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

Optimal. Leaf size=101 \[ \frac{3 b^4 \tanh ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{b x^2+c x^4}}\right )}{128 c^{5/2}}-\frac{3 b^2 \left (b+2 c x^2\right ) \sqrt{b x^2+c x^4}}{128 c^2}+\frac{\left (b+2 c x^2\right ) \left (b x^2+c x^4\right )^{3/2}}{16 c} \]

[Out]

(-3*b^2*(b + 2*c*x^2)*Sqrt[b*x^2 + c*x^4])/(128*c^2) + ((b + 2*c*x^2)*(b*x^2 + c
*x^4)^(3/2))/(16*c) + (3*b^4*ArcTanh[(Sqrt[c]*x^2)/Sqrt[b*x^2 + c*x^4]])/(128*c^
(5/2))

_______________________________________________________________________________________

Rubi [A]  time = 0.168027, antiderivative size = 101, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.235 \[ \frac{3 b^4 \tanh ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{b x^2+c x^4}}\right )}{128 c^{5/2}}-\frac{3 b^2 \left (b+2 c x^2\right ) \sqrt{b x^2+c x^4}}{128 c^2}+\frac{\left (b+2 c x^2\right ) \left (b x^2+c x^4\right )^{3/2}}{16 c} \]

Antiderivative was successfully verified.

[In]  Int[x*(b*x^2 + c*x^4)^(3/2),x]

[Out]

(-3*b^2*(b + 2*c*x^2)*Sqrt[b*x^2 + c*x^4])/(128*c^2) + ((b + 2*c*x^2)*(b*x^2 + c
*x^4)^(3/2))/(16*c) + (3*b^4*ArcTanh[(Sqrt[c]*x^2)/Sqrt[b*x^2 + c*x^4]])/(128*c^
(5/2))

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 14.4037, size = 92, normalized size = 0.91 \[ \frac{3 b^{4} \operatorname{atanh}{\left (\frac{\sqrt{c} x^{2}}{\sqrt{b x^{2} + c x^{4}}} \right )}}{128 c^{\frac{5}{2}}} - \frac{3 b^{2} \left (b + 2 c x^{2}\right ) \sqrt{b x^{2} + c x^{4}}}{128 c^{2}} + \frac{\left (b + 2 c x^{2}\right ) \left (b x^{2} + c x^{4}\right )^{\frac{3}{2}}}{16 c} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3*b**4*atanh(sqrt(c)*x**2/sqrt(b*x**2 + c*x**4))/(128*c**(5/2)) - 3*b**2*(b + 2*
c*x**2)*sqrt(b*x**2 + c*x**4)/(128*c**2) + (b + 2*c*x**2)*(b*x**2 + c*x**4)**(3/
2)/(16*c)

_______________________________________________________________________________________

Mathematica [A]  time = 0.144909, size = 114, normalized size = 1.13 \[ \frac{x \sqrt{b+c x^2} \left (3 b^4 \log \left (\sqrt{c} \sqrt{b+c x^2}+c x\right )+\sqrt{c} x \sqrt{b+c x^2} \left (-3 b^3+2 b^2 c x^2+24 b c^2 x^4+16 c^3 x^6\right )\right )}{128 c^{5/2} \sqrt{x^2 \left (b+c x^2\right )}} \]

Antiderivative was successfully verified.

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

[Out]

(x*Sqrt[b + c*x^2]*(Sqrt[c]*x*Sqrt[b + c*x^2]*(-3*b^3 + 2*b^2*c*x^2 + 24*b*c^2*x
^4 + 16*c^3*x^6) + 3*b^4*Log[c*x + Sqrt[c]*Sqrt[b + c*x^2]]))/(128*c^(5/2)*Sqrt[
x^2*(b + c*x^2)])

