\(\int \frac {(a+b x^n)^{3/2}}{c x} \, dx\) [377]

Optimal result

Integrand size = 18, antiderivative size = 73 \[ \int \frac {\left (a+b x^n\right )^{3/2}}{c x} \, dx=\frac {2 a \sqrt {a+b x^n}}{c n}+\frac {2 \left (a+b x^n\right )^{3/2}}{3 c n}-\frac {2 a^{3/2} \text {arctanh}\left (\frac {\sqrt {a+b x^n}}{\sqrt {a}}\right )}{c n} \]

[Out]

2/3*(a+b*x^n)^(3/2)/c/n-2*a^(3/2)*arctanh((a+b*x^n)^(1/2)/a^(1/2))/c/n+2*a*(a+b*x^n)^(1/2)/c/n

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.278, Rules used = {12, 272, 52, 65, 214} \[ \int \frac {\left (a+b x^n\right )^{3/2}}{c x} \, dx=-\frac {2 a^{3/2} \text {arctanh}\left (\frac {\sqrt {a+b x^n}}{\sqrt {a}}\right )}{c n}+\frac {2 a \sqrt {a+b x^n}}{c n}+\frac {2 \left (a+b x^n\right )^{3/2}}{3 c n} \]

[In]

Int[(a + b*x^n)^(3/2)/(c*x),x]

[Out]

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

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

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

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.79 \[ \int \frac {\left (a+b x^n\right )^{3/2}}{c x} \, dx=\frac {2 \sqrt {a+b x^n} \left (4 a+b x^n\right )-6 a^{3/2} \text {arctanh}\left (\frac {\sqrt {a+b x^n}}{\sqrt {a}}\right )}{3 c n} \]

[In]

Integrate[(a + b*x^n)^(3/2)/(c*x),x]

[Out]

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

Maple [A] (verified)

Time = 2.30 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.70

method	result	size
derivativedivides	\(\frac {\frac {2 \left (a +b \,x^{n}\right )^{\frac {3}{2}}}{3}+2 a \sqrt {a +b \,x^{n}}-2 a^{\frac {3}{2}} \operatorname {arctanh}\left (\frac {\sqrt {a +b \,x^{n}}}{\sqrt {a}}\right )}{c n}\)	\(51\)
default	\(\frac {\frac {2 \left (a +b \,x^{n}\right )^{\frac {3}{2}}}{3}+2 a \sqrt {a +b \,x^{n}}-2 a^{\frac {3}{2}} \operatorname {arctanh}\left (\frac {\sqrt {a +b \,x^{n}}}{\sqrt {a}}\right )}{c n}\)	\(51\)
risch	\(\frac {2 \left (b \,{\mathrm e}^{n \ln \left (x \right )}+4 a \right ) \sqrt {a +b \,{\mathrm e}^{n \ln \left (x \right )}}}{3 n c}-\frac {2 a^{\frac {3}{2}} \operatorname {arctanh}\left (\frac {\sqrt {a +b \,{\mathrm e}^{n \ln \left (x \right )}}}{\sqrt {a}}\right )}{n c}\)	\(59\)

[In]

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

[Out]

1/c/n*(2/3*(a+b*x^n)^(3/2)+2*a*(a+b*x^n)^(1/2)-2*a^(3/2)*arctanh((a+b*x^n)^(1/2)/a^(1/2)))

Fricas [A] (verification not implemented)

none

Time = 0.71 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.64 \[ \int \frac {\left (a+b x^n\right )^{3/2}}{c x} \, dx=\left [\frac {3 \, a^{\frac {3}{2}} \log \left (\frac {b x^{n} - 2 \, \sqrt {b x^{n} + a} \sqrt {a} + 2 \, a}{x^{n}}\right ) + 2 \, {\left (b x^{n} + 4 \, a\right )} \sqrt {b x^{n} + a}}{3 \, c n}, \frac {2 \, {\left (3 \, \sqrt {-a} a \arctan \left (\frac {\sqrt {b x^{n} + a} \sqrt {-a}}{a}\right ) + {\left (b x^{n} + 4 \, a\right )} \sqrt {b x^{n} + a}\right )}}{3 \, c n}\right ] \]

[In]

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

[Out]

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

Sympy [A] (verification not implemented)

Time = 1.58 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.21 \[ \int \frac {\left (a+b x^n\right )^{3/2}}{c x} \, dx=\frac {\frac {8 a^{\frac {3}{2}} \sqrt {1 + \frac {b x^{n}}{a}}}{3 n} + \frac {a^{\frac {3}{2}} \log {\left (\frac {b x^{n}}{a} \right )}}{n} - \frac {2 a^{\frac {3}{2}} \log {\left (\sqrt {1 + \frac {b x^{n}}{a}} + 1 \right )}}{n} + \frac {2 \sqrt {a} b x^{n} \sqrt {1 + \frac {b x^{n}}{a}}}{3 n}}{c} \]

[In]

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

[Out]

(8*a**(3/2)*sqrt(1 + b*x**n/a)/(3*n) + a**(3/2)*log(b*x**n/a)/n - 2*a**(3/2)*log(sqrt(1 + b*x**n/a) + 1)/n + 2
*sqrt(a)*b*x**n*sqrt(1 + b*x**n/a)/(3*n))/c

Maxima [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.00 \[ \int \frac {\left (a+b x^n\right )^{3/2}}{c x} \, dx=\frac {\frac {3 \, a^{\frac {3}{2}} \log \left (\frac {\sqrt {b x^{n} + a} - \sqrt {a}}{\sqrt {b x^{n} + a} + \sqrt {a}}\right )}{n} + \frac {2 \, {\left ({\left (b x^{n} + a\right )}^{\frac {3}{2}} + 3 \, \sqrt {b x^{n} + a} a\right )}}{n}}{3 \, c} \]

[In]

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

[Out]

1/3*(3*a^(3/2)*log((sqrt(b*x^n + a) - sqrt(a))/(sqrt(b*x^n + a) + sqrt(a)))/n + 2*((b*x^n + a)^(3/2) + 3*sqrt(
b*x^n + a)*a)/n)/c

Giac [F]

\[ \int \frac {\left (a+b x^n\right )^{3/2}}{c x} \, dx=\int { \frac {{\left (b x^{n} + a\right )}^{\frac {3}{2}}}{c x} \,d x } \]

[In]

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

[Out]

integrate((b*x^n + a)^(3/2)/(c*x), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (a+b x^n\right )^{3/2}}{c x} \, dx=\int \frac {{\left (a+b\,x^n\right )}^{3/2}}{c\,x} \,d x \]

[In]

int((a + b*x^n)^(3/2)/(c*x),x)

[Out]

int((a + b*x^n)^(3/2)/(c*x), x)