3.107 \(\int x^3 \left (A+B x^2\right ) \left (b x^2+c x^4\right )^{3/2} \, dx\)

Optimal. Leaf size=167 \[ \frac{b^5 (7 b B-12 A c) \tanh ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{b x^2+c x^4}}\right )}{1024 c^{9/2}}-\frac{b^3 \left (b+2 c x^2\right ) \sqrt{b x^2+c x^4} (7 b B-12 A c)}{1024 c^4}+\frac{b \left (b+2 c x^2\right ) \left (b x^2+c x^4\right )^{3/2} (7 b B-12 A c)}{384 c^3}-\frac{\left (b x^2+c x^4\right )^{5/2} \left (-12 A c+7 b B-10 B c x^2\right )}{120 c^2} \]

[Out]

-(b^3*(7*b*B - 12*A*c)*(b + 2*c*x^2)*Sqrt[b*x^2 + c*x^4])/(1024*c^4) + (b*(7*b*B
- 12*A*c)*(b + 2*c*x^2)*(b*x^2 + c*x^4)^(3/2))/(384*c^3) - ((7*b*B - 12*A*c - 1
0*B*c*x^2)*(b*x^2 + c*x^4)^(5/2))/(120*c^2) + (b^5*(7*b*B - 12*A*c)*ArcTanh[(Sqr
t[c]*x^2)/Sqrt[b*x^2 + c*x^4]])/(1024*c^(9/2))

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Rubi [A]  time = 0.437428, antiderivative size = 167, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.192 \[ \frac{b^5 (7 b B-12 A c) \tanh ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{b x^2+c x^4}}\right )}{1024 c^{9/2}}-\frac{b^3 \left (b+2 c x^2\right ) \sqrt{b x^2+c x^4} (7 b B-12 A c)}{1024 c^4}+\frac{b \left (b+2 c x^2\right ) \left (b x^2+c x^4\right )^{3/2} (7 b B-12 A c)}{384 c^3}-\frac{\left (b x^2+c x^4\right )^{5/2} \left (-12 A c+7 b B-10 B c x^2\right )}{120 c^2} \]

Antiderivative was successfully verified.

[In]  Int[x^3*(A + B*x^2)*(b*x^2 + c*x^4)^(3/2),x]

[Out]

-(b^3*(7*b*B - 12*A*c)*(b + 2*c*x^2)*Sqrt[b*x^2 + c*x^4])/(1024*c^4) + (b*(7*b*B
- 12*A*c)*(b + 2*c*x^2)*(b*x^2 + c*x^4)^(3/2))/(384*c^3) - ((7*b*B - 12*A*c - 1
0*B*c*x^2)*(b*x^2 + c*x^4)^(5/2))/(120*c^2) + (b^5*(7*b*B - 12*A*c)*ArcTanh[(Sqr
t[c]*x^2)/Sqrt[b*x^2 + c*x^4]])/(1024*c^(9/2))

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Rubi in Sympy [A]  time = 35.7382, size = 173, normalized size = 1.04 \[ \frac{B x^{2} \left (b x^{2} + c x^{4}\right )^{\frac{5}{2}}}{12 c} - \frac{b^{5} \left (12 A c - 7 B b\right ) \operatorname{atanh}{\left (\frac{\sqrt{c} x^{2}}{\sqrt{b x^{2} + c x^{4}}} \right )}}{1024 c^{\frac{9}{2}}} + \frac{b^{3} \left (b + 2 c x^{2}\right ) \left (12 A c - 7 B b\right ) \sqrt{b x^{2} + c x^{4}}}{1024 c^{4}} - \frac{b \left (b + 2 c x^{2}\right ) \left (12 A c - 7 B b\right ) \left (b x^{2} + c x^{4}\right )^{\frac{3}{2}}}{384 c^{3}} + \frac{\left (12 A c - 7 B b\right ) \left (b x^{2} + c x^{4}\right )^{\frac{5}{2}}}{120 c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**3*(B*x**2+A)*(c*x**4+b*x**2)**(3/2),x)

[Out]

