Optimal. Leaf size=225 \[ -\frac {(a B-(A c-a C) x) (d+e x)^2}{6 a c \left (a+c x^2\right )^3}-\frac {2 a e (4 A c d+2 a C d+a B e)+\left (3 a (A c+a C) e^2-c d (5 A c d+a C d+2 a B e)\right ) x}{24 a^2 c^2 \left (a+c x^2\right )^2}+\frac {\left (a (A c+a C) e^2+c d (5 A c d+a C d+2 a B e)\right ) x}{16 a^3 c^2 \left (a+c x^2\right )}+\frac {\left (a (A c+a C) e^2+c d (5 A c d+a C d+2 a B e)\right ) \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{16 a^{7/2} c^{5/2}} \]
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Rubi [A]
time = 0.20, antiderivative size = 225, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {1659, 792, 205,
211} \begin {gather*} \frac {\text {ArcTan}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right ) \left (c d (2 a B e+a C d+5 A c d)+a e^2 (a C+A c)\right )}{16 a^{7/2} c^{5/2}}+\frac {x \left (c d (2 a B e+a C d+5 A c d)+a e^2 (a C+A c)\right )}{16 a^3 c^2 \left (a+c x^2\right )}-\frac {x \left (3 a e^2 (a C+A c)-c d (2 a B e+a C d+5 A c d)\right )+2 a e (a B e+2 a C d+4 A c d)}{24 a^2 c^2 \left (a+c x^2\right )^2}-\frac {(d+e x)^2 (a B-x (A c-a C))}{6 a c \left (a+c x^2\right )^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 205
Rule 211
Rule 792
Rule 1659
Rubi steps
\begin {align*} \int \frac {(d+e x)^2 \left (A+B x+C x^2\right )}{\left (a+c x^2\right )^4} \, dx &=-\frac {(a B-(A c-a C) x) (d+e x)^2}{6 a c \left (a+c x^2\right )^3}-\frac {\int \frac {(d+e x) (-5 A c d-a C d-2 a B e-3 (A c+a C) e x)}{\left (a+c x^2\right )^3} \, dx}{6 a c}\\ &=-\frac {(a B-(A c-a C) x) (d+e x)^2}{6 a c \left (a+c x^2\right )^3}-\frac {2 a e (4 A c d+2 a C d+a B e)+\left (3 a (A c+a C) e^2-c d (5 A c d+a C d+2 a B e)\right ) x}{24 a^2 c^2 \left (a+c x^2\right )^2}+\frac {\left (a (A c+a C) e^2+c d (5 A c d+a C d+2 a B e)\right ) \int \frac {1}{\left (a+c x^2\right )^2} \, dx}{8 a^2 c^2}\\ &=-\frac {(a B-(A c-a C) x) (d+e x)^2}{6 a c \left (a+c x^2\right )^3}-\frac {2 a e (4 A c d+2 a C d+a B e)+\left (3 a (A c+a C) e^2-c d (5 A c d+a C d+2 a B e)\right ) x}{24 a^2 c^2 \left (a+c x^2\right )^2}+\frac {\left (a (A c+a C) e^2+c d (5 A c d+a C d+2 a B e)\right ) x}{16 a^3 c^2 \left (a+c x^2\right )}+\frac {\left (a (A c+a C) e^2+c d (5 A c d+a C d+2 a B e)\right ) \int \frac {1}{a+c x^2} \, dx}{16 a^3 c^2}\\ &=-\frac {(a B-(A c-a C) x) (d+e x)^2}{6 a c \left (a+c x^2\right )^3}-\frac {2 a e (4 A c d+2 a C d+a B e)+\left (3 a (A c+a C) e^2-c d (5 A c d+a C d+2 a B e)\right ) x}{24 a^2 c^2 \left (a+c x^2\right )^2}+\frac {\left (a (A c+a C) e^2+c d (5 A c d+a C d+2 a B e)\right ) x}{16 a^3 c^2 \left (a+c x^2\right )}+\frac {\left (a (A c+a C) e^2+c d (5 A c d+a C d+2 a B e)\right ) \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{16 a^{7/2} c^{5/2}}\\ \end {align*}
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Mathematica [A]
time = 0.11, size = 266, normalized size = 1.18 \begin {gather*} \frac {\left (A c \left (5 c d^2+a e^2\right )+a \left (a C e^2+c d (C d+2 B e)\right )\right ) x}{16 a^3 c^2 \left (a+c x^2\right )}+\frac {5 A c^2 d^2 x+a c \left (C d^2+e (2 B d+A e)\right ) x-a^2 e (12 C d+6 B e+7 C e x)}{24 a^2 c^2 \left (a+c x^2\right )^2}+\frac {A c^2 d^2 x+a^2 e (2 C d+B e+C e x)-a c \left (C d^2 x+A e (2 d+e x)+B d (d+2 e x)\right )}{6 a c^2 \left (a+c x^2\right )^3}+\frac {\left (A c \left (5 c d^2+a e^2\right )+a \left (a C e^2+c d (C d+2 B e)\right )\right ) \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{16 a^{7/2} c^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.10, size = 268, normalized size = 1.19
