Optimal. Leaf size=472 \[ \frac{(d+e x)^{m+1} \left (a \sqrt{c} d e m \left (a e^2 (5-2 m)+3 c d^2\right )-\sqrt{-a} \left (a^2 e^4 \left (m^2-4 m+3\right )+a c d^2 e^2 \left (-m^2-2 m+6\right )+3 c^2 d^4\right )\right ) \, _2F_1\left (1,m+1;m+2;\frac{\sqrt{c} (d+e x)}{\sqrt{c} d-\sqrt{-a} e}\right )}{16 a^3 (m+1) \left (\sqrt{c} d-\sqrt{-a} e\right ) \left (a e^2+c d^2\right )^2}+\frac{(d+e x)^{m+1} \left (\sqrt{-a} \left (a^2 e^4 \left (m^2-4 m+3\right )+a c d^2 e^2 \left (-m^2-2 m+6\right )+3 c^2 d^4\right )+a \sqrt{c} d e m \left (a e^2 (5-2 m)+3 c d^2\right )\right ) \, _2F_1\left (1,m+1;m+2;\frac{\sqrt{c} (d+e x)}{\sqrt{c} d+\sqrt{-a} e}\right )}{16 a^3 (m+1) \left (\sqrt{-a} e+\sqrt{c} d\right ) \left (a e^2+c d^2\right )^2}+\frac{(d+e x)^{m+1} \left (c d x \left (a e^2 (5-2 m)+3 c d^2\right )+a e \left (a e^2 (3-m)+c d^2 (m+1)\right )\right )}{8 a^2 \left (a+c x^2\right ) \left (a e^2+c d^2\right )^2}+\frac{(d+e x)^{m+1} (a e+c d x)}{4 a \left (a+c x^2\right )^2 \left (a e^2+c d^2\right )} \]
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Rubi [A] time = 0.81205, antiderivative size = 472, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.235, Rules used = {741, 823, 831, 68} \[ \frac{(d+e x)^{m+1} \left (a \sqrt{c} d e m \left (a e^2 (5-2 m)+3 c d^2\right )-\sqrt{-a} \left (a^2 e^4 \left (m^2-4 m+3\right )+a c d^2 e^2 \left (-m^2-2 m+6\right )+3 c^2 d^4\right )\right ) \, _2F_1\left (1,m+1;m+2;\frac{\sqrt{c} (d+e x)}{\sqrt{c} d-\sqrt{-a} e}\right )}{16 a^3 (m+1) \left (\sqrt{c} d-\sqrt{-a} e\right ) \left (a e^2+c d^2\right )^2}+\frac{(d+e x)^{m+1} \left (\sqrt{-a} \left (a^2 e^4 \left (m^2-4 m+3\right )+a c d^2 e^2 \left (-m^2-2 m+6\right )+3 c^2 d^4\right )+a \sqrt{c} d e m \left (a e^2 (5-2 m)+3 c d^2\right )\right ) \, _2F_1\left (1,m+1;m+2;\frac{\sqrt{c} (d+e x)}{\sqrt{c} d+\sqrt{-a} e}\right )}{16 a^3 (m+1) \left (\sqrt{-a} e+\sqrt{c} d\right ) \left (a e^2+c d^2\right )^2}+\frac{(d+e x)^{m+1} \left (c d x \left (a e^2 (5-2 m)+3 c d^2\right )+a e \left (a e^2 (3-m)+c d^2 (m+1)\right )\right )}{8 a^2 \left (a+c x^2\right ) \left (a e^2+c d^2\right )^2}+\frac{(d+e x)^{m+1} (a e+c d x)}{4 a \left (a+c x^2\right )^2 \left (a e^2+c d^2\right )} \]
Antiderivative was successfully verified.
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Rule 741
Rule 823
Rule 831
Rule 68
Rubi steps
\begin{align*} \int \frac{(d+e x)^m}{\left (a+c x^2\right )^3} \, dx &=\frac{(a e+c d x) (d+e x)^{1+m}}{4 a \left (c d^2+a e^2\right ) \left (a+c x^2\right )^2}-\frac{\int \frac{(d+e x)^m \left (-3 c d^2-a e^2 (3-m)-c d e (2-m) x\right )}{\left (a+c x^2\right )^2} \, dx}{4 a \left (c d^2+a e^2\right )}\\ &=\frac{(a e+c d x) (d+e x)^{1+m}}{4 a \left (c d^2+a e^2\right ) \left (a+c x^2\right )^2}+\frac{(d+e x)^{1+m} \left (a e \left (a e^2 (3-m)+c d^2 (1+m)\right )+c d \left (3 c d^2+a e^2 (5-2 m)\right ) x\right )}{8 a^2 \left (c d^2+a e^2\right )^2 \left (a+c x^2\right )}+\frac{\int \frac{(d+e x)^m \left (c \left (3 c^2 d^4+a c d^2 e^2 \left (6-2 m-m^2\right )+a^2 e^4 \left (3-4 m+m^2\right )\right )-c^2 d e \left (3 c d^2+a e^2 (5-2 m)\right ) m x\right )}{a+c x^2} \, dx}{8 a^2 c \left (c d^2+a e^2\right )^2}\\ &=\frac{(a e+c d x) (d+e x)^{1+m}}{4 a \left (c d^2+a e^2\right ) \left (a+c x^2\right )^2}+\frac{(d+e x)^{1+m} \left (a e \left (a e^2 (3-m)+c d^2 (1+m)\right )+c d \left (3 c d^2+a e^2 (5-2 m)\right ) x\right )}{8 a^2 \left (c d^2+a e^2\right )^2 \left (a+c x^2\right )}+\frac{\int \left (\frac{\left (a c^{3/2} d e \left (3 c d^2+a e^2 (5-2 m)\right ) m+\sqrt{-a} c \left (3 c^2 d^4+a c d^2 e^2 \left (6-2 m-m^2\right )+a^2 e^4 \left (3-4 m+m^2\right )\right )\right ) (d+e x)^m}{2 a \left (\sqrt{-a}-\sqrt{c} x\right )}+\frac{\left (-a c^{3/2} d e \left (3 c d^2+a e^2 (5-2 m)\right ) m+\sqrt{-a} c \left (3 c^2 d^4+a c d^2 e^2 \left (6-2 m-m^2\right )+a^2 e^4 \left (3-4 m+m^2\right )\right )\right ) (d+e x)^m}{2 a \left (\sqrt{-a}+\sqrt{c} x\right )}\right ) \, dx}{8 a^2 c \left (c d^2+a e^2\right )^2}\\ &=\frac{(a e+c d x) (d+e x)^{1+m}}{4 a \left (c d^2+a e^2\right ) \left (a+c x^2\right )^2}+\frac{(d+e x)^{1+m} \left (a e \left (a e^2 (3-m)+c d^2 (1+m)\right )+c d \left (3 c d^2+a e^2 (5-2 m)\right ) x\right )}{8 a^2 \left (c d^2+a e^2\right )^2 \left (a+c x^2\right )}-\frac{\left (a \sqrt{c} d e \left (3 c d^2+a e^2 (5-2 m)\right ) m-\sqrt{-a} \left (3 c^2 d^4+a c d^2 e^2 \left (6-2 m-m^2\right )+a^2 e^4 \left (3-4 m+m^2\right )\right )\right ) \int \frac{(d+e x)^m}{\sqrt{-a}+\sqrt{c} x} \, dx}{16 a^3 \left (c d^2+a e^2\right )^2}+\frac{\left (a \sqrt{c} d e \left (3 c d^2+a e^2 (5-2 m)\right ) m+\sqrt{-a} \left (3 c^2 d^4+a c d^2 e^2 \left (6-2 m-m^2\right )+a^2 e^4 \left (3-4 m+m^2\right )\right )\right ) \int \frac{(d+e x)^m}{\sqrt{-a}-\sqrt{c} x} \, dx}{16 a^3 \left (c d^2+a e^2\right )^2}\\ &=\frac{(a e+c d x) (d+e x)^{1+m}}{4 a \left (c d^2+a e^2\right ) \left (a+c x^2\right )^2}+\frac{(d+e x)^{1+m} \left (a e \left (a e^2 (3-m)+c d^2 (1+m)\right )+c d \left (3 c d^2+a e^2 (5-2 m)\right ) x\right )}{8 a^2 \left (c d^2+a e^2\right )^2 \left (a+c x^2\right )}+\frac{\left (a \sqrt{c} d e \left (3 c d^2+a e^2 (5-2 m)\right ) m-\sqrt{-a} \left (3 c^2 d^4+a c d^2 e^2 \left (6-2 m-m^2\right )+a^2 e^4 \left (3-4 m+m^2\right )\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 )}{16 a^3 \left (\sqrt{c} d-\sqrt{-a} e\right ) \left (c d^2+a e^2\right )^2 (1+m)}+\frac{\left (a \sqrt{c} d e \left (3 c d^2+a e^2 (5-2 m)\right ) m+\sqrt{-a} \left (3 c^2 d^4+a c d^2 e^2 \left (6-2 m-m^2\right )+a^2 e^4 \left (3-4 m+m^2\right )\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 )}{16 a^3 \left (\sqrt{c} d+\sqrt{-a} e\right ) \left (c d^2+a e^2\right )^2 (1+m)}\\ \end{align*}
Mathematica [A] time = 0.923583, size = 396, normalized size = 0.84 \[ \frac{(d+e x)^{m+1} \left (\frac{\frac{\left (\sqrt{-a} \left (-a^2 e^4 \left (m^2-4 m+3\right )+a c d^2 e^2 \left (m^2+2 m-6\right )-3 c^2 d^4\right )+a \sqrt{c} d e m \left (a e^2 (5-2 m)+3 c d^2\right )\right ) \, _2F_1\left (1,m+1;m+2;\frac{\sqrt{c} (d+e x)}{\sqrt{c} d-\sqrt{-a} e}\right )}{\sqrt{c} d-\sqrt{-a} e}+\frac{\left (\sqrt{-a} \left (a^2 e^4 \left (m^2-4 m+3\right )-a c d^2 e^2 \left (m^2+2 m-6\right )+3 c^2 d^4\right )+a \sqrt{c} d e m \left (a e^2 (5-2 m)+3 c d^2\right )\right ) \, _2F_1\left (1,m+1;m+2;\frac{\sqrt{c} (d+e x)}{\sqrt{c} d+\sqrt{-a} e}\right )}{\sqrt{-a} e+\sqrt{c} d}}{a (m+1)}+\frac{2 \left (-a^2 e^3 (m-3)+a c d e (d (m+1)+e (5-2 m) x)+3 c^2 d^3 x\right )}{a+c x^2}+\frac{4 a \left (a e^2+c d^2\right ) (a e+c d x)}{\left (a+c x^2\right )^2}\right )}{16 a^2 \left (a e^2+c d^2\right )^2} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.604, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( ex+d \right ) ^{m}}{ \left ( c{x}^{2}+a \right ) ^{3}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (e x + d\right )}^{m}}{{\left (c x^{2} + a\right )}^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (e x + d\right )}^{m}}{c^{3} x^{6} + 3 \, a c^{2} x^{4} + 3 \, a^{2} c x^{2} + a^{3}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (e x + d\right )}^{m}}{{\left (c x^{2} + a\right )}^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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