Optimal. Leaf size=144 \[ -\frac {3 b c-2 a d}{2 a^2 c (b c-a d) x}+\frac {b}{2 a (b c-a d) x \left (a+b x^2\right )}-\frac {b^{3/2} (3 b c-5 a d) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 a^{5/2} (b c-a d)^2}-\frac {d^{5/2} \tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{c^{3/2} (b c-a d)^2} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.13, antiderivative size = 144, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 4, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {483, 597, 536,
211} \begin {gather*} -\frac {b^{3/2} \text {ArcTan}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) (3 b c-5 a d)}{2 a^{5/2} (b c-a d)^2}-\frac {3 b c-2 a d}{2 a^2 c x (b c-a d)}-\frac {d^{5/2} \text {ArcTan}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{c^{3/2} (b c-a d)^2}+\frac {b}{2 a x \left (a+b x^2\right ) (b c-a d)} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 211
Rule 483
Rule 536
Rule 597
Rubi steps
\begin {align*} \int \frac {1}{x^2 \left (a+b x^2\right )^2 \left (c+d x^2\right )} \, dx &=\frac {b}{2 a (b c-a d) x \left (a+b x^2\right )}-\frac {\int \frac {-3 b c+2 a d-3 b d x^2}{x^2 \left (a+b x^2\right ) \left (c+d x^2\right )} \, dx}{2 a (b c-a d)}\\ &=-\frac {3 b c-2 a d}{2 a^2 c (b c-a d) x}+\frac {b}{2 a (b c-a d) x \left (a+b x^2\right )}+\frac {\int \frac {-3 b^2 c^2+2 a b c d+2 a^2 d^2-b d (3 b c-2 a d) x^2}{\left (a+b x^2\right ) \left (c+d x^2\right )} \, dx}{2 a^2 c (b c-a d)}\\ &=-\frac {3 b c-2 a d}{2 a^2 c (b c-a d) x}+\frac {b}{2 a (b c-a d) x \left (a+b x^2\right )}-\frac {d^3 \int \frac {1}{c+d x^2} \, dx}{c (b c-a d)^2}-\frac {\left (b^2 (3 b c-5 a d)\right ) \int \frac {1}{a+b x^2} \, dx}{2 a^2 (b c-a d)^2}\\ &=-\frac {3 b c-2 a d}{2 a^2 c (b c-a d) x}+\frac {b}{2 a (b c-a d) x \left (a+b x^2\right )}-\frac {b^{3/2} (3 b c-5 a d) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 a^{5/2} (b c-a d)^2}-\frac {d^{5/2} \tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{c^{3/2} (b c-a d)^2}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.14, size = 123, normalized size = 0.85 \begin {gather*} -\frac {1}{a^2 c x}+\frac {b^2 x}{2 a^2 (-b c+a d) \left (a+b x^2\right )}+\frac {b^{3/2} (-3 b c+5 a d) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 a^{5/2} (-b c+a d)^2}-\frac {d^{5/2} \tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{c^{3/2} (b c-a d)^2} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.17, size = 108, normalized size = 0.75
method | result | size |
