3.335 \(\int \frac{x^5}{\left (d+e x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}} \, dx\)

Optimal. Leaf size=167 \[ \frac{d^2 \tanh ^{-1}\left (\frac{-2 a e+x^2 (2 c d-b e)+b d}{2 \sqrt{a+b x^2+c x^4} \sqrt{a e^2-b d e+c d^2}}\right )}{2 \left (a e^2-b d e+c d^2\right )^{3/2}}-\frac{x^2 \left (-a b e-2 a c d+b^2 d\right )+a (b d-2 a e)}{\left (b^2-4 a c\right ) \sqrt{a+b x^2+c x^4} \left (a e^2-b d e+c d^2\right )} \]

[Out]

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

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Rubi [A]  time = 0.6344, antiderivative size = 167, 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 \[ \frac{d^2 \tanh ^{-1}\left (\frac{-2 a e+x^2 (2 c d-b e)+b d}{2 \sqrt{a+b x^2+c x^4} \sqrt{a e^2-b d e+c d^2}}\right )}{2 \left (a e^2-b d e+c d^2\right )^{3/2}}-\frac{x^2 \left (-a b e-2 a c d+b^2 d\right )+a (b d-2 a e)}{\left (b^2-4 a c\right ) \sqrt{a+b x^2+c x^4} \left (a e^2-b d e+c d^2\right )} \]

Antiderivative was successfully verified.

[In]  Int[x^5/((d + e*x^2)*(a + b*x^2 + c*x^4)^(3/2)),x]

[Out]

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

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Rubi in Sympy [A]  time = 105.888, size = 235, normalized size = 1.41 \[ - \frac{d^{2} \operatorname{atanh}{\left (\frac{2 a e - b d + x^{2} \left (b e - 2 c d\right )}{2 \sqrt{a + b x^{2} + c x^{4}} \sqrt{a e^{2} - b d e + c d^{2}}} \right )}}{2 \left (a e^{2} - b d e + c d^{2}\right )^{\frac{3}{2}}} + \frac{d^{2} \left (- 2 a c e + b^{2} e - b c d + c x^{2} \left (b e - 2 c d\right )\right )}{e^{2} \left (- 4 a c + b^{2}\right ) \sqrt{a + b x^{2} + c x^{4}} \left (a e^{2} - b d e + c d^{2}\right )} + \frac{d \left (2 b + 4 c x^{2}\right )}{2 e^{2} \left (- 4 a c + b^{2}\right ) \sqrt{a + b x^{2} + c x^{4}}} + \frac{4 a + 2 b x^{2}}{2 e \left (- 4 a c + b^{2}\right ) \sqrt{a + b x^{2} + c x^{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**5/(e*x**2+d)/(c*x**4+b*x**2+a)**(3/2),x)

[Out]

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

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Mathematica [A]  time = 0.735774, size = 204, normalized size = 1.22 \[ \frac{1}{2} \left (\frac{2 \left (-2 a^2 e+a b \left (d-e x^2\right )-2 a c d x^2+b^2 d x^2\right )}{\left (b^2-4 a c\right ) \sqrt{a+b x^2+c x^4} \left (e (b d-a e)-c d^2\right )}-\frac{d^2 \log \left (2 \sqrt{a+b x^2+c x^4} \sqrt{a e^2-b d e+c d^2}+2 a e-b d+b e x^2-2 c d x^2\right )}{\left (e (a e-b d)+c d^2\right )^{3/2}}+\frac{d^2 \log \left (d+e x^2\right )}{\left (e (a e-b d)+c d^2\right )^{3/2}}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[x^5/((d + e*x^2)*(a + b*x^2 + c*x^4)^(3/2)),x]

[Out]

