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

Optimal. Leaf size=294 \[ \frac{\sqrt{a^2+2 a b x^2+b^2 x^4} \sqrt{c+d x^2+e x} \left (2 a c e-x \left (8 b c^2-a e^2\right )\right )}{8 c^2 x^2 \left (a+b x^2\right )}-\frac{e \sqrt{a^2+2 a b x^2+b^2 x^4} \left (8 b c^2-a \left (4 c d-e^2\right )\right ) \tanh ^{-1}\left (\frac{2 c+e x}{2 \sqrt{c} \sqrt{c+d x^2+e x}}\right )}{16 c^{5/2} \left (a+b x^2\right )}+\frac{b \sqrt{d} \sqrt{a^2+2 a b x^2+b^2 x^4} \tanh ^{-1}\left (\frac{2 d x+e}{2 \sqrt{d} \sqrt{c+d x^2+e x}}\right )}{a+b x^2}-\frac{a \sqrt{a^2+2 a b x^2+b^2 x^4} \left (c+d x^2+e x\right )^{3/2}}{3 c x^3 \left (a+b x^2\right )} \]

[Out]

((2*a*c*e - (8*b*c^2 - a*e^2)*x)*Sqrt[c + e*x + d*x^2]*Sqrt[a^2 + 2*a*b*x^2 + b^
2*x^4])/(8*c^2*x^2*(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])/(3*c*x^3*(a + b*x^2)) + (b*Sqrt[d]*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])])/(a + b*x^2) - (e*(8*b*c
^2 - a*(4*c*d - e^2))*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]*ArcTanh[(2*c + e*x)/(2*Sqr
t[c]*Sqrt[c + e*x + d*x^2])])/(16*c^(5/2)*(a + b*x^2))

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Rubi [A]  time = 1.57956, 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 \[ \frac{\sqrt{a^2+2 a b x^2+b^2 x^4} \sqrt{c+d x^2+e x} \left (2 a c e-x \left (8 b c^2-a e^2\right )\right )}{8 c^2 x^2 \left (a+b x^2\right )}-\frac{e \sqrt{a^2+2 a b x^2+b^2 x^4} \left (8 b c^2-a \left (4 c d-e^2\right )\right ) \tanh ^{-1}\left (\frac{2 c+e x}{2 \sqrt{c} \sqrt{c+d x^2+e x}}\right )}{16 c^{5/2} \left (a+b x^2\right )}+\frac{b \sqrt{d} \sqrt{a^2+2 a b x^2+b^2 x^4} \tanh ^{-1}\left (\frac{2 d x+e}{2 \sqrt{d} \sqrt{c+d x^2+e x}}\right )}{a+b x^2}-\frac{a \sqrt{a^2+2 a b x^2+b^2 x^4} \left (c+d x^2+e x\right )^{3/2}}{3 c x^3 \left (a+b x^2\right )} \]

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^4,x]

[Out]

((2*a*c*e - (8*b*c^2 - a*e^2)*x)*Sqrt[c + e*x + d*x^2]*Sqrt[a^2 + 2*a*b*x^2 + b^
2*x^4])/(8*c^2*x^2*(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])/(3*c*x^3*(a + b*x^2)) + (b*Sqrt[d]*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])])/(a + b*x^2) - (e*(8*b*c
^2 - a*(4*c*d - e^2))*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]*ArcTanh[(2*c + e*x)/(2*Sqr
t[c]*Sqrt[c + e*x + d*x^2])])/(16*c^(5/2)*(a + b*x^2))

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{c + d x^{2} + e x} \sqrt{\left (a + b x^{2}\right )^{2}}}{x^{4}}\, dx \]

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**4,x)

[Out]

Integral(sqrt(c + d*x**2 + e*x)*sqrt((a + b*x**2)**2)/x**4, x)

