Optimal. Leaf size=104 \[ -\frac {c}{5 a x^5}+\frac {b c-a d}{3 a^2 x^3}-\frac {b^2 c-a b d+a^2 e}{a^3 x}-\frac {\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{a^{7/2} \sqrt {b}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.07, antiderivative size = 104, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {1816, 211}
\begin {gather*} \frac {b c-a d}{3 a^2 x^3}-\frac {a^2 e-a b d+b^2 c}{a^3 x}-\frac {\text {ArcTan}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{a^{7/2} \sqrt {b}}-\frac {c}{5 a x^5} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 211
Rule 1816
Rubi steps
\begin {align*} \int \frac {c+d x^2+e x^4+f x^6}{x^6 \left (a+b x^2\right )} \, dx &=\int \left (\frac {c}{a x^6}+\frac {-b c+a d}{a^2 x^4}+\frac {b^2 c-a b d+a^2 e}{a^3 x^2}+\frac {-b^3 c+a b^2 d-a^2 b e+a^3 f}{a^3 \left (a+b x^2\right )}\right ) \, dx\\ &=-\frac {c}{5 a x^5}+\frac {b c-a d}{3 a^2 x^3}-\frac {b^2 c-a b d+a^2 e}{a^3 x}+\frac {\left (-b^3 c+a b^2 d-a^2 b e+a^3 f\right ) \int \frac {1}{a+b x^2} \, dx}{a^3}\\ &=-\frac {c}{5 a x^5}+\frac {b c-a d}{3 a^2 x^3}-\frac {b^2 c-a b d+a^2 e}{a^3 x}-\frac {\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{a^{7/2} \sqrt {b}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.06, size = 103, normalized size = 0.99 \begin {gather*} -\frac {c}{5 a x^5}+\frac {b c-a d}{3 a^2 x^3}+\frac {-b^2 c+a b d-a^2 e}{a^3 x}+\frac {\left (-b^3 c+a b^2 d-a^2 b e+a^3 f\right ) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{a^{7/2} \sqrt {b}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.16, size = 94, normalized size = 0.90
method | result | size |
default | \(\frac {\left (a^{3} f -a^{2} b e +a \,b^{2} d -b^{3} c \right ) \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{a^{3} \sqrt {a b}}-\frac {c}{5 a \,x^{5}}-\frac {a d -b c}{3 a^{2} x^{3}}-\frac {a^{2} e -a b d +b^{2} c}{a^{3} x}\) | \(94\) |
risch | \(\frac {-\frac {\left (a^{2} e -a b d +b^{2} c \right ) x^{4}}{a^{3}}-\frac {\left (a d -b c \right ) x^{2}}{3 a^{2}}-\frac {c}{5 a}}{x^{5}}-\frac {\ln \left (-\sqrt {-a b}\, x +a \right ) f}{2 \sqrt {-a b}}+\frac {\ln \left (-\sqrt {-a b}\, x +a \right ) b e}{2 \sqrt {-a b}\, a}-\frac {\ln \left (-\sqrt {-a b}\, x +a \right ) b^{2} d}{2 \sqrt {-a b}\, a^{2}}+\frac {\ln \left (-\sqrt {-a b}\, x +a \right ) b^{3} c}{2 \sqrt {-a b}\, a^{3}}+\frac {\ln \left (-\sqrt {-a b}\, x -a \right ) f}{2 \sqrt {-a b}}-\frac {\ln \left (-\sqrt {-a b}\, x -a \right ) b e}{2 \sqrt {-a b}\, a}+\frac {\ln \left (-\sqrt {-a b}\, x -a \right ) b^{2} d}{2 \sqrt {-a b}\, a^{2}}-\frac {\ln \left (-\sqrt {-a b}\, x -a \right ) b^{3} c}{2 \sqrt {-a b}\, a^{3}}\) | \(261\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.51, size = 99, normalized size = 0.95 \begin {gather*} -\frac {{\left (b^{3} c - a b^{2} d - a^{3} f + a^{2} b e\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b} a^{3}} - \frac {15 \, {\left (b^{2} c - a b d + a^{2} e\right )} x^{4} + 3 \, a^{2} c - 5 \, {\left (a b c - a^{2} d\right )} x^{2}}{15 \, a^{3} x^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 9.33, size = 268, normalized size = 2.58 \begin {gather*} \left [-\frac {30 \, a^{3} b x^{4} e + 6 \, a^{3} b c + 30 \, {\left (a b^{3} c - a^{2} b^{2} d\right )} x^{4} - 10 \, {\left (a^{2} b^{2} c - a^{3} b d\right )} x^{2} + 15 \, {\left (a^{2} b x^{5} e + {\left (b^{3} c - a b^{2} d - a^{3} f\right )} x^{5}\right )} \sqrt {-a b} \log \left (\frac {b x^{2} + 2 \, \sqrt {-a b} x - a}{b x^{2} + a}\right )}{30 \, a^{4} b x^{5}}, -\frac {15 \, a^{3} b x^{4} e + 3 \, a^{3} b c + 15 \, {\left (a b^{3} c - a^{2} b^{2} d\right )} x^{4} - 5 \, {\left (a^{2} b^{2} c - a^{3} b d\right )} x^{2} + 15 \, {\left (a^{2} b x^{5} e + {\left (b^{3} c - a b^{2} d - a^{3} f\right )} x^{5}\right )} \sqrt {a b} \arctan \left (\frac {\sqrt {a b} x}{a}\right )}{15 \, a^{4} b x^{5}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A]
time = 2.53, size = 167, normalized size = 1.61 \begin {gather*} - \frac {\sqrt {- \frac {1}{a^{7} b}} \left (a^{3} f - a^{2} b e + a b^{2} d - b^{3} c\right ) \log {\left (- a^{4} \sqrt {- \frac {1}{a^{7} b}} + x \right )}}{2} + \frac {\sqrt {- \frac {1}{a^{7} b}} \left (a^{3} f - a^{2} b e + a b^{2} d - b^{3} c\right ) \log {\left (a^{4} \sqrt {- \frac {1}{a^{7} b}} + x \right )}}{2} + \frac {- 3 a^{2} c + x^{4} \left (- 15 a^{2} e + 15 a b d - 15 b^{2} c\right ) + x^{2} \left (- 5 a^{2} d + 5 a b c\right )}{15 a^{3} x^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 1.31, size = 105, normalized size = 1.01 \begin {gather*} -\frac {{\left (b^{3} c - a b^{2} d - a^{3} f + a^{2} b e\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b} a^{3}} - \frac {15 \, b^{2} c x^{4} - 15 \, a b d x^{4} + 15 \, a^{2} x^{4} e - 5 \, a b c x^{2} + 5 \, a^{2} d x^{2} + 3 \, a^{2} c}{15 \, a^{3} x^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 1.20, size = 94, normalized size = 0.90 \begin {gather*} -\frac {\frac {c}{5\,a}+\frac {x^2\,\left (a\,d-b\,c\right )}{3\,a^2}+\frac {x^4\,\left (e\,a^2-d\,a\,b+c\,b^2\right )}{a^3}}{x^5}-\frac {\mathrm {atan}\left (\frac {\sqrt {b}\,x}{\sqrt {a}}\right )\,\left (-f\,a^3+e\,a^2\,b-d\,a\,b^2+c\,b^3\right )}{a^{7/2}\,\sqrt {b}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________