3.17 \(\int \frac{\sqrt{c+e x+d x^2} \sqrt{a^2+2 a b x^2+b^2 x^4}}{x^3} \, dx\)

Optimal. Leaf size=288 \[ -\frac{\sqrt{a^2+2 a b x^2+b^2 x^4} \left (4 a c d-a e^2+8 b c^2\right ) \tanh ^{-1}\left (\frac{2 c+e x}{2 \sqrt{c} \sqrt{c+d x^2+e x}}\right )}{8 c^{3/2} \left (a+b x^2\right )}-\frac{a \sqrt{a^2+2 a b x^2+b^2 x^4} \left (c+d x^2+e x\right )^{3/2}}{2 c x^2 \left (a+b x^2\right )}+\frac{\sqrt{a^2+2 a b x^2+b^2 x^4} \sqrt{c+d x^2+e x} (2 x (a d+2 b c)+a e)}{4 c x \left (a+b x^2\right )}+\frac{b e \sqrt{a^2+2 a b x^2+b^2 x^4} \tanh ^{-1}\left (\frac{2 d x+e}{2 \sqrt{d} \sqrt{c+d x^2+e x}}\right )}{2 \sqrt{d} \left (a+b x^2\right )} \]

[Out]

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

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Rubi [A]  time = 1.60994, antiderivative size = 288, 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 \[ -\frac{\sqrt{a^2+2 a b x^2+b^2 x^4} \left (4 a c d-a e^2+8 b c^2\right ) \tanh ^{-1}\left (\frac{2 c+e x}{2 \sqrt{c} \sqrt{c+d x^2+e x}}\right )}{8 c^{3/2} \left (a+b x^2\right )}-\frac{a \sqrt{a^2+2 a b x^2+b^2 x^4} \left (c+d x^2+e x\right )^{3/2}}{2 c x^2 \left (a+b x^2\right )}+\frac{\sqrt{a^2+2 a b x^2+b^2 x^4} \sqrt{c+d x^2+e x} (2 x (a d+2 b c)+a e)}{4 c x \left (a+b x^2\right )}+\frac{b e \sqrt{a^2+2 a b x^2+b^2 x^4} \tanh ^{-1}\left (\frac{2 d x+e}{2 \sqrt{d} \sqrt{c+d x^2+e x}}\right )}{2 \sqrt{d} \left (a+b x^2\right )} \]

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^3,x]

[Out]

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

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

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**3,x)

[Out]

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

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Mathematica [A]  time = 0.533847, size = 203, normalized size = 0.7 \[ \frac{\sqrt{\left (a+b x^2\right )^2} \left (\sqrt{d} x^2 \log (x) \left (4 a c d-a e^2+8 b c^2\right )+\sqrt{d} x^2 \left (a \left (e^2-4 c d\right )-8 b c^2\right ) \log \left (2 \sqrt{c} \sqrt{c+x (d x+e)}+2 c+e x\right )+2 \sqrt{c} \left (\sqrt{d} \sqrt{c+x (d x+e)} \left (4 b c x^2-a (2 c+e x)\right )+2 b c e x^2 \log \left (2 \sqrt{d} \sqrt{c+x (d x+e)}+2 d x+e\right )\right )\right )}{8 c^{3/2} \sqrt{d} x^2 \left (a+b x^2\right )} \]

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^3,x]

[Out]

