Integrand size = 29, antiderivative size = 73 \[ \int \frac {x^5}{a c+b c x^3+d \sqrt {a+b x^3}} \, dx=\frac {x^3}{3 b c}-\frac {2 d \sqrt {a+b x^3}}{3 b^2 c^2}-\frac {2 \left (a c^2-d^2\right ) \log \left (d+c \sqrt {a+b x^3}\right )}{3 b^2 c^3} \]
Time = 0.05 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.90 \[ \int \frac {x^5}{a c+b c x^3+d \sqrt {a+b x^3}} \, dx=\frac {c \left (a c+b c x^3-2 d \sqrt {a+b x^3}\right )+\left (-2 a c^2+2 d^2\right ) \log \left (d+c \sqrt {a+b x^3}\right )}{3 b^2 c^3} \]
(c*(a*c + b*c*x^3 - 2*d*Sqrt[a + b*x^3]) + (-2*a*c^2 + 2*d^2)*Log[d + c*Sq rt[a + b*x^3]])/(3*b^2*c^3)
Time = 0.41 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.92, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.172, Rules used = {2586, 7267, 25, 476, 2009}
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.
\(\displaystyle \int \frac {x^5}{d \sqrt {a+b x^3}+a c+b c x^3} \, dx\) |
\(\Big \downarrow \) 2586 |
\(\displaystyle \frac {1}{3} \int \frac {x^3}{b c x^3+a c+d \sqrt {b x^3+a}}dx^3\) |
\(\Big \downarrow \) 7267 |
\(\displaystyle \frac {2 \int -\frac {a-x^6}{\sqrt {b x^3+a} c+d}d\sqrt {b x^3+a}}{3 b^2}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\frac {2 \int \frac {a-x^6}{\sqrt {b x^3+a} c+d}d\sqrt {b x^3+a}}{3 b^2}\) |
\(\Big \downarrow \) 476 |
\(\displaystyle -\frac {2 \int \left (\frac {d}{c^2}-\frac {\sqrt {b x^3+a}}{c}+\frac {a c^2-d^2}{c^2 \left (\sqrt {b x^3+a} c+d\right )}\right )d\sqrt {b x^3+a}}{3 b^2}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {2 \left (-\frac {d \sqrt {a+b x^3}}{c^2}-\frac {\left (a c^2-d^2\right ) \log \left (c \sqrt {a+b x^3}+d\right )}{c^3}+\frac {x^6}{2 c}\right )}{3 b^2}\) |
(2*(x^6/(2*c) - (d*Sqrt[a + b*x^3])/c^2 - ((a*c^2 - d^2)*Log[d + c*Sqrt[a + b*x^3]])/c^3))/(3*b^2)
3.6.53.3.1 Defintions of rubi rules used
Int[((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ ExpandIntegrand[(c + d*x)^n*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, n}, x] && IGtQ[p, 0]
Int[(x_)^(m_.)/((c_) + (d_.)*(x_)^(n_) + (e_.)*Sqrt[(a_) + (b_.)*(x_)^(n_)] ), x_Symbol] :> Simp[1/n Subst[Int[x^((m + 1)/n - 1)/(c + d*x + e*Sqrt[a + b*x]), x], x, x^n], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && EqQ[b*c - a* d, 0] && IntegerQ[(m + 1)/n]
Int[u_, x_Symbol] :> With[{lst = SubstForFractionalPowerOfLinear[u, x]}, Si mp[lst[[2]]*lst[[4]] Subst[Int[lst[[1]], x], x, lst[[3]]^(1/lst[[2]])], x ] /; !FalseQ[lst] && SubstForFractionalPowerQ[u, lst[[3]], x]]
Leaf count of result is larger than twice the leaf count of optimal. \(245\) vs. \(2(63)=126\).
