3.943 \(\int \frac{\left (a+b x^2+c x^4\right )^{3/2}}{x^7} \, dx\)

Optimal. Leaf size=163 \[ \frac{b \left (b^2-12 a c\right ) \tanh ^{-1}\left (\frac{2 a+b x^2}{2 \sqrt{a} \sqrt{a+b x^2+c x^4}}\right )}{32 a^{3/2}}-\frac{\left (x^2 \left (8 a c+b^2\right )+2 a b\right ) \sqrt{a+b x^2+c x^4}}{16 a x^4}+\frac{1}{2} c^{3/2} \tanh ^{-1}\left (\frac{b+2 c x^2}{2 \sqrt{c} \sqrt{a+b x^2+c x^4}}\right )-\frac{\left (a+b x^2+c x^4\right )^{3/2}}{6 x^6} \]

[Out]

-((2*a*b + (b^2 + 8*a*c)*x^2)*Sqrt[a + b*x^2 + c*x^4])/(16*a*x^4) - (a + b*x^2 +
c*x^4)^(3/2)/(6*x^6) + (b*(b^2 - 12*a*c)*ArcTanh[(2*a + b*x^2)/(2*Sqrt[a]*Sqrt[
a + b*x^2 + c*x^4])])/(32*a^(3/2)) + (c^(3/2)*ArcTanh[(b + 2*c*x^2)/(2*Sqrt[c]*S
qrt[a + b*x^2 + c*x^4])])/2

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Rubi [A]  time = 0.463887, antiderivative size = 163, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.35 \[ \frac{b \left (b^2-12 a c\right ) \tanh ^{-1}\left (\frac{2 a+b x^2}{2 \sqrt{a} \sqrt{a+b x^2+c x^4}}\right )}{32 a^{3/2}}-\frac{\left (x^2 \left (8 a c+b^2\right )+2 a b\right ) \sqrt{a+b x^2+c x^4}}{16 a x^4}+\frac{1}{2} c^{3/2} \tanh ^{-1}\left (\frac{b+2 c x^2}{2 \sqrt{c} \sqrt{a+b x^2+c x^4}}\right )-\frac{\left (a+b x^2+c x^4\right )^{3/2}}{6 x^6} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^2 + c*x^4)^(3/2)/x^7,x]

[Out]

-((2*a*b + (b^2 + 8*a*c)*x^2)*Sqrt[a + b*x^2 + c*x^4])/(16*a*x^4) - (a + b*x^2 +
c*x^4)^(3/2)/(6*x^6) + (b*(b^2 - 12*a*c)*ArcTanh[(2*a + b*x^2)/(2*Sqrt[a]*Sqrt[
a + b*x^2 + c*x^4])])/(32*a^(3/2)) + (c^(3/2)*ArcTanh[(b + 2*c*x^2)/(2*Sqrt[c]*S
qrt[a + b*x^2 + c*x^4])])/2

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Rubi in Sympy [A]  time = 36.1987, size = 146, normalized size = 0.9 \[ \frac{c^{\frac{3}{2}} \operatorname{atanh}{\left (\frac{b + 2 c x^{2}}{2 \sqrt{c} \sqrt{a + b x^{2} + c x^{4}}} \right )}}{2} - \frac{\left (a + b x^{2} + c x^{4}\right )^{\frac{3}{2}}}{6 x^{6}} - \frac{\left (a b + x^{2} \left (4 a c + \frac{b^{2}}{2}\right )\right ) \sqrt{a + b x^{2} + c x^{4}}}{8 a x^{4}} + \frac{b \left (- 12 a c + b^{2}\right ) \operatorname{atanh}{\left (\frac{2 a + b x^{2}}{2 \sqrt{a} \sqrt{a + b x^{2} + c x^{4}}} \right )}}{32 a^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((c*x**4+b*x**2+a)**(3/2)/x**7,x)

