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

Optimal. Leaf size=360 \[ \frac{\left (8 b^2 c e^3 (b d-3 a e)-192 c^3 d^2 e (b d-a e)+48 c^2 e^2 (b d-a e)^2+3 b^4 e^4+128 c^4 d^4\right ) \tanh ^{-1}\left (\frac{b+2 c x^2}{2 \sqrt{c} \sqrt{a+b x^2+c x^4}}\right )}{256 c^{5/2} e^5}-\frac{\sqrt{a+b x^2+c x^4} \left (-2 c e x^2 \left (-4 c e (2 b d-3 a e)-3 b^2 e^2+16 c^2 d^2\right )-16 c^2 d e (5 b d-4 a e)+4 b c e^2 (2 b d-3 a e)+3 b^3 e^3+64 c^3 d^3\right )}{128 c^2 e^4}-\frac{d \left (a e^2-b d e+c d^2\right )^{3/2} \tanh ^{-1}\left (\frac{-2 a e+x^2 (2 c d-b e)+b d}{2 \sqrt{a+b x^2+c x^4} \sqrt{a e^2-b d e+c d^2}}\right )}{2 e^5}-\frac{\left (a+b x^2+c x^4\right )^{3/2} \left (-3 b e+8 c d-6 c e x^2\right )}{48 c e^2} \]

[Out]

-((64*c^3*d^3 + 3*b^3*e^3 - 16*c^2*d*e*(5*b*d - 4*a*e) + 4*b*c*e^2*(2*b*d - 3*a*
e) - 2*c*e*(16*c^2*d^2 - 3*b^2*e^2 - 4*c*e*(2*b*d - 3*a*e))*x^2)*Sqrt[a + b*x^2
+ c*x^4])/(128*c^2*e^4) - ((8*c*d - 3*b*e - 6*c*e*x^2)*(a + b*x^2 + c*x^4)^(3/2)
)/(48*c*e^2) + ((128*c^4*d^4 + 3*b^4*e^4 + 8*b^2*c*e^3*(b*d - 3*a*e) - 192*c^3*d
^2*e*(b*d - a*e) + 48*c^2*e^2*(b*d - a*e)^2)*ArcTanh[(b + 2*c*x^2)/(2*Sqrt[c]*Sq
rt[a + b*x^2 + c*x^4])])/(256*c^(5/2)*e^5) - (d*(c*d^2 - b*d*e + a*e^2)^(3/2)*Ar
cTanh[(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^5)

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Rubi [A]  time = 1.49269, antiderivative size = 360, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.207 \[ \frac{\left (8 b^2 c e^3 (b d-3 a e)-192 c^3 d^2 e (b d-a e)+48 c^2 e^2 (b d-a e)^2+3 b^4 e^4+128 c^4 d^4\right ) \tanh ^{-1}\left (\frac{b+2 c x^2}{2 \sqrt{c} \sqrt{a+b x^2+c x^4}}\right )}{256 c^{5/2} e^5}-\frac{\sqrt{a+b x^2+c x^4} \left (-2 c e x^2 \left (-4 c e (2 b d-3 a e)-3 b^2 e^2+16 c^2 d^2\right )-16 c^2 d e (5 b d-4 a e)+4 b c e^2 (2 b d-3 a e)+3 b^3 e^3+64 c^3 d^3\right )}{128 c^2 e^4}-\frac{d \left (a e^2-b d e+c d^2\right )^{3/2} \tanh ^{-1}\left (\frac{-2 a e+x^2 (2 c d-b e)+b d}{2 \sqrt{a+b x^2+c x^4} \sqrt{a e^2-b d e+c d^2}}\right )}{2 e^5}-\frac{\left (a+b x^2+c x^4\right )^{3/2} \left (-3 b e+8 c d-6 c e x^2\right )}{48 c e^2} \]

Antiderivative was successfully verified.

