Optimal. Leaf size=165 \[ \frac {\tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right ) (a B e+a C d+5 A c d)}{16 a^{7/2} c^{3/2}}+\frac {x (a B e+a C d+5 A c d)}{16 a^3 c \left (a+c x^2\right )}-\frac {2 a e (a C+2 A c)-c x (a B e+a C d+5 A c d)}{24 a^2 c^2 \left (a+c x^2\right )^2}-\frac {(d+e x) (a B-x (A c-a C))}{6 a c \left (a+c x^2\right )^3} \]
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Rubi [A] time = 0.14, antiderivative size = 165, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {1645, 639, 199, 205} \[ -\frac {2 a e (a C+2 A c)-c x (a B e+a C d+5 A c d)}{24 a^2 c^2 \left (a+c x^2\right )^2}+\frac {\tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right ) (a B e+a C d+5 A c d)}{16 a^{7/2} c^{3/2}}+\frac {x (a B e+a C d+5 A c d)}{16 a^3 c \left (a+c x^2\right )}-\frac {(d+e x) (a B-x (A c-a C))}{6 a c \left (a+c x^2\right )^3} \]
Antiderivative was successfully verified.
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Rule 199
Rule 205
Rule 639
Rule 1645
Rubi steps
\begin {align*} \int \frac {(d+e x) \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)}{6 a c \left (a+c x^2\right )^3}-\frac {\int \frac {-5 A c d-a (C d+B e)-2 (2 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)}{6 a c \left (a+c x^2\right )^3}-\frac {2 a (2 A c+a C) e-c (5 A c d+a C d+a B e) x}{24 a^2 c^2 \left (a+c x^2\right )^2}+\frac {(5 A c d+a C d+a B e) \int \frac {1}{\left (a+c x^2\right )^2} \, dx}{8 a^2 c}\\ &=-\frac {(a B-(A c-a C) x) (d+e x)}{6 a c \left (a+c x^2\right )^3}-\frac {2 a (2 A c+a C) e-c (5 A c d+a C d+a B e) x}{24 a^2 c^2 \left (a+c x^2\right )^2}+\frac {(5 A c d+a C d+a B e) x}{16 a^3 c \left (a+c x^2\right )}+\frac {(5 A c d+a C d+a B e) \int \frac {1}{a+c x^2} \, dx}{16 a^3 c}\\ &=-\frac {(a B-(A c-a C) x) (d+e x)}{6 a c \left (a+c x^2\right )^3}-\frac {2 a (2 A c+a C) e-c (5 A c d+a C d+a B e) x}{24 a^2 c^2 \left (a+c x^2\right )^2}+\frac {(5 A c d+a C d+a B e) x}{16 a^3 c \left (a+c x^2\right )}+\frac {(5 A c d+a C d+a B e) \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{16 a^{7/2} c^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.13, size = 171, normalized size = 1.04 \[ \frac {\frac {8 a^{5/2} \left (a^2 C e-a c (A e+B (d+e x)+C d x)+A c^2 d x\right )}{\left (a+c x^2\right )^3}+\frac {2 a^{3/2} \left (-6 a^2 C e+a c x (B e+C d)+5 A c^2 d x\right )}{\left (a+c x^2\right )^2}+\frac {3 \sqrt {a} c x (a B e+a C d+5 A c d)}{a+c x^2}+3 \sqrt {c} \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right ) (a B e+a C d+5 A c d)}{48 a^{7/2} c^2} \]
Antiderivative was successfully verified.
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fricas [B] time = 1.20, size = 636, normalized size = 3.85 \[ \left [-\frac {24 \, C a^{4} c e x^{2} + 16 \, B a^{4} c d - 6 \, {\left (B a^{2} c^{3} e + {\left (C a^{2} c^{3} + 5 \, A a c^{4}\right )} d\right )} x^{5} - 16 \, {\left (B a^{3} c^{2} e + {\left (C a^{3} c^{2} + 5 \, A a^{2} c^{3}\right )} d\right )} x^{3} + 3 \, {\left ({\left (B a c^{3} e + {\left (C a c^{3} + 5 \, A c^{4}\right )} d\right )} x^{6} + B a^{4} e + 3 \, {\left (B a^{2} c^{2} e + {\left (C a^{2} c^{2} + 5 \, A a c^{3}\right )} d\right )} x^{4} + 3 \, {\left (B a^{3} c e + {\left (C a^{3} c + 5 \, A a^{2} c^{2}\right )} d\right )} x^{2} + {\left (C a^{4} + 5 \, A a^{3} c\right )} d\right )} \sqrt {-a c} \log \left (\frac {c x^{2} - 2 \, \sqrt {-a c} x - a}{c x^{2} + a}\right ) + 8 \, {\left (C a^{5} + 2 \, A a^{4} c\right )} e + 6 \, {\left (B a^{4} c e + {\left (C a^{4} c - 11 \, A a^{3} c^{2}\right )} d\right )} x}{96 \, {\left (a^{4} c^{5} x^{6} + 3 \, a^{5} c^{4} x^{4} + 3 \, a^{6} c^{3} x^{2} + a^{7} c^{2}\right )}}, -\frac {12 \, C a^{4} c e x^{2} + 8 \, B a^{4} c d - 3 \, {\left (B a^{2} c^{3} e + {\left (C a^{2} c^{3} + 5 \, A a c^{4}\right )} d\right )} x^{5} - 8 \, {\left (B a^{3} c^{2} e + {\left (C a^{3} c^{2} + 5 \, A a^{2} c^{3}\right )} d\right )} x^{3} - 3 \, {\left ({\left (B a c^{3} e + {\left (C a c^{3} + 5 \, A c^{4}\right )} d\right )} x^{6} + B a^{4} e + 3 \, {\left (B a^{2} c^{2} e + {\left (C a^{2} c^{2} + 5 \, A a c^{3}\right )} d\right )} x^{4} + 3 \, {\left (B a^{3} c e + {\left (C a^{3} c + 5 \, A a^{2} c^{2}\right )} d\right )} x^{2} + {\left (C a^{4} + 5 \, A a^{3} c\right )} d\right )} \sqrt {a c} \arctan \left (\frac {\sqrt {a c} x}{a}\right ) + 4 \, {\left (C a^{5} + 2 \, A a^{4} c\right )} e + 3 \, {\left (B a^{4} c e + {\left (C a^{4} c - 11 \, A a^{3} c^{2}\right )} d\right )} x}{48 \, {\left (a^{4} c^{5} x^{6} + 3 \, a^{5} c^{4} x^{4} + 3 \, a^{6} c^{3} x^{2} + a^{7} c^{2}\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 194, normalized size = 1.18 \[ \frac {{\left (C a d + 5 \, A c d + B a e\right )} \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{16 \, \sqrt {a c} a^{3} c} + \frac {3 \, C a c^{3} d x^{5} + 15 \, A c^{4} d x^{5} + 3 \, B a c^{3} x^{5} e + 8 \, C a^{2} c^{2} d x^{3} + 40 \, A a c^{3} d x^{3} + 8 \, B a^{2} c^{2} x^{3} e - 12 \, C a^{3} c x^{2} e - 3 \, C a^{3} c d x + 33 \, A a^{2} c^{2} d x - 3 \, B a^{3} c x e - 8 \, B a^{3} c d - 4 \, C a^{4} e - 8 \, A a^{3} c e}{48 \, {\left (c x^{2} + a\right )}^{3} a^{3} c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 182, normalized size = 1.10 \[ \frac {5 A d \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{16 \sqrt {a c}\, a^{3}}+\frac {B e \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{16 \sqrt {a c}\, a^{2} c}+\frac {C d \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{16 \sqrt {a c}\, a^{2} c}+\frac {-\frac {C e \,x^{2}}{4 c}+\frac {\left (5 A c d +B a e +C a d \right ) c \,x^{5}}{16 a^{3}}+\frac {\left (5 A c d +B a e +C a d \right ) x^{3}}{6 a^{2}}+\frac {\left (11 A c d -B a e -C a d \right ) x}{16 a c}-\frac {2 A c e +2 B c d +a C e}{12 c^{2}}}{\left (c \,x^{2}+a \right )^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.98, size = 208, normalized size = 1.26 \[ -\frac {12 \, C a^{3} c e x^{2} + 8 \, B a^{3} c d - 3 \, {\left (B a c^{3} e + {\left (C a c^{3} + 5 \, A c^{4}\right )} d\right )} x^{5} - 8 \, {\left (B a^{2} c^{2} e + {\left (C a^{2} c^{2} + 5 \, A a c^{3}\right )} d\right )} x^{3} + 4 \, {\left (C a^{4} + 2 \, A a^{3} c\right )} e + 3 \, {\left (B a^{3} c e + {\left (C a^{3} c - 11 \, A a^{2} c^{2}\right )} d\right )} x}{48 \, {\left (a^{3} c^{5} x^{6} + 3 \, a^{4} c^{4} x^{4} + 3 \, a^{5} c^{3} x^{2} + a^{6} c^{2}\right )}} + \frac {{\left (B a e + {\left (C a + 5 \, A c\right )} d\right )} \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{16 \, \sqrt {a c} a^{3} c} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.94, size = 164, normalized size = 0.99 \[ \frac {\mathrm {atan}\left (\frac {\sqrt {c}\,x}{\sqrt {a}}\right )\,\left (5\,A\,c\,d+B\,a\,e+C\,a\,d\right )}{16\,a^{7/2}\,c^{3/2}}-\frac {\frac {2\,A\,c\,e+2\,B\,c\,d+C\,a\,e}{12\,c^2}-\frac {x^3\,\left (5\,A\,c\,d+B\,a\,e+C\,a\,d\right )}{6\,a^2}+\frac {C\,e\,x^2}{4\,c}+\frac {x\,\left (B\,a\,e-11\,A\,c\,d+C\,a\,d\right )}{16\,a\,c}-\frac {c\,x^5\,\left (5\,A\,c\,d+B\,a\,e+C\,a\,d\right )}{16\,a^3}}{a^3+3\,a^2\,c\,x^2+3\,a\,c^2\,x^4+c^3\,x^6} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 139.97, size = 298, normalized size = 1.81 \[ - \frac {\sqrt {- \frac {1}{a^{7} c^{3}}} \left (5 A c d + B a e + C a d\right ) \log {\left (- a^{4} c \sqrt {- \frac {1}{a^{7} c^{3}}} + x \right )}}{32} + \frac {\sqrt {- \frac {1}{a^{7} c^{3}}} \left (5 A c d + B a e + C a d\right ) \log {\left (a^{4} c \sqrt {- \frac {1}{a^{7} c^{3}}} + x \right )}}{32} + \frac {- 8 A a^{3} c e - 8 B a^{3} c d - 4 C a^{4} e - 12 C a^{3} c e x^{2} + x^{5} \left (15 A c^{4} d + 3 B a c^{3} e + 3 C a c^{3} d\right ) + x^{3} \left (40 A a c^{3} d + 8 B a^{2} c^{2} e + 8 C a^{2} c^{2} d\right ) + x \left (33 A a^{2} c^{2} d - 3 B a^{3} c e - 3 C a^{3} c d\right )}{48 a^{6} c^{2} + 144 a^{5} c^{3} x^{2} + 144 a^{4} c^{4} x^{4} + 48 a^{3} c^{5} x^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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