3.2.88 \(\int x^2 (a+b \tanh ^{-1}(c \sqrt {x})) \, dx\) [188]

Optimal. Leaf size=75 \[ \frac {b \sqrt {x}}{3 c^5}+\frac {b x^{3/2}}{9 c^3}+\frac {b x^{5/2}}{15 c}-\frac {b \tanh ^{-1}\left (c \sqrt {x}\right )}{3 c^6}+\frac {1}{3} x^3 \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right ) \]

[Out]

1/9*b*x^(3/2)/c^3+1/15*b*x^(5/2)/c-1/3*b*arctanh(c*x^(1/2))/c^6+1/3*x^3*(a+b*arctanh(c*x^(1/2)))+1/3*b*x^(1/2)
/c^5

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Rubi [A]
time = 0.02, antiderivative size = 75, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {6037, 52, 65, 212} \begin {gather*} \frac {1}{3} x^3 \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )-\frac {b \tanh ^{-1}\left (c \sqrt {x}\right )}{3 c^6}+\frac {b \sqrt {x}}{3 c^5}+\frac {b x^{3/2}}{9 c^3}+\frac {b x^{5/2}}{15 c} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^2*(a + b*ArcTanh[c*Sqrt[x]]),x]

[Out]

(b*Sqrt[x])/(3*c^5) + (b*x^(3/2))/(9*c^3) + (b*x^(5/2))/(15*c) - (b*ArcTanh[c*Sqrt[x]])/(3*c^6) + (x^3*(a + b*
ArcTanh[c*Sqrt[x]]))/3

Rule 52

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^n/(b*(
m + n + 1))), x] + Dist[n*((b*c - a*d)/(b*(m + n + 1))), Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a
, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinearQ[a, b, c, d, m, n, x]

Rule 65

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 6037

Int[((a_.) + ArcTanh[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)*((a + b*ArcTanh[c*
x^n])^p/(m + 1)), x] - Dist[b*c*n*(p/(m + 1)), Int[x^(m + n)*((a + b*ArcTanh[c*x^n])^(p - 1)/(1 - c^2*x^(2*n))
), x], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0] && (EqQ[p, 1] || (EqQ[n, 1] && IntegerQ[m])) && NeQ[m, -1
]

Rubi steps

\begin {align*} \int x^2 \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right ) \, dx &=\frac {1}{3} x^3 \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )-\frac {1}{6} (b c) \int \frac {x^{5/2}}{1-c^2 x} \, dx\\ &=\frac {b x^{5/2}}{15 c}+\frac {1}{3} x^3 \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )-\frac {b \int \frac {x^{3/2}}{1-c^2 x} \, dx}{6 c}\\ &=\frac {b x^{3/2}}{9 c^3}+\frac {b x^{5/2}}{15 c}+\frac {1}{3} x^3 \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )-\frac {b \int \frac {\sqrt {x}}{1-c^2 x} \, dx}{6 c^3}\\ &=\frac {b \sqrt {x}}{3 c^5}+\frac {b x^{3/2}}{9 c^3}+\frac {b x^{5/2}}{15 c}+\frac {1}{3} x^3 \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )-\frac {b \int \frac {1}{\sqrt {x} \left (1-c^2 x\right )} \, dx}{6 c^5}\\ &=\frac {b \sqrt {x}}{3 c^5}+\frac {b x^{3/2}}{9 c^3}+\frac {b x^{5/2}}{15 c}+\frac {1}{3} x^3 \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )-\frac {b \text {Subst}\left (\int \frac {1}{1-c^2 x^2} \, dx,x,\sqrt {x}\right )}{3 c^5}\\ &=\frac {b \sqrt {x}}{3 c^5}+\frac {b x^{3/2}}{9 c^3}+\frac {b x^{5/2}}{15 c}-\frac {b \tanh ^{-1}\left (c \sqrt {x}\right )}{3 c^6}+\frac {1}{3} x^3 \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )\\ \end {align*}

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Mathematica [A]
time = 0.04, size = 101, normalized size = 1.35 \begin {gather*} \frac {b \sqrt {x}}{3 c^5}+\frac {b x^{3/2}}{9 c^3}+\frac {b x^{5/2}}{15 c}+\frac {a x^3}{3}+\frac {1}{3} b x^3 \tanh ^{-1}\left (c \sqrt {x}\right )+\frac {b \log \left (1-c \sqrt {x}\right )}{6 c^6}-\frac {b \log \left (1+c \sqrt {x}\right )}{6 c^6} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(b*Sqrt[x])/(3*c^5) + (b*x^(3/2))/(9*c^3) + (b*x^(5/2))/(15*c) + (a*x^3)/3 + (b*x^3*ArcTanh[c*Sqrt[x]])/3 + (b
*Log[1 - c*Sqrt[x]])/(6*c^6) - (b*Log[1 + c*Sqrt[x]])/(6*c^6)

