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

Optimal. Leaf size=89 \[ \frac{3 b^4 \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{64 c^{5/2}}-\frac{3 b^2 (b+2 c x) \sqrt{b x+c x^2}}{64 c^2}+\frac{(b+2 c x) \left (b x+c x^2\right )^{3/2}}{8 c} \]

[Out]

(-3*b^2*(b + 2*c*x)*Sqrt[b*x + c*x^2])/(64*c^2) + ((b + 2*c*x)*(b*x + c*x^2)^(3/
2))/(8*c) + (3*b^4*ArcTanh[(Sqrt[c]*x)/Sqrt[b*x + c*x^2]])/(64*c^(5/2))

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Rubi [A]  time = 0.0680799, antiderivative size = 89, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ \frac{3 b^4 \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{64 c^{5/2}}-\frac{3 b^2 (b+2 c x) \sqrt{b x+c x^2}}{64 c^2}+\frac{(b+2 c x) \left (b x+c x^2\right )^{3/2}}{8 c} \]

Antiderivative was successfully verified.

[In]  Int[(b*x + c*x^2)^(3/2),x]

[Out]

(-3*b^2*(b + 2*c*x)*Sqrt[b*x + c*x^2])/(64*c^2) + ((b + 2*c*x)*(b*x + c*x^2)^(3/
2))/(8*c) + (3*b^4*ArcTanh[(Sqrt[c]*x)/Sqrt[b*x + c*x^2]])/(64*c^(5/2))

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Rubi in Sympy [A]  time = 7.33404, size = 82, normalized size = 0.92 \[ \frac{3 b^{4} \operatorname{atanh}{\left (\frac{\sqrt{c} x}{\sqrt{b x + c x^{2}}} \right )}}{64 c^{\frac{5}{2}}} - \frac{3 b^{2} \left (b + 2 c x\right ) \sqrt{b x + c x^{2}}}{64 c^{2}} + \frac{\left (b + 2 c x\right ) \left (b x + c x^{2}\right )^{\frac{3}{2}}}{8 c} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3*b**4*atanh(sqrt(c)*x/sqrt(b*x + c*x**2))/(64*c**(5/2)) - 3*b**2*(b + 2*c*x)*sq
rt(b*x + c*x**2)/(64*c**2) + (b + 2*c*x)*(b*x + c*x**2)**(3/2)/(8*c)

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Mathematica [A]  time = 0.105572, size = 100, normalized size = 1.12 \[ \frac{\sqrt{x (b+c x)} \left (\frac{3 b^4 \log \left (\sqrt{c} \sqrt{b+c x}+c \sqrt{x}\right )}{\sqrt{x} \sqrt{b+c x}}+\sqrt{c} \left (-3 b^3+2 b^2 c x+24 b c^2 x^2+16 c^3 x^3\right )\right )}{64 c^{5/2}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[x*(b + c*x)]*(Sqrt[c]*(-3*b^3 + 2*b^2*c*x + 24*b*c^2*x^2 + 16*c^3*x^3) + (
3*b^4*Log[c*Sqrt[x] + Sqrt[c]*Sqrt[b + c*x]])/(Sqrt[x]*Sqrt[b + c*x])))/(64*c^(5
/2))

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Maple [A]  time = 0.005, size = 95, normalized size = 1.1 \[{\frac{2\,cx+b}{8\,c} \left ( c{x}^{2}+bx \right ) ^{{\frac{3}{2}}}}-{\frac{3\,{b}^{2}x}{32\,c}\sqrt{c{x}^{2}+bx}}-{\frac{3\,{b}^{3}}{64\,{c}^{2}}\sqrt{c{x}^{2}+bx}}+{\frac{3\,{b}^{4}}{128}\ln \left ({1 \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx} \right ){c}^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^2+b*x)^(3/2),x)

[Out]

1/8*(2*c*x+b)*(c*x^2+b*x)^(3/2)/c-3/32*b^2/c*(c*x^2+b*x)^(1/2)*x-3/64*b^3/c^2*(c
*x^2+b*x)^(1/2)+3/128*b^4/c^(5/2)*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((c*x^2 + b*x)^(3/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.230082, size = 1, normalized size = 0.01 \[ \left [\frac{3 \, b^{4} \log \left ({\left (2 \, c x + b\right )} \sqrt{c} + 2 \, \sqrt{c x^{2} + b x} c\right ) + 2 \,{\left (16 \, c^{3} x^{3} + 24 \, b c^{2} x^{2} + 2 \, b^{2} c x - 3 \, b^{3}\right )} \sqrt{c x^{2} + b x} \sqrt{c}}{128 \, c^{\frac{5}{2}}}, \frac{3 \, b^{4} \arctan \left (\frac{\sqrt{c x^{2} + b x} \sqrt{-c}}{c x}\right ) +{\left (16 \, c^{3} x^{3} + 24 \, b c^{2} x^{2} + 2 \, b^{2} c x - 3 \, b^{3}\right )} \sqrt{c x^{2} + b x} \sqrt{-c}}{64 \, \sqrt{-c} c^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/128*(3*b^4*log((2*c*x + b)*sqrt(c) + 2*sqrt(c*x^2 + b*x)*c) + 2*(16*c^3*x^3 +
24*b*c^2*x^2 + 2*b^2*c*x - 3*b^3)*sqrt(c*x^2 + b*x)*sqrt(c))/c^(5/2), 1/64*(3*b
^4*arctan(sqrt(c*x^2 + b*x)*sqrt(-c)/(c*x)) + (16*c^3*x^3 + 24*b*c^2*x^2 + 2*b^2
*c*x - 3*b^3)*sqrt(c*x^2 + b*x)*sqrt(-c))/(sqrt(-c)*c^2)]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral((b*x + c*x**2)**(3/2), x)

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GIAC/XCAS [A]  time = 0.222378, size = 112, normalized size = 1.26 \[ -\frac{3 \, b^{4}{\rm ln}\left ({\left | -2 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )} \sqrt{c} - b \right |}\right )}{128 \, c^{\frac{5}{2}}} + \frac{1}{64} \, \sqrt{c x^{2} + b x}{\left (2 \,{\left (4 \,{\left (2 \, c x + 3 \, b\right )} x + \frac{b^{2}}{c}\right )} x - \frac{3 \, b^{3}}{c^{2}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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