Time = 0.16 (sec) , antiderivative size = 246, normalized size of antiderivative = 3.37
method | result | size |
default | \(d \left (-\frac {2 a \sqrt {b \,x^{3}+a}}{3 b^{2} d^{2}}+\frac {\left (a \,c^{2}-d^{2}\right ) \left (d \ln \left (c \sqrt {b \,x^{3}+a}-d \right )-d \ln \left (d +c \sqrt {b \,x^{3}+a}\right )+2 c \sqrt {b \,x^{3}+a}\right )}{3 b^{2} d^{2} c^{3}}\right )-a c \left (\frac {\left (a \,c^{2}-d^{2}\right ) \ln \left (b \,c^{2} x^{3}+a \,c^{2}-d^{2}\right )}{3 b^{2} d^{2} c^{2}}-\frac {a \ln \left (b \,x^{3}+a \right )}{3 b^{2} d^{2}}\right )-b c \left (-\frac {x^{3}}{3 b^{2} c^{2}}+\frac {\left (-a^{2} c^{4}+2 a \,c^{2} d^{2}-d^{4}\right ) \ln \left (b \,c^{2} x^{3}+a \,c^{2}-d^{2}\right )}{3 d^{2} c^{4} b^{3}}+\frac {a^{2} \ln \left (b \,x^{3}+a \right )}{3 b^{3} d^{2}}\right )\) | \(246\) |
elliptic | \(\frac {\sqrt {b \,x^{3}+a}\, \left (d +c \sqrt {b \,x^{3}+a}\right ) \left (c \left (\frac {x^{3}}{3 b \,c^{2}}+\frac {\left (-a \,c^{2}+d^{2}\right ) \ln \left (b \,c^{2} x^{3}+a \,c^{2}-d^{2}\right )}{3 b^{2} c^{4}}\right )-\frac {2 d \sqrt {b \,x^{3}+a}}{3 b^{2} c^{2}}+\frac {i \sqrt {2}\, \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\operatorname {RootOf}\left (b \,c^{2} \textit {\_Z}^{3}+a \,c^{2}-d^{2}\right )}{\sum }\frac {\left (a \,c^{2}-d^{2}\right ) \left (-b^{2} a \right )^{\frac {1}{3}} \sqrt {2}\, \sqrt {\frac {i b \left (2 x +\frac {\left (-b^{2} a \right )^{\frac {1}{3}}-i \sqrt {3}\, \left (-b^{2} a \right )^{\frac {1}{3}}}{b}\right )}{\left (-b^{2} a \right )^{\frac {1}{3}}}}\, \sqrt {\frac {b \left (x -\frac {\left (-b^{2} a \right )^{\frac {1}{3}}}{b}\right )}{-3 \left (-b^{2} a \right )^{\frac {1}{3}}+i \sqrt {3}\, \left (-b^{2} a \right )^{\frac {1}{3}}}}\, \sqrt {-\frac {i b \left (2 x +\frac {\left (-b^{2} a \right )^{\frac {1}{3}}+i \sqrt {3}\, \left (-b^{2} a \right )^{\frac {1}{3}}}{b}\right )}{2 \left (-b^{2} a \right )^{\frac {1}{3}}}}\, \left (i \left (-b^{2} a \right )^{\frac {1}{3}} \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha b -i \left (-b^{2} a \right )^{\frac {2}{3}} \sqrt {3}+2 \underline {\hspace {1.25 ex}}\alpha ^{2} b^{2}-\left (-b^{2} a \right )^{\frac {1}{3}} \underline {\hspace {1.25 ex}}\alpha b -\left (-b^{2} a \right )^{\frac {2}{3}}\right ) \Pi \left (\frac {\sqrt {3}\, \sqrt {\frac {i \left (x +\frac {\left (-b^{2} a \right )^{\frac {1}{3}}}{2 b}-\frac {i \sqrt {3}\, \left (-b^{2} a \right )^{\frac {1}{3}}}{2 b}\right ) \sqrt {3}\, b}{\left (-b^{2} a \right )^{\frac {1}{3}}}}}{3}, -\frac {c^{2} \left (2 i \left (-b^{2} a \right )^{\frac {1}{3}} \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha ^{2} b -i \left (-b^{2} a \right )^{\frac {2}{3}} \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha +i \sqrt {3}\, a b -3 \left (-b^{2} a \right )^{\frac {2}{3}} \underline {\hspace {1.25 ex}}\alpha -3 a b \right )}{2 b \,d^{2}}, \sqrt {\frac {i \sqrt {3}\, \left (-b^{2} a \right )^{\frac {1}{3}}}{b \left (-\frac {3 \left (-b^{2} a \right )^{\frac {1}{3}}}{2 b}+\frac {i \sqrt {3}\, \left (-b^{2} a \right )^{\frac {1}{3}}}{2 b}\right )}}\right )}{2 \sqrt {b \,x^{3}+a}}\right )}{3 d \,b^{4} c^{2}}\right )}{a c +b c \,x^{3}+d \sqrt {b \,x^{3}+a}}\) | \(557\) |
