3.2360 \(\int \frac{x^6}{\sqrt{a+b x+c x^2}} \, dx\)

Optimal. Leaf size=261 \[ -\frac{\left (7 b \left (528 a^2 c^2-680 a b^2 c+165 b^4\right )-2 c x \left (400 a^2 c^2-1176 a b^2 c+385 b^4\right )\right ) \sqrt{a+b x+c x^2}}{2560 c^6}+\frac{\left (-320 a^3 c^3+1680 a^2 b^2 c^2-1260 a b^4 c+231 b^6\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{1024 c^{13/2}}-\frac{b x^2 \left (77 b^2-156 a c\right ) \sqrt{a+b x+c x^2}}{320 c^4}+\frac{x^3 \left (99 b^2-100 a c\right ) \sqrt{a+b x+c x^2}}{480 c^3}-\frac{11 b x^4 \sqrt{a+b x+c x^2}}{60 c^2}+\frac{x^5 \sqrt{a+b x+c x^2}}{6 c} \]

[Out]

-(b*(77*b^2 - 156*a*c)*x^2*Sqrt[a + b*x + c*x^2])/(320*c^4) + ((99*b^2 - 100*a*c
)*x^3*Sqrt[a + b*x + c*x^2])/(480*c^3) - (11*b*x^4*Sqrt[a + b*x + c*x^2])/(60*c^
2) + (x^5*Sqrt[a + b*x + c*x^2])/(6*c) - ((7*b*(165*b^4 - 680*a*b^2*c + 528*a^2*
c^2) - 2*c*(385*b^4 - 1176*a*b^2*c + 400*a^2*c^2)*x)*Sqrt[a + b*x + c*x^2])/(256
0*c^6) + ((231*b^6 - 1260*a*b^4*c + 1680*a^2*b^2*c^2 - 320*a^3*c^3)*ArcTanh[(b +
2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/(1024*c^(13/2))

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Rubi [A]  time = 0.989215, antiderivative size = 261, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.278 \[ -\frac{\left (7 b \left (528 a^2 c^2-680 a b^2 c+165 b^4\right )-2 c x \left (400 a^2 c^2-1176 a b^2 c+385 b^4\right )\right ) \sqrt{a+b x+c x^2}}{2560 c^6}+\frac{\left (-320 a^3 c^3+1680 a^2 b^2 c^2-1260 a b^4 c+231 b^6\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{1024 c^{13/2}}-\frac{b x^2 \left (77 b^2-156 a c\right ) \sqrt{a+b x+c x^2}}{320 c^4}+\frac{x^3 \left (99 b^2-100 a c\right ) \sqrt{a+b x+c x^2}}{480 c^3}-\frac{11 b x^4 \sqrt{a+b x+c x^2}}{60 c^2}+\frac{x^5 \sqrt{a+b x+c x^2}}{6 c} \]

Antiderivative was successfully verified.

[In]  Int[x^6/Sqrt[a + b*x + c*x^2],x]

[Out]

-(b*(77*b^2 - 156*a*c)*x^2*Sqrt[a + b*x + c*x^2])/(320*c^4) + ((99*b^2 - 100*a*c
)*x^3*Sqrt[a + b*x + c*x^2])/(480*c^3) - (11*b*x^4*Sqrt[a + b*x + c*x^2])/(60*c^
2) + (x^5*Sqrt[a + b*x + c*x^2])/(6*c) - ((7*b*(165*b^4 - 680*a*b^2*c + 528*a^2*
c^2) - 2*c*(385*b^4 - 1176*a*b^2*c + 400*a^2*c^2)*x)*Sqrt[a + b*x + c*x^2])/(256
0*c^6) + ((231*b^6 - 1260*a*b^4*c + 1680*a^2*b^2*c^2 - 320*a^3*c^3)*ArcTanh[(b +
2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/(1024*c^(13/2))