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Mathematica [A]  time = 1.34092, size = 210, normalized size = 0.71 \[ \frac{\sqrt{\left (a+b x^2\right )^2} \left (3 e x^3 \log (x) \left (a \left (e^2-4 c d\right )+8 b c^2\right )-3 e x^3 \left (a \left (e^2-4 c d\right )+8 b c^2\right ) \log \left (2 \sqrt{c} \sqrt{c+x (d x+e)}+2 c+e x\right )-2 \sqrt{c} \left (\sqrt{c+x (d x+e)} \left (a \left (8 c^2+2 c x (4 d x+e)-3 e^2 x^2\right )+24 b c^2 x^2\right )-24 b c^2 \sqrt{d} x^3 \log \left (2 \sqrt{d} \sqrt{c+x (d x+e)}+2 d x+e\right )\right )\right )}{48 c^{5/2} x^3 \left (a+b x^2\right )} \]

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^4,x]

[Out]

(Sqrt[(a + b*x^2)^2]*(3*e*(8*b*c^2 + a*(-4*c*d + e^2))*x^3*Log[x] - 3*e*(8*b*c^2
+ a*(-4*c*d + e^2))*x^3*Log[2*c + e*x + 2*Sqrt[c]*Sqrt[c + x*(e + d*x)]] - 2*Sq
rt[c]*(Sqrt[c + x*(e + d*x)]*(24*b*c^2*x^2 + a*(8*c^2 - 3*e^2*x^2 + 2*c*x*(e + 4
*d*x))) - 24*b*c^2*Sqrt[d]*x^3*Log[e + 2*d*x + 2*Sqrt[d]*Sqrt[c + x*(e + d*x)]])
))/(48*c^(5/2)*x^3*(a + b*x^2))

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Maple [A]  time = 0.018, size = 390, normalized size = 1.3 \[{\frac{1}{ \left ( 48\,b{x}^{2}+48\,a \right ){x}^{3}}\sqrt{ \left ( b{x}^{2}+a \right ) ^{2}} \left ( 48\,b\sqrt{d}\ln \left ( 1/2\,{\frac{2\,\sqrt{d{x}^{2}+ex+c}\sqrt{d}+2\,dx+e}{\sqrt{d}}} \right ){x}^{3}{c}^{11/2}+6\,a{e}^{2}d\sqrt{d{x}^{2}+ex+c}{x}^{4}{c}^{5/2}+48\,bd\sqrt{d{x}^{2}+ex+c}{x}^{4}{c}^{9/2}-24\,be\ln \left ({\frac{2\,c+ex+2\,\sqrt{c}\sqrt{d{x}^{2}+ex+c}}{x}} \right ){x}^{3}{c}^{5}-6\,a{e}^{2} \left ( d{x}^{2}+ex+c \right ) ^{3/2}{x}^{2}{c}^{5/2}-48\,b \left ( d{x}^{2}+ex+c \right ) ^{3/2}{x}^{2}{c}^{9/2}-12\,aed\sqrt{d{x}^{2}+ex+c}{x}^{3}{c}^{7/2}+6\,a{e}^{3}\sqrt{d{x}^{2}+ex+c}{x}^{3}{c}^{5/2}+48\,be\sqrt{d{x}^{2}+ex+c}{x}^{3}{c}^{9/2}+12\,aed\ln \left ({\frac{2\,c+ex+2\,\sqrt{c}\sqrt{d{x}^{2}+ex+c}}{x}} \right ){x}^{3}{c}^{4}+12\,ae \left ( d{x}^{2}+ex+c \right ) ^{3/2}x{c}^{7/2}-3\,a{e}^{3}\ln \left ({\frac{2\,c+ex+2\,\sqrt{c}\sqrt{d{x}^{2}+ex+c}}{x}} \right ){x}^{3}{c}^{3}-16\,a \left ( d{x}^{2}+ex+c \right ) ^{3/2}{c}^{9/2} \right ){c}^{-{\frac{11}{2}}}} \]

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^4,x)

[Out]