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

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Maple [A]  time = 0.016, size = 341, normalized size = 1.2 \[{\frac{1}{ \left ( 8\,b{x}^{2}+8\,a \right ){x}^{2}}\sqrt{ \left ( b{x}^{2}+a \right ) ^{2}} \left ( -8\,b{c}^{4}\ln \left ({\frac{2\,c+ex+2\,\sqrt{c}\sqrt{d{x}^{2}+ex+c}}{x}} \right ){x}^{2}\sqrt{d}-2\,ae{d}^{3/2}\sqrt{d{x}^{2}+ex+c}{x}^{3}{c}^{3/2}+4\,be\ln \left ( 1/2\,{\frac{2\,\sqrt{d{x}^{2}+ex+c}\sqrt{d}+2\,dx+e}{\sqrt{d}}} \right ){x}^{2}{c}^{7/2}-4\,a{d}^{3/2}\ln \left ({\frac{2\,c+ex+2\,\sqrt{c}\sqrt{d{x}^{2}+ex+c}}{x}} \right ){x}^{2}{c}^{3}+2\,ae \left ( d{x}^{2}+ex+c \right ) ^{3/2}x{c}^{3/2}\sqrt{d}+4\,a{d}^{3/2}\sqrt{d{x}^{2}+ex+c}{x}^{2}{c}^{5/2}-2\,a{e}^{2}\sqrt{d{x}^{2}+ex+c}{x}^{2}{c}^{3/2}\sqrt{d}+8\,b\sqrt{d{x}^{2}+ex+c}{x}^{2}{c}^{7/2}\sqrt{d}+a{e}^{2}\ln \left ({\frac{1}{x} \left ( 2\,c+ex+2\,\sqrt{c}\sqrt{d{x}^{2}+ex+c} \right ) } \right ){x}^{2}{c}^{2}\sqrt{d}-4\,a \left ( d{x}^{2}+ex+c \right ) ^{3/2}{c}^{5/2}\sqrt{d} \right ){c}^{-{\frac{7}{2}}}{\frac{1}{\sqrt{d}}}} \]

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^3,x)

[Out]

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

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.581191, 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^3,x, algorithm="fricas")

[Out]

[1/16*(4*b*c^(3/2)*e*x^2*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)) - (8*b*c^2 + 4*a*c*d - a*e^2)*sqrt(d)*x^2*lo
g((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*(4*b*c*x^2 - a*e*x - 2*a*c)*sqrt(d*x^2 + e*x + c)*sqrt(c)*s
qrt(d))/(c^(3/2)*sqrt(d)*x^2), 1/16*(8*b*c^(3/2)*e*x^2*arctan(1/2*(2*d*x + e)*sq
rt(-d)/(sqrt(d*x^2 + e*x + c)*d)) - (8*b*c^2 + 4*a*c*d - a*e^2)*sqrt(-d)*x^2*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*(4*b*c*x^2 - a*e*x - 2*a*c)*sqrt(d*x^2 + e*x + c)*sqrt(c)*sq
rt(-d))/(c^(3/2)*sqrt(-d)*x^2), 1/8*(2*b*sqrt(-c)*c*e*x^2*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)) - (8*b*c^2
+ 4*a*c*d - a*e^2)*sqrt(d)*x^2*arctan(1/2*(e*x + 2*c)*sqrt(-c)/(sqrt(d*x^2 + e*x
+ c)*c)) + 2*(4*b*c*x^2 - a*e*x - 2*a*c)*sqrt(d*x^2 + e*x + c)*sqrt(-c)*sqrt(d)
)/(sqrt(-c)*c*sqrt(d)*x^2), 1/8*(4*b*sqrt(-c)*c*e*x^2*arctan(1/2*(2*d*x + e)*sqr
t(-d)/(sqrt(d*x^2 + e*x + c)*d)) - (8*b*c^2 + 4*a*c*d - a*e^2)*sqrt(-d)*x^2*arct
an(1/2*(e*x + 2*c)*sqrt(-c)/(sqrt(d*x^2 + e*x + c)*c)) + 2*(4*b*c*x^2 - a*e*x -
2*a*c)*sqrt(d*x^2 + e*x + c)*sqrt(-c)*sqrt(-d))/(sqrt(-c)*c*sqrt(-d)*x^2)]

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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**3,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.68612, size = 4, normalized size = 0.01 \[ \mathit{sage}_{0} x \]

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^3,x, algorithm="giac")

[Out]

sage0*x