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

Optimal. Leaf size=175 \[ \frac{x \left (d+e x^2\right )^{3/2} \left (48 a e^2-8 b d e+3 c d^2\right )}{192 e^2}+\frac{d x \sqrt{d+e x^2} \left (48 a e^2-8 b d e+3 c d^2\right )}{128 e^2}+\frac{d^2 \tanh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d+e x^2}}\right ) \left (48 a e^2-8 b d e+3 c d^2\right )}{128 e^{5/2}}-\frac{x \left (d+e x^2\right )^{5/2} (3 c d-8 b e)}{48 e^2}+\frac{c x^3 \left (d+e x^2\right )^{5/2}}{8 e} \]

[Out]

(d*(3*c*d^2 - 8*b*d*e + 48*a*e^2)*x*Sqrt[d + e*x^2])/(128*e^2) + ((3*c*d^2 - 8*b
*d*e + 48*a*e^2)*x*(d + e*x^2)^(3/2))/(192*e^2) - ((3*c*d - 8*b*e)*x*(d + e*x^2)
^(5/2))/(48*e^2) + (c*x^3*(d + e*x^2)^(5/2))/(8*e) + (d^2*(3*c*d^2 - 8*b*d*e + 4
8*a*e^2)*ArcTanh[(Sqrt[e]*x)/Sqrt[d + e*x^2]])/(128*e^(5/2))

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Rubi [A]  time = 0.244707, antiderivative size = 175, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208 \[ \frac{x \left (d+e x^2\right )^{3/2} \left (48 a e^2-8 b d e+3 c d^2\right )}{192 e^2}+\frac{d x \sqrt{d+e x^2} \left (48 a e^2-8 b d e+3 c d^2\right )}{128 e^2}+\frac{d^2 \tanh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d+e x^2}}\right ) \left (48 a e^2-8 b d e+3 c d^2\right )}{128 e^{5/2}}-\frac{x \left (d+e x^2\right )^{5/2} (3 c d-8 b e)}{48 e^2}+\frac{c x^3 \left (d+e x^2\right )^{5/2}}{8 e} \]

Antiderivative was successfully verified.

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

[Out]

(d*(3*c*d^2 - 8*b*d*e + 48*a*e^2)*x*Sqrt[d + e*x^2])/(128*e^2) + ((3*c*d^2 - 8*b
*d*e + 48*a*e^2)*x*(d + e*x^2)^(3/2))/(192*e^2) - ((3*c*d - 8*b*e)*x*(d + e*x^2)
^(5/2))/(48*e^2) + (c*x^3*(d + e*x^2)^(5/2))/(8*e) + (d^2*(3*c*d^2 - 8*b*d*e + 4
8*a*e^2)*ArcTanh[(Sqrt[e]*x)/Sqrt[d + e*x^2]])/(128*e^(5/2))

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Rubi in Sympy [A]  time = 23.3841, size = 170, normalized size = 0.97 \[ \frac{c x^{3} \left (d + e x^{2}\right )^{\frac{5}{2}}}{8 e} + \frac{d^{2} \left (48 a e^{2} - 8 b d e + 3 c d^{2}\right ) \operatorname{atanh}{\left (\frac{\sqrt{e} x}{\sqrt{d + e x^{2}}} \right )}}{128 e^{\frac{5}{2}}} + \frac{d x \sqrt{d + e x^{2}} \left (48 a e^{2} - 8 b d e + 3 c d^{2}\right )}{128 e^{2}} + \frac{x \left (d + e x^{2}\right )^{\frac{5}{2}} \left (8 b e - 3 c d\right )}{48 e^{2}} + \frac{x \left (d + e x^{2}\right )^{\frac{3}{2}} \left (48 a e^{2} - 8 b d e + 3 c d^{2}\right )}{192 e^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

c*x**3*(d + e*x**2)**(5/2)/(8*e) + d**2*(48*a*e**2 - 8*b*d*e + 3*c*d**2)*atanh(s
qrt(e)*x/sqrt(d + e*x**2))/(128*e**(5/2)) + d*x*sqrt(d + e*x**2)*(48*a*e**2 - 8*
b*d*e + 3*c*d**2)/(128*e**2) + x*(d + e*x**2)**(5/2)*(8*b*e - 3*c*d)/(48*e**2) +
x*(d + e*x**2)**(3/2)*(48*a*e**2 - 8*b*d*e + 3*c*d**2)/(192*e**2)

