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

Optimal. Leaf size=145 \[ \frac{x \sqrt{a+b x^2} \left (5 a^2 f-6 a b e+8 b^2 d\right )}{16 b^3}+\frac{\tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right ) \left (-5 a^3 f+6 a^2 b e-8 a b^2 d+16 b^3 c\right )}{16 b^{7/2}}+\frac{x^3 \sqrt{a+b x^2} (6 b e-5 a f)}{24 b^2}+\frac{f x^5 \sqrt{a+b x^2}}{6 b} \]

[Out]

((8*b^2*d - 6*a*b*e + 5*a^2*f)*x*Sqrt[a + b*x^2])/(16*b^3) + ((6*b*e - 5*a*f)*x^
3*Sqrt[a + b*x^2])/(24*b^2) + (f*x^5*Sqrt[a + b*x^2])/(6*b) + ((16*b^3*c - 8*a*b
^2*d + 6*a^2*b*e - 5*a^3*f)*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/(16*b^(7/2))

_______________________________________________________________________________________

Rubi [A]  time = 0.271066, antiderivative size = 145, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.138 \[ \frac{x \sqrt{a+b x^2} \left (5 a^2 f-6 a b e+8 b^2 d\right )}{16 b^3}+\frac{\tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right ) \left (-5 a^3 f+6 a^2 b e-8 a b^2 d+16 b^3 c\right )}{16 b^{7/2}}+\frac{x^3 \sqrt{a+b x^2} (6 b e-5 a f)}{24 b^2}+\frac{f x^5 \sqrt{a+b x^2}}{6 b} \]

Antiderivative was successfully verified.

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

[Out]

((8*b^2*d - 6*a*b*e + 5*a^2*f)*x*Sqrt[a + b*x^2])/(16*b^3) + ((6*b*e - 5*a*f)*x^
3*Sqrt[a + b*x^2])/(24*b^2) + (f*x^5*Sqrt[a + b*x^2])/(6*b) + ((16*b^3*c - 8*a*b
^2*d + 6*a^2*b*e - 5*a^3*f)*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/(16*b^(7/2))

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 47.3924, size = 138, normalized size = 0.95 \[ \frac{f x^{5} \sqrt{a + b x^{2}}}{6 b} - \frac{x^{3} \sqrt{a + b x^{2}} \left (5 a f - 6 b e\right )}{24 b^{2}} + \frac{x \sqrt{a + b x^{2}} \left (a \left (5 a f - 6 b e\right ) + 8 b^{2} d\right )}{16 b^{3}} - \frac{\left (5 a^{3} f - 6 a^{2} b e + 8 a b^{2} d - 16 b^{3} c\right ) \operatorname{atanh}{\left (\frac{\sqrt{b} x}{\sqrt{a + b x^{2}}} \right )}}{16 b^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

f*x**5*sqrt(a + b*x**2)/(6*b) - x**3*sqrt(a + b*x**2)*(5*a*f - 6*b*e)/(24*b**2)
+ x*sqrt(a + b*x**2)*(a*(5*a*f - 6*b*e) + 8*b**2*d)/(16*b**3) - (5*a**3*f - 6*a*
*2*b*e + 8*a*b**2*d - 16*b**3*c)*atanh(sqrt(b)*x/sqrt(a + b*x**2))/(16*b**(7/2))

_______________________________________________________________________________________

Mathematica [A]  time = 0.147923, size = 121, normalized size = 0.83 \[ \frac{\sqrt{b} x \sqrt{a+b x^2} \left (15 a^2 f-2 a b \left (9 e+5 f x^2\right )+4 b^2 \left (6 d+3 e x^2+2 f x^4\right )\right )+3 \log \left (\sqrt{b} \sqrt{a+b x^2}+b x\right ) \left (-5 a^3 f+6 a^2 b e-8 a b^2 d+16 b^3 c\right )}{48 b^{7/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(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]*(15*a^2*f - 2*a*b*(9*e + 5*f*x^2) + 4*b^2*(6*d + 3*e*
x^2 + 2*f*x^4)) + 3*(16*b^3*c - 8*a*b^2*d + 6*a^2*b*e - 5*a^3*f)*Log[b*x + Sqrt[
b]*Sqrt[a + b*x^2]])/(48*b^(7/2))