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

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Maple [B]  time = 0.019, size = 613, normalized size = 3.7 \[ -{\frac{b{x}^{2}}{e \left ( 4\,ac-{b}^{2} \right ) }{\frac{1}{\sqrt{c{x}^{4}+b{x}^{2}+a}}}}-2\,{\frac{a}{e\sqrt{c{x}^{4}+b{x}^{2}+a} \left ( 4\,ac-{b}^{2} \right ) }}-2\,{\frac{cd{x}^{2}}{{e}^{2} \left ( 4\,ac-{b}^{2} \right ) \sqrt{c{x}^{4}+b{x}^{2}+a}}}-{\frac{bd}{{e}^{2} \left ( 4\,ac-{b}^{2} \right ) }{\frac{1}{\sqrt{c{x}^{4}+b{x}^{2}+a}}}}-2\,{\frac{c{d}^{2}}{{e}^{2} \left ( e\sqrt{-4\,ac+{b}^{2}}-be+2\,cd \right ) \left ( -4\,ac+{b}^{2} \right ) }\sqrt{ \left ({x}^{2}-1/2\,{\frac{-b+\sqrt{-4\,ac+{b}^{2}}}{c}} \right ) ^{2}c+\sqrt{-4\,ac+{b}^{2}} \left ({x}^{2}-1/2\,{\frac{-b+\sqrt{-4\,ac+{b}^{2}}}{c}} \right ) } \left ({x}^{2}+1/2\,{\frac{b}{c}}-1/2\,{\frac{\sqrt{-4\,ac+{b}^{2}}}{c}} \right ) ^{-1}}+2\,{\frac{c{d}^{2}}{{e}^{2} \left ( e\sqrt{-4\,ac+{b}^{2}}+be-2\,cd \right ) \left ( -4\,ac+{b}^{2} \right ) }\sqrt{ \left ({x}^{2}+1/2\,{\frac{b+\sqrt{-4\,ac+{b}^{2}}}{c}} \right ) ^{2}c-\sqrt{-4\,ac+{b}^{2}} \left ({x}^{2}+1/2\,{\frac{b+\sqrt{-4\,ac+{b}^{2}}}{c}} \right ) } \left ({x}^{2}+1/2\,{\frac{\sqrt{-4\,ac+{b}^{2}}}{c}}+1/2\,{\frac{b}{c}} \right ) ^{-1}}+2\,{\frac{c{d}^{2}}{e \left ( e\sqrt{-4\,ac+{b}^{2}}-be+2\,cd \right ) \left ( e\sqrt{-4\,ac+{b}^{2}}+be-2\,cd \right ) }\ln \left ({1 \left ( 2\,{\frac{a{e}^{2}-bde+c{d}^{2}}{{e}^{2}}}+{\frac{be-2\,cd}{e} \left ({x}^{2}+{\frac{d}{e}} \right ) }+2\,\sqrt{{\frac{a{e}^{2}-bde+c{d}^{2}}{{e}^{2}}}}\sqrt{ \left ({x}^{2}+{\frac{d}{e}} \right ) ^{2}c+{\frac{be-2\,cd}{e} \left ({x}^{2}+{\frac{d}{e}} \right ) }+{\frac{a{e}^{2}-bde+c{d}^{2}}{{e}^{2}}}} \right ) \left ({x}^{2}+{\frac{d}{e}} \right ) ^{-1}} \right ){\frac{1}{\sqrt{{\frac{a{e}^{2}-bde+c{d}^{2}}{{e}^{2}}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^5/(e*x^2+d)/(c*x^4+b*x^2+a)^(3/2),x)

[Out]

-1/e/(c*x^4+b*x^2+a)^(1/2)/(4*a*c-b^2)*x^2*b-2/e/(c*x^4+b*x^2+a)^(1/2)/(4*a*c-b^
2)*a-2/e^2*d/(4*a*c-b^2)/(c*x^4+b*x^2+a)^(1/2)*x^2*c-1/e^2*d/(4*a*c-b^2)/(c*x^4+
b*x^2+a)^(1/2)*b-2*d^2/e^2*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^2/e^2*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^2/e*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. \[ \int \frac{x^{5}}{{\left (c x^{4} + b x^{2} + a\right )}^{\frac{3}{2}}{\left (e x^{2} + d\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^5/((c*x^4 + b*x^2 + a)^(3/2)*(e*x^2 + d)),x, algorithm="maxima")

[Out]

integrate(x^5/((c*x^4 + b*x^2 + a)^(3/2)*(e*x^2 + d)), x)