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Rubi in Sympy [A]  time = 83.7256, size = 255, normalized size = 0.98 \[ - \frac{11 b x^{4} \sqrt{a + b x + c x^{2}}}{60 c^{2}} - \frac{b x^{2} \left (- 156 a c + 77 b^{2}\right ) \sqrt{a + b x + c x^{2}}}{320 c^{4}} + \frac{x^{5} \sqrt{a + b x + c x^{2}}}{6 c} + \frac{x^{3} \left (- 100 a c + 99 b^{2}\right ) \sqrt{a + b x + c x^{2}}}{480 c^{3}} - \frac{\left (\frac{63 b \left (528 a^{2} c^{2} - 680 a b^{2} c + 165 b^{4}\right )}{32} - \frac{9 c x \left (400 a^{2} c^{2} - 1176 a b^{2} c + 385 b^{4}\right )}{16}\right ) \sqrt{a + b x + c x^{2}}}{720 c^{6}} + \frac{\left (- 320 a^{3} c^{3} + 1680 a^{2} b^{2} c^{2} - 1260 a b^{4} c + 231 b^{6}\right ) \operatorname{atanh}{\left (\frac{b + 2 c x}{2 \sqrt{c} \sqrt{a + b x + c x^{2}}} \right )}}{1024 c^{\frac{13}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-11*b*x**4*sqrt(a + b*x + c*x**2)/(60*c**2) - b*x**2*(-156*a*c + 77*b**2)*sqrt(a
+ b*x + c*x**2)/(320*c**4) + x**5*sqrt(a + b*x + c*x**2)/(6*c) + x**3*(-100*a*c
+ 99*b**2)*sqrt(a + b*x + c*x**2)/(480*c**3) - (63*b*(528*a**2*c**2 - 680*a*b**
2*c + 165*b**4)/32 - 9*c*x*(400*a**2*c**2 - 1176*a*b**2*c + 385*b**4)/16)*sqrt(a
+ b*x + c*x**2)/(720*c**6) + (-320*a**3*c**3 + 1680*a**2*b**2*c**2 - 1260*a*b**
4*c + 231*b**6)*atanh((b + 2*c*x)/(2*sqrt(c)*sqrt(a + b*x + c*x**2)))/(1024*c**(
13/2))

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Mathematica [A]  time = 0.571494, size = 193, normalized size = 0.74 \[ \frac{2 \sqrt{c} \sqrt{a+x (b+c x)} \left (-16 b c^2 \left (693 a^2-234 a c x^2+88 c^2 x^4\right )+160 c^3 x \left (15 a^2-10 a c x^2+8 c^2 x^4\right )+168 b^3 c \left (85 a-11 c x^2\right )+144 b^2 c^2 x \left (11 c x^2-49 a\right )-3465 b^5+2310 b^4 c x\right )+15 \left (-320 a^3 c^3+1680 a^2 b^2 c^2-1260 a b^4 c+231 b^6\right ) \log \left (2 \sqrt{c} \sqrt{a+x (b+c x)}+b+2 c x\right )}{15360 c^{13/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^6/Sqrt[a + b*x + c*x^2],x]

[Out]

(2*Sqrt[c]*Sqrt[a + x*(b + c*x)]*(-3465*b^5 + 2310*b^4*c*x + 168*b^3*c*(85*a - 1
1*c*x^2) + 144*b^2*c^2*x*(-49*a + 11*c*x^2) + 160*c^3*x*(15*a^2 - 10*a*c*x^2 + 8
*c^2*x^4) - 16*b*c^2*(693*a^2 - 234*a*c*x^2 + 88*c^2*x^4)) + 15*(231*b^6 - 1260*
a*b^4*c + 1680*a^2*b^2*c^2 - 320*a^3*c^3)*Log[b + 2*c*x + 2*Sqrt[c]*Sqrt[a + x*(
b + c*x)]])/(15360*c^(13/2))

