3.317 \(\int \frac{\left (a+b x^2+c x^4\right )^{3/2}}{x \left (d+e x^2\right )} \, dx\)

Optimal. Leaf size=350 \[ -\frac{a^{3/2} \tanh ^{-1}\left (\frac{2 a+b x^2}{2 \sqrt{a} \sqrt{a+b x^2+c x^4}}\right )}{2 d}+\frac{\left (-12 c d e (b d-a e)+b e^2 (3 b d-4 a e)+8 c^2 d^3\right ) \tanh ^{-1}\left (\frac{b+2 c x^2}{2 \sqrt{c} \sqrt{a+b x^2+c x^4}}\right )}{16 \sqrt{c} d e^3}-\frac{\sqrt{a+b x^2+c x^4} \left (-e (5 b d-4 a e)+4 c d^2-2 c d e x^2\right )}{8 d e^2}-\frac{\left (a e^2-b d e+c d^2\right )^{3/2} \tanh ^{-1}\left (\frac{-2 a e+x^2 (2 c d-b e)+b d}{2 \sqrt{a+b x^2+c x^4} \sqrt{a e^2-b d e+c d^2}}\right )}{2 d e^3}+\frac{a \sqrt{a+b x^2+c x^4}}{2 d}+\frac{a b \tanh ^{-1}\left (\frac{b+2 c x^2}{2 \sqrt{c} \sqrt{a+b x^2+c x^4}}\right )}{4 \sqrt{c} d} \]

[Out]

(a*Sqrt[a + b*x^2 + c*x^4])/(2*d) - ((4*c*d^2 - e*(5*b*d - 4*a*e) - 2*c*d*e*x^2)
*Sqrt[a + b*x^2 + c*x^4])/(8*d*e^2) - (a^(3/2)*ArcTanh[(2*a + b*x^2)/(2*Sqrt[a]*
Sqrt[a + b*x^2 + c*x^4])])/(2*d) + (a*b*ArcTanh[(b + 2*c*x^2)/(2*Sqrt[c]*Sqrt[a
+ b*x^2 + c*x^4])])/(4*Sqrt[c]*d) + ((8*c^2*d^3 + b*e^2*(3*b*d - 4*a*e) - 12*c*d
*e*(b*d - a*e))*ArcTanh[(b + 2*c*x^2)/(2*Sqrt[c]*Sqrt[a + b*x^2 + c*x^4])])/(16*
Sqrt[c]*d*e^3) - ((c*d^2 - b*d*e + a*e^2)^(3/2)*ArcTanh[(b*d - 2*a*e + (2*c*d -
b*e)*x^2)/(2*Sqrt[c*d^2 - b*d*e + a*e^2]*Sqrt[a + b*x^2 + c*x^4])])/(2*d*e^3)

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Rubi [A]  time = 1.27014, antiderivative size = 350, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 8, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.276 \[ -\frac{a^{3/2} \tanh ^{-1}\left (\frac{2 a+b x^2}{2 \sqrt{a} \sqrt{a+b x^2+c x^4}}\right )}{2 d}+\frac{\left (-12 c d e (b d-a e)+b e^2 (3 b d-4 a e)+8 c^2 d^3\right ) \tanh ^{-1}\left (\frac{b+2 c x^2}{2 \sqrt{c} \sqrt{a+b x^2+c x^4}}\right )}{16 \sqrt{c} d e^3}-\frac{\sqrt{a+b x^2+c x^4} \left (-e (5 b d-4 a e)+4 c d^2-2 c d e x^2\right )}{8 d e^2}-\frac{\left (a e^2-b d e+c d^2\right )^{3/2} \tanh ^{-1}\left (\frac{-2 a e+x^2 (2 c d-b e)+b d}{2 \sqrt{a+b x^2+c x^4} \sqrt{a e^2-b d e+c d^2}}\right )}{2 d e^3}+\frac{a \sqrt{a+b x^2+c x^4}}{2 d}+\frac{a b \tanh ^{-1}\left (\frac{b+2 c x^2}{2 \sqrt{c} \sqrt{a+b x^2+c x^4}}\right )}{4 \sqrt{c} d} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^2 + c*x^4)^(3/2)/(x*(d + e*x^2)),x]

