3.89 \(\int x^7 \left (A+B x^2\right ) \sqrt{b x^2+c x^4} \, dx\)

Optimal. Leaf size=218 \[ -\frac{7 b^5 (3 b B-4 A c) \tanh ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{b x^2+c x^4}}\right )}{1024 c^{11/2}}+\frac{7 b^3 \left (b+2 c x^2\right ) \sqrt{b x^2+c x^4} (3 b B-4 A c)}{1024 c^5}-\frac{7 b^2 \left (b x^2+c x^4\right )^{3/2} (3 b B-4 A c)}{384 c^4}+\frac{7 b x^2 \left (b x^2+c x^4\right )^{3/2} (3 b B-4 A c)}{320 c^3}-\frac{x^4 \left (b x^2+c x^4\right )^{3/2} (3 b B-4 A c)}{40 c^2}+\frac{B x^6 \left (b x^2+c x^4\right )^{3/2}}{12 c} \]

[Out]

(7*b^3*(3*b*B - 4*A*c)*(b + 2*c*x^2)*Sqrt[b*x^2 + c*x^4])/(1024*c^5) - (7*b^2*(3
*b*B - 4*A*c)*(b*x^2 + c*x^4)^(3/2))/(384*c^4) + (7*b*(3*b*B - 4*A*c)*x^2*(b*x^2
+ c*x^4)^(3/2))/(320*c^3) - ((3*b*B - 4*A*c)*x^4*(b*x^2 + c*x^4)^(3/2))/(40*c^2
) + (B*x^6*(b*x^2 + c*x^4)^(3/2))/(12*c) - (7*b^5*(3*b*B - 4*A*c)*ArcTanh[(Sqrt[
c]*x^2)/Sqrt[b*x^2 + c*x^4]])/(1024*c^(11/2))

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Rubi [A]  time = 0.723507, antiderivative size = 218, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.269 \[ -\frac{7 b^5 (3 b B-4 A c) \tanh ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{b x^2+c x^4}}\right )}{1024 c^{11/2}}+\frac{7 b^3 \left (b+2 c x^2\right ) \sqrt{b x^2+c x^4} (3 b B-4 A c)}{1024 c^5}-\frac{7 b^2 \left (b x^2+c x^4\right )^{3/2} (3 b B-4 A c)}{384 c^4}+\frac{7 b x^2 \left (b x^2+c x^4\right )^{3/2} (3 b B-4 A c)}{320 c^3}-\frac{x^4 \left (b x^2+c x^4\right )^{3/2} (3 b B-4 A c)}{40 c^2}+\frac{B x^6 \left (b x^2+c x^4\right )^{3/2}}{12 c} \]

Antiderivative was successfully verified.

[In]  Int[x^7*(A + B*x^2)*Sqrt[b*x^2 + c*x^4],x]

[Out]

(7*b^3*(3*b*B - 4*A*c)*(b + 2*c*x^2)*Sqrt[b*x^2 + c*x^4])/(1024*c^5) - (7*b^2*(3
*b*B - 4*A*c)*(b*x^2 + c*x^4)^(3/2))/(384*c^4) + (7*b*(3*b*B - 4*A*c)*x^2*(b*x^2
+ c*x^4)^(3/2))/(320*c^3) - ((3*b*B - 4*A*c)*x^4*(b*x^2 + c*x^4)^(3/2))/(40*c^2
) + (B*x^6*(b*x^2 + c*x^4)^(3/2))/(12*c) - (7*b^5*(3*b*B - 4*A*c)*ArcTanh[(Sqrt[
c]*x^2)/Sqrt[b*x^2 + c*x^4]])/(1024*c^(11/2))

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Rubi in Sympy [A]  time = 42.9977, size = 211, normalized size = 0.97 \[ \frac{B x^{6} \left (b x^{2} + c x^{4}\right )^{\frac{3}{2}}}{12 c} + \frac{7 b^{5} \left (4 A c - 3 B b\right ) \operatorname{atanh}{\left (\frac{\sqrt{c} x^{2}}{\sqrt{b x^{2} + c x^{4}}} \right )}}{1024 c^{\frac{11}{2}}} - \frac{7 b^{3} \left (b + 2 c x^{2}\right ) \left (4 A c - 3 B b\right ) \sqrt{b x^{2} + c x^{4}}}{1024 c^{5}} + \frac{7 b^{2} \left (4 A c - 3 B b\right ) \left (b x^{2} + c x^{4}\right )^{\frac{3}{2}}}{384 c^{4}} - \frac{7 b x^{2} \left (4 A c - 3 B b\right ) \left (b x^{2} + c x^{4}\right )^{\frac{3}{2}}}{320 c^{3}} + \frac{x^{4} \left (4 A c - 3 B b\right ) \left (b x^{2} + c x^{4}\right )^{\frac{3}{2}}}{40 c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

