Optimal. Leaf size=240 \[ -\frac {(b e-a f) x \left (c+d x^2\right )^2}{6 e f \left (e+f x^2\right )^3}-\frac {(d e (5 b e+a f)-c f (b e+5 a f)) x \left (c+d x^2\right )}{24 e^2 f^2 \left (e+f x^2\right )^2}-\frac {\left (a f \left (3 d^2 e^2+4 c d e f-15 c^2 f^2\right )+b e \left (15 d^2 e^2-4 c d e f-3 c^2 f^2\right )\right ) x}{48 e^3 f^3 \left (e+f x^2\right )}+\frac {\left (b e \left (5 d^2 e^2+2 c d e f+c^2 f^2\right )+a f \left (d^2 e^2+2 c d e f+5 c^2 f^2\right )\right ) \tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )}{16 e^{7/2} f^{7/2}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.18, antiderivative size = 240, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {540, 393, 211}
\begin {gather*} \frac {\text {ArcTan}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right ) \left (a f \left (5 c^2 f^2+2 c d e f+d^2 e^2\right )+b e \left (c^2 f^2+2 c d e f+5 d^2 e^2\right )\right )}{16 e^{7/2} f^{7/2}}-\frac {x \left (a f \left (-15 c^2 f^2+4 c d e f+3 d^2 e^2\right )+b e \left (-3 c^2 f^2-4 c d e f+15 d^2 e^2\right )\right )}{48 e^3 f^3 \left (e+f x^2\right )}-\frac {x \left (c+d x^2\right ) (d e (a f+5 b e)-c f (5 a f+b e))}{24 e^2 f^2 \left (e+f x^2\right )^2}-\frac {x \left (c+d x^2\right )^2 (b e-a f)}{6 e f \left (e+f x^2\right )^3} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 211
Rule 393
Rule 540
Rubi steps
\begin {align*} \int \frac {\left (a+b x^2\right ) \left (c+d x^2\right )^2}{\left (e+f x^2\right )^4} \, dx &=-\frac {(b e-a f) x \left (c+d x^2\right )^2}{6 e f \left (e+f x^2\right )^3}-\frac {\int \frac {\left (c+d x^2\right ) \left (-c (b e+5 a f)-d (5 b e+a f) x^2\right )}{\left (e+f x^2\right )^3} \, dx}{6 e f}\\ &=-\frac {(b e-a f) x \left (c+d x^2\right )^2}{6 e f \left (e+f x^2\right )^3}-\frac {(d e (5 b e+a f)-c f (b e+5 a f)) x \left (c+d x^2\right )}{24 e^2 f^2 \left (e+f x^2\right )^2}+\frac {\int \frac {c (d e (5 b e+a f)+3 c f (b e+5 a f))+d (b e (15 d e+c f)+a f (3 d e+5 c f)) x^2}{\left (e+f x^2\right )^2} \, dx}{24 e^2 f^2}\\ &=-\frac {(b e-a f) x \left (c+d x^2\right )^2}{6 e f \left (e+f x^2\right )^3}-\frac {(d e (5 b e+a f)-c f (b e+5 a f)) x \left (c+d x^2\right )}{24 e^2 f^2 \left (e+f x^2\right )^2}-\frac {\left (a f \left (3 d^2 e^2+4 c d e f-15 c^2 f^2\right )+b e \left (15 d^2 e^2-4 c d e f-3 c^2 f^2\right )\right ) x}{48 e^3 f^3 \left (e+f x^2\right )}+\frac {\left (b e \left (5 