[Out]

(a*Sqrt[a + b*x^2 + c*x^4])/(2*d) - ((4*c*d^2 - e*(5*b*d - 4*a*e) - 2*c*d*e*x^2)
*Sqrt[a + b*x^2 + c*x^4])/(8*d*e^2) - (a^(3/2)*ArcTanh[(2*a + b*x^2)/(2*Sqrt[a]*
Sqrt[a + b*x^2 + c*x^4])])/(2*d) + (a*b*ArcTanh[(b + 2*c*x^2)/(2*Sqrt[c]*Sqrt[a
+ b*x^2 + c*x^4])])/(4*Sqrt[c]*d) + ((8*c^2*d^3 + b*e^2*(3*b*d - 4*a*e) - 12*c*d
*e*(b*d - a*e))*ArcTanh[(b + 2*c*x^2)/(2*Sqrt[c]*Sqrt[a + b*x^2 + c*x^4])])/(16*
Sqrt[c]*d*e^3) - ((c*d^2 - b*d*e + a*e^2)^(3/2)*ArcTanh[(b*d - 2*a*e + (2*c*d -
b*e)*x^2)/(2*Sqrt[c*d^2 - b*d*e + a*e^2]*Sqrt[a + b*x^2 + c*x^4])])/(2*d*e^3)

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Rubi in Sympy [A]  time = 130.837, size = 332, normalized size = 0.95 \[ - \frac{a^{\frac{3}{2}} \operatorname{atanh}{\left (\frac{2 a + b x^{2}}{2 \sqrt{a} \sqrt{a + b x^{2} + c x^{4}}} \right )}}{2 d} + \frac{a b \operatorname{atanh}{\left (\frac{b + 2 c x^{2}}{2 \sqrt{c} \sqrt{a + b x^{2} + c x^{4}}} \right )}}{4 \sqrt{c} d} + \frac{a \sqrt{a + b x^{2} + c x^{4}}}{2 d} - \frac{\sqrt{a + b x^{2} + c x^{4}} \left (2 a e^{2} - \frac{5 b d e}{2} + 2 c d^{2} - c d e x^{2}\right )}{4 d e^{2}} + \frac{\left (a e^{2} - b d e + c d^{2}\right )^{\frac{3}{2}} \operatorname{atanh}{\left (\frac{2 a e - b d + x^{2} \left (b e - 2 c d\right )}{2 \sqrt{a + b x^{2} + c x^{4}} \sqrt{a e^{2} - b d e + c d^{2}}} \right )}}{2 d e^{3}} + \frac{\left (- 4 a b e^{3} + 12 a c d e^{2} + 3 b^{2} d e^{2} - 12 b c d^{2} e + 8 c^{2} d^{3}\right ) \operatorname{atanh}{\left (\frac{b + 2 c x^{2}}{2 \sqrt{c} \sqrt{a + b x^{2} + c x^{4}}} \right )}}{16 \sqrt{c} d e^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((c*x**4+b*x**2+a)**(3/2)/x/(e*x**2+d),x)

[Out]

