Optimal. Leaf size=472 \[ \frac {(a e+c d x) (d+e x)^{1+m}}{4 a \left (c d^2+a e^2\right ) \left (a+c x^2\right )^2}+\frac {(d+e x)^{1+m} \left (a e \left (a e^2 (3-m)+c d^2 (1+m)\right )+c d \left (3 c d^2+a e^2 (5-2 m)\right ) x\right )}{8 a^2 \left (c d^2+a e^2\right )^2 \left (a+c x^2\right )}+\frac {\left (a \sqrt {c} d e \left (3 c d^2+a e^2 (5-2 m)\right ) m-\sqrt {-a} \left (3 c^2 d^4+a c d^2 e^2 \left (6-2 m-m^2\right )+a^2 e^4 \left (3-4 m+m^2\right )\right )\right ) (d+e x)^{1+m} \, _2F_1\left (1,1+m;2+m;\frac {\sqrt {c} (d+e x)}{\sqrt {c} d-\sqrt {-a} e}\right )}{16 a^3 \left (\sqrt {c} d-\sqrt {-a} e\right ) \left (c d^2+a e^2\right )^2 (1+m)}+\frac {\left (a \sqrt {c} d e \left (3 c d^2+a e^2 (5-2 m)\right ) m+\sqrt {-a} \left (3 c^2 d^4+a c d^2 e^2 \left (6-2 m-m^2\right )+a^2 e^4 \left (3-4 m+m^2\right )\right )\right ) (d+e x)^{1+m} \, _2F_1\left (1,1+m;2+m;\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {-a} e}\right )}{16 a^3 \left (\sqrt {c} d+\sqrt {-a} e\right ) \left (c d^2+a e^2\right )^2 (1+m)} \]
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Rubi [A]
time = 0.56, antiderivative size = 472, normalized size of antiderivative = 1.00, 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 = {755, 837, 845,
70} \begin {gather*} \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} \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} (a e+c d x)}{4 a \left (a+c x^2\right )^2 \left (a e^2+c d^2\right )} \end {gather*}
Antiderivative was successfully verified.
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Rule 70
Rule 755
Rule 837
Rule 845
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*}
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Mathematica [A]
time = 0.77, size = 396, normalized size = 0.84 \begin {gather*} \frac {(d+e x)^{1+m} \left (\frac {4 a \left (c d^2+a e^2\right ) (a e+c d x)}{\left (a+c x^2\right )^2}+\frac {2 \left (-a^2 e^3 (-3+m)+3 c^2 d^3 x+a c d e (d (1+m)+e (5-2 m) x)\right )}{a+c x^2}+\frac {\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^2 e^4 \left (3-4 m+m^2\right )+a c d^2 e^2 \left (-6+2 m+m^2\right )\right )\right ) \, _2F_1\left (1,1+m;2+m;\frac {\sqrt {c} (d+e x)}{\sqrt {c} d-\sqrt {-a} e}\right )}{\sqrt {c} d-\sqrt {-a} e}+\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^2 e^4 \left (3-4 m+m^2\right )-a c d^2 e^2 \left (-6+2 m+m^2\right )\right )\right ) \, _2F_1\left (1,1+m;2+m;\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {-a} e}\right )}{\sqrt {c} d+\sqrt {-a} e}}{a (1+m)}\right )}{16 a^2 \left (c d^2+a e^2\right )^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.16, size = 0, normalized size = 0.00 \[\int \frac {\left (e x +d \right )^{m}}{\left (c \,x^{2}+a \right )^{3}}\, dx\]
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {{\left (d+e\,x\right )}^m}{{\left (c\,x^2+a\right )}^3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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