Optimal. Leaf size=150 \[ \frac {\left (4 a c+b^2\right ) \tanh ^{-1}\left (\frac {b+2 c x^3}{2 \sqrt {c} \sqrt {a+b x^3+c x^6}}\right )}{8 \sqrt {c}}-\frac {\left (a+b x^3+c x^6\right )^{3/2}}{3 x^3}+\frac {1}{4} \left (3 b+2 c x^3\right ) \sqrt {a+b x^3+c x^6}-\frac {1}{2} \sqrt {a} b \tanh ^{-1}\left (\frac {2 a+b x^3}{2 \sqrt {a} \sqrt {a+b x^3+c x^6}}\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.17, antiderivative size = 150, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.350, Rules used = {1357, 732, 814, 843, 621, 206, 724} \[ \frac {\left (4 a c+b^2\right ) \tanh ^{-1}\left (\frac {b+2 c x^3}{2 \sqrt {c} \sqrt {a+b x^3+c x^6}}\right )}{8 \sqrt {c}}-\frac {\left (a+b x^3+c x^6\right )^{3/2}}{3 x^3}+\frac {1}{4} \left (3 b+2 c x^3\right ) \sqrt {a+b x^3+c x^6}-\frac {1}{2} \sqrt {a} b \tanh ^{-1}\left (\frac {2 a+b x^3}{2 \sqrt {a} \sqrt {a+b x^3+c x^6}}\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 206
Rule 621
Rule 724
Rule 732
Rule 814
Rule 843
Rule 1357
Rubi steps
\begin {align*} \int \frac {\left (a+b x^3+c x^6\right )^{3/2}}{x^4} \, dx &=\frac {1}{3} \operatorname {Subst}\left (\int \frac {\left (a+b x+c x^2\right )^{3/2}}{x^2} \, dx,x,x^3\right )\\ &=-\frac {\left (a+b x^3+c x^6\right )^{3/2}}{3 x^3}+\frac {1}{2} \operatorname {Subst}\left (\int \frac {(b+2 c x) \sqrt {a+b x+c x^2}}{x} \, dx,x,x^3\right )\\ &=\frac {1}{4} \left (3 b+2 c x^3\right ) \sqrt {a+b x^3+c x^6}-\frac {\left (a+b x^3+c x^6\right )^{3/2}}{3 x^3}-\frac {\operatorname {Subst}\left (\int \frac {-4 a b c-c \left (b^2+4 a c\right ) x}{x \sqrt {a+b x+c x^2}} \, dx,x,x^3\right )}{8 c}\\ &=\frac {1}{4} \left (3 b+2 c x^3\right ) \sqrt {a+b x^3+c x^6}-\frac {\left (a+b x^3+c x^6\right )^{3/2}}{3 x^3}+\frac {1}{2} (a b) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {a+b x+c x^2}} \, dx,x,x^3\right )-\frac {1}{8} \left (-b^2-4 a c\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {a+b x+c x^2}} \, dx,x,x^3\right )\\ &=\frac {1}{4} \left (3 b+2 c x^3\right ) \sqrt {a+b x^3+c x^6}-\frac {\left (a+b x^3+c x^6\right )^{3/2}}{3 x^3}-(a b) \operatorname {Subst}\left (\int \frac {1}{4 a-x^2} \, dx,x,\frac {2 a+b x^3}{\sqrt {a+b x^3+c x^6}}\right )-\frac {1}{4} \left (-b^2-4 a c\right ) \operatorname {Subst}\left (\int \frac {1}{4 c-x^2} \, dx,x,\frac {b+2 c x^3}{\sqrt {a+b x^3+c x^6}}\right )\\ &=\frac {1}{4} \left (3 b+2 c x^3\right ) \sqrt {a+b x^3+c x^6}-\frac {\left (a+b x^3+c x^6\right )^{3/2}}{3 x^3}-\frac {1}{2} \sqrt {a} b \tanh ^{-1}\left (\frac {2 a+b x^3}{2 \sqrt {a} \sqrt {a+b x^3+c x^6}}\right )+\frac {\left (b^2+4 a c\right ) \tanh ^{-1}\left (\frac {b+2 c x^3}{2 \sqrt {c} \sqrt {a+b x^3+c x^6}}\right )}{8 \sqrt {c}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.11, size = 134, normalized size = 0.89 \[ \frac {1}{24} \left (\frac {3 \left (4 a c+b^2\right ) \tanh ^{-1}\left (\frac {b+2 c x^3}{2 \sqrt {c} \sqrt {a+b x^3+c x^6}}\right )}{\sqrt {c}}+\frac {2 \sqrt {a+b x^3+c x^6} \left (-4 a+5 b x^3+2 c x^6\right )}{x^3}-12 \sqrt {a} b \tanh ^{-1}\left (\frac {2 a+b x^3}{2 \sqrt {a} \sqrt {a+b x^3+c x^6}}\right )\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 1.14, size = 713, normalized size = 4.75 \[ \left [\frac {12 \, \sqrt {a} b c x^{3} \log \left (-\frac {{\left (b^{2} + 4 \, a c\right )} x^{6} + 8 \, a b x^{3} - 4 \, \sqrt {c x^{6} + b x^{3} + a} {\left (b x^{3} + 2 \, a\right )} \sqrt {a} + 8 \, a^{2}}{x^{6}}\right ) + 3 \, {\left (b^{2} + 4 \, a c\right )} \sqrt {c} x^{3} \log \left (-8 \, c^{2} x^{6} - 8 \, b c x^{3} - b^{2} - 4 \, \sqrt {c x^{6} + b x^{3} + a} {\left (2 \, c x^{3} + b\right )} \sqrt {c} - 4 \, a c\right ) + 4 \, {\left (2 \, c^{2} x^{6} + 5 \, b c x^{3} - 4 \, a c\right )} \sqrt {c x^{6} + b x^{3} + a}}{48 \, c x^{3}}, \frac {6 \, \sqrt {a} b c x^{3} \log \left (-\frac {{\left (b^{2} + 4 \, a c\right )} x^{6} + 8 \, a b x^{3} - 4 \, \sqrt {c x^{6} + b x^{3} + a} {\left (b x^{3} + 2 \, a\right )} \sqrt {a} + 8 \, a^{2}}{x^{6}}\right ) - 3 \, {\left (b^{2} + 4 \, a c\right )} \sqrt {-c} x^{3} \arctan \left (\frac {\sqrt {c x^{6} + b x^{3} + a} {\left (2 \, c x^{3} + b\right )} \sqrt {-c}}{2 \, {\left (c^{2} x^{6} + b c x^{3} + a c\right )}}\right ) + 2 \, {\left (2 \, c^{2} x^{6} + 5 \, b c x^{3} - 4 \, a c\right )} \sqrt {c x^{6} + b x^{3} + a}}{24 \, c x^{3}}, \frac {24 \, \sqrt {-a} b c x^{3} \arctan \left (\frac {\sqrt {c x^{6} + b x^{3} + a} {\left (b x^{3} + 2 \, a\right )} \sqrt {-a}}{2 \, {\left (a c x^{6} + a b x^{3} + a^{2}\right )}}\right ) + 3 \, {\left (b^{2} + 4 \, a c\right )} \sqrt {c} x^{3} \log \left (-8 \, c^{2} x^{6} - 8 \, b c x^{3} - b^{2} - 4 \, \sqrt {c x^{6} + b x^{3} + a} {\left (2 \, c x^{3} + b\right )} \sqrt {c} - 4 \, a c\right ) + 4 \, {\left (2 \, c^{2} x^{6} + 5 \, b c x^{3} - 4 \, a c\right )} \sqrt {c x^{6} + b x^{3} + a}}{48 \, c x^{3}}, \frac {12 \, \sqrt {-a} b c x^{3} \arctan \left (\frac {\sqrt {c x^{6} + b x^{3} + a} {\left (b x^{3} + 2 \, a\right )} \sqrt {-a}}{2 \, {\left (a c x^{6} + a b x^{3} + a^{2}\right )}}\right ) - 3 \, {\left (b^{2} + 4 \, a c\right )} \sqrt {-c} x^{3} \arctan \left (\frac {\sqrt {c x^{6} + b x^{3} + a} {\left (2 \, c x^{3} + b\right )} \sqrt {-c}}{2 \, {\left (c^{2} x^{6} + b c x^{3} + a c\right )}}\right ) + 2 \, {\left (2 \, c^{2} x^{6} + 5 \, b c x^{3} - 4 \, a c\right )} \sqrt {c x^{6} + b x^{3} + a}}{24 \, c x^{3}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (c x^{6} + b x^{3} + a\right )}^{\frac {3}{2}}}{x^{4}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.06, size = 0, normalized size = 0.00 \[ \int \frac {\left (c \,x^{6}+b \,x^{3}+a \right )^{\frac {3}{2}}}{x^{4}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (c\,x^6+b\,x^3+a\right )}^{3/2}}{x^4} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (a + b x^{3} + c x^{6}\right )^{\frac {3}{2}}}{x^{4}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________