-a**(3/2)*atanh((2*a + b*x**2)/(2*sqrt(a)*sqrt(a + b*x**2 + c*x**4)))/(2*d) + a*
b*atanh((b + 2*c*x**2)/(2*sqrt(c)*sqrt(a + b*x**2 + c*x**4)))/(4*sqrt(c)*d) + a*
sqrt(a + b*x**2 + c*x**4)/(2*d) - sqrt(a + b*x**2 + c*x**4)*(2*a*e**2 - 5*b*d*e/
2 + 2*c*d**2 - c*d*e*x**2)/(4*d*e**2) + (a*e**2 - b*d*e + c*d**2)**(3/2)*atanh((
2*a*e - b*d + x**2*(b*e - 2*c*d))/(2*sqrt(a + b*x**2 + c*x**4)*sqrt(a*e**2 - b*d
*e + c*d**2)))/(2*d*e**3) + (-4*a*b*e**3 + 12*a*c*d*e**2 + 3*b**2*d*e**2 - 12*b*
c*d**2*e + 8*c**2*d**3)*atanh((b + 2*c*x**2)/(2*sqrt(c)*sqrt(a + b*x**2 + c*x**4
)))/(16*sqrt(c)*d*e**3)

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Mathematica [A]  time = 2.25722, size = 292, normalized size = 0.83 \[ \frac{-\frac{8 a^{3/2} e^3 \log \left (2 \sqrt{a} \sqrt{a+b x^2+c x^4}+2 a+b x^2\right )}{d}+\frac{8 a^{3/2} e^3 \log \left (x^2\right )}{d}+\frac{\left (12 c e (a e-b d)+3 b^2 e^2+8 c^2 d^2\right ) \log \left (2 \sqrt{c} \sqrt{a+b x^2+c x^4}+b+2 c x^2\right )}{\sqrt{c}}+\frac{8 \left (e (a e-b d)+c d^2\right )^{3/2} \log \left (2 \sqrt{a+b x^2+c x^4} \sqrt{a e^2-b d e+c d^2}+2 a e-b d+b e x^2-2 c d x^2\right )}{d}-\frac{8 \log \left (d+e x^2\right ) \left (e (a e-b d)+c d^2\right )^{3/2}}{d}+2 e \sqrt{a+b x^2+c x^4} \left (5 b e-4 c d+2 c e x^2\right )}{16 e^3} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x^2 + c*x^4)^(3/2)/(x*(d + e*x^2)),x]

[Out]

(2*e*(-4*c*d + 5*b*e + 2*c*e*x^2)*Sqrt[a + b*x^2 + c*x^4] + (8*a^(3/2)*e^3*Log[x
^2])/d - (8*(c*d^2 + e*(-(b*d) + a*e))^(3/2)*Log[d + e*x^2])/d - (8*a^(3/2)*e^3*
Log[2*a + b*x^2 + 2*Sqrt[a]*Sqrt[a + b*x^2 + c*x^4]])/d + ((8*c^2*d^2 + 3*b^2*e^
2 + 12*c*e*(-(b*d) + a*e))*Log[b + 2*c*x^2 + 2*Sqrt[c]*Sqrt[a + b*x^2 + c*x^4]])
/Sqrt[c] + (8*(c*d^2 + e*(-(b*d) + a*e))^(3/2)*Log[-(b*d) + 2*a*e - 2*c*d*x^2 +
b*e*x^2 + 2*Sqrt[c*d^2 - b*d*e + a*e^2]*Sqrt[a + b*x^2 + c*x^4]])/d)/(16*e^3)

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Maple [B]  time = 0.017, size = 1270, normalized size = 3.6 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^4+b*x^2+a)^(3/2)/x/(e*x^2+d),x)

[Out]