_______________________________________________________________________________________

Maple [A]  time = 0.01, size = 203, normalized size = 1.4 \[{c\ln \left ( x\sqrt{b}+\sqrt{b{x}^{2}+a} \right ){\frac{1}{\sqrt{b}}}}+{\frac{dx}{2\,b}\sqrt{b{x}^{2}+a}}-{\frac{ad}{2}\ln \left ( x\sqrt{b}+\sqrt{b{x}^{2}+a} \right ){b}^{-{\frac{3}{2}}}}+{\frac{e{x}^{3}}{4\,b}\sqrt{b{x}^{2}+a}}-{\frac{3\,aex}{8\,{b}^{2}}\sqrt{b{x}^{2}+a}}+{\frac{3\,{a}^{2}e}{8}\ln \left ( x\sqrt{b}+\sqrt{b{x}^{2}+a} \right ){b}^{-{\frac{5}{2}}}}+{\frac{f{x}^{5}}{6\,b}\sqrt{b{x}^{2}+a}}-{\frac{5\,af{x}^{3}}{24\,{b}^{2}}\sqrt{b{x}^{2}+a}}+{\frac{5\,{a}^{2}fx}{16\,{b}^{3}}\sqrt{b{x}^{2}+a}}-{\frac{5\,{a}^{3}f}{16}\ln \left ( x\sqrt{b}+\sqrt{b{x}^{2}+a} \right ){b}^{-{\frac{7}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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)/sqrt(b*x^2 + a),x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.281155, size = 1, normalized size = 0.01 \[ \left [\frac{2 \,{\left (8 \, b^{2} f x^{5} + 2 \,{\left (6 \, b^{2} e - 5 \, a b f\right )} x^{3} + 3 \,{\left (8 \, b^{2} d - 6 \, a b e + 5 \, a^{2} f\right )} x\right )} \sqrt{b x^{2} + a} \sqrt{b} - 3 \,{\left (16 \, b^{3} c - 8 \, a b^{2} d + 6 \, a^{2} b e - 5 \, a^{3} f\right )} \log \left (2 \, \sqrt{b x^{2} + a} b x -{\left (2 \, b x^{2} + a\right )} \sqrt{b}\right )}{96 \, b^{\frac{7}{2}}}, \frac{{\left (8 \, b^{2} f x^{5} + 2 \,{\left (6 \, b^{2} e - 5 \, a b f\right )} x^{3} + 3 \,{\left (8 \, b^{2} d - 6 \, a b e + 5 \, a^{2} f\right )} x\right )} \sqrt{b x^{2} + a} \sqrt{-b} + 3 \,{\left (16 \, b^{3} c - 8 \, a b^{2} d + 6 \, a^{2} b e - 5 \, a^{3} f\right )} \arctan \left (\frac{\sqrt{-b} x}{\sqrt{b x^{2} + a}}\right )}{48 \, \sqrt{-b} b^{3}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/96*(2*(8*b^2*f*x^5 + 2*(6*b^2*e - 5*a*b*f)*x^3 + 3*(8*b^2*d - 6*a*b*e + 5*a^2
*f)*x)*sqrt(b*x^2 + a)*sqrt(b) - 3*(16*b^3*c - 8*a*b^2*d + 6*a^2*b*e - 5*a^3*f)*
log(2*sqrt(b*x^2 + a)*b*x - (2*b*x^2 + a)*sqrt(b)))/b^(7/2), 1/48*((8*b^2*f*x^5
+ 2*(6*b^2*e - 5*a*b*f)*x^3 + 3*(8*b^2*d - 6*a*b*e + 5*a^2*f)*x)*sqrt(b*x^2 + a)
*sqrt(-b) + 3*(16*b^3*c - 8*a*b^2*d + 6*a^2*b*e - 5*a^3*f)*arctan(sqrt(-b)*x/sqr
t(b*x^2 + a)))/(sqrt(-b)*b^3)]

