3.1 \(\int x^3 \sqrt{b x+c x^2} \, dx\)

Optimal. Leaf size=131 \[ \frac{7 b^5 \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{128 c^{9/2}}-\frac{7 b^3 (b+2 c x) \sqrt{b x+c x^2}}{128 c^4}+\frac{7 b^2 \left (b x+c x^2\right )^{3/2}}{48 c^3}-\frac{7 b x \left (b x+c x^2\right )^{3/2}}{40 c^2}+\frac{x^2 \left (b x+c x^2\right )^{3/2}}{5 c} \]

[Out]

(-7*b^3*(b + 2*c*x)*Sqrt[b*x + c*x^2])/(128*c^4) + (7*b^2*(b*x + c*x^2)^(3/2))/(
48*c^3) - (7*b*x*(b*x + c*x^2)^(3/2))/(40*c^2) + (x^2*(b*x + c*x^2)^(3/2))/(5*c)
+ (7*b^5*ArcTanh[(Sqrt[c]*x)/Sqrt[b*x + c*x^2]])/(128*c^(9/2))

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Rubi [A]  time = 0.179804, antiderivative size = 131, 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 \[ \frac{7 b^5 \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{128 c^{9/2}}-\frac{7 b^3 (b+2 c x) \sqrt{b x+c x^2}}{128 c^4}+\frac{7 b^2 \left (b x+c x^2\right )^{3/2}}{48 c^3}-\frac{7 b x \left (b x+c x^2\right )^{3/2}}{40 c^2}+\frac{x^2 \left (b x+c x^2\right )^{3/2}}{5 c} \]

Antiderivative was successfully verified.

[In]  Int[x^3*Sqrt[b*x + c*x^2],x]

[Out]

(-7*b^3*(b + 2*c*x)*Sqrt[b*x + c*x^2])/(128*c^4) + (7*b^2*(b*x + c*x^2)^(3/2))/(
48*c^3) - (7*b*x*(b*x + c*x^2)^(3/2))/(40*c^2) + (x^2*(b*x + c*x^2)^(3/2))/(5*c)
+ (7*b^5*ArcTanh[(Sqrt[c]*x)/Sqrt[b*x + c*x^2]])/(128*c^(9/2))

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Rubi in Sympy [A]  time = 19.1581, size = 122, normalized size = 0.93 \[ \frac{7 b^{5} \operatorname{atanh}{\left (\frac{\sqrt{c} x}{\sqrt{b x + c x^{2}}} \right )}}{128 c^{\frac{9}{2}}} - \frac{7 b^{3} \left (b + 2 c x\right ) \sqrt{b x + c x^{2}}}{128 c^{4}} + \frac{7 b^{2} \left (b x + c x^{2}\right )^{\frac{3}{2}}}{48 c^{3}} - \frac{7 b x \left (b x + c x^{2}\right )^{\frac{3}{2}}}{40 c^{2}} + \frac{x^{2} \left (b x + c x^{2}\right )^{\frac{3}{2}}}{5 c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**3*(c*x**2+b*x)**(1/2),x)

[Out]

7*b**5*atanh(sqrt(c)*x/sqrt(b*x + c*x**2))/(128*c**(9/2)) - 7*b**3*(b + 2*c*x)*s
qrt(b*x + c*x**2)/(128*c**4) + 7*b**2*(b*x + c*x**2)**(3/2)/(48*c**3) - 7*b*x*(b
*x + c*x**2)**(3/2)/(40*c**2) + x**2*(b*x + c*x**2)**(3/2)/(5*c)

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Mathematica [A]  time = 0.135966, size = 111, normalized size = 0.85 \[ \frac{\sqrt{x (b+c x)} \left (\frac{105 b^5 \log \left (\sqrt{c} \sqrt{b+c x}+c \sqrt{x}\right )}{\sqrt{x} \sqrt{b+c x}}+\sqrt{c} \left (-105 b^4+70 b^3 c x-56 b^2 c^2 x^2+48 b c^3 x^3+384 c^4 x^4\right )\right )}{1920 c^{9/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^3*Sqrt[b*x + c*x^2],x]

[Out]

(Sqrt[x*(b + c*x)]*(Sqrt[c]*(-105*b^4 + 70*b^3*c*x - 56*b^2*c^2*x^2 + 48*b*c^3*x
^3 + 384*c^4*x^4) + (105*b^5*Log[c*Sqrt[x] + Sqrt[c]*Sqrt[b + c*x]])/(Sqrt[x]*Sq
rt[b + c*x])))/(1920*c^(9/2))

