3.1970 \(\int \frac{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}{(d+e x)^{7/2}} \, dx\)

Optimal. Leaf size=41 \[ -\frac{2 \left (a-\frac{c d^2}{e^2}\right )}{3 (d+e x)^{3/2}}-\frac{2 c d}{e^2 \sqrt{d+e x}} \]

[Out]

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

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Rubi [A]  time = 0.0648942, antiderivative size = 41, 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 \[ -\frac{2 \left (a-\frac{c d^2}{e^2}\right )}{3 (d+e x)^{3/2}}-\frac{2 c d}{e^2 \sqrt{d+e x}} \]

Antiderivative was successfully verified.

[In]  Int[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)/(d + e*x)^(7/2),x]

[Out]

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

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Rubi in Sympy [A]  time = 15.0731, size = 44, normalized size = 1.07 \[ - \frac{2 c d}{e^{2} \sqrt{d + e x}} - \frac{2 \left (a e^{2} - c d^{2}\right )}{3 e^{2} \left (d + e x\right )^{\frac{3}{2}}} \]

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)**(7/2),x)

[Out]

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

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Mathematica [A]  time = 0.0421334, size = 33, normalized size = 0.8 \[ -\frac{2 \left (a e^2+c d (2 d+3 e x)\right )}{3 e^2 (d+e x)^{3/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)/(d + e*x)^(7/2),x]

[Out]

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

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Maple [A]  time = 0.005, size = 31, normalized size = 0.8 \[ -{\frac{6\,cdex+2\,a{e}^{2}+4\,c{d}^{2}}{3\,{e}^{2}} \left ( ex+d \right ) ^{-{\frac{3}{2}}}} \]

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)^(7/2),x)

[Out]

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

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Maxima [A]  time = 0.755289, size = 45, normalized size = 1.1 \[ -\frac{2 \,{\left (3 \,{\left (e x + d\right )} c d - c d^{2} + a e^{2}\right )}}{3 \,{\left (e x + d\right )}^{\frac{3}{2}} e^{2}} \]

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)^(7/2),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.220007, size = 54, normalized size = 1.32 \[ -\frac{2 \,{\left (3 \, c d e x + 2 \, c d^{2} + a e^{2}\right )}}{3 \,{\left (e^{3} x + d e^{2}\right )} \sqrt{e x + d}} \]

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)^(7/2),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 8.8414, size = 126, normalized size = 3.07 \[ \begin{cases} - \frac{2 a e^{2}}{3 d e^{2} \sqrt{d + e x} + 3 e^{3} x \sqrt{d + e x}} - \frac{4 c d^{2}}{3 d e^{2} \sqrt{d + e x} + 3 e^{3} x \sqrt{d + e x}} - \frac{6 c d e x}{3 d e^{2} \sqrt{d + e x} + 3 e^{3} x \sqrt{d + e x}} & \text{for}\: e \neq 0 \\\frac{c x^{2}}{2 d^{\frac{3}{2}}} & \text{otherwise} \end{cases} \]

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)**(7/2),x)

[Out]

Piecewise((-2*a*e**2/(3*d*e**2*sqrt(d + e*x) + 3*e**3*x*sqrt(d + e*x)) - 4*c*d**
2/(3*d*e**2*sqrt(d + e*x) + 3*e**3*x*sqrt(d + e*x)) - 6*c*d*e*x/(3*d*e**2*sqrt(d
+ e*x) + 3*e**3*x*sqrt(d + e*x)), Ne(e, 0)), (c*x**2/(2*d**(3/2)), True))

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GIAC/XCAS [A]  time = 0.203087, size = 63, normalized size = 1.54 \[ -\frac{2 \,{\left (3 \,{\left (x e + d\right )}^{2} c d -{\left (x e + d\right )} c d^{2} +{\left (x e + d\right )} a e^{2}\right )} e^{\left (-2\right )}}{3 \,{\left (x e + d\right )}^{\frac{5}{2}}} \]

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)^(7/2),x, algorithm="giac")

[Out]

-2/3*(3*(x*e + d)^2*c*d - (x*e + d)*c*d^2 + (x*e + d)*a*e^2)*e^(-2)/(x*e + d)^(5
/2)