3/16/e*b^2/c^(1/2)*ln((1/2*b+c*x^2)/c^(1/2)+(c*x^4+b*x^2+a)^(1/2))+3/4/e*a*c^(1/
2)*ln((1/2*b+c*x^2)/c^(1/2)+(c*x^4+b*x^2+a)^(1/2))-1/2*d/e^2*c*(c*x^4+b*x^2+a)^(
1/2)+1/2*d^2/e^3*c^(3/2)*ln((1/2*b+c*x^2)/c^(1/2)+(c*x^4+b*x^2+a)^(1/2))+1/2/d/(
(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*ln((2*(a*e^2-b*d*e+c*d^2)/e^2+(b*e-2*c*d)/e*(x^2+
d/e)+2*((a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*((x^2+d/e)^2*c+(b*e-2*c*d)/e*(x^2+d/e)+(a
*e^2-b*d*e+c*d^2)/e^2)^(1/2))/(x^2+d/e))*a^2+1/4/e*x^2*c*(c*x^4+b*x^2+a)^(1/2)-3
/4*d/e^2*b*c^(1/2)*ln((1/2*b+c*x^2)/c^(1/2)+(c*x^4+b*x^2+a)^(1/2))-1/e/((a*e^2-b
*d*e+c*d^2)/e^2)^(1/2)*ln((2*(a*e^2-b*d*e+c*d^2)/e^2+(b*e-2*c*d)/e*(x^2+d/e)+2*(
(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*((x^2+d/e)^2*c+(b*e-2*c*d)/e*(x^2+d/e)+(a*e^2-b*d
*e+c*d^2)/e^2)^(1/2))/(x^2+d/e))*a*b+1/2*d/e^2/((a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*l
n((2*(a*e^2-b*d*e+c*d^2)/e^2+(b*e-2*c*d)/e*(x^2+d/e)+2*((a*e^2-b*d*e+c*d^2)/e^2)
^(1/2)*((x^2+d/e)^2*c+(b*e-2*c*d)/e*(x^2+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2))/(x
^2+d/e))*b^2+1/2*d^3/e^4/((a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*ln((2*(a*e^2-b*d*e+c*d^
2)/e^2+(b*e-2*c*d)/e*(x^2+d/e)+2*((a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*((x^2+d/e)^2*c+
(b*e-2*c*d)/e*(x^2+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2))/(x^2+d/e))*c^2+d/e^2/((a
*e^2-b*d*e+c*d^2)/e^2)^(1/2)*ln((2*(a*e^2-b*d*e+c*d^2)/e^2+(b*e-2*c*d)/e*(x^2+d/
e)+2*((a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*((x^2+d/e)^2*c+(b*e-2*c*d)/e*(x^2+d/e)+(a*e
^2-b*d*e+c*d^2)/e^2)^(1/2))/(x^2+d/e))*a*c-d^2/e^3/((a*e^2-b*d*e+c*d^2)/e^2)^(1/
2)*ln((2*(a*e^2-b*d*e+c*d^2)/e^2+(b*e-2*c*d)/e*(x^2+d/e)+2*((a*e^2-b*d*e+c*d^2)/
e^2)^(1/2)*((x^2+d/e)^2*c+(b*e-2*c*d)/e*(x^2+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)
)/(x^2+d/e))*b*c+5/8/e*b*(c*x^4+b*x^2+a)^(1/2)-1/2/d*a^(3/2)*ln((2*a+b*x^2+2*a^(
1/2)*(c*x^4+b*x^2+a)^(1/2))/x^2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (c x^{4} + b x^{2} + a\right )}^{\frac{3}{2}}}{{\left (e x^{2} + d\right )} x}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^4 + b*x^2 + a)^(3/2)/((e*x^2 + d)*x),x, algorithm="maxima")

[Out]

integrate((c*x^4 + b*x^2 + a)^(3/2)/((e*x^2 + d)*x), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^4 + b*x^2 + a)^(3/2)/((e*x^2 + d)*x),x, algorithm="fricas")

[Out]

Timed out

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (a + b x^{2} + c x^{4}\right )^{\frac{3}{2}}}{x \left (d + e x^{2}\right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x**4+b*x**2+a)**(3/2)/x/(e*x**2+d),x)

[Out]

Integral((a + b*x**2 + c*x**4)**(3/2)/(x*(d + e*x**2)), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (c x^{4} + b x^{2} + a\right )}^{\frac{3}{2}}}{{\left (e x^{2} + d\right )} x}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^4 + b*x^2 + a)^(3/2)/((e*x^2 + d)*x),x, algorithm="giac")

[Out]

integrate((c*x^4 + b*x^2 + a)^(3/2)/((e*x^2 + d)*x), x)