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

Optimal. Leaf size=159 \[ \frac{c x^2 (b d-2 a e)+a (2 c d-b e)}{\left (b^2-4 a c\right ) \sqrt{a+b x^2+c x^4} \left (a e^2-b d e+c d^2\right )}-\frac{d e \tanh ^{-1}\left (\frac{-2 a e+x^2 (2 c d-b e)+b d}{2 \sqrt{a+b x^2+c x^4} \sqrt{a e^2-b d e+c d^2}}\right )}{2 \left (a e^2-b d e+c d^2\right )^{3/2}} \]

[Out]

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

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Rubi [A]  time = 0.521329, antiderivative size = 159, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.172 \[ \frac{c x^2 (b d-2 a e)+a (2 c d-b e)}{\left (b^2-4 a c\right ) \sqrt{a+b x^2+c x^4} \left (a e^2-b d e+c d^2\right )}-\frac{d e \tanh ^{-1}\left (\frac{-2 a e+x^2 (2 c d-b e)+b d}{2 \sqrt{a+b x^2+c x^4} \sqrt{a e^2-b d e+c d^2}}\right )}{2 \left (a e^2-b d e+c d^2\right )^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 64.4128, size = 143, normalized size = 0.9 \[ \frac{d e \operatorname{atanh}{\left (\frac{2 a e - b d + x^{2} \left (b e - 2 c d\right )}{2 \sqrt{a + b x^{2} + c x^{4}} \sqrt{a e^{2} - b d e + c d^{2}}} \right )}}{2 \left (a e^{2} - b d e + c d^{2}\right )^{\frac{3}{2}}} - \frac{a \left (b e - 2 c d\right ) + c x^{2} \left (2 a e - b d\right )}{\left (- 4 a c + b^{2}\right ) \sqrt{a + b x^{2} + c x^{4}} \left (a e^{2} - b d e + c d^{2}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

d*e*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*(a*e**2 - b*d*e + c*d**2)**(3/2)) - (a*(b*e - 2*c*d
) + c*x**2*(2*a*e - b*d))/((-4*a*c + b**2)*sqrt(a + b*x**2 + c*x**4)*(a*e**2 - b
*d*e + c*d**2))

