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

Optimal. Leaf size=134 \[ \frac{7 b^6 \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{512 c^{9/2}}-\frac{7 b^4 (b+2 c x) \sqrt{b x+c x^2}}{512 c^4}+\frac{7 b^2 (b+2 c x) \left (b x+c x^2\right )^{3/2}}{192 c^3}-\frac{7 b \left (b x+c x^2\right )^{5/2}}{60 c^2}+\frac{x \left (b x+c x^2\right )^{5/2}}{6 c} \]

[Out]

(-7*b^4*(b + 2*c*x)*Sqrt[b*x + c*x^2])/(512*c^4) + (7*b^2*(b + 2*c*x)*(b*x + c*x
^2)^(3/2))/(192*c^3) - (7*b*(b*x + c*x^2)^(5/2))/(60*c^2) + (x*(b*x + c*x^2)^(5/
2))/(6*c) + (7*b^6*ArcTanh[(Sqrt[c]*x)/Sqrt[b*x + c*x^2]])/(512*c^(9/2))

_______________________________________________________________________________________

Rubi [A]  time = 0.154254, antiderivative size = 134, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.294 \[ \frac{7 b^6 \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{512 c^{9/2}}-\frac{7 b^4 (b+2 c x) \sqrt{b x+c x^2}}{512 c^4}+\frac{7 b^2 (b+2 c x) \left (b x+c x^2\right )^{3/2}}{192 c^3}-\frac{7 b \left (b x+c x^2\right )^{5/2}}{60 c^2}+\frac{x \left (b x+c x^2\right )^{5/2}}{6 c} \]

Antiderivative was successfully verified.

[In]  Int[x^2*(b*x + c*x^2)^(3/2),x]

[Out]

(-7*b^4*(b + 2*c*x)*Sqrt[b*x + c*x^2])/(512*c^4) + (7*b^2*(b + 2*c*x)*(b*x + c*x
^2)^(3/2))/(192*c^3) - (7*b*(b*x + c*x^2)^(5/2))/(60*c^2) + (x*(b*x + c*x^2)^(5/
2))/(6*c) + (7*b^6*ArcTanh[(Sqrt[c]*x)/Sqrt[b*x + c*x^2]])/(512*c^(9/2))

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 18.0601, size = 126, normalized size = 0.94 \[ \frac{7 b^{6} \operatorname{atanh}{\left (\frac{\sqrt{c} x}{\sqrt{b x + c x^{2}}} \right )}}{512 c^{\frac{9}{2}}} - \frac{7 b^{4} \left (b + 2 c x\right ) \sqrt{b x + c x^{2}}}{512 c^{4}} + \frac{7 b^{2} \left (b + 2 c x\right ) \left (b x + c x^{2}\right )^{\frac{3}{2}}}{192 c^{3}} - \frac{7 b \left (b x + c x^{2}\right )^{\frac{5}{2}}}{60 c^{2}} + \frac{x \left (b x + c x^{2}\right )^{\frac{5}{2}}}{6 c} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

7*b**6*atanh(sqrt(c)*x/sqrt(b*x + c*x**2))/(512*c**(9/2)) - 7*b**4*(b + 2*c*x)*s
qrt(b*x + c*x**2)/(512*c**4) + 7*b**2*(b + 2*c*x)*(b*x + c*x**2)**(3/2)/(192*c**
3) - 7*b*(b*x + c*x**2)**(5/2)/(60*c**2) + x*(b*x + c*x**2)**(5/2)/(6*c)

_______________________________________________________________________________________

Mathematica [A]  time = 0.141493, size = 122, normalized size = 0.91 \[ \frac{\sqrt{x (b+c x)} \left (\frac{105 b^6 \log \left (\sqrt{c} \sqrt{b+c x}+c \sqrt{x}\right )}{\sqrt{x} \sqrt{b+c x}}+\sqrt{c} \left (-105 b^5+70 b^4 c x-56 b^3 c^2 x^2+48 b^2 c^3 x^3+1664 b c^4 x^4+1280 c^5 x^5\right )\right )}{7680 c^{9/2}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[x*(b + c*x)]*(Sqrt[c]*(-105*b^5 + 70*b^4*c*x - 56*b^3*c^2*x^2 + 48*b^2*c^3
*x^3 + 1664*b*c^4*x^4 + 1280*c^5*x^5) + (105*b^6*Log[c*Sqrt[x] + Sqrt[c]*Sqrt[b
+ c*x]])/(Sqrt[x]*Sqrt[b + c*x])))/(7680*c^(9/2))