method | result | size |
default | \(\frac {\frac {\left (A a c \,e^{2}+5 A \,c^{2} d^{2}+2 a c d e B +a^{2} C \,e^{2}+C a c \,d^{2}\right ) x^{5}}{16 a^{3}}+\frac {\left (A a c \,e^{2}+5 A \,c^{2} d^{2}+2 a c d e B -a^{2} C \,e^{2}+C a c \,d^{2}\right ) x^{3}}{6 a^{2} c}-\frac {e \left (B e +2 C d \right ) x^{2}}{4 c}-\frac {\left (A a c \,e^{2}-11 A \,c^{2} d^{2}+2 a c d e B +a^{2} C \,e^{2}+C a c \,d^{2}\right ) x}{16 c^{2} a}-\frac {4 c d e A +B a \,e^{2}+2 B c \,d^{2}+2 a d e C}{12 c^{2}}}{\left (c \,x^{2}+a \right )^{3}}+\frac {\left (A a c \,e^{2}+5 A \,c^{2} d^{2}+2 a c d e B +a^{2} C \,e^{2}+C a c \,d^{2}\right ) \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{16 a^{3} c^{2} \sqrt {a c}}\) | \(268\) |
risch | \(\frac {\frac {\left (A a c \,e^{2}+5 A \,c^{2} d^{2}+2 a c d e B +a^{2} C \,e^{2}+C a c \,d^{2}\right ) x^{5}}{16 a^{3}}+\frac {\left (A a c \,e^{2}+5 A \,c^{2} d^{2}+2 a c d e B -a^{2} C \,e^{2}+C a c \,d^{2}\right ) x^{3}}{6 a^{2} c}-\frac {e \left (B e +2 C d \right ) x^{2}}{4 c}-\frac {\left (A a c \,e^{2}-11 A \,c^{2} d^{2}+2 a c d e B +a^{2} C \,e^{2}+C a c \,d^{2}\right ) x}{16 c^{2} a}-\frac {4 c d e A +B a \,e^{2}+2 B c \,d^{2}+2 a d e C}{12 c^{2}}}{\left (c \,x^{2}+a \right )^{3}}-\frac {\ln \left (c x +\sqrt {-a c}\right ) A \,e^{2}}{32 \sqrt {-a c}\, c \,a^{2}}-\frac {5 \ln \left (c x +\sqrt {-a c}\right ) A \,d^{2}}{32 \sqrt {-a c}\, a^{3}}-\frac {\ln \left (c x +\sqrt {-a c}\right ) d e B}{16 \sqrt {-a c}\, c \,a^{2}}-\frac {\ln \left (c x +\sqrt {-a c}\right ) C \,e^{2}}{32 \sqrt {-a c}\, c^{2} a}-\frac {\ln \left (c x +\sqrt {-a c}\right ) C \,d^{2}}{32 \sqrt {-a c}\, c \,a^{2}}+\frac {\ln \left (-c x +\sqrt {-a c}\right ) A \,e^{2}}{32 \sqrt {-a c}\, c \,a^{2}}+\frac {5 \ln \left (-c x +\sqrt {-a c}\right ) A \,d^{2}}{32 \sqrt {-a c}\, a^{3}}+\frac {\ln \left (-c x +\sqrt {-a c}\right ) d e B}{16 \sqrt {-a c}\, c \,a^{2}}+\frac {\ln \left (-c x +\sqrt {-a c}\right ) C \,e^{2}}{32 \sqrt {-a c}\, c^{2} a}+\frac {\ln \left (-c x +\sqrt {-a c}\right ) C \,d^{2}}{32 \sqrt {-a c}\, c \,a^{2}}\) | \(494\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.50, size = 324, normalized size = 1.44 \begin {gather*} -\frac {8 \, B a^{3} c d^{2} - 3 \, {\left (2 \, B a c^{3} d e + C a^{2} c^{2} e^{2} + A a c^{3} e^{2} + {\left (C a c^{3} + 5 \, A c^{4}\right )} d^{2}\right )} x^{5} + 4 \, B a^{4} e^{2} - 8 \, {\left (2 \, B a^{2} c^{2} d e - C a^{3} c e^{2} + A a^{2} c^{2} e^{2} + {\left (C a^{2} c^{2} + 5 \, A a c^{3}\right )} d^{2}\right )} x^{3} + 12 \, {\left (2 \, C a^{3} c d e + B a^{3} c e^{2}\right )} x^{2} + 8 \, {\left (C a^{4} e + 2 \, A a^{3} c e\right )} d + 3 \, {\left (2 \, B a^{3} c d e + C a^{4} e^{2} + A a^{3} c e^{2} + {\left (C a^{3} c - 11 \, A a^{2} c^{2}\right )} d^{2}\right )} x}{48 \, {\left (a^{3} c^{5} x^{6} + 3 \, a^{4} c^{4} x^{4} + 3 \, a^{5} c^{3} x^{2} + a^{6} c^{2}\right )}} + \frac {{\left (2 \, B a c d e + C a^{2} e^{2} + A a c e^{2} + {\left (C a c + 5 \, A c^{2}\right )} d^{2}\right )} \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{16 \, \sqrt {a c} a^{3} c^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 524 vs.
