3.212 \(\int \frac{\left (a+b x^3+c x^6\right )^{3/2}}{x^{19}} \, dx\)

Optimal. Leaf size=216 \[ -\frac{\left (b^2-4 a c\right )^2 \left (7 b^2-4 a c\right ) \tanh ^{-1}\left (\frac{2 a+b x^3}{2 \sqrt{a} \sqrt{a+b x^3+c x^6}}\right )}{3072 a^{9/2}}+\frac{\left (b^2-4 a c\right ) \left (7 b^2-4 a c\right ) \left (2 a+b x^3\right ) \sqrt{a+b x^3+c x^6}}{1536 a^4 x^6}-\frac{\left (7 b^2-4 a c\right ) \left (2 a+b x^3\right ) \left (a+b x^3+c x^6\right )^{3/2}}{576 a^3 x^{12}}+\frac{7 b \left (a+b x^3+c x^6\right )^{5/2}}{180 a^2 x^{15}}-\frac{\left (a+b x^3+c x^6\right )^{5/2}}{18 a x^{18}} \]

[Out]

((b^2 - 4*a*c)*(7*b^2 - 4*a*c)*(2*a + b*x^3)*Sqrt[a + b*x^3 + c*x^6])/(1536*a^4*
x^6) - ((7*b^2 - 4*a*c)*(2*a + b*x^3)*(a + b*x^3 + c*x^6)^(3/2))/(576*a^3*x^12)
- (a + b*x^3 + c*x^6)^(5/2)/(18*a*x^18) + (7*b*(a + b*x^3 + c*x^6)^(5/2))/(180*a
^2*x^15) - ((b^2 - 4*a*c)^2*(7*b^2 - 4*a*c)*ArcTanh[(2*a + b*x^3)/(2*Sqrt[a]*Sqr
t[a + b*x^3 + c*x^6])])/(3072*a^(9/2))

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Rubi [A]  time = 0.452807, antiderivative size = 216, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3 \[ -\frac{\left (b^2-4 a c\right )^2 \left (7 b^2-4 a c\right ) \tanh ^{-1}\left (\frac{2 a+b x^3}{2 \sqrt{a} \sqrt{a+b x^3+c x^6}}\right )}{3072 a^{9/2}}+\frac{\left (b^2-4 a c\right ) \left (7 b^2-4 a c\right ) \left (2 a+b x^3\right ) \sqrt{a+b x^3+c x^6}}{1536 a^4 x^6}-\frac{\left (7 b^2-4 a c\right ) \left (2 a+b x^3\right ) \left (a+b x^3+c x^6\right )^{3/2}}{576 a^3 x^{12}}+\frac{7 b \left (a+b x^3+c x^6\right )^{5/2}}{180 a^2 x^{15}}-\frac{\left (a+b x^3+c x^6\right )^{5/2}}{18 a x^{18}} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^3 + c*x^6)^(3/2)/x^19,x]

[Out]

((b^2 - 4*a*c)*(7*b^2 - 4*a*c)*(2*a + b*x^3)*Sqrt[a + b*x^3 + c*x^6])/(1536*a^4*
x^6) - ((7*b^2 - 4*a*c)*(2*a + b*x^3)*(a + b*x^3 + c*x^6)^(3/2))/(576*a^3*x^12)
- (a + b*x^3 + c*x^6)^(5/2)/(18*a*x^18) + (7*b*(a + b*x^3 + c*x^6)^(5/2))/(180*a
^2*x^15) - ((b^2 - 4*a*c)^2*(7*b^2 - 4*a*c)*ArcTanh[(2*a + b*x^3)/(2*Sqrt[a]*Sqr
t[a + b*x^3 + c*x^6])])/(3072*a^(9/2))

