3.2361 \(\int \frac{x^5}{\sqrt{a+b x+c x^2}} \, dx\)

Optimal. Leaf size=202 \[ -\frac{b \left (240 a^2 c^2-280 a b^2 c+63 b^4\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{256 c^{11/2}}+\frac{\left (1024 a^2 c^2-14 b c x \left (45 b^2-92 a c\right )-2940 a b^2 c+945 b^4\right ) \sqrt{a+b x+c x^2}}{1920 c^5}+\frac{x^2 \left (63 b^2-64 a c\right ) \sqrt{a+b x+c x^2}}{240 c^3}-\frac{9 b x^3 \sqrt{a+b x+c x^2}}{40 c^2}+\frac{x^4 \sqrt{a+b x+c x^2}}{5 c} \]

[Out]

((63*b^2 - 64*a*c)*x^2*Sqrt[a + b*x + c*x^2])/(240*c^3) - (9*b*x^3*Sqrt[a + b*x
+ c*x^2])/(40*c^2) + (x^4*Sqrt[a + b*x + c*x^2])/(5*c) + ((945*b^4 - 2940*a*b^2*
c + 1024*a^2*c^2 - 14*b*c*(45*b^2 - 92*a*c)*x)*Sqrt[a + b*x + c*x^2])/(1920*c^5)
- (b*(63*b^4 - 280*a*b^2*c + 240*a^2*c^2)*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a
+ b*x + c*x^2])])/(256*c^(11/2))

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Rubi [A]  time = 0.615902, antiderivative size = 202, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.278 \[ -\frac{b \left (240 a^2 c^2-280 a b^2 c+63 b^4\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{256 c^{11/2}}+\frac{\left (1024 a^2 c^2-14 b c x \left (45 b^2-92 a c\right )-2940 a b^2 c+945 b^4\right ) \sqrt{a+b x+c x^2}}{1920 c^5}+\frac{x^2 \left (63 b^2-64 a c\right ) \sqrt{a+b x+c x^2}}{240 c^3}-\frac{9 b x^3 \sqrt{a+b x+c x^2}}{40 c^2}+\frac{x^4 \sqrt{a+b x+c x^2}}{5 c} \]

Antiderivative was successfully verified.

[In]  Int[x^5/Sqrt[a + b*x + c*x^2],x]

[Out]

((63*b^2 - 64*a*c)*x^2*Sqrt[a + b*x + c*x^2])/(240*c^3) - (9*b*x^3*Sqrt[a + b*x
+ c*x^2])/(40*c^2) + (x^4*Sqrt[a + b*x + c*x^2])/(5*c) + ((945*b^4 - 2940*a*b^2*
c + 1024*a^2*c^2 - 14*b*c*(45*b^2 - 92*a*c)*x)*Sqrt[a + b*x + c*x^2])/(1920*c^5)
- (b*(63*b^4 - 280*a*b^2*c + 240*a^2*c^2)*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a
+ b*x + c*x^2])])/(256*c^(11/2))

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Rubi in Sympy [A]  time = 58.1052, size = 199, normalized size = 0.99 \[ - \frac{9 b x^{3} \sqrt{a + b x + c x^{2}}}{40 c^{2}} - \frac{b \left (240 a^{2} c^{2} - 280 a b^{2} c + 63 b^{4}\right ) \operatorname{atanh}{\left (\frac{b + 2 c x}{2 \sqrt{c} \sqrt{a + b x + c x^{2}}} \right )}}{256 c^{\frac{11}{2}}} + \frac{x^{4} \sqrt{a + b x + c x^{2}}}{5 c} + \frac{x^{2} \left (- 64 a c + 63 b^{2}\right ) \sqrt{a + b x + c x^{2}}}{240 c^{3}} + \frac{\sqrt{a + b x + c x^{2}} \left (64 a^{2} c^{2} - \frac{735 a b^{2} c}{4} + \frac{945 b^{4}}{16} - \frac{7 b c x \left (- 92 a c + 45 b^{2}\right )}{8}\right )}{120 c^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**5/(c*x**2+b*x+a)**(1/2),x)

[Out]

-9*b*x**3*sqrt(a + b*x + c*x**2)/(40*c**2) - b*(240*a**2*c**2 - 280*a*b**2*c + 6
3*b**4)*atanh((b + 2*c*x)/(2*sqrt(c)*sqrt(a + b*x + c*x**2)))/(256*c**(11/2)) +
x**4*sqrt(a + b*x + c*x**2)/(5*c) + x**2*(-64*a*c + 63*b**2)*sqrt(a + b*x + c*x*
*2)/(240*c**3) + sqrt(a + b*x + c*x**2)*(64*a**2*c**2 - 735*a*b**2*c/4 + 945*b**
4/16 - 7*b*c*x*(-92*a*c + 45*b**2)/8)/(120*c**5)

