Optimal. Leaf size=176 \[ -\frac{x^3 (3 A b-5 a C)}{6 a b^2}+\frac{x (3 A b-5 a C)}{2 b^3}-\frac{\sqrt{a} (3 A b-5 a C) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 b^{7/2}}-\frac{x^4 \left (a \left (B-\frac{a D}{b}\right )-x (A b-a C)\right )}{2 a b \left (a+b x^2\right )}+\frac{x^2 (2 b B-3 a D)}{2 b^3}-\frac{a (2 b B-3 a D) \log \left (a+b x^2\right )}{2 b^4}+\frac{D x^4}{4 b^2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.267927, antiderivative size = 176, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.179, Rules used = {1804, 1802, 635, 205, 260} \[ -\frac{x^3 (3 A b-5 a C)}{6 a b^2}+\frac{x (3 A b-5 a C)}{2 b^3}-\frac{\sqrt{a} (3 A b-5 a C) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 b^{7/2}}-\frac{x^4 \left (a \left (B-\frac{a D}{b}\right )-x (A b-a C)\right )}{2 a b \left (a+b x^2\right )}+\frac{x^2 (2 b B-3 a D)}{2 b^3}-\frac{a (2 b B-3 a D) \log \left (a+b x^2\right )}{2 b^4}+\frac{D x^4}{4 b^2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 1804
Rule 1802
Rule 635
Rule 205
Rule 260
Rubi steps
\begin{align*} \int \frac{x^4 \left (A+B x+C x^2+D x^3\right )}{\left (a+b x^2\right )^2} \, dx &=-\frac{x^4 \left (a \left (B-\frac{a D}{b}\right )-(A b-a C) x\right )}{2 a b \left (a+b x^2\right )}-\frac{\int \frac{x^3 \left (-4 a \left (B-\frac{a D}{b}\right )+(3 A b-5 a C) x-2 a D x^2\right )}{a+b x^2} \, dx}{2 a b}\\ &=-\frac{x^4 \left (a \left (B-\frac{a D}{b}\right )-(A b-a C) x\right )}{2 a b \left (a+b x^2\right )}-\frac{\int \left (-\frac{a (3 A b-5 a C)}{b^2}-\frac{2 a (2 b B-3 a D) x}{b^2}+\frac{(3 A b-5 a C) x^2}{b}-\frac{2 a D x^3}{b}+\frac{a^2 (3 A b-5 a C)+2 a^2 (2 b B-3 a D) x}{b^2 \left (a+b x^2\right )}\right ) \, dx}{2 a b}\\ &=\frac{(3 A b-5 a C) x}{2 b^3}+\frac{(2 b B-3 a D) x^2}{2 b^3}-\frac{(3 A b-5 a C) x^3}{6 a b^2}+\frac{D x^4}{4 b^2}-\frac{x^4 \left (a \left (B-\frac{a D}{b}\right )-(A b-a C) x\right )}{2 a b \left (a+b x^2\right )}-\frac{\int \frac{a^2 (3 A b-5 a C)+2 a^2 (2 b B-3 a D) x}{a+b x^2} \, dx}{2 a b^3}\\ &=\frac{(3 A b-5 a C) x}{2 b^3}+\frac{(2 b B-3 a D) x^2}{2 b^3}-\frac{(3 A b-5 a C) x^3}{6 a b^2}+\frac{D x^4}{4 b^2}-\frac{x^4 \left (a \left (B-\frac{a D}{b}\right )-(A b-a C) x\right )}{2 a b \left (a+b x^2\right )}-\frac{(a (3 A b-5 a C)) \int \frac{1}{a+b x^2} \, dx}{2 b^3}-\frac{(a (2 b B-3 a D)) \int \frac{x}{a+b x^2} \, dx}{b^3}\\ &=\frac{(3 A b-5 a C) x}{2 b^3}+\frac{(2 b B-3 a D) x^2}{2 b^3}-\frac{(3 A b-5 a C) x^3}{6 a b^2}+\frac{D x^4}{4 b^2}-\frac{x^4 \left (a \left (B-\frac{a D}{b}\right )-(A b-a C) x\right )}{2 a b \left (a+b x^2\right )}-\frac{\sqrt{a} (3 A b-5 a C) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 b^{7/2}}-\frac{a (2 b B-3 a D) \log \left (a+b x^2\right )}{2 b^4}\\ \end{align*}
