3.1972 \(\int \frac{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}{(d+e x)^{11/2}} \, dx\)

Optimal. Leaf size=43 \[ -\frac{2 \left (a-\frac{c d^2}{e^2}\right )}{7 (d+e x)^{7/2}}-\frac{2 c d}{5 e^2 (d+e x)^{5/2}} \]

[Out]

(-2*(a - (c*d^2)/e^2))/(7*(d + e*x)^(7/2)) - (2*c*d)/(5*e^2*(d + e*x)^(5/2))

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Rubi [A]  time = 0.0645204, 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 \[ -\frac{2 \left (a-\frac{c d^2}{e^2}\right )}{7 (d+e x)^{7/2}}-\frac{2 c d}{5 e^2 (d+e x)^{5/2}} \]

Antiderivative was successfully verified.

[In]  Int[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)/(d + e*x)^(11/2),x]

[Out]

(-2*(a - (c*d^2)/e^2))/(7*(d + e*x)^(7/2)) - (2*c*d)/(5*e^2*(d + e*x)^(5/2))

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Rubi in Sympy [A]  time = 15.2769, size = 46, normalized size = 1.07 \[ - \frac{2 c d}{5 e^{2} \left (d + e x\right )^{\frac{5}{2}}} - \frac{2 \left (a e^{2} - c d^{2}\right )}{7 e^{2} \left (d + e x\right )^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)/(e*x+d)**(11/2),x)

[Out]

-2*c*d/(5*e**2*(d + e*x)**(5/2)) - 2*(a*e**2 - c*d**2)/(7*e**2*(d + e*x)**(7/2))

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Mathematica [A]  time = 0.0447426, size = 34, normalized size = 0.79 \[ -\frac{2 \left (5 a e^2+c d (2 d+7 e x)\right )}{35 e^2 (d+e x)^{7/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)/(d + e*x)^(11/2),x]

[Out]

(-2*(5*a*e^2 + c*d*(2*d + 7*e*x)))/(35*e^2*(d + e*x)^(7/2))

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Maple [A]  time = 0.006, size = 32, normalized size = 0.7 \[ -{\frac{14\,cdex+10\,a{e}^{2}+4\,c{d}^{2}}{35\,{e}^{2}} \left ( ex+d \right ) ^{-{\frac{7}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)/(e*x+d)^(11/2),x)

[Out]

-2/35/(e*x+d)^(7/2)*(7*c*d*e*x+5*a*e^2+2*c*d^2)/e^2

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Maxima [A]  time = 0.759117, size = 46, normalized size = 1.07 \[ -\frac{2 \,{\left (7 \,{\left (e x + d\right )} c d - 5 \, c d^{2} + 5 \, a e^{2}\right )}}{35 \,{\left (e x + d\right )}^{\frac{7}{2}} e^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)/(e*x + d)^(11/2),x, algorithm="maxima")

[Out]

-2/35*(7*(e*x + d)*c*d - 5*c*d^2 + 5*a*e^2)/((e*x + d)^(7/2)*e^2)

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Fricas [A]  time = 0.222483, size = 85, normalized size = 1.98 \[ -\frac{2 \,{\left (7 \, c d e x + 2 \, c d^{2} + 5 \, a e^{2}\right )}}{35 \,{\left (e^{5} x^{3} + 3 \, d e^{4} x^{2} + 3 \, d^{2} e^{3} x + d^{3} e^{2}\right )} \sqrt{e x + d}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)/(e*x + d)^(11/2),x, algorithm="fricas")

[Out]

-2/35*(7*c*d*e*x + 2*c*d^2 + 5*a*e^2)/((e^5*x^3 + 3*d*e^4*x^2 + 3*d^2*e^3*x + d^
3*e^2)*sqrt(e*x + d))

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Sympy [A]  time = 39.0533, size = 248, normalized size = 5.77 \[ \begin{cases} - \frac{10 a e^{2}}{35 d^{3} e^{2} \sqrt{d + e x} + 105 d^{2} e^{3} x \sqrt{d + e x} + 105 d e^{4} x^{2} \sqrt{d + e x} + 35 e^{5} x^{3} \sqrt{d + e x}} - \frac{4 c d^{2}}{35 d^{3} e^{2} \sqrt{d + e x} + 105 d^{2} e^{3} x \sqrt{d + e x} + 105 d e^{4} x^{2} \sqrt{d + e x} + 35 e^{5} x^{3} \sqrt{d + e x}} - \frac{14 c d e x}{35 d^{3} e^{2} \sqrt{d + e x} + 105 d^{2} e^{3} x \sqrt{d + e x} + 105 d e^{4} x^{2} \sqrt{d + e x} + 35 e^{5} x^{3} \sqrt{d + e x}} & \text{for}\: e \neq 0 \\\frac{c x^{2}}{2 d^{\frac{7}{2}}} & \text{otherwise} \end{cases} \]

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)**(11/2),x)

[Out]

Piecewise((-10*a*e**2/(35*d**3*e**2*sqrt(d + e*x) + 105*d**2*e**3*x*sqrt(d + e*x
) + 105*d*e**4*x**2*sqrt(d + e*x) + 35*e**5*x**3*sqrt(d + e*x)) - 4*c*d**2/(35*d
**3*e**2*sqrt(d + e*x) + 105*d**2*e**3*x*sqrt(d + e*x) + 105*d*e**4*x**2*sqrt(d
+ e*x) + 35*e**5*x**3*sqrt(d + e*x)) - 14*c*d*e*x/(35*d**3*e**2*sqrt(d + e*x) +
105*d**2*e**3*x*sqrt(d + e*x) + 105*d*e**4*x**2*sqrt(d + e*x) + 35*e**5*x**3*sqr
t(d + e*x)), Ne(e, 0)), (c*x**2/(2*d**(7/2)), True))

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GIAC/XCAS [A]  time = 0.202316, size = 65, normalized size = 1.51 \[ -\frac{2 \,{\left (7 \,{\left (x e + d\right )}^{2} c d - 5 \,{\left (x e + d\right )} c d^{2} + 5 \,{\left (x e + d\right )} a e^{2}\right )} e^{\left (-2\right )}}{35 \,{\left (x e + d\right )}^{\frac{9}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)/(e*x + d)^(11/2),x, algorithm="giac")

[Out]

-2/35*(7*(x*e + d)^2*c*d - 5*(x*e + d)*c*d^2 + 5*(x*e + d)*a*e^2)*e^(-2)/(x*e +
d)^(9/2)