B*x**2*(b*x**2 + c*x**4)**(5/2)/(12*c) - b**5*(12*A*c - 7*B*b)*atanh(sqrt(c)*x**
2/sqrt(b*x**2 + c*x**4))/(1024*c**(9/2)) + b**3*(b + 2*c*x**2)*(12*A*c - 7*B*b)*
sqrt(b*x**2 + c*x**4)/(1024*c**4) - b*(b + 2*c*x**2)*(12*A*c - 7*B*b)*(b*x**2 +
c*x**4)**(3/2)/(384*c**3) + (12*A*c - 7*B*b)*(b*x**2 + c*x**4)**(5/2)/(120*c**2)

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Mathematica [A]  time = 0.364703, size = 192, normalized size = 1.15 \[ \frac{x \sqrt{b+c x^2} \left (15 b^5 (7 b B-12 A c) \log \left (\sqrt{c} \sqrt{b+c x^2}+c x\right )+\sqrt{c} x \sqrt{b+c x^2} \left (10 b^4 c \left (18 A+7 B x^2\right )-8 b^3 c^2 x^2 \left (15 A+7 B x^2\right )+48 b^2 c^3 x^4 \left (2 A+B x^2\right )+64 b c^4 x^6 \left (33 A+26 B x^2\right )+256 c^5 x^8 \left (6 A+5 B x^2\right )-105 b^5 B\right )\right )}{15360 c^{9/2} \sqrt{x^2 \left (b+c x^2\right )}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^3*(A + B*x^2)*(b*x^2 + c*x^4)^(3/2),x]

[Out]

(x*Sqrt[b + c*x^2]*(Sqrt[c]*x*Sqrt[b + c*x^2]*(-105*b^5*B + 48*b^2*c^3*x^4*(2*A
+ B*x^2) + 256*c^5*x^8*(6*A + 5*B*x^2) - 8*b^3*c^2*x^2*(15*A + 7*B*x^2) + 10*b^4
*c*(18*A + 7*B*x^2) + 64*b*c^4*x^6*(33*A + 26*B*x^2)) + 15*b^5*(7*b*B - 12*A*c)*
Log[c*x + Sqrt[c]*Sqrt[b + c*x^2]]))/(15360*c^(9/2)*Sqrt[x^2*(b + c*x^2)])

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Maple [A]  time = 0.018, size = 291, normalized size = 1.7 \[{\frac{1}{15360\,{x}^{3}} \left ( c{x}^{4}+b{x}^{2} \right ) ^{{\frac{3}{2}}} \left ( 1280\,B{x}^{7} \left ( c{x}^{2}+b \right ) ^{5/2}{c}^{15/2}+1536\,A{x}^{5} \left ( c{x}^{2}+b \right ) ^{5/2}{c}^{15/2}-896\,Bb{x}^{5} \left ( c{x}^{2}+b \right ) ^{5/2}{c}^{13/2}-960\,Ab{x}^{3} \left ( c{x}^{2}+b \right ) ^{5/2}{c}^{13/2}+560\,B{b}^{2}{x}^{3} \left ( c{x}^{2}+b \right ) ^{5/2}{c}^{11/2}+480\,A{b}^{2}x \left ( c{x}^{2}+b \right ) ^{5/2}{c}^{11/2}-280\,B{b}^{3}x \left ( c{x}^{2}+b \right ) ^{5/2}{c}^{9/2}-120\,A{b}^{3}x \left ( c{x}^{2}+b \right ) ^{3/2}{c}^{11/2}+70\,B{b}^{4}x \left ( c{x}^{2}+b \right ) ^{3/2}{c}^{9/2}-180\,A{b}^{4}x\sqrt{c{x}^{2}+b}{c}^{11/2}+105\,B{b}^{5}x\sqrt{c{x}^{2}+b}{c}^{9/2}-180\,A{b}^{5}\ln \left ( \sqrt{c}x+\sqrt{c{x}^{2}+b} \right ){c}^{5}+105\,B{b}^{6}\ln \left ( \sqrt{c}x+\sqrt{c{x}^{2}+b} \right ){c}^{4} \right ) \left ( c{x}^{2}+b \right ) ^{-{\frac{3}{2}}}{c}^{-{\frac{17}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^3*(B*x^2+A)*(c*x^4+b*x^2)^(3/2),x)