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Mathematica [A]  time = 0.447692, size = 153, normalized size = 0.76 \[ \frac{2 \sqrt{c} \sqrt{a+x (b+c x)} \left (128 c^2 \left (8 a^2-4 a c x^2+3 c^2 x^4\right )+84 b^2 c \left (6 c x^2-35 a\right )+8 b c^2 x \left (161 a-54 c x^2\right )+945 b^4-630 b^3 c x\right )-15 \left (240 a^2 b c^2-280 a b^3 c+63 b^5\right ) \log \left (2 \sqrt{c} \sqrt{a+x (b+c x)}+b+2 c x\right )}{3840 c^{11/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^5/Sqrt[a + b*x + c*x^2],x]

[Out]

(2*Sqrt[c]*Sqrt[a + x*(b + c*x)]*(945*b^4 - 630*b^3*c*x + 8*b*c^2*x*(161*a - 54*
c*x^2) + 84*b^2*c*(-35*a + 6*c*x^2) + 128*c^2*(8*a^2 - 4*a*c*x^2 + 3*c^2*x^4)) -
15*(63*b^5 - 280*a*b^3*c + 240*a^2*b*c^2)*Log[b + 2*c*x + 2*Sqrt[c]*Sqrt[a + x*
(b + c*x)]])/(3840*c^(11/2))

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Maple [A]  time = 0.014, size = 290, normalized size = 1.4 \[{\frac{{x}^{4}}{5\,c}\sqrt{c{x}^{2}+bx+a}}-{\frac{9\,b{x}^{3}}{40\,{c}^{2}}\sqrt{c{x}^{2}+bx+a}}+{\frac{21\,{b}^{2}{x}^{2}}{80\,{c}^{3}}\sqrt{c{x}^{2}+bx+a}}-{\frac{21\,{b}^{3}x}{64\,{c}^{4}}\sqrt{c{x}^{2}+bx+a}}+{\frac{63\,{b}^{4}}{128\,{c}^{5}}\sqrt{c{x}^{2}+bx+a}}-{\frac{63\,{b}^{5}}{256}\ln \left ({1 \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx+a} \right ){c}^{-{\frac{11}{2}}}}+{\frac{35\,a{b}^{3}}{32}\ln \left ({1 \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx+a} \right ){c}^{-{\frac{9}{2}}}}-{\frac{49\,a{b}^{2}}{32\,{c}^{4}}\sqrt{c{x}^{2}+bx+a}}+{\frac{161\,abx}{240\,{c}^{3}}\sqrt{c{x}^{2}+bx+a}}-{\frac{15\,{a}^{2}b}{16}\ln \left ({1 \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx+a} \right ){c}^{-{\frac{7}{2}}}}-{\frac{4\,a{x}^{2}}{15\,{c}^{2}}\sqrt{c{x}^{2}+bx+a}}+{\frac{8\,{a}^{2}}{15\,{c}^{3}}\sqrt{c{x}^{2}+bx+a}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^5/(c*x^2+b*x+a)^(1/2),x)

[Out]