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Rubi in Sympy [A]  time = 48.431, size = 201, normalized size = 0.93 \[ - \frac{\left (a + b x^{3} + c x^{6}\right )^{\frac{5}{2}}}{18 a x^{18}} + \frac{7 b \left (a + b x^{3} + c x^{6}\right )^{\frac{5}{2}}}{180 a^{2} x^{15}} - \frac{\left (2 a + b x^{3}\right ) \left (- 4 a c + 7 b^{2}\right ) \left (a + b x^{3} + c x^{6}\right )^{\frac{3}{2}}}{576 a^{3} x^{12}} + \frac{\left (2 a + b x^{3}\right ) \left (- 4 a c + b^{2}\right ) \left (- 4 a c + 7 b^{2}\right ) \sqrt{a + b x^{3} + c x^{6}}}{1536 a^{4} x^{6}} - \frac{\left (- 4 a c + b^{2}\right )^{2} \left (- 4 a c + 7 b^{2}\right ) \operatorname{atanh}{\left (\frac{2 a + b x^{3}}{2 \sqrt{a} \sqrt{a + b x^{3} + c x^{6}}} \right )}}{3072 a^{\frac{9}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((c*x**6+b*x**3+a)**(3/2)/x**19,x)

[Out]

-(a + b*x**3 + c*x**6)**(5/2)/(18*a*x**18) + 7*b*(a + b*x**3 + c*x**6)**(5/2)/(1
80*a**2*x**15) - (2*a + b*x**3)*(-4*a*c + 7*b**2)*(a + b*x**3 + c*x**6)**(3/2)/(
576*a**3*x**12) + (2*a + b*x**3)*(-4*a*c + b**2)*(-4*a*c + 7*b**2)*sqrt(a + b*x*
*3 + c*x**6)/(1536*a**4*x**6) - (-4*a*c + b**2)**2*(-4*a*c + 7*b**2)*atanh((2*a
+ b*x**3)/(2*sqrt(a)*sqrt(a + b*x**3 + c*x**6)))/(3072*a**(9/2))

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Mathematica [A]  time = 0.352849, size = 206, normalized size = 0.95 \[ \frac{15 \left (b^2-4 a c\right )^2 \left (7 b^2-4 a c\right ) \left (\log \left (x^3\right )-\log \left (2 \sqrt{a} \sqrt{a+b x^3+c x^6}+2 a+b x^3\right )\right )-\frac{2 \sqrt{a} \sqrt{a+b x^3+c x^6} \left (1280 a^5+64 a^4 \left (26 b x^3+35 c x^6\right )+48 a^3 x^6 \left (b^2+6 b c x^3+10 c^2 x^6\right )-8 a^2 b x^9 \left (7 b^2+54 b c x^3+162 c^2 x^6\right )+10 a b^3 x^{12} \left (7 b+76 c x^3\right )-105 b^5 x^{15}\right )}{x^{18}}}{46080 a^{9/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x^3 + c*x^6)^(3/2)/x^19,x]

[Out]

((-2*Sqrt[a]*Sqrt[a + b*x^3 + c*x^6]*(1280*a^5 - 105*b^5*x^15 + 10*a*b^3*x^12*(7
*b + 76*c*x^3) + 64*a^4*(26*b*x^3 + 35*c*x^6) + 48*a^3*x^6*(b^2 + 6*b*c*x^3 + 10
*c^2*x^6) - 8*a^2*b*x^9*(7*b^2 + 54*b*c*x^3 + 162*c^2*x^6)))/x^18 + 15*(b^2 - 4*
a*c)^2*(7*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]]))/(46080*a^(9/2))

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Maple [F]  time = 0.089, size = 0, normalized size = 0. \[ \int{\frac{1}{{x}^{19}} \left ( c{x}^{6}+b{x}^{3}+a \right ) ^{{\frac{3}{2}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^6+b*x^3+a)^(3/2)/x^19,x)

[Out]

