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

Optimal. Leaf size=482 \[ -\frac{\left (16 b c^2 e^3 \left (3 a^2 e^2-3 a b d e+b^2 d^2\right )+6 b^3 c e^4 (b d-4 a e)-384 c^4 d^3 e (b d-a e)+96 c^3 d e^2 (b d-a e)^2+3 b^5 e^5+256 c^5 d^5\right ) \tanh ^{-1}\left (\frac{b+2 c x^2}{2 \sqrt{c} \sqrt{a+b x^2+c x^4}}\right )}{512 c^{7/2} e^6}+\frac{\left (a+b x^2+c x^4\right )^{3/2} \left (-3 b^2 e^2-6 c e x^2 (b e+2 c d)-6 b c d e+16 c^2 d^2\right )}{96 c^2 e^3}+\frac{\sqrt{a+b x^2+c x^4} \left (-2 c e x^2 \left (-8 c^2 d e (2 b d-3 a e)-6 b c e^2 (b d-2 a e)-3 b^3 e^3+32 c^3 d^3\right )+6 b^2 c e^3 (b d-2 a e)-32 c^3 d^2 e (5 b d-4 a e)+8 b c^2 d e^2 (2 b d-3 a e)+3 b^4 e^4+128 c^4 d^4\right )}{256 c^3 e^5}+\frac{d^2 \left (a e^2-b d e+c d^2\right )^{3/2} \tanh ^{-1}\left (\frac{-2 a e+x^2 (2 c d-b e)+b d}{2 \sqrt{a+b x^2+c x^4} \sqrt{a e^2-b d e+c d^2}}\right )}{2 e^6}+\frac{\left (a+b x^2+c x^4\right )^{5/2}}{10 c e} \]

[Out]

((128*c^4*d^4 + 3*b^4*e^4 - 32*c^3*d^2*e*(5*b*d - 4*a*e) + 8*b*c^2*d*e^2*(2*b*d
- 3*a*e) + 6*b^2*c*e^3*(b*d - 2*a*e) - 2*c*e*(32*c^3*d^3 - 3*b^3*e^3 - 8*c^2*d*e
*(2*b*d - 3*a*e) - 6*b*c*e^2*(b*d - 2*a*e))*x^2)*Sqrt[a + b*x^2 + c*x^4])/(256*c
^3*e^5) + ((16*c^2*d^2 - 6*b*c*d*e - 3*b^2*e^2 - 6*c*e*(2*c*d + b*e)*x^2)*(a + b
*x^2 + c*x^4)^(3/2))/(96*c^2*e^3) + (a + b*x^2 + c*x^4)^(5/2)/(10*c*e) - ((256*c
^5*d^5 + 3*b^5*e^5 + 6*b^3*c*e^4*(b*d - 4*a*e) - 384*c^4*d^3*e*(b*d - a*e) + 96*
c^3*d*e^2*(b*d - a*e)^2 + 16*b*c^2*e^3*(b^2*d^2 - 3*a*b*d*e + 3*a^2*e^2))*ArcTan
h[(b + 2*c*x^2)/(2*Sqrt[c]*Sqrt[a + b*x^2 + c*x^4])])/(512*c^(7/2)*e^6) + (d^2*(
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*e^6)

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Rubi [A]  time = 2.15979, antiderivative size = 482, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 7, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.241 \[ -\frac{\left (16 b c^2 e^3 \left (3 a^2 e^2-3 a b d e+b^2 d^2\right )+6 b^3 c e^4 (b d-4 a e)-384 c^4 d^3 e (b d-a e)+96 c^3 d e^2 (b d-a e)^2+3 b^5 e^5+256 c^5 d^5\right ) \tanh ^{-1}\left (\frac{b+2 c x^2}{2 \sqrt{c} \sqrt{a+b x^2+c x^4}}\right )}{512 c^{7/2} e^6}+\frac{\left (a+b x^2+c x^4\right )^{3/2} \left (-3 b^2 e^2-6 c e x^2 (b e+2 c d)-6 b c d e+16 c^2 d^2\right )}{96 c^2 e^3}+\frac{\sqrt{a+b x^2+c x^4} \left (-2 c e x^2 \left (-8 c^2 d e (2 b d-3 a e)-6 b c e^2 (b d-2 a e)-3 b^3 e^3+32 c^3 d^3\right )+6 b^2 c e^3 (b d-2 a e)-32 c^3 d^2 e (5 b d-4 a e)+8 b c^2 d e^2 (2 b d-3 a e)+3 b^4 e^4+128 c^4 d^4\right )}{256 c^3 e^5}+\frac{d^2 \left (a e^2-b d e+c d^2\right )^{3/2} \tanh ^{-1}\left (\frac{-2 a e+x^2 (2 c d-b e)+b d}{2 \sqrt{a+b x^2+c x^4} \sqrt{a e^2-b d e+c d^2}}\right )}{2 e^6}+\frac{\left (a+b x^2+c x^4\right )^{5/2}}{10 c e} \]

Antiderivative was successfully verified.