\(2 (211) = 422\).
time = 0.41, size = 1069, normalized size = 4.75 \begin {gather*} \left [-\frac {16 \, B a^{4} c^{2} d^{2} - 6 \, {\left (C a^{2} c^{4} + 5 \, A a c^{5}\right )} d^{2} x^{5} - 16 \, {\left (C a^{3} c^{3} + 5 \, A a^{2} c^{4}\right )} d^{2} x^{3} + 6 \, {\left (C a^{4} c^{2} - 11 \, A a^{3} c^{3}\right )} d^{2} x + 3 \, {\left ({\left (C a c^{4} + 5 \, A c^{5}\right )} d^{2} x^{6} + 3 \, {\left (C a^{2} c^{3} + 5 \, A a c^{4}\right )} d^{2} x^{4} + 3 \, {\left (C a^{3} c^{2} + 5 \, A a^{2} c^{3}\right )} d^{2} x^{2} + {\left (C a^{4} c + 5 \, A a^{3} c^{2}\right )} d^{2} + {\left ({\left (C a^{2} c^{3} + A a c^{4}\right )} x^{6} + C a^{5} + A a^{4} c + 3 \, {\left (C a^{3} c^{2} + A a^{2} c^{3}\right )} x^{4} + 3 \, {\left (C a^{4} c + A a^{3} c^{2}\right )} x^{2}\right )} e^{2} + 2 \, {\left (B a c^{4} d x^{6} + 3 \, B a^{2} c^{3} d x^{4} + 3 \, B a^{3} c^{2} d x^{2} + B a^{4} c d\right )} e\right )} \sqrt {-a c} \log \left (\frac {c x^{2} - 2 \, \sqrt {-a c} x - a}{c x^{2} + a}\right ) + 2 \, {\left (12 \, B a^{4} c^{2} x^{2} + 4 \, B a^{5} c - 3 \, {\left (C a^{3} c^{3} + A a^{2} c^{4}\right )} x^{5} + 8 \, {\left (C a^{4} c^{2} - A a^{3} c^{3}\right )} x^{3} + 3 \, {\left (C a^{5} c + A a^{4} c^{2}\right )} x\right )} e^{2} - 4 \, {\left (3 \, B a^{2} c^{4} d x^{5} + 8 \, B a^{3} c^{3} d x^{3} - 12 \, C a^{4} c^{2} d x^{2} - 3 \, B a^{4} c^{2} d x - 4 \, {\left (C a^{5} c + 2 \, A a^{4} c^{2}\right )} d\right )} e}{96 \, {\left (a^{4} c^{6} x^{6} + 3 \, a^{5} c^{5} x^{4} + 3 \, a^{6} c^{4} x^{2} + a^{7} c^{3}\right )}}, -\frac {8 \, B a^{4} c^{2} d^{2} - 3 \, {\left (C a^{2} c^{4} + 5 \, A a c^{5}\right )} d^{2} x^{5} - 8 \, {\left (C a^{3} c^{3} + 5 \, A a^{2} c^{4}\right )} d^{2} x^{3} + 3 \, {\left (C a^{4} c^{2} - 11 \, A a^{3} c^{3}\right )} d^{2} x - 3 \, {\left ({\left (C a c^{4} + 5 \, A c^{5}\right )} d^{2} x^{6} + 3 \, {\left (C a^{2} c^{3} + 5 \, A a c^{4}\right )} d^{2} x^{4} + 3 \, {\left (C a^{3} c^{2} + 5 \, A a^{2} c^{3}\right )} d^{2} x^{2} + {\left (C a^{4} c + 5 \, A a^{3} c^{2}\right )} d^{2} + {\left ({\left (C a^{2} c^{3} + A a c^{4}\right )} x^{6} + C a^{5} + A a^{4} c + 3 \, {\left (C a^{3} c^{2} + A a^{2} c^{3}\right )} x^{4} + 3 \, {\left (C a^{4} c + A a^{3} c^{2}\right )} x^{2}\right )} e^{2} + 2 \, {\left (B a c^{4} d x^{6} + 3 \, B a^{2} c^{3} d x^{4} + 3 \, B a^{3} c^{2} d x^{2} + B a^{4} c d\right )} e\right )} \sqrt {a c} \arctan \left (\frac {\sqrt {a c} x}{a}\right ) + {\left (12 \, B a^{4} c^{2} x^{2} + 