int((c*x^6+b*x^3+a)^(3/2)/x^19,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((c*x^6 + b*x^3 + a)^(3/2)/x^19,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.337772, size = 1, normalized size = 0. \[ \left [-\frac{15 \,{\left (7 \, b^{6} - 60 \, a b^{4} c + 144 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}\right )} x^{18} \log \left (-\frac{4 \, \sqrt{c x^{6} + b x^{3} + a}{\left (a b x^{3} + 2 \, a^{2}\right )} +{\left ({\left (b^{2} + 4 \, a c\right )} x^{6} + 8 \, a b x^{3} + 8 \, a^{2}\right )} \sqrt{a}}{x^{6}}\right ) - 4 \,{\left ({\left (105 \, b^{5} - 760 \, a b^{3} c + 1296 \, a^{2} b c^{2}\right )} x^{15} - 2 \,{\left (35 \, a b^{4} - 216 \, a^{2} b^{2} c + 240 \, a^{3} c^{2}\right )} x^{12} + 8 \,{\left (7 \, a^{2} b^{3} - 36 \, a^{3} b c\right )} x^{9} - 1664 \, a^{4} b x^{3} - 16 \,{\left (3 \, a^{3} b^{2} + 140 \, a^{4} c\right )} x^{6} - 1280 \, a^{5}\right )} \sqrt{c x^{6} + b x^{3} + a} \sqrt{a}}{92160 \, a^{\frac{9}{2}} x^{18}}, -\frac{15 \,{\left (7 \, b^{6} - 60 \, a b^{4} c + 144 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}\right )} x^{18} \arctan \left (\frac{{\left (b x^{3} + 2 \, a\right )} \sqrt{-a}}{2 \, \sqrt{c x^{6} + b x^{3} + a} a}\right ) - 2 \,{\left ({\left (105 \, b^{5} - 760 \, a b^{3} c + 1296 \, a^{2} b c^{2}\right )} x^{15} - 2 \,{\left (35 \, a b^{4} - 216 \, a^{2} b^{2} c + 240 \, a^{3} c^{2}\right )} x^{12} + 8 \,{\left (7 \, a^{2} b^{3} - 36 \, a^{3} b c\right )} x^{9} - 1664 \, a^{4} b x^{3} - 16 \,{\left (3 \, a^{3} b^{2} + 140 \, a^{4} c\right )} x^{6} - 1280 \, a^{5}\right )} \sqrt{c x^{6} + b x^{3} + a} \sqrt{-a}}{46080 \, \sqrt{-a} a^{4} x^{18}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^6 + b*x^3 + a)^(3/2)/x^19,x, algorithm="fricas")

[Out]

[-1/92160*(15*(7*b^6 - 60*a*b^4*c + 144*a^2*b^2*c^2 - 64*a^3*c^3)*x^18*log(-(4*s
qrt(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) - 4*((105*b^5 - 760*a*b^3*c + 1296*a^2*b*c^2)*x^15 - 2*(35*a*b^
4 - 216*a^2*b^2*c + 240*a^3*c^2)*x^12 + 8*(7*a^2*b^3 - 36*a^3*b*c)*x^9 - 1664*a^
4*b*x^3 - 16*(3*a^3*b^2 + 140*a^4*c)*x^6 - 1280*a^5)*sqrt(c*x^6 + b*x^3 + a)*sqr
t(a))/(a^(9/2)*x^18), -1/46080*(15*(7*b^6 - 60*a*b^4*c + 144*a^2*b^2*c^2 - 64*a^
3*c^3)*x^18*arctan(1/2*(b*x^3 + 2*a)*sqrt(-a)/(sqrt(c*x^6 + b*x^3 + a)*a)) - 2*(
(105*b^5 - 760*a*b^3*c + 1296*a^2*b*c^2)*x^15 - 2*(35*a*b^4 - 216*a^2*b^2*c + 24
0*a^3*c^2)*x^12 + 8*(7*a^2*b^3 - 36*a^3*b*c)*x^9 - 1664*a^4*b*x^3 - 16*(3*a^3*b^
2 + 140*a^4*c)*x^6 - 1280*a^5)*sqrt(c*x^6 + b*x^3 + a)*sqrt(-a))/(sqrt(-a)*a^4*x
^18)]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (a + b x^{3} + c x^{6}\right )^{\frac{3}{2}}}{x^{19}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x**6+b*x**3+a)**(3/2)/x**19,x)

[Out]

Integral((a + b*x**3 + c*x**6)**(3/2)/x**19, x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (c x^{6} + b x^{3} + a\right )}^{\frac{3}{2}}}{x^{19}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^6 + b*x^3 + a)^(3/2)/x^19,x, algorithm="giac")

[Out]

integrate((c*x^6 + b*x^3 + a)^(3/2)/x^19, x)