Integrand size = 19, antiderivative size = 70 \[ \int \frac {\sqrt {a+b x^3}}{(c x)^{5/2}} \, dx=-\frac {2 \sqrt {a+b x^3}}{3 c (c x)^{3/2}}+\frac {2 \sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} (c x)^{3/2}}{c^{3/2} \sqrt {a+b x^3}}\right )}{3 c^{5/2}} \] Output:
-2/3*(b*x^3+a)^(1/2)/c/(c*x)^(3/2)+2/3*b^(1/2)*arctanh(b^(1/2)*(c*x)^(3/2) /c^(3/2)/(b*x^3+a)^(1/2))/c^(5/2)
Time = 0.17 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.89 \[ \int \frac {\sqrt {a+b x^3}}{(c x)^{5/2}} \, dx=\frac {x \left (-2 \sqrt {a+b x^3}+2 \sqrt {b} x^{3/2} \log \left (\sqrt {b} x^{3/2}+\sqrt {a+b x^3}\right )\right )}{3 (c x)^{5/2}} \] Input:
Integrate[Sqrt[a + b*x^3]/(c*x)^(5/2),x]
Output:
(x*(-2*Sqrt[a + b*x^3] + 2*Sqrt[b]*x^(3/2)*Log[Sqrt[b]*x^(3/2) + Sqrt[a + b*x^3]]))/(3*(c*x)^(5/2))
Time = 0.37 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.01, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {809, 851, 807, 224, 219}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {\sqrt {a+b x^3}}{(c x)^{5/2}} \, dx\) |
\(\Big \downarrow \) 809 |
\(\displaystyle \frac {b \int \frac {\sqrt {c x}}{\sqrt {b x^3+a}}dx}{c^3}-\frac {2 \sqrt {a+b x^3}}{3 c (c x)^{3/2}}\) |
\(\Big \downarrow \) 851 |
\(\displaystyle \frac {2 b \int \frac {c x}{\sqrt {b x^3+a}}d\sqrt {c x}}{c^4}-\frac {2 \sqrt {a+b x^3}}{3 c (c x)^{3/2}}\) |
\(\Big \downarrow \) 807 |
\(\displaystyle \frac {2 b \int \frac {1}{\sqrt {a+\frac {b x}{c^2}}}d(c x)^{3/2}}{3 c^4}-\frac {2 \sqrt {a+b x^3}}{3 c (c x)^{3/2}}\) |
\(\Big \downarrow \) 224 |
\(\displaystyle \frac {2 b \int \frac {1}{1-\frac {b x}{c^2}}d\frac {(c x)^{3/2}}{\sqrt {a+\frac {b x}{c^2}}}}{3 c^4}-\frac {2 \sqrt {a+b x^3}}{3 c (c x)^{3/2}}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {2 \sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} (c x)^{3/2}}{c^{3/2} \sqrt {a+\frac {b x}{c^2}}}\right )}{3 c^{5/2}}-\frac {2 \sqrt {a+b x^3}}{3 c (c x)^{3/2}}\) |
Input:
Int[Sqrt[a + b*x^3]/(c*x)^(5/2),x]
Output:
(-2*Sqrt[a + b*x^3])/(3*c*(c*x)^(3/2)) + (2*Sqrt[b]*ArcTanh[(Sqrt[b]*(c*x) ^(3/2))/(c^(3/2)*Sqrt[a + (b*x)/c^2])])/(3*c^(5/2))
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] && (Gt Q[a, 0] || LtQ[b, 0])
Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a, b}, x] && !GtQ[a, 0]
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = GCD[m + 1, n]}, Simp[1/k Subst[Int[x^((m + 1)/k - 1)*(a + b*x^(n/k))^p, x], x, x^k], x] /; k != 1] /; FreeQ[{a, b, p}, x] && IGtQ[n, 0] && IntegerQ[m]
