3.152 \(\int \frac{x^2 \left (c+d x^2+e x^4+f x^6\right )}{\sqrt{a+b x^2}} \, dx\)

Optimal. Leaf size=194 \[ \frac{x^3 \sqrt{a+b x^2} \left (35 a^2 f-40 a b e+48 b^2 d\right )}{192 b^3}-\frac{a \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right ) \left (-35 a^3 f+40 a^2 b e-48 a b^2 d+64 b^3 c\right )}{128 b^{9/2}}+\frac{x \sqrt{a+b x^2} \left (-35 a^3 f+40 a^2 b e-48 a b^2 d+64 b^3 c\right )}{128 b^4}+\frac{x^5 \sqrt{a+b x^2} (8 b e-7 a f)}{48 b^2}+\frac{f x^7 \sqrt{a+b x^2}}{8 b} \]

[Out]

((64*b^3*c - 48*a*b^2*d + 40*a^2*b*e - 35*a^3*f)*x*Sqrt[a + b*x^2])/(128*b^4) +
((48*b^2*d - 40*a*b*e + 35*a^2*f)*x^3*Sqrt[a + b*x^2])/(192*b^3) + ((8*b*e - 7*a
*f)*x^5*Sqrt[a + b*x^2])/(48*b^2) + (f*x^7*Sqrt[a + b*x^2])/(8*b) - (a*(64*b^3*c
- 48*a*b^2*d + 40*a^2*b*e - 35*a^3*f)*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/(12
8*b^(9/2))

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Rubi [A]  time = 0.532242, antiderivative size = 194, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.156 \[ \frac{x^3 \sqrt{a+b x^2} \left (35 a^2 f-40 a b e+48 b^2 d\right )}{192 b^3}-\frac{a \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right ) \left (-35 a^3 f+40 a^2 b e-48 a b^2 d+64 b^3 c\right )}{128 b^{9/2}}+\frac{x \sqrt{a+b x^2} \left (-35 a^3 f+40 a^2 b e-48 a b^2 d+64 b^3 c\right )}{128 b^4}+\frac{x^5 \sqrt{a+b x^2} (8 b e-7 a f)}{48 b^2}+\frac{f x^7 \sqrt{a+b x^2}}{8 b} \]

Antiderivative was successfully verified.

[In]  Int[(x^2*(c + d*x^2 + e*x^4 + f*x^6))/Sqrt[a + b*x^2],x]

[Out]

((64*b^3*c - 48*a*b^2*d + 40*a^2*b*e - 35*a^3*f)*x*Sqrt[a + b*x^2])/(128*b^4) +
((48*b^2*d - 40*a*b*e + 35*a^2*f)*x^3*Sqrt[a + b*x^2])/(192*b^3) + ((8*b*e - 7*a
*f)*x^5*Sqrt[a + b*x^2])/(48*b^2) + (f*x^7*Sqrt[a + b*x^2])/(8*b) - (a*(64*b^3*c
- 48*a*b^2*d + 40*a^2*b*e - 35*a^3*f)*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/(12
8*b^(9/2))

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Rubi in Sympy [A]  time = 70.0503, size = 187, normalized size = 0.96 \[ \frac{a \left (a \left (35 a^{2} f - 40 a b e + 48 b^{2} d\right ) - 64 b^{3} c\right ) \operatorname{atanh}{\left (\frac{\sqrt{b} x}{\sqrt{a + b x^{2}}} \right )}}{128 b^{\frac{9}{2}}} + \frac{f x^{7} \sqrt{a + b x^{2}}}{8 b} - \frac{x^{5} \sqrt{a + b x^{2}} \left (7 a f - 8 b e\right )}{48 b^{2}} + \frac{x^{3} \sqrt{a + b x^{2}} \left (35 a^{2} f - 40 a b e + 48 b^{2} d\right )}{192 b^{3}} - \frac{x \sqrt{a + b x^{2}} \left (a \left (35 a^{2} f - 40 a b e + 48 b^{2} d\right ) - 64 b^{3} c\right )}{128 b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**2*(f*x**6+e*x**4+d*x**2+c)/(b*x**2+a)**(1/2),x)

[Out]

a*(a*(35*a**2*f - 40*a*b*e + 48*b**2*d) - 64*b**3*c)*atanh(sqrt(b)*x/sqrt(a + b*
x**2))/(128*b**(9/2)) + f*x**7*sqrt(a + b*x**2)/(8*b) - x**5*sqrt(a + b*x**2)*(7
*a*f - 8*b*e)/(48*b**2) + x**3*sqrt(a + b*x**2)*(35*a**2*f - 40*a*b*e + 48*b**2*
d)/(192*b**3) - x*sqrt(a + b*x**2)*(a*(35*a**2*f - 40*a*b*e + 48*b**2*d) - 64*b*
*3*c)/(128*b**4)