[Out]

c**(3/2)*atanh((b + 2*c*x**2)/(2*sqrt(c)*sqrt(a + b*x**2 + c*x**4)))/2 - (a + b*
x**2 + c*x**4)**(3/2)/(6*x**6) - (a*b + x**2*(4*a*c + b**2/2))*sqrt(a + b*x**2 +
c*x**4)/(8*a*x**4) + b*(-12*a*c + b**2)*atanh((2*a + b*x**2)/(2*sqrt(a)*sqrt(a
+ b*x**2 + c*x**4)))/(32*a**(3/2))

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Mathematica [A]  time = 0.597306, size = 169, normalized size = 1.04 \[ \frac{16 a^{3/2} c^{3/2} \log \left (2 \sqrt{c} \sqrt{a+x^2 \left (b+c x^2\right )}+b+2 c x^2\right )-\log \left (x^2\right ) \left (b^3-12 a b c\right )+\left (b^3-12 a b c\right ) \log \left (2 \sqrt{a} \sqrt{a+x^2 \left (b+c x^2\right )}+2 a+b x^2\right )}{32 a^{3/2}}+\sqrt{a+b x^2+c x^4} \left (\frac{-32 a c-3 b^2}{48 a x^2}-\frac{a}{6 x^6}-\frac{7 b}{24 x^4}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x^2 + c*x^4)^(3/2)/x^7,x]

[Out]

(-a/(6*x^6) - (7*b)/(24*x^4) + (-3*b^2 - 32*a*c)/(48*a*x^2))*Sqrt[a + b*x^2 + c*
x^4] + (-((b^3 - 12*a*b*c)*Log[x^2]) + (b^3 - 12*a*b*c)*Log[2*a + b*x^2 + 2*Sqrt
[a]*Sqrt[a + x^2*(b + c*x^2)]] + 16*a^(3/2)*c^(3/2)*Log[b + 2*c*x^2 + 2*Sqrt[c]*
Sqrt[a + x^2*(b + c*x^2)]])/(32*a^(3/2))

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Maple [A]  time = 0.025, size = 202, normalized size = 1.2 \[{\frac{1}{2}{c}^{{\frac{3}{2}}}\ln \left ({1 \left ({\frac{b}{2}}+c{x}^{2} \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{4}+b{x}^{2}+a} \right ) }-{\frac{a}{6\,{x}^{6}}\sqrt{c{x}^{4}+b{x}^{2}+a}}-{\frac{7\,b}{24\,{x}^{4}}\sqrt{c{x}^{4}+b{x}^{2}+a}}-{\frac{{b}^{2}}{16\,a{x}^{2}}\sqrt{c{x}^{4}+b{x}^{2}+a}}+{\frac{{b}^{3}}{32}\ln \left ({\frac{1}{{x}^{2}} \left ( 2\,a+b{x}^{2}+2\,\sqrt{a}\sqrt{c{x}^{4}+b{x}^{2}+a} \right ) } \right ){a}^{-{\frac{3}{2}}}}-{\frac{3\,bc}{8}\ln \left ({\frac{1}{{x}^{2}} \left ( 2\,a+b{x}^{2}+2\,\sqrt{a}\sqrt{c{x}^{4}+b{x}^{2}+a} \right ) } \right ){\frac{1}{\sqrt{a}}}}-{\frac{2\,c}{3\,{x}^{2}}\sqrt{c{x}^{4}+b{x}^{2}+a}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^4+b*x^2+a)^(3/2)/x^7,x)

[Out]

1/2*c^(3/2)*ln((1/2*b+c*x^2)/c^(1/2)+(c*x^4+b*x^2+a)^(1/2))-1/6*a/x^6*(c*x^4+b*x
^2+a)^(1/2)-7/24*b/x^4*(c*x^4+b*x^2+a)^(1/2)-1/16/a*b^2/x^2*(c*x^4+b*x^2+a)^(1/2
)+1/32/a^(3/2)*b^3*ln((2*a+b*x^2+2*a^(1/2)*(c*x^4+b*x^2+a)^(1/2))/x^2)-3/8/a^(1/
2)*b*c*ln((2*a+b*x^2+2*a^(1/2)*(c*x^4+b*x^2+a)^(1/2))/x^2)-2/3*c/x^2*(c*x^4+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((c*x^4 + b*x^2 + a)^(3/2)/x^7,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.377694, size = 1, normalized size = 0.01 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^4 + b*x^2 + a)^(3/2)/x^7,x, algorithm="fricas")

