Optimal. Leaf size=361 \[ \frac {(d+e x)^{m+1} \left (a e m (a B e+A c d)-\sqrt {-a} \sqrt {c} \left (A \left (a e^2 (1-m)+c d^2\right )+a B d e m\right )\right ) \, _2F_1\left (1,m+1;m+2;\frac {\sqrt {c} (d+e x)}{\sqrt {c} d-\sqrt {-a} e}\right )}{4 a^2 \sqrt {c} (m+1) \left (\sqrt {c} d-\sqrt {-a} e\right ) \left (a e^2+c d^2\right )}+\frac {(d+e x)^{m+1} \left (\sqrt {-a} \sqrt {c} \left (A \left (a e^2 (1-m)+c d^2\right )+a B d e m\right )+a e m (a B e+A c d)\right ) \, _2F_1\left (1,m+1;m+2;\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {-a} e}\right )}{4 a^2 \sqrt {c} (m+1) \left (\sqrt {-a} e+\sqrt {c} d\right ) \left (a e^2+c d^2\right )}-\frac {(d+e x)^{m+1} (a (B d-A e)-x (a B e+A c d))}{2 a \left (a+c x^2\right ) \left (a e^2+c d^2\right )} \]
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Rubi [A] time = 0.58, antiderivative size = 359, normalized size of antiderivative = 0.99, number of steps used = 5, number of rules used = 3, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {823, 831, 68} \[ \frac {(d+e x)^{m+1} \left (a e m (a B e+A c d)-\sqrt {-a} \sqrt {c} \left (a A e^2 (1-m)+a B d e m+A c d^2\right )\right ) \, _2F_1\left (1,m+1;m+2;\frac {\sqrt {c} (d+e x)}{\sqrt {c} d-\sqrt {-a} e}\right )}{4 a^2 \sqrt {c} (m+1) \left (\sqrt {c} d-\sqrt {-a} e\right ) \left (a e^2+c d^2\right )}+\frac {(d+e x)^{m+1} \left (\sqrt {-a} \sqrt {c} \left (a A e^2 (1-m)+a B d e m+A c d^2\right )+a e m (a B e+A c d)\right ) \, _2F_1\left (1,m+1;m+2;\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {-a} e}\right )}{4 a^2 \sqrt {c} (m+1) \left (\sqrt {-a} e+\sqrt {c} d\right ) \left (a e^2+c d^2\right )}-\frac {(d+e x)^{m+1} (a (B d-A e)-x (a B e+A c d))}{2 a \left (a+c x^2\right ) \left (a e^2+c d^2\right )} \]
Antiderivative was successfully verified.
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Rule 68
Rule 823
Rule 831
Rubi steps
\begin {align*} \int \frac {(A+B x) (d+e x)^m}{\left (a+c x^2\right )^2} \, dx &=-\frac {(d+e x)^{1+m} (a (B d-A e)-(A c d+a B e) x)}{2 a \left (c d^2+a e^2\right ) \left (a+c x^2\right )}-\frac {\int \frac {(d+e x)^m \left (-c \left (A c d^2+a A e^2 (1-m)+a B d e m\right )+c e (A c d+a B e) m x\right )}{a+c x^2} \, dx}{2 a c \left (c d^2+a e^2\right )}\\ &=-\frac {(d+e x)^{1+m} (a (B d-A e)-(A c d+a B e) x)}{2 a \left (c d^2+a e^2\right ) \left (a+c x^2\right )}-\frac {\int \left (\frac {\left (-a \sqrt {c} e (A c d+a B e) m-\sqrt {-a} c \left (A c d^2+a A e^2 (1-m)+a B d e m\right )\right ) (d+e x)^m}{2 a \left (\sqrt {-a}-\sqrt {c} x\right )}+\frac {\left (a \sqrt {c} e (A c d+a B e) m-\sqrt {-a} c \left (A c d^2+a A e^2 (1-m)+a B d e m\right )\right ) (d+e x)^m}{2 a \left (\sqrt {-a}+\sqrt {c} x\right )}\right ) \, dx}{2 a c \left (c d^2+a e^2\right )}\\ &=-\frac {(d+e x)^{1+m} (a (B d-A e)-(A c d+a B e) x)}{2 a \left (c d^2+a e^2\right ) \left (a+c x^2\right )}-\frac {\left (a e (A c d+a B e) m-\sqrt {-a} \sqrt {c} \left (A c d^2+a A e^2 (1-m)+a B d e m\right )\right ) \int \frac {(d+e x)^m}{\sqrt {-a}+\sqrt {c} x} \, dx}{4 a^2 \sqrt {c} \left (c d^2+a e^2\right )}+\frac {\left (a e (A c d+a B e) m+\sqrt {-a} \sqrt {c} \left (A c d^2+a A e^2 (1-m)+a B d e m\right )\right ) \int \frac {(d+e x)^m}{\sqrt {-a}-\sqrt {c} x} \, dx}{4 a^2 \sqrt {c} \left (c d^2+a e^2\right )}\\ &=-\frac {(d+e x)^{1+m} (a (B d-A e)-(A c d+a B e) x)}{2 a \left (c d^2+a e^2\right ) \left (a+c x^2\right )}+\frac {\left (a e (A c d+a B e) m-\sqrt {-a} \sqrt {c} \left (A c d^2+a A e^2 (1-m)+a B d e m\right )\right ) (d+e x)^{1+m} \, _2F_1\left (1,1+m;2+m;\frac {\sqrt {c} (d+e x)}{\sqrt {c} d-\sqrt {-a} e}\right )}{4 a^2 \sqrt {c} \left (\sqrt {c} d-\sqrt {-a} e\right ) \left (c d^2+a e^2\right ) (1+m)}+\frac {\left (a e (A c d+a B e) m+\sqrt {-a} \sqrt {c} \left (A c d^2+a A e^2 (1-m)+a B d e m\right )\right ) (d+e x)^{1+m} \, _2F_1\left (1,1+m;2+m;\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {-a} e}\right )}{4 a^2 \sqrt {c} \left (\sqrt {c} d+\sqrt {-a} e\right ) \left (c d^2+a e^2\right ) (1+m)}\\ \end {align*}
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Mathematica [A] time = 0.75, size = 310, normalized size = 0.86 \[ \frac {(d+e x)^{m+1} \left (\frac {\sqrt {c} \left (a e m (a B e+A c d)-\sqrt {-a} \sqrt {c} \left (-a A e^2 (m-1)+a B d e m+A c d^2\right )\right ) \, _2F_1\left (1,m+1;m+2;\frac {\sqrt {c} (d+e x)}{\sqrt {c} d-\sqrt {-a} e}\right )}{a (m+1) \left (\sqrt {c} d-\sqrt {-a} e\right )}+\frac {\sqrt {c} \left (\sqrt {-a} \sqrt {c} \left (-a A e^2 (m-1)+a B d e m+A c d^2\right )+a e m (a B e+A c d)\right ) \, _2F_1\left (1,m+1;m+2;\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {-a} e}\right )}{a (m+1) \left (\sqrt {-a} e+\sqrt {c} d\right )}+\frac {2 c (a (A e-B d+B e x)+A c d x)}{a+c x^2}\right )}{4 a c \left (a e^2+c d^2\right )} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.23, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (B x + A\right )} {\left (e x + d\right )}^{m}}{c^{2} x^{4} + 2 \, a c x^{2} + a^{2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (B x + A\right )} {\left (e x + d\right )}^{m}}{{\left (c x^{2} + a\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 1.20, size = 0, normalized size = 0.00 \[ \int \frac {\left (B x +A \right ) \left (e x +d \right )^{m}}{\left (c \,x^{2}+a \right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (B x + A\right )} {\left (e x + d\right )}^{m}}{{\left (c x^{2} + a\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {\left (A+B\,x\right )\,{\left (d+e\,x\right )}^m}{{\left (c\,x^2+a\right )}^2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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