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c* x)^(m + 1)*((a + b*x^n)^p/(c*(m + 1))), x] - Simp[b*n*(p/(c^n*(m + 1))) I nt[(c*x)^(m + n)*(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b, c}, x] && IGtQ [n, 0] && GtQ[p, 0] && LtQ[m, -1] && !ILtQ[(m + n*p + n + 1)/n, 0] && IntB inomialQ[a, b, c, n, m, p, x]
Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = Denominator[m]}, Simp[k/c Subst[Int[x^(k*(m + 1) - 1)*(a + b*(x^(k*n)/c^ n))^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && FractionQ[m] && IntBinomialQ[a, b, c, n, m, p, x]
Time = 0.88 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.19
method | result | size |
risch | \(-\frac {2 \sqrt {b \,x^{3}+a}}{3 x \,c^{2} \sqrt {c x}}+\frac {2 b \,\operatorname {arctanh}\left (\frac {\sqrt {c x \left (b \,x^{3}+a \right )}}{x^{2} \sqrt {b c}}\right ) \sqrt {c x \left (b \,x^{3}+a \right )}}{3 \sqrt {b c}\, c^{2} \sqrt {c x}\, \sqrt {b \,x^{3}+a}}\) | \(83\) |
default | \(\frac {2 \sqrt {b \,x^{3}+a}\, \left (b c \,\operatorname {arctanh}\left (\frac {\sqrt {c x \left (b \,x^{3}+a \right )}}{x^{2} \sqrt {b c}}\right ) x^{2}-\sqrt {c x \left (b \,x^{3}+a \right )}\, \sqrt {b c}\right )}{3 x \,c^{2} \sqrt {c x}\, \sqrt {c x \left (b \,x^{3}+a \right )}\, \sqrt {b c}}\) | \(88\) |
elliptic | \(\text {Expression too large to display}\) | \(1034\) |
Input:
int((b*x^3+a)^(1/2)/(c*x)^(5/2),x,method=_RETURNVERBOSE)
Output:
-2/3*(b*x^3+a)^(1/2)/x/c^2/(c*x)^(1/2)+2/3*b/(b*c)^(1/2)*arctanh((c*x*(b*x ^3+a))^(1/2)/x^2/(b*c)^(1/2))/c^2*(c*x*(b*x^3+a))^(1/2)/(c*x)^(1/2)/(b*x^3 +a)^(1/2)
Time = 0.16 (sec) , antiderivative size = 167, normalized size of antiderivative = 2.39 \[ \int \frac {\sqrt {a+b x^3}}{(c x)^{5/2}} \, dx=\left [\frac {c x^{2} \sqrt {\frac {b}{c}} \log \left (-8 \, b^{2} x^{6} - 8 \, a b x^{3} - 4 \, {\left (2 \, b x^{4} + a x\right )} \sqrt {b x^{3} + a} \sqrt {c x} \sqrt {\frac {b}{c}} - a^{2}\right ) - 4 \, \sqrt {b x^{3} + a} \sqrt {c x}}{6 \, c^{3} x^{2}}, -\frac {c x^{2} \sqrt {-\frac {b}{c}} \arctan \left (\frac {2 \, \sqrt {b x^{3} + a} \sqrt {c x} x \sqrt {-\frac {b}{c}}}{2 \, b x^{3} + a}\right ) + 2 \, \sqrt {b x^{3} + a} \sqrt {c x}}{3 \, c^{3} x^{2}}\right ] \] Input:
integrate((b*x^3+a)^(1/2)/(c*x)^(5/2),x, algorithm="fricas")
Output:
[1/6*(c*x^2*sqrt(b/c)*log(-8*b^2*x^6 - 8*a*b*x^3 - 4*(2*b*x^4 + a*x)*sqrt( b*x^3 + a)*sqrt(c*x)*sqrt(b/c) - a^2) - 4*sqrt(b*x^3 + a)*sqrt(c*x))/(c^3* x^2), -1/3*(c*x^2*sqrt(-b/c)*arctan(2*sqrt(b*x^3 + a)*sqrt(c*x)*x*sqrt(-b/ c)/(2*b*x^3 + a)) + 2*sqrt(b*x^3 + a)*sqrt(c*x))/(c^3*x^2)]
