3.569 \(\int \frac{(d+e x)^2}{\left (a+c x^2\right )^{5/2}} \, dx\)

Optimal. Leaf size=58 \[ \frac{x (d+e x)^2}{3 a \left (a+c x^2\right )^{3/2}}-\frac{2 d (a e-c d x)}{3 a^2 c \sqrt{a+c x^2}} \]

[Out]

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

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Rubi [A]  time = 0.0717018, antiderivative size = 58, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105 \[ \frac{x (d+e x)^2}{3 a \left (a+c x^2\right )^{3/2}}-\frac{2 d (a e-c d x)}{3 a^2 c \sqrt{a+c x^2}} \]

Antiderivative was successfully verified.

[In]  Int[(d + e*x)^2/(a + c*x^2)^(5/2),x]

[Out]

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

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Rubi in Sympy [A]  time = 8.8404, size = 49, normalized size = 0.84 \[ \frac{x \left (d + e x\right )^{2}}{3 a \left (a + c x^{2}\right )^{\frac{3}{2}}} - \frac{2 d \left (a e - c d x\right )}{3 a^{2} c \sqrt{a + c x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x+d)**2/(c*x**2+a)**(5/2),x)

[Out]

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

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Mathematica [A]  time = 0.0609068, size = 57, normalized size = 0.98 \[ \frac{-2 a^2 d e+a c x \left (3 d^2+e^2 x^2\right )+2 c^2 d^2 x^3}{3 a^2 c \left (a+c x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(d + e*x)^2/(a + c*x^2)^(5/2),x]

[Out]

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

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Maple [A]  time = 0.01, size = 55, normalized size = 1. \[ -{\frac{-ac{e}^{2}{x}^{3}-2\,{c}^{2}{d}^{2}{x}^{3}-3\,{d}^{2}xac+2\,de{a}^{2}}{3\,{a}^{2}c} \left ( c{x}^{2}+a \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)^2/(c*x^2+a)^(5/2),x)

[Out]

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

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Maxima [A]  time = 0.712615, size = 124, normalized size = 2.14 \[ \frac{2 \, d^{2} x}{3 \, \sqrt{c x^{2} + a} a^{2}} + \frac{d^{2} x}{3 \,{\left (c x^{2} + a\right )}^{\frac{3}{2}} a} - \frac{e^{2} x}{3 \,{\left (c x^{2} + a\right )}^{\frac{3}{2}} c} + \frac{e^{2} x}{3 \, \sqrt{c x^{2} + a} a c} - \frac{2 \, d e}{3 \,{\left (c x^{2} + a\right )}^{\frac{3}{2}} c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^2/(c*x^2 + a)^(5/2),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.223967, size = 101, normalized size = 1.74 \[ \frac{{\left (3 \, a c d^{2} x - 2 \, a^{2} d e +{\left (2 \, c^{2} d^{2} + a c e^{2}\right )} x^{3}\right )} \sqrt{c x^{2} + a}}{3 \,{\left (a^{2} c^{3} x^{4} + 2 \, a^{3} c^{2} x^{2} + a^{4} c\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^2/(c*x^2 + a)^(5/2),x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (d + e x\right )^{2}}{\left (a + c x^{2}\right )^{\frac{5}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x+d)**2/(c*x**2+a)**(5/2),x)

[Out]

Integral((d + e*x)**2/(a + c*x**2)**(5/2), x)

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GIAC/XCAS [A]  time = 0.217132, size = 74, normalized size = 1.28 \[ \frac{{\left (\frac{3 \, d^{2}}{a} + \frac{{\left (2 \, c^{2} d^{2} + a c e^{2}\right )} x^{2}}{a^{2} c}\right )} x - \frac{2 \, d e}{c}}{3 \,{\left (c x^{2} + a\right )}^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^2/(c*x^2 + a)^(5/2),x, algorithm="giac")

[Out]

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