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Fricas [A]  time = 0.633866, 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^5/((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)*(a*b*d - 2*a^2*e - (a*b*e - (b^2 - 2*a*c)*d)*x^
2)*sqrt(c*d^2 - b*d*e + a*e^2) - ((b^2*c - 4*a*c^2)*d^2*x^4 + (b^3 - 4*a*b*c)*d^
2*x^2 + (a*b^2 - 4*a^2*c)*d^2)*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)*(a*b*d - 2*a^2*
e - (a*b*e - (b^2 - 2*a*c)*d)*x^2)*sqrt(-c*d^2 + b*d*e - a*e^2) + ((b^2*c - 4*a*
c^2)*d^2*x^4 + (b^3 - 4*a*b*c)*d^2*x^2 + (a*b^2 - 4*a^2*c)*d^2)*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. \[ \int \frac{x^{5}}{\left (d + e x^{2}\right ) \left (a + b x^{2} + c x^{4}\right )^{\frac{3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**5/(e*x**2+d)/(c*x**4+b*x**2+a)**(3/2),x)

[Out]

Integral(x**5/((d + e*x**2)*(a + b*x**2 + c*x**4)**(3/2)), x)

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GIAC/XCAS [A]  time = 0.351573, size = 536, normalized size = 3.21 \[ \frac{d^{2} \arctan \left (-\frac{{\left (\sqrt{c} x^{2} - \sqrt{c x^{4} + b x^{2} + a}\right )} e + \sqrt{c} d}{\sqrt{-c d^{2} + b d e - a e^{2}}}\right )}{{\left (c d^{2} - b d e + a e^{2}\right )} \sqrt{-c d^{2} + b d e - a e^{2}}} - \frac{\frac{{\left (b^{4} c d^{3} - 6 \, a b^{2} c^{2} d^{3} + 8 \, a^{2} c^{3} d^{3} - b^{5} d^{2} e + 5 \, a b^{3} c d^{2} e - 4 \, a^{2} b c^{2} d^{2} e + 2 \, a b^{4} d e^{2} - 10 \, a^{2} b^{2} c d e^{2} + 8 \, a^{3} c^{2} d e^{2} - a^{2} b^{3} e^{3} + 4 \, a^{3} b c e^{3}\right )} x^{2}}{a b^{4} c^{2} - 8 \, a^{2} b^{2} c^{3} + 16 \, a^{3} c^{4}} + \frac{a b^{3} c d^{3} - 4 \, a^{2} b c^{2} d^{3} - a b^{4} d^{2} e + 2 \, a^{2} b^{2} c d^{2} e + 8 \, a^{3} c^{2} d^{2} e + 3 \, a^{2} b^{3} d e^{2} - 12 \, a^{3} b c d e^{2} - 2 \, a^{3} b^{2} e^{3} + 8 \, a^{4} c e^{3}}{a b^{4} c^{2} - 8 \, a^{2} b^{2} c^{3} + 16 \, a^{3} c^{4}}}{32 \, \sqrt{c x^{4} + b x^{2} + a}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^5/((c*x^4 + b*x^2 + a)^(3/2)*(e*x^2 + d)),x, algorithm="giac")

[Out]

d^2*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))/((c*d^2 - b*d*e + a*e^2)*sqrt(-c*d^2 + b*d*e - a*e^2)) - 1/32*
((b^4*c*d^3 - 6*a*b^2*c^2*d^3 + 8*a^2*c^3*d^3 - b^5*d^2*e + 5*a*b^3*c*d^2*e - 4*
a^2*b*c^2*d^2*e + 2*a*b^4*d*e^2 - 10*a^2*b^2*c*d*e^2 + 8*a^3*c^2*d*e^2 - a^2*b^3
*e^3 + 4*a^3*b*c*e^3)*x^2/(a*b^4*c^2 - 8*a^2*b^2*c^3 + 16*a^3*c^4) + (a*b^3*c*d^
3 - 4*a^2*b*c^2*d^3 - a*b^4*d^2*e + 2*a^2*b^2*c*d^2*e + 8*a^3*c^2*d^2*e + 3*a^2*
b^3*d*e^2 - 12*a^3*b*c*d*e^2 - 2*a^3*b^2*e^3 + 8*a^4*c*e^3)/(a*b^4*c^2 - 8*a^2*b
^2*c^3 + 16*a^3*c^4))/sqrt(c*x^4 + b*x^2 + a)