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Maple [A]  time = 0.018, size = 394, normalized size = 1.5 \[{\frac{{x}^{5}}{6\,c}\sqrt{c{x}^{2}+bx+a}}-{\frac{11\,b{x}^{4}}{60\,{c}^{2}}\sqrt{c{x}^{2}+bx+a}}+{\frac{33\,{b}^{2}{x}^{3}}{160\,{c}^{3}}\sqrt{c{x}^{2}+bx+a}}-{\frac{77\,{b}^{3}{x}^{2}}{320\,{c}^{4}}\sqrt{c{x}^{2}+bx+a}}+{\frac{77\,{b}^{4}x}{256\,{c}^{5}}\sqrt{c{x}^{2}+bx+a}}-{\frac{231\,{b}^{5}}{512\,{c}^{6}}\sqrt{c{x}^{2}+bx+a}}+{\frac{231\,{b}^{6}}{1024}\ln \left ({1 \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx+a} \right ){c}^{-{\frac{13}{2}}}}-{\frac{315\,a{b}^{4}}{256}\ln \left ({1 \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx+a} \right ){c}^{-{\frac{11}{2}}}}+{\frac{119\,a{b}^{3}}{64\,{c}^{5}}\sqrt{c{x}^{2}+bx+a}}-{\frac{147\,a{b}^{2}x}{160\,{c}^{4}}\sqrt{c{x}^{2}+bx+a}}+{\frac{105\,{a}^{2}{b}^{2}}{64}\ln \left ({1 \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx+a} \right ){c}^{-{\frac{9}{2}}}}+{\frac{39\,ab{x}^{2}}{80\,{c}^{3}}\sqrt{c{x}^{2}+bx+a}}-{\frac{231\,{a}^{2}b}{160\,{c}^{4}}\sqrt{c{x}^{2}+bx+a}}-{\frac{5\,a{x}^{3}}{24\,{c}^{2}}\sqrt{c{x}^{2}+bx+a}}+{\frac{5\,{a}^{2}x}{16\,{c}^{3}}\sqrt{c{x}^{2}+bx+a}}-{\frac{5\,{a}^{3}}{16}\ln \left ({1 \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx+a} \right ){c}^{-{\frac{7}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^6/(c*x^2+b*x+a)^(1/2),x)

[Out]