1/5*x^4*(c*x^2+b*x+a)^(1/2)/c-9/40*b*x^3*(c*x^2+b*x+a)^(1/2)/c^2+21/80*b^2/c^3*x
^2*(c*x^2+b*x+a)^(1/2)-21/64*b^3/c^4*x*(c*x^2+b*x+a)^(1/2)+63/128*b^4/c^5*(c*x^2
+b*x+a)^(1/2)-63/256*b^5/c^(11/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))+35
/32*b^3/c^(9/2)*a*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))-49/32*b^2/c^4*a*(c
*x^2+b*x+a)^(1/2)+161/240*b/c^3*a*x*(c*x^2+b*x+a)^(1/2)-15/16*b/c^(7/2)*a^2*ln((
1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))-4/15*a/c^2*x^2*(c*x^2+b*x+a)^(1/2)+8/15*
a^2/c^3*(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^5/sqrt(c*x^2 + b*x + a),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.273945, size = 1, normalized size = 0. \[ \left [\frac{4 \,{\left (384 \, c^{4} x^{4} - 432 \, b c^{3} x^{3} + 945 \, b^{4} - 2940 \, a b^{2} c + 1024 \, a^{2} c^{2} + 8 \,{\left (63 \, b^{2} c^{2} - 64 \, a c^{3}\right )} x^{2} - 14 \,{\left (45 \, b^{3} c - 92 \, a b c^{2}\right )} x\right )} \sqrt{c x^{2} + b x + a} \sqrt{c} + 15 \,{\left (63 \, b^{5} - 280 \, a b^{3} c + 240 \, a^{2} b c^{2}\right )} \log \left (4 \,{\left (2 \, c^{2} x + b c\right )} \sqrt{c x^{2} + b x + a} -{\left (8 \, c^{2} x^{2} + 8 \, b c x + b^{2} + 4 \, a c\right )} \sqrt{c}\right )}{7680 \, c^{\frac{11}{2}}}, \frac{2 \,{\left (384 \, c^{4} x^{4} - 432 \, b c^{3} x^{3} + 945 \, b^{4} - 2940 \, a b^{2} c + 1024 \, a^{2} c^{2} + 8 \,{\left (63 \, b^{2} c^{2} - 64 \, a c^{3}\right )} x^{2} - 14 \,{\left (45 \, b^{3} c - 92 \, a b c^{2}\right )} x\right )} \sqrt{c x^{2} + b x + a} \sqrt{-c} - 15 \,{\left (63 \, b^{5} - 280 \, a b^{3} c + 240 \, a^{2} b c^{2}\right )} \arctan \left (\frac{{\left (2 \, c x + b\right )} \sqrt{-c}}{2 \, \sqrt{c x^{2} + b x + a} c}\right )}{3840 \, \sqrt{-c} c^{5}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^5/sqrt(c*x^2 + b*x + a),x, algorithm="fricas")

[Out]

[1/7680*(4*(384*c^4*x^4 - 432*b*c^3*x^3 + 945*b^4 - 2940*a*b^2*c + 1024*a^2*c^2
+ 8*(63*b^2*c^2 - 64*a*c^3)*x^2 - 14*(45*b^3*c - 92*a*b*c^2)*x)*sqrt(c*x^2 + b*x
+ a)*sqrt(c) + 15*(63*b^5 - 280*a*b^3*c + 240*a^2*b*c^2)*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^(11/2),
1/3840*(2*(384*c^4*x^4 - 432*b*c^3*x^3 + 945*b^4 - 2940*a*b^2*c + 1024*a^2*c^2 +
8*(63*b^2*c^2 - 64*a*c^3)*x^2 - 14*(45*b^3*c - 92*a*b*c^2)*x)*sqrt(c*x^2 + b*x
+ a)*sqrt(-c) - 15*(63*b^5 - 280*a*b^3*c + 240*a^2*b*c^2)*arctan(1/2*(2*c*x + b)
*sqrt(-c)/(sqrt(c*x^2 + b*x + a)*c)))/(sqrt(-c)*c^5)]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{5}}{\sqrt{a + b x + c x^{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**5/(c*x**2+b*x+a)**(1/2),x)

[Out]

Integral(x**5/sqrt(a + b*x + c*x**2), x)

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GIAC/XCAS [A]  time = 0.225008, size = 217, normalized size = 1.07 \[ \frac{1}{1920} \, \sqrt{c x^{2} + b x + a}{\left (2 \,{\left (4 \,{\left (6 \, x{\left (\frac{8 \, x}{c} - \frac{9 \, b}{c^{2}}\right )} + \frac{63 \, b^{2} c^{2} - 64 \, a c^{3}}{c^{5}}\right )} x - \frac{7 \,{\left (45 \, b^{3} c - 92 \, a b c^{2}\right )}}{c^{5}}\right )} x + \frac{945 \, b^{4} - 2940 \, a b^{2} c + 1024 \, a^{2} c^{2}}{c^{5}}\right )} + \frac{{\left (63 \, b^{5} - 280 \, a b^{3} c + 240 \, a^{2} b c^{2}\right )}{\rm ln}\left ({\left | -2 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )} \sqrt{c} - b \right |}\right )}{256 \, c^{\frac{11}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^5/sqrt(c*x^2 + b*x + a),x, algorithm="giac")

[Out]

1/1920*sqrt(c*x^2 + b*x + a)*(2*(4*(6*x*(8*x/c - 9*b/c^2) + (63*b^2*c^2 - 64*a*c
^3)/c^5)*x - 7*(45*b^3*c - 92*a*b*c^2)/c^5)*x + (945*b^4 - 2940*a*b^2*c + 1024*a
^2*c^2)/c^5) + 1/256*(63*b^5 - 280*a*b^3*c + 240*a^2*b*c^2)*ln(abs(-2*(sqrt(c)*x
- sqrt(c*x^2 + b*x + a))*sqrt(c) - b))/c^(11/2)