4 \, B a^{5} c - 3 \, {\left (C a^{3} c^{3} + A a^{2} c^{4}\right )} x^{5} + 8 \, {\left (C a^{4} c^{2} - A a^{3} c^{3}\right )} x^{3} + 3 \, {\left (C a^{5} c + A a^{4} c^{2}\right )} x\right )} e^{2} - 2 \, {\left (3 \, B a^{2} c^{4} d x^{5} + 8 \, B a^{3} c^{3} d x^{3} - 12 \, C a^{4} c^{2} d x^{2} - 3 \, B a^{4} c^{2} d x - 4 \, {\left (C a^{5} c + 2 \, A a^{4} c^{2}\right )} d\right )} e}{48 \, {\left (a^{4} c^{6} x^{6} + 3 \, a^{5} c^{5} x^{4} + 3 \, a^{6} c^{4} x^{2} + a^{7} c^{3}\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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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.
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Giac [A]
time = 3.01, size = 328, normalized size = 1.46 \begin {gather*} \frac {{\left (C a c d^{2} + 5 \, A c^{2} d^{2} + 2 \, B a c d e + C a^{2} e^{2} + A a c e^{2}\right )} \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{16 \, \sqrt {a c} a^{3} c^{2}} + \frac {3 \, C a c^{3} d^{2} x^{5} + 15 \, A c^{4} d^{2} x^{5} + 6 \, B a c^{3} d x^{5} e + 3 \, C a^{2} c^{2} x^{5} e^{2} + 3 \, A a c^{3} x^{5} e^{2} + 8 \, C a^{2} c^{2} d^{2} x^{3} + 40 \, A a c^{3} d^{2} x^{3} + 16 \, B a^{2} c^{2} d x^{3} e - 8 \, C a^{3} c x^{3} e^{2} + 8 \, A a^{2} c^{2} x^{3} e^{2} - 24 \, C a^{3} c d x^{2} e - 3 \, C a^{3} c d^{2} x + 33 \, A a^{2} c^{2} d^{2} x - 12 \, B a^{3} c x^{2} e^{2} - 6 \, B a^{3} c d x e - 8 \, B a^{3} c d^{2} - 3 \, C a^{4} x e^{2} - 3 \, A a^{3} c x e^{2} - 8 \, C a^{4} d e - 16 \, A a^{3} c d e - 4 \, B a^{4} e^{2}}{48 \, {\left (c x^{2} + a\right )}^{3} a^{3} c^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.23, size = 287, normalized size = 1.28 \begin {gather*} \frac {\mathrm {atan}\left (\frac {\sqrt {c}\,x}{\sqrt {a}}\right )\,\left (C\,a^2\,e^2+C\,a\,c\,d^2+2\,B\,a\,c\,d\,e+A\,a\,c\,e^2+5\,A\,c^2\,d^2\right )}{16\,a^{7/2}\,c^{5/2}}-\frac {\frac {B\,a\,e^2+2\,B\,c\,d^2+4\,A\,c\,d\,e+2\,C\,a\,d\,e}{12\,c^2}-\frac {x^5\,\left (C\,a^2\,e^2+C\,a\,c\,d^2+2\,B\,a\,c\,d\,e+A\,a\,c\,e^2+5\,A\,c^2\,d^2\right )}{16\,a^3}+\frac {x^2\,\left (B\,e^2+2\,C\,d\,e\right )}{4\,c}+\frac {x\,\left (C\,a^2\,e^2+C\,a\,c\,d^2+2\,B\,a\,c\,d\,e+A\,a\,c\,e^2-11\,A\,c^2\,d^2\right )}{16\,a\,c^2}-\frac {x^3\,\left (-C\,a^2\,e^2+C\,a\,c\,d^2+2\,B\,a\,c\,d\,e+A\,a\,c\,e^2+5\,A\,c^2\,d^2\right )}{6\,a^2\,c}}{a^3+3\,a^2\,c\,x^2+3\,a\,c^2\,x^4+c^3\,x^6} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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