default | \(\frac {b^{2} \left (\frac {\left (\frac {a d}{2}-\frac {b c}{2}\right ) x}{b \,x^{2}+a}+\frac {\left (5 a d -3 b c \right ) \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \sqrt {a b}}\right )}{a^{2} \left (a d -b c \right )^{2}}-\frac {d^{3} \arctan \left (\frac {d x}{\sqrt {c d}}\right )}{c \left (a d -b c \right )^{2} \sqrt {c d}}-\frac {1}{a^{2} c x}\) | \(108\) |
risch | \(\frac {-\frac {b \left (2 a d -3 b c \right ) x^{2}}{2 a^{2} c \left (a d -b c \right )}-\frac {1}{a c}}{x \left (b \,x^{2}+a \right )}+\frac {5 \sqrt {-a b}\, b \ln \left (\left (-4 a^{5} b^{2} d^{4}-4 a^{4} b^{3} c \,d^{3}-4 a^{3} b^{4} c^{2} d^{2}+21 a^{2} b^{5} c^{3} d -9 a \,b^{6} c^{4}\right ) x +10 \left (-a b \right )^{\frac {3}{2}} a^{4} d^{4}+4 \left (-a b \right )^{\frac {3}{2}} a^{3} b c \,d^{3}+4 \left (-a b \right )^{\frac {3}{2}} a^{2} b^{2} c^{2} d^{2}-21 \left (-a b \right )^{\frac {3}{2}} a \,b^{3} c^{3} d +9 \left (-a b \right )^{\frac {3}{2}} b^{4} c^{4}+6 \sqrt {-a b}\, a^{5} b \,d^{4}\right ) d}{4 a^{2} \left (a d -b c \right )^{2}}-\frac {3 \sqrt {-a b}\, b^{2} \ln \left (\left (-4 a^{5} b^{2} d^{4}-4 a^{4} b^{3} c \,d^{3}-4 a^{3} b^{4} c^{2} d^{2}+21 a^{2} b^{5} c^{3} d -9 a \,b^{6} c^{4}\right ) x +10 \left (-a b \right )^{\frac {3}{2}} a^{4} d^{4}+4 \left (-a b \right )^{\frac {3}{2}} a^{3} b c \,d^{3}+4 \left (-a b \right )^{\frac {3}{2}} a^{2} b^{2} c^{2} d^{2}-21 \left (-a b \right )^{\frac {3}{2}} a \,b^{3} c^{3} d +9 \left (-a b \right )^{\frac {3}{2}} b^{4} c^{4}+6 \sqrt {-a b}\, a^{5} b \,d^{4}\right ) c}{4 a^{3} \left (a d -b c \right )^{2}}-\frac {5 \sqrt {-a b}\, b \ln \left (\left (-4 a^{5} b^{2} d^{4}-4 a^{4} b^{3} c \,d^{3}-4 a^{3} b^{4} c^{2} d^{2}+21 a^{2} b^{5} c^{3} d -9 a \,b^{6} c^{4}\right ) x -10 \left (-a b \right )^{\frac {3}{2}} a^{4} d^{4}-4 \left (-a b \right )^{\frac {3}{2}} a^{3} b c \,d^{3}-4 \left (-a b \right )^{\frac {3}{2}} a^{2} b^{2} c^{2} d^{2}+21 \left (-a b \right )^{\frac {3}{2}} a \,b^{3} c^{3} d -9 \left (-a b \right )^{\frac {3}{2}} b^{4} c^{4}-6 \sqrt {-a b}\, a^{5} b \,d^{4}\right ) d}{4 a^{2} \left (a d -b c \right )^{2}}+\frac {3 \sqrt {-a b}\, b^{2} \ln \left (\left (-4 a^{5} b^{2} d^{4}-4 a^{4} b^{3} c \,d^{3}-4 a^{3} b^{4} c^{2} d^{2}+21 a^{2} b^{5} c^{3} d -9 a \,b^{6} c^{4}\right ) x -10 \left (-a b \right )^{\frac {3}{2}} a^{4} d^{4}-4 \left (-a b \right )^{\frac {3}{2}} a^{3} b c \,d^{3}-4 \left (-a b \right )^{\frac {3}{2}} a^{2} b^{2} c^{2} d^{2}+21 \left (-a b \right )^{\frac {3}{2}} a \,b^{3} c^{3} d -9 \left (-a b \right )^{\frac {3}{2}} b^{4} c^{4}-6 \sqrt {-a b}\, a^{5} b \,d^{4}\right ) c}{4 a^{3} \left (a d -b c \right )^{2}}+\frac {\left (\munderset {\textit {\_R} =\RootOf \left (\left (d^{4} c^{3} a^{4}-4 a^{3} b \,c^{4} d^{3}+6 a^{2} b^{2} c^{5} d^{2}-4 a \,b^{3} c^{6} d +b^{4} c^{7}\right ) \textit {\_Z}^{2}+d^{5}\right )}{\sum }\textit {\_R} \ln \left (\left (\left (12 a^{13} c^{3} d^{8}-88 a^{12} b \,c^{4} d^{7}+288 a^{11} b^{2} c^{5} d^{6}-552 a^{10} b^{3} c^{6} d^{5}+680 a^{9} b^{4} c^{7} d^{4}-552 a^{8} b^{5} c^{8} d^{3}+288 a^{7} b^{6} c^{9} d^{2}-88 a^{6} b^{7} c^{10} d +12 a^{5} b^{8} c^{11}\right ) \textit {\_R}^{4}+\left (8 a^{9} d^{9}-32 a^{8} b c \,d^{8}+52 a^{7} b^{2} c^{2} d^{7}+35 a^{6} b^{3} c^{3} d^{6}-328 a^{5} b^{4} c^{4} d^{5}+652 a^{4} b^{5} c^{5} d^{4}-680 a^{3} b^{6} c^{6} d^{3}+407 a^{2} b^{7} c^{7} d^{2}-132 a \,b^{8} c^{8} d +18 b^{9} c^{9}\right ) \textit {\_R}^{2}+50 a^{2} b^{3} d^{7}-60 a \,b^{4} c \,d^{6}+18 b^{5} c^{2} d^{5}\right ) x +\left (4 a^{11} c^{2} d^{8}-16 a^{10} b \,c^{3} d^{7}+24 a^{9} b^{2} c^{4} d^{6}-26 a^{8} b^{3} c^{5} d^{5}+50 a^{7} b^{4} c^{6} d^{4}-84 a^{6} b^{5} c^{7} d^{3}+76 a^{5} b^{6} c^{8} d^{2}-34 a^{4} b^{7} c^{9} d +6 a^{3} b^{8} c^{10}\right ) \textit {\_R}^{3}+\left (15 a^{4} b^{3} c^{2} d^{6}-24 a^{3} b^{4} c^{3} d^{5}+9 a^{2} b^{5} c^{4} d^{4}\right ) \textit {\_R} \right )\right )}{2}\) | \(1302\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.53, size = 178, normalized size = 1.24 \begin {gather*} -\frac {d^{3} \arctan \left (\frac {d x}{\sqrt {c d}}\right )}{{\left (b^{2} c^{3} - 2 \, a b c^{2} d + a^{2} c d^{2}\right )} \sqrt {c d}} - \frac {{\left (3 \, b^{3} c - 5 \, a b^{2} d\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \, {\left (a^{2} b^{2} c^{2} - 2 \, a^{3} b c d + a^{4} d^{2}\right )} \sqrt {a b}} - \frac {2 \, a b c - 2 \, a^{2} d + {\left (3 \, b^{2} c - 2 \, a b d\right )} x^{2}}{2 \, {\left ({\left (a^{2} b^{2} c^{2} - a^{3} b c d\right )} x^{3} + {\left (a^{3} b c^{2} - a^{4} c d\right )} x\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 1.97, size = 1003, normalized size = 6.97 \begin {gather*} \left [-\frac {4 \, a b^{2} c^{2} - 8 \, a^{2} b c d + 4 \, a^{3} d^{2} + 2 \, {\left (3 \, b^{3} c^{2} - 5 \, a b^{2} c d + 2 \, a^{2} b d^{2}\right )} x^{2} + {\left ({\left (3 \, b^{3} c^{2} - 5 \, a b^{2} c d\right )} x^{3} + {\left (3 \, a b^{2} c^{2} - 5 \, a^{2} b c d\right )} x\right )} \sqrt {-\frac {b}{a}} \log \left (\frac {b x^{2} + 2 \, a x \sqrt {-\frac {b}{a}} - a}{b x^{2} + a}\right ) - 2 \, {\left (a^{2} b d^{2} x^{3} + a^{3} d^{2} x\right )} \sqrt {-\frac {d}{c}} \log \left (\frac {d x^{2} - 2 \, c x \sqrt {-\frac {d}{c}} - c}{d x^{2} + c}\right )}{4 \, {\left ({\left (a^{2} b^{3} c^{3} - 2 \, a^{3} b^{2} c^{2} d + a^{4} b c d^{2}\right )} x^{3} + {\left (a^{3} b^{2} c^{3} - 2 \, a^{4} b c^{2} d + a^{5} c d^{2}\right )} x\right )}}, -\frac {4 \, a b^{2} c^{2} - 8 \, a^{2} b c d + 4 \, a^{3} d^{2} + 2 \, {\left (3 \, b^{3} c^{2} - 5 \, a b^{2} c d + 2 \, a^{2} b