Time = 1.88 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.31 \[ \int \frac {\sqrt {a+b x^3}}{(c x)^{5/2}} \, dx=- \frac {2 \sqrt {a}}{3 c^{\frac {5}{2}} x^{\frac {3}{2}} \sqrt {1 + \frac {b x^{3}}{a}}} + \frac {2 \sqrt {b} \operatorname {asinh}{\left (\frac {\sqrt {b} x^{\frac {3}{2}}}{\sqrt {a}} \right )}}{3 c^{\frac {5}{2}}} - \frac {2 b x^{\frac {3}{2}}}{3 \sqrt {a} c^{\frac {5}{2}} \sqrt {1 + \frac {b x^{3}}{a}}} \] Input:
integrate((b*x**3+a)**(1/2)/(c*x)**(5/2),x)
Output:
-2*sqrt(a)/(3*c**(5/2)*x**(3/2)*sqrt(1 + b*x**3/a)) + 2*sqrt(b)*asinh(sqrt (b)*x**(3/2)/sqrt(a))/(3*c**(5/2)) - 2*b*x**(3/2)/(3*sqrt(a)*c**(5/2)*sqrt (1 + b*x**3/a))
\[ \int \frac {\sqrt {a+b x^3}}{(c x)^{5/2}} \, dx=\int { \frac {\sqrt {b x^{3} + a}}{\left (c x\right )^{\frac {5}{2}}} \,d x } \] Input:
integrate((b*x^3+a)^(1/2)/(c*x)^(5/2),x, algorithm="maxima")
Output:
integrate(sqrt(b*x^3 + a)/(c*x)^(5/2), x)
Time = 0.13 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.34 \[ \int \frac {\sqrt {a+b x^3}}{(c x)^{5/2}} \, dx=-\frac {2 \, {\left ({\left (\frac {b \arctan \left (\frac {\sqrt {b c + \frac {a c}{x^{3}}}}{\sqrt {-b c}}\right )}{\sqrt {-b c}} + \frac {\sqrt {b c + \frac {a c}{x^{3}}}}{c}\right )} c - \frac {b c \arctan \left (\frac {\sqrt {b c}}{\sqrt {-b c}}\right ) + \sqrt {b c} \sqrt {-b c}}{\sqrt {-b c}}\right )} {\left | c \right |}^{2}}{3 \, c^{5}} \] Input:
integrate((b*x^3+a)^(1/2)/(c*x)^(5/2),x, algorithm="giac")
Output:
-2/3*((b*arctan(sqrt(b*c + a*c/x^3)/sqrt(-b*c))/sqrt(-b*c) + sqrt(b*c + a* c/x^3)/c)*c - (b*c*arctan(sqrt(b*c)/sqrt(-b*c)) + sqrt(b*c)*sqrt(-b*c))/sq rt(-b*c))*abs(c)^2/c^5
Timed out. \[ \int \frac {\sqrt {a+b x^3}}{(c x)^{5/2}} \, dx=\int \frac {\sqrt {b\,x^3+a}}{{\left (c\,x\right )}^{5/2}} \,d x \] Input:
int((a + b*x^3)^(1/2)/(c*x)^(5/2),x)
Output:
int((a + b*x^3)^(1/2)/(c*x)^(5/2), x)
Time = 0.29 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.01 \[ \int \frac {\sqrt {a+b x^3}}{(c x)^{5/2}} \, dx=\frac {\sqrt {c}\, \left (-2 \sqrt {b \,x^{3}+a}-\sqrt {x}\, \sqrt {b}\, \mathrm {log}\left (\sqrt {b \,x^{3}+a}-\sqrt {x}\, \sqrt {b}\, x \right ) x +\sqrt {x}\, \sqrt {b}\, \mathrm {log}\left (\sqrt {b \,x^{3}+a}+\sqrt {x}\, \sqrt {b}\, x \right ) x \right )}{3 \sqrt {x}\, c^{3} x} \] Input:
int((b*x^3+a)^(1/2)/(c*x)^(5/2),x)
Output:
(sqrt(c)*( - 2*sqrt(a + b*x**3) - sqrt(x)*sqrt(b)*log(sqrt(a + b*x**3) - s qrt(x)*sqrt(b)*x)*x + sqrt(x)*sqrt(b)*log(sqrt(a + b*x**3) + sqrt(x)*sqrt( b)*x)*x))/(3*sqrt(x)*c**3*x)