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Maple [A]
time = 0.07, size = 78, normalized size = 1.04

method	result	size
derivativedivides	\(\frac {\frac {c^{6} x^{3} a}{3}+\frac {b \,c^{6} x^{3} \arctanh \left (c \sqrt {x}\right )}{3}+\frac {b \,c^{5} x^{\frac {5}{2}}}{15}+\frac {b \,c^{3} x^{\frac {3}{2}}}{9}+\frac {b c \sqrt {x}}{3}+\frac {b \ln \left (c \sqrt {x}-1\right )}{6}-\frac {b \ln \left (1+c \sqrt {x}\right )}{6}}{c^{6}}\)	\(78\)
default	\(\frac {\frac {c^{6} x^{3} a}{3}+\frac {b \,c^{6} x^{3} \arctanh \left (c \sqrt {x}\right )}{3}+\frac {b \,c^{5} x^{\frac {5}{2}}}{15}+\frac {b \,c^{3} x^{\frac {3}{2}}}{9}+\frac {b c \sqrt {x}}{3}+\frac {b \ln \left (c \sqrt {x}-1\right )}{6}-\frac {b \ln \left (1+c \sqrt {x}\right )}{6}}{c^{6}}\)	\(78\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*(a+b*arctanh(c*x^(1/2))),x,method=_RETURNVERBOSE)

[Out]

2/c^6*(1/6*c^6*x^3*a+1/6*b*c^6*x^3*arctanh(c*x^(1/2))+1/30*b*c^5*x^(5/2)+1/18*b*c^3*x^(3/2)+1/6*b*c*x^(1/2)+1/
12*b*ln(c*x^(1/2)-1)-1/12*b*ln(1+c*x^(1/2)))

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Maxima [A]
time = 0.27, size = 78, normalized size = 1.04 \begin {gather*} \frac {1}{3} \, a x^{3} + \frac {1}{90} \, {\left (30 \, x^{3} \operatorname {artanh}\left (c \sqrt {x}\right ) + c {\left (\frac {2 \, {\left (3 \, c^{4} x^{\frac {5}{2}} + 5 \, c^{2} x^{\frac {3}{2}} + 15 \, \sqrt {x}\right )}}{c^{6}} - \frac {15 \, \log \left (c \sqrt {x} + 1\right )}{c^{7}} + \frac {15 \, \log \left (c \sqrt {x} - 1\right )}{c^{7}}\right )}\right )} b \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(a+b*arctanh(c*x^(1/2))),x, algorithm="maxima")

[Out]

1/3*a*x^3 + 1/90*(30*x^3*arctanh(c*sqrt(x)) + c*(2*(3*c^4*x^(5/2) + 5*c^2*x^(3/2) + 15*sqrt(x))/c^6 - 15*log(c
*sqrt(x) + 1)/c^7 + 15*log(c*sqrt(x) - 1)/c^7))*b

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Fricas [A]
time = 0.35, size = 80, normalized size = 1.07 \begin {gather*} \frac {30 \, a c^{6} x^{3} + 15 \, {\left (b c^{6} x^{3} - b\right )} \log \left (-\frac {c^{2} x + 2 \, c \sqrt {x} + 1}{c^{2} x - 1}\right ) + 2 \, {\left (3 \, b c^{5} x^{2} + 5 \, b c^{3} x + 15 \, b c\right )} \sqrt {x}}{90 \, c^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(a+b*arctanh(c*x^(1/2))),x, algorithm="fricas")

[Out]

1/90*(30*a*c^6*x^3 + 15*(b*c^6*x^3 - b)*log(-(c^2*x + 2*c*sqrt(x) + 1)/(c^2*x - 1)) + 2*(3*b*c^5*x^2 + 5*b*c^3
*x + 15*b*c)*sqrt(x))/c^6

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int x^{2} \left (a + b \operatorname {atanh}{\left (c \sqrt {x} \right )}\right )\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**2*(a+b*atanh(c*x**(1/2))),x)

[Out]

