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

Optimal. Leaf size=215 \[ \frac{x \left (d+e x^2\right )^{5/2} \left (80 a e^2-10 b d e+3 c d^2\right )}{480 e^2}+\frac{d x \left (d+e x^2\right )^{3/2} \left (80 a e^2-10 b d e+3 c d^2\right )}{384 e^2}+\frac{d^2 x \sqrt{d+e x^2} \left (80 a e^2-10 b d e+3 c d^2\right )}{256 e^2}+\frac{d^3 \tanh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d+e x^2}}\right ) \left (80 a e^2-10 b d e+3 c d^2\right )}{256 e^{5/2}}-\frac{x \left (d+e x^2\right )^{7/2} (3 c d-10 b e)}{80 e^2}+\frac{c x^3 \left (d+e x^2\right )^{7/2}}{10 e} \]

[Out]

(d^2*(3*c*d^2 - 10*b*d*e + 80*a*e^2)*x*Sqrt[d + e*x^2])/(256*e^2) + (d*(3*c*d^2
- 10*b*d*e + 80*a*e^2)*x*(d + e*x^2)^(3/2))/(384*e^2) + ((3*c*d^2 - 10*b*d*e + 8
0*a*e^2)*x*(d + e*x^2)^(5/2))/(480*e^2) - ((3*c*d - 10*b*e)*x*(d + e*x^2)^(7/2))
/(80*e^2) + (c*x^3*(d + e*x^2)^(7/2))/(10*e) + (d^3*(3*c*d^2 - 10*b*d*e + 80*a*e
^2)*ArcTanh[(Sqrt[e]*x)/Sqrt[d + e*x^2]])/(256*e^(5/2))

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Rubi [A]  time = 0.302854, antiderivative size = 215, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208 \[ \frac{x \left (d+e x^2\right )^{5/2} \left (80 a e^2-10 b d e+3 c d^2\right )}{480 e^2}+\frac{d x \left (d+e x^2\right )^{3/2} \left (80 a e^2-10 b d e+3 c d^2\right )}{384 e^2}+\frac{d^2 x \sqrt{d+e x^2} \left (80 a e^2-10 b d e+3 c d^2\right )}{256 e^2}+\frac{d^3 \tanh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d+e x^2}}\right ) \left (80 a e^2-10 b d e+3 c d^2\right )}{256 e^{5/2}}-\frac{x \left (d+e x^2\right )^{7/2} (3 c d-10 b e)}{80 e^2}+\frac{c x^3 \left (d+e x^2\right )^{7/2}}{10 e} \]

Antiderivative was successfully verified.

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

[Out]

(d^2*(3*c*d^2 - 10*b*d*e + 80*a*e^2)*x*Sqrt[d + e*x^2])/(256*e^2) + (d*(3*c*d^2
- 10*b*d*e + 80*a*e^2)*x*(d + e*x^2)^(3/2))/(384*e^2) + ((3*c*d^2 - 10*b*d*e + 8
0*a*e^2)*x*(d + e*x^2)^(5/2))/(480*e^2) - ((3*c*d - 10*b*e)*x*(d + e*x^2)^(7/2))
/(80*e^2) + (c*x^3*(d + e*x^2)^(7/2))/(10*e) + (d^3*(3*c*d^2 - 10*b*d*e + 80*a*e
^2)*ArcTanh[(Sqrt[e]*x)/Sqrt[d + e*x^2]])/(256*e^(5/2))

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Rubi in Sympy [A]  time = 27.2144, size = 211, normalized size = 0.98 \[ \frac{c x^{3} \left (d + e x^{2}\right )^{\frac{7}{2}}}{10 e} + \frac{d^{3} \left (80 a e^{2} - 10 b d e + 3 c d^{2}\right ) \operatorname{atanh}{\left (\frac{\sqrt{e} x}{\sqrt{d + e x^{2}}} \right )}}{256 e^{\frac{5}{2}}} + \frac{d^{2} x \sqrt{d + e x^{2}} \left (80 a e^{2} - 10 b d e + 3 c d^{2}\right )}{256 e^{2}} + \frac{d x \left (d + e x^{2}\right )^{\frac{3}{2}} \left (80 a e^{2} - 10 b d e + 3 c d^{2}\right )}{384 e^{2}} + \frac{x \left (d + e x^{2}\right )^{\frac{7}{2}} \left (10 b e - 3 c d\right )}{80 e^{2}} + \frac{x \left (d + e x^{2}\right )^{\frac{5}{2}} \left (80 a e^{2} - 10 b d e + 3 c d^{2}\right )}{480 e^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

