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

Optimal. Leaf size=148 \[ -\frac{3 b^4 (b B-2 A c) \tanh ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{b x^2+c x^4}}\right )}{256 c^{7/2}}+\frac{3 b^2 \left (b+2 c x^2\right ) \sqrt{b x^2+c x^4} (b B-2 A c)}{256 c^3}-\frac{\left (b+2 c x^2\right ) \left (b x^2+c x^4\right )^{3/2} (b B-2 A c)}{32 c^2}+\frac{B \left (b x^2+c x^4\right )^{5/2}}{10 c} \]

[Out]

(3*b^2*(b*B - 2*A*c)*(b + 2*c*x^2)*Sqrt[b*x^2 + c*x^4])/(256*c^3) - ((b*B - 2*A*
c)*(b + 2*c*x^2)*(b*x^2 + c*x^4)^(3/2))/(32*c^2) + (B*(b*x^2 + c*x^4)^(5/2))/(10
*c) - (3*b^4*(b*B - 2*A*c)*ArcTanh[(Sqrt[c]*x^2)/Sqrt[b*x^2 + c*x^4]])/(256*c^(7
/2))

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Rubi [A]  time = 0.33281, antiderivative size = 148, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208 \[ -\frac{3 b^4 (b B-2 A c) \tanh ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{b x^2+c x^4}}\right )}{256 c^{7/2}}+\frac{3 b^2 \left (b+2 c x^2\right ) \sqrt{b x^2+c x^4} (b B-2 A c)}{256 c^3}-\frac{\left (b+2 c x^2\right ) \left (b x^2+c x^4\right )^{3/2} (b B-2 A c)}{32 c^2}+\frac{B \left (b x^2+c x^4\right )^{5/2}}{10 c} \]

Antiderivative was successfully verified.

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

[Out]

(3*b^2*(b*B - 2*A*c)*(b + 2*c*x^2)*Sqrt[b*x^2 + c*x^4])/(256*c^3) - ((b*B - 2*A*
c)*(b + 2*c*x^2)*(b*x^2 + c*x^4)^(3/2))/(32*c^2) + (B*(b*x^2 + c*x^4)^(5/2))/(10
*c) - (3*b^4*(b*B - 2*A*c)*ArcTanh[(Sqrt[c]*x^2)/Sqrt[b*x^2 + c*x^4]])/(256*c^(7
/2))

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Rubi in Sympy [A]  time = 26.425, size = 138, normalized size = 0.93 \[ \frac{B \left (b x^{2} + c x^{4}\right )^{\frac{5}{2}}}{10 c} + \frac{3 b^{4} \left (2 A c - B b\right ) \operatorname{atanh}{\left (\frac{\sqrt{c} x^{2}}{\sqrt{b x^{2} + c x^{4}}} \right )}}{256 c^{\frac{7}{2}}} - \frac{3 b^{2} \left (b + 2 c x^{2}\right ) \left (2 A c - B b\right ) \sqrt{b x^{2} + c x^{4}}}{256 c^{3}} + \frac{\left (b + 2 c x^{2}\right ) \left (2 A c - B b\right ) \left (b x^{2} + c x^{4}\right )^{\frac{3}{2}}}{32 c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

B*(b*x**2 + c*x**4)**(5/2)/(10*c) + 3*b**4*(2*A*c - B*b)*atanh(sqrt(c)*x**2/sqrt
(b*x**2 + c*x**4))/(256*c**(7/2)) - 3*b**2*(b + 2*c*x**2)*(2*A*c - B*b)*sqrt(b*x
**2 + c*x**4)/(256*c**3) + (b + 2*c*x**2)*(2*A*c - B*b)*(b*x**2 + c*x**4)**(3/2)
/(32*c**2)

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Mathematica [A]  time = 0.343324, size = 166, normalized size = 1.12 \[ \frac{x \left (\sqrt{c} x \left (b+c x^2\right ) \left (-10 b^3 c \left (3 A+B x^2\right )+4 b^2 c^2 x^2 \left (5 A+2 B x^2\right )+16 b c^3 x^4 \left (15 A+11 B x^2\right )+32 c^4 x^6 \left (5 A+4 B x^2\right )+15 b^4 B\right )-15 b^4 \sqrt{b+c x^2} (b B-2 A c) \log \left (\sqrt{c} \sqrt{b+c x^2}+c x\right )\right )}{1280 c^{7/2} \sqrt{x^2 \left (b+c x^2\right )}} \]

Antiderivative was successfully verified.