Integral(x**2*(a + b*atanh(c*sqrt(x))), x)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 301 vs. \(2 (55) = 110\).
time = 0.44, size = 301, normalized size = 4.01 \begin {gather*} \frac {1}{3} \, a x^{3} + \frac {2}{45} \, b c {\left (\frac {\frac {45 \, {\left (c \sqrt {x} + 1\right )}^{4}}{{\left (c \sqrt {x} - 1\right )}^{4}} - \frac {90 \, {\left (c \sqrt {x} + 1\right )}^{3}}{{\left (c \sqrt {x} - 1\right )}^{3}} + \frac {140 \, {\left (c \sqrt {x} + 1\right )}^{2}}{{\left (c \sqrt {x} - 1\right )}^{2}} - \frac {70 \, {\left (c \sqrt {x} + 1\right )}}{c \sqrt {x} - 1} + 23}{c^{7} {\left (\frac {c \sqrt {x} + 1}{c \sqrt {x} - 1} - 1\right )}^{5}} + \frac {15 \, {\left (\frac {3 \, {\left (c \sqrt {x} + 1\right )}^{5}}{{\left (c \sqrt {x} - 1\right )}^{5}} + \frac {10 \, {\left (c \sqrt {x} + 1\right )}^{3}}{{\left (c \sqrt {x} - 1\right )}^{3}} + \frac {3 \, {\left (c \sqrt {x} + 1\right )}}{c \sqrt {x} - 1}\right )} \log \left (-\frac {\frac {c {\left (\frac {c \sqrt {x} + 1}{c \sqrt {x} - 1} + 1\right )}}{\frac {{\left (c \sqrt {x} + 1\right )} c}{c \sqrt {x} - 1} - c} + 1}{\frac {c {\left (\frac {c \sqrt {x} + 1}{c \sqrt {x} - 1} + 1\right )}}{\frac {{\left (c \sqrt {x} + 1\right )} c}{c \sqrt {x} - 1} - c} - 1}\right )}{c^{7} {\left (\frac {c \sqrt {x} + 1}{c \sqrt {x} - 1} - 1\right )}^{6}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(a+b*arctanh(c*x^(1/2))),x, algorithm="giac")

[Out]

1/3*a*x^3 + 2/45*b*c*((45*(c*sqrt(x) + 1)^4/(c*sqrt(x) - 1)^4 - 90*(c*sqrt(x) + 1)^3/(c*sqrt(x) - 1)^3 + 140*(
c*sqrt(x) + 1)^2/(c*sqrt(x) - 1)^2 - 70*(c*sqrt(x) + 1)/(c*sqrt(x) - 1) + 23)/(c^7*((c*sqrt(x) + 1)/(c*sqrt(x)
- 1) - 1)^5) + 15*(3*(c*sqrt(x) + 1)^5/(c*sqrt(x) - 1)^5 + 10*(c*sqrt(x) + 1)^3/(c*sqrt(x) - 1)^3 + 3*(c*sqrt
(x) + 1)/(c*sqrt(x) - 1))*log(-(c*((c*sqrt(x) + 1)/(c*sqrt(x) - 1) + 1)/((c*sqrt(x) + 1)*c/(c*sqrt(x) - 1) - c
) + 1)/(c*((c*sqrt(x) + 1)/(c*sqrt(x) - 1) + 1)/((c*sqrt(x) + 1)*c/(c*sqrt(x) - 1) - c) - 1))/(c^7*((c*sqrt(x)
+ 1)/(c*sqrt(x) - 1) - 1)^6))

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Mupad [B]
time = 1.22, size = 58, normalized size = 0.77 \begin {gather*} \frac {a\,x^3}{3}+\frac {\frac {b\,c^3\,x^{3/2}}{9}-\frac {b\,\mathrm {atanh}\left (c\,\sqrt {x}\right )}{3}+\frac {b\,c^5\,x^{5/2}}{15}+\frac {b\,c\,\sqrt {x}}{3}}{c^6}+\frac {b\,x^3\,\mathrm {atanh}\left (c\,\sqrt {x}\right )}{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*(a + b*atanh(c*x^(1/2))),x)

[Out]

(a*x^3)/3 + ((b*c^3*x^(3/2))/9 - (b*atanh(c*x^(1/2)))/3 + (b*c^5*x^(5/2))/15 + (b*c*x^(1/2))/3)/c^6 + (b*x^3*a
tanh(c*x^(1/2)))/3

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