B*x**6*(b*x**2 + c*x**4)**(3/2)/(12*c) + 7*b**5*(4*A*c - 3*B*b)*atanh(sqrt(c)*x*
*2/sqrt(b*x**2 + c*x**4))/(1024*c**(11/2)) - 7*b**3*(b + 2*c*x**2)*(4*A*c - 3*B*
b)*sqrt(b*x**2 + c*x**4)/(1024*c**5) + 7*b**2*(4*A*c - 3*B*b)*(b*x**2 + c*x**4)*
*(3/2)/(384*c**4) - 7*b*x**2*(4*A*c - 3*B*b)*(b*x**2 + c*x**4)**(3/2)/(320*c**3)
+ x**4*(4*A*c - 3*B*b)*(b*x**2 + c*x**4)**(3/2)/(40*c**2)

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Mathematica [A]  time = 0.428454, size = 188, normalized size = 0.86 \[ \frac{x \left (\sqrt{c} x \left (b+c x^2\right ) \left (-210 b^4 c \left (2 A+B x^2\right )+56 b^3 c^2 x^2 \left (5 A+3 B x^2\right )-16 b^2 c^3 x^4 \left (14 A+9 B x^2\right )+64 b c^4 x^6 \left (3 A+2 B x^2\right )+256 c^5 x^8 \left (6 A+5 B x^2\right )+315 b^5 B\right )-105 b^5 \sqrt{b+c x^2} (3 b B-4 A c) \log \left (\sqrt{c} \sqrt{b+c x^2}+c x\right )\right )}{15360 c^{11/2} \sqrt{x^2 \left (b+c x^2\right )}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^7*(A + B*x^2)*Sqrt[b*x^2 + c*x^4],x]

[Out]

(x*(Sqrt[c]*x*(b + c*x^2)*(315*b^5*B - 210*b^4*c*(2*A + B*x^2) + 64*b*c^4*x^6*(3
*A + 2*B*x^2) + 56*b^3*c^2*x^2*(5*A + 3*B*x^2) + 256*c^5*x^8*(6*A + 5*B*x^2) - 1
6*b^2*c^3*x^4*(14*A + 9*B*x^2)) - 105*b^5*(3*b*B - 4*A*c)*Sqrt[b + c*x^2]*Log[c*
x + Sqrt[c]*Sqrt[b + c*x^2]]))/(15360*c^(11/2)*Sqrt[x^2*(b + c*x^2)])