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Mathematica [A]  time = 0.219571, size = 152, normalized size = 0.78 \[ \frac{3 a \log \left (\sqrt{b} \sqrt{a+b x^2}+b x\right ) \left (35 a^3 f-40 a^2 b e+48 a b^2 d-64 b^3 c\right )+\sqrt{b} x \sqrt{a+b x^2} \left (-105 a^3 f+10 a^2 b \left (12 e+7 f x^2\right )-8 a b^2 \left (18 d+10 e x^2+7 f x^4\right )+16 b^3 \left (12 c+6 d x^2+4 e x^4+3 f x^6\right )\right )}{384 b^{9/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(x^2*(c + d*x^2 + e*x^4 + f*x^6))/Sqrt[a + b*x^2],x]

[Out]

(Sqrt[b]*x*Sqrt[a + b*x^2]*(-105*a^3*f + 10*a^2*b*(12*e + 7*f*x^2) - 8*a*b^2*(18
*d + 10*e*x^2 + 7*f*x^4) + 16*b^3*(12*c + 6*d*x^2 + 4*e*x^4 + 3*f*x^6)) + 3*a*(-
64*b^3*c + 48*a*b^2*d - 40*a^2*b*e + 35*a^3*f)*Log[b*x + Sqrt[b]*Sqrt[a + b*x^2]
])/(384*b^(9/2))

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Maple [A]  time = 0.014, size = 284, normalized size = 1.5 \[{\frac{d{x}^{3}}{4\,b}\sqrt{b{x}^{2}+a}}-{\frac{3\,adx}{8\,{b}^{2}}\sqrt{b{x}^{2}+a}}+{\frac{3\,{a}^{2}d}{8}\ln \left ( x\sqrt{b}+\sqrt{b{x}^{2}+a} \right ){b}^{-{\frac{5}{2}}}}+{\frac{cx}{2\,b}\sqrt{b{x}^{2}+a}}-{\frac{ac}{2}\ln \left ( x\sqrt{b}+\sqrt{b{x}^{2}+a} \right ){b}^{-{\frac{3}{2}}}}+{\frac{e{x}^{5}}{6\,b}\sqrt{b{x}^{2}+a}}-{\frac{5\,ae{x}^{3}}{24\,{b}^{2}}\sqrt{b{x}^{2}+a}}+{\frac{5\,{a}^{2}ex}{16\,{b}^{3}}\sqrt{b{x}^{2}+a}}-{\frac{5\,e{a}^{3}}{16}\ln \left ( x\sqrt{b}+\sqrt{b{x}^{2}+a} \right ){b}^{-{\frac{7}{2}}}}+{\frac{f{x}^{7}}{8\,b}\sqrt{b{x}^{2}+a}}-{\frac{7\,af{x}^{5}}{48\,{b}^{2}}\sqrt{b{x}^{2}+a}}+{\frac{35\,{a}^{2}f{x}^{3}}{192\,{b}^{3}}\sqrt{b{x}^{2}+a}}-{\frac{35\,{a}^{3}fx}{128\,{b}^{4}}\sqrt{b{x}^{2}+a}}+{\frac{35\,f{a}^{4}}{128}\ln \left ( x\sqrt{b}+\sqrt{b{x}^{2}+a} \right ){b}^{-{\frac{9}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^2*(f*x^6+e*x^4+d*x^2+c)/(b*x^2+a)^(1/2),x)

[Out]

