Optimal. Leaf size=97 \[ \frac {\tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right ) (a B e+a C d+A c d)}{2 a^{3/2} c^{3/2}}-\frac {(d+e x) (a B-x (A c-a C))}{2 a c \left (a+c x^2\right )}+\frac {C e \log \left (a+c x^2\right )}{2 c^2} \]
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Rubi [A] time = 0.08, antiderivative size = 97, 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, 635, 205, 260} \[ \frac {\tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right ) (a B e+a C d+A c d)}{2 a^{3/2} c^{3/2}}-\frac {(d+e x) (a B-x (A c-a C))}{2 a c \left (a+c x^2\right )}+\frac {C e \log \left (a+c x^2\right )}{2 c^2} \]
Antiderivative was successfully verified.
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Rule 205
Rule 260
Rule 635
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 )^2} \, dx &=-\frac {(a B-(A c-a C) x) (d+e x)}{2 a c \left (a+c x^2\right )}-\frac {\int \frac {-A c d-a (C d+B e)-2 a C e x}{a+c x^2} \, dx}{2 a c}\\ &=-\frac {(a B-(A c-a C) x) (d+e x)}{2 a c \left (a+c x^2\right )}+\frac {(C e) \int \frac {x}{a+c x^2} \, dx}{c}+\frac {(A c d+a C d+a B e) \int \frac {1}{a+c x^2} \, dx}{2 a c}\\ &=-\frac {(a B-(A c-a C) x) (d+e x)}{2 a c \left (a+c x^2\right )}+\frac {(A c d+a C d+a B e) \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{2 a^{3/2} c^{3/2}}+\frac {C e \log \left (a+c x^2\right )}{2 c^2}\\ \end {align*}
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Mathematica [A] time = 0.09, size = 102, normalized size = 1.05 \[ \frac {\frac {\sqrt {c} \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right ) (a B e+a C d+A c d)}{a^{3/2}}+\frac {a^2 C e-a c (A e+B (d+e x)+C d x)+A c^2 d x}{a \left (a+c x^2\right )}+C e \log \left (a+c x^2\right )}{2 c^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.97, size = 337, normalized size = 3.47 \[ \left [-\frac {2 \, B a^{2} c d + {\left (B a^{2} e + {\left (B a c e + {\left (C a c + A c^{2}\right )} d\right )} x^{2} + {\left (C a^{2} + A a c\right )} d\right )} \sqrt {-a c} \log \left (\frac {c x^{2} - 2 \, \sqrt {-a c} x - a}{c x^{2} + a}\right ) - 2 \, {\left (C a^{3} - A a^{2} c\right )} e + 2 \, {\left (B a^{2} c e + {\left (C a^{2} c - A a c^{2}\right )} d\right )} x - 2 \, {\left (C a^{2} c e x^{2} + C a^{3} e\right )} \log \left (c x^{2} + a\right )}{4 \, {\left (a^{2} c^{3} x^{2} + a^{3} c^{2}\right )}}, -\frac {B a^{2} c d - {\left (B a^{2} e + {\left (B a c e + {\left (C a c + A c^{2}\right )} d\right )} x^{2} + {\left (C a^{2} + A a c\right )} d\right )} \sqrt {a c} \arctan \left (\frac {\sqrt {a c} x}{a}\right ) - {\left (C a^{3} - A a^{2} c\right )} e + {\left (B a^{2} c e + {\left (C a^{2} c - A a c^{2}\right )} d\right )} x - {\left (C a^{2} c e x^{2} + C a^{3} e\right )} \log \left (c x^{2} + a\right )}{2 \, {\left (a^{2} c^{3} x^{2} + a^{3} c^{2}\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.22, size = 112, normalized size = 1.15 \[ \frac {C e \log \left (c x^{2} + a\right )}{2 \, c^{2}} + \frac {{\left (C a d + A c d + B a e\right )} \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{2 \, \sqrt {a c} a c} - \frac {{\left (C a d - A c d + B a e\right )} x + \frac {B a c d - C a^{2} e + A a c e}{c}}{2 \, {\left (c x^{2} + a\right )} a c} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 134, normalized size = 1.38 \[ \frac {A d \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{2 \sqrt {a c}\, a}+\frac {B e \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{2 \sqrt {a c}\, c}+\frac {C d \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{2 \sqrt {a c}\, c}+\frac {C e \ln \left (c \,x^{2}+a \right )}{2 c^{2}}+\frac {\frac {\left (A c d -B a e -C a d \right ) x}{2 a c}-\frac {A c e +B c d -a C e}{2 c^{2}}}{c \,x^{2}+a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.97, size = 113, normalized size = 1.16 \[ \frac {C e \log \left (c x^{2} + a\right )}{2 \, c^{2}} - \frac {B a c d - {\left (C a^{2} - A a c\right )} e + {\left (B a c e + {\left (C a c - A c^{2}\right )} d\right )} x}{2 \, {\left (a c^{3} x^{2} + a^{2} c^{2}\right )}} + \frac {{\left (B a e + {\left (C a + A c\right )} d\right )} \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{2 \, \sqrt {a c} a c} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.14, size = 191, normalized size = 1.97 \[ \frac {C\,e\,\ln \left (c\,x^2+a\right )}{2\,c^2}-\frac {B\,d}{2\,\left (c^2\,x^2+a\,c\right )}-\frac {B\,e\,x}{2\,\left (c^2\,x^2+a\,c\right )}-\frac {C\,d\,x}{2\,\left (c^2\,x^2+a\,c\right )}-\frac {A\,e}{2\,\left (c^2\,x^2+a\,c\right )}+\frac {C\,a\,e}{2\,\left (c^3\,x^2+a\,c^2\right )}+\frac {A\,d\,x}{2\,\left (a^2+c\,a\,x^2\right )}+\frac {A\,d\,\mathrm {atan}\left (\frac {\sqrt {c}\,x}{\sqrt {a}}\right )}{2\,a^{3/2}\,\sqrt {c}}+\frac {B\,e\,\mathrm {atan}\left (\frac {\sqrt {c}\,x}{\sqrt {a}}\right )}{2\,\sqrt {a}\,c^{3/2}}+\frac {C\,d\,\mathrm {atan}\left (\frac {\sqrt {c}\,x}{\sqrt {a}}\right )}{2\,\sqrt {a}\,c^{3/2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 6.41, size = 318, normalized size = 3.28 \[ \left (\frac {C e}{2 c^{2}} - \frac {\sqrt {- a^{3} c^{5}} \left (A c d + B a e + C a d\right )}{4 a^{3} c^{4}}\right ) \log {\left (x + \frac {- 2 C a^{2} e + 4 a^{2} c^{2} \left (\frac {C e}{2 c^{2}} - \frac {\sqrt {- a^{3} c^{5}} \left (A c d + B a e + C a d\right )}{4 a^{3} c^{4}}\right )}{A c^{2} d + B a c e + C a c d} \right )} + \left (\frac {C e}{2 c^{2}} + \frac {\sqrt {- a^{3} c^{5}} \left (A c d + B a e + C a d\right )}{4 a^{3} c^{4}}\right ) \log {\left (x + \frac {- 2 C a^{2} e + 4 a^{2} c^{2} \left (\frac {C e}{2 c^{2}} + \frac {\sqrt {- a^{3} c^{5}} \left (A c d + B a e + C a d\right )}{4 a^{3} c^{4}}\right )}{A c^{2} d + B a c e + C a c d} \right )} + \frac {- A a c e - B a c d + C a^{2} e + x \left (A c^{2} d - B a c e - C a c d\right )}{2 a^{2} c^{2} + 2 a c^{3} x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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