d^2 e^2+2 c d e f+c^2 f^2\right )+a f \left (d^2 e^2+2 c d e f+5 c^2 f^2\right )\right ) \int \frac {1}{e+f x^2} \, dx}{16 e^3 f^3}\\ &=-\frac {(b e-a f) x \left (c+d x^2\right )^2}{6 e f \left (e+f x^2\right )^3}-\frac {(d e (5 b e+a f)-c f (b e+5 a f)) x \left (c+d x^2\right )}{24 e^2 f^2 \left (e+f x^2\right )^2}-\frac {\left (a f \left (3 d^2 e^2+4 c d e f-15 c^2 f^2\right )+b e \left (15 d^2 e^2-4 c d e f-3 c^2 f^2\right )\right ) x}{48 e^3 f^3 \left (e+f x^2\right )}+\frac {\left (b e \left (5 d^2 e^2+2 c d e f+c^2 f^2\right )+a f \left (d^2 e^2+2 c d e f+5 c^2 f^2\right )\right ) \tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )}{16 e^{7/2} f^{7/2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.11, size = 242, normalized size = 1.01 \begin {gather*} -\frac {(b e-a f) (d e-c f)^2 x}{6 e f^3 \left (e+f x^2\right )^3}+\frac {(d e-c f) (b e (13 d e-c f)-a f (7 d e+5 c f)) x}{24 e^2 f^3 \left (e+f x^2\right )^2}+\frac {\left (b e \left (-11 d^2 e^2+2 c d e f+c^2 f^2\right )+a f \left (d^2 e^2+2 c d e f+5 c^2 f^2\right )\right ) x}{16 e^3 f^3 \left (e+f x^2\right )}+\frac {\left (b e \left (5 d^2 e^2+2 c d e f+c^2 f^2\right )+a f \left (d^2 e^2+2 c d e f+5 c^2 f^2\right )\right ) \tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )}{16 e^{7/2} f^{7/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.16, size = 289, normalized size = 1.20
method | result | size |
default | \(\frac {\frac {\left (5 c^{2} a \,f^{3}+2 a c d e \,f^{2}+a \,d^{2} e^{2} f +b \,c^{2} e \,f^{2}+2 b c d \,e^{2} f -11 b \,d^{2} e^{3}\right ) x^{5}}{16 e^{3} f}+\frac {\left (5 c^{2} a \,f^{3}+2 a c d e \,f^{2}-a \,d^{2} e^{2} f +b \,c^{2} e \,f^{2}-2 b c d \,e^{2} f -5 b \,d^{2} e^{3}\right ) x^{3}}{6 e^{2} f^{2}}+\frac {\left (11 c^{2} a \,f^{3}-2 a c d e \,f^{2}-a \,d^{2} e^{2} f -b \,c^{2} e \,f^{2}-2 b c d \,e^{2} f -5 b \,d^{2} e^{3}\right ) x}{16 f^{3} e}}{\left (f \,x^{2}+e \right )^{3}}+\frac {\left (5 c^{2} a \,f^{3}+2 a c d e \,f^{2}+a \,d^{2} e^{2} f +b \,c^{2} e \,f^{2}+2 b c d \,e^{2} f +5 b \,d^{2} e^{3}\right ) \arctan \left (\frac {f x}{\sqrt {f e}}\right )}{16 e^{3} f^{3} \sqrt {f e}}\) | \(289\) |