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

[Out]

((128*c^4*d^4 + 3*b^4*e^4 - 32*c^3*d^2*e*(5*b*d - 4*a*e) + 8*b*c^2*d*e^2*(2*b*d
- 3*a*e) + 6*b^2*c*e^3*(b*d - 2*a*e) - 2*c*e*(32*c^3*d^3 - 3*b^3*e^3 - 8*c^2*d*e
*(2*b*d - 3*a*e) - 6*b*c*e^2*(b*d - 2*a*e))*x^2)*Sqrt[a + b*x^2 + c*x^4])/(256*c
^3*e^5) + ((16*c^2*d^2 - 6*b*c*d*e - 3*b^2*e^2 - 6*c*e*(2*c*d + b*e)*x^2)*(a + b
*x^2 + c*x^4)^(3/2))/(96*c^2*e^3) + (a + b*x^2 + c*x^4)^(5/2)/(10*c*e) - ((256*c
^5*d^5 + 3*b^5*e^5 + 6*b^3*c*e^4*(b*d - 4*a*e) - 384*c^4*d^3*e*(b*d - a*e) + 96*
c^3*d*e^2*(b*d - a*e)^2 + 16*b*c^2*e^3*(b^2*d^2 - 3*a*b*d*e + 3*a^2*e^2))*ArcTan
h[(b + 2*c*x^2)/(2*Sqrt[c]*Sqrt[a + b*x^2 + c*x^4])])/(512*c^(7/2)*e^6) + (d^2*(
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*e^6)

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Rubi in Sympy [A]  time = 178.164, size = 551, normalized size = 1.14 \[ - \frac{b \left (b + 2 c x^{2}\right ) \left (a + b x^{2} + c x^{4}\right )^{\frac{3}{2}}}{32 c^{2} e} + \frac{3 b \left (b + 2 c x^{2}\right ) \left (- 4 a c + b^{2}\right ) \sqrt{a + b x^{2} + c x^{4}}}{256 c^{3} e} - \frac{3 b \left (- 4 a c + b^{2}\right )^{2} \operatorname{atanh}{\left (\frac{b + 2 c x^{2}}{2 \sqrt{c} \sqrt{a + b x^{2} + c x^{4}}} \right )}}{512 c^{\frac{7}{2}} e} + \frac{d^{2} \left (a + b x^{2} + c x^{4}\right )^{\frac{3}{2}}}{6 e^{3}} - \frac{d^{2} \left (a e^{2} - b d e + c d^{2}\right )^{\frac{3}{2}} \operatorname{atanh}{\left (\frac{2 a e - b d + x^{2} \left (b e - 2 c d\right )}{2 \sqrt{a + b x^{2} + c x^{4}} \sqrt{a e^{2} - b d e + c d^{2}}} \right )}}{2 e^{6}} + \frac{d^{2} \sqrt{a + b x^{2} + c x^{4}} \left (4 a c e^{2} + \frac{b^{2} e^{2}}{2} - 5 b c d e + 4 c^{2} d^{2} + c e x^{2} \left (b e - 2 c d\right )\right )}{8 c e^{5}} - \frac{d \left (b + 2 c x^{2}\right ) \left (a + b x^{2} + c x^{4}\right )^{\frac{3}{2}}}{16 c e^{2}} + \frac{\left (a + b x^{2} + c x^{4}\right )^{\frac{5}{2}}}{10 c e} + \frac{3 d \left (b + 2 c x^{2}\right ) \left (- 4 a c + b^{2}\right ) \sqrt{a + b x^{2} + c x^{4}}}{128 c^{2} e^{2}} - \frac{d^{2} \left (b e - 2 c d\right ) \left (- 12 a c e^{2} + b^{2} e^{2} + 8 b c d e - 8 c^{2} d^{2}\right ) \operatorname{atanh}{\left (\frac{b + 2 c x^{2}}{2 \sqrt{c} \sqrt{a + b x^{2} + c x^{4}}} \right )}}{32 c^{\frac{3}{2}} e^{6}} - \frac{3 d \left (- 4 a c + b^{2}\right )^{2} \operatorname{atanh}{\left (\frac{b + 2 c x^{2}}{2 \sqrt{c} \sqrt{a + b x^{2} + c x^{4}}} \right )}}{256 c^{\frac{5}{2}} e^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-b*(b + 2*c*x**2)*(a + b*x**2 + c*x**4)**(3/2)/(32*c**2*e) + 3*b*(b + 2*c*x**2)*
(-4*a*c + b**2)*sqrt(a + b*x**2 + c*x**4)/(256*c**3*e) - 3*b*(-4*a*c + b**2)**2*
atanh((b + 2*c*x**2)/(2*sqrt(c)*sqrt(a + b*x**2 + c*x**4)))/(512*c**(7/2)*e) + d
**2*(a + b*x**2 + c*x**4)**(3/2)/(6*e**3) - d**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*e**6) + d**2*sqrt(a + b*x**2 + c*x**4)*(4*a*c*e**2 +
b**2*e**2/2 - 5*b*c*d*e + 4*c**2*d**2 + c*e*x**2*(b*e - 2*c*d))/(8*c*e**5) - d*(
b + 2*c*x**2)*(a + b*x**2 + c*x**4)**(3/2)/(16*c*e**2) + (a + b*x**2 + c*x**4)**
(5/2)/(10*c*e) + 3*d*(b + 2*c*x**2)*(-4*a*c + b**2)*sqrt(a + b*x**2 + c*x**4)/(1
28*c**2*e**2) - d**2*(b*e - 2*c*d)*(-12*a*c*e**2 + b**2*e**2 + 8*b*c*d*e - 8*c**
2*d**2)*atanh((b + 2*c*x**2)/(2*sqrt(c)*sqrt(a + b*x**2 + c*x**4)))/(32*c**(3/2)
*e**6) - 3*d*(-4*a*c + b**2)**2*atanh((b + 2*c*x**2)/(2*sqrt(c)*sqrt(a + b*x**2
+ c*x**4)))/(256*c**(5/2)*e**2)

