3.1979 \(\int \frac{a d e+(c d^2+a e^2) x+c d e x^2}{(d+e x)^{3/2}} \, dx\)

Optimal. Leaf size=41 \[ 2 \sqrt{d+e x} \left (a-\frac{c d^2}{e^2}\right )+\frac{2 c d (d+e x)^{3/2}}{3 e^2} \]

[Out]

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

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Rubi [A]  time = 0.024613, antiderivative size = 41, normalized size of antiderivative = 1., 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, 43} \[ 2 \sqrt{d+e x} \left (a-\frac{c d^2}{e^2}\right )+\frac{2 c d (d+e x)^{3/2}}{3 e^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

2*(a - (c*d^2)/e^2)*Sqrt[d + e*x] + (2*c*d*(d + e*x)^(3/2))/(3*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 43

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

Mathematica [A]  time = 0.0208033, size = 33, normalized size = 0.8 \[ \frac{2 \sqrt{d+e x} \left (3 a e^2+c d (e x-2 d)\right )}{3 e^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.042, size = 31, normalized size = 0.8 \begin{align*}{\frac{2\,cdex+6\,a{e}^{2}-4\,c{d}^{2}}{3\,{e}^{2}}\sqrt{ex+d}} \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)/(e*x+d)^(3/2),x)

[Out]

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

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Maxima [A]  time = 0.989093, size = 50, normalized size = 1.22 \begin{align*} \frac{2 \,{\left ({\left (e x + d\right )}^{\frac{3}{2}} c d - 3 \,{\left (c d^{2} - a e^{2}\right )} \sqrt{e x + d}\right )}}{3 \, e^{2}} \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)/(e*x+d)^(3/2),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 2.13835, size = 72, normalized size = 1.76 \begin{align*} \frac{2 \,{\left (c d e x - 2 \, c d^{2} + 3 \, a e^{2}\right )} \sqrt{e x + d}}{3 \, e^{2}} \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)/(e*x+d)^(3/2),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 9.61783, size = 124, normalized size = 3.02 \begin{align*} \begin{cases} - \frac{\frac{2 a d e}{\sqrt{d + e x}} + 2 a e \left (- \frac{d}{\sqrt{d + e x}} - \sqrt{d + e x}\right ) + \frac{2 c d^{2} \left (- \frac{d}{\sqrt{d + e x}} - \sqrt{d + e x}\right )}{e} + \frac{2 c d \left (\frac{d^{2}}{\sqrt{d + e x}} + 2 d \sqrt{d + e x} - \frac{\left (d + e x\right )^{\frac{3}{2}}}{3}\right )}{e}}{e} & \text{for}\: e \neq 0 \\\frac{c \sqrt{d} x^{2}}{2} & \text{otherwise} \end{cases} \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)/(e*x+d)**(3/2),x)

[Out]

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

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Giac [A]  time = 1.16205, size = 63, normalized size = 1.54 \begin{align*} \frac{2}{3} \,{\left ({\left (x e + d\right )}^{\frac{3}{2}} c d e^{4} - 3 \, \sqrt{x e + d} c d^{2} e^{4} + 3 \, \sqrt{x e + d} a e^{6}\right )} e^{\left (-6\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)/(e*x+d)^(3/2),x, algorithm="giac")

[Out]

2/3*((x*e + d)^(3/2)*c*d*e^4 - 3*sqrt(x*e + d)*c*d^2*e^4 + 3*sqrt(x*e + d)*a*e^6)*e^(-6)