = 1.52, size = 38, normalized size = 0.97 \begin {gather*} 2 \, \sqrt {x e + d} c d e^{\left (-2\right )} + \frac {2 \, {\left (c d^{2} - a e^{2}\right )} e^{\left (-2\right )}}{\sqrt {x e + d}} \end {gather*}

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)/(e*x+d)^(5/2),x, algorithm="giac")

[Out]

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

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Mupad [B]
time = 0.05, size = 30, normalized size = 0.77 \begin {gather*} \frac {4\,c\,d^2+2\,c\,x\,d\,e-2\,a\,e^2}{e^2\,\sqrt {d+e\,x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

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