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

[Out]

(x*(Sqrt[c]*x*(b + c*x^2)*(15*b^4*B - 10*b^3*c*(3*A + B*x^2) + 4*b^2*c^2*x^2*(5*
A + 2*B*x^2) + 32*c^4*x^6*(5*A + 4*B*x^2) + 16*b*c^3*x^4*(15*A + 11*B*x^2)) - 15
*b^4*(b*B - 2*A*c)*Sqrt[b + c*x^2]*Log[c*x + Sqrt[c]*Sqrt[b + c*x^2]]))/(1280*c^
(7/2)*Sqrt[x^2*(b + c*x^2)])

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Maple [A]  time = 0.016, size = 249, normalized size = 1.7 \[{\frac{1}{1280\,{x}^{3}} \left ( c{x}^{4}+b{x}^{2} \right ) ^{{\frac{3}{2}}} \left ( 128\,B{x}^{5} \left ( c{x}^{2}+b \right ) ^{5/2}{c}^{11/2}+160\,A{x}^{3} \left ( c{x}^{2}+b \right ) ^{5/2}{c}^{11/2}-80\,Bb{x}^{3} \left ( c{x}^{2}+b \right ) ^{5/2}{c}^{9/2}-80\,Abx \left ( c{x}^{2}+b \right ) ^{5/2}{c}^{9/2}+40\,B{b}^{2}x \left ( c{x}^{2}+b \right ) ^{5/2}{c}^{7/2}+20\,A{b}^{2}x \left ( c{x}^{2}+b \right ) ^{3/2}{c}^{9/2}-10\,B{b}^{3}x \left ( c{x}^{2}+b \right ) ^{3/2}{c}^{7/2}+30\,A{b}^{3}x\sqrt{c{x}^{2}+b}{c}^{9/2}-15\,B{b}^{4}x\sqrt{c{x}^{2}+b}{c}^{7/2}+30\,A{b}^{4}\ln \left ( \sqrt{c}x+\sqrt{c{x}^{2}+b} \right ){c}^{4}-15\,B{b}^{5}\ln \left ( \sqrt{c}x+\sqrt{c{x}^{2}+b} \right ){c}^{3} \right ) \left ( c{x}^{2}+b \right ) ^{-{\frac{3}{2}}}{c}^{-{\frac{13}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/1280*(c*x^4+b*x^2)^(3/2)*(128*B*x^5*(c*x^2+b)^(5/2)*c^(11/2)+160*A*x^3*(c*x^2+
b)^(5/2)*c^(11/2)-80*B*b*x^3*(c*x^2+b)^(5/2)*c^(9/2)-80*A*b*x*(c*x^2+b)^(5/2)*c^
(9/2)+40*B*b^2*x*(c*x^2+b)^(5/2)*c^(7/2)+20*A*b^2*x*(c*x^2+b)^(3/2)*c^(9/2)-10*B
*b^3*x*(c*x^2+b)^(3/2)*c^(7/2)+30*A*b^3*x*(c*x^2+b)^(1/2)*c^(9/2)-15*B*b^4*x*(c*
x^2+b)^(1/2)*c^(7/2)+30*A*b^4*ln(c^(1/2)*x+(c*x^2+b)^(1/2))*c^4-15*B*b^5*ln(c^(1
/2)*x+(c*x^2+b)^(1/2))*c^3)/x^3/(c*x^2+b)^(3/2)/c^(13/2)

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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,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.376439, size = 1, normalized size = 0.01 \[ \left [-\frac{15 \,{\left (B b^{5} - 2 \, A b^{4} c\right )} \sqrt{c} \log \left (-{\left (2 \, c x^{2} + b\right )} \sqrt{c} - 2 \, \sqrt{c x^{4} + b x^{2}} c\right ) - 2 \,{\left (128 \, B c^{5} x^{8} + 16 \,{\left (11 \, B b c^{4} + 10 \, A c^{5}\right )} x^{6} + 15 \, B b^{4} c - 30 \, A b^{3} c^{2} + 8 \,{\left (B b^{2} c^{3} + 30 \, A b c^{4}\right )} x^{4} - 10 \,{\left (B b^{3} c^{2} - 2 \, A b^{2} c^{3}\right )} x^{2}\right )} \sqrt{c x^{4} + b x^{2}}}{2560 \, c^{4}}, \frac{15 \,{\left (B b^{5} - 2 \, A b^{4} c\right )} \sqrt{-c} \arctan \left (\frac{\sqrt{-c} x^{2}}{\sqrt{c x^{4} + b x^{2}}}\right ) +{\left (128 \, B c^{5} x^{8} + 16 \,{\left (11 \, B b c^{4} + 10 \, A c^{5}\right )} x^{6} + 15 \, B b^{4} c - 30 \, A b^{3} c^{2} + 8 \,{\left (B b^{2} c^{3} + 30 \, A b c^{4}\right )} x^{4} - 10 \,{\left (B b^{3} c^{2} - 2 \, A b^{2} c^{3}\right )} x^{2}\right )} \sqrt{c x^{4} + b x^{2}}}{1280 \, c^{4}}\right ] \]

