3.221 \(\int x^2 (a+b \tanh ^{-1}(c x^{3/2}))^2 \, dx\)

Optimal. Leaf size=101 \[ -\frac{\left (a+b \tanh ^{-1}\left (c x^{3/2}\right )\right )^2}{3 c^2}+\frac{2 a b x^{3/2}}{3 c}+\frac{1}{3} x^3 \left (a+b \tanh ^{-1}\left (c x^{3/2}\right )\right )^2+\frac{b^2 \log \left (1-c^2 x^3\right )}{3 c^2}+\frac{2 b^2 x^{3/2} \tanh ^{-1}\left (c x^{3/2}\right )}{3 c} \]

[Out]

(2*a*b*x^(3/2))/(3*c) + (2*b^2*x^(3/2)*ArcTanh[c*x^(3/2)])/(3*c) - (a + b*ArcTanh[c*x^(3/2)])^2/(3*c^2) + (x^3
*(a + b*ArcTanh[c*x^(3/2)])^2)/3 + (b^2*Log[1 - c^2*x^3])/(3*c^2)

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Rubi [F]  time = 0.0244631, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int x^2 \left (a+b \tanh ^{-1}\left (c x^{3/2}\right )\right )^2 \, dx \]

Verification is Not applicable to the result.

[In]

Int[x^2*(a + b*ArcTanh[c*x^(3/2)])^2,x]

[Out]

Defer[Int][x^2*(a + b*ArcTanh[c*x^(3/2)])^2, x]

Rubi steps

\begin{align*} \int x^2 \left (a+b \tanh ^{-1}\left (c x^{3/2}\right )\right )^2 \, dx &=\int x^2 \left (a+b \tanh ^{-1}\left (c x^{3/2}\right )\right )^2 \, dx\\ \end{align*}

Mathematica [A]  time = 0.0928518, size = 122, normalized size = 1.21 \[ \frac{a^2 c^2 x^3+2 a b c x^{3/2}+b (a+b) \log \left (1-c x^{3/2}\right )-a b \log \left (c x^{3/2}+1\right )+2 b c x^{3/2} \tanh ^{-1}\left (c x^{3/2}\right ) \left (a c x^{3/2}+b\right )+b^2 \left (c^2 x^3-1\right ) \tanh ^{-1}\left (c x^{3/2}\right )^2+b^2 \log \left (c x^{3/2}+1\right )}{3 c^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*a*b*c*x^(3/2) + a^2*c^2*x^3 + 2*b*c*x^(3/2)*(b + a*c*x^(3/2))*ArcTanh[c*x^(3/2)] + b^2*(-1 + c^2*x^3)*ArcTa
nh[c*x^(3/2)]^2 + b*(a + b)*Log[1 - c*x^(3/2)] - a*b*Log[1 + c*x^(3/2)] + b^2*Log[1 + c*x^(3/2)])/(3*c^2)