[Out]

1/15360*(c*x^4+b*x^2)^(3/2)*(1280*B*x^7*(c*x^2+b)^(5/2)*c^(15/2)+1536*A*x^5*(c*x
^2+b)^(5/2)*c^(15/2)-896*B*b*x^5*(c*x^2+b)^(5/2)*c^(13/2)-960*A*b*x^3*(c*x^2+b)^
(5/2)*c^(13/2)+560*B*b^2*x^3*(c*x^2+b)^(5/2)*c^(11/2)+480*A*b^2*x*(c*x^2+b)^(5/2
)*c^(11/2)-280*B*b^3*x*(c*x^2+b)^(5/2)*c^(9/2)-120*A*b^3*x*(c*x^2+b)^(3/2)*c^(11
/2)+70*B*b^4*x*(c*x^2+b)^(3/2)*c^(9/2)-180*A*b^4*x*(c*x^2+b)^(1/2)*c^(11/2)+105*
B*b^5*x*(c*x^2+b)^(1/2)*c^(9/2)-180*A*b^5*ln(c^(1/2)*x+(c*x^2+b)^(1/2))*c^5+105*
B*b^6*ln(c^(1/2)*x+(c*x^2+b)^(1/2))*c^4)/x^3/(c*x^2+b)^(3/2)/c^(17/2)

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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((c*x^4 + b*x^2)^(3/2)*(B*x^2 + A)*x^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.578956, size = 1, normalized size = 0.01 \[ \left [-\frac{15 \,{\left (7 \, B b^{6} - 12 \, A b^{5} c\right )} \sqrt{c} \log \left (-{\left (2 \, c x^{2} + b\right )} \sqrt{c} + 2 \, \sqrt{c x^{4} + b x^{2}} c\right ) - 2 \,{\left (1280 \, B c^{6} x^{10} + 128 \,{\left (13 \, B b c^{5} + 12 \, A c^{6}\right )} x^{8} - 105 \, B b^{5} c + 180 \, A b^{4} c^{2} + 48 \,{\left (B b^{2} c^{4} + 44 \, A b c^{5}\right )} x^{6} - 8 \,{\left (7 \, B b^{3} c^{3} - 12 \, A b^{2} c^{4}\right )} x^{4} + 10 \,{\left (7 \, B b^{4} c^{2} - 12 \, A b^{3} c^{3}\right )} x^{2}\right )} \sqrt{c x^{4} + b x^{2}}}{30720 \, c^{5}}, -\frac{15 \,{\left (7 \, B b^{6} - 12 \, A b^{5} c\right )} \sqrt{-c} \arctan \left (\frac{\sqrt{-c} x^{2}}{\sqrt{c x^{4} + b x^{2}}}\right ) -{\left (1280 \, B c^{6} x^{10} + 128 \,{\left (13 \, B b c^{5} + 12 \, A c^{6}\right )} x^{8} - 105 \, B b^{5} c + 180 \, A b^{4} c^{2} + 48 \,{\left (B b^{2} c^{4} + 44 \, A b c^{5}\right )} x^{6} - 8 \,{\left (7 \, B b^{3} c^{3} - 12 \, A b^{2} c^{4}\right )} x^{4} + 10 \,{\left (7 \, B b^{4} c^{2} - 12 \, A b^{3} c^{3}\right )} x^{2}\right )} \sqrt{c x^{4} + b x^{2}}}{15360 \, c^{5}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^4 + b*x^2)^(3/2)*(B*x^2 + A)*x^3,x, algorithm="fricas")

[Out]