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Mathematica [A]  time = 0.197433, size = 146, normalized size = 0.83 \[ \sqrt{d+e x^2} \left (\frac{x^3 \left (48 a e^2+56 b d e+3 c d^2\right )}{192 e}-\frac{d x \left (-80 a e^2-8 b d e+3 c d^2\right )}{128 e^2}+\frac{1}{48} x^5 (8 b e+9 c d)+\frac{1}{8} c e x^7\right )+\frac{d^2 \log \left (\sqrt{e} \sqrt{d+e x^2}+e x\right ) \left (48 a e^2-8 b d e+3 c d^2\right )}{128 e^{5/2}} \]

Antiderivative was successfully verified.

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

[Out]

Sqrt[d + e*x^2]*(-(d*(3*c*d^2 - 8*b*d*e - 80*a*e^2)*x)/(128*e^2) + ((3*c*d^2 + 5
6*b*d*e + 48*a*e^2)*x^3)/(192*e) + ((9*c*d + 8*b*e)*x^5)/48 + (c*e*x^7)/8) + (d^
2*(3*c*d^2 - 8*b*d*e + 48*a*e^2)*Log[e*x + Sqrt[e]*Sqrt[d + e*x^2]])/(128*e^(5/2
))

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Maple [A]  time = 0.014, size = 229, normalized size = 1.3 \[{\frac{ax}{4} \left ( e{x}^{2}+d \right ) ^{{\frac{3}{2}}}}+{\frac{3\,adx}{8}\sqrt{e{x}^{2}+d}}+{\frac{3\,a{d}^{2}}{8}\ln \left ( x\sqrt{e}+\sqrt{e{x}^{2}+d} \right ){\frac{1}{\sqrt{e}}}}+{\frac{bx}{6\,e} \left ( e{x}^{2}+d \right ) ^{{\frac{5}{2}}}}-{\frac{bdx}{24\,e} \left ( e{x}^{2}+d \right ) ^{{\frac{3}{2}}}}-{\frac{xb{d}^{2}}{16\,e}\sqrt{e{x}^{2}+d}}-{\frac{b{d}^{3}}{16}\ln \left ( x\sqrt{e}+\sqrt{e{x}^{2}+d} \right ){e}^{-{\frac{3}{2}}}}+{\frac{c{x}^{3}}{8\,e} \left ( e{x}^{2}+d \right ) ^{{\frac{5}{2}}}}-{\frac{cdx}{16\,{e}^{2}} \left ( e{x}^{2}+d \right ) ^{{\frac{5}{2}}}}+{\frac{c{d}^{2}x}{64\,{e}^{2}} \left ( e{x}^{2}+d \right ) ^{{\frac{3}{2}}}}+{\frac{3\,c{d}^{3}x}{128\,{e}^{2}}\sqrt{e{x}^{2}+d}}+{\frac{3\,c{d}^{4}}{128}\ln \left ( x\sqrt{e}+\sqrt{e{x}^{2}+d} \right ){e}^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/4*a*x*(e*x^2+d)^(3/2)+3/8*a*d*x*(e*x^2+d)^(1/2)+3/8*a*d^2/e^(1/2)*ln(x*e^(1/2)
+(e*x^2+d)^(1/2))+1/6*b*x*(e*x^2+d)^(5/2)/e-1/24*b*d/e*x*(e*x^2+d)^(3/2)-1/16*b*