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Maple [A]  time = 0.046, size = 295, normalized size = 1.4 \[{\frac{1}{15360\,x}\sqrt{c{x}^{4}+b{x}^{2}} \left ( 1280\,B{x}^{9} \left ( c{x}^{2}+b \right ) ^{3/2}{c}^{19/2}+1536\,A{x}^{7} \left ( c{x}^{2}+b \right ) ^{3/2}{c}^{19/2}-1152\,Bb{x}^{7} \left ( c{x}^{2}+b \right ) ^{3/2}{c}^{17/2}-1344\,Ab{x}^{5} \left ( c{x}^{2}+b \right ) ^{3/2}{c}^{17/2}+1008\,B{b}^{2}{x}^{5} \left ( c{x}^{2}+b \right ) ^{3/2}{c}^{15/2}+1120\,A{b}^{2}{x}^{3} \left ( c{x}^{2}+b \right ) ^{3/2}{c}^{15/2}-840\,B{b}^{3}{x}^{3} \left ( c{x}^{2}+b \right ) ^{3/2}{c}^{13/2}-840\,A{b}^{3}x \left ( c{x}^{2}+b \right ) ^{3/2}{c}^{13/2}+630\,B{b}^{4}x \left ( c{x}^{2}+b \right ) ^{3/2}{c}^{11/2}+420\,A{b}^{4}x\sqrt{c{x}^{2}+b}{c}^{13/2}-315\,B{b}^{5}x\sqrt{c{x}^{2}+b}{c}^{11/2}+420\,A{b}^{5}\ln \left ( \sqrt{c}x+\sqrt{c{x}^{2}+b} \right ){c}^{6}-315\,B{b}^{6}\ln \left ( \sqrt{c}x+\sqrt{c{x}^{2}+b} \right ){c}^{5} \right ){\frac{1}{\sqrt{c{x}^{2}+b}}}{c}^{-{\frac{21}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/15360*(c*x^4+b*x^2)^(1/2)*(1280*B*x^9*(c*x^2+b)^(3/2)*c^(19/2)+1536*A*x^7*(c*x
^2+b)^(3/2)*c^(19/2)-1152*B*b*x^7*(c*x^2+b)^(3/2)*c^(17/2)-1344*A*b*x^5*(c*x^2+b
)^(3/2)*c^(17/2)+1008*B*b^2*x^5*(c*x^2+b)^(3/2)*c^(15/2)+1120*A*b^2*x^3*(c*x^2+b
)^(3/2)*c^(15/2)-840*B*b^3*x^3*(c*x^2+b)^(3/2)*c^(13/2)-840*A*b^3*x*(c*x^2+b)^(3
/2)*c^(13/2)+630*B*b^4*x*(c*x^2+b)^(3/2)*c^(11/2)+420*A*b^4*x*(c*x^2+b)^(1/2)*c^
(13/2)-315*B*b^5*x*(c*x^2+b)^(1/2)*c^(11/2)+420*A*b^5*ln(c^(1/2)*x+(c*x^2+b)^(1/
2))*c^6-315*B*b^6*ln(c^(1/2)*x+(c*x^2+b)^(1/2))*c^5)/x/(c*x^2+b)^(1/2)/c^(21/2)

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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)*(B*x^2 + A)*x^7,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.581304, size = 1, normalized size = 0. \[ \left [-\frac{105 \,{\left (3 \, B b^{6} - 4 \, A b^{5} c\right )} \sqrt{c} \log \left (-{\left (2 \, c x^{2} + b\right )} \sqrt{c} - 2 \, \sqrt{c x^{4} + b x^{2}} c\right ) - 2 \,{\left (1280 \, B c^{6} x^{10} + 128 \,{\left (B b c^{5} + 12 \, A c^{6}\right )} x^{8} + 315 \, B b^{5} c - 420 \, A b^{4} c^{2} - 48 \,{\left (3 \, B b^{2} c^{4} - 4 \, A b c^{5}\right )} x^{6} + 56 \,{\left (3 \, B b^{3} c^{3} - 4 \, A b^{2} c^{4}\right )} x^{4} - 70 \,{\left (3 \, B b^{4} c^{2} - 4 \, A b^{3} c^{3}\right )} x^{2}\right )} \sqrt{c x^{4} + b x^{2}}}{30720 \, c^{6}}, \frac{105 \,{\left (3 \, B b^{6} - 4 \, A b^{5} c\right )} \sqrt{-c} \arctan \left (\frac{\sqrt{-c} x^{2}}{\sqrt{c x^{4} + b x^{2}}}\right ) +{\left (1280 \, B c^{6} x^{10} + 128 \,{\left (B b c^{5} + 12 \, A c^{6}\right )} x^{8} + 315 \, B b^{5} c - 420 \, A b^{4} c^{2} - 48 \,{\left (3 \, B b^{2} c^{4} - 4 \, A b c^{5}\right )} x^{6} + 56 \,{\left (3 \, B b^{3} c^{3} - 4 \, A b^{2} c^{4}\right )} x^{4} - 70 \,{\left (3 \, B b^{4} c^{2} - 4 \, A b^{3} c^{3}\right )} x^{2}\right )} \sqrt{c x^{4} + b x^{2}}}{15360 \, c^{6}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[-1/30720*(105*(3*B*b^6 - 4*A*b^5*c)*sqrt(c)*log(-(2*c*x^2 + b)*sqrt(c) - 2*sqrt
(c*x^4 + b*x^2)*c) - 2*(1280*B*c^6*x^10 + 128*(B*b*c^5 + 12*A*c^6)*x^8 + 315*B*b
^5*c - 420*A*b^4*c^2 - 48*(3*B*b^2*c^4 - 4*A*b*c^5)*x^6 + 56*(3*B*b^3*c^3 - 4*A*
b^2*c^4)*x^4 - 70*(3*B*b^4*c^2 - 4*A*b^3*c^3)*x^2)*sqrt(c*x^4 + b*x^2))/c^6, 1/1
5360*(105*(3*B*b^6 - 4*A*b^5*c)*sqrt(-c)*arctan(sqrt(-c)*x^2/sqrt(c*x^4 + b*x^2)
) + (1280*B*c^6*x^10 + 128*(B*b*c^5 + 12*A*c^6)*x^8 + 315*B*b^5*c - 420*A*b^4*c^
2 - 48*(3*B*b^2*c^4 - 4*A*b*c^5)*x^6 + 56*(3*B*b^3*c^3 - 4*A*b^2*c^4)*x^4 - 70*(
3*B*b^4*c^2 - 4*A*b^3*c^3)*x^2)*sqrt(c*x^4 + b*x^2))/c^6]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int x^{7} \sqrt{x^{2} \left (b + c x^{2}\right )} \left (A + B x^{2}\right )\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(x**7*sqrt(x**2*(b + c*x**2))*(A + B*x**2), x)

