Optimal. Leaf size=142 \[ \frac {d (2 b c-a d) \left (2 b^2 c^2-2 a b c d+a^2 d^2\right ) x}{b^4}+\frac {d^2 \left (6 b^2 c^2-4 a b c d+a^2 d^2\right ) x^3}{3 b^3}+\frac {d^3 (4 b c-a d) x^5}{5 b^2}+\frac {d^4 x^7}{7 b}+\frac {(b c-a d)^4 \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {a} b^{9/2}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.07, antiderivative size = 142, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {398, 211}
\begin {gather*} \frac {d x (2 b c-a d) \left (a^2 d^2-2 a b c d+2 b^2 c^2\right )}{b^4}+\frac {d^2 x^3 \left (a^2 d^2-4 a b c d+6 b^2 c^2\right )}{3 b^3}+\frac {\text {ArcTan}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) (b c-a d)^4}{\sqrt {a} b^{9/2}}+\frac {d^3 x^5 (4 b c-a d)}{5 b^2}+\frac {d^4 x^7}{7 b} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 211
Rule 398
Rubi steps
\begin {align*} \int \frac {\left (c+d x^2\right )^4}{a+b x^2} \, dx &=\int \left (\frac {d (2 b c-a d) \left (2 b^2 c^2-2 a b c d+a^2 d^2\right )}{b^4}+\frac {d^2 \left (6 b^2 c^2-4 a b c d+a^2 d^2\right ) x^2}{b^3}+\frac {d^3 (4 b c-a d) x^4}{b^2}+\frac {d^4 x^6}{b}+\frac {b^4 c^4-4 a b^3 c^3 d+6 a^2 b^2 c^2 d^2-4 a^3 b c d^3+a^4 d^4}{b^4 \left (a+b x^2\right )}\right ) \, dx\\ &=\frac {d (2 b c-a d) \left (2 b^2 c^2-2 a b c d+a^2 d^2\right ) x}{b^4}+\frac {d^2 \left (6 b^2 c^2-4 a b c d+a^2 d^2\right ) x^3}{3 b^3}+\frac {d^3 (4 b c-a d) x^5}{5 b^2}+\frac {d^4 x^7}{7 b}+\frac {(b c-a d)^4 \int \frac {1}{a+b x^2} \, dx}{b^4}\\ &=\frac {d (2 b c-a d) \left (2 b^2 c^2-2 a b c d+a^2 d^2\right ) x}{b^4}+\frac {d^2 \left (6 b^2 c^2-4 a b c d+a^2 d^2\right ) x^3}{3 b^3}+\frac {d^3 (4 b c-a d) x^5}{5 b^2}+\frac {d^4 x^7}{7 b}+\frac {(b c-a d)^4 \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {a} b^{9/2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.06, size = 136, normalized size = 0.96 \begin {gather*} \frac {d x \left (-105 a^3 d^3+35 a^2 b d^2 \left (12 c+d x^2\right )-7 a b^2 d \left (90 c^2+20 c d x^2+3 d^2 x^4\right )+3 b^3 \left (140 c^3+70 c^2 d x^2+28 c d^2 x^4+5 d^3 x^6\right )\right )}{105 b^4}+\frac {(b c-a d)^4 \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {a} b^{9/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.13, size = 196, normalized size = 1.38
method | result | size |
default | \(-\frac {d \left (-\frac {d^{3} x^{7} b^{3}}{7}+\frac {\left (\left (a d -2 b c \right ) b^{2} d^{2}-2 b^{3} c \,d^{2}\right ) x^{5}}{5}+\frac {\left (2 \left (a d -2 b c \right ) b^{2} c d -b d \left (a^{2} d^{2}-2 a b c d +2 b^{2} c^{2}\right )\right ) x^{3}}{3}+\left (a d -2 b c \right ) \left (a^{2} d^{2}-2 a b c d +2 b^{2} c^{2}\right ) x \right )}{b^{4}}+\frac {\left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right ) \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{b^{4} \sqrt {a b}}\) | \(196\) |
