3.8.86 \(\int \frac {(a+c x^4)^{3/2}}{x} \, dx\) [786]

Optimal. Leaf size=59 \[ \frac {1}{2} a \sqrt {a+c x^4}+\frac {1}{6} \left (a+c x^4\right )^{3/2}-\frac {1}{2} a^{3/2} \tanh ^{-1}\left (\frac {\sqrt {a+c x^4}}{\sqrt {a}}\right ) \]

[Out]

1/6*(c*x^4+a)^(3/2)-1/2*a^(3/2)*arctanh((c*x^4+a)^(1/2)/a^(1/2))+1/2*a*(c*x^4+a)^(1/2)

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Rubi [A]
time = 0.03, antiderivative size = 59, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {272, 52, 65, 214} \begin {gather*} -\frac {1}{2} a^{3/2} \tanh ^{-1}\left (\frac {\sqrt {a+c x^4}}{\sqrt {a}}\right )+\frac {1}{2} a \sqrt {a+c x^4}+\frac {1}{6} \left (a+c x^4\right )^{3/2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + c*x^4)^(3/2)/x,x]

[Out]

(a*Sqrt[a + c*x^4])/2 + (a + c*x^4)^(3/2)/6 - (a^(3/2)*ArcTanh[Sqrt[a + c*x^4]/Sqrt[a]])/2

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 214

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

Rule 272

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rubi steps

\begin {align*} \int \frac {\left (a+c x^4\right )^{3/2}}{x} \, dx &=\frac {1}{4} \text {Subst}\left (\int \frac {(a+c x)^{3/2}}{x} \, dx,x,x^4\right )\\ &=\frac {1}{6} \left (a+c x^4\right )^{3/2}+\frac {1}{4} a \text {Subst}\left (\int \frac {\sqrt {a+c x}}{x} \, dx,x,x^4\right )\\ &=\frac {1}{2} a \sqrt {a+c x^4}+\frac {1}{6} \left (a+c x^4\right )^{3/2}+\frac {1}{4} a^2 \text {Subst}\left (\int \frac {1}{x \sqrt {a+c x}} \, dx,x,x^4\right )\\ &=\frac {1}{2} a \sqrt {a+c x^4}+\frac {1}{6} \left (a+c x^4\right )^{3/2}+\frac {a^2 \text {Subst}\left (\int \frac {1}{-\frac {a}{c}+\frac {x^2}{c}} \, dx,x,\sqrt {a+c x^4}\right )}{2 c}\\ &=\frac {1}{2} a \sqrt {a+c x^4}+\frac {1}{6} \left (a+c x^4\right )^{3/2}-\frac {1}{2} a^{3/2} \tanh ^{-1}\left (\frac {\sqrt {a+c x^4}}{\sqrt {a}}\right )\\ \end {align*}

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Mathematica [A]
time = 0.04, size = 52, normalized size = 0.88 \begin {gather*} \frac {1}{6} \sqrt {a+c x^4} \left (4 a+c x^4\right )-\frac {1}{2} a^{3/2} \tanh ^{-1}\left (\frac {\sqrt {a+c x^4}}{\sqrt {a}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + c*x^4)^(3/2)/x,x]

[Out]

(Sqrt[a + c*x^4]*(4*a + c*x^4))/6 - (a^(3/2)*ArcTanh[Sqrt[a + c*x^4]/Sqrt[a]])/2

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Maple [A]
time = 0.15, size = 57, normalized size = 0.97

method	result	size
default	\(\frac {c \,x^{4} \sqrt {x^{4} c +a}}{6}+\frac {2 a \sqrt {x^{4} c +a}}{3}-\frac {a^{\frac {3}{2}} \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {x^{4} c +a}}{x^{2}}\right )}{2}\)	\(57\)
elliptic	\(\frac {c \,x^{4} \sqrt {x^{4} c +a}}{6}+\frac {2 a \sqrt {x^{4} c +a}}{3}-\frac {a^{\frac {3}{2}} \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {x^{4} c +a}}{x^{2}}\right )}{2}\)	\(57\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^4+a)^(3/2)/x,x,method=_RETURNVERBOSE)

[Out]

1/6*c*x^4*(c*x^4+a)^(1/2)+2/3*a*(c*x^4+a)^(1/2)-1/2*a^(3/2)*ln((2*a+2*a^(1/2)*(c*x^4+a)^(1/2))/x^2)