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GIAC/XCAS [A]  time = 0.223384, size = 333, normalized size = 1.53 \[ \frac{1}{15360} \,{\left (2 \,{\left (4 \,{\left (2 \,{\left (8 \,{\left (10 \, B x^{2}{\rm sign}\left (x\right ) + \frac{B b c^{9}{\rm sign}\left (x\right ) + 12 \, A c^{10}{\rm sign}\left (x\right )}{c^{10}}\right )} x^{2} - \frac{3 \,{\left (3 \, B b^{2} c^{8}{\rm sign}\left (x\right ) - 4 \, A b c^{9}{\rm sign}\left (x\right )\right )}}{c^{10}}\right )} x^{2} + \frac{7 \,{\left (3 \, B b^{3} c^{7}{\rm sign}\left (x\right ) - 4 \, A b^{2} c^{8}{\rm sign}\left (x\right )\right )}}{c^{10}}\right )} x^{2} - \frac{35 \,{\left (3 \, B b^{4} c^{6}{\rm sign}\left (x\right ) - 4 \, A b^{3} c^{7}{\rm sign}\left (x\right )\right )}}{c^{10}}\right )} x^{2} + \frac{105 \,{\left (3 \, B b^{5} c^{5}{\rm sign}\left (x\right ) - 4 \, A b^{4} c^{6}{\rm sign}\left (x\right )\right )}}{c^{10}}\right )} \sqrt{c x^{2} + b} x + \frac{7 \,{\left (3 \, B b^{6}{\rm sign}\left (x\right ) - 4 \, A b^{5} c{\rm sign}\left (x\right )\right )}{\rm ln}\left ({\left | -\sqrt{c} x + \sqrt{c x^{2} + b} \right |}\right )}{1024 \, c^{\frac{11}{2}}} - \frac{7 \,{\left (3 \, B b^{6}{\rm ln}\left (\sqrt{b}\right ) - 4 \, A b^{5} c{\rm ln}\left (\sqrt{b}\right )\right )}{\rm sign}\left (x\right )}{1024 \, c^{\frac{11}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/15360*(2*(4*(2*(8*(10*B*x^2*sign(x) + (B*b*c^9*sign(x) + 12*A*c^10*sign(x))/c^
10)*x^2 - 3*(3*B*b^2*c^8*sign(x) - 4*A*b*c^9*sign(x))/c^10)*x^2 + 7*(3*B*b^3*c^7
*sign(x) - 4*A*b^2*c^8*sign(x))/c^10)*x^2 - 35*(3*B*b^4*c^6*sign(x) - 4*A*b^3*c^
7*sign(x))/c^10)*x^2 + 105*(3*B*b^5*c^5*sign(x) - 4*A*b^4*c^6*sign(x))/c^10)*sqr
t(c*x^2 + b)*x + 7/1024*(3*B*b^6*sign(x) - 4*A*b^5*c*sign(x))*ln(abs(-sqrt(c)*x
+ sqrt(c*x^2 + b)))/c^(11/2) - 7/1024*(3*B*b^6*ln(sqrt(b)) - 4*A*b^5*c*ln(sqrt(b
)))*sign(x)/c^(11/2)