[-1/30720*(15*(7*B*b^6 - 12*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*(13*B*b*c^5 + 12*A*c^6)*x^8 - 105*
B*b^5*c + 180*A*b^4*c^2 + 48*(B*b^2*c^4 + 44*A*b*c^5)*x^6 - 8*(7*B*b^3*c^3 - 12*
A*b^2*c^4)*x^4 + 10*(7*B*b^4*c^2 - 12*A*b^3*c^3)*x^2)*sqrt(c*x^4 + b*x^2))/c^5,
-1/15360*(15*(7*B*b^6 - 12*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*(13*B*b*c^5 + 12*A*c^6)*x^8 - 105*B*b^5*c + 180*A
*b^4*c^2 + 48*(B*b^2*c^4 + 44*A*b*c^5)*x^6 - 8*(7*B*b^3*c^3 - 12*A*b^2*c^4)*x^4
+ 10*(7*B*b^4*c^2 - 12*A*b^3*c^3)*x^2)*sqrt(c*x^4 + b*x^2))/c^5]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**3*(B*x**2+A)*(c*x**4+b*x**2)**(3/2),x)

[Out]

Integral(x**3*(x**2*(b + c*x**2))**(3/2)*(A + B*x**2), x)

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GIAC/XCAS [A]  time = 0.222713, size = 335, normalized size = 2.01 \[ \frac{1}{15360} \,{\left (2 \,{\left (4 \,{\left (2 \,{\left (8 \,{\left (10 \, B c x^{2}{\rm sign}\left (x\right ) + \frac{13 \, B b c^{10}{\rm sign}\left (x\right ) + 12 \, A c^{11}{\rm sign}\left (x\right )}{c^{10}}\right )} x^{2} + \frac{3 \,{\left (B b^{2} c^{9}{\rm sign}\left (x\right ) + 44 \, A b c^{10}{\rm sign}\left (x\right )\right )}}{c^{10}}\right )} x^{2} - \frac{7 \, B b^{3} c^{8}{\rm sign}\left (x\right ) - 12 \, A b^{2} c^{9}{\rm sign}\left (x\right )}{c^{10}}\right )} x^{2} + \frac{5 \,{\left (7 \, B b^{4} c^{7}{\rm sign}\left (x\right ) - 12 \, A b^{3} c^{8}{\rm sign}\left (x\right )\right )}}{c^{10}}\right )} x^{2} - \frac{15 \,{\left (7 \, B b^{5} c^{6}{\rm sign}\left (x\right ) - 12 \, A b^{4} c^{7}{\rm sign}\left (x\right )\right )}}{c^{10}}\right )} \sqrt{c x^{2} + b} x - \frac{{\left (7 \, B b^{6}{\rm sign}\left (x\right ) - 12 \, A b^{5} c{\rm sign}\left (x\right )\right )}{\rm ln}\left ({\left | -\sqrt{c} x + \sqrt{c x^{2} + b} \right |}\right )}{1024 \, c^{\frac{9}{2}}} + \frac{{\left (7 \, B b^{6}{\rm ln}\left (\sqrt{b}\right ) - 12 \, A b^{5} c{\rm ln}\left (\sqrt{b}\right )\right )}{\rm sign}\left (x\right )}{1024 \, c^{\frac{9}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^4 + b*x^2)^(3/2)*(B*x^2 + A)*x^3,x, algorithm="giac")

[Out]

1/15360*(2*(4*(2*(8*(10*B*c*x^2*sign(x) + (13*B*b*c^10*sign(x) + 12*A*c^11*sign(
x))/c^10)*x^2 + 3*(B*b^2*c^9*sign(x) + 44*A*b*c^10*sign(x))/c^10)*x^2 - (7*B*b^3
*c^8*sign(x) - 12*A*b^2*c^9*sign(x))/c^10)*x^2 + 5*(7*B*b^4*c^7*sign(x) - 12*A*b
^3*c^8*sign(x))/c^10)*x^2 - 15*(7*B*b^5*c^6*sign(x) - 12*A*b^4*c^7*sign(x))/c^10
)*sqrt(c*x^2 + b)*x - 1/1024*(7*B*b^6*sign(x) - 12*A*b^5*c*sign(x))*ln(abs(-sqrt
(c)*x + sqrt(c*x^2 + b)))/c^(9/2) + 1/1024*(7*B*b^6*ln(sqrt(b)) - 12*A*b^5*c*ln(
sqrt(b)))*sign(x)/c^(9/2)