_______________________________________________________________________________________

Sympy [A]  time = 19.264, size = 362, normalized size = 2.5 \[ \frac{5 a^{\frac{5}{2}} f x}{16 b^{3} \sqrt{1 + \frac{b x^{2}}{a}}} - \frac{3 a^{\frac{3}{2}} e x}{8 b^{2} \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{5 a^{\frac{3}{2}} f x^{3}}{48 b^{2} \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{\sqrt{a} d x \sqrt{1 + \frac{b x^{2}}{a}}}{2 b} - \frac{\sqrt{a} e x^{3}}{8 b \sqrt{1 + \frac{b x^{2}}{a}}} - \frac{\sqrt{a} f x^{5}}{24 b \sqrt{1 + \frac{b x^{2}}{a}}} - \frac{5 a^{3} f \operatorname{asinh}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{16 b^{\frac{7}{2}}} + \frac{3 a^{2} e \operatorname{asinh}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{8 b^{\frac{5}{2}}} - \frac{a d \operatorname{asinh}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{2 b^{\frac{3}{2}}} + c \left (\begin{cases} \frac{\sqrt{- \frac{a}{b}} \operatorname{asin}{\left (x \sqrt{- \frac{b}{a}} \right )}}{\sqrt{a}} & \text{for}\: a > 0 \wedge b < 0 \\\frac{\sqrt{\frac{a}{b}} \operatorname{asinh}{\left (x \sqrt{\frac{b}{a}} \right )}}{\sqrt{a}} & \text{for}\: a > 0 \wedge b > 0 \\\frac{\sqrt{- \frac{a}{b}} \operatorname{acosh}{\left (x \sqrt{- \frac{b}{a}} \right )}}{\sqrt{- a}} & \text{for}\: b > 0 \wedge a < 0 \end{cases}\right ) + \frac{e x^{5}}{4 \sqrt{a} \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{f x^{7}}{6 \sqrt{a} \sqrt{1 + \frac{b x^{2}}{a}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

5*a**(5/2)*f*x/(16*b**3*sqrt(1 + b*x**2/a)) - 3*a**(3/2)*e*x/(8*b**2*sqrt(1 + b*
x**2/a)) + 5*a**(3/2)*f*x**3/(48*b**2*sqrt(1 + b*x**2/a)) + sqrt(a)*d*x*sqrt(1 +
b*x**2/a)/(2*b) - sqrt(a)*e*x**3/(8*b*sqrt(1 + b*x**2/a)) - sqrt(a)*f*x**5/(24*
b*sqrt(1 + b*x**2/a)) - 5*a**3*f*asinh(sqrt(b)*x/sqrt(a))/(16*b**(7/2)) + 3*a**2
*e*asinh(sqrt(b)*x/sqrt(a))/(8*b**(5/2)) - a*d*asinh(sqrt(b)*x/sqrt(a))/(2*b**(3
/2)) + c*Piecewise((sqrt(-a/b)*asin(x*sqrt(-b/a))/sqrt(a), (a > 0) & (b < 0)), (
sqrt(a/b)*asinh(x*sqrt(b/a))/sqrt(a), (a > 0) & (b > 0)), (sqrt(-a/b)*acosh(x*sq
rt(-b/a))/sqrt(-a), (b > 0) & (a < 0))) + e*x**5/(4*sqrt(a)*sqrt(1 + b*x**2/a))
+ f*x**7/(6*sqrt(a)*sqrt(1 + b*x**2/a))

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.22088, size = 174, normalized size = 1.2 \[ \frac{1}{48} \,{\left (2 \,{\left (\frac{4 \, f x^{2}}{b} - \frac{5 \, a b^{3} f - 6 \, b^{4} e}{b^{5}}\right )} x^{2} + \frac{3 \,{\left (8 \, b^{4} d + 5 \, a^{2} b^{2} f - 6 \, a b^{3} e\right )}}{b^{5}}\right )} \sqrt{b x^{2} + a} x - \frac{{\left (16 \, b^{3} c - 8 \, a b^{2} d - 5 \, a^{3} f + 6 \, a^{2} b e\right )}{\rm ln}\left ({\left | -\sqrt{b} x + \sqrt{b x^{2} + a} \right |}\right )}{16 \, b^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/48*(2*(4*f*x^2/b - (5*a*b^3*f - 6*b^4*e)/b^5)*x^2 + 3*(8*b^4*d + 5*a^2*b^2*f -
6*a*b^3*e)/b^5)*sqrt(b*x^2 + a)*x - 1/16*(16*b^3*c - 8*a*b^2*d - 5*a^3*f + 6*a^
2*b*e)*ln(abs(-sqrt(b)*x + sqrt(b*x^2 + a)))/b^(7/2)