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

Optimal. Leaf size=110 \[ -\frac{3 b^5 \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{128 c^{7/2}}+\frac{3 b^3 (b+2 c x) \sqrt{b x+c x^2}}{128 c^3}-\frac{b (b+2 c x) \left (b x+c x^2\right )^{3/2}}{16 c^2}+\frac{\left (b x+c x^2\right )^{5/2}}{5 c} \]

[Out]

(3*b^3*(b + 2*c*x)*Sqrt[b*x + c*x^2])/(128*c^3) - (b*(b + 2*c*x)*(b*x + c*x^2)^(
3/2))/(16*c^2) + (b*x + c*x^2)^(5/2)/(5*c) - (3*b^5*ArcTanh[(Sqrt[c]*x)/Sqrt[b*x
+ c*x^2]])/(128*c^(7/2))

_______________________________________________________________________________________

Rubi [A]  time = 0.103964, antiderivative size = 110, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267 \[ -\frac{3 b^5 \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{128 c^{7/2}}+\frac{3 b^3 (b+2 c x) \sqrt{b x+c x^2}}{128 c^3}-\frac{b (b+2 c x) \left (b x+c x^2\right )^{3/2}}{16 c^2}+\frac{\left (b x+c x^2\right )^{5/2}}{5 c} \]

Antiderivative was successfully verified.

[In]  Int[x*(b*x + c*x^2)^(3/2),x]

[Out]

(3*b^3*(b + 2*c*x)*Sqrt[b*x + c*x^2])/(128*c^3) - (b*(b + 2*c*x)*(b*x + c*x^2)^(
3/2))/(16*c^2) + (b*x + c*x^2)^(5/2)/(5*c) - (3*b^5*ArcTanh[(Sqrt[c]*x)/Sqrt[b*x
+ c*x^2]])/(128*c^(7/2))

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 12.4817, size = 100, normalized size = 0.91 \[ - \frac{3 b^{5} \operatorname{atanh}{\left (\frac{\sqrt{c} x}{\sqrt{b x + c x^{2}}} \right )}}{128 c^{\frac{7}{2}}} + \frac{3 b^{3} \left (b + 2 c x\right ) \sqrt{b x + c x^{2}}}{128 c^{3}} - \frac{b \left (b + 2 c x\right ) \left (b x + c x^{2}\right )^{\frac{3}{2}}}{16 c^{2}} + \frac{\left (b x + c x^{2}\right )^{\frac{5}{2}}}{5 c} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-3*b**5*atanh(sqrt(c)*x/sqrt(b*x + c*x**2))/(128*c**(7/2)) + 3*b**3*(b + 2*c*x)*
sqrt(b*x + c*x**2)/(128*c**3) - b*(b + 2*c*x)*(b*x + c*x**2)**(3/2)/(16*c**2) +
(b*x + c*x**2)**(5/2)/(5*c)

_______________________________________________________________________________________

Mathematica [A]  time = 0.123335, size = 111, normalized size = 1.01 \[ \frac{\sqrt{x (b+c x)} \left (\sqrt{c} \left (15 b^4-10 b^3 c x+8 b^2 c^2 x^2+176 b c^3 x^3+128 c^4 x^4\right )-\frac{15 b^5 \log \left (\sqrt{c} \sqrt{b+c x}+c \sqrt{x}\right )}{\sqrt{x} \sqrt{b+c x}}\right )}{640 c^{7/2}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[x*(b + c*x)]*(Sqrt[c]*(15*b^4 - 10*b^3*c*x + 8*b^2*c^2*x^2 + 176*b*c^3*x^3
+ 128*c^4*x^4) - (15*b^5*Log[c*Sqrt[x] + Sqrt[c]*Sqrt[b + c*x]])/(Sqrt[x]*Sqrt[
b + c*x])))/(640*c^(7/2))