c*x**3*(d + e*x**2)**(7/2)/(10*e) + d**3*(80*a*e**2 - 10*b*d*e + 3*c*d**2)*atanh
(sqrt(e)*x/sqrt(d + e*x**2))/(256*e**(5/2)) + d**2*x*sqrt(d + e*x**2)*(80*a*e**2
- 10*b*d*e + 3*c*d**2)/(256*e**2) + d*x*(d + e*x**2)**(3/2)*(80*a*e**2 - 10*b*d
*e + 3*c*d**2)/(384*e**2) + x*(d + e*x**2)**(7/2)*(10*b*e - 3*c*d)/(80*e**2) + x
*(d + e*x**2)**(5/2)*(80*a*e**2 - 10*b*d*e + 3*c*d**2)/(480*e**2)

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Mathematica [A]  time = 0.383082, size = 177, normalized size = 0.82 \[ \sqrt{d+e x^2} \left (\frac{1}{480} x^5 \left (80 a e^2+170 b d e+93 c d^2\right )+\frac{d x^3 \left (208 a e^2+118 b d e+3 c d^2\right )}{384 e}-\frac{d^2 x \left (-176 a e^2-10 b d e+3 c d^2\right )}{256 e^2}+\frac{1}{80} e x^7 (10 b e+21 c d)+\frac{1}{10} c e^2 x^9\right )+\frac{d^3 \log \left (\sqrt{e} \sqrt{d+e x^2}+e x\right ) \left (80 a e^2-10 b d e+3 c d^2\right )}{256 e^{5/2}} \]

Antiderivative was successfully verified.

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

[Out]

Sqrt[d + e*x^2]*(-(d^2*(3*c*d^2 - 10*b*d*e - 176*a*e^2)*x)/(256*e^2) + (d*(3*c*d
^2 + 118*b*d*e + 208*a*e^2)*x^3)/(384*e) + ((93*c*d^2 + 170*b*d*e + 80*a*e^2)*x^
5)/480 + (e*(21*c*d + 10*b*e)*x^7)/80 + (c*e^2*x^9)/10) + (d^3*(3*c*d^2 - 10*b*d
*e + 80*a*e^2)*Log[e*x + Sqrt[e]*Sqrt[d + e*x^2]])/(256*e^(5/2))