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Maple [A]  time = 0.008, size = 129, normalized size = 1. \[{\frac{{x}^{2}}{5\,c} \left ( c{x}^{2}+bx \right ) ^{{\frac{3}{2}}}}-{\frac{7\,bx}{40\,{c}^{2}} \left ( c{x}^{2}+bx \right ) ^{{\frac{3}{2}}}}+{\frac{7\,{b}^{2}}{48\,{c}^{3}} \left ( c{x}^{2}+bx \right ) ^{{\frac{3}{2}}}}-{\frac{7\,{b}^{3}x}{64\,{c}^{3}}\sqrt{c{x}^{2}+bx}}-{\frac{7\,{b}^{4}}{128\,{c}^{4}}\sqrt{c{x}^{2}+bx}}+{\frac{7\,{b}^{5}}{256}\ln \left ({1 \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx} \right ){c}^{-{\frac{9}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^3*(c*x^2+b*x)^(1/2),x)

[Out]

1/5*x^2*(c*x^2+b*x)^(3/2)/c-7/40*b*x*(c*x^2+b*x)^(3/2)/c^2+7/48*b^2*(c*x^2+b*x)^
(3/2)/c^3-7/64*b^3/c^3*(c*x^2+b*x)^(1/2)*x-7/128*b^4/c^4*(c*x^2+b*x)^(1/2)+7/256
*b^5/c^(9/2)*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^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.227606, size = 1, normalized size = 0.01 \[ \left [\frac{105 \, b^{5} \log \left ({\left (2 \, c x + b\right )} \sqrt{c} + 2 \, \sqrt{c x^{2} + b x} c\right ) + 2 \,{\left (384 \, c^{4} x^{4} + 48 \, b c^{3} x^{3} - 56 \, b^{2} c^{2} x^{2} + 70 \, b^{3} c x - 105 \, b^{4}\right )} \sqrt{c x^{2} + b x} \sqrt{c}}{3840 \, c^{\frac{9}{2}}}, \frac{105 \, b^{5} \arctan \left (\frac{\sqrt{c x^{2} + b x} \sqrt{-c}}{c x}\right ) +{\left (384 \, c^{4} x^{4} + 48 \, b c^{3} x^{3} - 56 \, b^{2} c^{2} x^{2} + 70 \, b^{3} c x - 105 \, b^{4}\right )} \sqrt{c x^{2} + b x} \sqrt{-c}}{1920 \, \sqrt{-c} c^{4}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/3840*(105*b^5*log((2*c*x + b)*sqrt(c) + 2*sqrt(c*x^2 + b*x)*c) + 2*(384*c^4*x
^4 + 48*b*c^3*x^3 - 56*b^2*c^2*x^2 + 70*b^3*c*x - 105*b^4)*sqrt(c*x^2 + b*x)*sqr
t(c))/c^(9/2), 1/1920*(105*b^5*arctan(sqrt(c*x^2 + b*x)*sqrt(-c)/(c*x)) + (384*c
^4*x^4 + 48*b*c^3*x^3 - 56*b^2*c^2*x^2 + 70*b^3*c*x - 105*b^4)*sqrt(c*x^2 + b*x)
*sqrt(-c))/(sqrt(-c)*c^4)]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**3*(c*x**2+b*x)**(1/2),x)

[Out]

Integral(x**3*sqrt(x*(b + c*x)), x)

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GIAC/XCAS [A]  time = 0.220647, size = 131, normalized size = 1. \[ \frac{1}{1920} \, \sqrt{c x^{2} + b x}{\left (2 \,{\left (4 \,{\left (6 \,{\left (8 \, x + \frac{b}{c}\right )} x - \frac{7 \, b^{2}}{c^{2}}\right )} x + \frac{35 \, b^{3}}{c^{3}}\right )} x - \frac{105 \, b^{4}}{c^{4}}\right )} - \frac{7 \, b^{5}{\rm ln}\left ({\left | -2 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )} \sqrt{c} - b \right |}\right )}{256 \, c^{\frac{9}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/1920*sqrt(c*x^2 + b*x)*(2*(4*(6*(8*x + b/c)*x - 7*b^2/c^2)*x + 35*b^3/c^3)*x -
105*b^4/c^4) - 7/256*b^5*ln(abs(-2*(sqrt(c)*x - sqrt(c*x^2 + b*x))*sqrt(c) - b)
)/c^(9/2)