d^2/e*x*(e*x^2+d)^(1/2)-1/16*b*d^3/e^(3/2)*ln(x*e^(1/2)+(e*x^2+d)^(1/2))+1/8*c*x
^3*(e*x^2+d)^(5/2)/e-1/16*c*d/e^2*x*(e*x^2+d)^(5/2)+1/64*c*d^2/e^2*x*(e*x^2+d)^(
3/2)+3/128*c*d^3/e^2*x*(e*x^2+d)^(1/2)+3/128*c*d^4/e^(5/2)*ln(x*e^(1/2)+(e*x^2+d
)^(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)*(e*x^2 + d)^(3/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.364653, size = 1, normalized size = 0.01 \[ \left [\frac{2 \,{\left (48 \, c e^{3} x^{7} + 8 \,{\left (9 \, c d e^{2} + 8 \, b e^{3}\right )} x^{5} + 2 \,{\left (3 \, c d^{2} e + 56 \, b d e^{2} + 48 \, a e^{3}\right )} x^{3} - 3 \,{\left (3 \, c d^{3} - 8 \, b d^{2} e - 80 \, a d e^{2}\right )} x\right )} \sqrt{e x^{2} + d} \sqrt{e} + 3 \,{\left (3 \, c d^{4} - 8 \, b d^{3} e + 48 \, a d^{2} e^{2}\right )} \log \left (-2 \, \sqrt{e x^{2} + d} e x -{\left (2 \, e x^{2} + d\right )} \sqrt{e}\right )}{768 \, e^{\frac{5}{2}}}, \frac{{\left (48 \, c e^{3} x^{7} + 8 \,{\left (9 \, c d e^{2} + 8 \, b e^{3}\right )} x^{5} + 2 \,{\left (3 \, c d^{2} e + 56 \, b d e^{2} + 48 \, a e^{3}\right )} x^{3} - 3 \,{\left (3 \, c d^{3} - 8 \, b d^{2} e - 80 \, a d e^{2}\right )} x\right )} \sqrt{e x^{2} + d} \sqrt{-e} + 3 \,{\left (3 \, c d^{4} - 8 \, b d^{3} e + 48 \, a d^{2} e^{2}\right )} \arctan \left (\frac{\sqrt{-e} x}{\sqrt{e x^{2} + d}}\right )}{384 \, \sqrt{-e} e^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/768*(2*(48*c*e^3*x^7 + 8*(9*c*d*e^2 + 8*b*e^3)*x^5 + 2*(3*c*d^2*e + 56*b*d*e^
2 + 48*a*e^3)*x^3 - 3*(3*c*d^3 - 8*b*d^2*e - 80*a*d*e^2)*x)*sqrt(e*x^2 + d)*sqrt
(e) + 3*(3*c*d^4 - 8*b*d^3*e + 48*a*d^2*e^2)*log(-2*sqrt(e*x^2 + d)*e*x - (2*e*x
^2 + d)*sqrt(e)))/e^(5/2), 1/384*((48*c*e^3*x^7 + 8*(9*c*d*e^2 + 8*b*e^3)*x^5 +
2*(3*c*d^2*e + 56*b*d*e^2 + 48*a*e^3)*x^3 - 3*(3*c*d^3 - 8*b*d^2*e - 80*a*d*e^2)
*x)*sqrt(e*x^2 + d)*sqrt(-e) + 3*(3*c*d^4 - 8*b*d^3*e + 48*a*d^2*e^2)*arctan(sqr
t(-e)*x/sqrt(e*x^2 + d)))/(sqrt(-e)*e^2)]