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,x, algorithm="fricas")

[Out]

[-1/2560*(15*(B*b^5 - 2*A*b^4*c)*sqrt(c)*log(-(2*c*x^2 + b)*sqrt(c) - 2*sqrt(c*x
^4 + b*x^2)*c) - 2*(128*B*c^5*x^8 + 16*(11*B*b*c^4 + 10*A*c^5)*x^6 + 15*B*b^4*c
- 30*A*b^3*c^2 + 8*(B*b^2*c^3 + 30*A*b*c^4)*x^4 - 10*(B*b^3*c^2 - 2*A*b^2*c^3)*x
^2)*sqrt(c*x^4 + b*x^2))/c^4, 1/1280*(15*(B*b^5 - 2*A*b^4*c)*sqrt(-c)*arctan(sqr
t(-c)*x^2/sqrt(c*x^4 + b*x^2)) + (128*B*c^5*x^8 + 16*(11*B*b*c^4 + 10*A*c^5)*x^6
+ 15*B*b^4*c - 30*A*b^3*c^2 + 8*(B*b^2*c^3 + 30*A*b*c^4)*x^4 - 10*(B*b^3*c^2 -
2*A*b^2*c^3)*x^2)*sqrt(c*x^4 + b*x^2))/c^4]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.22073, size = 282, normalized size = 1.91 \[ \frac{1}{1280} \,{\left (2 \,{\left (4 \,{\left (2 \,{\left (8 \, B c x^{2}{\rm sign}\left (x\right ) + \frac{11 \, B b c^{8}{\rm sign}\left (x\right ) + 10 \, A c^{9}{\rm sign}\left (x\right )}{c^{8}}\right )} x^{2} + \frac{B b^{2} c^{7}{\rm sign}\left (x\right ) + 30 \, A b c^{8}{\rm sign}\left (x\right )}{c^{8}}\right )} x^{2} - \frac{5 \,{\left (B b^{3} c^{6}{\rm sign}\left (x\right ) - 2 \, A b^{2} c^{7}{\rm sign}\left (x\right )\right )}}{c^{8}}\right )} x^{2} + \frac{15 \,{\left (B b^{4} c^{5}{\rm sign}\left (x\right ) - 2 \, A b^{3} c^{6}{\rm sign}\left (x\right )\right )}}{c^{8}}\right )} \sqrt{c x^{2} + b} x + \frac{3 \,{\left (B b^{5}{\rm sign}\left (x\right ) - 2 \, A b^{4} c{\rm sign}\left (x\right )\right )}{\rm ln}\left ({\left | -\sqrt{c} x + \sqrt{c x^{2} + b} \right |}\right )}{256 \, c^{\frac{7}{2}}} - \frac{3 \,{\left (B b^{5}{\rm ln}\left (\sqrt{b}\right ) - 2 \, A b^{4} c{\rm ln}\left (\sqrt{b}\right )\right )}{\rm sign}\left (x\right )}{256 \, c^{\frac{7}{2}}} \]

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,x, algorithm="giac")

[Out]

1/1280*(2*(4*(2*(8*B*c*x^2*sign(x) + (11*B*b*c^8*sign(x) + 10*A*c^9*sign(x))/c^8
)*x^2 + (B*b^2*c^7*sign(x) + 30*A*b*c^8*sign(x))/c^8)*x^2 - 5*(B*b^3*c^6*sign(x)
- 2*A*b^2*c^7*sign(x))/c^8)*x^2 + 15*(B*b^4*c^5*sign(x) - 2*A*b^3*c^6*sign(x))/
c^8)*sqrt(c*x^2 + b)*x + 3/256*(B*b^5*sign(x) - 2*A*b^4*c*sign(x))*ln(abs(-sqrt(
c)*x + sqrt(c*x^2 + b)))/c^(7/2) - 3/256*(B*b^5*ln(sqrt(b)) - 2*A*b^4*c*ln(sqrt(
b)))*sign(x)/c^(7/2)