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Maple [A]  time = 0.014, size = 283, normalized size = 1.3 \[{\frac{ax}{6} \left ( e{x}^{2}+d \right ) ^{{\frac{5}{2}}}}+{\frac{5\,adx}{24} \left ( e{x}^{2}+d \right ) ^{{\frac{3}{2}}}}+{\frac{5\,a{d}^{2}x}{16}\sqrt{e{x}^{2}+d}}+{\frac{5\,a{d}^{3}}{16}\ln \left ( x\sqrt{e}+\sqrt{e{x}^{2}+d} \right ){\frac{1}{\sqrt{e}}}}+{\frac{bx}{8\,e} \left ( e{x}^{2}+d \right ) ^{{\frac{7}{2}}}}-{\frac{bdx}{48\,e} \left ( e{x}^{2}+d \right ) ^{{\frac{5}{2}}}}-{\frac{5\,xb{d}^{2}}{192\,e} \left ( e{x}^{2}+d \right ) ^{{\frac{3}{2}}}}-{\frac{5\,{d}^{3}bx}{128\,e}\sqrt{e{x}^{2}+d}}-{\frac{5\,{d}^{4}b}{128}\ln \left ( x\sqrt{e}+\sqrt{e{x}^{2}+d} \right ){e}^{-{\frac{3}{2}}}}+{\frac{c{x}^{3}}{10\,e} \left ( e{x}^{2}+d \right ) ^{{\frac{7}{2}}}}-{\frac{3\,cdx}{80\,{e}^{2}} \left ( e{x}^{2}+d \right ) ^{{\frac{7}{2}}}}+{\frac{c{d}^{2}x}{160\,{e}^{2}} \left ( e{x}^{2}+d \right ) ^{{\frac{5}{2}}}}+{\frac{c{d}^{3}x}{128\,{e}^{2}} \left ( e{x}^{2}+d \right ) ^{{\frac{3}{2}}}}+{\frac{3\,c{d}^{4}x}{256\,{e}^{2}}\sqrt{e{x}^{2}+d}}+{\frac{3\,c{d}^{5}}{256}\ln \left ( x\sqrt{e}+\sqrt{e{x}^{2}+d} \right ){e}^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/6*a*x*(e*x^2+d)^(5/2)+5/24*a*d*x*(e*x^2+d)^(3/2)+5/16*a*d^2*x*(e*x^2+d)^(1/2)+
5/16*a*d^3/e^(1/2)*ln(x*e^(1/2)+(e*x^2+d)^(1/2))+1/8*b*x*(e*x^2+d)^(7/2)/e-1/48*
b*d/e*x*(e*x^2+d)^(5/2)-5/192*b*d^2/e*x*(e*x^2+d)^(3/2)-5/128*b*d^3/e*x*(e*x^2+d
)^(1/2)-5/128*b*d^4/e^(3/2)*ln(x*e^(1/2)+(e*x^2+d)^(1/2))+1/10*c*x^3*(e*x^2+d)^(
7/2)/e-3/80*c*d/e^2*x*(e*x^2+d)^(7/2)+1/160*c*d^2/e^2*x*(e*x^2+d)^(5/2)+1/128*c*
d^3/e^2*x*(e*x^2+d)^(3/2)+3/256*c*d^4/e^2*x*(e*x^2+d)^(1/2)+3/256*c*d^5/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)^(5/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.494495, size = 1, normalized size = 0. \[ \left [\frac{2 \,{\left (384 \, c e^{4} x^{9} + 48 \,{\left (21 \, c d e^{3} + 10 \, b e^{4}\right )} x^{7} + 8 \,{\left (93 \, c d^{2} e^{2} + 170 \, b d e^{3} + 80 \, a e^{4}\right )} x^{5} + 10 \,{\left (3 \, c d^{3} e + 118 \, b d^{2} e^{2} + 208 \, a d e^{3}\right )} x^{3} - 15 \,{\left (3 \, c d^{4} - 10 \, b d^{3} e - 176 \, a d^{2} e^{2}\right )} x\right )} \sqrt{e x^{2} + d} \sqrt{e} + 15 \,{\left (3 \, c d^{5} - 10 \, b d^{4} e + 80 \, a d^{3} e^{2}\right )} \log \left (-2 \, \sqrt{e x^{2} + d} e x -{\left (2 \, e x^{2} + d\right )} \sqrt{e}\right )}{7680 \, e^{\frac{5}{2}}}, \frac{{\left (384 \, c e^{4} x^{9} + 48 \,{\left (21 \, c d e^{3} + 10 \, b e^{4}\right )} x^{7} + 8 \,{\left (93 \, c d^{2} e^{2} + 170 \, b d e^{3} + 80 \, a e^{4}\right )} x^{5} + 10 \,{\left (3 \, c d^{3} e + 118 \, b d^{2} e^{2} + 208 \, a d e^{3}\right )} x^{3} - 15 \,{\left (3 \, c d^{4} - 10 \, b d^{3} e - 176 \, a d^{2} e^{2}\right )} x\right )} \sqrt{e x^{2} + d} \sqrt{-e} + 15 \,{\left (3 \, c d^{5} - 10 \, b d^{4} e + 80 \, a d^{3} e^{2}\right )} \arctan \left (\frac{\sqrt{-e} x}{\sqrt{e x^{2} + d}}\right )}{3840 \, \sqrt{-e} e^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/7680*(2*(384*c*e^4*x^9 + 48*(21*c*d*e^3 + 10*b*e^4)*x^7 + 8*(93*c*d^2*e^2 + 1
70*b*d*e^3 + 80*a*e^4)*x^5 + 10*(3*c*d^3*e + 118*b*d^2*e^2 + 208*a*d*e^3)*x^3 -
15*(3*c*d^4 - 10*b*d^3*e - 176*a*d^2*e^2)*x)*sqrt(e*x^2 + d)*sqrt(e) + 15*(3*c*d
^5 - 10*b*d^4*e + 80*a*d^3*e^2)*log(-2*sqrt(e*x^2 + d)*e*x - (2*e*x^2 + d)*sqrt(
e)))/e^(5/2), 1/3840*((384*c*e^4*x^9 + 48*(21*c*d*e^3 + 10*b*e^4)*x^7 + 8*(93*c*
d^2*e^2 + 170*b*d*e^3 + 80*a*e^4)*x^5 + 10*(3*c*d^3*e + 118*b*d^2*e^2 + 208*a*d*
e^3)*x^3 - 15*(3*c*d^4 - 10*b*d^3*e - 176*a*d^2*e^2)*x)*sqrt(e*x^2 + d)*sqrt(-e)
+ 15*(3*c*d^5 - 10*b*d^4*e + 80*a*d^3*e^2)*arctan(sqrt(-e)*x/sqrt(e*x^2 + d)))/
(sqrt(-e)*e^2)]

