3.149 \(\int \frac{c+d x^2+e x^4+f x^6}{x^7 \sqrt{a+b x^2}} \, dx\)

Optimal. Leaf size=146 \[ \frac{\sqrt{a+b x^2} (5 b c-6 a d)}{24 a^2 x^4}-\frac{\sqrt{a+b x^2} \left (8 a^2 e-6 a b d+5 b^2 c\right )}{16 a^3 x^2}+\frac{\tanh ^{-1}\left (\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right ) \left (-16 a^3 f+8 a^2 b e-6 a b^2 d+5 b^3 c\right )}{16 a^{7/2}}-\frac{c \sqrt{a+b x^2}}{6 a x^6} \]

[Out]

-(c*Sqrt[a + b*x^2])/(6*a*x^6) + ((5*b*c - 6*a*d)*Sqrt[a + b*x^2])/(24*a^2*x^4)
- ((5*b^2*c - 6*a*b*d + 8*a^2*e)*Sqrt[a + b*x^2])/(16*a^3*x^2) + ((5*b^3*c - 6*a
*b^2*d + 8*a^2*b*e - 16*a^3*f)*ArcTanh[Sqrt[a + b*x^2]/Sqrt[a]])/(16*a^(7/2))

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Rubi [A]  time = 0.59972, antiderivative size = 146, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.188 \[ \frac{\sqrt{a+b x^2} (5 b c-6 a d)}{24 a^2 x^4}-\frac{\sqrt{a+b x^2} \left (8 a^2 e-6 a b d+5 b^2 c\right )}{16 a^3 x^2}+\frac{\tanh ^{-1}\left (\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right ) \left (-16 a^3 f+8 a^2 b e-6 a b^2 d+5 b^3 c\right )}{16 a^{7/2}}-\frac{c \sqrt{a+b x^2}}{6 a x^6} \]

Antiderivative was successfully verified.

[In]  Int[(c + d*x^2 + e*x^4 + f*x^6)/(x^7*Sqrt[a + b*x^2]),x]

[Out]

-(c*Sqrt[a + b*x^2])/(6*a*x^6) + ((5*b*c - 6*a*d)*Sqrt[a + b*x^2])/(24*a^2*x^4)
- ((5*b^2*c - 6*a*b*d + 8*a^2*e)*Sqrt[a + b*x^2])/(16*a^3*x^2) + ((5*b^3*c - 6*a
*b^2*d + 8*a^2*b*e - 16*a^3*f)*ArcTanh[Sqrt[a + b*x^2]/Sqrt[a]])/(16*a^(7/2))

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Rubi in Sympy [A]  time = 152.04, size = 141, normalized size = 0.97 \[ - \frac{c \sqrt{a + b x^{2}}}{6 a x^{6}} - \frac{\sqrt{a + b x^{2}} \left (6 a d - 5 b c\right )}{24 a^{2} x^{4}} - \frac{\sqrt{a + b x^{2}} \left (8 a^{2} e - 6 a b d + 5 b^{2} c\right )}{16 a^{3} x^{2}} - \frac{\left (16 a^{3} f - 8 a^{2} b e + 6 a b^{2} d - 5 b^{3} c\right ) \operatorname{atanh}{\left (\frac{\sqrt{a + b x^{2}}}{\sqrt{a}} \right )}}{16 a^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((f*x**6+e*x**4+d*x**2+c)/x**7/(b*x**2+a)**(1/2),x)

[Out]

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

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Mathematica [A]  time = 0.272577, size = 166, normalized size = 1.14 \[ \sqrt{a+b x^2} \left (\frac{5 b c-6 a d}{24 a^2 x^4}+\frac{-8 a^2 e+6 a b d-5 b^2 c}{16 a^3 x^2}-\frac{c}{6 a x^6}\right )-\frac{\log \left (\sqrt{a} \sqrt{a+b x^2}+a\right ) \left (16 a^3 f-8 a^2 b e+6 a b^2 d-5 b^3 c\right )}{16 a^{7/2}}+\frac{\log (x) \left (16 a^3 f-8 a^2 b e+6 a b^2 d-5 b^3 c\right )}{16 a^{7/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(c + d*x^2 + e*x^4 + f*x^6)/(x^7*Sqrt[a + b*x^2]),x]