1/4*d*x^3/b*(b*x^2+a)^(1/2)-3/8*d*a/b^2*x*(b*x^2+a)^(1/2)+3/8*d*a^2/b^(5/2)*ln(x
*b^(1/2)+(b*x^2+a)^(1/2))+1/2*c*x/b*(b*x^2+a)^(1/2)-1/2*c*a/b^(3/2)*ln(x*b^(1/2)
+(b*x^2+a)^(1/2))+1/6*e*x^5/b*(b*x^2+a)^(1/2)-5/24*e*a/b^2*x^3*(b*x^2+a)^(1/2)+5
/16*e*a^2/b^3*x*(b*x^2+a)^(1/2)-5/16*e*a^3/b^(7/2)*ln(x*b^(1/2)+(b*x^2+a)^(1/2))
+1/8*f*x^7*(b*x^2+a)^(1/2)/b-7/48*f*a/b^2*x^5*(b*x^2+a)^(1/2)+35/192*f*a^2/b^3*x
^3*(b*x^2+a)^(1/2)-35/128*f*a^3/b^4*x*(b*x^2+a)^(1/2)+35/128*f*a^4/b^(9/2)*ln(x*
b^(1/2)+(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((f*x^6 + e*x^4 + d*x^2 + c)*x^2/sqrt(b*x^2 + a),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.37357, size = 1, normalized size = 0.01 \[ \left [\frac{2 \,{\left (48 \, b^{3} f x^{7} + 8 \,{\left (8 \, b^{3} e - 7 \, a b^{2} f\right )} x^{5} + 2 \,{\left (48 \, b^{3} d - 40 \, a b^{2} e + 35 \, a^{2} b f\right )} x^{3} + 3 \,{\left (64 \, b^{3} c - 48 \, a b^{2} d + 40 \, a^{2} b e - 35 \, a^{3} f\right )} x\right )} \sqrt{b x^{2} + a} \sqrt{b} - 3 \,{\left (64 \, a b^{3} c - 48 \, a^{2} b^{2} d + 40 \, a^{3} b e - 35 \, a^{4} f\right )} \log \left (-2 \, \sqrt{b x^{2} + a} b x -{\left (2 \, b x^{2} + a\right )} \sqrt{b}\right )}{768 \, b^{\frac{9}{2}}}, \frac{{\left (48 \, b^{3} f x^{7} + 8 \,{\left (8 \, b^{3} e - 7 \, a b^{2} f\right )} x^{5} + 2 \,{\left (48 \, b^{3} d - 40 \, a b^{2} e + 35 \, a^{2} b f\right )} x^{3} + 3 \,{\left (64 \, b^{3} c - 48 \, a b^{2} d + 40 \, a^{2} b e - 35 \, a^{3} f\right )} x\right )} \sqrt{b x^{2} + a} \sqrt{-b} - 3 \,{\left (64 \, a b^{3} c - 48 \, a^{2} b^{2} d + 40 \, a^{3} b e - 35 \, a^{4} f\right )} \arctan \left (\frac{\sqrt{-b} x}{\sqrt{b x^{2} + a}}\right )}{384 \, \sqrt{-b} b^{4}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((f*x^6 + e*x^4 + d*x^2 + c)*x^2/sqrt(b*x^2 + a),x, algorithm="fricas")

[Out]

[1/768*(2*(48*b^3*f*x^7 + 8*(8*b^3*e - 7*a*b^2*f)*x^5 + 2*(48*b^3*d - 40*a*b^2*e
+ 35*a^2*b*f)*x^3 + 3*(64*b^3*c - 48*a*b^2*d + 40*a^2*b*e - 35*a^3*f)*x)*sqrt(b
*x^2 + a)*sqrt(b) - 3*(64*a*b^3*c - 48*a^2*b^2*d + 40*a^3*b*e - 35*a^4*f)*log(-2
*sqrt(b*x^2 + a)*b*x - (2*b*x^2 + a)*sqrt(b)))/b^(9/2), 1/384*((48*b^3*f*x^7 + 8
*(8*b^3*e - 7*a*b^2*f)*x^5 + 2*(48*b^3*d - 40*a*b^2*e + 35*a^2*b*f)*x^3 + 3*(64*
b^3*c - 48*a*b^2*d + 40*a^2*b*e - 35*a^3*f)*x)*sqrt(b*x^2 + a)*sqrt(-b) - 3*(64*
a*b^3*c - 48*a^2*b^2*d + 40*a^3*b*e - 35*a^4*f)*arctan(sqrt(-b)*x/sqrt(b*x^2 + a
)))/(sqrt(-b)*b^4)]

