3.560 \(\int \frac{(d+e x)^2}{\left (a+c x^2\right )^{3/2}} \, dx\)

Optimal. Leaf size=83 \[ \frac{e^2 \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )}{c^{3/2}}-\frac{d e \sqrt{a+c x^2}}{a c}-\frac{(d+e x) (a e-c d x)}{a c \sqrt{a+c x^2}} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.107459, antiderivative size = 83, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21 \[ \frac{e^2 \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )}{c^{3/2}}-\frac{d e \sqrt{a+c x^2}}{a c}-\frac{(d+e x) (a e-c d x)}{a c \sqrt{a+c x^2}} \]

Antiderivative was successfully verified.

[In]  Int[(d + e*x)^2/(a + c*x^2)^(3/2),x]

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 16.0368, size = 70, normalized size = 0.84 \[ \frac{e^{2} \operatorname{atanh}{\left (\frac{\sqrt{c} x}{\sqrt{a + c x^{2}}} \right )}}{c^{\frac{3}{2}}} - \frac{d e \sqrt{a + c x^{2}}}{a c} - \frac{\left (d + e x\right ) \left (a e - c d x\right )}{a c \sqrt{a + c x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x+d)**2/(c*x**2+a)**(3/2),x)

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.0871806, size = 69, normalized size = 0.83 \[ \frac{e^2 \log \left (\sqrt{c} \sqrt{a+c x^2}+c x\right )}{c^{3/2}}+\frac{-2 a d e-a e^2 x+c d^2 x}{a c \sqrt{a+c x^2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(d + e*x)^2/(a + c*x^2)^(3/2),x]

[Out]

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

_______________________________________________________________________________________

Maple [A]  time = 0.009, size = 76, normalized size = 0.9 \[{\frac{{d}^{2}x}{a}{\frac{1}{\sqrt{c{x}^{2}+a}}}}-{\frac{{e}^{2}x}{c}{\frac{1}{\sqrt{c{x}^{2}+a}}}}+{{e}^{2}\ln \left ( \sqrt{c}x+\sqrt{c{x}^{2}+a} \right ){c}^{-{\frac{3}{2}}}}-2\,{\frac{de}{c\sqrt{c{x}^{2}+a}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)^2/(c*x^2+a)^(3/2),x)

[Out]

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

_______________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^2/(c*x^2 + a)^(3/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.240434, size = 1, normalized size = 0.01 \[ \left [-\frac{2 \,{\left (2 \, a d e -{\left (c d^{2} - a e^{2}\right )} x\right )} \sqrt{c x^{2} + a} \sqrt{c} -{\left (a c e^{2} x^{2} + a^{2} e^{2}\right )} \log \left (-2 \, \sqrt{c x^{2} + a} c x -{\left (2 \, c x^{2} + a\right )} \sqrt{c}\right )}{2 \,{\left (a c^{2} x^{2} + a^{2} c\right )} \sqrt{c}}, -\frac{{\left (2 \, a d e -{\left (c d^{2} - a e^{2}\right )} x\right )} \sqrt{c x^{2} + a} \sqrt{-c} -{\left (a c e^{2} x^{2} + a^{2} e^{2}\right )} \arctan \left (\frac{\sqrt{-c} x}{\sqrt{c x^{2} + a}}\right )}{{\left (a c^{2} x^{2} + a^{2} c\right )} \sqrt{-c}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^2/(c*x^2 + a)^(3/2),x, algorithm="fricas")

[Out]

[-1/2*(2*(2*a*d*e - (c*d^2 - a*e^2)*x)*sqrt(c*x^2 + a)*sqrt(c) - (a*c*e^2*x^2 +
a^2*e^2)*log(-2*sqrt(c*x^2 + a)*c*x - (2*c*x^2 + a)*sqrt(c)))/((a*c^2*x^2 + a^2*
c)*sqrt(c)), -((2*a*d*e - (c*d^2 - a*e^2)*x)*sqrt(c*x^2 + a)*sqrt(-c) - (a*c*e^2
*x^2 + a^2*e^2)*arctan(sqrt(-c)*x/sqrt(c*x^2 + a)))/((a*c^2*x^2 + a^2*c)*sqrt(-c
))]

_______________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (d + e x\right )^{2}}{\left (a + c x^{2}\right )^{\frac{3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x+d)**2/(c*x**2+a)**(3/2),x)

[Out]

Integral((d + e*x)**2/(a + c*x**2)**(3/2), x)

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.220806, size = 93, normalized size = 1.12 \[ -\frac{\frac{2 \, d e}{c} - \frac{{\left (c^{2} d^{2} - a c e^{2}\right )} x}{a c^{2}}}{\sqrt{c x^{2} + a}} - \frac{e^{2}{\rm ln}\left ({\left | -\sqrt{c} x + \sqrt{c x^{2} + a} \right |}\right )}{c^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^2/(c*x^2 + a)^(3/2),x, algorithm="giac")

[Out]

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