3.520 \(\int \frac{\sqrt{a+c x^2}}{(d+e x)^3} \, dx\)

Optimal. Leaf size=103 \[ -\frac{\sqrt{a+c x^2} (a e-c d x)}{2 (d+e x)^2 \left (a e^2+c d^2\right )}-\frac{a c \tanh ^{-1}\left (\frac{a e-c d x}{\sqrt{a+c x^2} \sqrt{a e^2+c d^2}}\right )}{2 \left (a e^2+c d^2\right )^{3/2}} \]

[Out]

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

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Rubi [A]  time = 0.124793, antiderivative size = 103, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158 \[ -\frac{\sqrt{a+c x^2} (a e-c d x)}{2 (d+e x)^2 \left (a e^2+c d^2\right )}-\frac{a c \tanh ^{-1}\left (\frac{a e-c d x}{\sqrt{a+c x^2} \sqrt{a e^2+c d^2}}\right )}{2 \left (a e^2+c d^2\right )^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 14.6173, size = 94, normalized size = 0.91 \[ - \frac{a c \operatorname{atanh}{\left (\frac{a e - c d x}{\sqrt{a + c x^{2}} \sqrt{a e^{2} + c d^{2}}} \right )}}{2 \left (a e^{2} + c d^{2}\right )^{\frac{3}{2}}} - \frac{\sqrt{a + c x^{2}} \left (2 a e - 2 c d x\right )}{4 \left (d + e x\right )^{2} \left (a e^{2} + c d^{2}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((c*x**2+a)**(1/2)/(e*x+d)**3,x)

[Out]

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

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Mathematica [A]  time = 0.162328, size = 127, normalized size = 1.23 \[ \frac{\sqrt{a+c x^2} \sqrt{a e^2+c d^2} (c d x-a e)-a c (d+e x)^2 \log \left (\sqrt{a+c x^2} \sqrt{a e^2+c d^2}+a e-c d x\right )+a c (d+e x)^2 \log (d+e x)}{2 (d+e x)^2 \left (a e^2+c d^2\right )^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.017, size = 1174, normalized size = 11.4 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^2+a)^(1/2)/(e*x+d)^3,x)

[Out]

-1/2/e/(a*e^2+c*d^2)/(d/e+x)^2*(c*(d/e+x)^2-2*c*d/e*(d/e+x)+(a*e^2+c*d^2)/e^2)^(
3/2)-1/2*c*d/(a*e^2+c*d^2)^2/(d/e+x)*(c*(d/e+x)^2-2*c*d/e*(d/e+x)+(a*e^2+c*d^2)/
e^2)^(3/2)-1/2/e*c^2*d^2/(a*e^2+c*d^2)^2*(c*(d/e+x)^2-2*c*d/e*(d/e+x)+(a*e^2+c*d
^2)/e^2)^(1/2)+1/2/e^2*c^(5/2)*d^3/(a*e^2+c*d^2)^2*ln((-c*d/e+c*(d/e+x))/c^(1/2)
+(c*(d/e+x)^2-2*c*d/e*(d/e+x)+(a*e^2+c*d^2)/e^2)^(1/2))+1/2/e*c^2*d^2/(a*e^2+c*d
^2)^2/((a*e^2+c*d^2)/e^2)^(1/2)*ln((2*(a*e^2+c*d^2)/e^2-2*c*d/e*(d/e+x)+2*((a*e^
2+c*d^2)/e^2)^(1/2)*(c*(d/e+x)^2-2*c*d/e*(d/e+x)+(a*e^2+c*d^2)/e^2)^(1/2))/(d/e+
x))*a+1/2/e^3*c^3*d^4/(a*e^2+c*d^2)^2/((a*e^2+c*d^2)/e^2)^(1/2)*ln((2*(a*e^2+c*d
^2)/e^2-2*c*d/e*(d/e+x)+2*((a*e^2+c*d^2)/e^2)^(1/2)*(c*(d/e+x)^2-2*c*d/e*(d/e+x)
+(a*e^2+c*d^2)/e^2)^(1/2))/(d/e+x))+1/2*c^2*d/(a*e^2+c*d^2)^2*(c*(d/e+x)^2-2*c*d
/e*(d/e+x)+(a*e^2+c*d^2)/e^2)^(1/2)*x+1/2*c^(3/2)*d/(a*e^2+c*d^2)^2*ln((-c*d/e+c
*(d/e+x))/c^(1/2)+(c*(d/e+x)^2-2*c*d/e*(d/e+x)+(a*e^2+c*d^2)/e^2)^(1/2))*a+1/2/e
/(a*e^2+c*d^2)*c*(c*(d/e+x)^2-2*c*d/e*(d/e+x)+(a*e^2+c*d^2)/e^2)^(1/2)-1/2/e^2/(
a*e^2+c*d^2)*c^(3/2)*d*ln((-c*d/e+c*(d/e+x))/c^(1/2)+(c*(d/e+x)^2-2*c*d/e*(d/e+x
)+(a*e^2+c*d^2)/e^2)^(1/2))-1/2/e/(a*e^2+c*d^2)*c/((a*e^2+c*d^2)/e^2)^(1/2)*ln((
2*(a*e^2+c*d^2)/e^2-2*c*d/e*(d/e+x)+2*((a*e^2+c*d^2)/e^2)^(1/2)*(c*(d/e+x)^2-2*c
*d/e*(d/e+x)+(a*e^2+c*d^2)/e^2)^(1/2))/(d/e+x))*a-1/2/e^3/(a*e^2+c*d^2)*c^2/((a*
e^2+c*d^2)/e^2)^(1/2)*ln((2*(a*e^2+c*d^2)/e^2-2*c*d/e*(d/e+x)+2*((a*e^2+c*d^2)/e
^2)^(1/2)*(c*(d/e+x)^2-2*c*d/e*(d/e+x)+(a*e^2+c*d^2)/e^2)^(1/2))/(d/e+x))*d^2