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Maxima [A]
time = 0.51, size = 61, normalized size = 1.03 \begin {gather*} \frac {1}{4} \, a^{\frac {3}{2}} \log \left (\frac {\sqrt {c x^{4} + a} - \sqrt {a}}{\sqrt {c x^{4} + a} + \sqrt {a}}\right ) + \frac {1}{6} \, {\left (c x^{4} + a\right )}^{\frac {3}{2}} + \frac {1}{2} \, \sqrt {c x^{4} + a} a \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^4+a)^(3/2)/x,x, algorithm="maxima")

[Out]

1/4*a^(3/2)*log((sqrt(c*x^4 + a) - sqrt(a))/(sqrt(c*x^4 + a) + sqrt(a))) + 1/6*(c*x^4 + a)^(3/2) + 1/2*sqrt(c*
x^4 + a)*a

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Fricas [A]
time = 0.36, size = 103, normalized size = 1.75 \begin {gather*} \left [\frac {1}{4} \, a^{\frac {3}{2}} \log \left (\frac {c x^{4} - 2 \, \sqrt {c x^{4} + a} \sqrt {a} + 2 \, a}{x^{4}}\right ) + \frac {1}{6} \, {\left (c x^{4} + 4 \, a\right )} \sqrt {c x^{4} + a}, \frac {1}{2} \, \sqrt {-a} a \arctan \left (\frac {\sqrt {c x^{4} + a} \sqrt {-a}}{a}\right ) + \frac {1}{6} \, {\left (c x^{4} + 4 \, a\right )} \sqrt {c x^{4} + a}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^4+a)^(3/2)/x,x, algorithm="fricas")

[Out]

[1/4*a^(3/2)*log((c*x^4 - 2*sqrt(c*x^4 + a)*sqrt(a) + 2*a)/x^4) + 1/6*(c*x^4 + 4*a)*sqrt(c*x^4 + a), 1/2*sqrt(
-a)*a*arctan(sqrt(c*x^4 + a)*sqrt(-a)/a) + 1/6*(c*x^4 + 4*a)*sqrt(c*x^4 + a)]

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Sympy [A]
time = 1.07, size = 80, normalized size = 1.36 \begin {gather*} \frac {2 a^{\frac {3}{2}} \sqrt {1 + \frac {c x^{4}}{a}}}{3} + \frac {a^{\frac {3}{2}} \log {\left (\frac {c x^{4}}{a} \right )}}{4} - \frac {a^{\frac {3}{2}} \log {\left (\sqrt {1 + \frac {c x^{4}}{a}} + 1 \right )}}{2} + \frac {\sqrt {a} c x^{4} \sqrt {1 + \frac {c x^{4}}{a}}}{6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x**4+a)**(3/2)/x,x)

[Out]

2*a**(3/2)*sqrt(1 + c*x**4/a)/3 + a**(3/2)*log(c*x**4/a)/4 - a**(3/2)*log(sqrt(1 + c*x**4/a) + 1)/2 + sqrt(a)*
c*x**4*sqrt(1 + c*x**4/a)/6

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Giac [A]
time = 0.97, size = 50, normalized size = 0.85 \begin {gather*} \frac {a^{2} \arctan \left (\frac {\sqrt {c x^{4} + a}}{\sqrt {-a}}\right )}{2 \, \sqrt {-a}} + \frac {1}{6} \, {\left (c x^{4} + a\right )}^{\frac {3}{2}} + \frac {1}{2} \, \sqrt {c x^{4} + a} a \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^4+a)^(3/2)/x,x, algorithm="giac")

[Out]

1/2*a^2*arctan(sqrt(c*x^4 + a)/sqrt(-a))/sqrt(-a) + 1/6*(c*x^4 + a)^(3/2) + 1/2*sqrt(c*x^4 + a)*a

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Mupad [B]
time = 1.17, size = 43, normalized size = 0.73 \begin {gather*} \frac {a\,\sqrt {c\,x^4+a}}{2}-\frac {a^{3/2}\,\mathrm {atanh}\left (\frac {\sqrt {c\,x^4+a}}{\sqrt {a}}\right )}{2}+\frac {{\left (c\,x^4+a\right )}^{3/2}}{6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + c*x^4)^(3/2)/x,x)

[Out]

(a*(a + c*x^4)^(1/2))/2 - (a^(3/2)*atanh((a + c*x^4)^(1/2)/a^(1/2)))/2 + (a + c*x^4)^(3/2)/6

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