3.17.73 \(\int \frac {(a d e+(c d^2+a e^2) x+c d e x^2)^2}{(d+e x)^{9/2}} \, dx\)

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

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Rubi [A]  time = 0.04, antiderivative size = 79, normalized size of antiderivative = 1.00, 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} \begin {gather*} \frac {4 c d \left (c d^2-a e^2\right )}{e^3 \sqrt {d+e x}}-\frac {2 \left (c d^2-a e^2\right )^2}{3 e^3 (d+e x)^{3/2}}+\frac {2 c^2 d^2 \sqrt {d+e x}}{e^3} \end {gather*}

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)^(9/2),x]

[Out]

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

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])

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]

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)^{9/2}} \, dx &=\int \frac {(a e+c d x)^2}{(d+e x)^{5/2}} \, dx\\ &=\int \left (\frac {\left (-c d^2+a e^2\right )^2}{e^2 (d+e x)^{5/2}}-\frac {2 c d \left (c d^2-a e^2\right )}{e^2 (d+e x)^{3/2}}+\frac {c^2 d^2}{e^2 \sqrt {d+e x}}\right ) \, dx\\ &=-\frac {2 \left (c d^2-a e^2\right )^2}{3 e^3 (d+e x)^{3/2}}+\frac {4 c d \left (c d^2-a e^2\right )}{e^3 \sqrt {d+e x}}+\frac {2 c^2 d^2 \sqrt {d+e x}}{e^3}\\ \end {align*}

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Mathematica [A]  time = 0.04, size = 68, normalized size = 0.86 \begin {gather*} \frac {-2 a^2 e^4-4 a c d e^2 (2 d+3 e x)+2 c^2 d^2 \left (8 d^2+12 d e x+3 e^2 x^2\right )}{3 e^3 (d+e x)^{3/2}} \end {gather*}

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)^(9/2),x]

[Out]

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

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IntegrateAlgebraic [A]  time = 0.09, size = 84, normalized size = 1.06 \begin {gather*} \frac {2 \left (-a^2 e^4+2 a c d^2 e^2-6 a c d e^2 (d+e x)-c^2 d^4+6 c^2 d^3 (d+e x)+3 c^2 d^2 (d+e x)^2\right )}{3 e^3 (d+e x)^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^2/(d + e*x)^(9/2),x]

[Out]

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

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fricas [A]  time = 0.40, size = 95, normalized size = 1.20 \begin {gather*} \frac {2 \, {\left (3 \, c^{2} d^{2} e^{2} x^{2} + 8 \, c^{2} d^{4} - 4 \, a c d^{2} e^{2} - a^{2} e^{4} + 6 \, {\left (2 \, c^{2} d^{3} e - a c d e^{3}\right )} x\right )} \sqrt {e x + d}}{3 \, {\left (e^{5} x^{2} + 2 \, d e^{4} x + d^{2} e^{3}\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)^2/(e*x+d)^(9/2),x, algorithm="fricas")

[Out]

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

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giac [A]  time = 0.22, size = 111, normalized size = 1.41 \begin {gather*} 2 \, \sqrt {x e + d} c^{2} d^{2} e^{\left (-3\right )} + \frac {2 \, {\left (6 \, {\left (x e + d\right )}^{3} c^{2} d^{3} - {\left (x e + d\right )}^{2} c^{2} d^{4} - 6 \, {\left (x e + d\right )}^{3} a c d e^{2} + 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 )}}{3 \, {\left (x e + d\right )}^{\frac {7}{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)^2/(e*x+d)^(9/2),x, algorithm="giac")

[Out]

2*sqrt(x*e + d)*c^2*d^2*e^(-3) + 2/3*(6*(x*e + d)^3*c^2*d^3 - (x*e + d)^2*c^2*d^4 - 6*(x*e + d)^3*a*c*d*e^2 +
2*(x*e + d)^2*a*c*d^2*e^2 - (x*e + d)^2*a^2*e^4)*e^(-3)/(x*e + d)^(7/2)