[Out]

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

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Maple [A]  time = 0.016, size = 238, normalized size = 1.6 \[ -{\frac{c}{6\,a{x}^{6}}\sqrt{b{x}^{2}+a}}+{\frac{5\,bc}{24\,{a}^{2}{x}^{4}}\sqrt{b{x}^{2}+a}}-{\frac{5\,{b}^{2}c}{16\,{a}^{3}{x}^{2}}\sqrt{b{x}^{2}+a}}+{\frac{5\,{b}^{3}c}{16}\ln \left ({\frac{1}{x} \left ( 2\,a+2\,\sqrt{a}\sqrt{b{x}^{2}+a} \right ) } \right ){a}^{-{\frac{7}{2}}}}-{\frac{d}{4\,a{x}^{4}}\sqrt{b{x}^{2}+a}}+{\frac{3\,bd}{8\,{a}^{2}{x}^{2}}\sqrt{b{x}^{2}+a}}-{\frac{3\,{b}^{2}d}{8}\ln \left ({\frac{1}{x} \left ( 2\,a+2\,\sqrt{a}\sqrt{b{x}^{2}+a} \right ) } \right ){a}^{-{\frac{5}{2}}}}-{\frac{e}{2\,a{x}^{2}}\sqrt{b{x}^{2}+a}}+{\frac{be}{2}\ln \left ({\frac{1}{x} \left ( 2\,a+2\,\sqrt{a}\sqrt{b{x}^{2}+a} \right ) } \right ){a}^{-{\frac{3}{2}}}}-{f\ln \left ({\frac{1}{x} \left ( 2\,a+2\,\sqrt{a}\sqrt{b{x}^{2}+a} \right ) } \right ){\frac{1}{\sqrt{a}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((f*x^6+e*x^4+d*x^2+c)/x^7/(b*x^2+a)^(1/2),x)

[Out]

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

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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)/(sqrt(b*x^2 + a)*x^7),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.314747, size = 1, normalized size = 0.01 \[ \left [-\frac{3 \,{\left (5 \, b^{3} c - 6 \, a b^{2} d + 8 \, a^{2} b e - 16 \, a^{3} f\right )} x^{6} \log \left (-\frac{{\left (b x^{2} + 2 \, a\right )} \sqrt{a} - 2 \, \sqrt{b x^{2} + a} a}{x^{2}}\right ) + 2 \,{\left (3 \,{\left (5 \, b^{2} c - 6 \, a b d + 8 \, a^{2} e\right )} x^{4} + 8 \, a^{2} c - 2 \,{\left (5 \, a b c - 6 \, a^{2} d\right )} x^{2}\right )} \sqrt{b x^{2} + a} \sqrt{a}}{96 \, a^{\frac{7}{2}} x^{6}}, \frac{3 \,{\left (5 \, b^{3} c - 6 \, a b^{2} d + 8 \, a^{2} b e - 16 \, a^{3} f\right )} x^{6} \arctan \left (\frac{\sqrt{-a}}{\sqrt{b x^{2} + a}}\right ) -{\left (3 \,{\left (5 \, b^{2} c - 6 \, a b d + 8 \, a^{2} e\right )} x^{4} + 8 \, a^{2} c - 2 \,{\left (5 \, a b c - 6 \, a^{2} d\right )} x^{2}\right )} \sqrt{b x^{2} + a} \sqrt{-a}}{48 \, \sqrt{-a} a^{3} x^{6}}\right ] \]

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^7),x, algorithm="fricas")

[Out]

[-1/96*(3*(5*b^3*c - 6*a*b^2*d + 8*a^2*b*e - 16*a^3*f)*x^6*log(-((b*x^2 + 2*a)*s
qrt(a) - 2*sqrt(b*x^2 + a)*a)/x^2) + 2*(3*(5*b^2*c - 6*a*b*d + 8*a^2*e)*x^4 + 8*
a^2*c - 2*(5*a*b*c - 6*a^2*d)*x^2)*sqrt(b*x^2 + a)*sqrt(a))/(a^(7/2)*x^6), 1/48*
(3*(5*b^3*c - 6*a*b^2*d + 8*a^2*b*e - 16*a^3*f)*x^6*arctan(sqrt(-a)/sqrt(b*x^2 +
a)) - (3*(5*b^2*c - 6*a*b*d + 8*a^2*e)*x^4 + 8*a^2*c - 2*(5*a*b*c - 6*a^2*d)*x^
2)*sqrt(b*x^2 + a)*sqrt(-a))/(sqrt(-a)*a^3*x^6)]

