3.241 \(\int \frac{1}{x^7 \left (a+b x^3+c x^6\right )^{3/2}} \, dx\)

Optimal. Leaf size=198 \[ -\frac{\left (5 b^2-4 a c\right ) \tanh ^{-1}\left (\frac{2 a+b x^3}{2 \sqrt{a} \sqrt{a+b x^3+c x^6}}\right )}{8 a^{7/2}}+\frac{b \left (15 b^2-52 a c\right ) \sqrt{a+b x^3+c x^6}}{12 a^3 x^3 \left (b^2-4 a c\right )}-\frac{\left (5 b^2-12 a c\right ) \sqrt{a+b x^3+c x^6}}{6 a^2 x^6 \left (b^2-4 a c\right )}+\frac{2 \left (-2 a c+b^2+b c x^3\right )}{3 a x^6 \left (b^2-4 a c\right ) \sqrt{a+b x^3+c x^6}} \]

[Out]

(2*(b^2 - 2*a*c + b*c*x^3))/(3*a*(b^2 - 4*a*c)*x^6*Sqrt[a + b*x^3 + c*x^6]) - ((
5*b^2 - 12*a*c)*Sqrt[a + b*x^3 + c*x^6])/(6*a^2*(b^2 - 4*a*c)*x^6) + (b*(15*b^2
- 52*a*c)*Sqrt[a + b*x^3 + c*x^6])/(12*a^3*(b^2 - 4*a*c)*x^3) - ((5*b^2 - 4*a*c)
*ArcTanh[(2*a + b*x^3)/(2*Sqrt[a]*Sqrt[a + b*x^3 + c*x^6])])/(8*a^(7/2))

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Rubi [A]  time = 0.495767, antiderivative size = 198, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3 \[ -\frac{\left (5 b^2-4 a c\right ) \tanh ^{-1}\left (\frac{2 a+b x^3}{2 \sqrt{a} \sqrt{a+b x^3+c x^6}}\right )}{8 a^{7/2}}+\frac{b \left (15 b^2-52 a c\right ) \sqrt{a+b x^3+c x^6}}{12 a^3 x^3 \left (b^2-4 a c\right )}-\frac{\left (5 b^2-12 a c\right ) \sqrt{a+b x^3+c x^6}}{6 a^2 x^6 \left (b^2-4 a c\right )}+\frac{2 \left (-2 a c+b^2+b c x^3\right )}{3 a x^6 \left (b^2-4 a c\right ) \sqrt{a+b x^3+c x^6}} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^7*(a + b*x^3 + c*x^6)^(3/2)),x]

[Out]

(2*(b^2 - 2*a*c + b*c*x^3))/(3*a*(b^2 - 4*a*c)*x^6*Sqrt[a + b*x^3 + c*x^6]) - ((
5*b^2 - 12*a*c)*Sqrt[a + b*x^3 + c*x^6])/(6*a^2*(b^2 - 4*a*c)*x^6) + (b*(15*b^2
- 52*a*c)*Sqrt[a + b*x^3 + c*x^6])/(12*a^3*(b^2 - 4*a*c)*x^3) - ((5*b^2 - 4*a*c)
*ArcTanh[(2*a + b*x^3)/(2*Sqrt[a]*Sqrt[a + b*x^3 + c*x^6])])/(8*a^(7/2))

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Rubi in Sympy [A]  time = 57.0319, size = 184, normalized size = 0.93 \[ \frac{2 \left (- 2 a c + b^{2} + b c x^{3}\right )}{3 a x^{6} \left (- 4 a c + b^{2}\right ) \sqrt{a + b x^{3} + c x^{6}}} - \frac{\left (- 12 a c + 5 b^{2}\right ) \sqrt{a + b x^{3} + c x^{6}}}{6 a^{2} x^{6} \left (- 4 a c + b^{2}\right )} + \frac{b \left (- 52 a c + 15 b^{2}\right ) \sqrt{a + b x^{3} + c x^{6}}}{12 a^{3} x^{3} \left (- 4 a c + b^{2}\right )} - \frac{\left (- 4 a c + 5 b^{2}\right ) \operatorname{atanh}{\left (\frac{2 a + b x^{3}}{2 \sqrt{a} \sqrt{a + b x^{3} + c x^{6}}} \right )}}{8 a^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**7/(c*x**6+b*x**3+a)**(3/2),x)