d^{2}\right )} x^{2} + 4 \, {\left (a^{2} b d^{2} x^{3} + a^{3} d^{2} x\right )} \sqrt {\frac {d}{c}} \arctan \left (x \sqrt {\frac {d}{c}}\right ) + {\left ({\left (3 \, b^{3} c^{2} - 5 \, a b^{2} c d\right )} x^{3} + {\left (3 \, a b^{2} c^{2} - 5 \, a^{2} b c d\right )} x\right )} \sqrt {-\frac {b}{a}} \log \left (\frac {b x^{2} + 2 \, a x \sqrt {-\frac {b}{a}} - a}{b x^{2} + a}\right )}{4 \, {\left ({\left (a^{2} b^{3} c^{3} - 2 \, a^{3} b^{2} c^{2} d + a^{4} b c d^{2}\right )} x^{3} + {\left (a^{3} b^{2} c^{3} - 2 \, a^{4} b c^{2} d + a^{5} c d^{2}\right )} x\right )}}, -\frac {2 \, a b^{2} c^{2} - 4 \, a^{2} b c d + 2 \, a^{3} d^{2} + {\left (3 \, b^{3} c^{2} - 5 \, a b^{2} c d + 2 \, a^{2} b d^{2}\right )} x^{2} + {\left ({\left (3 \, b^{3} c^{2} - 5 \, a b^{2} c d\right )} x^{3} + {\left (3 \, a b^{2} c^{2} - 5 \, a^{2} b c d\right )} x\right )} \sqrt {\frac {b}{a}} \arctan \left (x \sqrt {\frac {b}{a}}\right ) - {\left (a^{2} b d^{2} x^{3} + a^{3} d^{2} x\right )} \sqrt {-\frac {d}{c}} \log \left (\frac {d x^{2} - 2 \, c x \sqrt {-\frac {d}{c}} - c}{d x^{2} + c}\right )}{2 \, {\left ({\left (a^{2} b^{3} c^{3} - 2 \, a^{3} b^{2} c^{2} d + a^{4} b c d^{2}\right )} x^{3} + {\left (a^{3} b^{2} c^{3} - 2 \, a^{4} b c^{2} d + a^{5} c d^{2}\right )} x\right )}}, -\frac {2 \, a b^{2} c^{2} - 4 \, a^{2} b c d + 2 \, a^{3} d^{2} + {\left (3 \, b^{3} c^{2} - 5 \, a b^{2} c d + 2 \, a^{2} b d^{2}\right )} x^{2} + {\left ({\left (3 \, b^{3} c^{2} - 5 \, a b^{2} c d\right )} x^{3} + {\left (3 \, a b^{2} c^{2} - 5 \, a^{2} b c d\right )} x\right )} \sqrt {\frac {b}{a}} \arctan \left (x \sqrt {\frac {b}{a}}\right ) + 2 \, {\left (a^{2} b d^{2} x^{3} + a^{3} d^{2} x\right )} \sqrt {\frac {d}{c}} \arctan \left (x \sqrt {\frac {d}{c}}\right )}{2 \, {\left ({\left (a^{2} b^{3} c^{3} - 2 \, a^{3} b^{2} c^{2} d + a^{4} b c d^{2}\right )} x^{3} + {\left (a^{3} b^{2} c^{3} - 2 \, a^{4} b c^{2} d + a^{5} c d^{2}\right )} x\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 0.74, size = 164, normalized size = 1.14 \begin {gather*} -\frac {d^{3} \arctan \left (\frac {d x}{\sqrt {c d}}\right )}{{\left (b^{2} c^{3} - 2 \, a b c^{2} d + a^{2} c d^{2}\right )} \sqrt {c d}} - \frac {{\left (3 \, b^{3} c - 5 \, a b^{2} d\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \, {\left (a^{2} b^{2} c^{2} - 2 \, a^{3} b c d + a^{4} d^{2}\right )} \sqrt {a b}} - \frac {3 \, b^{2} c x^{2} - 2 \, a b d x^{2} + 2 \, a b c - 2 \, a^{2} d}{2 \, {\left (a^{2} b c^{2} - a^{3} c d\right )} {\left (b x^{3} + a x\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 0.56, size = 2400, normalized size = 16.67 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________