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Sympy [A]  time = 166.516, size = 505, normalized size = 2.35 \[ \frac{a d^{\frac{5}{2}} x \sqrt{1 + \frac{e x^{2}}{d}}}{2} + \frac{3 a d^{\frac{5}{2}} x}{16 \sqrt{1 + \frac{e x^{2}}{d}}} + \frac{35 a d^{\frac{3}{2}} e x^{3}}{48 \sqrt{1 + \frac{e x^{2}}{d}}} + \frac{17 a \sqrt{d} e^{2} x^{5}}{24 \sqrt{1 + \frac{e x^{2}}{d}}} + \frac{5 a d^{3} \operatorname{asinh}{\left (\frac{\sqrt{e} x}{\sqrt{d}} \right )}}{16 \sqrt{e}} + \frac{a e^{3} x^{7}}{6 \sqrt{d} \sqrt{1 + \frac{e x^{2}}{d}}} + \frac{5 b d^{\frac{7}{2}} x}{128 e \sqrt{1 + \frac{e x^{2}}{d}}} + \frac{133 b d^{\frac{5}{2}} x^{3}}{384 \sqrt{1 + \frac{e x^{2}}{d}}} + \frac{127 b d^{\frac{3}{2}} e x^{5}}{192 \sqrt{1 + \frac{e x^{2}}{d}}} + \frac{23 b \sqrt{d} e^{2} x^{7}}{48 \sqrt{1 + \frac{e x^{2}}{d}}} - \frac{5 b d^{4} \operatorname{asinh}{\left (\frac{\sqrt{e} x}{\sqrt{d}} \right )}}{128 e^{\frac{3}{2}}} + \frac{b e^{3} x^{9}}{8 \sqrt{d} \sqrt{1 + \frac{e x^{2}}{d}}} - \frac{3 c d^{\frac{9}{2}} x}{256 e^{2} \sqrt{1 + \frac{e x^{2}}{d}}} - \frac{c d^{\frac{7}{2}} x^{3}}{256 e \sqrt{1 + \frac{e x^{2}}{d}}} + \frac{129 c d^{\frac{5}{2}} x^{5}}{640 \sqrt{1 + \frac{e x^{2}}{d}}} + \frac{73 c d^{\frac{3}{2}} e x^{7}}{160 \sqrt{1 + \frac{e x^{2}}{d}}} + \frac{29 c \sqrt{d} e^{2} x^{9}}{80 \sqrt{1 + \frac{e x^{2}}{d}}} + \frac{3 c d^{5} \operatorname{asinh}{\left (\frac{\sqrt{e} x}{\sqrt{d}} \right )}}{256 e^{\frac{5}{2}}} + \frac{c e^{3} x^{11}}{10 \sqrt{d} \sqrt{1 + \frac{e x^{2}}{d}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a*d**(5/2)*x*sqrt(1 + e*x**2/d)/2 + 3*a*d**(5/2)*x/(16*sqrt(1 + e*x**2/d)) + 35*
a*d**(3/2)*e*x**3/(48*sqrt(1 + e*x**2/d)) + 17*a*sqrt(d)*e**2*x**5/(24*sqrt(1 +
e*x**2/d)) + 5*a*d**3*asinh(sqrt(e)*x/sqrt(d))/(16*sqrt(e)) + a*e**3*x**7/(6*sqr
t(d)*sqrt(1 + e*x**2/d)) + 5*b*d**(7/2)*x/(128*e*sqrt(1 + e*x**2/d)) + 133*b*d**
(5/2)*x**3/(384*sqrt(1 + e*x**2/d)) + 127*b*d**(3/2)*e*x**5/(192*sqrt(1 + e*x**2
/d)) + 23*b*sqrt(d)*e**2*x**7/(48*sqrt(1 + e*x**2/d)) - 5*b*d**4*asinh(sqrt(e)*x
/sqrt(d))/(128*e**(3/2)) + b*e**3*x**9/(8*sqrt(d)*sqrt(1 + e*x**2/d)) - 3*c*d**(
9/2)*x/(256*e**2*sqrt(1 + e*x**2/d)) - c*d**(7/2)*x**3/(256*e*sqrt(1 + e*x**2/d)
) + 129*c*d**(5/2)*x**5/(640*sqrt(1 + e*x**2/d)) + 73*c*d**(3/2)*e*x**7/(160*sqr
t(1 + e*x**2/d)) + 29*c*sqrt(d)*e**2*x**9/(80*sqrt(1 + e*x**2/d)) + 3*c*d**5*asi
nh(sqrt(e)*x/sqrt(d))/(256*e**(5/2)) + c*e**3*x**11/(10*sqrt(d)*sqrt(1 + e*x**2/
d))