[Out]

[1/192*(48*a^(3/2)*c^(3/2)*x^6*log(-8*c^2*x^4 - 8*b*c*x^2 - b^2 - 4*sqrt(c*x^4 +
b*x^2 + a)*(2*c*x^2 + b)*sqrt(c) - 4*a*c) - 3*(b^3 - 12*a*b*c)*x^6*log((4*sqrt(
c*x^4 + b*x^2 + a)*(a*b*x^2 + 2*a^2) - ((b^2 + 4*a*c)*x^4 + 8*a*b*x^2 + 8*a^2)*s
qrt(a))/x^4) - 4*((3*b^2 + 32*a*c)*x^4 + 14*a*b*x^2 + 8*a^2)*sqrt(c*x^4 + b*x^2
+ a)*sqrt(a))/(a^(3/2)*x^6), 1/192*(96*a^(3/2)*sqrt(-c)*c*x^6*arctan(1/2*(2*c*x^
2 + b)/(sqrt(c*x^4 + b*x^2 + a)*sqrt(-c))) - 3*(b^3 - 12*a*b*c)*x^6*log((4*sqrt(
c*x^4 + b*x^2 + a)*(a*b*x^2 + 2*a^2) - ((b^2 + 4*a*c)*x^4 + 8*a*b*x^2 + 8*a^2)*s
qrt(a))/x^4) - 4*((3*b^2 + 32*a*c)*x^4 + 14*a*b*x^2 + 8*a^2)*sqrt(c*x^4 + b*x^2
+ a)*sqrt(a))/(a^(3/2)*x^6), 1/96*(24*sqrt(-a)*a*c^(3/2)*x^6*log(-8*c^2*x^4 - 8*
b*c*x^2 - b^2 - 4*sqrt(c*x^4 + b*x^2 + a)*(2*c*x^2 + b)*sqrt(c) - 4*a*c) + 3*(b^
3 - 12*a*b*c)*x^6*arctan(1/2*(b*x^2 + 2*a)*sqrt(-a)/(sqrt(c*x^4 + b*x^2 + a)*a))
- 2*((3*b^2 + 32*a*c)*x^4 + 14*a*b*x^2 + 8*a^2)*sqrt(c*x^4 + b*x^2 + a)*sqrt(-a
))/(sqrt(-a)*a*x^6), 1/96*(48*sqrt(-a)*a*sqrt(-c)*c*x^6*arctan(1/2*(2*c*x^2 + b)
/(sqrt(c*x^4 + b*x^2 + a)*sqrt(-c))) + 3*(b^3 - 12*a*b*c)*x^6*arctan(1/2*(b*x^2
+ 2*a)*sqrt(-a)/(sqrt(c*x^4 + b*x^2 + a)*a)) - 2*((3*b^2 + 32*a*c)*x^4 + 14*a*b*
x^2 + 8*a^2)*sqrt(c*x^4 + b*x^2 + a)*sqrt(-a))/(sqrt(-a)*a*x^6)]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (a + b x^{2} + c x^{4}\right )^{\frac{3}{2}}}{x^{7}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x**4+b*x**2+a)**(3/2)/x**7,x)

[Out]

Integral((a + b*x**2 + c*x**4)**(3/2)/x**7, x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (c x^{4} + b x^{2} + a\right )}^{\frac{3}{2}}}{x^{7}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^4 + b*x^2 + a)^(3/2)/x^7,x, algorithm="giac")

[Out]

integrate((c*x^4 + b*x^2 + a)^(3/2)/x^7, x)