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Mathematica [A]  time = 1.10718, size = 492, normalized size = 1.02 \[ \frac{\frac{2 e \sqrt{a+b x^2+c x^4} \left (12 c^2 e^2 \left (32 a^2 e^2+2 a b e \left (7 e x^2-25 d\right )+b^2 \left (20 d^2-5 d e x^2+2 e^2 x^4\right )\right )-30 b^2 c e^3 \left (10 a e-3 b d+b e x^2\right )-16 c^3 e \left (a e \left (-160 d^2+75 d e x^2-48 e^2 x^4\right )+b \left (150 d^3-70 d^2 e x^2+45 d e^2 x^4-33 e^3 x^6\right )\right )+45 b^4 e^4+32 c^4 \left (60 d^4-30 d^3 e x^2+20 d^2 e^2 x^4-15 d e^3 x^6+12 e^4 x^8\right )\right )}{c^3}-\frac{15 \left (16 b c^2 e^3 \left (3 a^2 e^2-3 a b d e+b^2 d^2\right )+6 b^3 c e^4 (b d-4 a e)-384 c^4 d^3 e (b d-a e)+96 c^3 d e^2 (b d-a e)^2+3 b^5 e^5+256 c^5 d^5\right ) \log \left (2 \sqrt{c} \sqrt{a+b x^2+c x^4}+b+2 c x^2\right )}{c^{7/2}}-3840 d^2 \left (e (a e-b d)+c d^2\right )^{3/2} \log \left (2 \sqrt{a+b x^2+c x^4} \sqrt{a e^2-b d e+c d^2}+2 a e-b d+b e x^2-2 c d x^2\right )+3840 d^2 \log \left (d+e x^2\right ) \left (e (a e-b d)+c d^2\right )^{3/2}}{7680 e^6} \]

Antiderivative was successfully verified.

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

[Out]

((2*e*Sqrt[a + b*x^2 + c*x^4]*(45*b^4*e^4 - 30*b^2*c*e^3*(-3*b*d + 10*a*e + b*e*
x^2) + 32*c^4*(60*d^4 - 30*d^3*e*x^2 + 20*d^2*e^2*x^4 - 15*d*e^3*x^6 + 12*e^4*x^
8) + 12*c^2*e^2*(32*a^2*e^2 + 2*a*b*e*(-25*d + 7*e*x^2) + b^2*(20*d^2 - 5*d*e*x^
2 + 2*e^2*x^4)) - 16*c^3*e*(a*e*(-160*d^2 + 75*d*e*x^2 - 48*e^2*x^4) + b*(150*d^
3 - 70*d^2*e*x^2 + 45*d*e^2*x^4 - 33*e^3*x^6))))/c^3 + 3840*d^2*(c*d^2 + e*(-(b*
d) + a*e))^(3/2)*Log[d + e*x^2] - (15*(256*c^5*d^5 + 3*b^5*e^5 + 6*b^3*c*e^4*(b*
d - 4*a*e) - 384*c^4*d^3*e*(b*d - a*e) + 96*c^3*d*e^2*(b*d - a*e)^2 + 16*b*c^2*e
^3*(b^2*d^2 - 3*a*b*d*e + 3*a^2*e^2))*Log[b + 2*c*x^2 + 2*Sqrt[c]*Sqrt[a + b*x^2
+ c*x^4]])/c^(7/2) - 3840*d^2*(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]
])/(7680*e^6)

