Optimal. Leaf size=197 \[ -\frac {3 \left (b^2-4 a c\right )^2 \tanh ^{-1}\left (\frac {x (2 a+b x)}{2 \sqrt {a} \sqrt {a x^2+b x^3+c x^4}}\right )}{128 a^{5/2}}+\frac {b \left (3 b^2-20 a c\right ) \sqrt {a x^2+b x^3+c x^4}}{64 a^2 x^2}-\frac {\left (b^2-12 a c\right ) \sqrt {a x^2+b x^3+c x^4}}{32 a x^3}-\frac {(b+6 c x) \sqrt {a x^2+b x^3+c x^4}}{8 x^4}-\frac {\left (a x^2+b x^3+c x^4\right )^{3/2}}{4 x^7} \]
________________________________________________________________________________________
Rubi [A] time = 0.36, antiderivative size = 197, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {1920, 1941, 1951, 12, 1904, 206} \begin {gather*} \frac {b \left (3 b^2-20 a c\right ) \sqrt {a x^2+b x^3+c x^4}}{64 a^2 x^2}-\frac {3 \left (b^2-4 a c\right )^2 \tanh ^{-1}\left (\frac {x (2 a+b x)}{2 \sqrt {a} \sqrt {a x^2+b x^3+c x^4}}\right )}{128 a^{5/2}}-\frac {\left (b^2-12 a c\right ) \sqrt {a x^2+b x^3+c x^4}}{32 a x^3}-\frac {\left (a x^2+b x^3+c x^4\right )^{3/2}}{4 x^7}-\frac {(b+6 c x) \sqrt {a x^2+b x^3+c x^4}}{8 x^4} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 206
Rule 1904
Rule 1920
Rule 1941
Rule 1951
Rubi steps
\begin {align*} \int \frac {\left (a x^2+b x^3+c x^4\right )^{3/2}}{x^8} \, dx &=-\frac {\left (a x^2+b x^3+c x^4\right )^{3/2}}{4 x^7}+\frac {3}{8} \int \frac {(b+2 c x) \sqrt {a x^2+b x^3+c x^4}}{x^5} \, dx\\ &=-\frac {(b+6 c x) \sqrt {a x^2+b x^3+c x^4}}{8 x^4}-\frac {\left (a x^2+b x^3+c x^4\right )^{3/2}}{4 x^7}+\frac {1}{16} \int \frac {b^2-12 a c-4 b c x}{x^2 \sqrt {a x^2+b x^3+c x^4}} \, dx\\ &=-\frac {\left (b^2-12 a c\right ) \sqrt {a x^2+b x^3+c x^4}}{32 a x^3}-\frac {(b+6 c x) \sqrt {a x^2+b x^3+c x^4}}{8 x^4}-\frac {\left (a x^2+b x^3+c x^4\right )^{3/2}}{4 x^7}-\frac {\int \frac {\frac {1}{2} b \left (3 b^2-20 a c\right )+c \left (b^2-12 a c\right ) x}{x \sqrt {a x^2+b x^3+c x^4}} \, dx}{32 a}\\ &=-\frac {\left (b^2-12 a c\right ) \sqrt {a x^2+b x^3+c x^4}}{32 a x^3}+\frac {b \left (3 b^2-20 a c\right ) \sqrt {a x^2+b x^3+c x^4}}{64 a^2 x^2}-\frac {(b+6 c x) \sqrt {a x^2+b x^3+c x^4}}{8 x^4}-\frac {\left (a x^2+b x^3+c x^4\right )^{3/2}}{4 x^7}+\frac {\int \frac {3 \left (b^2-4 a c\right )^2}{4 \sqrt {a x^2+b x^3+c x^4}} \, dx}{32 a^2}\\ &=-\frac {\left (b^2-12 a c\right ) \sqrt {a x^2+b x^3+c x^4}}{32 a x^3}+\frac {b \left (3 b^2-20 a c\right ) \sqrt {a x^2+b x^3+c x^4}}{64 a^2 x^2}-\frac {(b+6 c x) \sqrt {a x^2+b x^3+c x^4}}{8 x^4}-\frac {\left (a x^2+b x^3+c x^4\right )^{3/2}}{4 x^7}+\frac {\left (3 \left (b^2-4 a c\right )^2\right ) \int \frac {1}{\sqrt {a x^2+b x^3+c x^4}} \, dx}{128 a^2}\\ &=-\frac {\left (b^2-12 a c\right ) \sqrt {a x^2+b x^3+c x^4}}{32 a x^3}+\frac {b \left (3 b^2-20 a c\right ) \sqrt {a x^2+b x^3+c x^4}}{64 a^2 x^2}-\frac {(b+6 c x) \sqrt {a x^2+b x^3+c x^4}}{8 x^4}-\frac {\left (a x^2+b x^3+c x^4\right )^{3/2}}{4 x^7}-\frac {\left (3 \left (b^2-4 a c\right )^2\right ) \operatorname {Subst}\left (\int \frac {1}{4 a-x^2} \, dx,x,\frac {x (2 a+b x)}{\sqrt {a x^2+b x^3+c x^4}}\right )}{64 a^2}\\ &=-\frac {\left (b^2-12 a c\right ) \sqrt {a x^2+b x^3+c x^4}}{32 a x^3}+\frac {b \left (3 b^2-20 a c\right ) \sqrt {a x^2+b x^3+c x^4}}{64 a^2 x^2}-\frac {(b+6 c x) \sqrt {a x^2+b x^3+c x^4}}{8 x^4}-\frac {\left (a x^2+b x^3+c x^4\right )^{3/2}}{4 x^7}-\frac {3 \left (b^2-4 a c\right )^2 \tanh ^{-1}\left (\frac {x (2 a+b x)}{2 \sqrt {a} \sqrt {a x^2+b x^3+c x^4}}\right )}{128 a^{5/2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.12, size = 141, normalized size = 0.72 \begin {gather*} -\frac {\sqrt {x^2 (a+x (b+c x))} \left (2 \sqrt {a} (2 a+b x) \sqrt {a+x (b+c x)} \left (8 a^2+4 a x (2 b+5 c x)-3 b^2 x^2\right )+3 x^4 \left (b^2-4 a c\right )^2 \tanh ^{-1}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+x (b+c x)}}\right )\right )}{128 a^{5/2} x^5 \sqrt {a+x (b+c x)}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [A] time = 2.11, size = 175, normalized size = 0.89 \begin {gather*} \frac {3 \left (16 a^2 c^2-8 a b^2 c+b^4\right ) \log \left (2 \sqrt {a} \sqrt {a x^2+b x^3+c x^4}-2 a x-b x^2\right )}{128 a^{5/2}}-\frac {3 \log (x) \left (16 a^2 c^2-8 a b^2 c+b^4\right )}{64 a^{5/2}}+\frac {\sqrt {a x^2+b x^3+c x^4} \left (-16 a^3-24 a^2 b x-40 a^2 c x^2-2 a b^2 x^2-20 a b c x^3+3 b^3 x^3\right )}{64 a^2 x^5} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 1.63, size = 332, normalized size = 1.69 \begin {gather*} \left [\frac {3 \, {\left (b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}\right )} \sqrt {a} x^{5} \log \left (-\frac {8 \, a b x^{2} + {\left (b^{2} + 4 \, a c\right )} x^{3} + 8 \, a^{2} x - 4 \, \sqrt {c x^{4} + b x^{3} + a x^{2}} {\left (b x + 2 \, a\right )} \sqrt {a}}{x^{3}}\right ) - 4 \, {\left (24 \, a^{3} b x + 16 \, a^{4} - {\left (3 \, a b^{3} - 20 \, a^{2} b c\right )} x^{3} + 2 \, {\left (a^{2} b^{2} + 20 \, a^{3} c\right )} x^{2}\right )} \sqrt {c x^{4} + b x^{3} + a x^{2}}}{256 \, a^{3} x^{5}}, \frac {3 \, {\left (b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}\right )} \sqrt {-a} x^{5} \arctan \left (\frac {\sqrt {c x^{4} + b x^{3} + a x^{2}} {\left (b x + 2 \, a\right )} \sqrt {-a}}{2 \, {\left (a c x^{3} + a b x^{2} + a^{2} x\right )}}\right ) - 2 \, {\left (24 \, a^{3} b x + 16 \, a^{4} - {\left (3 \, a b^{3} - 20 \, a^{2} b c\right )} x^{3} + 2 \, {\left (a^{2} b^{2} + 20 \, a^{3} c\right )} x^{2}\right )} \sqrt {c x^{4} + b x^{3} + a x^{2}}}{128 \, a^{3} x^{5}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 0.01, size = 501, normalized size = 2.54 \begin {gather*} -\frac {\left (c \,x^{4}+b \,x^{3}+a \,x^{2}\right )^{\frac {3}{2}} \left (48 a^{\frac {7}{2}} c^{2} x^{4} \ln \left (\frac {b x +2 a +2 \sqrt {c \,x^{2}+b x +a}\, \sqrt {a}}{x}\right )-24 a^{\frac {5}{2}} b^{2} c \,x^{4} \ln \left (\frac {b x +2 a +2 \sqrt {c \,x^{2}+b x +a}\, \sqrt {a}}{x}\right )+3 a^{\frac {3}{2}} b^{4} x^{4} \ln \left (\frac {b x +2 a +2 \sqrt {c \,x^{2}+b x +a}\, \sqrt {a}}{x}\right )+24 \sqrt {c \,x^{2}+b x +a}\, a^{2} b \,c^{2} x^{5}-6 \sqrt {c \,x^{2}+b x +a}\, a \,b^{3} c \,x^{5}-48 \sqrt {c \,x^{2}+b x +a}\, a^{3} c^{2} x^{4}+36 \sqrt {c \,x^{2}+b x +a}\, a^{2} b^{2} c \,x^{4}-6 \sqrt {c \,x^{2}+b x +a}\, a \,b^{4} x^{4}+24 \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} a b \,c^{2} x^{5}-2 \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} b^{3} c \,x^{5}-16 \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} a^{2} c^{2} x^{4}+20 \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} a \,b^{2} c \,x^{4}-2 \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} b^{4} x^{4}-24 \left (c \,x^{2}+b x +a \right )^{\frac {5}{2}} a b c \,x^{3}+2 \left (c \,x^{2}+b x +a \right )^{\frac {5}{2}} b^{3} x^{3}+16 \left (c \,x^{2}+b x +a \right )^{\frac {5}{2}} a^{2} c \,x^{2}+4 \left (c \,x^{2}+b x +a \right )^{\frac {5}{2}} a \,b^{2} x^{2}-16 \left (c \,x^{2}+b x +a \right )^{\frac {5}{2}} a^{2} b x +32 \left (c \,x^{2}+b x +a \right )^{\frac {5}{2}} a^{3}\right )}{128 \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} a^{4} x^{7}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (c x^{4} + b x^{3} + a x^{2}\right )}^{\frac {3}{2}}}{x^{8}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (c\,x^4+b\,x^3+a\,x^2\right )}^{3/2}}{x^8} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (x^{2} \left (a + b x + c x^{2}\right )\right )^{\frac {3}{2}}}{x^{8}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________