3.16 \(\int \frac{\sqrt{c+e x+d x^2} \sqrt{a^2+2 a b x^2+b^2 x^4}}{x^2} \, dx\)

Optimal. Leaf size=294 \[ \frac{\sqrt{a^2+2 a b x^2+b^2 x^4} \left (8 a d^2+4 b c d-b e^2\right ) \tanh ^{-1}\left (\frac{2 d x+e}{2 \sqrt{d} \sqrt{c+d x^2+e x}}\right )}{8 d^{3/2} \left (a+b x^2\right )}-\frac{a \sqrt{a^2+2 a b x^2+b^2 x^4} \left (c+d x^2+e x\right )^{3/2}}{c x \left (a+b x^2\right )}+\frac{\sqrt{a^2+2 a b x^2+b^2 x^4} \sqrt{c+d x^2+e x} (e (4 a d+b c)+2 d x (2 a d+b c))}{4 c d \left (a+b x^2\right )}-\frac{a e \sqrt{a^2+2 a b x^2+b^2 x^4} \tanh ^{-1}\left (\frac{2 c+e x}{2 \sqrt{c} \sqrt{c+d x^2+e x}}\right )}{2 \sqrt{c} \left (a+b x^2\right )} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 1.54367, antiderivative size = 294, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 40, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.175 \[ \frac{\sqrt{a^2+2 a b x^2+b^2 x^4} \left (8 a d^2+4 b c d-b e^2\right ) \tanh ^{-1}\left (\frac{2 d x+e}{2 \sqrt{d} \sqrt{c+d x^2+e x}}\right )}{8 d^{3/2} \left (a+b x^2\right )}-\frac{a \sqrt{a^2+2 a b x^2+b^2 x^4} \left (c+d x^2+e x\right )^{3/2}}{c x \left (a+b x^2\right )}+\frac{\sqrt{a^2+2 a b x^2+b^2 x^4} \sqrt{c+d x^2+e x} (e (4 a d+b c)+2 d x (2 a d+b c))}{4 c d \left (a+b x^2\right )}-\frac{a e \sqrt{a^2+2 a b x^2+b^2 x^4} \tanh ^{-1}\left (\frac{2 c+e x}{2 \sqrt{c} \sqrt{c+d x^2+e x}}\right )}{2 \sqrt{c} \left (a+b x^2\right )} \]

Antiderivative was successfully verified.

[In]  Int[(Sqrt[c + e*x + d*x^2]*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/x^2,x]

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.449775, size = 178, normalized size = 0.61 \[ \frac{\sqrt{\left (a+b x^2\right )^2} \left (\sqrt{c} \left (x \left (8 a d^2+4 b c d-b e^2\right ) \log \left (2 \sqrt{d} \sqrt{c+x (d x+e)}+2 d x+e\right )+2 \sqrt{d} \sqrt{c+x (d x+e)} (b x (2 d x+e)-4 a d)\right )-4 a d^{3/2} e x \log \left (2 \sqrt{c} \sqrt{c+x (d x+e)}+2 c+e x\right )+4 a d^{3/2} e x \log (x)\right )}{8 \sqrt{c} d^{3/2} x \left (a+b x^2\right )} \]

Antiderivative was successfully verified.

[In]  Integrate[(Sqrt[c + e*x + d*x^2]*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/x^2,x]

[Out]