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

[Out]

-((64*c^3*d^3 + 3*b^3*e^3 - 16*c^2*d*e*(5*b*d - 4*a*e) + 4*b*c*e^2*(2*b*d - 3*a*
e) - 2*c*e*(16*c^2*d^2 - 3*b^2*e^2 - 4*c*e*(2*b*d - 3*a*e))*x^2)*Sqrt[a + b*x^2
+ c*x^4])/(128*c^2*e^4) - ((8*c*d - 3*b*e - 6*c*e*x^2)*(a + b*x^2 + c*x^4)^(3/2)
)/(48*c*e^2) + ((128*c^4*d^4 + 3*b^4*e^4 + 8*b^2*c*e^3*(b*d - 3*a*e) - 192*c^3*d
^2*e*(b*d - a*e) + 48*c^2*e^2*(b*d - a*e)^2)*ArcTanh[(b + 2*c*x^2)/(2*Sqrt[c]*Sq
rt[a + b*x^2 + c*x^4])])/(256*c^(5/2)*e^5) - (d*(c*d^2 - b*d*e + a*e^2)^(3/2)*Ar
cTanh[(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^5)

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Rubi in Sympy [A]  time = 127.517, size = 388, normalized size = 1.08 \[ \frac{d \left (a e^{2} - b d e + c d^{2}\right )^{\frac{3}{2}} \operatorname{atanh}{\left (\frac{2 a e - b d + x^{2} \left (b e - 2 c d\right )}{2 \sqrt{a + b x^{2} + c x^{4}} \sqrt{a e^{2} - b d e + c d^{2}}} \right )}}{2 e^{5}} + \frac{\left (a + b x^{2} + c x^{4}\right )^{\frac{3}{2}} \left (\frac{3 b e}{2} - 4 c d + 3 c e x^{2}\right )}{24 c e^{2}} - \frac{\sqrt{a + b x^{2} + c x^{4}} \left (- 3 a b c e^{3} + 16 a c^{2} d e^{2} + \frac{3 b^{3} e^{3}}{4} + 2 b^{2} c d e^{2} - 20 b c^{2} d^{2} e + 16 c^{3} d^{3} + \frac{c e x^{2} \left (3 b^{2} e^{2} - 16 c^{2} d^{2} - 4 c e \left (3 a e - 2 b d\right )\right )}{2}\right )}{32 c^{2} e^{4}} + \frac{\left (- c d e \left (b e - 2 c d\right ) \left (4 a c e + b \left (3 b e - 8 c d\right )\right ) + \left (\frac{b^{2} e^{2}}{4} - 2 c^{2} d^{2} - c e \left (a e - b d\right )\right ) \left (3 b^{2} e^{2} - 16 c^{2} d^{2} - 4 c e \left (3 a e - 2 b d\right )\right )\right ) \operatorname{atanh}{\left (\frac{b + 2 c x^{2}}{2 \sqrt{c} \sqrt{a + b x^{2} + c x^{4}}} \right )}}{64 c^{\frac{5}{2}} e^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

d*(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*s
qrt(a + b*x**2 + c*x**4)*sqrt(a*e**2 - b*d*e + c*d**2)))/(2*e**5) + (a + b*x**2
+ c*x**4)**(3/2)*(3*b*e/2 - 4*c*d + 3*c*e*x**2)/(24*c*e**2) - sqrt(a + b*x**2 +
c*x**4)*(-3*a*b*c*e**3 + 16*a*c**2*d*e**2 + 3*b**3*e**3/4 + 2*b**2*c*d*e**2 - 20
*b*c**2*d**2*e + 16*c**3*d**3 + c*e*x**2*(3*b**2*e**2 - 16*c**2*d**2 - 4*c*e*(3*
a*e - 2*b*d))/2)/(32*c**2*e**4) + (-c*d*e*(b*e - 2*c*d)*(4*a*c*e + b*(3*b*e - 8*
c*d)) + (b**2*e**2/4 - 2*c**2*d**2 - c*e*(a*e - b*d))*(3*b**2*e**2 - 16*c**2*d**
2 - 4*c*e*(3*a*e - 2*b*d)))*atanh((b + 2*c*x**2)/(2*sqrt(c)*sqrt(a + b*x**2 + c*
x**4)))/(64*c**(5/2)*e**5)

