Optimal. Leaf size=112 \[ \frac {\tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right ) \left (a e (A e+2 B d)+A c d^2\right )}{2 a^{3/2} c^{3/2}}-\frac {(d+e x) (a (A e+B d)-x (A c d-a B e))}{2 a c \left (a+c x^2\right )}+\frac {B e^2 \log \left (a+c x^2\right )}{2 c^2} \]
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Rubi [A] time = 0.07, antiderivative size = 112, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {819, 635, 205, 260} \begin {gather*} \frac {\tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right ) \left (a e (A e+2 B d)+A c d^2\right )}{2 a^{3/2} c^{3/2}}-\frac {(d+e x) (a (A e+B d)-x (A c d-a B e))}{2 a c \left (a+c x^2\right )}+\frac {B e^2 \log \left (a+c x^2\right )}{2 c^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 205
Rule 260
Rule 635
Rule 819
Rubi steps
\begin {align*} \int \frac {(A+B x) (d+e x)^2}{\left (a+c x^2\right )^2} \, dx &=-\frac {(d+e x) (a (B d+A e)-(A c d-a B e) x)}{2 a c \left (a+c x^2\right )}+\frac {\int \frac {A c d^2+a e (2 B d+A e)+2 a B e^2 x}{a+c x^2} \, dx}{2 a c}\\ &=-\frac {(d+e x) (a (B d+A e)-(A c d-a B e) x)}{2 a c \left (a+c x^2\right )}+\frac {\left (B e^2\right ) \int \frac {x}{a+c x^2} \, dx}{c}+\frac {\left (A c d^2+a e (2 B d+A e)\right ) \int \frac {1}{a+c x^2} \, dx}{2 a c}\\ &=-\frac {(d+e x) (a (B d+A e)-(A c d-a B e) x)}{2 a c \left (a+c x^2\right )}+\frac {\left (A c d^2+a e (2 B d+A e)\right ) \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{2 a^{3/2} c^{3/2}}+\frac {B e^2 \log \left (a+c x^2\right )}{2 c^2}\\ \end {align*}
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Mathematica [A] time = 0.10, size = 119, normalized size = 1.06 \begin {gather*} \frac {\frac {\sqrt {c} \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right ) \left (a A e^2+2 a B d e+A c d^2\right )}{a^{3/2}}+\frac {a^2 B e^2-a c (A e (2 d+e x)+B d (d+2 e x))+A c^2 d^2 x}{a \left (a+c x^2\right )}+B e^2 \log \left (a+c x^2\right )}{2 c^2} \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)^2}{\left (a+c x^2\right )^2} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.42, size = 384, normalized size = 3.43 \begin {gather*} \left [-\frac {2 \, B a^{2} c d^{2} + 4 \, A a^{2} c d e - 2 \, B a^{3} e^{2} + {\left (A a c d^{2} + 2 \, B a^{2} d e + A a^{2} e^{2} + {\left (A c^{2} d^{2} + 2 \, B a c d e + A a c e^{2}\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^{2} - 2 \, B a^{2} c d e - A a^{2} c e^{2}\right )} x - 2 \, {\left (B a^{2} c e^{2} x^{2} + B a^{3} e^{2}\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^{2} + 2 \, A a^{2} c d e - B a^{3} e^{2} - {\left (A a c d^{2} + 2 \, B a^{2} d e + A a^{2} e^{2} + {\left (A c^{2} d^{2} + 2 \, B a c d e + A a c e^{2}\right )} x^{2}\right )} \sqrt {a c} \arctan \left (\frac {\sqrt {a c} x}{a}\right ) - {\left (A a c^{2} d^{2} - 2 \, B a^{2} c d e - A a^{2} c e^{2}\right )} x - {\left (B a^{2} c e^{2} x^{2} + B a^{3} e^{2}\right )} \log \left (c x^{2} + a\right )}{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 = 0.16, size = 127, normalized size = 1.13 \begin {gather*} \frac {B e^{2} \log \left (c x^{2} + a\right )}{2 \, c^{2}} + \frac {{\left (A c d^{2} + 2 \, B a d e + A a e^{2}\right )} \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{2 \, \sqrt {a c} a c} + \frac {{\left (A c d^{2} - 2 \, B a d e - A a e^{2}\right )} x - \frac {B a c d^{2} + 2 \, A a c d e - B a^{2} e^{2}}{c}}{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 = 151, normalized size = 1.35 \begin {gather*} \frac {A \,d^{2} \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{2 \sqrt {a c}\, a}+\frac {A \,e^{2} \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{2 \sqrt {a c}\, c}+\frac {B d e \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{\sqrt {a c}\, c}+\frac {B \,e^{2} \ln \left (c \,x^{2}+a \right )}{2 c^{2}}+\frac {-\frac {\left (A a \,e^{2}-A c \,d^{2}+2 a B d e \right ) x}{2 a c}-\frac {2 A c d e -B a \,e^{2}+B c \,d^{2}}{2 c^{2}}}{c \,x^{2}+a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.28, size = 130, normalized size = 1.16 \begin {gather*} \frac {B e^{2} \log \left (c x^{2} + a\right )}{2 \, c^{2}} - \frac {B a c d^{2} + 2 \, A a c d e - B a^{2} e^{2} - {\left (A c^{2} d^{2} - 2 \, B a c d e - A a c e^{2}\right )} x}{2 \, {\left (a c^{3} x^{2} + a^{2} c^{2}\right )}} + \frac {{\left (A c d^{2} + 2 \, B a d e + A a e^{2}\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 = 1.80, size = 203, normalized size = 1.81 \begin {gather*} \frac {B\,a\,e^2}{2\,\left (c^3\,x^2+a\,c^2\right )}-\frac {B\,d^2}{2\,\left (c^2\,x^2+a\,c\right )}-\frac {A\,d\,e}{c^2\,x^2+a\,c}+\frac {A\,d^2\,x}{2\,\left (a^2+c\,a\,x^2\right )}-\frac {A\,e^2\,x}{2\,\left (c^2\,x^2+a\,c\right )}+\frac {B\,e^2\,\ln \left (c\,x^2+a\right )}{2\,c^2}+\frac {A\,d^2\,\mathrm {atan}\left (\frac {\sqrt {c}\,x}{\sqrt {a}}\right )}{2\,a^{3/2}\,\sqrt {c}}+\frac {A\,e^2\,\mathrm {atan}\left (\frac {\sqrt {c}\,x}{\sqrt {a}}\right )}{2\,\sqrt {a}\,c^{3/2}}-\frac {B\,d\,e\,x}{c^2\,x^2+a\,c}+\frac {B\,d\,e\,\mathrm {atan}\left (\frac {\sqrt {c}\,x}{\sqrt {a}}\right )}{\sqrt {a}\,c^{3/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 2.96, size = 382, normalized size = 3.41 \begin {gather*} \left (\frac {B e^{2}}{2 c^{2}} - \frac {\sqrt {- a^{3} c^{5}} \left (A a e^{2} + A c d^{2} + 2 B a d e\right )}{4 a^{3} c^{4}}\right ) \log {\left (x + \frac {- 2 B a^{2} e^{2} + 4 a^{2} c^{2} \left (\frac {B e^{2}}{2 c^{2}} - \frac {\sqrt {- a^{3} c^{5}} \left (A a e^{2} + A c d^{2} + 2 B a d e\right )}{4 a^{3} c^{4}}\right )}{A a c e^{2} + A c^{2} d^{2} + 2 B a c d e} \right )} + \left (\frac {B e^{2}}{2 c^{2}} + \frac {\sqrt {- a^{3} c^{5}} \left (A a e^{2} + A c d^{2} + 2 B a d e\right )}{4 a^{3} c^{4}}\right ) \log {\left (x + \frac {- 2 B a^{2} e^{2} + 4 a^{2} c^{2} \left (\frac {B e^{2}}{2 c^{2}} + \frac {\sqrt {- a^{3} c^{5}} \left (A a e^{2} + A c d^{2} + 2 B a d e\right )}{4 a^{3} c^{4}}\right )}{A a c e^{2} + A c^{2} d^{2} + 2 B a c d e} \right )} + \frac {- 2 A a c d e + B a^{2} e^{2} - B a c d^{2} + x \left (- A a c e^{2} + A c^{2} d^{2} - 2 B a c d e\right )}{2 a^{2} c^{2} + 2 a c^{3} x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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