Optimal. Leaf size=142 \[ \frac {\tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right ) \left (a A e^2+2 a B d e+3 A c d^2\right )}{8 a^{5/2} c^{3/2}}-\frac {2 a e (a B e+2 A c d)-c x \left (-a A e^2+2 a B d e+3 A c d^2\right )}{8 a^2 c^2 \left (a+c x^2\right )}-\frac {(d+e x)^2 (a B-A c x)}{4 a c \left (a+c x^2\right )^2} \]
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Rubi [A] time = 0.13, antiderivative size = 142, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {821, 778, 205} \begin {gather*} -\frac {2 a e (a B e+2 A c d)-c x \left (-a A e^2+2 a B d e+3 A c d^2\right )}{8 a^2 c^2 \left (a+c x^2\right )}+\frac {\tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right ) \left (a A e^2+2 a B d e+3 A c d^2\right )}{8 a^{5/2} c^{3/2}}-\frac {(d+e x)^2 (a B-A c x)}{4 a c \left (a+c x^2\right )^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 205
Rule 778
Rule 821
Rubi steps
\begin {align*} \int \frac {(A+B x) (d+e x)^2}{\left (a+c x^2\right )^3} \, dx &=-\frac {(a B-A c x) (d+e x)^2}{4 a c \left (a+c x^2\right )^2}+\frac {\int \frac {(d+e x) (3 A c d+2 a B e+A c e x)}{\left (a+c x^2\right )^2} \, dx}{4 a c}\\ &=-\frac {(a B-A c x) (d+e x)^2}{4 a c \left (a+c x^2\right )^2}-\frac {2 a e (2 A c d+a B e)-c \left (3 A c d^2+2 a B d e-a A e^2\right ) x}{8 a^2 c^2 \left (a+c x^2\right )}+\frac {\left (3 A c d^2+2 a B d e+a A e^2\right ) \int \frac {1}{a+c x^2} \, dx}{8 a^2 c}\\ &=-\frac {(a B-A c x) (d+e x)^2}{4 a c \left (a+c x^2\right )^2}-\frac {2 a e (2 A c d+a B e)-c \left (3 A c d^2+2 a B d e-a A e^2\right ) x}{8 a^2 c^2 \left (a+c x^2\right )}+\frac {\left (3 A c d^2+2 a B d e+a A e^2\right ) \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 = 0.11, size = 158, normalized size = 1.11 \begin {gather*} \frac {\frac {\sqrt {a} \left (-4 a^2 B e^2+a c e x (A e+2 B d)+3 A c^2 d^2 x\right )}{a+c x^2}+\frac {2 a^{3/2} \left (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\right )}{\left (a+c x^2\right )^2}+\sqrt {c} \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right ) \left (a A e^2+2 a B d e+3 A c d^2\right )}{8 a^{5/2} 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 )^3} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [B] time = 0.43, size = 537, normalized size = 3.78 \begin {gather*} \left [-\frac {8 \, B a^{3} c e^{2} x^{2} + 4 \, B a^{3} c d^{2} + 8 \, A a^{3} c d e + 4 \, B a^{4} e^{2} - 2 \, {\left (3 \, A a c^{3} d^{2} + 2 \, B a^{2} c^{2} d e + A a^{2} c^{2} e^{2}\right )} x^{3} + {\left (3 \, A a^{2} c d^{2} + 2 \, B a^{3} d e + A a^{3} e^{2} + {\left (3 \, A c^{3} d^{2} + 2 \, B a c^{2} d e + A a c^{2} e^{2}\right )} x^{4} + 2 \, {\left (3 \, A a c^{2} d^{2} + 2 \, B a^{2} c d e + A a^{2} 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 (5 \, A a^{2} c^{2} d^{2} - 2 \, B a^{3} c d e - A a^{3} c e^{2}\right )} x}{16 \, {\left (a^{3} c^{4} x^{4} + 2 \, a^{4} c^{3} x^{2} + a^{5} c^{2}\right )}}, -\frac {4 \, B a^{3} c e^{2} x^{2} + 2 \, B a^{3} c d^{2} + 4 \, A a^{3} c d e + 2 \, B a^{4} e^{2} - {\left (3 \, A a c^{3} d^{2} + 2 \, B a^{2} c^{2} d e + A a^{2} c^{2} e^{2}\right )} x^{3} - {\left (3 \, A a^{2} c d^{2} + 2 \, B a^{3} d e + A a^{3} e^{2} + {\left (3 \, A c^{3} d^{2} + 2 \, B a c^{2} d