[Out]

2*(-2*a*c + b**2 + b*c*x**3)/(3*a*x**6*(-4*a*c + b**2)*sqrt(a + b*x**3 + c*x**6)
) - (-12*a*c + 5*b**2)*sqrt(a + b*x**3 + c*x**6)/(6*a**2*x**6*(-4*a*c + b**2)) +
b*(-52*a*c + 15*b**2)*sqrt(a + b*x**3 + c*x**6)/(12*a**3*x**3*(-4*a*c + b**2))
- (-4*a*c + 5*b**2)*atanh((2*a + b*x**3)/(2*sqrt(a)*sqrt(a + b*x**3 + c*x**6)))/
(8*a**(7/2))

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Mathematica [A]  time = 0.350352, size = 159, normalized size = 0.8 \[ \frac{\left (5 b^2-4 a c\right ) \left (\log \left (x^3\right )-\log \left (2 \sqrt{a} \sqrt{a+b x^3+c x^6}+2 a+b x^3\right )\right )}{8 a^{7/2}}+\frac{\sqrt{a+b x^3+c x^6} \left (\frac{8 \left (2 a^2 c^2-4 a b^2 c-3 a b c^2 x^3+b^4+b^3 c x^3\right )}{\left (b^2-4 a c\right ) \left (a+b x^3+c x^6\right )}-\frac{2 a}{x^6}+\frac{7 b}{x^3}\right )}{12 a^3} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^7*(a + b*x^3 + c*x^6)^(3/2)),x]

[Out]

(Sqrt[a + b*x^3 + c*x^6]*((-2*a)/x^6 + (7*b)/x^3 + (8*(b^4 - 4*a*b^2*c + 2*a^2*c
^2 + b^3*c*x^3 - 3*a*b*c^2*x^3))/((b^2 - 4*a*c)*(a + b*x^3 + c*x^6))))/(12*a^3)
+ ((5*b^2 - 4*a*c)*(Log[x^3] - Log[2*a + b*x^3 + 2*Sqrt[a]*Sqrt[a + b*x^3 + c*x^
6]]))/(8*a^(7/2))

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Maple [F]  time = 0.074, size = 0, normalized size = 0. \[ \int{\frac{1}{{x}^{7}} \left ( c{x}^{6}+b{x}^{3}+a \right ) ^{-{\frac{3}{2}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^7/(c*x^6+b*x^3+a)^(3/2),x)

[Out]

int(1/x^7/(c*x^6+b*x^3+a)^(3/2),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(1/((c*x^6 + b*x^3 + a)^(3/2)*x^7),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.33167, size = 1, normalized size = 0.01 \[ \left [\frac{4 \,{\left ({\left (15 \, b^{3} c - 52 \, a b c^{2}\right )} x^{9} +{\left (15 \, b^{4} - 62 \, a b^{2} c + 24 \, a^{2} c^{2}\right )} x^{6} - 2 \, a^{2} b^{2} + 8 \, a^{3} c + 5 \,{\left (a b^{3} - 4 \, a^{2} b c\right )} x^{3}\right )} \sqrt{c x^{6} + b x^{3} + a} \sqrt{a} - 3 \,{\left ({\left (5 \, b^{4} c - 24 \, a b^{2} c^{2} + 16 \, a^{2} c^{3}\right )} x^{12} +{\left (5 \, b^{5} - 24 \, a b^{3} c + 16 \, a^{2} b c^{2}\right )} x^{9} +{\left (5 \, a b^{4} - 24 \, a^{2} b^{2} c + 16 \, a^{3} c^{2}\right )} x^{6}\right )} \log \left (-\frac{4 \, \sqrt{c x^{6} + b x^{3} + a}{\left (a b x^{3} + 2 \, a^{2}\right )} +{\left ({\left (b^{2} + 4 \, a c\right )} x^{6} + 8 \, a b x^{3} + 8 \, a^{2}\right )} \sqrt{a}}{x^{6}}\right )}{48 \,{\left ({\left (a^{3} b^{2} c - 4 \, a^{4} c^{2}\right )} x^{12} +{\left (a^{3} b^{3} - 4 \, a^{4} b c\right )} x^{9} +{\left (a^{4} b^{2} - 4 \, a^{5} c\right )} x^{6}\right )} \sqrt{a}}, \frac{2 \,{\left ({\left (15 \, b^{3} c - 52 \, a b c^{2}\right )} x^{9} +{\left (15 \, b^{4} - 62 \, a b^{2} c + 24 \, a^{2} c^{2}\right )} x^{6} - 2 \, a^{2} b^{2} + 8 \, a^{3} c + 5 \,{\left (a b^{3} - 4 \, a^{2} b c\right )} x^{3}\right )} \sqrt{c x^{6} + b x^{3} + a} \sqrt{-a} - 3 \,{\left ({\left (5 \, b^{4} c - 24 \, a b^{2} c^{2} + 16 \, a^{2} c^{3}\right )} x^{12} +{\left (5 \, b^{5} - 24 \, a b^{3} c + 16 \, a^{2} b c^{2}\right )} x^{9} +{\left (5 \, a b^{4} - 24 \, a^{2} b^{2} c + 16 \, a^{3} c^{2}\right )} x^{6}\right )} \arctan \left (\frac{{\left (b x^{3} + 2 \, a\right )} \sqrt{-a}}{2 \, \sqrt{c x^{6} + b x^{3} + a} a}\right )}{24 \,{\left ({\left (a^{3} b^{2} c - 4 \, a^{4} c^{2}\right )} x^{12} +{\left (a^{3} b^{3} - 4 \, a^{4} b c\right )} x^{9} +{\left (a^{4} b^{2} - 4 \, a^{5} c\right )} x^{6}\right )} \sqrt{-a}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((c*x^6 + b*x^3 + a)^(3/2)*x^7),x, algorithm="fricas")