_______________________________________________________________________________________

Maple [A]  time = 0.011, size = 122, normalized size = 1.2 \[{\frac{1}{128\,{x}^{3}} \left ( c{x}^{4}+b{x}^{2} \right ) ^{{\frac{3}{2}}} \left ( 16\,{x}^{3} \left ( c{x}^{2}+b \right ) ^{5/2}{c}^{3/2}-8\, \left ( c{x}^{2}+b \right ) ^{5/2}\sqrt{c}xb+2\, \left ( c{x}^{2}+b \right ) ^{3/2}\sqrt{c}x{b}^{2}+3\,\sqrt{c{x}^{2}+b}\sqrt{c}x{b}^{3}+3\,\ln \left ( x\sqrt{c}+\sqrt{c{x}^{2}+b} \right ){b}^{4} \right ) \left ( c{x}^{2}+b \right ) ^{-{\frac{3}{2}}}{c}^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x*(c*x^4+b*x^2)^(3/2),x)

[Out]

1/128*(c*x^4+b*x^2)^(3/2)*(16*x^3*(c*x^2+b)^(5/2)*c^(3/2)-8*(c*x^2+b)^(5/2)*c^(1
/2)*x*b+2*(c*x^2+b)^(3/2)*c^(1/2)*x*b^2+3*(c*x^2+b)^(1/2)*c^(1/2)*x*b^3+3*ln(x*c
^(1/2)+(c*x^2+b)^(1/2))*b^4)/x^3/(c*x^2+b)^(3/2)/c^(5/2)

_______________________________________________________________________________________

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)*x,x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.29652, size = 1, normalized size = 0.01 \[ \left [\frac{3 \, b^{4} \sqrt{c} \log \left (-{\left (2 \, c x^{2} + b\right )} \sqrt{c} - 2 \, \sqrt{c x^{4} + b x^{2}} c\right ) + 2 \,{\left (16 \, c^{4} x^{6} + 24 \, b c^{3} x^{4} + 2 \, b^{2} c^{2} x^{2} - 3 \, b^{3} c\right )} \sqrt{c x^{4} + b x^{2}}}{256 \, c^{3}}, -\frac{3 \, b^{4} \sqrt{-c} \arctan \left (\frac{\sqrt{-c} x^{2}}{\sqrt{c x^{4} + b x^{2}}}\right ) -{\left (16 \, c^{4} x^{6} + 24 \, b c^{3} x^{4} + 2 \, b^{2} c^{2} x^{2} - 3 \, b^{3} c\right )} \sqrt{c x^{4} + b x^{2}}}{128 \, c^{3}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/256*(3*b^4*sqrt(c)*log(-(2*c*x^2 + b)*sqrt(c) - 2*sqrt(c*x^4 + b*x^2)*c) + 2*
(16*c^4*x^6 + 24*b*c^3*x^4 + 2*b^2*c^2*x^2 - 3*b^3*c)*sqrt(c*x^4 + b*x^2))/c^3,
-1/128*(3*b^4*sqrt(-c)*arctan(sqrt(-c)*x^2/sqrt(c*x^4 + b*x^2)) - (16*c^4*x^6 +
24*b*c^3*x^4 + 2*b^2*c^2*x^2 - 3*b^3*c)*sqrt(c*x^4 + b*x^2))/c^3]

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.278967, size = 135, normalized size = 1.34 \[ \frac{3 \, b^{4}{\rm ln}\left (\sqrt{b}\right ){\rm sign}\left (x\right )}{128 \, c^{\frac{5}{2}}} - \frac{3 \, b^{4}{\rm ln}\left ({\left | -\sqrt{c} x + \sqrt{c x^{2} + b} \right |}\right ){\rm sign}\left (x\right )}{128 \, c^{\frac{5}{2}}} + \frac{1}{128} \,{\left (2 \,{\left (4 \,{\left (2 \, c x^{2}{\rm sign}\left (x\right ) + 3 \, b{\rm sign}\left (x\right )\right )} x^{2} + \frac{b^{2}{\rm sign}\left (x\right )}{c}\right )} x^{2} - \frac{3 \, b^{3}{\rm sign}\left (x\right )}{c^{2}}\right )} \sqrt{c x^{2} + b} x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3/128*b^4*ln(sqrt(b))*sign(x)/c^(5/2) - 3/128*b^4*ln(abs(-sqrt(c)*x + sqrt(c*x^2
+ b)))*sign(x)/c^(5/2) + 1/128*(2*(4*(2*c*x^2*sign(x) + 3*b*sign(x))*x^2 + b^2*
sign(x)/c)*x^2 - 3*b^3*sign(x)/c^2)*sqrt(c*x^2 + b)*x