3.920 \(\int x^7 \sqrt{a+b x^2+c x^4} \, dx\)

Optimal. Leaf size=171 \[ \frac{b \left (7 b^2-12 a c\right ) \left (b^2-4 a c\right ) \tanh ^{-1}\left (\frac{b+2 c x^2}{2 \sqrt{c} \sqrt{a+b x^2+c x^4}}\right )}{512 c^{9/2}}-\frac{b \left (7 b^2-12 a c\right ) \left (b+2 c x^2\right ) \sqrt{a+b x^2+c x^4}}{256 c^4}+\frac{\left (-32 a c+35 b^2-42 b c x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}}{480 c^3}+\frac{x^4 \left (a+b x^2+c x^4\right )^{3/2}}{10 c} \]

[Out]

-(b*(7*b^2 - 12*a*c)*(b + 2*c*x^2)*Sqrt[a + b*x^2 + c*x^4])/(256*c^4) + (x^4*(a
+ b*x^2 + c*x^4)^(3/2))/(10*c) + ((35*b^2 - 32*a*c - 42*b*c*x^2)*(a + b*x^2 + c*
x^4)^(3/2))/(480*c^3) + (b*(7*b^2 - 12*a*c)*(b^2 - 4*a*c)*ArcTanh[(b + 2*c*x^2)/
(2*Sqrt[c]*Sqrt[a + b*x^2 + c*x^4])])/(512*c^(9/2))

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Rubi [A]  time = 0.387127, antiderivative size = 171, 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 \[ \frac{b \left (7 b^2-12 a c\right ) \left (b^2-4 a c\right ) \tanh ^{-1}\left (\frac{b+2 c x^2}{2 \sqrt{c} \sqrt{a+b x^2+c x^4}}\right )}{512 c^{9/2}}-\frac{b \left (7 b^2-12 a c\right ) \left (b+2 c x^2\right ) \sqrt{a+b x^2+c x^4}}{256 c^4}+\frac{\left (-32 a c+35 b^2-42 b c x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}}{480 c^3}+\frac{x^4 \left (a+b x^2+c x^4\right )^{3/2}}{10 c} \]

Antiderivative was successfully verified.

[In]  Int[x^7*Sqrt[a + b*x^2 + c*x^4],x]

[Out]

-(b*(7*b^2 - 12*a*c)*(b + 2*c*x^2)*Sqrt[a + b*x^2 + c*x^4])/(256*c^4) + (x^4*(a
+ b*x^2 + c*x^4)^(3/2))/(10*c) + ((35*b^2 - 32*a*c - 42*b*c*x^2)*(a + b*x^2 + c*
x^4)^(3/2))/(480*c^3) + (b*(7*b^2 - 12*a*c)*(b^2 - 4*a*c)*ArcTanh[(b + 2*c*x^2)/
(2*Sqrt[c]*Sqrt[a + b*x^2 + c*x^4])])/(512*c^(9/2))

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Rubi in Sympy [A]  time = 27.2212, size = 163, normalized size = 0.95 \[ - \frac{b \left (b + 2 c x^{2}\right ) \left (- 12 a c + 7 b^{2}\right ) \sqrt{a + b x^{2} + c x^{4}}}{256 c^{4}} + \frac{b \left (- 12 a c + 7 b^{2}\right ) \left (- 4 a c + b^{2}\right ) \operatorname{atanh}{\left (\frac{b + 2 c x^{2}}{2 \sqrt{c} \sqrt{a + b x^{2} + c x^{4}}} \right )}}{512 c^{\frac{9}{2}}} + \frac{x^{4} \left (a + b x^{2} + c x^{4}\right )^{\frac{3}{2}}}{10 c} + \frac{\left (a + b x^{2} + c x^{4}\right )^{\frac{3}{2}} \left (- 8 a c + \frac{35 b^{2}}{4} - \frac{21 b c x^{2}}{2}\right )}{120 c^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**7*(c*x**4+b*x**2+a)**(1/2),x)

[Out]

-b*(b + 2*c*x**2)*(-12*a*c + 7*b**2)*sqrt(a + b*x**2 + c*x**4)/(256*c**4) + b*(-
12*a*c + 7*b**2)*(-4*a*c + b**2)*atanh((b + 2*c*x**2)/(2*sqrt(c)*sqrt(a + b*x**2
+ c*x**4)))/(512*c**(9/2)) + x**4*(a + b*x**2 + c*x**4)**(3/2)/(10*c) + (a + b*
x**2 + c*x**4)**(3/2)*(-8*a*c + 35*b**2/4 - 21*b*c*x**2/2)/(120*c**3)