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Sympy [A]  time = 86.7126, size = 303, normalized size = 2.08 \[ - \frac{c}{6 \sqrt{b} x^{7} \sqrt{\frac{a}{b x^{2}} + 1}} - \frac{d}{4 \sqrt{b} x^{5} \sqrt{\frac{a}{b x^{2}} + 1}} + \frac{\sqrt{b} c}{24 a x^{5} \sqrt{\frac{a}{b x^{2}} + 1}} + \frac{\sqrt{b} d}{8 a x^{3} \sqrt{\frac{a}{b x^{2}} + 1}} - \frac{\sqrt{b} e \sqrt{\frac{a}{b x^{2}} + 1}}{2 a x} - \frac{5 b^{\frac{3}{2}} c}{48 a^{2} x^{3} \sqrt{\frac{a}{b x^{2}} + 1}} + \frac{3 b^{\frac{3}{2}} d}{8 a^{2} x \sqrt{\frac{a}{b x^{2}} + 1}} - \frac{5 b^{\frac{5}{2}} c}{16 a^{3} x \sqrt{\frac{a}{b x^{2}} + 1}} - \frac{f \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{b} x} \right )}}{\sqrt{a}} + \frac{b e \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{b} x} \right )}}{2 a^{\frac{3}{2}}} - \frac{3 b^{2} d \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{b} x} \right )}}{8 a^{\frac{5}{2}}} + \frac{5 b^{3} c \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{b} x} \right )}}{16 a^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((f*x**6+e*x**4+d*x**2+c)/x**7/(b*x**2+a)**(1/2),x)

[Out]

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

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GIAC/XCAS [A]  time = 0.229471, size = 313, normalized size = 2.14 \[ -\frac{\frac{3 \,{\left (5 \, b^{4} c - 6 \, a b^{3} d - 16 \, a^{3} b f + 8 \, a^{2} b^{2} e\right )} \arctan \left (\frac{\sqrt{b x^{2} + a}}{\sqrt{-a}}\right )}{\sqrt{-a} a^{3}} + \frac{15 \,{\left (b x^{2} + a\right )}^{\frac{5}{2}} b^{4} c - 40 \,{\left (b x^{2} + a\right )}^{\frac{3}{2}} a b^{4} c + 33 \, \sqrt{b x^{2} + a} a^{2} b^{4} c - 18 \,{\left (b x^{2} + a\right )}^{\frac{5}{2}} a b^{3} d + 48 \,{\left (b x^{2} + a\right )}^{\frac{3}{2}} a^{2} b^{3} d - 30 \, \sqrt{b x^{2} + a} a^{3} b^{3} d + 24 \,{\left (b x^{2} + a\right )}^{\frac{5}{2}} a^{2} b^{2} e - 48 \,{\left (b x^{2} + a\right )}^{\frac{3}{2}} a^{3} b^{2} e + 24 \, \sqrt{b x^{2} + a} a^{4} b^{2} e}{a^{3} b^{3} x^{6}}}{48 \, b} \]

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^7),x, algorithm="giac")

[Out]

-1/48*(3*(5*b^4*c - 6*a*b^3*d - 16*a^3*b*f + 8*a^2*b^2*e)*arctan(sqrt(b*x^2 + a)
/sqrt(-a))/(sqrt(-a)*a^3) + (15*(b*x^2 + a)^(5/2)*b^4*c - 40*(b*x^2 + a)^(3/2)*a
*b^4*c + 33*sqrt(b*x^2 + a)*a^2*b^4*c - 18*(b*x^2 + a)^(5/2)*a*b^3*d + 48*(b*x^2
+ a)^(3/2)*a^2*b^3*d - 30*sqrt(b*x^2 + a)*a^3*b^3*d + 24*(b*x^2 + a)^(5/2)*a^2*
b^2*e - 48*(b*x^2 + a)^(3/2)*a^3*b^2*e + 24*sqrt(b*x^2 + a)*a^4*b^2*e)/(a^3*b^3*
x^6))/b