e + A a c^{2} e^{2}\right )} x^{4} + 2 \, {\left (3 \, A a c^{2} d^{2} + 2 \, B a^{2} c d e + A a^{2} c e^{2}\right )} x^{2}\right )} \sqrt {a c} \arctan \left (\frac {\sqrt {a c} x}{a}\right ) - {\left (5 \, A a^{2} c^{2} d^{2} - 2 \, B a^{3} c d e - A a^{3} c e^{2}\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 = 169, normalized size = 1.19 \begin {gather*} \frac {{\left (3 \, A c d^{2} + 2 \, B a d e + A a e^{2}\right )} \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{8 \, \sqrt {a c} a^{2} c} + \frac {3 \, A c^{3} d^{2} x^{3} + 2 \, B a c^{2} d x^{3} e + A a c^{2} x^{3} e^{2} + 5 \, A a c^{2} d^{2} x - 4 \, B a^{2} c x^{2} e^{2} - 2 \, B a^{2} c d x e - 2 \, B a^{2} c d^{2} - A a^{2} c x e^{2} - 4 \, A a^{2} c d e - 2 \, B a^{3} e^{2}}{8 \, {\left (c x^{2} + a\right )}^{2} a^{2} c^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 180, normalized size = 1.27 \begin {gather*} \frac {A \,e^{2} \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{8 \sqrt {a c}\, a c}+\frac {3 A \,d^{2} \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{8 \sqrt {a c}\, a^{2}}+\frac {B d e \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{4 \sqrt {a c}\, a c}+\frac {-\frac {B \,e^{2} x^{2}}{2 c}+\frac {\left (A a \,e^{2}+3 A c \,d^{2}+2 a B d e \right ) x^{3}}{8 a^{2}}-\frac {\left (A a \,e^{2}-5 A c \,d^{2}+2 a B d e \right ) x}{8 a c}-\frac {2 A c d e +B a \,e^{2}+B c \,d^{2}}{4 c^{2}}}{\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 = 1.18, size = 184, normalized size = 1.30 \begin {gather*} -\frac {4 \, B a^{2} c e^{2} x^{2} + 2 \, B a^{2} c d^{2} + 4 \, A a^{2} c d e + 2 \, B a^{3} e^{2} - {\left (3 \, A c^{3} d^{2} + 2 \, B a c^{2} d e + A a c^{2} e^{2}\right )} x^{3} - {\left (5 \, A a c^{2} d^{2} - 2 \, B a^{2} c d e - A a^{2} c e^{2}\right )} x}{8 \, {\left (a^{2} c^{4} x^{4} + 2 \, a^{3} c^{3} x^{2} + a^{4} c^{2}\right )}} + \frac {{\left (3 \, A c d^{2} + 2 \, B a d e + A a e^{2}\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 = 1.81, size = 154, normalized size = 1.08 \begin {gather*} \frac {\mathrm {atan}\left (\frac {\sqrt {c}\,x}{\sqrt {a}}\right )\,\left (3\,A\,c\,d^2+2\,B\,a\,d\,e+A\,a\,e^2\right )}{8\,a^{5/2}\,c^{3/2}}-\frac {\frac {B\,c\,d^2+2\,A\,c\,d\,e+B\,a\,e^2}{4\,c^2}-\frac {x^3\,\left (3\,A\,c\,d^2+2\,B\,a\,d\,e+A\,a\,e^2\right )}{8\,a^2}+\frac {x\,\left (-5\,A\,c\,d^2+2\,B\,a\,d\,e+A\,a\,e^2\right )}{8\,a\,c}+\frac {B\,e^2\,x^2}{2\,c}}{a^2+2\,a\,c\,x^2+c^2\,x^4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 7.84, size = 274, normalized size = 1.93 \begin {gather*} - \frac {\sqrt {- \frac {1}{a^{5} c^{3}}} \left (A a e^{2} + 3 A c d^{2} + 2 B a d 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 (A a e^{2} + 3 A c d^{2} + 2 B a d e\right ) \log {\left (a^{3} c \sqrt {- \frac {1}{a^{5} c^{3}}} + x \right )}}{16} + \frac {- 4 A a^{2} c d e - 2 B a^{3} e^{2} - 2 B a^{2} c d^{2} - 4 B a^{2} c e^{2} x^{2} + x^{3} \left (A a c^{2} e^{2} + 3 A c^{3} d^{2} + 2 B a c^{2} d e\right ) + x \left (- A a^{2} c e^{2} + 5 A a c^{2} d^{2} - 2 B a^{2} c d e\right )}{8 a^{4} c^{2} + 16 a^{3} c^{3} x^{2} + 8 a^{2} c^{4} x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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