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

Optimal. Leaf size=110 \[ \frac{c d \tanh ^{-1}\left (\frac{a e-c d x}{\sqrt{a+c x^2} \sqrt{a e^2+c d^2}}\right )}{e^2 \sqrt{a e^2+c d^2}}-\frac{\sqrt{a+c x^2}}{e (d+e x)}+\frac{\sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )}{e^2} \]

[Out]

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

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Rubi [A]  time = 0.224798, antiderivative size = 110, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.263 \[ \frac{c d \tanh ^{-1}\left (\frac{a e-c d x}{\sqrt{a+c x^2} \sqrt{a e^2+c d^2}}\right )}{e^2 \sqrt{a e^2+c d^2}}-\frac{\sqrt{a+c x^2}}{e (d+e x)}+\frac{\sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )}{e^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 22.5291, size = 97, normalized size = 0.88 \[ \frac{\sqrt{c} \operatorname{atanh}{\left (\frac{\sqrt{c} x}{\sqrt{a + c x^{2}}} \right )}}{e^{2}} + \frac{c d \operatorname{atanh}{\left (\frac{a e - c d x}{\sqrt{a + c x^{2}} \sqrt{a e^{2} + c d^{2}}} \right )}}{e^{2} \sqrt{a e^{2} + c d^{2}}} - \frac{\sqrt{a + c x^{2}}}{e \left (d + e x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.209523, size = 134, normalized size = 1.22 \[ \frac{\frac{c d \log \left (\sqrt{a+c x^2} \sqrt{a e^2+c d^2}+a e-c d x\right )}{\sqrt{a e^2+c d^2}}-\frac{c d \log (d+e x)}{\sqrt{a e^2+c d^2}}-\frac{e \sqrt{a+c x^2}}{d+e x}+\sqrt{c} \log \left (\sqrt{c} \sqrt{a+c x^2}+c x\right )}{e^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.014, size = 649, normalized size = 5.9 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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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)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.352025, size = 1, normalized size = 0.01 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/2*(sqrt(c*d^2 + a*e^2)*(e*x + d)*sqrt(c)*log(-2*c*x^2 - 2*sqrt(c*x^2 + a)*sqr
t(c)*x - a) - 2*sqrt(c*d^2 + a*e^2)*sqrt(c*x^2 + a)*e + (c*d*e*x + 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)*sqrt(c*x^2 + a))/(e^2*x^2
+ 2*d*e*x + d^2)))/((e^3*x + d*e^2)*sqrt(c*d^2 + a*e^2)), 1/2*(sqrt(-c*d^2 - a*
e^2)*(e*x + d)*sqrt(c)*log(-2*c*x^2 - 2*sqrt(c*x^2 + a)*sqrt(c)*x - a) - 2*sqrt(
-c*d^2 - a*e^2)*sqrt(c*x^2 + a)*e - 2*(c*d*e*x + c*d^2)*arctan(sqrt(-c*d^2 - a*e
^2)*(c*d*x - a*e)/((c*d^2 + a*e^2)*sqrt(c*x^2 + a))))/((e^3*x + d*e^2)*sqrt(-c*d
^2 - a*e^2)), 1/2*(2*sqrt(c*d^2 + a*e^2)*(e*x + d)*sqrt(-c)*arctan(c*x/(sqrt(c*x
^2 + a)*sqrt(-c))) - 2*sqrt(c*d^2 + a*e^2)*sqrt(c*x^2 + a)*e + (c*d*e*x + 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)*sqrt(c*x^2 + a))/(
e^2*x^2 + 2*d*e*x + d^2)))/((e^3*x + d*e^2)*sqrt(c*d^2 + a*e^2)), (sqrt(-c*d^2 -
a*e^2)*(e*x + d)*sqrt(-c)*arctan(c*x/(sqrt(c*x^2 + a)*sqrt(-c))) - sqrt(-c*d^2
- a*e^2)*sqrt(c*x^2 + a)*e - (c*d*e*x + c*d^2)*arctan(sqrt(-c*d^2 - a*e^2)*(c*d*
x - a*e)/((c*d^2 + a*e^2)*sqrt(c*x^2 + a))))/((e^3*x + d*e^2)*sqrt(-c*d^2 - a*e^
2))]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: TypeError