risch | \(\frac {d^{4} x^{7}}{7 b}-\frac {d^{4} x^{5} a}{5 b^{2}}+\frac {4 d^{3} x^{5} c}{5 b}-\frac {4 d^{3} x^{3} a c}{3 b^{2}}+\frac {2 d^{2} x^{3} c^{2}}{b}+\frac {d^{4} x^{3} a^{2}}{3 b^{3}}-\frac {d^{4} a^{3} x}{b^{4}}+\frac {4 d^{3} a^{2} c x}{b^{3}}-\frac {6 d^{2} a \,c^{2} x}{b^{2}}+\frac {4 d \,c^{3} x}{b}-\frac {\ln \left (b x +\sqrt {-a b}\right ) a^{4} d^{4}}{2 b^{4} \sqrt {-a b}}+\frac {2 \ln \left (b x +\sqrt {-a b}\right ) a^{3} c \,d^{3}}{b^{3} \sqrt {-a b}}-\frac {3 \ln \left (b x +\sqrt {-a b}\right ) a^{2} c^{2} d^{2}}{b^{2} \sqrt {-a b}}+\frac {2 \ln \left (b x +\sqrt {-a b}\right ) a \,c^{3} d}{b \sqrt {-a b}}-\frac {\ln \left (b x +\sqrt {-a b}\right ) c^{4}}{2 \sqrt {-a b}}+\frac {\ln \left (-b x +\sqrt {-a b}\right ) a^{4} d^{4}}{2 b^{4} \sqrt {-a b}}-\frac {2 \ln \left (-b x +\sqrt {-a b}\right ) a^{3} c \,d^{3}}{b^{3} \sqrt {-a b}}+\frac {3 \ln \left (-b x +\sqrt {-a b}\right ) a^{2} c^{2} d^{2}}{b^{2} \sqrt {-a b}}-\frac {2 \ln \left (-b x +\sqrt {-a b}\right ) a \,c^{3} d}{b \sqrt {-a b}}+\frac {\ln \left (-b x +\sqrt {-a b}\right ) c^{4}}{2 \sqrt {-a b}}\) | \(405\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.51, size = 187, normalized size = 1.32 \begin {gather*} \frac {{\left (b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b} b^{4}} + \frac {15 \, b^{3} d^{4} x^{7} + 21 \, {\left (4 \, b^{3} c d^{3} - a b^{2} d^{4}\right )} x^{5} + 35 \, {\left (6 \, b^{3} c^{2} d^{2} - 4 \, a b^{2} c d^{3} + a^{2} b d^{4}\right )} x^{3} + 105 \, {\left (4 \, b^{3} c^{3} d - 6 \, a b^{2} c^{2} d^{2} + 4 \, a^{2} b c d^{3} - a^{3} d^{4}\right )} x}{105 \, b^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.44, size = 428, normalized size = 3.01 \begin {gather*} \left [\frac {30 \, a b^{4} d^{4} x^{7} + 42 \, {\left (4 \, a b^{4} c d^{3} - a^{2} b^{3} d^{4}\right )} x^{5} + 70 \, {\left (6 \, a b^{4} c^{2} d^{2} - 4 \, a^{2} b^{3} c d^{3} + a^{3} b^{2} d^{4}\right )} x^{3} - 105 \, {\left (b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}\right )} \sqrt {-a b} \log \left (\frac {b x^{2} - 2 \, \sqrt {-a b} x - a}{b x^{2} + a}\right ) + 210 \, {\left (4 \, a b^{4} c^{3} d - 6 \, a^{2} b^{3} c^{2} d^{2} + 4 \, a^{3} b^{2} c d^{3} - a^{4} b d^{4}\right )} x}{210 \, a b^{5}}, \frac {15 \, a b^{4} d^{4} x^{7} + 21 \, {\left (4 \, a b^{4} c d^{3} - a^{2} b^{3} d^{4}\right )} x^{5} + 35 \, {\left (6 \, a b^{4} c^{2} d^{2} - 4 \, a^{2} b^{3} c d^{3} + a^{3} b^{2} d^{4}\right )} x^{3} + 105 \, {\left (b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}\right )} \sqrt {a b} \arctan \left (\frac {\sqrt {a b} x}{a}\right ) + 105 \, {\left (4 \, a b^{4} c^{3} d - 6 \, a^{2} b^{3} c^{2} d^{2} + 4 \, a^{3} b^{2} c d^{3} - a^{4} b d^{4}\right )} x}{105 \, a b^{5}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 326 vs.