Mathematica [A] time = 0.125929, size = 139, normalized size = 0.79 \[ \frac{\frac{6 a \left (a^2 D-a b (B+C x)+A b^2 x\right )}{a+b x^2}+12 b x (A b-2 a C)+6 \sqrt{a} \sqrt{b} (5 a C-3 A b) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )+6 b x^2 (b B-2 a D)+6 a (3 a D-2 b B) \log \left (a+b x^2\right )+4 b^2 C x^3+3 b^2 D x^4}{12 b^4} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.01, size = 201, normalized size = 1.1 \begin{align*}{\frac{D{x}^{4}}{4\,{b}^{2}}}+{\frac{C{x}^{3}}{3\,{b}^{2}}}+{\frac{B{x}^{2}}{2\,{b}^{2}}}-{\frac{D{x}^{2}a}{{b}^{3}}}+{\frac{Ax}{{b}^{2}}}-2\,{\frac{aCx}{{b}^{3}}}+{\frac{aAx}{2\,{b}^{2} \left ( b{x}^{2}+a \right ) }}-{\frac{{a}^{2}Cx}{2\,{b}^{3} \left ( b{x}^{2}+a \right ) }}-{\frac{{a}^{2}B}{2\,{b}^{3} \left ( b{x}^{2}+a \right ) }}+{\frac{{a}^{3}D}{2\,{b}^{4} \left ( b{x}^{2}+a \right ) }}-{\frac{\ln \left ( b{x}^{2}+a \right ) Ba}{{b}^{3}}}+{\frac{3\,{a}^{2}\ln \left ( b{x}^{2}+a \right ) D}{2\,{b}^{4}}}-{\frac{3\,Aa}{2\,{b}^{2}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}+{\frac{5\,{a}^{2}C}{2\,{b}^{3}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [B] time = 3.68118, size = 333, normalized size = 1.89 \begin{align*} \frac{C x^{3}}{3 b^{2}} + \frac{D x^{4}}{4 b^{2}} + \left (\frac{a \left (- 2 B b + 3 D a\right )}{2 b^{4}} - \frac{\sqrt{- a b^{9}} \left (- 3 A b + 5 C a\right )}{4 b^{8}}\right ) \log{\left (x + \frac{4 B a b - 6 D a^{2} + 4 b^{4} \left (\frac{a \left (- 2 B b + 3 D a\right )}{2 b^{4}} - \frac{\sqrt{- a b^{9}} \left (- 3 A b + 5 C a\right )}{4 b^{8}}\right )}{- 3 A b^{2} + 5 C a b} \right )} + \left (\frac{a \left (- 2 B b + 3 D a\right )}{2 b^{4}} + \frac{\sqrt{- a b^{9}} \left (- 3 A b + 5 C a\right )}{4 b^{8}}\right ) \log{\left (x + \frac{4 B a b - 6 D a^{2} + 4 b^{4} \left (\frac{a \left (- 2 B b + 3 D a\right )}{2 b^{4}} + \frac{\sqrt{- a b^{9}} \left (- 3 A b + 5 C a\right )}{4 b^{8}}\right )}{- 3 A b^{2} + 5 C a b} \right )} - \frac{B a^{2} b - D a^{3} + x \left (- A a b^{2} + C a^{2} b\right )}{2 a b^{4} + 2 b^{5} x^{2}} - \frac{x^{2} \left (- B b + 2 D a\right )}{2 b^{3}} - \frac{x \left (- A b + 2 C a\right )}{b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.18027, size = 215, normalized size = 1.22 \begin{align*} \frac{{\left (5 \, C a^{2} - 3 \, A a b\right )} \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{2 \, \sqrt{a b} b^{3}} + \frac{{\left (3 \, D a^{2} - 2 \, B a b\right )} \log \left (b x^{2} + a\right )}{2 \, b^{4}} + \frac{D a^{3} - B a^{2} b -{\left (C a^{2} b - A a b^{2}\right )} x}{2 \,{\left (b x^{2} + a\right )} b^{4}} + \frac{3 \, D b^{6} x^{4} + 4 \, C b^{6} x^{3} - 12 \, D a b^{5} x^{2} + 6 \, B b^{6} x^{2} - 24 \, C a b^{5} x + 12 \, A b^{6} x}{12 \, b^{8}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]