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maple [A]  time = 0.05, size = 72, normalized size = 0.91 \begin {gather*} -\frac {2 \left (-3 c^{2} d^{2} e^{2} x^{2}+6 a c d \,e^{3} x -12 c^{2} d^{3} e x +a^{2} e^{4}+4 a c \,d^{2} e^{2}-8 c^{2} d^{4}\right )}{3 \left (e x +d \right )^{\frac {3}{2}} e^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^2/(e*x+d)^(9/2),x)

[Out]

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

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maxima [A]  time = 1.13, size = 84, normalized size = 1.06 \begin {gather*} \frac {2 \, {\left (\frac {3 \, \sqrt {e x + d} c^{2} d^{2}}{e^{2}} - \frac {c^{2} d^{4} - 2 \, a c d^{2} e^{2} + a^{2} e^{4} - 6 \, {\left (c^{2} d^{3} - a c d e^{2}\right )} {\left (e x + d\right )}}{{\left (e x + d\right )}^{\frac {3}{2}} e^{2}}\right )}}{3 \, e} \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)^2/(e*x+d)^(9/2),x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 0.63, size = 80, normalized size = 1.01 \begin {gather*} -\frac {2\,a^2\,e^4+2\,c^2\,d^4-6\,c^2\,d^2\,{\left (d+e\,x\right )}^2-12\,c^2\,d^3\,\left (d+e\,x\right )-4\,a\,c\,d^2\,e^2+12\,a\,c\,d\,e^2\,\left (d+e\,x\right )}{3\,e^3\,{\left (d+e\,x\right )}^{3/2}} \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)^2/(d + e*x)^(9/2),x)

[Out]

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

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sympy [A]  time = 9.46, size = 264, normalized size = 3.34 \begin {gather*} \begin {cases} - \frac {2 a^{2} e^{4}}{3 d e^{3} \sqrt {d + e x} + 3 e^{4} x \sqrt {d + e x}} - \frac {8 a c d^{2} e^{2}}{3 d e^{3} \sqrt {d + e x} + 3 e^{4} x \sqrt {d + e x}} - \frac {12 a c d e^{3} x}{3 d e^{3} \sqrt {d + e x} + 3 e^{4} x \sqrt {d + e x}} + \frac {16 c^{2} d^{4}}{3 d e^{3} \sqrt {d + e x} + 3 e^{4} x \sqrt {d + e x}} + \frac {24 c^{2} d^{3} e x}{3 d e^{3} \sqrt {d + e x} + 3 e^{4} x \sqrt {d + e x}} + \frac {6 c^{2} d^{2} e^{2} x^{2}}{3 d e^{3} \sqrt {d + e x} + 3 e^{4} x \sqrt {d + e x}} & \text {for}\: e \neq 0 \\\frac {c^{2} x^{3}}{3 \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)**2/(e*x+d)**(9/2),x)

[Out]

Piecewise((-2*a**2*e**4/(3*d*e**3*sqrt(d + e*x) + 3*e**4*x*sqrt(d + e*x)) - 8*a*c*d**2*e**2/(3*d*e**3*sqrt(d +
e*x) + 3*e**4*x*sqrt(d + e*x)) - 12*a*c*d*e**3*x/(3*d*e**3*sqrt(d + e*x) + 3*e**4*x*sqrt(d + e*x)) + 16*c**2*
d**4/(3*d*e**3*sqrt(d + e*x) + 3*e**4*x*sqrt(d + e*x)) + 24*c**2*d**3*e*x/(3*d*e**3*sqrt(d + e*x) + 3*e**4*x*s
qrt(d + e*x)) + 6*c**2*d**2*e**2*x**2/(3*d*e**3*sqrt(d + e*x) + 3*e**4*x*sqrt(d + e*x)), Ne(e, 0)), (c**2*x**3
/(3*sqrt(d)), True))

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