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

Optimal. Leaf size=51 \[ \frac{2 d x}{3 a^2 \sqrt{a+c x^2}}-\frac{a e-c d x}{3 a c \left (a+c x^2\right )^{3/2}} \]

[Out]

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

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Rubi [A]  time = 0.0439669, antiderivative size = 51, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118 \[ \frac{2 d x}{3 a^2 \sqrt{a+c x^2}}-\frac{a e-c d x}{3 a c \left (a+c x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 4.9629, size = 42, normalized size = 0.82 \[ - \frac{a e - c d x}{3 a c \left (a + c x^{2}\right )^{\frac{3}{2}}} + \frac{2 d x}{3 a^{2} \sqrt{a + c x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0359072, size = 43, normalized size = 0.84 \[ \frac{-a^2 e+3 a c d x+2 c^2 d x^3}{3 a^2 c \left (a+c x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.006, size = 39, normalized size = 0.8 \[ -{\frac{-2\,{c}^{2}d{x}^{3}-3\,dxac+e{a}^{2}}{3\,{a}^{2}c} \left ( c{x}^{2}+a \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)/(c*x^2+a)^(5/2),x)

[Out]

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

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Maxima [A]  time = 0.71359, size = 65, normalized size = 1.27 \[ \frac{2 \, d x}{3 \, \sqrt{c x^{2} + a} a^{2}} + \frac{d x}{3 \,{\left (c x^{2} + a\right )}^{\frac{3}{2}} a} - \frac{e}{3 \,{\left (c x^{2} + a\right )}^{\frac{3}{2}} c} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.22462, size = 84, normalized size = 1.65 \[ \frac{{\left (2 \, c^{2} d x^{3} + 3 \, a c d x - a^{2} e\right )} \sqrt{c x^{2} + a}}{3 \,{\left (a^{2} c^{3} x^{4} + 2 \, a^{3} c^{2} x^{2} + a^{4} c\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*(2*c^2*d*x^3 + 3*a*c*d*x - a^2*e)*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 [A]  time = 37.3949, size = 146, normalized size = 2.86 \[ d \left (\frac{3 a x}{3 a^{\frac{7}{2}} \sqrt{1 + \frac{c x^{2}}{a}} + 3 a^{\frac{5}{2}} c x^{2} \sqrt{1 + \frac{c x^{2}}{a}}} + \frac{2 c x^{3}}{3 a^{\frac{7}{2}} \sqrt{1 + \frac{c x^{2}}{a}} + 3 a^{\frac{5}{2}} c x^{2} \sqrt{1 + \frac{c x^{2}}{a}}}\right ) + e \left (\begin{cases} - \frac{1}{3 a c \sqrt{a + c x^{2}} + 3 c^{2} x^{2} \sqrt{a + c x^{2}}} & \text{for}\: c \neq 0 \\\frac{x^{2}}{2 a^{\frac{5}{2}}} & \text{otherwise} \end{cases}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

d*(3*a*x/(3*a**(7/2)*sqrt(1 + c*x**2/a) + 3*a**(5/2)*c*x**2*sqrt(1 + c*x**2/a))
+ 2*c*x**3/(3*a**(7/2)*sqrt(1 + c*x**2/a) + 3*a**(5/2)*c*x**2*sqrt(1 + c*x**2/a)
)) + e*Piecewise((-1/(3*a*c*sqrt(a + c*x**2) + 3*c**2*x**2*sqrt(a + c*x**2)), Ne
(c, 0)), (x**2/(2*a**(5/2)), True))

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GIAC/XCAS [A]  time = 0.217286, size = 51, normalized size = 1. \[ \frac{{\left (\frac{2 \, c d x^{2}}{a^{2}} + \frac{3 \, d}{a}\right )} x - \frac{e}{c}}{3 \,{\left (c x^{2} + a\right )}^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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