1/48*((b*x^2+a)^2)^(1/2)*(48*b*d^(1/2)*ln(1/2*(2*(d*x^2+e*x+c)^(1/2)*d^(1/2)+2*d
*x+e)/d^(1/2))*x^3*c^(11/2)+6*a*e^2*d*(d*x^2+e*x+c)^(1/2)*x^4*c^(5/2)+48*b*d*(d*
x^2+e*x+c)^(1/2)*x^4*c^(9/2)-24*b*e*ln((2*c+e*x+2*c^(1/2)*(d*x^2+e*x+c)^(1/2))/x
)*x^3*c^5-6*a*e^2*(d*x^2+e*x+c)^(3/2)*x^2*c^(5/2)-48*b*(d*x^2+e*x+c)^(3/2)*x^2*c
^(9/2)-12*a*e*d*(d*x^2+e*x+c)^(1/2)*x^3*c^(7/2)+6*a*e^3*(d*x^2+e*x+c)^(1/2)*x^3*
c^(5/2)+48*b*e*(d*x^2+e*x+c)^(1/2)*x^3*c^(9/2)+12*a*e*d*ln((2*c+e*x+2*c^(1/2)*(d
*x^2+e*x+c)^(1/2))/x)*x^3*c^4+12*a*e*(d*x^2+e*x+c)^(3/2)*x*c^(7/2)-3*a*e^3*ln((2
*c+e*x+2*c^(1/2)*(d*x^2+e*x+c)^(1/2))/x)*x^3*c^3-16*a*(d*x^2+e*x+c)^(3/2)*c^(9/2
))/(b*x^2+a)/x^3/c^(11/2)

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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^4,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.578588, 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^4,x, algorithm="fricas")

[Out]

[1/96*(48*b*c^(5/2)*sqrt(d)*x^3*log(8*d^2*x^2 + 8*d*e*x + 4*sqrt(d*x^2 + e*x + c
)*(2*d*x + e)*sqrt(d) + 4*c*d + e^2) + 3*(a*e^3 + 4*(2*b*c^2 - a*c*d)*e)*x^3*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*(2*a*c*e*x + 8*a*c^2 + (24*b*c^2 + 8*a*c*d - 3*a*e^2)*x^2)*
sqrt(d*x^2 + e*x + c)*sqrt(c))/(c^(5/2)*x^3), 1/96*(96*b*c^(5/2)*sqrt(-d)*x^3*ar
ctan(1/2*(2*d*x + e)/(sqrt(d*x^2 + e*x + c)*sqrt(-d))) + 3*(a*e^3 + 4*(2*b*c^2 -
a*c*d)*e)*x^3*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*(2*a*c*e*x + 8*a*c^2 + (24*b*c^2 + 8*a*c*
d - 3*a*e^2)*x^2)*sqrt(d*x^2 + e*x + c)*sqrt(c))/(c^(5/2)*x^3), 1/48*(24*b*sqrt(
-c)*c^2*sqrt(d)*x^3*log(8*d^2*x^2 + 8*d*e*x + 4*sqrt(d*x^2 + e*x + c)*(2*d*x + e
)*sqrt(d) + 4*c*d + e^2) - 3*(a*e^3 + 4*(2*b*c^2 - a*c*d)*e)*x^3*arctan(1/2*(e*x
+ 2*c)*sqrt(-c)/(sqrt(d*x^2 + e*x + c)*c)) - 2*(2*a*c*e*x + 8*a*c^2 + (24*b*c^2
+ 8*a*c*d - 3*a*e^2)*x^2)*sqrt(d*x^2 + e*x + c)*sqrt(-c))/(sqrt(-c)*c^2*x^3), 1
/48*(48*b*sqrt(-c)*c^2*sqrt(-d)*x^3*arctan(1/2*(2*d*x + e)/(sqrt(d*x^2 + e*x + c
)*sqrt(-d))) - 3*(a*e^3 + 4*(2*b*c^2 - a*c*d)*e)*x^3*arctan(1/2*(e*x + 2*c)*sqrt
(-c)/(sqrt(d*x^2 + e*x + c)*c)) - 2*(2*a*c*e*x + 8*a*c^2 + (24*b*c^2 + 8*a*c*d -
3*a*e^2)*x^2)*sqrt(d*x^2 + e*x + c)*sqrt(-c))/(sqrt(-c)*c^2*x^3)]

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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**4,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.705428, 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^4,x, algorithm="giac")

[Out]

sage0*x