Optimal. Leaf size=147 \[ \frac {x \left (c-\frac {a \left (a^2 f-a b e+b^2 d\right )}{b^3}\right )}{4 a \left (a+b x^2\right )^2}+\frac {x \left (9 a^3 f-5 a^2 b e+a b^2 d+3 b^3 c\right )}{8 a^2 b^3 \left (a+b x^2\right )}+\frac {\tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) \left (-15 a^3 f+3 a^2 b e+a b^2 d+3 b^3 c\right )}{8 a^{5/2} b^{7/2}}+\frac {f x}{b^3} \]
________________________________________________________________________________________
Rubi [A] time = 0.15, antiderivative size = 147, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {1814, 1157, 388, 205} \begin {gather*} \frac {x \left (-5 a^2 b e+9 a^3 f+a b^2 d+3 b^3 c\right )}{8 a^2 b^3 \left (a+b x^2\right )}+\frac {x \left (c-\frac {a \left (a^2 f-a b e+b^2 d\right )}{b^3}\right )}{4 a \left (a+b x^2\right )^2}+\frac {\tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) \left (3 a^2 b e-15 a^3 f+a b^2 d+3 b^3 c\right )}{8 a^{5/2} b^{7/2}}+\frac {f x}{b^3} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 205
Rule 388
Rule 1157
Rule 1814
Rubi steps
\begin {align*} \int \frac {c+d x^2+e x^4+f x^6}{\left (a+b x^2\right )^3} \, dx &=\frac {\left (c-\frac {a \left (b^2 d-a b e+a^2 f\right )}{b^3}\right ) x}{4 a \left (a+b x^2\right )^2}-\frac {\int \frac {-\frac {3 b^3 c+a b^2 d-a^2 b e+a^3 f}{b^3}-\frac {4 a (b e-a f) x^2}{b^2}-\frac {4 a f x^4}{b}}{\left (a+b x^2\right )^2} \, dx}{4 a}\\ &=\frac {\left (c-\frac {a \left (b^2 d-a b e+a^2 f\right )}{b^3}\right ) x}{4 a \left (a+b x^2\right )^2}+\frac {\left (3 b^3 c+a b^2 d-5 a^2 b e+9 a^3 f\right ) x}{8 a^2 b^3 \left (a+b x^2\right )}+\frac {\int \frac {\frac {3 b^3 c+a b^2 d+3 a^2 b e-7 a^3 f}{b^3}+\frac {8 a^2 f x^2}{b^2}}{a+b x^2} \, dx}{8 a^2}\\ &=\frac {f x}{b^3}+\frac {\left (c-\frac {a \left (b^2 d-a b e+a^2 f\right )}{b^3}\right ) x}{4 a \left (a+b x^2\right )^2}+\frac {\left (3 b^3 c+a b^2 d-5 a^2 b e+9 a^3 f\right ) x}{8 a^2 b^3 \left (a+b x^2\right )}+\frac {\left (3 b^3 c+a b^2 d+3 a^2 b e-15 a^3 f\right ) \int \frac {1}{a+b x^2} \, dx}{8 a^2 b^3}\\ &=\frac {f x}{b^3}+\frac {\left (c-\frac {a \left (b^2 d-a b e+a^2 f\right )}{b^3}\right ) x}{4 a \left (a+b x^2\right )^2}+\frac {\left (3 b^3 c+a b^2 d-5 a^2 b e+9 a^3 f\right ) x}{8 a^2 b^3 \left (a+b x^2\right )}+\frac {\left (3 b^3 c+a b^2 d+3 a^2 b e-15 a^3 f\right ) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{8 a^{5/2} b^{7/2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.12, size = 141, normalized size = 0.96 \begin {gather*} \frac {\tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) \left (-15 a^3 f+3 a^2 b e+a b^2 d+3 b^3 c\right )}{8 a^{5/2} b^{7/2}}+\frac {x \left (15 a^4 f+a^3 b \left (25 f x^2-3 e\right )-a^2 b^2 \left (d+5 e x^2-8 f x^4\right )+a b^3 \left (5 c+d x^2\right )+3 b^4 c x^2\right )}{8 a^2 b^3 \left (a+b x^2\right )^2} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {c+d x^2+e x^4+f x^6}{\left (a+b x^2\right )^3} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 1.17, size = 504, normalized size = 3.43 \begin {gather*} \left [\frac {16 \, a^{3} b^{3} f x^{5} + 2 \, {\left (3 \, a b^{5} c + a^{2} b^{4} d - 5 \, a^{3} b^{3} e + 25 \, a^{4} b^{2} f\right )} x^{3} + {\left (3 \, a^{2} b^{3} c + a^{3} b^{2} d + 3 \, a^{4} b e - 15 \, a^{5} f + {\left (3 \, b^{5} c + a b^{4} d + 3 \, a^{2} b^{3} e - 15 \, a^{3} b^{2} f\right )} x^{4} + 2 \, {\left (3 \, a b^{4} c + a^{2} b^{3} d + 3 \, a^{3} b^{2} e - 15 \, a^{4} b f\right )} x^{2}\right )} \sqrt {-a b} \log \left (\frac {b x^{2} + 2 \, \sqrt {-a b} x - a}{b x^{2} + a}\right ) + 2 \, {\left (5 \, a^{2} b^{4} c - a^{3} b^{3} d - 3 \, a^{4} b^{2} e + 15 \, a^{5} b f\right )} x}{16 \, {\left (a^{3} b^{6} x^{4} + 2 \, a^{4} b^{5} x^{2} + a^{5} b^{4}\right )}}, \frac {8 \, a^{3} b^{3} f x^{5} + {\left (3 \, a b^{5} c + a^{2} b^{4} d - 5 \, a^{3} b^{3} e + 25 \, a^{4} b^{2} f\right )} x^{3} + {\left (3 \, a^{2} b^{3} c + a^{3} b^{2} d + 3 \, a^{4} b e - 15 \, a^{5} f + {\left (3 \, b^{5} c + a b^{4} d + 3 \, a^{2} b^{3} e - 15 \, a^{3} b^{2} f\right )} x^{4} + 2 \, {\left (3 \, a b^{4} c + a^{2} b^{3} d + 3 \, a^{3} b^{2} e - 15 \, a^{4} b f\right )} x^{2}\right )} \sqrt {a b} \arctan \left (\frac {\sqrt {a b} x}{a}\right ) + {\left (5 \, a^{2} b^{4} c - a^{3} b^{3} d - 3 \, a^{4} b^{2} e + 15 \, a^{5} b f\right )} x}{8 \, {\left (a^{3} b^{6} x^{4} + 2 \, a^{4} b^{5} x^{2} + a^{5} b^{4}\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.46, size = 149, normalized size = 1.01 \begin {gather*} \frac {f x}{b^{3}} + \frac {{\left (3 \, b^{3} c + a b^{2} d - 15 \, a^{3} f + 3 \, a^{2} b e\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{8 \, \sqrt {a b} a^{2} b^{3}} + \frac {3 \, b^{4} c x^{3} + a b^{3} d x^{3} + 9 \, a^{3} b f x^{3} - 5 \, a^{2} b^{2} x^{3} e + 5 \, a b^{3} c x - a^{2} b^{2} d x + 7 \, a^{4} f x - 3 \, a^{3} b x e}{8 \, {\left (b x^{2} + a\right )}^{2} a^{2} b^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.01, size = 234, normalized size = 1.59 \begin {gather*} \frac {9 a f \,x^{3}}{8 \left (b \,x^{2}+a \right )^{2} b^{2}}+\frac {d \,x^{3}}{8 \left (b \,x^{2}+a \right )^{2} a}+\frac {3 b c \,x^{3}}{8 \left (b \,x^{2}+a \right )^{2} a^{2}}-\frac {5 e \,x^{3}}{8 \left (b \,x^{2}+a \right )^{2} b}+\frac {7 a^{2} f x}{8 \left (b \,x^{2}+a \right )^{2} b^{3}}-\frac {3 a e x}{8 \left (b \,x^{2}+a \right )^{2} b^{2}}+\frac {5 c x}{8 \left (b \,x^{2}+a \right )^{2} a}-\frac {d x}{8 \left (b \,x^{2}+a \right )^{2} b}-\frac {15 a f \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{8 \sqrt {a b}\, b^{3}}+\frac {d \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{8 \sqrt {a b}\, a b}+\frac {3 c \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{8 \sqrt {a b}\, a^{2}}+\frac {3 e \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{8 \sqrt {a b}\, b^{2}}+\frac {f x}{b^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 3.00, size = 154, normalized size = 1.05 \begin {gather*} \frac {{\left (3 \, b^{4} c + a b^{3} d - 5 \, a^{2} b^{2} e + 9 \, a^{3} b f\right )} x^{3} + {\left (5 \, a b^{3} c - a^{2} b^{2} d - 3 \, a^{3} b e + 7 \, a^{4} f\right )} x}{8 \, {\left (a^{2} b^{5} x^{4} + 2 \, a^{3} b^{4} x^{2} + a^{4} b^{3}\right )}} + \frac {f x}{b^{3}} + \frac {{\left (3 \, b^{3} c + a b^{2} d + 3 \, a^{2} b e - 15 \, a^{3} f\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{8 \, \sqrt {a b} a^{2} b^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 1.05, size = 148, normalized size = 1.01 \begin {gather*} \frac {\frac {x\,\left (7\,f\,a^3-3\,e\,a^2\,b-d\,a\,b^2+5\,c\,b^3\right )}{8\,a}+\frac {x^3\,\left (9\,f\,a^3\,b-5\,e\,a^2\,b^2+d\,a\,b^3+3\,c\,b^4\right )}{8\,a^2}}{a^2\,b^3+2\,a\,b^4\,x^2+b^5\,x^4}+\frac {f\,x}{b^3}+\frac {\mathrm {atan}\left (\frac {\sqrt {b}\,x}{\sqrt {a}}\right )\,\left (-15\,f\,a^3+3\,e\,a^2\,b+d\,a\,b^2+3\,c\,b^3\right )}{8\,a^{5/2}\,b^{7/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 10.09, size = 243, normalized size = 1.65 \begin {gather*} \frac {\sqrt {- \frac {1}{a^{5} b^{7}}} \left (15 a^{3} f - 3 a^{2} b e - a b^{2} d - 3 b^{3} c\right ) \log {\left (- a^{3} b^{3} \sqrt {- \frac {1}{a^{5} b^{7}}} + x \right )}}{16} - \frac {\sqrt {- \frac {1}{a^{5} b^{7}}} \left (15 a^{3} f - 3 a^{2} b e - a b^{2} d - 3 b^{3} c\right ) \log {\left (a^{3} b^{3} \sqrt {- \frac {1}{a^{5} b^{7}}} + x \right )}}{16} + \frac {x^{3} \left (9 a^{3} b f - 5 a^{2} b^{2} e + a b^{3} d + 3 b^{4} c\right ) + x \left (7 a^{4} f - 3 a^{3} b e - a^{2} b^{2} d + 5 a b^{3} c\right )}{8 a^{4} b^{3} + 16 a^{3} b^{4} x^{2} + 8 a^{2} b^{5} x^{4}} + \frac {f x}{b^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________