_______________________________________________________________________________________

Maple [A]  time = 0.008, size = 146, normalized size = 1.1 \[{\frac{x}{6\,c} \left ( c{x}^{2}+bx \right ) ^{{\frac{5}{2}}}}-{\frac{7\,b}{60\,{c}^{2}} \left ( c{x}^{2}+bx \right ) ^{{\frac{5}{2}}}}+{\frac{7\,{b}^{2}x}{96\,{c}^{2}} \left ( c{x}^{2}+bx \right ) ^{{\frac{3}{2}}}}+{\frac{7\,{b}^{3}}{192\,{c}^{3}} \left ( c{x}^{2}+bx \right ) ^{{\frac{3}{2}}}}-{\frac{7\,{b}^{4}x}{256\,{c}^{3}}\sqrt{c{x}^{2}+bx}}-{\frac{7\,{b}^{5}}{512\,{c}^{4}}\sqrt{c{x}^{2}+bx}}+{\frac{7\,{b}^{6}}{1024}\ln \left ({1 \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx} \right ){c}^{-{\frac{9}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^2*(c*x^2+b*x)^(3/2),x)

[Out]

1/6*x*(c*x^2+b*x)^(5/2)/c-7/60*b*(c*x^2+b*x)^(5/2)/c^2+7/96*b^2/c^2*(c*x^2+b*x)^
(3/2)*x+7/192*b^3/c^3*(c*x^2+b*x)^(3/2)-7/256*b^4/c^3*(c*x^2+b*x)^(1/2)*x-7/512*
b^5/c^4*(c*x^2+b*x)^(1/2)+7/1024*b^6/c^(9/2)*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^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.232875, size = 1, normalized size = 0.01 \[ \left [\frac{105 \, b^{6} \log \left ({\left (2 \, c x + b\right )} \sqrt{c} + 2 \, \sqrt{c x^{2} + b x} c\right ) + 2 \,{\left (1280 \, c^{5} x^{5} + 1664 \, b c^{4} x^{4} + 48 \, b^{2} c^{3} x^{3} - 56 \, b^{3} c^{2} x^{2} + 70 \, b^{4} c x - 105 \, b^{5}\right )} \sqrt{c x^{2} + b x} \sqrt{c}}{15360 \, c^{\frac{9}{2}}}, \frac{105 \, b^{6} \arctan \left (\frac{\sqrt{c x^{2} + b x} \sqrt{-c}}{c x}\right ) +{\left (1280 \, c^{5} x^{5} + 1664 \, b c^{4} x^{4} + 48 \, b^{2} c^{3} x^{3} - 56 \, b^{3} c^{2} x^{2} + 70 \, b^{4} c x - 105 \, b^{5}\right )} \sqrt{c x^{2} + b x} \sqrt{-c}}{7680 \, \sqrt{-c} c^{4}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/15360*(105*b^6*log((2*c*x + b)*sqrt(c) + 2*sqrt(c*x^2 + b*x)*c) + 2*(1280*c^5
*x^5 + 1664*b*c^4*x^4 + 48*b^2*c^3*x^3 - 56*b^3*c^2*x^2 + 70*b^4*c*x - 105*b^5)*
sqrt(c*x^2 + b*x)*sqrt(c))/c^(9/2), 1/7680*(105*b^6*arctan(sqrt(c*x^2 + b*x)*sqr
t(-c)/(c*x)) + (1280*c^5*x^5 + 1664*b*c^4*x^4 + 48*b^2*c^3*x^3 - 56*b^3*c^2*x^2
+ 70*b^4*c*x - 105*b^5)*sqrt(c*x^2 + b*x)*sqrt(-c))/(sqrt(-c)*c^4)]

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.222318, size = 146, normalized size = 1.09 \[ -\frac{7 \, b^{6}{\rm ln}\left ({\left | -2 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )} \sqrt{c} - b \right |}\right )}{1024 \, c^{\frac{9}{2}}} + \frac{1}{7680} \, \sqrt{c x^{2} + b x}{\left (2 \,{\left (4 \,{\left (2 \,{\left (8 \,{\left (10 \, c x + 13 \, b\right )} x + \frac{3 \, b^{2}}{c}\right )} x - \frac{7 \, b^{3}}{c^{2}}\right )} x + \frac{35 \, b^{4}}{c^{3}}\right )} x - \frac{105 \, b^{5}}{c^{4}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-7/1024*b^6*ln(abs(-2*(sqrt(c)*x - sqrt(c*x^2 + b*x))*sqrt(c) - b))/c^(9/2) + 1/
7680*sqrt(c*x^2 + b*x)*(2*(4*(2*(8*(10*c*x + 13*b)*x + 3*b^2/c)*x - 7*b^3/c^2)*x
+ 35*b^4/c^3)*x - 105*b^5/c^4)