_______________________________________________________________________________________

Maple [A]  time = 0.007, size = 126, normalized size = 1.2 \[{\frac{1}{5\,c} \left ( c{x}^{2}+bx \right ) ^{{\frac{5}{2}}}}-{\frac{bx}{8\,c} \left ( c{x}^{2}+bx \right ) ^{{\frac{3}{2}}}}-{\frac{{b}^{2}}{16\,{c}^{2}} \left ( c{x}^{2}+bx \right ) ^{{\frac{3}{2}}}}+{\frac{3\,{b}^{3}x}{64\,{c}^{2}}\sqrt{c{x}^{2}+bx}}+{\frac{3\,{b}^{4}}{128\,{c}^{3}}\sqrt{c{x}^{2}+bx}}-{\frac{3\,{b}^{5}}{256}\ln \left ({1 \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx} \right ){c}^{-{\frac{7}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x*(c*x^2+b*x)^(3/2),x)

[Out]

1/5*(c*x^2+b*x)^(5/2)/c-1/8*b/c*x*(c*x^2+b*x)^(3/2)-1/16*b^2/c^2*(c*x^2+b*x)^(3/
2)+3/64*b^3/c^2*(c*x^2+b*x)^(1/2)*x+3/128*b^4/c^3*(c*x^2+b*x)^(1/2)-3/256*b^5/c^
(7/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x)^(1/2))

_______________________________________________________________________________________

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^2 + b*x)^(3/2)*x,x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.229778, size = 1, normalized size = 0.01 \[ \left [\frac{15 \, b^{5} \log \left ({\left (2 \, c x + b\right )} \sqrt{c} - 2 \, \sqrt{c x^{2} + b x} c\right ) + 2 \,{\left (128 \, c^{4} x^{4} + 176 \, b c^{3} x^{3} + 8 \, b^{2} c^{2} x^{2} - 10 \, b^{3} c x + 15 \, b^{4}\right )} \sqrt{c x^{2} + b x} \sqrt{c}}{1280 \, c^{\frac{7}{2}}}, -\frac{15 \, b^{5} \arctan \left (\frac{\sqrt{c x^{2} + b x} \sqrt{-c}}{c x}\right ) -{\left (128 \, c^{4} x^{4} + 176 \, b c^{3} x^{3} + 8 \, b^{2} c^{2} x^{2} - 10 \, b^{3} c x + 15 \, b^{4}\right )} \sqrt{c x^{2} + b x} \sqrt{-c}}{640 \, \sqrt{-c} c^{3}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/1280*(15*b^5*log((2*c*x + b)*sqrt(c) - 2*sqrt(c*x^2 + b*x)*c) + 2*(128*c^4*x^
4 + 176*b*c^3*x^3 + 8*b^2*c^2*x^2 - 10*b^3*c*x + 15*b^4)*sqrt(c*x^2 + b*x)*sqrt(
c))/c^(7/2), -1/640*(15*b^5*arctan(sqrt(c*x^2 + b*x)*sqrt(-c)/(c*x)) - (128*c^4*
x^4 + 176*b*c^3*x^3 + 8*b^2*c^2*x^2 - 10*b^3*c*x + 15*b^4)*sqrt(c*x^2 + b*x)*sqr
t(-c))/(sqrt(-c)*c^3)]

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(x*(x*(b + c*x))**(3/2), x)

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.222455, size = 128, normalized size = 1.16 \[ \frac{3 \, b^{5}{\rm ln}\left ({\left | -2 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )} \sqrt{c} - b \right |}\right )}{256 \, c^{\frac{7}{2}}} + \frac{1}{640} \, \sqrt{c x^{2} + b x}{\left (2 \,{\left (4 \,{\left (2 \,{\left (8 \, c x + 11 \, b\right )} x + \frac{b^{2}}{c}\right )} x - \frac{5 \, b^{3}}{c^{2}}\right )} x + \frac{15 \, b^{4}}{c^{3}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3/256*b^5*ln(abs(-2*(sqrt(c)*x - sqrt(c*x^2 + b*x))*sqrt(c) - b))/c^(7/2) + 1/64
0*sqrt(c*x^2 + b*x)*(2*(4*(2*(8*c*x + 11*b)*x + b^2/c)*x - 5*b^3/c^2)*x + 15*b^4
/c^3)