3.1988 \(\int \frac{(a d e+(c d^2+a e^2) x+c d e x^2)^2}{(d+e x)^{7/2}} \, dx\)

Optimal. Leaf size=79 \[ -\frac{4 c d \sqrt{d+e x} \left (c d^2-a e^2\right )}{e^3}-\frac{2 \left (c d^2-a e^2\right )^2}{e^3 \sqrt{d+e x}}+\frac{2 c^2 d^2 (d+e x)^{3/2}}{3 e^3} \]

[Out]

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

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Rubi [A]  time = 0.0410338, antiderivative size = 79, 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} \[ -\frac{4 c d \sqrt{d+e x} \left (c d^2-a e^2\right )}{e^3}-\frac{2 \left (c d^2-a e^2\right )^2}{e^3 \sqrt{d+e x}}+\frac{2 c^2 d^2 (d+e x)^{3/2}}{3 e^3} \]

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)^(7/2),x]

[Out]

(-2*(c*d^2 - a*e^2)^2)/(e^3*Sqrt[d + e*x]) - (4*c*d*(c*d^2 - a*e^2)*Sqrt[d + e*x])/e^3 + (2*c^2*d^2*(d + e*x)^
(3/2))/(3*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)^{7/2}} \, dx &=\int \frac{(a e+c d x)^2}{(d+e x)^{3/2}} \, dx\\ &=\int \left (\frac{\left (-c d^2+a e^2\right )^2}{e^2 (d+e x)^{3/2}}-\frac{2 c d \left (c d^2-a e^2\right )}{e^2 \sqrt{d+e x}}+\frac{c^2 d^2 \sqrt{d+e x}}{e^2}\right ) \, dx\\ &=-\frac{2 \left (c d^2-a e^2\right )^2}{e^3 \sqrt{d+e x}}-\frac{4 c d \left (c d^2-a e^2\right ) \sqrt{d+e x}}{e^3}+\frac{2 c^2 d^2 (d+e x)^{3/2}}{3 e^3}\\ \end{align*}

Mathematica [A]  time = 0.0354436, size = 65, normalized size = 0.82 \[ \frac{2 \left (-3 a^2 e^4+6 a c d e^2 (2 d+e x)+c^2 d^2 \left (-8 d^2-4 d e x+e^2 x^2\right )\right )}{3 e^3 \sqrt{d+e x}} \]

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)^(7/2),x]

[Out]

(2*(-3*a^2*e^4 + 6*a*c*d*e^2*(2*d + e*x) + c^2*d^2*(-8*d^2 - 4*d*e*x + e^2*x^2)))/(3*e^3*Sqrt[d + e*x])

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Maple [A]  time = 0.044, size = 73, normalized size = 0.9 \begin{align*} -{\frac{-2\,{c}^{2}{d}^{2}{x}^{2}{e}^{2}-12\,acd{e}^{3}x+8\,{c}^{2}{d}^{3}ex+6\,{a}^{2}{e}^{4}-24\,ac{d}^{2}{e}^{2}+16\,{c}^{2}{d}^{4}}{3\,{e}^{3}}{\frac{1}{\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)^(7/2),x)

[Out]

-2/3/(e*x+d)^(1/2)*(-c^2*d^2*e^2*x^2-6*a*c*d*e^3*x+4*c^2*d^3*e*x+3*a^2*e^4-12*a*c*d^2*e^2+8*c^2*d^4)/e^3

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Maxima [A]  time = 1.02826, size = 117, normalized size = 1.48 \begin{align*} \frac{2 \,{\left (\frac{{\left (e x + d\right )}^{\frac{3}{2}} c^{2} d^{2} - 6 \,{\left (c^{2} d^{3} - a c d e^{2}\right )} \sqrt{e x + d}}{e^{2}} - \frac{3 \,{\left (c^{2} d^{4} - 2 \, a c d^{2} e^{2} + a^{2} e^{4}\right )}}{\sqrt{e x + d} e^{2}}\right )}}{3 \, 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)^2/(e*x+d)^(7/2),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.83272, size = 173, normalized size = 2.19 \begin{align*} \frac{2 \,{\left (c^{2} d^{2} e^{2} x^{2} - 8 \, c^{2} d^{4} + 12 \, a c d^{2} e^{2} - 3 \, a^{2} e^{4} - 2 \,{\left (2 \, c^{2} d^{3} e - 3 \, a c d e^{3}\right )} x\right )} \sqrt{e x + d}}{3 \,{\left (e^{4} x + d e^{3}\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)^(7/2),x, algorithm="fricas")

[Out]

2/3*(c^2*d^2*e^2*x^2 - 8*c^2*d^4 + 12*a*c*d^2*e^2 - 3*a^2*e^4 - 2*(2*c^2*d^3*e - 3*a*c*d*e^3)*x)*sqrt(e*x + d)
/(e^4*x + d*e^3)

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Sympy [A]  time = 5.87905, size = 133, normalized size = 1.68 \begin{align*} \begin{cases} - \frac{2 a^{2} e}{\sqrt{d + e x}} + \frac{8 a c d^{2}}{e \sqrt{d + e x}} + \frac{4 a c d x}{\sqrt{d + e x}} - \frac{16 c^{2} d^{4}}{3 e^{3} \sqrt{d + e x}} - \frac{8 c^{2} d^{3} x}{3 e^{2} \sqrt{d + e x}} + \frac{2 c^{2} d^{2} x^{2}}{3 e \sqrt{d + e x}} & \text{for}\: e \neq 0 \\\frac{c^{2} \sqrt{d} 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)**(7/2),x)

[Out]

Piecewise((-2*a**2*e/sqrt(d + e*x) + 8*a*c*d**2/(e*sqrt(d + e*x)) + 4*a*c*d*x/sqrt(d + e*x) - 16*c**2*d**4/(3*
e**3*sqrt(d + e*x)) - 8*c**2*d**3*x/(3*e**2*sqrt(d + e*x)) + 2*c**2*d**2*x**2/(3*e*sqrt(d + e*x)), Ne(e, 0)),
(c**2*sqrt(d)*x**3/3, True))

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Giac [A]  time = 1.13009, size = 155, normalized size = 1.96 \begin{align*} \frac{2}{3} \,{\left ({\left (x e + d\right )}^{\frac{3}{2}} c^{2} d^{2} e^{6} - 6 \, \sqrt{x e + d} c^{2} d^{3} e^{6} + 6 \, \sqrt{x e + d} a c d e^{8}\right )} e^{\left (-9\right )} - \frac{2 \,{\left ({\left (x e + d\right )}^{2} c^{2} d^{4} - 2 \,{\left (x e + d\right )}^{2} a c d^{2} e^{2} +{\left (x e + d\right )}^{2} a^{2} e^{4}\right )} e^{\left (-3\right )}}{{\left (x e + d\right )}^{\frac{5}{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)^2/(e*x+d)^(7/2),x, algorithm="giac")

[Out]

2/3*((x*e + d)^(3/2)*c^2*d^2*e^6 - 6*sqrt(x*e + d)*c^2*d^3*e^6 + 6*sqrt(x*e + d)*a*c*d*e^8)*e^(-9) - 2*((x*e +
d)^2*c^2*d^4 - 2*(x*e + d)^2*a*c*d^2*e^2 + (x*e + d)^2*a^2*e^4)*e^(-3)/(x*e + d)^(5/2)