d*(-2/3*a/b^2/d^2*(b*x^3+a)^(1/2)+1/3*(a*c^2-d^2)/b^2/d^2*(d*ln(c*(b*x^3+a )^(1/2)-d)-d*ln(d+c*(b*x^3+a)^(1/2))+2*c*(b*x^3+a)^(1/2))/c^3)-a*c*(1/3*(a *c^2-d^2)/b^2/d^2/c^2*ln(b*c^2*x^3+a*c^2-d^2)-1/3*a/b^2/d^2*ln(b*x^3+a))-b *c*(-1/3/b^2/c^2*x^3+1/3*(-a^2*c^4+2*a*c^2*d^2-d^4)/d^2/c^4/b^3*ln(b*c^2*x ^3+a*c^2-d^2)+1/3*a^2/b^3/d^2*ln(b*x^3+a))
Time = 0.28 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.62 \[ \int \frac {x^5}{a c+b c x^3+d \sqrt {a+b x^3}} \, dx=\frac {b c^{2} x^{3} - 2 \, \sqrt {b x^{3} + a} c d - {\left (a c^{2} - d^{2}\right )} \log \left (b c^{2} x^{3} + a c^{2} - d^{2}\right ) - {\left (a c^{2} - d^{2}\right )} \log \left (\sqrt {b x^{3} + a} c + d\right ) + {\left (a c^{2} - d^{2}\right )} \log \left (\sqrt {b x^{3} + a} c - d\right )}{3 \, b^{2} c^{3}} \]
1/3*(b*c^2*x^3 - 2*sqrt(b*x^3 + a)*c*d - (a*c^2 - d^2)*log(b*c^2*x^3 + a*c ^2 - d^2) - (a*c^2 - d^2)*log(sqrt(b*x^3 + a)*c + d) + (a*c^2 - d^2)*log(s qrt(b*x^3 + a)*c - d))/(b^2*c^3)
Time = 1.63 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.26 \[ \int \frac {x^5}{a c+b c x^3+d \sqrt {a+b x^3}} \, dx=\begin {cases} \frac {2 \left (\frac {a + b x^{3}}{6 c} - \frac {d \sqrt {a + b x^{3}}}{3 c^{2}} - \frac {\left (a c^{2} - d^{2}\right ) \left (\begin {cases} \frac {\sqrt {a + b x^{3}}}{d} & \text {for}\: c = 0 \\\frac {\log {\left (c \sqrt {a + b x^{3}} + d \right )}}{c} & \text {otherwise} \end {cases}\right )}{3 c^{2}}\right )}{b^{2}} & \text {for}\: b \neq 0 \\\frac {x^{6}}{2 \cdot \left (3 \sqrt {a} d + 3 a c\right )} & \text {otherwise} \end {cases} \]
Piecewise((2*((a + b*x**3)/(6*c) - d*sqrt(a + b*x**3)/(3*c**2) - (a*c**2 - d**2)*Piecewise((sqrt(a + b*x**3)/d, Eq(c, 0)), (log(c*sqrt(a + b*x**3) + d)/c, True))/(3*c**2))/b**2, Ne(b, 0)), (x**6/(2*(3*sqrt(a)*d + 3*a*c)), True))
Time = 0.18 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.85 \[ \int \frac {x^5}{a c+b c x^3+d \sqrt {a+b x^3}} \, dx=\frac {\frac {{\left (b x^{3} + a\right )} c - 2 \, \sqrt {b x^{3} + a} d}{c^{2}} - \frac {2 \, {\left (a c^{2} - d^{2}\right )} \log \left (\sqrt {b x^{3} + a} c + d\right )}{c^{3}}}{3 \, b^{2}} \]
1/3*(((b*x^3 + a)*c - 2*sqrt(b*x^3 + a)*d)/c^2 - 2*(a*c^2 - d^2)*log(sqrt( b*x^3 + a)*c + d)/c^3)/b^2
Time = 0.33 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.99 \[ \int \frac {x^5}{a c+b c x^3+d \sqrt {a+b x^3}} \, dx=-\frac {\frac {2 \, {\left (a c^{2} - d^{2}\right )} \log \left ({\left | \sqrt {b x^{3} + a} c + d \right |}\right )}{b c^{3}} - \frac {{\left (b x^{3} + a\right )} b c - 2 \, \sqrt {b x^{3} + a} b d}{b^{2} c^{2}}}{3 \, b} \]
-1/3*(2*(a*c^2 - d^2)*log(abs(sqrt(b*x^3 + a)*c + d))/(b*c^3) - ((b*x^3 + a)*b*c - 2*sqrt(b*x^3 + a)*b*d)/(b^2*c^2))/b
Time = 18.04 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.63 \[ \int \frac {x^5}{a c+b c x^3+d \sqrt {a+b x^3}} \, dx=\frac {x^3}{3\,b\,c}-\frac {2\,d\,\sqrt {b\,x^3+a}}{3\,b^2\,c^2}+\frac {\ln \left (\frac {d-c\,\sqrt {b\,x^3+a}}{d+c\,\sqrt {b\,x^3+a}}\right )\,\left (a\,c^2-d^2\right )}{3\,b^2\,c^3}-\frac {\ln \left (b\,c^2\,x^3+a\,c^2-d^2\right )\,\left (a\,c^2-d^2\right )}{3\,b^2\,c^3} \]