3.2 \(\int x^2 \sqrt{b x+c x^2} \, dx\)

Optimal. Leaf size=105 \[ -\frac{5 b^4 \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{64 c^{7/2}}+\frac{5 b^2 (b+2 c x) \sqrt{b x+c x^2}}{64 c^3}-\frac{5 b \left (b x+c x^2\right )^{3/2}}{24 c^2}+\frac{x \left (b x+c x^2\right )^{3/2}}{4 c} \]

[Out]

(5*b^2*(b + 2*c*x)*Sqrt[b*x + c*x^2])/(64*c^3) - (5*b*(b*x + c*x^2)^(3/2))/(24*c
^2) + (x*(b*x + c*x^2)^(3/2))/(4*c) - (5*b^4*ArcTanh[(Sqrt[c]*x)/Sqrt[b*x + c*x^
2]])/(64*c^(7/2))

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Rubi [A]  time = 0.117302, antiderivative size = 105, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.294 \[ -\frac{5 b^4 \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{64 c^{7/2}}+\frac{5 b^2 (b+2 c x) \sqrt{b x+c x^2}}{64 c^3}-\frac{5 b \left (b x+c x^2\right )^{3/2}}{24 c^2}+\frac{x \left (b x+c x^2\right )^{3/2}}{4 c} \]

Antiderivative was successfully verified.

[In]  Int[x^2*Sqrt[b*x + c*x^2],x]

[Out]

(5*b^2*(b + 2*c*x)*Sqrt[b*x + c*x^2])/(64*c^3) - (5*b*(b*x + c*x^2)^(3/2))/(24*c
^2) + (x*(b*x + c*x^2)^(3/2))/(4*c) - (5*b^4*ArcTanh[(Sqrt[c]*x)/Sqrt[b*x + c*x^
2]])/(64*c^(7/2))

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Rubi in Sympy [A]  time = 14.1051, size = 97, normalized size = 0.92 \[ - \frac{5 b^{4} \operatorname{atanh}{\left (\frac{\sqrt{c} x}{\sqrt{b x + c x^{2}}} \right )}}{64 c^{\frac{7}{2}}} + \frac{5 b^{2} \left (b + 2 c x\right ) \sqrt{b x + c x^{2}}}{64 c^{3}} - \frac{5 b \left (b x + c x^{2}\right )^{\frac{3}{2}}}{24 c^{2}} + \frac{x \left (b x + c x^{2}\right )^{\frac{3}{2}}}{4 c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**2*(c*x**2+b*x)**(1/2),x)

[Out]

-5*b**4*atanh(sqrt(c)*x/sqrt(b*x + c*x**2))/(64*c**(7/2)) + 5*b**2*(b + 2*c*x)*s
qrt(b*x + c*x**2)/(64*c**3) - 5*b*(b*x + c*x**2)**(3/2)/(24*c**2) + x*(b*x + c*x
**2)**(3/2)/(4*c)

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Mathematica [A]  time = 0.106459, size = 100, normalized size = 0.95 \[ \frac{\sqrt{x (b+c x)} \left (\sqrt{c} \left (15 b^3-10 b^2 c x+8 b c^2 x^2+48 c^3 x^3\right )-\frac{15 b^4 \log \left (\sqrt{c} \sqrt{b+c x}+c \sqrt{x}\right )}{\sqrt{x} \sqrt{b+c x}}\right )}{192 c^{7/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^2*Sqrt[b*x + c*x^2],x]

[Out]

(Sqrt[x*(b + c*x)]*(Sqrt[c]*(15*b^3 - 10*b^2*c*x + 8*b*c^2*x^2 + 48*c^3*x^3) - (
15*b^4*Log[c*Sqrt[x] + Sqrt[c]*Sqrt[b + c*x]])/(Sqrt[x]*Sqrt[b + c*x])))/(192*c^
(7/2))

