3.1978 \(\int \frac{a d e+(c d^2+a e^2) x+c d e x^2}{\sqrt{d+e x}} \, dx\)

Optimal. Leaf size=43 \[ \frac{2}{3} (d+e x)^{3/2} \left (a-\frac{c d^2}{e^2}\right )+\frac{2 c d (d+e x)^{5/2}}{5 e^2} \]

[Out]

(2*(a - (c*d^2)/e^2)*(d + e*x)^(3/2))/3 + (2*c*d*(d + e*x)^(5/2))/(5*e^2)

________________________________________________________________________________________

Rubi [A]  time = 0.0191914, antiderivative size = 43, 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 = {626, 43} \[ \frac{2}{3} (d+e x)^{3/2} \left (a-\frac{c d^2}{e^2}\right )+\frac{2 c d (d+e x)^{5/2}}{5 e^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(a - (c*d^2)/e^2)*(d + e*x)^(3/2))/3 + (2*c*d*(d + e*x)^(5/2))/(5*e^2)

Rule 626

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

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

Mathematica [A]  time = 0.0224801, size = 34, normalized size = 0.79 \[ \frac{2 (d+e x)^{3/2} \left (5 a e^2+c d (3 e x-2 d)\right )}{15 e^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(d + e*x)^(3/2)*(5*a*e^2 + c*d*(-2*d + 3*e*x)))/(15*e^2)

________________________________________________________________________________________

Maple [A]  time = 0.041, size = 32, normalized size = 0.7 \begin{align*}{\frac{6\,cdex+10\,a{e}^{2}-4\,c{d}^{2}}{15\,{e}^{2}} \left ( ex+d \right ) ^{{\frac{3}{2}}}} \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)^(1/2),x)

[Out]

2/15*(e*x+d)^(3/2)*(3*c*d*e*x+5*a*e^2-2*c*d^2)/e^2

________________________________________________________________________________________

Maxima [B]  time = 0.997965, size = 122, normalized size = 2.84 \begin{align*} \frac{2 \,{\left (15 \, \sqrt{e x + d} a d e + \frac{{\left (3 \,{\left (e x + d\right )}^{\frac{5}{2}} - 10 \,{\left (e x + d\right )}^{\frac{3}{2}} d + 15 \, \sqrt{e x + d} d^{2}\right )} c d}{e} + \frac{5 \,{\left (c d^{2} + a e^{2}\right )}{\left ({\left (e x + d\right )}^{\frac{3}{2}} - 3 \, \sqrt{e x + d} d\right )}}{e}\right )}}{15 \, e} \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)^(1/2),x, algorithm="maxima")

[Out]

2/15*(15*sqrt(e*x + d)*a*d*e + (3*(e*x + d)^(5/2) - 10*(e*x + d)^(3/2)*d + 15*sqrt(e*x + d)*d^2)*c*d/e + 5*(c*
d^2 + a*e^2)*((e*x + d)^(3/2) - 3*sqrt(e*x + d)*d)/e)/e

________________________________________________________________________________________

Fricas [A]  time = 2.1406, size = 116, normalized size = 2.7 \begin{align*} \frac{2 \,{\left (3 \, c d e^{2} x^{2} - 2 \, c d^{3} + 5 \, a d e^{2} +{\left (c d^{2} e + 5 \, a e^{3}\right )} x\right )} \sqrt{e x + d}}{15 \, 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)^(1/2),x, algorithm="fricas")

[Out]

2/15*(3*c*d*e^2*x^2 - 2*c*d^3 + 5*a*d*e^2 + (c*d^2*e + 5*a*e^3)*x)*sqrt(e*x + d)/e^2

________________________________________________________________________________________

Sympy [A]  time = 22.5958, size = 221, normalized size = 5.14 \begin{align*} \begin{cases} - \frac{\frac{2 a d^{2} e}{\sqrt{d + e x}} + 4 a d e \left (- \frac{d}{\sqrt{d + e x}} - \sqrt{d + e x}\right ) + 2 a e \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 ) + \frac{2 c d^{3} \left (- \frac{d}{\sqrt{d + e x}} - \sqrt{d + e x}\right )}{e} + \frac{4 c d^{2} \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} + \frac{2 c d \left (- \frac{d^{3}}{\sqrt{d + e x}} - 3 d^{2} \sqrt{d + e x} + d \left (d + e x\right )^{\frac{3}{2}} - \frac{\left (d + e x\right )^{\frac{5}{2}}}{5}\right )}{e}}{e} & \text{for}\: e \neq 0 \\\frac{c d^{\frac{3}{2}} 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)**(1/2),x)

[Out]

Piecewise((-(2*a*d**2*e/sqrt(d + e*x) + 4*a*d*e*(-d/sqrt(d + e*x) - sqrt(d + e*x)) + 2*a*e*(d**2/sqrt(d + e*x)
+ 2*d*sqrt(d + e*x) - (d + e*x)**(3/2)/3) + 2*c*d**3*(-d/sqrt(d + e*x) - sqrt(d + e*x))/e + 4*c*d**2*(d**2/sq
rt(d + e*x) + 2*d*sqrt(d + e*x) - (d + e*x)**(3/2)/3)/e + 2*c*d*(-d**3/sqrt(d + e*x) - 3*d**2*sqrt(d + e*x) +
d*(d + e*x)**(3/2) - (d + e*x)**(5/2)/5)/e)/e, Ne(e, 0)), (c*d**(3/2)*x**2/2, True))

________________________________________________________________________________________

Giac [B]  time = 1.1204, size = 151, normalized size = 3.51 \begin{align*} \frac{2}{15} \,{\left (5 \,{\left ({\left (x e + d\right )}^{\frac{3}{2}} - 3 \, \sqrt{x e + d} d\right )} c d^{2} e^{\left (-1\right )} +{\left (3 \,{\left (x e + d\right )}^{\frac{5}{2}} - 10 \,{\left (x e + d\right )}^{\frac{3}{2}} d + 15 \, \sqrt{x e + d} d^{2}\right )} c d e^{\left (-1\right )} + 15 \, \sqrt{x e + d} a d e + 5 \,{\left ({\left (x e + d\right )}^{\frac{3}{2}} - 3 \, \sqrt{x e + d} d\right )} a e\right )} e^{\left (-1\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)^(1/2),x, algorithm="giac")

[Out]

2/15*(5*((x*e + d)^(3/2) - 3*sqrt(x*e + d)*d)*c*d^2*e^(-1) + (3*(x*e + d)^(5/2) - 10*(x*e + d)^(3/2)*d + 15*sq
rt(x*e + d)*d^2)*c*d*e^(-1) + 15*sqrt(x*e + d)*a*d*e + 5*((x*e + d)^(3/2) - 3*sqrt(x*e + d)*d)*a*e)*e^(-1)