Optimal. Leaf size=165 \[ \frac {3 x \left (b^2-4 a c\right )^2 \sqrt {a+b x+c x^2} \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{128 c^{5/2} \sqrt {a x^2+b x^3+c x^4}}-\frac {3 \left (b^2-4 a c\right ) (b+2 c x) \sqrt {a x^2+b x^3+c x^4}}{64 c^2 x}+\frac {(b+2 c x) \left (a x^2+b x^3+c x^4\right )^{3/2}}{8 c x^3} \]
________________________________________________________________________________________
Rubi [A] time = 0.13, antiderivative size = 165, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {1918, 1914, 621, 206} \begin {gather*} -\frac {3 \left (b^2-4 a c\right ) (b+2 c x) \sqrt {a x^2+b x^3+c x^4}}{64 c^2 x}+\frac {3 x \left (b^2-4 a c\right )^2 \sqrt {a+b x+c x^2} \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{128 c^{5/2} \sqrt {a x^2+b x^3+c x^4}}+\frac {(b+2 c x) \left (a x^2+b x^3+c x^4\right )^{3/2}}{8 c x^3} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 206
Rule 621
Rule 1914
Rule 1918
Rubi steps
\begin {align*} \int \frac {\left (a x^2+b x^3+c x^4\right )^{3/2}}{x^3} \, dx &=\frac {(b+2 c x) \left (a x^2+b x^3+c x^4\right )^{3/2}}{8 c x^3}-\frac {\left (3 \left (b^2-4 a c\right )\right ) \int \frac {\sqrt {a x^2+b x^3+c x^4}}{x} \, dx}{16 c}\\ &=-\frac {3 \left (b^2-4 a c\right ) (b+2 c x) \sqrt {a x^2+b x^3+c x^4}}{64 c^2 x}+\frac {(b+2 c x) \left (a x^2+b x^3+c x^4\right )^{3/2}}{8 c x^3}+\frac {\left (3 \left (b^2-4 a c\right )^2\right ) \int \frac {x}{\sqrt {a x^2+b x^3+c x^4}} \, dx}{128 c^2}\\ &=-\frac {3 \left (b^2-4 a c\right ) (b+2 c x) \sqrt {a x^2+b x^3+c x^4}}{64 c^2 x}+\frac {(b+2 c x) \left (a x^2+b x^3+c x^4\right )^{3/2}}{8 c x^3}+\frac {\left (3 \left (b^2-4 a c\right )^2 x \sqrt {a+b x+c x^2}\right ) \int \frac {1}{\sqrt {a+b x+c x^2}} \, dx}{128 c^2 \sqrt {a x^2+b x^3+c x^4}}\\ &=-\frac {3 \left (b^2-4 a c\right ) (b+2 c x) \sqrt {a x^2+b x^3+c x^4}}{64 c^2 x}+\frac {(b+2 c x) \left (a x^2+b x^3+c x^4\right )^{3/2}}{8 c x^3}+\frac {\left (3 \left (b^2-4 a c\right )^2 x \sqrt {a+b x+c x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{4 c-x^2} \, dx,x,\frac {b+2 c x}{\sqrt {a+b x+c x^2}}\right )}{64 c^2 \sqrt {a x^2+b x^3+c x^4}}\\ &=-\frac {3 \left (b^2-4 a c\right ) (b+2 c x) \sqrt {a x^2+b x^3+c x^4}}{64 c^2 x}+\frac {(b+2 c x) \left (a x^2+b x^3+c x^4\right )^{3/2}}{8 c x^3}+\frac {3 \left (b^2-4 a c\right )^2 x \sqrt {a+b x+c x^2} \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{128 c^{5/2} \sqrt {a x^2+b x^3+c x^4}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.06, size = 132, normalized size = 0.80 \begin {gather*} \frac {x \sqrt {a+x (b+c x)} \left (2 \sqrt {c} (b+2 c x) \sqrt {a+x (b+c x)} \left (4 c \left (5 a+2 c x^2\right )-3 b^2+8 b c x\right )+3 \left (b^2-4 a c\right )^2 \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+x (b+c x)}}\right )\right )}{128 c^{5/2} \sqrt {x^2 (a+x (b+c x))}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [A] time = 1.58, size = 143, normalized size = 0.87 \begin {gather*} \frac {\left (20 a b c+40 a c^2 x-3 b^3+2 b^2 c x+24 b c^2 x^2+16 c^3 x^3\right ) \sqrt {a x^2+b x^3+c x^4}}{64 c^2 x}-\frac {3 \left (16 a^2 c^2-8 a b^2 c+b^4\right ) \tanh ^{-1}\left (\frac {\sqrt {c} x^2}{\sqrt {a} x-\sqrt {a x^2+b x^3+c x^4}}\right )}{64 c^{5/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 1.31, size = 320, normalized size = 1.94 \begin {gather*} \left [\frac {3 \, {\left (b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}\right )} \sqrt {c} x \log \left (-\frac {8 \, c^{2} x^{3} + 8 \, b c x^{2} + 4 \, \sqrt {c x^{4} + b x^{3} + a x^{2}} {\left (2 \, c x + b\right )} \sqrt {c} + {\left (b^{2} + 4 \, a c\right )} x}{x}\right ) + 4 \, {\left (16 \, c^{4} x^{3} + 24 \, b c^{3} x^{2} - 3 \, b^{3} c + 20 \, a b c^{2} + 2 \, {\left (b^{2} c^{2} + 20 \, a c^{3}\right )} x\right )} \sqrt {c x^{4} + b x^{3} + a x^{2}}}{256 \, c^{3} x}, -\frac {3 \, {\left (b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}\right )} \sqrt {-c} x \arctan \left (\frac {\sqrt {c x^{4} + b x^{3} + a x^{2}} {\left (2 \, c x + b\right )} \sqrt {-c}}{2 \, {\left (c^{2} x^{3} + b c x^{2} + a c x\right )}}\right ) - 2 \, {\left (16 \, c^{4} x^{3} + 24 \, b c^{3} x^{2} - 3 \, b^{3} c + 20 \, a b c^{2} + 2 \, {\left (b^{2} c^{2} + 20 \, a c^{3}\right )} x\right )} \sqrt {c x^{4} + b x^{3} + a x^{2}}}{128 \, c^{3} x}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 1.01, size = 232, normalized size = 1.41 \begin {gather*} \frac {1}{64} \, \sqrt {c x^{2} + b x + a} {\left (2 \, {\left (4 \, {\left (2 \, c x \mathrm {sgn}\relax (x) + 3 \, b \mathrm {sgn}\relax (x)\right )} x + \frac {b^{2} c^{2} \mathrm {sgn}\relax (x) + 20 \, a c^{3} \mathrm {sgn}\relax (x)}{c^{3}}\right )} x - \frac {3 \, b^{3} c \mathrm {sgn}\relax (x) - 20 \, a b c^{2} \mathrm {sgn}\relax (x)}{c^{3}}\right )} - \frac {3 \, {\left (b^{4} \mathrm {sgn}\relax (x) - 8 \, a b^{2} c \mathrm {sgn}\relax (x) + 16 \, a^{2} c^{2} \mathrm {sgn}\relax (x)\right )} \log \left ({\left | -2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} \sqrt {c} - b \right |}\right )}{128 \, c^{\frac {5}{2}}} + \frac {{\left (3 \, b^{4} \log \left ({\left | -b + 2 \, \sqrt {a} \sqrt {c} \right |}\right ) - 24 \, a b^{2} c \log \left ({\left | -b + 2 \, \sqrt {a} \sqrt {c} \right |}\right ) + 48 \, a^{2} c^{2} \log \left ({\left | -b + 2 \, \sqrt {a} \sqrt {c} \right |}\right ) + 6 \, \sqrt {a} b^{3} \sqrt {c} - 40 \, a^{\frac {3}{2}} b c^{\frac {3}{2}}\right )} \mathrm {sgn}\relax (x)}{128 \, c^{\frac {5}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.00, size = 265, normalized size = 1.61 \begin {gather*} \frac {\left (c \,x^{4}+b \,x^{3}+a \,x^{2}\right )^{\frac {3}{2}} \left (48 a^{2} c^{3} \ln \left (\frac {2 c x +b +2 \sqrt {c \,x^{2}+b x +a}\, \sqrt {c}}{2 \sqrt {c}}\right )-24 a \,b^{2} c^{2} \ln \left (\frac {2 c x +b +2 \sqrt {c \,x^{2}+b x +a}\, \sqrt {c}}{2 \sqrt {c}}\right )+3 b^{4} c \ln \left (\frac {2 c x +b +2 \sqrt {c \,x^{2}+b x +a}\, \sqrt {c}}{2 \sqrt {c}}\right )+48 \sqrt {c \,x^{2}+b x +a}\, a \,c^{\frac {7}{2}} x -12 \sqrt {c \,x^{2}+b x +a}\, b^{2} c^{\frac {5}{2}} x +24 \sqrt {c \,x^{2}+b x +a}\, a b \,c^{\frac {5}{2}}-6 \sqrt {c \,x^{2}+b x +a}\, b^{3} c^{\frac {3}{2}}+32 \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} c^{\frac {7}{2}} x +16 \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} b \,c^{\frac {5}{2}}\right )}{128 \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} c^{\frac {7}{2}} x^{3}} \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^{3}}\,{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^3} \,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^{3}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________