Integrand size = 17, antiderivative size = 91 \[ \int \sqrt {a+b \left (c x^3\right )^{3/2}} \, dx=\frac {4}{13} x \sqrt {a+b \left (c x^3\right )^{3/2}}+\frac {9 a x \sqrt {1+\frac {b \left (c x^3\right )^{3/2}}{a}} \operatorname {Hypergeometric2F1}\left (\frac {2}{9},\frac {1}{2},\frac {11}{9},-\frac {b \left (c x^3\right )^{3/2}}{a}\right )}{13 \sqrt {a+b \left (c x^3\right )^{3/2}}} \]
4/13*x*(a+b*(c*x^3)^(3/2))^(1/2)+9/13*a*x*hypergeom([2/9, 1/2],[11/9],-b*( c*x^3)^(3/2)/a)*(1+b*(c*x^3)^(3/2)/a)^(1/2)/(a+b*(c*x^3)^(3/2))^(1/2)
\[ \int \sqrt {a+b \left (c x^3\right )^{3/2}} \, dx=\int \sqrt {a+b \left (c x^3\right )^{3/2}} \, dx \]
Time = 0.24 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.29, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.294, Rules used = {787, 774, 811, 889, 888}
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.
\(\displaystyle \int \sqrt {a+b \left (c x^3\right )^{3/2}} \, dx\) |
\(\Big \downarrow \) 787 |
\(\displaystyle \int \sqrt {a+b c^{3/2} x^{9/2}}dx\) |
\(\Big \downarrow \) 774 |
\(\displaystyle 2 \int \frac {\sqrt {c x^3} \sqrt {\frac {b \left (c x^3\right )^{9/2}}{c^3 x^9}+a}}{\sqrt {c} x}d\frac {\sqrt {c x^3}}{\sqrt {c} x}\) |
\(\Big \downarrow \) 811 |
\(\displaystyle 2 \left (\frac {9}{13} a \int \frac {\sqrt {c x^3}}{\sqrt {c} x \sqrt {\frac {b \left (c x^3\right )^{9/2}}{c^3 x^9}+a}}d\frac {\sqrt {c x^3}}{\sqrt {c} x}+\frac {2}{13} x \sqrt {a+\frac {b \left (c x^3\right )^{9/2}}{c^3 x^9}}\right )\) |
\(\Big \downarrow \) 889 |
\(\displaystyle 2 \left (\frac {9 a \sqrt {\frac {b \left (c x^3\right )^{9/2}}{a c^3 x^9}+1} \int \frac {\sqrt {c x^3}}{\sqrt {c} x \sqrt {\frac {b \left (c x^3\right )^{9/2}}{a c^3 x^9}+1}}d\frac {\sqrt {c x^3}}{\sqrt {c} x}}{13 \sqrt {a+\frac {b \left (c x^3\right )^{9/2}}{c^3 x^9}}}+\frac {2}{13} x \sqrt {a+\frac {b \left (c x^3\right )^{9/2}}{c^3 x^9}}\right )\) |
\(\Big \downarrow \) 888 |
\(\displaystyle 2 \left (\frac {9 a x \sqrt {\frac {b \left (c x^3\right )^{9/2}}{a c^3 x^9}+1} \operatorname {Hypergeometric2F1}\left (\frac {2}{9},\frac {1}{2},\frac {11}{9},-\frac {b \left (c x^3\right )^{9/2}}{a c^3 x^9}\right )}{26 \sqrt {a+\frac {b \left (c x^3\right )^{9/2}}{c^3 x^9}}}+\frac {2}{13} x \sqrt {a+\frac {b \left (c x^3\right )^{9/2}}{c^3 x^9}}\right )\) |
2*((2*x*Sqrt[a + (b*(c*x^3)^(9/2))/(c^3*x^9)])/13 + (9*a*x*Sqrt[1 + (b*(c* x^3)^(9/2))/(a*c^3*x^9)]*Hypergeometric2F1[2/9, 1/2, 11/9, -((b*(c*x^3)^(9 /2))/(a*c^3*x^9))])/(26*Sqrt[a + (b*(c*x^3)^(9/2))/(c^3*x^9)]))
3.30.76.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = Denominator[n]}, Simp[k Subst[Int[x^(k - 1)*(a + b*x^(k*n))^p, x], x, x^(1/k)], x]] /; Fre eQ[{a, b, p}, x] && FractionQ[n]
Int[((a_) + (b_.)*((c_.)*(x_)^(q_.))^(n_))^(p_), x_Symbol] :> With[{k = Den ominator[n]}, Subst[Int[(a + b*c^n*x^(n*q))^p, x], x^(1/k), (c*x^q)^(1/k)/( c^(1/k)*(x^(1/k))^(q - 1))]] /; FreeQ[{a, b, c, p, q}, x] && FractionQ[n]
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c* x)^(m + 1)*((a + b*x^n)^p/(c*(m + n*p + 1))), x] + Simp[a*n*(p/(m + n*p + 1 )) Int[(c*x)^m*(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b, c, m}, x] && I GtQ[n, 0] && GtQ[p, 0] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b, c, n, m , p, x]
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^p *((c*x)^(m + 1)/(c*(m + 1)))*Hypergeometric2F1[-p, (m + 1)/n, (m + 1)/n + 1 , (-b)*(x^n/a)], x] /; FreeQ[{a, b, c, m, n, p}, x] && !IGtQ[p, 0] && (ILt Q[p, 0] || GtQ[a, 0])
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^I ntPart[p]*((a + b*x^n)^FracPart[p]/(1 + b*(x^n/a))^FracPart[p]) Int[(c*x) ^m*(1 + b*(x^n/a))^p, x], x] /; FreeQ[{a, b, c, m, n, p}, x] && !IGtQ[p, 0 ] && !(ILtQ[p, 0] || GtQ[a, 0])
\[\int \sqrt {a +b \left (c \,x^{3}\right )^{\frac {3}{2}}}d x\]
\[ \int \sqrt {a+b \left (c x^3\right )^{3/2}} \, dx=\int { \sqrt {\left (c x^{3}\right )^{\frac {3}{2}} b + a} \,d x } \]
\[ \int \sqrt {a+b \left (c x^3\right )^{3/2}} \, dx=\int \sqrt {a + b \left (c x^{3}\right )^{\frac {3}{2}}}\, dx \]
\[ \int \sqrt {a+b \left (c x^3\right )^{3/2}} \, dx=\int { \sqrt {\left (c x^{3}\right )^{\frac {3}{2}} b + a} \,d x } \]
\[ \int \sqrt {a+b \left (c x^3\right )^{3/2}} \, dx=\int { \sqrt {\left (c x^{3}\right )^{\frac {3}{2}} b + a} \,d x } \]
Timed out. \[ \int \sqrt {a+b \left (c x^3\right )^{3/2}} \, dx=\int \sqrt {a+b\,{\left (c\,x^3\right )}^{3/2}} \,d x \]