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Mathematica [A]  time = 0.16346, size = 162, normalized size = 0.95 \[ \frac{15 b \left (7 b^2-12 a c\right ) \left (b^2-4 a c\right ) \log \left (2 \sqrt{c} \sqrt{a+b x^2+c x^4}+b+2 c x^2\right )-2 \sqrt{c} \sqrt{a+b x^2+c x^4} \left (-128 c^2 \left (-2 a^2+a c x^4+3 c^2 x^8\right )+b^2 \left (56 c^2 x^4-460 a c\right )+8 b c^2 x^2 \left (29 a-6 c x^4\right )+105 b^4-70 b^3 c x^2\right )}{7680 c^{9/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^7*Sqrt[a + b*x^2 + c*x^4],x]

[Out]

(-2*Sqrt[c]*Sqrt[a + b*x^2 + c*x^4]*(105*b^4 - 70*b^3*c*x^2 + 8*b*c^2*x^2*(29*a
- 6*c*x^4) + b^2*(-460*a*c + 56*c^2*x^4) - 128*c^2*(-2*a^2 + a*c*x^4 + 3*c^2*x^8
)) + 15*b*(7*b^2 - 12*a*c)*(b^2 - 4*a*c)*Log[b + 2*c*x^2 + 2*Sqrt[c]*Sqrt[a + b*
x^2 + c*x^4]])/(7680*c^(9/2))

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Maple [A]  time = 0.046, size = 296, normalized size = 1.7 \[{\frac{{x}^{4}}{10\,c} \left ( c{x}^{4}+b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}-{\frac{7\,b{x}^{2}}{80\,{c}^{2}} \left ( c{x}^{4}+b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}+{\frac{7\,{b}^{2}}{96\,{c}^{3}} \left ( c{x}^{4}+b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}-{\frac{7\,{b}^{3}{x}^{2}}{128\,{c}^{3}}\sqrt{c{x}^{4}+b{x}^{2}+a}}-{\frac{7\,{b}^{4}}{256\,{c}^{4}}\sqrt{c{x}^{4}+b{x}^{2}+a}}-{\frac{5\,a{b}^{3}}{64}\ln \left ({1 \left ({\frac{b}{2}}+c{x}^{2} \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{4}+b{x}^{2}+a} \right ){c}^{-{\frac{7}{2}}}}+{\frac{7\,{b}^{5}}{512}\ln \left ({1 \left ({\frac{b}{2}}+c{x}^{2} \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{4}+b{x}^{2}+a} \right ){c}^{-{\frac{9}{2}}}}+{\frac{3\,ab{x}^{2}}{32\,{c}^{2}}\sqrt{c{x}^{4}+b{x}^{2}+a}}+{\frac{3\,a{b}^{2}}{64\,{c}^{3}}\sqrt{c{x}^{4}+b{x}^{2}+a}}+{\frac{3\,{a}^{2}b}{32}\ln \left ({1 \left ({\frac{b}{2}}+c{x}^{2} \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{4}+b{x}^{2}+a} \right ){c}^{-{\frac{5}{2}}}}-{\frac{a}{15\,{c}^{2}} \left ( c{x}^{4}+b{x}^{2}+a \right ) ^{{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^7*(c*x^4+b*x^2+a)^(1/2),x)

[Out]

1/10*x^4*(c*x^4+b*x^2+a)^(3/2)/c-7/80*b/c^2*x^2*(c*x^4+b*x^2+a)^(3/2)+7/96*b^2/c
^3*(c*x^4+b*x^2+a)^(3/2)-7/128*b^3/c^3*(c*x^4+b*x^2+a)^(1/2)*x^2-7/256*b^4/c^4*(
c*x^4+b*x^2+a)^(1/2)-5/64*b^3/c^(7/2)*ln((1/2*b+c*x^2)/c^(1/2)+(c*x^4+b*x^2+a)^(
1/2))*a+7/512*b^5/c^(9/2)*ln((1/2*b+c*x^2)/c^(1/2)+(c*x^4+b*x^2+a)^(1/2))+3/32*b
/c^2*a*(c*x^4+b*x^2+a)^(1/2)*x^2+3/64*b^2/c^3*a*(c*x^4+b*x^2+a)^(1/2)+3/32*b/c^(
5/2)*a^2*ln((1/2*b+c*x^2)/c^(1/2)+(c*x^4+b*x^2+a)^(1/2))-1/15/c^2*a*(c*x^4+b*x^2
+a)^(3/2)

