3.1987 \(\int \frac{(a d e+(c d^2+a e^2) x+c d e x^2)^2}{(d+e x)^{5/2}} \, dx\)

Optimal. Leaf size=81 \[ -\frac{4 c d (d+e x)^{3/2} \left (c d^2-a e^2\right )}{3 e^3}+\frac{2 \sqrt{d+e x} \left (c d^2-a e^2\right )^2}{e^3}+\frac{2 c^2 d^2 (d+e x)^{5/2}}{5 e^3} \]

[Out]

(2*(c*d^2 - a*e^2)^2*Sqrt[d + e*x])/e^3 - (4*c*d*(c*d^2 - a*e^2)*(d + e*x)^(3/2))/(3*e^3) + (2*c^2*d^2*(d + e*
x)^(5/2))/(5*e^3)

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Rubi [A]  time = 0.0392762, antiderivative size = 81, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 37, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.054, Rules used = {626, 43} \[ -\frac{4 c d (d+e x)^{3/2} \left (c d^2-a e^2\right )}{3 e^3}+\frac{2 \sqrt{d+e x} \left (c d^2-a e^2\right )^2}{e^3}+\frac{2 c^2 d^2 (d+e x)^{5/2}}{5 e^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(c*d^2 - a*e^2)^2*Sqrt[d + e*x])/e^3 - (4*c*d*(c*d^2 - a*e^2)*(d + e*x)^(3/2))/(3*e^3) + (2*c^2*d^2*(d + e*
x)^(5/2))/(5*e^3)

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{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2}{(d+e x)^{5/2}} \, dx &=\int \frac{(a e+c d x)^2}{\sqrt{d+e x}} \, dx\\ &=\int \left (\frac{\left (-c d^2+a e^2\right )^2}{e^2 \sqrt{d+e x}}-\frac{2 c d \left (c d^2-a e^2\right ) \sqrt{d+e x}}{e^2}+\frac{c^2 d^2 (d+e x)^{3/2}}{e^2}\right ) \, dx\\ &=\frac{2 \left (c d^2-a e^2\right )^2 \sqrt{d+e x}}{e^3}-\frac{4 c d \left (c d^2-a e^2\right ) (d+e x)^{3/2}}{3 e^3}+\frac{2 c^2 d^2 (d+e x)^{5/2}}{5 e^3}\\ \end{align*}

Mathematica [A]  time = 0.0392344, size = 66, normalized size = 0.81 \[ \frac{2 \sqrt{d+e x} \left (15 a^2 e^4+10 a c d e^2 (e x-2 d)+c^2 d^2 \left (8 d^2-4 d e x+3 e^2 x^2\right )\right )}{15 e^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[d + e*x]*(15*a^2*e^4 + 10*a*c*d*e^2*(-2*d + e*x) + c^2*d^2*(8*d^2 - 4*d*e*x + 3*e^2*x^2)))/(15*e^3)

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Maple [A]  time = 0.045, size = 73, normalized size = 0.9 \begin{align*}{\frac{6\,{c}^{2}{d}^{2}{x}^{2}{e}^{2}+20\,acd{e}^{3}x-8\,{c}^{2}{d}^{3}ex+30\,{a}^{2}{e}^{4}-40\,ac{d}^{2}{e}^{2}+16\,{c}^{2}{d}^{4}}{15\,{e}^{3}}\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)^2/(e*x+d)^(5/2),x)

[Out]

2/15*(e*x+d)^(1/2)*(3*c^2*d^2*e^2*x^2+10*a*c*d*e^3*x-4*c^2*d^3*e*x+15*a^2*e^4-20*a*c*d^2*e^2+8*c^2*d^4)/e^3

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Maxima [A]  time = 0.988448, size = 108, normalized size = 1.33 \begin{align*} \frac{2 \,{\left (3 \,{\left (e x + d\right )}^{\frac{5}{2}} c^{2} d^{2} - 10 \,{\left (c^{2} d^{3} - a c d e^{2}\right )}{\left (e x + d\right )}^{\frac{3}{2}} + 15 \,{\left (c^{2} d^{4} - 2 \, a c d^{2} e^{2} + a^{2} e^{4}\right )} \sqrt{e x + d}\right )}}{15 \, e^{3}} \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)^2/(e*x+d)^(5/2),x, algorithm="maxima")

[Out]

2/15*(3*(e*x + d)^(5/2)*c^2*d^2 - 10*(c^2*d^3 - a*c*d*e^2)*(e*x + d)^(3/2) + 15*(c^2*d^4 - 2*a*c*d^2*e^2 + a^2
*e^4)*sqrt(e*x + d))/e^3

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Fricas [A]  time = 1.85779, size = 162, normalized size = 2. \begin{align*} \frac{2 \,{\left (3 \, c^{2} d^{2} e^{2} x^{2} + 8 \, c^{2} d^{4} - 20 \, a c d^{2} e^{2} + 15 \, a^{2} e^{4} - 2 \,{\left (2 \, c^{2} d^{3} e - 5 \, a c d e^{3}\right )} x\right )} \sqrt{e x + d}}{15 \, e^{3}} \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)^2/(e*x+d)^(5/2),x, algorithm="fricas")

[Out]

2/15*(3*c^2*d^2*e^2*x^2 + 8*c^2*d^4 - 20*a*c*d^2*e^2 + 15*a^2*e^4 - 2*(2*c^2*d^3*e - 5*a*c*d*e^3)*x)*sqrt(e*x
+ d)/e^3

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Sympy [A]  time = 40.0993, size = 236, normalized size = 2.91 \begin{align*} \begin{cases} - \frac{\frac{2 a^{2} d e^{2}}{\sqrt{d + e x}} + 2 a^{2} e^{2} \left (- \frac{d}{\sqrt{d + e x}} - \sqrt{d + e x}\right ) + 4 a c d^{2} \left (- \frac{d}{\sqrt{d + e x}} - \sqrt{d + e x}\right ) + 4 a 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 ) + \frac{2 c^{2} d^{3} \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^{2}} + \frac{2 c^{2} d^{2} \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^{2}}}{e} & \text{for}\: e \neq 0 \\\frac{c^{2} d^{\frac{3}{2}} x^{3}}{3} & \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)**2/(e*x+d)**(5/2),x)

[Out]

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

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Giac [A]  time = 1.18047, size = 143, normalized size = 1.77 \begin{align*} \frac{2}{15} \,{\left (3 \,{\left (x e + d\right )}^{\frac{5}{2}} c^{2} d^{2} e^{12} - 10 \,{\left (x e + d\right )}^{\frac{3}{2}} c^{2} d^{3} e^{12} + 15 \, \sqrt{x e + d} c^{2} d^{4} e^{12} + 10 \,{\left (x e + d\right )}^{\frac{3}{2}} a c d e^{14} - 30 \, \sqrt{x e + d} a c d^{2} e^{14} + 15 \, \sqrt{x e + d} a^{2} e^{16}\right )} e^{\left (-15\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)^2/(e*x+d)^(5/2),x, algorithm="giac")

[Out]

2/15*(3*(x*e + d)^(5/2)*c^2*d^2*e^12 - 10*(x*e + d)^(3/2)*c^2*d^3*e^12 + 15*sqrt(x*e + d)*c^2*d^4*e^12 + 10*(x
*e + d)^(3/2)*a*c*d*e^14 - 30*sqrt(x*e + d)*a*c*d^2*e^14 + 15*sqrt(x*e + d)*a^2*e^16)*e^(-15)