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

Optimal. Leaf size=124 \[ -\frac{3 b^5 \tanh ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{b x^2+c x^4}}\right )}{256 c^{7/2}}+\frac{3 b^3 \left (b+2 c x^2\right ) \sqrt{b x^2+c x^4}}{256 c^3}-\frac{b \left (b+2 c x^2\right ) \left (b x^2+c x^4\right )^{3/2}}{32 c^2}+\frac{\left (b x^2+c x^4\right )^{5/2}}{10 c} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.24973, antiderivative size = 124, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.263 \[ -\frac{3 b^5 \tanh ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{b x^2+c x^4}}\right )}{256 c^{7/2}}+\frac{3 b^3 \left (b+2 c x^2\right ) \sqrt{b x^2+c x^4}}{256 c^3}-\frac{b \left (b+2 c x^2\right ) \left (b x^2+c x^4\right )^{3/2}}{32 c^2}+\frac{\left (b x^2+c x^4\right )^{5/2}}{10 c} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 21.5317, size = 112, normalized size = 0.9 \[ - \frac{3 b^{5} \operatorname{atanh}{\left (\frac{\sqrt{c} x^{2}}{\sqrt{b x^{2} + c x^{4}}} \right )}}{256 c^{\frac{7}{2}}} + \frac{3 b^{3} \left (b + 2 c x^{2}\right ) \sqrt{b x^{2} + c x^{4}}}{256 c^{3}} - \frac{b \left (b + 2 c x^{2}\right ) \left (b x^{2} + c x^{4}\right )^{\frac{3}{2}}}{32 c^{2}} + \frac{\left (b x^{2} + c x^{4}\right )^{\frac{5}{2}}}{10 c} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.172593, size = 125, normalized size = 1.01 \[ \frac{x \sqrt{b+c x^2} \left (\sqrt{c} x \sqrt{b+c x^2} \left (15 b^4-10 b^3 c x^2+8 b^2 c^2 x^4+176 b c^3 x^6+128 c^4 x^8\right )-15 b^5 \log \left (\sqrt{c} \sqrt{b+c x^2}+c x\right )\right )}{1280 c^{7/2} \sqrt{x^2 \left (b+c x^2\right )}} \]

Antiderivative was successfully verified.

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

[Out]

(x*Sqrt[b + c*x^2]*(Sqrt[c]*x*Sqrt[b + c*x^2]*(15*b^4 - 10*b^3*c*x^2 + 8*b^2*c^2
*x^4 + 176*b*c^3*x^6 + 128*c^4*x^8) - 15*b^5*Log[c*x + Sqrt[c]*Sqrt[b + c*x^2]])
)/(1280*c^(7/2)*Sqrt[x^2*(b + c*x^2)])

_______________________________________________________________________________________

Maple [A]  time = 0.015, size = 142, normalized size = 1.2 \[ -{\frac{1}{1280\,{x}^{3}} \left ( c{x}^{4}+b{x}^{2} \right ) ^{{\frac{3}{2}}} \left ( -128\,{x}^{5} \left ( c{x}^{2}+b \right ) ^{5/2}{c}^{5/2}+80\, \left ( c{x}^{2}+b \right ) ^{5/2}{c}^{3/2}{x}^{3}b-40\, \left ( c{x}^{2}+b \right ) ^{5/2}\sqrt{c}x{b}^{2}+10\, \left ( c{x}^{2}+b \right ) ^{3/2}\sqrt{c}x{b}^{3}+15\,\sqrt{c{x}^{2}+b}\sqrt{c}x{b}^{4}+15\,\ln \left ( x\sqrt{c}+\sqrt{c{x}^{2}+b} \right ){b}^{5} \right ) \left ( c{x}^{2}+b \right ) ^{-{\frac{3}{2}}}{c}^{-{\frac{7}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^3*(c*x^4+b*x^2)^(3/2),x)

[Out]

-1/1280*(c*x^4+b*x^2)^(3/2)*(-128*x^5*(c*x^2+b)^(5/2)*c^(5/2)+80*(c*x^2+b)^(5/2)
*c^(3/2)*x^3*b-40*(c*x^2+b)^(5/2)*c^(1/2)*x*b^2+10*(c*x^2+b)^(3/2)*c^(1/2)*x*b^3
+15*(c*x^2+b)^(1/2)*c^(1/2)*x*b^4+15*ln(x*c^(1/2)+(c*x^2+b)^(1/2))*b^5)/x^3/(c*x
^2+b)^(3/2)/c^(7/2)

_______________________________________________________________________________________

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^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.335906, size = 1, normalized size = 0.01 \[ \left [\frac{15 \, b^{5} \sqrt{c} \log \left (-{\left (2 \, c x^{2} + b\right )} \sqrt{c} + 2 \, \sqrt{c x^{4} + b x^{2}} c\right ) + 2 \,{\left (128 \, c^{5} x^{8} + 176 \, b c^{4} x^{6} + 8 \, b^{2} c^{3} x^{4} - 10 \, b^{3} c^{2} x^{2} + 15 \, b^{4} c\right )} \sqrt{c x^{4} + b x^{2}}}{2560 \, c^{4}}, \frac{15 \, b^{5} \sqrt{-c} \arctan \left (\frac{\sqrt{-c} x^{2}}{\sqrt{c x^{4} + b x^{2}}}\right ) +{\left (128 \, c^{5} x^{8} + 176 \, b c^{4} x^{6} + 8 \, b^{2} c^{3} x^{4} - 10 \, b^{3} c^{2} x^{2} + 15 \, b^{4} c\right )} \sqrt{c x^{4} + b x^{2}}}{1280 \, c^{4}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/2560*(15*b^5*sqrt(c)*log(-(2*c*x^2 + b)*sqrt(c) + 2*sqrt(c*x^4 + b*x^2)*c) +
2*(128*c^5*x^8 + 176*b*c^4*x^6 + 8*b^2*c^3*x^4 - 10*b^3*c^2*x^2 + 15*b^4*c)*sqrt
(c*x^4 + b*x^2))/c^4, 1/1280*(15*b^5*sqrt(-c)*arctan(sqrt(-c)*x^2/sqrt(c*x^4 + b
*x^2)) + (128*c^5*x^8 + 176*b*c^4*x^6 + 8*b^2*c^3*x^4 - 10*b^3*c^2*x^2 + 15*b^4*
c)*sqrt(c*x^4 + b*x^2))/c^4]

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.279853, size = 157, normalized size = 1.27 \[ -\frac{3 \, b^{5}{\rm ln}\left (\sqrt{b}\right ){\rm sign}\left (x\right )}{256 \, c^{\frac{7}{2}}} + \frac{3 \, b^{5}{\rm ln}\left ({\left | -\sqrt{c} x + \sqrt{c x^{2} + b} \right |}\right ){\rm sign}\left (x\right )}{256 \, c^{\frac{7}{2}}} + \frac{1}{1280} \,{\left (2 \,{\left (4 \,{\left (2 \,{\left (8 \, c x^{2}{\rm sign}\left (x\right ) + 11 \, b{\rm sign}\left (x\right )\right )} x^{2} + \frac{b^{2}{\rm sign}\left (x\right )}{c}\right )} x^{2} - \frac{5 \, b^{3}{\rm sign}\left (x\right )}{c^{2}}\right )} x^{2} + \frac{15 \, b^{4}{\rm sign}\left (x\right )}{c^{3}}\right )} \sqrt{c x^{2} + b} x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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