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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(sqrt(c*x^4 + b*x^2 + a)*x^7,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.307548, size = 1, normalized size = 0.01 \[ \left [\frac{4 \,{\left (384 \, c^{4} x^{8} + 48 \, b c^{3} x^{6} - 8 \,{\left (7 \, b^{2} c^{2} - 16 \, a c^{3}\right )} x^{4} - 105 \, b^{4} + 460 \, a b^{2} c - 256 \, a^{2} c^{2} + 2 \,{\left (35 \, b^{3} c - 116 \, a b c^{2}\right )} x^{2}\right )} \sqrt{c x^{4} + b x^{2} + a} \sqrt{c} + 15 \,{\left (7 \, b^{5} - 40 \, a b^{3} c + 48 \, a^{2} b c^{2}\right )} \log \left (-4 \, \sqrt{c x^{4} + b x^{2} + a}{\left (2 \, c^{2} x^{2} + b c\right )} -{\left (8 \, c^{2} x^{4} + 8 \, b c x^{2} + b^{2} + 4 \, a c\right )} \sqrt{c}\right )}{15360 \, c^{\frac{9}{2}}}, \frac{2 \,{\left (384 \, c^{4} x^{8} + 48 \, b c^{3} x^{6} - 8 \,{\left (7 \, b^{2} c^{2} - 16 \, a c^{3}\right )} x^{4} - 105 \, b^{4} + 460 \, a b^{2} c - 256 \, a^{2} c^{2} + 2 \,{\left (35 \, b^{3} c - 116 \, a b c^{2}\right )} x^{2}\right )} \sqrt{c x^{4} + b x^{2} + a} \sqrt{-c} + 15 \,{\left (7 \, b^{5} - 40 \, a b^{3} c + 48 \, a^{2} b c^{2}\right )} \arctan \left (\frac{{\left (2 \, c x^{2} + b\right )} \sqrt{-c}}{2 \, \sqrt{c x^{4} + b x^{2} + a} c}\right )}{7680 \, \sqrt{-c} c^{4}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/15360*(4*(384*c^4*x^8 + 48*b*c^3*x^6 - 8*(7*b^2*c^2 - 16*a*c^3)*x^4 - 105*b^4
+ 460*a*b^2*c - 256*a^2*c^2 + 2*(35*b^3*c - 116*a*b*c^2)*x^2)*sqrt(c*x^4 + b*x^
2 + a)*sqrt(c) + 15*(7*b^5 - 40*a*b^3*c + 48*a^2*b*c^2)*log(-4*sqrt(c*x^4 + b*x^
2 + a)*(2*c^2*x^2 + b*c) - (8*c^2*x^4 + 8*b*c*x^2 + b^2 + 4*a*c)*sqrt(c)))/c^(9/
2), 1/7680*(2*(384*c^4*x^8 + 48*b*c^3*x^6 - 8*(7*b^2*c^2 - 16*a*c^3)*x^4 - 105*b
^4 + 460*a*b^2*c - 256*a^2*c^2 + 2*(35*b^3*c - 116*a*b*c^2)*x^2)*sqrt(c*x^4 + b*
x^2 + a)*sqrt(-c) + 15*(7*b^5 - 40*a*b^3*c + 48*a^2*b*c^2)*arctan(1/2*(2*c*x^2 +
b)*sqrt(-c)/(sqrt(c*x^4 + b*x^2 + a)*c)))/(sqrt(-c)*c^4)]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int x^{7} \sqrt{a + b x^{2} + c x^{4}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**7*(c*x**4+b*x**2+a)**(1/2),x)

[Out]

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

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GIAC/XCAS [A]  time = 0.303072, size = 248, normalized size = 1.45 \[ \frac{1}{3840} \, \sqrt{c x^{4} + b x^{2} + a}{\left (2 \,{\left (4 \,{\left (6 \,{\left (8 \, x^{2} + \frac{b}{c}\right )} x^{2} - \frac{7 \, b^{2} c^{5} - 16 \, a c^{6}}{c^{7}}\right )} x^{2} + \frac{35 \, b^{3} c^{4} - 116 \, a b c^{5}}{c^{7}}\right )} x^{2} - \frac{105 \, b^{4} c^{3} - 460 \, a b^{2} c^{4} + 256 \, a^{2} c^{5}}{c^{7}}\right )} - \frac{{\left (7 \, b^{5} c^{3} - 40 \, a b^{3} c^{4} + 48 \, a^{2} b c^{5}\right )}{\rm ln}\left ({\left | -2 \,{\left (\sqrt{c} x^{2} - \sqrt{c x^{4} + b x^{2} + a}\right )} \sqrt{c} - b \right |}\right )}{512 \, c^{\frac{15}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3840*sqrt(c*x^4 + b*x^2 + a)*(2*(4*(6*(8*x^2 + b/c)*x^2 - (7*b^2*c^5 - 16*a*c^
6)/c^7)*x^2 + (35*b^3*c^4 - 116*a*b*c^5)/c^7)*x^2 - (105*b^4*c^3 - 460*a*b^2*c^4
+ 256*a^2*c^5)/c^7) - 1/512*(7*b^5*c^3 - 40*a*b^3*c^4 + 48*a^2*b*c^5)*ln(abs(-2
*(sqrt(c)*x^2 - sqrt(c*x^4 + b*x^2 + a))*sqrt(c) - b))/c^(15/2)