risch | \(\frac {\frac {\left (5 c^{2} a \,f^{3}+2 a c d e \,f^{2}+a \,d^{2} e^{2} f +b \,c^{2} e \,f^{2}+2 b c d \,e^{2} f -11 b \,d^{2} e^{3}\right ) x^{5}}{16 e^{3} f}+\frac {\left (5 c^{2} a \,f^{3}+2 a c d e \,f^{2}-a \,d^{2} e^{2} f +b \,c^{2} e \,f^{2}-2 b c d \,e^{2} f -5 b \,d^{2} e^{3}\right ) x^{3}}{6 e^{2} f^{2}}+\frac {\left (11 c^{2} a \,f^{3}-2 a c d e \,f^{2}-a \,d^{2} e^{2} f -b \,c^{2} e \,f^{2}-2 b c d \,e^{2} f -5 b \,d^{2} e^{3}\right ) x}{16 f^{3} e}}{\left (f \,x^{2}+e \right )^{3}}-\frac {5 \ln \left (f x +\sqrt {-f e}\right ) c^{2} a}{32 \sqrt {-f e}\, e^{3}}-\frac {\ln \left (f x +\sqrt {-f e}\right ) a c d}{16 \sqrt {-f e}\, f \,e^{2}}-\frac {\ln \left (f x +\sqrt {-f e}\right ) a \,d^{2}}{32 \sqrt {-f e}\, f^{2} e}-\frac {\ln \left (f x +\sqrt {-f e}\right ) b \,c^{2}}{32 \sqrt {-f e}\, f \,e^{2}}-\frac {\ln \left (f x +\sqrt {-f e}\right ) b c d}{16 \sqrt {-f e}\, f^{2} e}-\frac {5 \ln \left (f x +\sqrt {-f e}\right ) b \,d^{2}}{32 \sqrt {-f e}\, f^{3}}+\frac {5 \ln \left (-f x +\sqrt {-f e}\right ) c^{2} a}{32 \sqrt {-f e}\, e^{3}}+\frac {\ln \left (-f x +\sqrt {-f e}\right ) a c d}{16 \sqrt {-f e}\, f \,e^{2}}+\frac {\ln \left (-f x +\sqrt {-f e}\right ) a \,d^{2}}{32 \sqrt {-f e}\, f^{2} e}+\frac {\ln \left (-f x +\sqrt {-f e}\right ) b \,c^{2}}{32 \sqrt {-f e}\, f \,e^{2}}+\frac {\ln \left (-f x +\sqrt {-f e}\right ) b c d}{16 \sqrt {-f e}\, f^{2} e}+\frac {5 \ln \left (-f x +\sqrt {-f e}\right ) b \,d^{2}}{32 \sqrt {-f e}\, f^{3}}\) | \(550\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.51, size = 301, normalized size = 1.25 \begin {gather*} \frac {3 \, {\left (5 \, a c^{2} f^{5} - 11 \, b d^{2} f^{2} e^{3} + {\left (b c^{2} e + 2 \, a c d e\right )} f^{4} + {\left (2 \, b c d e^{2} + a d^{2} e^{2}\right )} f^{3}\right )} x^{5} + 8 \, {\left (5 \, a c^{2} f^{4} e - 5 \, b d^{2} f e^{4} + {\left (b c^{2} e^{2} + 2 \, a c d e^{2}\right )} f^{3} - {\left (2 \, b c d e^{3} + a d^{2} e^{3}\right )} f^{2}\right )} x^{3} + 3 \, {\left (11 \, a c^{2} f^{3} e^{2} - 5 \, b d^{2} e^{5} - {\left (b c^{2} e^{3} + 2 \, a c d e^{3}\right )} f^{2} - {\left (2 \, b c d e^{4} + a d^{2} e^{4}\right )} f\right )} x}{48 \, {\left (f^{6} x^{6} e^{3} + 3 \, f^{5} x^{4} e^{4} + 3 \, f^{4} x^{2} e^{5} + f^{3} e^{6}\right )}} + \frac {{\left (5 \, a c^{2} f^{3} + 5 \, b d^{2} e^{3} + {\left (b c^{2} e + 2 \, a c d e\right )} f^{2} + {\left (2 \, b c d e^{2} + a d^{2} e^{2}\right )} f\right )} \arctan \left (\sqrt {f} x e^{\left (-\frac {1}{2}\right )}\right ) e^{\left (-\frac {7}{2}\right )}}{16 \, f^{\frac {7}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 502 vs.
\(2 (228) = 456\).
