Optimal. Leaf size=79 \[ \frac {\tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right ) (a B e+A c d)}{2 a^{3/2} c^{3/2}}-\frac {a (A e+B d)-x (A c d-a B e)}{2 a c \left (a+c x^2\right )} \]
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Rubi [A] time = 0.03, antiderivative size = 79, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {778, 205} \begin {gather*} \frac {\tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right ) (a B e+A c d)}{2 a^{3/2} c^{3/2}}-\frac {a (A e+B d)-x (A c d-a B e)}{2 a c \left (a+c x^2\right )} \end {gather*}
Antiderivative was successfully verified.
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Rule 205
Rule 778
Rubi steps
\begin {align*} \int \frac {(A+B x) (d+e x)}{\left (a+c x^2\right )^2} \, dx &=-\frac {a (B d+A e)-(A c d-a B e) x}{2 a c \left (a+c x^2\right )}+\frac {(A c d+a B e) \int \frac {1}{a+c x^2} \, dx}{2 a c}\\ &=-\frac {a (B d+A e)-(A c d-a B e) x}{2 a c \left (a+c x^2\right )}+\frac {(A c d+a B e) \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{2 a^{3/2} c^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 78, normalized size = 0.99 \begin {gather*} \frac {\tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right ) (a B e+A c d)}{2 a^{3/2} c^{3/2}}+\frac {-a A e-a B d-a B e x+A c d x}{2 a c \left (a+c x^2\right )} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(A+B x) (d+e x)}{\left (a+c x^2\right )^2} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.41, size = 225, normalized size = 2.85 \begin {gather*} \left [-\frac {2 \, B a^{2} c d + 2 \, A a^{2} c e + {\left (A a c d + B a^{2} e + {\left (A c^{2} d + B a c e\right )} x^{2}\right )} \sqrt {-a c} \log \left (\frac {c x^{2} - 2 \, \sqrt {-a c} x - a}{c x^{2} + a}\right ) - 2 \, {\left (A a c^{2} d - B a^{2} c e\right )} x}{4 \, {\left (a^{2} c^{3} x^{2} + a^{3} c^{2}\right )}}, -\frac {B a^{2} c d + A a^{2} c e - {\left (A a c d + B a^{2} e + {\left (A c^{2} d + B a c e\right )} x^{2}\right )} \sqrt {a c} \arctan \left (\frac {\sqrt {a c} x}{a}\right ) - {\left (A a c^{2} d - B a^{2} c e\right )} x}{2 \, {\left (a^{2} c^{3} x^{2} + a^{3} c^{2}\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 74, normalized size = 0.94 \begin {gather*} \frac {{\left (A c d + B a e\right )} \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{2 \, \sqrt {a c} a c} + \frac {A c d x - B a x e - B a d - A a e}{2 \, {\left (c x^{2} + a\right )} a c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 86, normalized size = 1.09 \begin {gather*} \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 {\frac {\left (A c d -a B e \right ) x}{2 a c}-\frac {A e +B d}{2 c}}{c \,x^{2}+a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.20, size = 72, normalized size = 0.91 \begin {gather*} -\frac {B a d + A a e - {\left (A c d - B a e\right )} x}{2 \, {\left (a c^{2} x^{2} + a^{2} c\right )}} + \frac {{\left (A c d + B a e\right )} \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{2 \, \sqrt {a c} a c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.77, size = 70, normalized size = 0.89 \begin {gather*} \frac {\mathrm {atan}\left (\frac {\sqrt {c}\,x}{\sqrt {a}}\right )\,\left (A\,c\,d+B\,a\,e\right )}{2\,a^{3/2}\,c^{3/2}}-\frac {\frac {A\,e+B\,d}{2\,c}-\frac {x\,\left (A\,c\,d-B\,a\,e\right )}{2\,a\,c}}{c\,x^2+a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.92, size = 133, normalized size = 1.68 \begin {gather*} - \frac {\sqrt {- \frac {1}{a^{3} c^{3}}} \left (A c d + B a e\right ) \log {\left (- a^{2} c \sqrt {- \frac {1}{a^{3} c^{3}}} + x \right )}}{4} + \frac {\sqrt {- \frac {1}{a^{3} c^{3}}} \left (A c d + B a e\right ) \log {\left (a^{2} c \sqrt {- \frac {1}{a^{3} c^{3}}} + x \right )}}{4} + \frac {- A a e - B a d + x \left (A c d - B a e\right )}{2 a^{2} c + 2 a c^{2} x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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