[Out]

[1/48*(4*((15*b^3*c - 52*a*b*c^2)*x^9 + (15*b^4 - 62*a*b^2*c + 24*a^2*c^2)*x^6 -
2*a^2*b^2 + 8*a^3*c + 5*(a*b^3 - 4*a^2*b*c)*x^3)*sqrt(c*x^6 + b*x^3 + a)*sqrt(a
) - 3*((5*b^4*c - 24*a*b^2*c^2 + 16*a^2*c^3)*x^12 + (5*b^5 - 24*a*b^3*c + 16*a^2
*b*c^2)*x^9 + (5*a*b^4 - 24*a^2*b^2*c + 16*a^3*c^2)*x^6)*log(-(4*sqrt(c*x^6 + b*
x^3 + a)*(a*b*x^3 + 2*a^2) + ((b^2 + 4*a*c)*x^6 + 8*a*b*x^3 + 8*a^2)*sqrt(a))/x^
6))/(((a^3*b^2*c - 4*a^4*c^2)*x^12 + (a^3*b^3 - 4*a^4*b*c)*x^9 + (a^4*b^2 - 4*a^
5*c)*x^6)*sqrt(a)), 1/24*(2*((15*b^3*c - 52*a*b*c^2)*x^9 + (15*b^4 - 62*a*b^2*c
+ 24*a^2*c^2)*x^6 - 2*a^2*b^2 + 8*a^3*c + 5*(a*b^3 - 4*a^2*b*c)*x^3)*sqrt(c*x^6
+ b*x^3 + a)*sqrt(-a) - 3*((5*b^4*c - 24*a*b^2*c^2 + 16*a^2*c^3)*x^12 + (5*b^5 -
24*a*b^3*c + 16*a^2*b*c^2)*x^9 + (5*a*b^4 - 24*a^2*b^2*c + 16*a^3*c^2)*x^6)*arc
tan(1/2*(b*x^3 + 2*a)*sqrt(-a)/(sqrt(c*x^6 + b*x^3 + a)*a)))/(((a^3*b^2*c - 4*a^
4*c^2)*x^12 + (a^3*b^3 - 4*a^4*b*c)*x^9 + (a^4*b^2 - 4*a^5*c)*x^6)*sqrt(-a))]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{x^{7} \left (a + b x^{3} + c x^{6}\right )^{\frac{3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**7/(c*x**6+b*x**3+a)**(3/2),x)

[Out]

Integral(1/(x**7*(a + b*x**3 + c*x**6)**(3/2)), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (c x^{6} + b x^{3} + a\right )}^{\frac{3}{2}} x^{7}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((c*x^6 + b*x^3 + a)^(3/2)*x^7),x, algorithm="giac")

[Out]

integrate(1/((c*x^6 + b*x^3 + a)^(3/2)*x^7), x)