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Sympy [A]  time = 34.2524, size = 444, normalized size = 2.29 \[ - \frac{35 a^{\frac{7}{2}} f x}{128 b^{4} \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{5 a^{\frac{5}{2}} e x}{16 b^{3} \sqrt{1 + \frac{b x^{2}}{a}}} - \frac{35 a^{\frac{5}{2}} f x^{3}}{384 b^{3} \sqrt{1 + \frac{b x^{2}}{a}}} - \frac{3 a^{\frac{3}{2}} d x}{8 b^{2} \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{5 a^{\frac{3}{2}} e x^{3}}{48 b^{2} \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{7 a^{\frac{3}{2}} f x^{5}}{192 b^{2} \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{\sqrt{a} c x \sqrt{1 + \frac{b x^{2}}{a}}}{2 b} - \frac{\sqrt{a} d x^{3}}{8 b \sqrt{1 + \frac{b x^{2}}{a}}} - \frac{\sqrt{a} e x^{5}}{24 b \sqrt{1 + \frac{b x^{2}}{a}}} - \frac{\sqrt{a} f x^{7}}{48 b \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{35 a^{4} f \operatorname{asinh}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{128 b^{\frac{9}{2}}} - \frac{5 a^{3} e \operatorname{asinh}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{16 b^{\frac{7}{2}}} + \frac{3 a^{2} d \operatorname{asinh}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{8 b^{\frac{5}{2}}} - \frac{a c \operatorname{asinh}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{2 b^{\frac{3}{2}}} + \frac{d x^{5}}{4 \sqrt{a} \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{e x^{7}}{6 \sqrt{a} \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{f x^{9}}{8 \sqrt{a} \sqrt{1 + \frac{b x^{2}}{a}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**2*(f*x**6+e*x**4+d*x**2+c)/(b*x**2+a)**(1/2),x)

[Out]

-35*a**(7/2)*f*x/(128*b**4*sqrt(1 + b*x**2/a)) + 5*a**(5/2)*e*x/(16*b**3*sqrt(1
+ b*x**2/a)) - 35*a**(5/2)*f*x**3/(384*b**3*sqrt(1 + b*x**2/a)) - 3*a**(3/2)*d*x
/(8*b**2*sqrt(1 + b*x**2/a)) + 5*a**(3/2)*e*x**3/(48*b**2*sqrt(1 + b*x**2/a)) +
7*a**(3/2)*f*x**5/(192*b**2*sqrt(1 + b*x**2/a)) + sqrt(a)*c*x*sqrt(1 + b*x**2/a)
/(2*b) - sqrt(a)*d*x**3/(8*b*sqrt(1 + b*x**2/a)) - sqrt(a)*e*x**5/(24*b*sqrt(1 +
b*x**2/a)) - sqrt(a)*f*x**7/(48*b*sqrt(1 + b*x**2/a)) + 35*a**4*f*asinh(sqrt(b)
*x/sqrt(a))/(128*b**(9/2)) - 5*a**3*e*asinh(sqrt(b)*x/sqrt(a))/(16*b**(7/2)) + 3
*a**2*d*asinh(sqrt(b)*x/sqrt(a))/(8*b**(5/2)) - a*c*asinh(sqrt(b)*x/sqrt(a))/(2*
b**(3/2)) + d*x**5/(4*sqrt(a)*sqrt(1 + b*x**2/a)) + e*x**7/(6*sqrt(a)*sqrt(1 + b
*x**2/a)) + f*x**9/(8*sqrt(a)*sqrt(1 + b*x**2/a))

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GIAC/XCAS [A]  time = 0.226074, size = 236, normalized size = 1.22 \[ \frac{1}{384} \,{\left (2 \,{\left (4 \,{\left (\frac{6 \, f x^{2}}{b} - \frac{7 \, a b^{5} f - 8 \, b^{6} e}{b^{7}}\right )} x^{2} + \frac{48 \, b^{6} d + 35 \, a^{2} b^{4} f - 40 \, a b^{5} e}{b^{7}}\right )} x^{2} + \frac{3 \,{\left (64 \, b^{6} c - 48 \, a b^{5} d - 35 \, a^{3} b^{3} f + 40 \, a^{2} b^{4} e\right )}}{b^{7}}\right )} \sqrt{b x^{2} + a} x + \frac{{\left (64 \, a b^{3} c - 48 \, a^{2} b^{2} d - 35 \, a^{4} f + 40 \, a^{3} b e\right )}{\rm ln}\left ({\left | -\sqrt{b} x + \sqrt{b x^{2} + a} \right |}\right )}{128 \, b^{\frac{9}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((f*x^6 + e*x^4 + d*x^2 + c)*x^2/sqrt(b*x^2 + a),x, algorithm="giac")

[Out]

1/384*(2*(4*(6*f*x^2/b - (7*a*b^5*f - 8*b^6*e)/b^7)*x^2 + (48*b^6*d + 35*a^2*b^4
*f - 40*a*b^5*e)/b^7)*x^2 + 3*(64*b^6*c - 48*a*b^5*d - 35*a^3*b^3*f + 40*a^2*b^4
*e)/b^7)*sqrt(b*x^2 + a)*x + 1/128*(64*a*b^3*c - 48*a^2*b^2*d - 35*a^4*f + 40*a^
3*b*e)*ln(abs(-sqrt(b)*x + sqrt(b*x^2 + a)))/b^(9/2)