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

_______________________________________________________________________________________

Maple [A]  time = 0.017, size = 304, normalized size = 1. \[{\frac{1}{ \left ( 8\,b{x}^{2}+8\,a \right ) x}\sqrt{ \left ( b{x}^{2}+a \right ) ^{2}} \left ( 8\,a{d}^{3}\ln \left ( 1/2\,{\frac{2\,\sqrt{d{x}^{2}+ex+c}\sqrt{d}+2\,dx+e}{\sqrt{d}}} \right ){c}^{3/2}x+8\,a{d}^{7/2}\sqrt{d{x}^{2}+ex+c}{x}^{2}\sqrt{c}+4\,b\sqrt{d{x}^{2}+ex+c}{x}^{2}{d}^{5/2}{c}^{3/2}+4\,b\ln \left ( 1/2\,{\frac{2\,\sqrt{d{x}^{2}+ex+c}\sqrt{d}+2\,dx+e}{\sqrt{d}}} \right ){c}^{5/2}{d}^{2}x-4\,ae\ln \left ({\frac{2\,c+ex+2\,\sqrt{c}\sqrt{d{x}^{2}+ex+c}}{x}} \right ){d}^{5/2}cx-8\,a \left ( d{x}^{2}+ex+c \right ) ^{3/2}{d}^{5/2}\sqrt{c}+8\,ae\sqrt{d{x}^{2}+ex+c}{d}^{5/2}x\sqrt{c}+2\,b\sqrt{d{x}^{2}+ex+c}e{d}^{3/2}{c}^{3/2}x-b\ln \left ({\frac{1}{2} \left ( 2\,\sqrt{d{x}^{2}+ex+c}\sqrt{d}+2\,dx+e \right ){\frac{1}{\sqrt{d}}}} \right ){e}^{2}d{c}^{{\frac{3}{2}}}x \right ){d}^{-{\frac{5}{2}}}{c}^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/8*((b*x^2+a)^2)^(1/2)*(8*a*d^3*ln(1/2*(2*(d*x^2+e*x+c)^(1/2)*d^(1/2)+2*d*x+e)/
d^(1/2))*c^(3/2)*x+8*a*d^(7/2)*(d*x^2+e*x+c)^(1/2)*x^2*c^(1/2)+4*b*(d*x^2+e*x+c)
^(1/2)*x^2*d^(5/2)*c^(3/2)+4*b*ln(1/2*(2*(d*x^2+e*x+c)^(1/2)*d^(1/2)+2*d*x+e)/d^
(1/2))*c^(5/2)*d^2*x-4*a*e*ln((2*c+e*x+2*c^(1/2)*(d*x^2+e*x+c)^(1/2))/x)*d^(5/2)
*c*x-8*a*(d*x^2+e*x+c)^(3/2)*d^(5/2)*c^(1/2)+8*a*e*(d*x^2+e*x+c)^(1/2)*d^(5/2)*x
*c^(1/2)+2*b*(d*x^2+e*x+c)^(1/2)*e*d^(3/2)*c^(3/2)*x-b*ln(1/2*(2*(d*x^2+e*x+c)^(
1/2)*d^(1/2)+2*d*x+e)/d^(1/2))*e^2*d*c^(3/2)*x)/(b*x^2+a)/d^(5/2)/c^(3/2)/x

_______________________________________________________________________________________

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

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.770168, size = 1, normalized size = 0. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/16*(4*a*d^(3/2)*e*x*log(-(4*(c*e*x + 2*c^2)*sqrt(d*x^2 + e*x + c) - (8*c*e*x
+ (4*c*d + e^2)*x^2 + 8*c^2)*sqrt(c))/x^2) - (4*b*c*d + 8*a*d^2 - b*e^2)*sqrt(c)
*x*log(-4*(2*d^2*x + d*e)*sqrt(d*x^2 + e*x + c) + (8*d^2*x^2 + 8*d*e*x + 4*c*d +
e^2)*sqrt(d)) + 4*(2*b*d*x^2 + b*e*x - 4*a*d)*sqrt(d*x^2 + e*x + c)*sqrt(c)*sqr
t(d))/(sqrt(c)*d^(3/2)*x), 1/8*(2*a*sqrt(-d)*d*e*x*log(-(4*(c*e*x + 2*c^2)*sqrt(
d*x^2 + e*x + c) - (8*c*e*x + (4*c*d + e^2)*x^2 + 8*c^2)*sqrt(c))/x^2) + (4*b*c*
d + 8*a*d^2 - b*e^2)*sqrt(c)*x*arctan(1/2*(2*d*x + e)*sqrt(-d)/(sqrt(d*x^2 + e*x
+ c)*d)) + 2*(2*b*d*x^2 + b*e*x - 4*a*d)*sqrt(d*x^2 + e*x + c)*sqrt(c)*sqrt(-d)
)/(sqrt(c)*sqrt(-d)*d*x), -1/16*(8*a*d^(3/2)*e*x*arctan(1/2*(e*x + 2*c)*sqrt(-c)
/(sqrt(d*x^2 + e*x + c)*c)) + (4*b*c*d + 8*a*d^2 - b*e^2)*sqrt(-c)*x*log(-4*(2*d
^2*x + d*e)*sqrt(d*x^2 + e*x + c) + (8*d^2*x^2 + 8*d*e*x + 4*c*d + e^2)*sqrt(d))
- 4*(2*b*d*x^2 + b*e*x - 4*a*d)*sqrt(d*x^2 + e*x + c)*sqrt(-c)*sqrt(d))/(sqrt(-
c)*d^(3/2)*x), -1/8*(4*a*sqrt(-d)*d*e*x*arctan(1/2*(e*x + 2*c)*sqrt(-c)/(sqrt(d*
x^2 + e*x + c)*c)) - (4*b*c*d + 8*a*d^2 - b*e^2)*sqrt(-c)*x*arctan(1/2*(2*d*x +
e)*sqrt(-d)/(sqrt(d*x^2 + e*x + c)*d)) - 2*(2*b*d*x^2 + b*e*x - 4*a*d)*sqrt(d*x^
2 + e*x + c)*sqrt(-c)*sqrt(-d))/(sqrt(-c)*sqrt(-d)*d*x)]

_______________________________________________________________________________________

Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

_______________________________________________________________________________________

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

[Out]

Exception raised: TypeError