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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(sqrt(c*x^2 + a)/(e*x + d)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.314998, size = 1, normalized size = 0.01 \[ \left [\frac{2 \, \sqrt{c d^{2} + a e^{2}}{\left (c d x - a e\right )} \sqrt{c x^{2} + a} +{\left (a c e^{2} x^{2} + 2 \, a c d e x + a c d^{2}\right )} \log \left (\frac{{\left (2 \, a c d e x - a c d^{2} - 2 \, a^{2} e^{2} -{\left (2 \, c^{2} d^{2} + a c e^{2}\right )} x^{2}\right )} \sqrt{c d^{2} + a e^{2}} + 2 \,{\left (a c d^{2} e + a^{2} e^{3} -{\left (c^{2} d^{3} + a c d e^{2}\right )} x\right )} \sqrt{c x^{2} + a}}{e^{2} x^{2} + 2 \, d e x + d^{2}}\right )}{4 \,{\left (c d^{4} + a d^{2} e^{2} +{\left (c d^{2} e^{2} + a e^{4}\right )} x^{2} + 2 \,{\left (c d^{3} e + a d e^{3}\right )} x\right )} \sqrt{c d^{2} + a e^{2}}}, \frac{\sqrt{-c d^{2} - a e^{2}}{\left (c d x - a e\right )} \sqrt{c x^{2} + a} +{\left (a c e^{2} x^{2} + 2 \, a c d e x + a c d^{2}\right )} \arctan \left (\frac{\sqrt{-c d^{2} - a e^{2}}{\left (c d x - a e\right )}}{{\left (c d^{2} + a e^{2}\right )} \sqrt{c x^{2} + a}}\right )}{2 \,{\left (c d^{4} + a d^{2} e^{2} +{\left (c d^{2} e^{2} + a e^{4}\right )} x^{2} + 2 \,{\left (c d^{3} e + a d e^{3}\right )} x\right )} \sqrt{-c d^{2} - a e^{2}}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/4*(2*sqrt(c*d^2 + a*e^2)*(c*d*x - a*e)*sqrt(c*x^2 + a) + (a*c*e^2*x^2 + 2*a*c
*d*e*x + a*c*d^2)*log(((2*a*c*d*e*x - a*c*d^2 - 2*a^2*e^2 - (2*c^2*d^2 + a*c*e^2
)*x^2)*sqrt(c*d^2 + a*e^2) + 2*(a*c*d^2*e + a^2*e^3 - (c^2*d^3 + a*c*d*e^2)*x)*s
qrt(c*x^2 + a))/(e^2*x^2 + 2*d*e*x + d^2)))/((c*d^4 + a*d^2*e^2 + (c*d^2*e^2 + a
*e^4)*x^2 + 2*(c*d^3*e + a*d*e^3)*x)*sqrt(c*d^2 + a*e^2)), 1/2*(sqrt(-c*d^2 - a*
e^2)*(c*d*x - a*e)*sqrt(c*x^2 + a) + (a*c*e^2*x^2 + 2*a*c*d*e*x + a*c*d^2)*arcta
n(sqrt(-c*d^2 - a*e^2)*(c*d*x - a*e)/((c*d^2 + a*e^2)*sqrt(c*x^2 + a))))/((c*d^4
+ a*d^2*e^2 + (c*d^2*e^2 + a*e^4)*x^2 + 2*(c*d^3*e + a*d*e^3)*x)*sqrt(-c*d^2 -
a*e^2))]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x**2+a)**(1/2)/(e*x+d)**3,x)

[Out]

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

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GIAC/XCAS [A]  time = 0.229049, size = 421, normalized size = 4.09 \[ -\frac{a c \arctan \left (\frac{{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )} e + \sqrt{c} d}{\sqrt{-c d^{2} - a e^{2}}}\right )}{{\left (c d^{2} + a e^{2}\right )} \sqrt{-c d^{2} - a e^{2}}} + \frac{2 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{3} c^{2} d^{2} e + 2 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{2} c^{\frac{5}{2}} d^{3} - 2 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )} a c^{2} d^{2} e -{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{2} a c^{\frac{3}{2}} d e^{2} +{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{3} a c e^{3} + a^{2} c^{\frac{3}{2}} d e^{2} +{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )} a^{2} c e^{3}}{{\left (c d^{2} e^{2} + a e^{4}\right )}{\left ({\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{2} e + 2 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )} \sqrt{c} d - a e\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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