Optimal. Leaf size=304 \[ -\frac {(d+e x)^{m+1} \left (\sqrt {-a} \sqrt {c} d e m+a e^2 (1-m)+c d^2\right ) \, _2F_1\left (1,m+1;m+2;\frac {\sqrt {c} (d+e x)}{\sqrt {c} d-\sqrt {-a} e}\right )}{4 (-a)^{3/2} (m+1) \left (\sqrt {c} d-\sqrt {-a} e\right ) \left (a e^2+c d^2\right )}+\frac {(d+e x)^{m+1} \left (-\sqrt {-a} \sqrt {c} d e m+a e^2 (1-m)+c d^2\right ) \, _2F_1\left (1,m+1;m+2;\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {-a} e}\right )}{4 (-a)^{3/2} (m+1) \left (\sqrt {-a} e+\sqrt {c} d\right ) \left (a e^2+c d^2\right )}+\frac {(d+e x)^{m+1} (a e+c d x)}{2 a \left (a+c x^2\right ) \left (a e^2+c d^2\right )} \]
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Rubi [A] time = 0.38, antiderivative size = 304, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {741, 831, 68} \[ -\frac {(d+e x)^{m+1} \left (\sqrt {-a} \sqrt {c} d e m+a e^2 (1-m)+c d^2\right ) \, _2F_1\left (1,m+1;m+2;\frac {\sqrt {c} (d+e x)}{\sqrt {c} d-\sqrt {-a} e}\right )}{4 (-a)^{3/2} (m+1) \left (\sqrt {c} d-\sqrt {-a} e\right ) \left (a e^2+c d^2\right )}+\frac {(d+e x)^{m+1} \left (-\sqrt {-a} \sqrt {c} d e m+a e^2 (1-m)+c d^2\right ) \, _2F_1\left (1,m+1;m+2;\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {-a} e}\right )}{4 (-a)^{3/2} (m+1) \left (\sqrt {-a} e+\sqrt {c} d\right ) \left (a e^2+c d^2\right )}+\frac {(d+e x)^{m+1} (a e+c d x)}{2 a \left (a+c x^2\right ) \left (a e^2+c d^2\right )} \]
Antiderivative was successfully verified.
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Rule 68
Rule 741
Rule 831
Rubi steps
\begin {align*} \int \frac {(d+e x)^m}{\left (a+c x^2\right )^2} \, dx &=\frac {(a e+c d x) (d+e x)^{1+m}}{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 d^2-a e^2 (1-m)+c d e m x\right )}{a+c x^2} \, dx}{2 a \left (c d^2+a e^2\right )}\\ &=\frac {(a e+c d x) (d+e x)^{1+m}}{2 a \left (c d^2+a e^2\right ) \left (a+c x^2\right )}-\frac {\int \left (\frac {\left (\sqrt {-a} \left (-c d^2-a e^2 (1-m)\right )-a \sqrt {c} d e m\right ) (d+e x)^m}{2 a \left (\sqrt {-a}-\sqrt {c} x\right )}+\frac {\left (\sqrt {-a} \left (-c d^2-a e^2 (1-m)\right )+a \sqrt {c} d e m\right ) (d+e x)^m}{2 a \left (\sqrt {-a}+\sqrt {c} x\right )}\right ) \, dx}{2 a \left (c d^2+a e^2\right )}\\ &=\frac {(a e+c d x) (d+e x)^{1+m}}{2 a \left (c d^2+a e^2\right ) \left (a+c x^2\right )}+\frac {\left (c d^2+a e^2 (1-m)-\sqrt {-a} \sqrt {c} d e m\right ) \int \frac {(d+e x)^m}{\sqrt {-a}-\sqrt {c} x} \, dx}{4 (-a)^{3/2} \left (c d^2+a e^2\right )}+\frac {\left (c d^2+a e^2 (1-m)+\sqrt {-a} \sqrt {c} d e m\right ) \int \frac {(d+e x)^m}{\sqrt {-a}+\sqrt {c} x} \, dx}{4 (-a)^{3/2} \left (c d^2+a e^2\right )}\\ &=\frac {(a e+c d x) (d+e x)^{1+m}}{2 a \left (c d^2+a e^2\right ) \left (a+c x^2\right )}-\frac {\left (c d^2+a e^2 (1-m)+\sqrt {-a} \sqrt {c} d e m\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)^{3/2} \left (\sqrt {c} d-\sqrt {-a} e\right ) \left (c d^2+a e^2\right ) (1+m)}+\frac {\left (c d^2+a e^2 (1-m)-\sqrt {-a} \sqrt {c} d e m\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)^{3/2} \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 = 0.43, size = 253, normalized size = 0.83 \[ \frac {(d+e x)^{m+1} \left (\frac {\left (\sqrt {-a} \sqrt {c} d e m-a e^2 (m-1)+c d^2\right ) \, _2F_1\left (1,m+1;m+2;\frac {\sqrt {c} (d+e x)}{\sqrt {c} d-\sqrt {-a} e}\right )}{\sqrt {-a} (m+1) \left (\sqrt {c} d-\sqrt {-a} e\right )}+\frac {\left (\sqrt {-a} \sqrt {c} d e m+a e^2 (m-1)-c d^2\right ) \, _2F_1\left (1,m+1;m+2;\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {-a} e}\right )}{\sqrt {-a} (m+1) \left (\sqrt {-a} e+\sqrt {c} d\right )}+\frac {2 (a e+c d x)}{a+c x^2}\right )}{4 a \left (a e^2+c d^2\right )} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.81, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (e x + d\right )}^{m}}{c^{2} x^{4} + 2 \, a c x^{2} + a^{2}}, x\right ) \]
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 \[ \int \frac {{\left (e x + d\right )}^{m}}{{\left (c x^{2} + a\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.83, size = 0, normalized size = 0.00 \[ \int \frac {\left (e x +d \right )^{m}}{\left (c \,x^{2}+a \right )^{2}}\, dx \]
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 \[ \int \frac {{\left (e x + d\right )}^{m}}{{\left (c x^{2} + a\right )}^{2}}\,{d x} \]
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 \[ \int \frac {{\left (d+e\,x\right )}^m}{{\left (c\,x^2+a\right )}^2} \,d x \]
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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