3.583 \(\int \frac{a+c x^2}{\sqrt{d+e x}} \, dx\)

Optimal. Leaf size=61 \[ \frac{2 \sqrt{d+e x} \left (a e^2+c d^2\right )}{e^3}+\frac{2 c (d+e x)^{5/2}}{5 e^3}-\frac{4 c d (d+e x)^{3/2}}{3 e^3} \]

[Out]

(2*(c*d^2 + a*e^2)*Sqrt[d + e*x])/e^3 - (4*c*d*(d + e*x)^(3/2))/(3*e^3) + (2*c*(
d + e*x)^(5/2))/(5*e^3)

_______________________________________________________________________________________

Rubi [A]  time = 0.0674393, antiderivative size = 61, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.059 \[ \frac{2 \sqrt{d+e x} \left (a e^2+c d^2\right )}{e^3}+\frac{2 c (d+e x)^{5/2}}{5 e^3}-\frac{4 c d (d+e x)^{3/2}}{3 e^3} \]

Antiderivative was successfully verified.

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

[Out]

(2*(c*d^2 + a*e^2)*Sqrt[d + e*x])/e^3 - (4*c*d*(d + e*x)^(3/2))/(3*e^3) + (2*c*(
d + e*x)^(5/2))/(5*e^3)

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 11.391, size = 58, normalized size = 0.95 \[ - \frac{4 c d \left (d + e x\right )^{\frac{3}{2}}}{3 e^{3}} + \frac{2 c \left (d + e x\right )^{\frac{5}{2}}}{5 e^{3}} + \frac{2 \sqrt{d + e x} \left (a e^{2} + c d^{2}\right )}{e^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((c*x**2+a)/(e*x+d)**(1/2),x)

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.035991, size = 44, normalized size = 0.72 \[ \frac{2 \sqrt{d+e x} \left (15 a e^2+c \left (8 d^2-4 d e x+3 e^2 x^2\right )\right )}{15 e^3} \]

Antiderivative was successfully verified.

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

[Out]

(2*Sqrt[d + e*x]*(15*a*e^2 + c*(8*d^2 - 4*d*e*x + 3*e^2*x^2)))/(15*e^3)

_______________________________________________________________________________________

Maple [A]  time = 0.006, size = 41, normalized size = 0.7 \[{\frac{6\,c{e}^{2}{x}^{2}-8\,cdex+30\,a{e}^{2}+16\,c{d}^{2}}{15\,{e}^{3}}\sqrt{ex+d}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^2+a)/(e*x+d)^(1/2),x)

[Out]

2/15*(e*x+d)^(1/2)*(3*c*e^2*x^2-4*c*d*e*x+15*a*e^2+8*c*d^2)/e^3

_______________________________________________________________________________________

Maxima [A]  time = 0.694781, size = 72, normalized size = 1.18 \[ \frac{2 \,{\left (15 \, \sqrt{e x + d} a + \frac{{\left (3 \,{\left (e x + d\right )}^{\frac{5}{2}} - 10 \,{\left (e x + d\right )}^{\frac{3}{2}} d + 15 \, \sqrt{e x + d} d^{2}\right )} c}{e^{2}}\right )}}{15 \, e} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/15*(15*sqrt(e*x + d)*a + (3*(e*x + d)^(5/2) - 10*(e*x + d)^(3/2)*d + 15*sqrt(e
*x + d)*d^2)*c/e^2)/e

_______________________________________________________________________________________

Fricas [A]  time = 0.206401, size = 54, normalized size = 0.89 \[ \frac{2 \,{\left (3 \, c e^{2} x^{2} - 4 \, c d e x + 8 \, c d^{2} + 15 \, a e^{2}\right )} \sqrt{e x + d}}{15 \, e^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/15*(3*c*e^2*x^2 - 4*c*d*e*x + 8*c*d^2 + 15*a*e^2)*sqrt(e*x + d)/e^3

_______________________________________________________________________________________

Sympy [A]  time = 6.06693, size = 150, normalized size = 2.46 \[ \begin{cases} - \frac{\frac{2 a d}{\sqrt{d + e x}} + 2 a \left (- \frac{d}{\sqrt{d + e x}} - \sqrt{d + e x}\right ) + \frac{2 c d \left (\frac{d^{2}}{\sqrt{d + e x}} + 2 d \sqrt{d + e x} - \frac{\left (d + e x\right )^{\frac{3}{2}}}{3}\right )}{e^{2}} + \frac{2 c \left (- \frac{d^{3}}{\sqrt{d + e x}} - 3 d^{2} \sqrt{d + e x} + d \left (d + e x\right )^{\frac{3}{2}} - \frac{\left (d + e x\right )^{\frac{5}{2}}}{5}\right )}{e^{2}}}{e} & \text{for}\: e \neq 0 \\\frac{a x + \frac{c x^{3}}{3}}{\sqrt{d}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x**2+a)/(e*x+d)**(1/2),x)

[Out]

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

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.211023, size = 82, normalized size = 1.34 \[ \frac{2}{15} \,{\left ({\left (3 \,{\left (x e + d\right )}^{\frac{5}{2}} e^{8} - 10 \,{\left (x e + d\right )}^{\frac{3}{2}} d e^{8} + 15 \, \sqrt{x e + d} d^{2} e^{8}\right )} c e^{\left (-10\right )} + 15 \, \sqrt{x e + d} a\right )} e^{\left (-1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/15*((3*(x*e + d)^(5/2)*e^8 - 10*(x*e + d)^(3/2)*d*e^8 + 15*sqrt(x*e + d)*d^2*e
^8)*c*e^(-10) + 15*sqrt(x*e + d)*a)*e^(-1)