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Maple [B]  time = 0.045, size = 2068, normalized size = 4.3 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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))*a*b-d^4/e^5/((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^5/e^6/((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+1/10/e*a^2/c*(c*x^4+b*x^2+a)^(1/2)+1/10/e*c*x^8*(c*x^4+b*x^2+a)^(1/2)+11/80/e*
b*x^6*(c*x^4+b*x^2+a)^(1/2)+3/256/e/c^3*b^4*(c*x^4+b*x^2+a)^(1/2)-3/512/e/c^(7/2
)*b^5*ln((1/2*b+c*x^2)/c^(1/2)+(c*x^4+b*x^2+a)^(1/2))+1/5/e*a*x^4*(c*x^4+b*x^2+a
)^(1/2)-5/8*d^3/e^4*b*(c*x^4+b*x^2+a)^(1/2)+2/3*d^2/e^3*(c*x^4+b*x^2+a)^(1/2)*a+
1/2*d^4/e^5*c*(c*x^4+b*x^2+a)^(1/2)-1/2*d^5/e^6*c^(3/2)*ln((1/2*b+c*x^2)/c^(1/2)
+(c*x^4+b*x^2+a)^(1/2))-1/128/e/c^2*b^3*x^2*(c*x^4+b*x^2+a)^(1/2)+3/64/e/c^(5/2)
*b^3*a*ln((1/2*b+c*x^2)/c^(1/2)+(c*x^4+b*x^2+a)^(1/2))+7/160/e/c*b*a*x^2*(c*x^4+
b*x^2+a)^(1/2)-1/64/e^2*d/c*b^2*x^2*(c*x^4+b*x^2+a)^(1/2)+3/32/e^2*d/c^(3/2)*b^2
*a*ln((1/2*b+c*x^2)/c^(1/2)+(c*x^4+b*x^2+a)^(1/2))-5/32/e^2*d/c*b*a*(c*x^4+b*x^2
+a)^(1/2)+3/8*d^2/e^3*a/c^(1/2)*ln((1/2*b+c*x^2)/c^(1/2)+(c*x^4+b*x^2+a)^(1/2))*
b-5/64/e/c^2*b^2*a*(c*x^4+b*x^2+a)^(1/2)-3/16/e^2*d*a^2*ln((1/2*b+c*x^2)/c^(1/2)
+(c*x^4+b*x^2+a)^(1/2))/c^(1/2)-1/8/e^2*d*c*x^6*(c*x^4+b*x^2+a)^(1/2)-3/32/e*a^2
*b/c^(3/2)*ln((1/2*b+c*x^2)/c^(1/2)+(c*x^4+b*x^2+a)^(1/2))+1/160/e/c*b^2*x^4*(c*
x^4+b*x^2+a)^(1/2)+7/24*d^2/e^3*x^2*(c*x^4+b*x^2+a)^(1/2)*b-1/4*d^3/e^4*x^2*c*(c
*x^4+b*x^2+a)^(1/2)+1/16*d^2/e^3*b^2/c*(c*x^4+b*x^2+a)^(1/2)-1/32*d^2/e^3*b^3/c^
(3/2)*ln((1/2*b+c*x^2)/c^(1/2)+(c*x^4+b*x^2+a)^(1/2))-3/16*d^3/e^4*b^2/c^(1/2)*l
n((1/2*b+c*x^2)/c^(1/2)+(c*x^4+b*x^2+a)^(1/2))-3/4*d^3/e^4*a*c^(1/2)*ln((1/2*b+c
*x^2)/c^(1/2)+(c*x^4+b*x^2+a)^(1/2))+3/4*d^4/e^5*b*c^(1/2)*ln((1/2*b+c*x^2)/c^(1
/2)+(c*x^4+b*x^2+a)^(1/2))+1/6*d^2/e^3*c*x^4*(c*x^4+b*x^2+a)^(1/2)-1/2*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))*a^2-1/2*d^4/e^5/((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^2-1/2*d^6/e^7/((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
-3/16/e^2*d*b*x^4*(c*x^4+b*x^2+a)^(1/2)+3/128/e^2*d/c^2*b^3*(c*x^4+b*x^2+a)^(1/2
)-3/256/e^2*d/c^(5/2)*b^4*ln((1/2*b+c*x^2)/c^(1/2)+(c*x^4+b*x^2+a)^(1/2))-5/16/e
^2*d*a*x^2*(c*x^4+b*x^2+a)^(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((c*x^4 + b*x^2 + a)^(3/2)*x^5/(e*x^2 + d),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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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)*x^5/(e*x^2 + d),x, algorithm="fricas")

[Out]

Timed out

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

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (c x^{4} + b x^{2} + a\right )}^{\frac{3}{2}} x^{5}}{e x^{2} + d}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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