time = 1.22, size = 1033, normalized size = 4.30 \begin {gather*} \left [\frac {30 \, a c^{2} f^{6} x^{5} e - 30 \, b d^{2} f x e^{6} - 3 \, {\left (5 \, a c^{2} f^{6} x^{6} + 5 \, b d^{2} e^{6} + {\left (15 \, b d^{2} f x^{2} + {\left (2 \, b c d + a d^{2}\right )} f\right )} e^{5} + {\left (15 \, b d^{2} f^{2} x^{4} + 3 \, {\left (2 \, b c d + a d^{2}\right )} f^{2} x^{2} + {\left (b c^{2} + 2 \, a c d\right )} f^{2}\right )} e^{4} + {\left (5 \, b d^{2} f^{3} x^{6} + 3 \, {\left (2 \, b c d + a d^{2}\right )} f^{3} x^{4} + 5 \, a c^{2} f^{3} + 3 \, {\left (b c^{2} + 2 \, a c d\right )} f^{3} x^{2}\right )} e^{3} + {\left ({\left (2 \, b c d + a d^{2}\right )} f^{4} x^{6} + 15 \, a c^{2} f^{4} x^{2} + 3 \, {\left (b c^{2} + 2 \, a c d\right )} f^{4} x^{4}\right )} e^{2} + {\left (15 \, a c^{2} f^{5} x^{4} + {\left (b c^{2} + 2 \, a c d\right )} f^{5} x^{6}\right )} e\right )} \sqrt {-f e} \log \left (\frac {f x^{2} - 2 \, \sqrt {-f e} x - e}{f x^{2} + e}\right ) - 2 \, {\left (40 \, b d^{2} f^{2} x^{3} + 3 \, {\left (2 \, b c d + a d^{2}\right )} f^{2} x\right )} e^{5} - 2 \, {\left (33 \, b d^{2} f^{3} x^{5} + 8 \, {\left (2 \, b c d + a d^{2}\right )} f^{3} x^{3} + 3 \, {\left (b c^{2} + 2 \, a c d\right )} f^{3} x\right )} e^{4} + 2 \, {\left (3 \, {\left (2 \, b c d + a d^{2}\right )} f^{4} x^{5} + 33 \, a c^{2} f^{4} x + 8 \, {\left (b c^{2} + 2 \, a c d\right )} f^{4} x^{3}\right )} e^{3} + 2 \, {\left (40 \, a c^{2} f^{5} x^{3} + 3 \, {\left (b c^{2} + 2 \, a c d\right )} f^{5} x^{5}\right )} e^{2}}{96 \, {\left (f^{7} x^{6} e^{4} + 3 \, f^{6} x^{4} e^{5} + 3 \, f^{5} x^{2} e^{6} + f^{4} e^{7}\right )}}, \frac {15 \, a c^{2} f^{6} x^{5} e - 15 \, b d^{2} f x e^{6} + 3 \, {\left (5 \, a c^{2} f^{6} x^{6} + 5 \, b d^{2} e^{6} + {\left (15 \, b d^{2} f x^{2} + {\left (2 \, b c d + a d^{2}\right )} f\right )} e^{5} + {\left (15 \, b d^{2} f^{2} x^{4} + 3 \, {\left (2 \, b c d + a d^{2}\right )} f^{2} x^{2} + {\left (b c^{2} + 2 \, a c d\right )} f^{2}\right )} e^{4} + {\left (5 \, b d^{2} f^{3} x^{6} + 3 \, {\left (2 \, b c d + a d^{2}\right )} f^{3} x^{4} + 5 \, a c^{2} f^{3} + 3 \, {\left (b c^{2} + 2 \, a c d\right )} f^{3} x^{2}\right )} e^{3} + {\left ({\left (2 \, b c d + a d^{2}\right )} f^{4} x^{6} + 15 \, a c^{2} f^{4} x^{2} + 3 \, {\left (b c^{2} + 2 \, a c d\right )} f^{4} x^{4}\right )} e^{2} + {\left (15 \, a c^{2} f^{5} x^{4} + {\left (b c^{2} + 2 \, a c d\right )} f^{5} x^{6}\right )} e\right )} \sqrt {f} \arctan \left (\sqrt {f} x e^{\left (-\frac {1}{2}\right )}\right ) e^{\frac {1}{2}} - {\left (40 \, b d^{2} f^{2} x^{3} + 3 \, {\left (2 \, b c d + a d^{2}\right )} f^{2} x\right )} e^{5} - {\left (33 \, b d^{2} f^{3} x^{5} + 8 \, {\left (2 \, b c d + a d^{2}\right )} f^{3} x^{3} + 3 \, {\left (b c^{2} + 2 \, a c d\right )} f^{3} x\right )} e^{4} + {\left (3 \, {\left (2 \, b c d + a d^{2}\right )} f^{4} x^{5} + 33 \, a c^{2} f^{4} x + 8 \, {\left (b c^{2} + 2 \, a c d\right )} f^{4} x^{3}\right )} e^{3} + {\left (40 \, a c^{2} f^{5} x^{3} + 3 \, {\left (b c^{2} + 2 \, a c d\right )} f^{5} x^{5}\right )} e^{2}}{48 \, {\left (f^{7} x^{6} e^{4} + 3 \, f^{6} x^{4} e^{5} + 3 \, f^{5} x^{2} e^{6} + f^{4} e^{7}\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 = 0.63, size = 311, normalized size = 1.30 \begin {gather*} \frac {{\left (5 \, a c^{2} f^{3} + b c^{2} f^{2} e + 2 \, a c d f^{2} e + 2 \, b c d f e^{2} + a d^{2} f e^{2} + 5 \, b d^{2} e^{3}\right )} \arctan \left (\sqrt {f} x e^{\left (-\frac {1}{2}\right )}\right ) e^{\left (-\frac {7}{2}\right )}}{16 \, f^{\frac {7}{2}}} + \frac {{\left (15 \, a c^{2} f^{5} x^{5} + 3 \, b c^{2} f^{4} x^{5} e + 6 \, a c d f^{4} x^{5} e + 6 \, b c d f^{3} x^{5} e^{2} + 3 \, a d^{2} f^{3} x^{5} e^{2} - 33 \, b d^{2} f^{2} x^{5} e^{3} + 40 \, a c^{2} f^{4} x^{3} e + 8 \, b c^{2} f^{3} x^{3} e^{2} + 16 \, a c d f^{3} x^{3} e^{2} - 16 \, b c d f^{2} x^{3} e^{3} - 8 \, a d^{2} f^{2} x^{3} e^{3} - 40 \, b d^{2} f x^{3} e^{4} + 33 \, a c^{2} f^{3} x e^{2} - 3 \, b c^{2} f^{2} x e^{3} - 6 \, a c d f^{2} x e^{3} - 6 \, b c d f x e^{4} - 3 \, a d^{2} f x e^{4} - 15 \, b d^{2} x e^{5}\right )} e^{\left (-3\right )}}{48 \, {\left (f x^{2} + e\right )}^{3} f^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 0.98, size = 303, normalized size = 1.26 \begin {gather*} \frac {\frac {x^3\,\left (b\,c^2\,e\,f^2+5\,a\,c^2\,f^3-2\,b\,c\,d\,e^2\,f+2\,a\,c\,d\,e\,f^2-5\,b\,d^2\,e^3-a\,d^2\,e^2\,f\right )}{6\,e^2\,f^2}-\frac {x\,\left (b\,c^2\,e\,f^2-11\,a\,c^2\,f^3+2\,b\,c\,d\,e^2\,f+2\,a\,c\,d\,e\,f^2+5\,b\,d^2\,e^3+a\,d^2\,e^2\,f\right )}{16\,e\,f^3}+\frac {x^5\,\left (b\,c^2\,e\,f^2+5\,a\,c^2\,f^3+2\,b\,c\,d\,e^2\,f+2\,a\,c\,d\,e\,f^2-11\,b\,d^2\,e^3+a\,d^2\,e^2\,f\right )}{16\,e^3\,f}}{e^3+3\,e^2\,f\,x^2+3\,e\,f^2\,x^4+f^3\,x^6}+\frac {\mathrm {atan}\left (\frac {\sqrt {f}\,x}{\sqrt {e}}\right )\,\left (b\,c^2\,e\,f^2+5\,a\,c^2\,f^3+2\,b\,c\,d\,e^2\,f+2\,a\,c\,d\,e\,f^2+5\,b\,d^2\,e^3+a\,d^2\,e^2\,f\right )}{16\,e^{7/2}\,f^{7/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________