1/6*x^5*(c*x^2+b*x+a)^(1/2)/c-11/60*b*x^4*(c*x^2+b*x+a)^(1/2)/c^2+33/160*b^2/c^3
*x^3*(c*x^2+b*x+a)^(1/2)-77/320*b^3/c^4*x^2*(c*x^2+b*x+a)^(1/2)+77/256*b^4/c^5*x
*(c*x^2+b*x+a)^(1/2)-231/512*b^5/c^6*(c*x^2+b*x+a)^(1/2)+231/1024*b^6/c^(13/2)*l
n((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))-315/256*b^4/c^(11/2)*a*ln((1/2*b+c*x)
/c^(1/2)+(c*x^2+b*x+a)^(1/2))+119/64*b^3/c^5*a*(c*x^2+b*x+a)^(1/2)-147/160*b^2/c
^4*a*x*(c*x^2+b*x+a)^(1/2)+105/64*b^2/c^(9/2)*a^2*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+
b*x+a)^(1/2))+39/80*b/c^3*a*x^2*(c*x^2+b*x+a)^(1/2)-231/160*b/c^4*a^2*(c*x^2+b*x
+a)^(1/2)-5/24*a/c^2*x^3*(c*x^2+b*x+a)^(1/2)+5/16*a^2/c^3*x*(c*x^2+b*x+a)^(1/2)-
5/16*a^3/c^(7/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+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(x^6/sqrt(c*x^2 + b*x + a),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.281067, size = 1, normalized size = 0. \[ \left [\frac{4 \,{\left (1280 \, c^{5} x^{5} - 1408 \, b c^{4} x^{4} - 3465 \, b^{5} + 14280 \, a b^{3} c - 11088 \, a^{2} b c^{2} + 16 \,{\left (99 \, b^{2} c^{3} - 100 \, a c^{4}\right )} x^{3} - 24 \,{\left (77 \, b^{3} c^{2} - 156 \, a b c^{3}\right )} x^{2} + 6 \,{\left (385 \, b^{4} c - 1176 \, a b^{2} c^{2} + 400 \, a^{2} c^{3}\right )} x\right )} \sqrt{c x^{2} + b x + a} \sqrt{c} - 15 \,{\left (231 \, b^{6} - 1260 \, a b^{4} c + 1680 \, a^{2} b^{2} c^{2} - 320 \, a^{3} c^{3}\right )} \log \left (4 \,{\left (2 \, c^{2} x + b c\right )} \sqrt{c x^{2} + b x + a} -{\left (8 \, c^{2} x^{2} + 8 \, b c x + b^{2} + 4 \, a c\right )} \sqrt{c}\right )}{30720 \, c^{\frac{13}{2}}}, \frac{2 \,{\left (1280 \, c^{5} x^{5} - 1408 \, b c^{4} x^{4} - 3465 \, b^{5} + 14280 \, a b^{3} c - 11088 \, a^{2} b c^{2} + 16 \,{\left (99 \, b^{2} c^{3} - 100 \, a c^{4}\right )} x^{3} - 24 \,{\left (77 \, b^{3} c^{2} - 156 \, a b c^{3}\right )} x^{2} + 6 \,{\left (385 \, b^{4} c - 1176 \, a b^{2} c^{2} + 400 \, a^{2} c^{3}\right )} x\right )} \sqrt{c x^{2} + b x + a} \sqrt{-c} + 15 \,{\left (231 \, b^{6} - 1260 \, a b^{4} c + 1680 \, a^{2} b^{2} c^{2} - 320 \, a^{3} c^{3}\right )} \arctan \left (\frac{{\left (2 \, c x + b\right )} \sqrt{-c}}{2 \, \sqrt{c x^{2} + b x + a} c}\right )}{15360 \, \sqrt{-c} c^{6}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/30720*(4*(1280*c^5*x^5 - 1408*b*c^4*x^4 - 3465*b^5 + 14280*a*b^3*c - 11088*a^
2*b*c^2 + 16*(99*b^2*c^3 - 100*a*c^4)*x^3 - 24*(77*b^3*c^2 - 156*a*b*c^3)*x^2 +
6*(385*b^4*c - 1176*a*b^2*c^2 + 400*a^2*c^3)*x)*sqrt(c*x^2 + b*x + a)*sqrt(c) -
15*(231*b^6 - 1260*a*b^4*c + 1680*a^2*b^2*c^2 - 320*a^3*c^3)*log(4*(2*c^2*x + b*
c)*sqrt(c*x^2 + b*x + a) - (8*c^2*x^2 + 8*b*c*x + b^2 + 4*a*c)*sqrt(c)))/c^(13/2
), 1/15360*(2*(1280*c^5*x^5 - 1408*b*c^4*x^4 - 3465*b^5 + 14280*a*b^3*c - 11088*
a^2*b*c^2 + 16*(99*b^2*c^3 - 100*a*c^4)*x^3 - 24*(77*b^3*c^2 - 156*a*b*c^3)*x^2
+ 6*(385*b^4*c - 1176*a*b^2*c^2 + 400*a^2*c^3)*x)*sqrt(c*x^2 + b*x + a)*sqrt(-c)
+ 15*(231*b^6 - 1260*a*b^4*c + 1680*a^2*b^2*c^2 - 320*a^3*c^3)*arctan(1/2*(2*c*
x + b)*sqrt(-c)/(sqrt(c*x^2 + b*x + a)*c)))/(sqrt(-c)*c^6)]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{6}}{\sqrt{a + b x + c x^{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.225052, size = 281, normalized size = 1.08 \[ \frac{1}{7680} \, \sqrt{c x^{2} + b x + a}{\left (2 \,{\left (4 \,{\left (2 \,{\left (8 \, x{\left (\frac{10 \, x}{c} - \frac{11 \, b}{c^{2}}\right )} + \frac{99 \, b^{2} c^{3} - 100 \, a c^{4}}{c^{6}}\right )} x - \frac{3 \,{\left (77 \, b^{3} c^{2} - 156 \, a b c^{3}\right )}}{c^{6}}\right )} x + \frac{3 \,{\left (385 \, b^{4} c - 1176 \, a b^{2} c^{2} + 400 \, a^{2} c^{3}\right )}}{c^{6}}\right )} x - \frac{21 \,{\left (165 \, b^{5} - 680 \, a b^{3} c + 528 \, a^{2} b c^{2}\right )}}{c^{6}}\right )} - \frac{{\left (231 \, b^{6} - 1260 \, a b^{4} c + 1680 \, a^{2} b^{2} c^{2} - 320 \, a^{3} c^{3}\right )}{\rm ln}\left ({\left | -2 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )} \sqrt{c} - b \right |}\right )}{1024 \, c^{\frac{13}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/7680*sqrt(c*x^2 + b*x + a)*(2*(4*(2*(8*x*(10*x/c - 11*b/c^2) + (99*b^2*c^3 - 1
00*a*c^4)/c^6)*x - 3*(77*b^3*c^2 - 156*a*b*c^3)/c^6)*x + 3*(385*b^4*c - 1176*a*b
^2*c^2 + 400*a^2*c^3)/c^6)*x - 21*(165*b^5 - 680*a*b^3*c + 528*a^2*b*c^2)/c^6) -
1/1024*(231*b^6 - 1260*a*b^4*c + 1680*a^2*b^2*c^2 - 320*a^3*c^3)*ln(abs(-2*(sqr
t(c)*x - sqrt(c*x^2 + b*x + a))*sqrt(c) - b))/c^(13/2)