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Maple [A]  time = 0.007, size = 107, normalized size = 1. \[{\frac{x}{4\,c} \left ( c{x}^{2}+bx \right ) ^{{\frac{3}{2}}}}-{\frac{5\,b}{24\,{c}^{2}} \left ( c{x}^{2}+bx \right ) ^{{\frac{3}{2}}}}+{\frac{5\,{b}^{2}x}{32\,{c}^{2}}\sqrt{c{x}^{2}+bx}}+{\frac{5\,{b}^{3}}{64\,{c}^{3}}\sqrt{c{x}^{2}+bx}}-{\frac{5\,{b}^{4}}{128}\ln \left ({1 \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx} \right ){c}^{-{\frac{7}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^2*(c*x^2+b*x)^(1/2),x)

[Out]

1/4*x*(c*x^2+b*x)^(3/2)/c-5/24*b*(c*x^2+b*x)^(3/2)/c^2+5/32*b^2/c^2*(c*x^2+b*x)^
(1/2)*x+5/64*b^3/c^3*(c*x^2+b*x)^(1/2)-5/128*b^4/c^(7/2)*ln((1/2*b+c*x)/c^(1/2)+
(c*x^2+b*x)^(1/2))

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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(sqrt(c*x^2 + b*x)*x^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.234667, size = 1, normalized size = 0.01 \[ \left [\frac{15 \, b^{4} \log \left ({\left (2 \, c x + b\right )} \sqrt{c} - 2 \, \sqrt{c x^{2} + b x} c\right ) + 2 \,{\left (48 \, c^{3} x^{3} + 8 \, b c^{2} x^{2} - 10 \, b^{2} c x + 15 \, b^{3}\right )} \sqrt{c x^{2} + b x} \sqrt{c}}{384 \, c^{\frac{7}{2}}}, -\frac{15 \, b^{4} \arctan \left (\frac{\sqrt{c x^{2} + b x} \sqrt{-c}}{c x}\right ) -{\left (48 \, c^{3} x^{3} + 8 \, b c^{2} x^{2} - 10 \, b^{2} c x + 15 \, b^{3}\right )} \sqrt{c x^{2} + b x} \sqrt{-c}}{192 \, \sqrt{-c} c^{3}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/384*(15*b^4*log((2*c*x + b)*sqrt(c) - 2*sqrt(c*x^2 + b*x)*c) + 2*(48*c^3*x^3
+ 8*b*c^2*x^2 - 10*b^2*c*x + 15*b^3)*sqrt(c*x^2 + b*x)*sqrt(c))/c^(7/2), -1/192*
(15*b^4*arctan(sqrt(c*x^2 + b*x)*sqrt(-c)/(c*x)) - (48*c^3*x^3 + 8*b*c^2*x^2 - 1
0*b^2*c*x + 15*b^3)*sqrt(c*x^2 + b*x)*sqrt(-c))/(sqrt(-c)*c^3)]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**2*(c*x**2+b*x)**(1/2),x)

[Out]

Integral(x**2*sqrt(x*(b + c*x)), x)

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GIAC/XCAS [A]  time = 0.2221, size = 115, normalized size = 1.1 \[ \frac{1}{192} \, \sqrt{c x^{2} + b x}{\left (2 \,{\left (4 \,{\left (6 \, x + \frac{b}{c}\right )} x - \frac{5 \, b^{2}}{c^{2}}\right )} x + \frac{15 \, b^{3}}{c^{3}}\right )} + \frac{5 \, b^{4}{\rm ln}\left ({\left | -2 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )} \sqrt{c} - b \right |}\right )}{128 \, c^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/192*sqrt(c*x^2 + b*x)*(2*(4*(6*x + b/c)*x - 5*b^2/c^2)*x + 15*b^3/c^3) + 5/128
*b^4*ln(abs(-2*(sqrt(c)*x - sqrt(c*x^2 + b*x))*sqrt(c) - b))/c^(7/2)