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Maple [B]  time = 0.049, size = 284, normalized size = 2.8 \begin{align*}{\frac{{x}^{3}{a}^{2}}{3}}+{\frac{{x}^{3}{b}^{2}}{3} \left ({\it Artanh} \left ( c{x}^{{\frac{3}{2}}} \right ) \right ) ^{2}}+{\frac{2\,{b}^{2}}{3\,c}{x}^{{\frac{3}{2}}}{\it Artanh} \left ( c{x}^{{\frac{3}{2}}} \right ) }+{\frac{{b}^{2}}{3\,{c}^{2}}{\it Artanh} \left ( c{x}^{{\frac{3}{2}}} \right ) \ln \left ( c{x}^{{\frac{3}{2}}}-1 \right ) }-{\frac{{b}^{2}}{3\,{c}^{2}}{\it Artanh} \left ( c{x}^{{\frac{3}{2}}} \right ) \ln \left ( c{x}^{{\frac{3}{2}}}+1 \right ) }-{\frac{{b}^{2}}{6\,{c}^{2}}\ln \left ( c{x}^{{\frac{3}{2}}}-1 \right ) \ln \left ({\frac{1}{2}}+{\frac{c}{2}{x}^{{\frac{3}{2}}}} \right ) }+{\frac{{b}^{2}}{12\,{c}^{2}} \left ( \ln \left ( c{x}^{{\frac{3}{2}}}-1 \right ) \right ) ^{2}}+{\frac{{b}^{2}}{3\,{c}^{2}}\ln \left ( c{x}^{{\frac{3}{2}}}-1 \right ) }+{\frac{{b}^{2}}{3\,{c}^{2}}\ln \left ( c{x}^{{\frac{3}{2}}}+1 \right ) }-{\frac{{b}^{2}}{6\,{c}^{2}}\ln \left ( -{\frac{c}{2}{x}^{{\frac{3}{2}}}}+{\frac{1}{2}} \right ) \ln \left ( c{x}^{{\frac{3}{2}}}+1 \right ) }+{\frac{{b}^{2}}{6\,{c}^{2}}\ln \left ( -{\frac{c}{2}{x}^{{\frac{3}{2}}}}+{\frac{1}{2}} \right ) \ln \left ({\frac{1}{2}}+{\frac{c}{2}{x}^{{\frac{3}{2}}}} \right ) }+{\frac{{b}^{2}}{12\,{c}^{2}} \left ( \ln \left ( c{x}^{{\frac{3}{2}}}+1 \right ) \right ) ^{2}}+{\frac{2\,ab{x}^{3}}{3}{\it Artanh} \left ( c{x}^{{\frac{3}{2}}} \right ) }+{\frac{2\,ab}{3\,c}{x}^{{\frac{3}{2}}}}+{\frac{ab}{3\,{c}^{2}}\ln \left ( c{x}^{{\frac{3}{2}}}-1 \right ) }-{\frac{ab}{3\,{c}^{2}}\ln \left ( c{x}^{{\frac{3}{2}}}+1 \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*(a+b*arctanh(c*x^(3/2)))^2,x)

[Out]

1/3*x^3*a^2+1/3*b^2*x^3*arctanh(c*x^(3/2))^2+2/3*b^2*x^(3/2)*arctanh(c*x^(3/2))/c+1/3/c^2*b^2*arctanh(c*x^(3/2
))*ln(c*x^(3/2)-1)-1/3/c^2*b^2*arctanh(c*x^(3/2))*ln(c*x^(3/2)+1)-1/6/c^2*b^2*ln(c*x^(3/2)-1)*ln(1/2+1/2*c*x^(
3/2))+1/12/c^2*b^2*ln(c*x^(3/2)-1)^2+1/3/c^2*b^2*ln(c*x^(3/2)-1)+1/3/c^2*b^2*ln(c*x^(3/2)+1)-1/6/c^2*b^2*ln(-1
/2*c*x^(3/2)+1/2)*ln(c*x^(3/2)+1)+1/6/c^2*b^2*ln(-1/2*c*x^(3/2)+1/2)*ln(1/2+1/2*c*x^(3/2))+1/12/c^2*b^2*ln(c*x
^(3/2)+1)^2+2/3*a*b*x^3*arctanh(c*x^(3/2))+2/3*a*b*x^(3/2)/c+1/3/c^2*a*b*ln(c*x^(3/2)-1)-1/3/c^2*a*b*ln(c*x^(3
/2)+1)