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GIAC/XCAS [A]  time = 0.287158, size = 243, normalized size = 1.13 \[ -\frac{1}{256} \,{\left (3 \, c d^{5} - 10 \, b d^{4} e + 80 \, a d^{3} e^{2}\right )} e^{\left (-\frac{5}{2}\right )}{\rm ln}\left ({\left | -x e^{\frac{1}{2}} + \sqrt{x^{2} e + d} \right |}\right ) + \frac{1}{3840} \,{\left (2 \,{\left (4 \,{\left (6 \,{\left (8 \, c x^{2} e^{2} +{\left (21 \, c d e^{9} + 10 \, b e^{10}\right )} e^{\left (-8\right )}\right )} x^{2} +{\left (93 \, c d^{2} e^{8} + 170 \, b d e^{9} + 80 \, a e^{10}\right )} e^{\left (-8\right )}\right )} x^{2} + 5 \,{\left (3 \, c d^{3} e^{7} + 118 \, b d^{2} e^{8} + 208 \, a d e^{9}\right )} e^{\left (-8\right )}\right )} x^{2} - 15 \,{\left (3 \, c d^{4} e^{6} - 10 \, b d^{3} e^{7} - 176 \, a d^{2} e^{8}\right )} e^{\left (-8\right )}\right )} \sqrt{x^{2} e + d} x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/256*(3*c*d^5 - 10*b*d^4*e + 80*a*d^3*e^2)*e^(-5/2)*ln(abs(-x*e^(1/2) + sqrt(x
^2*e + d))) + 1/3840*(2*(4*(6*(8*c*x^2*e^2 + (21*c*d*e^9 + 10*b*e^10)*e^(-8))*x^
2 + (93*c*d^2*e^8 + 170*b*d*e^9 + 80*a*e^10)*e^(-8))*x^2 + 5*(3*c*d^3*e^7 + 118*
b*d^2*e^8 + 208*a*d*e^9)*e^(-8))*x^2 - 15*(3*c*d^4*e^6 - 10*b*d^3*e^7 - 176*a*d^
2*e^8)*e^(-8))*sqrt(x^2*e + d)*x