3.20.89 \(\int \frac {(a d e+(c d^2+a e^2) x+c d e x^2)^2}{(d+e x)^{9/2}} \, dx\) [1989]

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

[Out]

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

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Rubi [A]
time = 0.03, 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 = {640, 45} \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 45

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 640

Int[((d_) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[(d + e*x)^(m + p)*(a
/d + (c/e)*x)^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 = 0.05, 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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Maple [A]
time = 0.72, size = 78, normalized size = 0.99

method	result	size
risch	\(\frac {2 c^{2} d^{2} \sqrt {e x +d}}{e^{3}}-\frac {2 \left (6 c d e x +e^{2} a +5 c \,d^{2}\right ) \left (e^{2} a -c \,d^{2}\right )}{3 e^{3} \left (e x +d \right )^{\frac {3}{2}}}\)	\(62\)
gosper	\(-\frac {2 \left (-3 e^{2} x^{2} c^{2} d^{2}+6 a c d \,e^{3} x -12 c^{2} d^{3} e x +a^{2} e^{4}+4 a c \,d^{2} e^{2}-8 c^{2} d^{4}\right )}{3 \left (e x +d \right )^{\frac {3}{2}} e^{3}}\)	\(72\)
trager	\(-\frac {2 \left (-3 e^{2} x^{2} c^{2} d^{2}+6 a c d \,e^{3} x -12 c^{2} d^{3} e x +a^{2} e^{4}+4 a c \,d^{2} e^{2}-8 c^{2} d^{4}\right )}{3 \left (e x +d \right )^{\frac {3}{2}} e^{3}}\)	\(72\)
derivativedivides	\(\frac {2 c^{2} d^{2} \sqrt {e x +d}-\frac {2 \left (a^{2} e^{4}-2 a c \,d^{2} e^{2}+c^{2} d^{4}\right )}{3 \left (e x +d \right )^{\frac {3}{2}}}-\frac {4 c d \left (e^{2} a -c \,d^{2}\right )}{\sqrt {e x +d}}}{e^{3}}\)	\(78\)
default	\(\frac {2 c^{2} d^{2} \sqrt {e x +d}-\frac {2 \left (a^{2} e^{4}-2 a c \,d^{2} e^{2}+c^{2} d^{4}\right )}{3 \left (e x +d \right )^{\frac {3}{2}}}-\frac {4 c d \left (e^{2} a -c \,d^{2}\right )}{\sqrt {e x +d}}}{e^{3}}\)	\(78\)

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)^(9/2),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.28, size = 81, normalized size = 1.03 \begin {gather*} \frac {2}{3} \, {\left (3 \, \sqrt {x e + d} c^{2} d^{2} e^{\left (-2\right )} - \frac {{\left (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 (x e + d\right )}\right )} e^{\left (-2\right )}}{{\left (x e + d\right )}^{\frac {3}{2}}}\right )} e^{\left (-1\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="maxima")

[Out]

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

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Fricas [A]
time = 3.34, size = 88, normalized size = 1.11 \begin {gather*} \frac {2 \, {\left (12 \, c^{2} d^{3} x e + 8 \, c^{2} d^{4} - 6 \, a c d x e^{3} - a^{2} e^{4} + {\left (3 \, c^{2} d^{2} x^{2} - 4 \, a c d^{2}\right )} e^{2}\right )} \sqrt {x e + d}}{3 \, {\left (x^{2} e^{5} + 2 \, d x e^{4} + 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*(12*c^2*d^3*x*e + 8*c^2*d^4 - 6*a*c*d*x*e^3 - a^2*e^4 + (3*c^2*d^2*x^2 - 4*a*c*d^2)*e^2)*sqrt(x*e + d)/(x^
2*e^5 + 2*d*x*e^4 + d^2*e^3)

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 264 vs. \(2 (73) = 146\).
time = 1.05, 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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Giac [A]
time = 1.26, size = 83, normalized size = 1.05 \begin {gather*} 2 \, \sqrt {x e + d} c^{2} d^{2} e^{\left (-3\right )} + \frac {2 \, {\left (6 \, {\left (x e + d\right )} c^{2} d^{3} - c^{2} d^{4} - 6 \, {\left (x e + d\right )} a c d e^{2} + 2 \, a c d^{2} e^{2} - a^{2} e^{4}\right )} e^{\left (-3\right )}}{3 \, {\left (x e + d\right )}^{\frac {3}{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)*c^2*d^3 - c^2*d^4 - 6*(x*e + d)*a*c*d*e^2 + 2*a*c*d^2*e^2 -
a^2*e^4)*e^(-3)/(x*e + d)^(3/2)

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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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