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Mathematica [A]  time = 0.510719, size = 225, normalized size = 1.42 \[ \frac{-d e \left (b^2-4 a c\right ) \sqrt{a+b x^2+c x^4} \log \left (2 \sqrt{a+b x^2+c x^4} \sqrt{a e^2-b d e+c d^2}+2 a e-b d+b e x^2-2 c d x^2\right )+d e \left (b^2-4 a c\right ) \sqrt{a+b x^2+c x^4} \log \left (d+e x^2\right )+2 \sqrt{a e^2-b d e+c d^2} \left (a b e-2 a c d+2 a c e x^2-b c d x^2\right )}{2 \left (4 a c-b^2\right ) \sqrt{a+b x^2+c x^4} \left (e (a e-b d)+c d^2\right )^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.016, size = 506, normalized size = 3.2 \[{\frac{2\,c{x}^{2}+b}{e \left ( 4\,ac-{b}^{2} \right ) }{\frac{1}{\sqrt{c{x}^{4}+b{x}^{2}+a}}}}+2\,{\frac{cd}{e \left ( e\sqrt{-4\,ac+{b}^{2}}-be+2\,cd \right ) \left ( -4\,ac+{b}^{2} \right ) }\sqrt{ \left ({x}^{2}-1/2\,{\frac{-b+\sqrt{-4\,ac+{b}^{2}}}{c}} \right ) ^{2}c+\sqrt{-4\,ac+{b}^{2}} \left ({x}^{2}-1/2\,{\frac{-b+\sqrt{-4\,ac+{b}^{2}}}{c}} \right ) } \left ({x}^{2}+1/2\,{\frac{b}{c}}-1/2\,{\frac{\sqrt{-4\,ac+{b}^{2}}}{c}} \right ) ^{-1}}-2\,{\frac{cd}{e \left ( e\sqrt{-4\,ac+{b}^{2}}+be-2\,cd \right ) \left ( -4\,ac+{b}^{2} \right ) }\sqrt{ \left ({x}^{2}+1/2\,{\frac{b+\sqrt{-4\,ac+{b}^{2}}}{c}} \right ) ^{2}c-\sqrt{-4\,ac+{b}^{2}} \left ({x}^{2}+1/2\,{\frac{b+\sqrt{-4\,ac+{b}^{2}}}{c}} \right ) } \left ({x}^{2}+1/2\,{\frac{\sqrt{-4\,ac+{b}^{2}}}{c}}+1/2\,{\frac{b}{c}} \right ) ^{-1}}-2\,{\frac{cd}{ \left ( e\sqrt{-4\,ac+{b}^{2}}-be+2\,cd \right ) \left ( e\sqrt{-4\,ac+{b}^{2}}+be-2\,cd \right ) }\ln \left ({1 \left ( 2\,{\frac{a{e}^{2}-bde+c{d}^{2}}{{e}^{2}}}+{\frac{be-2\,cd}{e} \left ({x}^{2}+{\frac{d}{e}} \right ) }+2\,\sqrt{{\frac{a{e}^{2}-bde+c{d}^{2}}{{e}^{2}}}}\sqrt{ \left ({x}^{2}+{\frac{d}{e}} \right ) ^{2}c+{\frac{be-2\,cd}{e} \left ({x}^{2}+{\frac{d}{e}} \right ) }+{\frac{a{e}^{2}-bde+c{d}^{2}}{{e}^{2}}}} \right ) \left ({x}^{2}+{\frac{d}{e}} \right ) ^{-1}} \right ){\frac{1}{\sqrt{{\frac{a{e}^{2}-bde+c{d}^{2}}{{e}^{2}}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/e*(2*c*x^2+b)/(4*a*c-b^2)/(c*x^4+b*x^2+a)^(1/2)+2*d/e*c/(e*(-4*a*c+b^2)^(1/2)-
b*e+2*c*d)/(-4*a*c+b^2)/(x^2+1/2*b/c-1/2/c*(-4*a*c+b^2)^(1/2))*((x^2-1/2/c*(-b+(
-4*a*c+b^2)^(1/2)))^2*c+(-4*a*c+b^2)^(1/2)*(x^2-1/2/c*(-b+(-4*a*c+b^2)^(1/2))))^
(1/2)-2*d/e*c/(e*(-4*a*c+b^2)^(1/2)+b*e-2*c*d)/(-4*a*c+b^2)/(x^2+1/2/c*(-4*a*c+b
^2)^(1/2)+1/2*b/c)*((x^2+1/2*(b+(-4*a*c+b^2)^(1/2))/c)^2*c-(-4*a*c+b^2)^(1/2)*(x
^2+1/2*(b+(-4*a*c+b^2)^(1/2))/c))^(1/2)-2*d*c/(e*(-4*a*c+b^2)^(1/2)-b*e+2*c*d)/(
e*(-4*a*c+b^2)^(1/2)+b*e-2*c*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))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.620768, size = 1, normalized size = 0.01 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.31888, size = 595, normalized size = 3.74 \[ -\frac{d \arctan \left (-\frac{{\left (\sqrt{c} x^{2} - \sqrt{c x^{4} + b x^{2} + a}\right )} e + \sqrt{c} d}{\sqrt{-c d^{2} + b d e - a e^{2}}}\right ) e}{{\left (c d^{2} - b d e + a e^{2}\right )} \sqrt{-c d^{2} + b d e - a e^{2}}} + \frac{\frac{{\left (b c^{2} d^{3} - b^{2} c d^{2} e - 2 \, a c^{2} d^{2} e + 3 \, a b c d e^{2} - 2 \, a^{2} c e^{3}\right )} x^{2}}{b^{2} c^{2} d^{4} - 4 \, a c^{3} d^{4} - 2 \, b^{3} c d^{3} e + 8 \, a b c^{2} d^{3} e + b^{4} d^{2} e^{2} - 2 \, a b^{2} c d^{2} e^{2} - 8 \, a^{2} c^{2} d^{2} e^{2} - 2 \, a b^{3} d e^{3} + 8 \, a^{2} b c d e^{3} + a^{2} b^{2} e^{4} - 4 \, a^{3} c e^{4}} + \frac{2 \, a c^{2} d^{3} - 3 \, a b c d^{2} e + a b^{2} d e^{2} + 2 \, a^{2} c d e^{2} - a^{2} b e^{3}}{b^{2} c^{2} d^{4} - 4 \, a c^{3} d^{4} - 2 \, b^{3} c d^{3} e + 8 \, a b c^{2} d^{3} e + b^{4} d^{2} e^{2} - 2 \, a b^{2} c d^{2} e^{2} - 8 \, a^{2} c^{2} d^{2} e^{2} - 2 \, a b^{3} d e^{3} + 8 \, a^{2} b c d e^{3} + a^{2} b^{2} e^{4} - 4 \, a^{3} c e^{4}}}{\sqrt{c x^{4} + b x^{2} + a}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-d*arctan(-((sqrt(c)*x^2 - sqrt(c*x^4 + b*x^2 + a))*e + sqrt(c)*d)/sqrt(-c*d^2 +
b*d*e - a*e^2))*e/((c*d^2 - b*d*e + a*e^2)*sqrt(-c*d^2 + b*d*e - a*e^2)) + ((b*
c^2*d^3 - b^2*c*d^2*e - 2*a*c^2*d^2*e + 3*a*b*c*d*e^2 - 2*a^2*c*e^3)*x^2/(b^2*c^
2*d^4 - 4*a*c^3*d^4 - 2*b^3*c*d^3*e + 8*a*b*c^2*d^3*e + b^4*d^2*e^2 - 2*a*b^2*c*
d^2*e^2 - 8*a^2*c^2*d^2*e^2 - 2*a*b^3*d*e^3 + 8*a^2*b*c*d*e^3 + a^2*b^2*e^4 - 4*
a^3*c*e^4) + (2*a*c^2*d^3 - 3*a*b*c*d^2*e + a*b^2*d*e^2 + 2*a^2*c*d*e^2 - a^2*b*
e^3)/(b^2*c^2*d^4 - 4*a*c^3*d^4 - 2*b^3*c*d^3*e + 8*a*b*c^2*d^3*e + b^4*d^2*e^2
- 2*a*b^2*c*d^2*e^2 - 8*a^2*c^2*d^2*e^2 - 2*a*b^3*d*e^3 + 8*a^2*b*c*d*e^3 + a^2*
b^2*e^4 - 4*a^3*c*e^4))/sqrt(c*x^4 + b*x^2 + a)