\(2 (136) = 272\).
time = 0.43, size = 326, normalized size = 2.30 \begin {gather*} x^{5} \left (- \frac {a d^{4}}{5 b^{2}} + \frac {4 c d^{3}}{5 b}\right ) + x^{3} \left (\frac {a^{2} d^{4}}{3 b^{3}} - \frac {4 a c d^{3}}{3 b^{2}} + \frac {2 c^{2} d^{2}}{b}\right ) + x \left (- \frac {a^{3} d^{4}}{b^{4}} + \frac {4 a^{2} c d^{3}}{b^{3}} - \frac {6 a c^{2} d^{2}}{b^{2}} + \frac {4 c^{3} d}{b}\right ) - \frac {\sqrt {- \frac {1}{a b^{9}}} \left (a d - b c\right )^{4} \log {\left (- \frac {a b^{4} \sqrt {- \frac {1}{a b^{9}}} \left (a d - b c\right )^{4}}{a^{4} d^{4} - 4 a^{3} b c d^{3} + 6 a^{2} b^{2} c^{2} d^{2} - 4 a b^{3} c^{3} d + b^{4} c^{4}} + x \right )}}{2} + \frac {\sqrt {- \frac {1}{a b^{9}}} \left (a d - b c\right )^{4} \log {\left (\frac {a b^{4} \sqrt {- \frac {1}{a b^{9}}} \left (a d - b c\right )^{4}}{a^{4} d^{4} - 4 a^{3} b c d^{3} + 6 a^{2} b^{2} c^{2} d^{2} - 4 a b^{3} c^{3} d + b^{4} c^{4}} + x \right )}}{2} + \frac {d^{4} x^{7}}{7 b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 1.07, size = 198, normalized size = 1.39 \begin {gather*} \frac {{\left (b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b} b^{4}} + \frac {15 \, b^{6} d^{4} x^{7} + 84 \, b^{6} c d^{3} x^{5} - 21 \, a b^{5} d^{4} x^{5} + 210 \, b^{6} c^{2} d^{2} x^{3} - 140 \, a b^{5} c d^{3} x^{3} + 35 \, a^{2} b^{4} d^{4} x^{3} + 420 \, b^{6} c^{3} d x - 630 \, a b^{5} c^{2} d^{2} x + 420 \, a^{2} b^{4} c d^{3} x - 105 \, a^{3} b^{3} d^{4} x}{105 \, b^{7}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 4.86, size = 216, normalized size = 1.52 \begin {gather*} x\,\left (\frac {4\,c^3\,d}{b}-\frac {a\,\left (\frac {a\,\left (\frac {a\,d^4}{b^2}-\frac {4\,c\,d^3}{b}\right )}{b}+\frac {6\,c^2\,d^2}{b}\right )}{b}\right )-x^5\,\left (\frac {a\,d^4}{5\,b^2}-\frac {4\,c\,d^3}{5\,b}\right )+x^3\,\left (\frac {a\,\left (\frac {a\,d^4}{b^2}-\frac {4\,c\,d^3}{b}\right )}{3\,b}+\frac {2\,c^2\,d^2}{b}\right )+\frac {d^4\,x^7}{7\,b}+\frac {\mathrm {atan}\left (\frac {\sqrt {b}\,x\,{\left (a\,d-b\,c\right )}^4}{\sqrt {a}\,\left (a^4\,d^4-4\,a^3\,b\,c\,d^3+6\,a^2\,b^2\,c^2\,d^2-4\,a\,b^3\,c^3\,d+b^4\,c^4\right )}\right )\,{\left (a\,d-b\,c\right )}^4}{\sqrt {a}\,b^{9/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________