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Sympy [A]  time = 85.9207, size = 413, normalized size = 2.36 \[ \frac{a d^{\frac{3}{2}} x \sqrt{1 + \frac{e x^{2}}{d}}}{2} + \frac{a d^{\frac{3}{2}} x}{8 \sqrt{1 + \frac{e x^{2}}{d}}} + \frac{3 a \sqrt{d} e x^{3}}{8 \sqrt{1 + \frac{e x^{2}}{d}}} + \frac{3 a d^{2} \operatorname{asinh}{\left (\frac{\sqrt{e} x}{\sqrt{d}} \right )}}{8 \sqrt{e}} + \frac{a e^{2} x^{5}}{4 \sqrt{d} \sqrt{1 + \frac{e x^{2}}{d}}} + \frac{b d^{\frac{5}{2}} x}{16 e \sqrt{1 + \frac{e x^{2}}{d}}} + \frac{17 b d^{\frac{3}{2}} x^{3}}{48 \sqrt{1 + \frac{e x^{2}}{d}}} + \frac{11 b \sqrt{d} e x^{5}}{24 \sqrt{1 + \frac{e x^{2}}{d}}} - \frac{b d^{3} \operatorname{asinh}{\left (\frac{\sqrt{e} x}{\sqrt{d}} \right )}}{16 e^{\frac{3}{2}}} + \frac{b e^{2} x^{7}}{6 \sqrt{d} \sqrt{1 + \frac{e x^{2}}{d}}} - \frac{3 c d^{\frac{7}{2}} x}{128 e^{2} \sqrt{1 + \frac{e x^{2}}{d}}} - \frac{c d^{\frac{5}{2}} x^{3}}{128 e \sqrt{1 + \frac{e x^{2}}{d}}} + \frac{13 c d^{\frac{3}{2}} x^{5}}{64 \sqrt{1 + \frac{e x^{2}}{d}}} + \frac{5 c \sqrt{d} e x^{7}}{16 \sqrt{1 + \frac{e x^{2}}{d}}} + \frac{3 c d^{4} \operatorname{asinh}{\left (\frac{\sqrt{e} x}{\sqrt{d}} \right )}}{128 e^{\frac{5}{2}}} + \frac{c e^{2} x^{9}}{8 \sqrt{d} \sqrt{1 + \frac{e x^{2}}{d}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a*d**(3/2)*x*sqrt(1 + e*x**2/d)/2 + a*d**(3/2)*x/(8*sqrt(1 + e*x**2/d)) + 3*a*sq
rt(d)*e*x**3/(8*sqrt(1 + e*x**2/d)) + 3*a*d**2*asinh(sqrt(e)*x/sqrt(d))/(8*sqrt(
e)) + a*e**2*x**5/(4*sqrt(d)*sqrt(1 + e*x**2/d)) + b*d**(5/2)*x/(16*e*sqrt(1 + e
*x**2/d)) + 17*b*d**(3/2)*x**3/(48*sqrt(1 + e*x**2/d)) + 11*b*sqrt(d)*e*x**5/(24
*sqrt(1 + e*x**2/d)) - b*d**3*asinh(sqrt(e)*x/sqrt(d))/(16*e**(3/2)) + b*e**2*x*
*7/(6*sqrt(d)*sqrt(1 + e*x**2/d)) - 3*c*d**(7/2)*x/(128*e**2*sqrt(1 + e*x**2/d))
- c*d**(5/2)*x**3/(128*e*sqrt(1 + e*x**2/d)) + 13*c*d**(3/2)*x**5/(64*sqrt(1 +
e*x**2/d)) + 5*c*sqrt(d)*e*x**7/(16*sqrt(1 + e*x**2/d)) + 3*c*d**4*asinh(sqrt(e)
*x/sqrt(d))/(128*e**(5/2)) + c*e**2*x**9/(8*sqrt(d)*sqrt(1 + e*x**2/d))

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GIAC/XCAS [A]  time = 0.269201, size = 196, normalized size = 1.12 \[ -\frac{1}{128} \,{\left (3 \, c d^{4} - 8 \, b d^{3} e + 48 \, a d^{2} e^{2}\right )} e^{\left (-\frac{5}{2}\right )}{\rm ln}\left ({\left | -x e^{\frac{1}{2}} + \sqrt{x^{2} e + d} \right |}\right ) + \frac{1}{384} \,{\left (2 \,{\left (4 \,{\left (6 \, c x^{2} e +{\left (9 \, c d e^{6} + 8 \, b e^{7}\right )} e^{\left (-6\right )}\right )} x^{2} +{\left (3 \, c d^{2} e^{5} + 56 \, b d e^{6} + 48 \, a e^{7}\right )} e^{\left (-6\right )}\right )} x^{2} - 3 \,{\left (3 \, c d^{3} e^{4} - 8 \, b d^{2} e^{5} - 80 \, a d e^{6}\right )} e^{\left (-6\right )}\right )} \sqrt{x^{2} e + d} x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/128*(3*c*d^4 - 8*b*d^3*e + 48*a*d^2*e^2)*e^(-5/2)*ln(abs(-x*e^(1/2) + sqrt(x^
2*e + d))) + 1/384*(2*(4*(6*c*x^2*e + (9*c*d*e^6 + 8*b*e^7)*e^(-6))*x^2 + (3*c*d
^2*e^5 + 56*b*d*e^6 + 48*a*e^7)*e^(-6))*x^2 - 3*(3*c*d^3*e^4 - 8*b*d^2*e^5 - 80*
a*d*e^6)*e^(-6))*sqrt(x^2*e + d)*x