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Maxima [B]  time = 1.00881, size = 251, normalized size = 2.49 \begin{align*} \frac{1}{3} \, b^{2} x^{3} \operatorname{artanh}\left (c x^{\frac{3}{2}}\right )^{2} + \frac{1}{3} \, a^{2} x^{3} + \frac{1}{3} \,{\left (2 \, x^{3} \operatorname{artanh}\left (c x^{\frac{3}{2}}\right ) + c{\left (\frac{2 \, x^{\frac{3}{2}}}{c^{2}} - \frac{\log \left (c x^{\frac{3}{2}} + 1\right )}{c^{3}} + \frac{\log \left (c x^{\frac{3}{2}} - 1\right )}{c^{3}}\right )}\right )} a b + \frac{1}{12} \,{\left (4 \, c{\left (\frac{2 \, x^{\frac{3}{2}}}{c^{2}} - \frac{\log \left (c x^{\frac{3}{2}} + 1\right )}{c^{3}} + \frac{\log \left (c x^{\frac{3}{2}} - 1\right )}{c^{3}}\right )} \operatorname{artanh}\left (c x^{\frac{3}{2}}\right ) - \frac{2 \,{\left (\log \left (c x^{\frac{3}{2}} - 1\right ) - 2\right )} \log \left (c x^{\frac{3}{2}} + 1\right ) - \log \left (c x^{\frac{3}{2}} + 1\right )^{2} - \log \left (c x^{\frac{3}{2}} - 1\right )^{2} - 4 \, \log \left (c x^{\frac{3}{2}} - 1\right )}{c^{2}}\right )} b^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(a+b*arctanh(c*x^(3/2)))^2,x, algorithm="maxima")

[Out]

1/3*b^2*x^3*arctanh(c*x^(3/2))^2 + 1/3*a^2*x^3 + 1/3*(2*x^3*arctanh(c*x^(3/2)) + c*(2*x^(3/2)/c^2 - log(c*x^(3
/2) + 1)/c^3 + log(c*x^(3/2) - 1)/c^3))*a*b + 1/12*(4*c*(2*x^(3/2)/c^2 - log(c*x^(3/2) + 1)/c^3 + log(c*x^(3/2
) - 1)/c^3)*arctanh(c*x^(3/2)) - (2*(log(c*x^(3/2) - 1) - 2)*log(c*x^(3/2) + 1) - log(c*x^(3/2) + 1)^2 - log(c
*x^(3/2) - 1)^2 - 4*log(c*x^(3/2) - 1))/c^2)*b^2

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Fricas [B]  time = 1.99773, size = 402, normalized size = 3.98 \begin{align*} \frac{4 \, a^{2} c^{2} x^{3} + 8 \, a b c x^{\frac{3}{2}} +{\left (b^{2} c^{2} x^{3} - b^{2}\right )} \log \left (-\frac{c^{2} x^{3} + 2 \, c x^{\frac{3}{2}} + 1}{c^{2} x^{3} - 1}\right )^{2} + 4 \,{\left (a b c^{2} - a b + b^{2}\right )} \log \left (c x^{\frac{3}{2}} + 1\right ) - 4 \,{\left (a b c^{2} - a b - b^{2}\right )} \log \left (c x^{\frac{3}{2}} - 1\right ) + 4 \,{\left (a b c^{2} x^{3} + b^{2} c x^{\frac{3}{2}} - a b c^{2}\right )} \log \left (-\frac{c^{2} x^{3} + 2 \, c x^{\frac{3}{2}} + 1}{c^{2} x^{3} - 1}\right )}{12 \, c^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(a+b*arctanh(c*x^(3/2)))^2,x, algorithm="fricas")

[Out]

1/12*(4*a^2*c^2*x^3 + 8*a*b*c*x^(3/2) + (b^2*c^2*x^3 - b^2)*log(-(c^2*x^3 + 2*c*x^(3/2) + 1)/(c^2*x^3 - 1))^2
+ 4*(a*b*c^2 - a*b + b^2)*log(c*x^(3/2) + 1) - 4*(a*b*c^2 - a*b - b^2)*log(c*x^(3/2) - 1) + 4*(a*b*c^2*x^3 + b
^2*c*x^(3/2) - a*b*c^2)*log(-(c^2*x^3 + 2*c*x^(3/2) + 1)/(c^2*x^3 - 1)))/c^2

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**2*(a+b*atanh(c*x**(3/2)))**2,x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b \operatorname{artanh}\left (c x^{\frac{3}{2}}\right ) + a\right )}^{2} x^{2}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(a+b*arctanh(c*x^(3/2)))^2,x, algorithm="giac")

[Out]

integrate((b*arctanh(c*x^(3/2)) + a)^2*x^2, x)