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Mathematica [A]  time = 0.473627, size = 376, normalized size = 1.04 \[ \frac{-2 \sqrt{c} e \sqrt{a+b x^2+c x^4} \left (-8 c^2 e \left (a e \left (15 e x^2-32 d\right )+b \left (30 d^2-14 d e x^2+9 e^2 x^4\right )\right )-6 b c e^2 \left (10 a e-4 b d+b e x^2\right )+9 b^3 e^3+16 c^3 \left (12 d^3-6 d^2 e x^2+4 d e^2 x^4-3 e^3 x^6\right )\right )+3 \left (8 b^2 c e^3 (b d-3 a e)-192 c^3 d^2 e (b d-a e)+48 c^2 e^2 (b d-a e)^2+3 b^4 e^4+128 c^4 d^4\right ) \log \left (2 \sqrt{c} \sqrt{a+b x^2+c x^4}+b+2 c x^2\right )+384 c^{5/2} d \left (e (a e-b d)+c d^2\right )^{3/2} \log \left (2 \sqrt{a+b x^2+c x^4} \sqrt{a e^2-b d e+c d^2}+2 a e-b d+b e x^2-2 c d x^2\right )-384 c^{5/2} d \log \left (d+e x^2\right ) \left (e (a e-b d)+c d^2\right )^{3/2}}{768 c^{5/2} e^5} \]

Antiderivative was successfully verified.

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

[Out]

(-2*Sqrt[c]*e*Sqrt[a + b*x^2 + c*x^4]*(9*b^3*e^3 - 6*b*c*e^2*(-4*b*d + 10*a*e +
b*e*x^2) + 16*c^3*(12*d^3 - 6*d^2*e*x^2 + 4*d*e^2*x^4 - 3*e^3*x^6) - 8*c^2*e*(a*
e*(-32*d + 15*e*x^2) + b*(30*d^2 - 14*d*e*x^2 + 9*e^2*x^4))) - 384*c^(5/2)*d*(c*
d^2 + e*(-(b*d) + a*e))^(3/2)*Log[d + e*x^2] + 3*(128*c^4*d^4 + 3*b^4*e^4 + 8*b^
2*c*e^3*(b*d - 3*a*e) - 192*c^3*d^2*e*(b*d - a*e) + 48*c^2*e^2*(b*d - a*e)^2)*Lo
g[b + 2*c*x^2 + 2*Sqrt[c]*Sqrt[a + b*x^2 + c*x^4]] + 384*c^(5/2)*d*(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]])/(768*c^(5/2)*e^5)

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Maple [B]  time = 0.017, size = 1696, normalized size = 4.7 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3/16/e*b*x^4*(c*x^4+b*x^2+a)^(1/2)-3/128/e/c^2*b^3*(c*x^4+b*x^2+a)^(1/2)+3/256/e
/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*a*x^2*(c*x^4
+b*x^2+a)^(1/2)+3/16/e*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*c*x^6*(c*x^4+b*x^2+a)^(1/2)+5/8*d^2/e^3*b*(c*x^4+b*x^2+a)^(1/2)-2/3*d/e
^2*(c*x^4+b*x^2+a)^(1/2)*a-1/2*d^3/e^4*c*(c*x^4+b*x^2+a)^(1/2)+1/2*d^4/e^5*c^(3/
2)*ln((1/2*b+c*x^2)/c^(1/2)+(c*x^4+b*x^2+a)^(1/2))-3/4*d^3/e^4*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/e^2*c*x^4*(c*x^4+b*x^2+a)^(1/2)-7
/24*d/e^2*x^2*(c*x^4+b*x^2+a)^(1/2)*b-1/16*d/e^2*b^2/c*(c*x^4+b*x^2+a)^(1/2)+1/3
2*d/e^2*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^2/e^3
*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*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))-3/32/e/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/c*b*a*(c*x^4+b*x^2+a)^(1/2)+1/64/
e/c*b^2*x^2*(c*x^4+b*x^2+a)^(1/2)+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))*b^2+1/2*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))*c^2+1/4*d^2/e^
3*x^2*c*(c*x^4+b*x^2+a)^(1/2)-3/8*d/e^2*a/c^(1/2)*ln((1/2*b+c*x^2)/c^(1/2)+(c*x^
4+b*x^2+a)^(1/2))*b-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*b+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*c-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*c+1/2*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^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^3/(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^3/(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**3*(c*x**4+b*x**2+a)**(3/2)/(e*x**2+d),x)

[Out]

Timed out

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GIAC/XCAS [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: TypeError