Optimal. Leaf size=110 \[ \frac {\tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right ) (a B e+3 A c d)}{8 a^{5/2} c^{3/2}}+\frac {x (a B e+3 A c d)}{8 a^2 c \left (a+c x^2\right )}-\frac {a (A e+B d)-x (A c d-a B e)}{4 a c \left (a+c x^2\right )^2} \]
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Rubi [A] time = 0.04, antiderivative size = 110, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {778, 199, 205} \begin {gather*} \frac {\tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right ) (a B e+3 A c d)}{8 a^{5/2} c^{3/2}}+\frac {x (a B e+3 A c d)}{8 a^2 c \left (a+c x^2\right )}-\frac {a (A e+B d)-x (A c d-a B e)}{4 a c \left (a+c x^2\right )^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 199
Rule 205
Rule 778
Rubi steps
\begin {align*} \int \frac {(A+B x) (d+e x)}{\left (a+c x^2\right )^3} \, dx &=-\frac {a (B d+A e)-(A c d-a B e) x}{4 a c \left (a+c x^2\right )^2}+\frac {(3 A c d+a B e) \int \frac {1}{\left (a+c x^2\right )^2} \, dx}{4 a c}\\ &=-\frac {a (B d+A e)-(A c d-a B e) x}{4 a c \left (a+c x^2\right )^2}+\frac {(3 A c d+a B e) x}{8 a^2 c \left (a+c x^2\right )}+\frac {(3 A c d+a B e) \int \frac {1}{a+c x^2} \, dx}{8 a^2 c}\\ &=-\frac {a (B d+A e)-(A c d-a B e) x}{4 a c \left (a+c x^2\right )^2}+\frac {(3 A c d+a B e) x}{8 a^2 c \left (a+c x^2\right )}+\frac {(3 A c d+a B e) \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{8 a^{5/2} c^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.08, size = 101, normalized size = 0.92 \begin {gather*} \frac {\tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right ) (a B e+3 A c d)}{8 a^{5/2} c^{3/2}}+\frac {-a^2 (2 A e+2 B d+B e x)+a c x \left (5 A d+B e x^2\right )+3 A c^2 d x^3}{8 a^2 c \left (a+c x^2\right )^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)}{\left (a+c x^2\right )^3} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.43, size = 355, normalized size = 3.23 \begin {gather*} \left [-\frac {4 \, B a^{3} c d + 4 \, A a^{3} c e - 2 \, {\left (3 \, A a c^{3} d + B a^{2} c^{2} e\right )} x^{3} + {\left (3 \, A a^{2} c d + B a^{3} e + {\left (3 \, A c^{3} d + B a c^{2} e\right )} x^{4} + 2 \, {\left (3 \, A a c^{2} d + B a^{2} 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 (5 \, A a^{2} c^{2} d - B a^{3} c e\right )} x}{16 \, {\left (a^{3} c^{4} x^{4} + 2 \, a^{4} c^{3} x^{2} + a^{5} c^{2}\right )}}, -\frac {2 \, B a^{3} c d + 2 \, A a^{3} c e - {\left (3 \, A a c^{3} d + B a^{2} c^{2} e\right )} x^{3} - {\left (3 \, A a^{2} c d + B a^{3} e + {\left (3 \, A c^{3} d + B a c^{2} e\right )} x^{4} + 2 \, {\left (3 \, A a c^{2} d + B a^{2} c e\right )} x^{2}\right )} \sqrt {a c} \arctan \left (\frac {\sqrt {a c} x}{a}\right ) - {\left (5 \, A a^{2} c^{2} d - B a^{3} c e\right )} x}{8 \, {\left (a^{3} c^{4} x^{4} + 2 \, a^{4} c^{3} x^{2} + a^{5} 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 = 102, normalized size = 0.93 \begin {gather*} \frac {{\left (3 \, A c d + B a e\right )} \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{8 \, \sqrt {a c} a^{2} c} + \frac {3 \, A c^{2} d x^{3} + B a c x^{3} e + 5 \, A a c d x - B a^{2} x e - 2 \, B a^{2} d - 2 \, A a^{2} e}{8 \, {\left (c x^{2} + a\right )}^{2} a^{2} c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 108, normalized size = 0.98 \begin {gather*} \frac {3 A d \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{8 \sqrt {a c}\, a^{2}}+\frac {B e \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{8 \sqrt {a c}\, a c}+\frac {\frac {\left (3 A c d +a B e \right ) x^{3}}{8 a^{2}}+\frac {\left (5 A c d -a B e \right ) x}{8 a c}-\frac {A e +B d}{4 c}}{\left (c \,x^{2}+a \right )^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.12, size = 114, normalized size = 1.04 \begin {gather*} -\frac {2 \, B a^{2} d + 2 \, A a^{2} e - {\left (3 \, A c^{2} d + B a c e\right )} x^{3} - {\left (5 \, A a c d - B a^{2} e\right )} x}{8 \, {\left (a^{2} c^{3} x^{4} + 2 \, a^{3} c^{2} x^{2} + a^{4} c\right )}} + \frac {{\left (3 \, A c d + B a e\right )} \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{8 \, \sqrt {a c} a^{2} c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.77, size = 100, normalized size = 0.91 \begin {gather*} \frac {\frac {x^3\,\left (3\,A\,c\,d+B\,a\,e\right )}{8\,a^2}-\frac {A\,e+B\,d}{4\,c}+\frac {x\,\left (5\,A\,c\,d-B\,a\,e\right )}{8\,a\,c}}{a^2+2\,a\,c\,x^2+c^2\,x^4}+\frac {\mathrm {atan}\left (\frac {\sqrt {c}\,x}{\sqrt {a}}\right )\,\left (3\,A\,c\,d+B\,a\,e\right )}{8\,a^{5/2}\,c^{3/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.93, size = 180, normalized size = 1.64 \begin {gather*} - \frac {\sqrt {- \frac {1}{a^{5} c^{3}}} \left (3 A c d + B a e\right ) \log {\left (- a^{3} c \sqrt {- \frac {1}{a^{5} c^{3}}} + x \right )}}{16} + \frac {\sqrt {- \frac {1}{a^{5} c^{3}}} \left (3 A c d + B a e\right ) \log {\left (a^{3} c \sqrt {- \frac {1}{a^{5} c^{3}}} + x \right )}}{16} + \frac {- 2 A a^{2} e - 2 B a^{2} d + x^{3} \left (3 A c^{2} d + B a c e\right ) + x \left (5 A a c d - B a^{2} e\right )}{8 a^{4} c + 16 a^{3} c^{2} x^{2} + 8 a^{2} c^{3} x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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