Optimal. Leaf size=73 \[ \frac {(A b-a C) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {a} b^{3/2}}+\frac {(b B-a D) \log \left (a+b x^2\right )}{2 b^2}+\frac {C x}{b}+\frac {D x^2}{2 b} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.07, antiderivative size = 73, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {1810, 635, 205, 260} \[ \frac {(A b-a C) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {a} b^{3/2}}+\frac {(b B-a D) \log \left (a+b x^2\right )}{2 b^2}+\frac {C x}{b}+\frac {D x^2}{2 b} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 205
Rule 260
Rule 635
Rule 1810
Rubi steps
\begin {align*} \int \frac {A+B x+C x^2+D x^3}{a+b x^2} \, dx &=\int \left (\frac {C}{b}+\frac {D x}{b}+\frac {A b-a C+(b B-a D) x}{b \left (a+b x^2\right )}\right ) \, dx\\ &=\frac {C x}{b}+\frac {D x^2}{2 b}+\frac {\int \frac {A b-a C+(b B-a D) x}{a+b x^2} \, dx}{b}\\ &=\frac {C x}{b}+\frac {D x^2}{2 b}+\frac {(A b-a C) \int \frac {1}{a+b x^2} \, dx}{b}+\frac {(b B-a D) \int \frac {x}{a+b x^2} \, dx}{b}\\ &=\frac {C x}{b}+\frac {D x^2}{2 b}+\frac {(A b-a C) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {a} b^{3/2}}+\frac {(b B-a D) \log \left (a+b x^2\right )}{2 b^2}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.04, size = 68, normalized size = 0.93 \[ \frac {\frac {2 \sqrt {b} (A b-a C) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {a}}+(b B-a D) \log \left (a+b x^2\right )+b x (2 C+D x)}{2 b^2} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.68, size = 157, normalized size = 2.15 \[ \left [\frac {D a b x^{2} + 2 \, C a b x + {\left (C a - A b\right )} \sqrt {-a b} \log \left (\frac {b x^{2} - 2 \, \sqrt {-a b} x - a}{b x^{2} + a}\right ) - {\left (D a^{2} - B a b\right )} \log \left (b x^{2} + a\right )}{2 \, a b^{2}}, \frac {D a b x^{2} + 2 \, C a b x - 2 \, {\left (C a - A b\right )} \sqrt {a b} \arctan \left (\frac {\sqrt {a b} x}{a}\right ) - {\left (D a^{2} - B a b\right )} \log \left (b x^{2} + a\right )}{2 \, a b^{2}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.46, size = 66, normalized size = 0.90 \[ -\frac {{\left (C a - A b\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b} b} - \frac {{\left (D a - B b\right )} \log \left (b x^{2} + a\right )}{2 \, b^{2}} + \frac {D b x^{2} + 2 \, C b x}{2 \, b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.00, size = 83, normalized size = 1.14 \[ \frac {A \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b}}-\frac {C a \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b}\, b}+\frac {D x^{2}}{2 b}+\frac {B \ln \left (b \,x^{2}+a \right )}{2 b}+\frac {C x}{b}-\frac {D a \ln \left (b \,x^{2}+a \right )}{2 b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 2.93, size = 64, normalized size = 0.88 \[ -\frac {{\left (C a - A b\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b} b} + \frac {D x^{2} + 2 \, C x}{2 \, b} - \frac {{\left (D a - B b\right )} \log \left (b x^{2} + a\right )}{2 \, b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 1.42, size = 79, normalized size = 1.08 \[ \frac {B\,\ln \left (b\,x^2+a\right )}{2\,b}-\frac {\left (a\,\ln \left (b\,x^2+a\right )-b\,x^2\right )\,D}{2\,b^2}+\frac {C\,x}{b}+\frac {A\,\mathrm {atan}\left (\frac {\sqrt {b}\,x}{\sqrt {a}}\right )}{\sqrt {a}\,\sqrt {b}}-\frac {C\,\sqrt {a}\,\mathrm {atan}\left (\frac {\sqrt {b}\,x}{\sqrt {a}}\right )}{b^{3/2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [B] time = 0.88, size = 219, normalized size = 3.00 \[ \frac {C x}{b} + \frac {D x^{2}}{2 b} + \left (- \frac {- B b + D a}{2 b^{2}} - \frac {\sqrt {- a b^{5}} \left (- A b + C a\right )}{2 a b^{4}}\right ) \log {\left (x + \frac {B a b - D a^{2} - 2 a b^{2} \left (- \frac {- B b + D a}{2 b^{2}} - \frac {\sqrt {- a b^{5}} \left (- A b + C a\right )}{2 a b^{4}}\right )}{- A b^{2} + C a b} \right )} + \left (- \frac {- B b + D a}{2 b^{2}} + \frac {\sqrt {- a b^{5}} \left (- A b + C a\right )}{2 a b^{4}}\right ) \log {\left (x + \frac {B a b - D a^{2} - 2 a b^{2} \left (- \frac {- B b + D a}{2 b^{2}} + \frac {\sqrt {- a b^{5}} \left (- A b + C a\right )}{2 a b^{4}}\right )}{- A b^{2} + C a b} \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________