Optimal. Leaf size=440 \[ -\frac {\left (-\sqrt {b^2-4 a c}+b+2 c x\right ) (d+e x)^{m+3} \left (a+b x+c x^2\right )^{-\frac {m}{2}-2} \left (4 c e (a e-b d (m+1))+b^2 e^2 m+4 c^2 d^2 (m+1)\right ) \left (\frac {\left (\sqrt {b^2-4 a c}+b+2 c x\right ) \left (2 c d-e \left (b-\sqrt {b^2-4 a c}\right )\right )}{\left (-\sqrt {b^2-4 a c}+b+2 c x\right ) \left (2 c d-e \left (\sqrt {b^2-4 a c}+b\right )\right )}\right )^{\frac {m+4}{2}} \, _2F_1\left (m+3,\frac {m+4}{2};m+4;-\frac {4 c \sqrt {b^2-4 a c} (d+e x)}{\left (2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e\right ) \left (b+2 c x-\sqrt {b^2-4 a c}\right )}\right )}{4 (m+1) (m+3) \left (2 c d-e \left (b-\sqrt {b^2-4 a c}\right )\right ) \left (a e^2-b d e+c d^2\right )^2}+\frac {e (d+e x)^{m+1} \left (a+b x+c x^2\right )^{-\frac {m}{2}-1}}{(m+1) \left (a e^2-b d e+c d^2\right )}+\frac {e m (2 c d-b e) (d+e x)^{m+2} \left (a+b x+c x^2\right )^{-\frac {m}{2}-1}}{2 (m+1) (m+2) \left (a e^2-b d e+c d^2\right )^2} \]
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Rubi [A] time = 0.39, antiderivative size = 440, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {744, 806, 726} \[ -\frac {\left (-\sqrt {b^2-4 a c}+b+2 c x\right ) (d+e x)^{m+3} \left (a+b x+c x^2\right )^{-\frac {m}{2}-2} \left (4 c e (a e-b d (m+1))+b^2 e^2 m+4 c^2 d^2 (m+1)\right ) \left (\frac {\left (\sqrt {b^2-4 a c}+b+2 c x\right ) \left (2 c d-e \left (b-\sqrt {b^2-4 a c}\right )\right )}{\left (-\sqrt {b^2-4 a c}+b+2 c x\right ) \left (2 c d-e \left (\sqrt {b^2-4 a c}+b\right )\right )}\right )^{\frac {m+4}{2}} \, _2F_1\left (m+3,\frac {m+4}{2};m+4;-\frac {4 c \sqrt {b^2-4 a c} (d+e x)}{\left (2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e\right ) \left (b+2 c x-\sqrt {b^2-4 a c}\right )}\right )}{4 (m+1) (m+3) \left (2 c d-e \left (b-\sqrt {b^2-4 a c}\right )\right ) \left (a e^2-b d e+c d^2\right )^2}+\frac {e (d+e x)^{m+1} \left (a+b x+c x^2\right )^{-\frac {m}{2}-1}}{(m+1) \left (a e^2-b d e+c d^2\right )}+\frac {e m (2 c d-b e) (d+e x)^{m+2} \left (a+b x+c x^2\right )^{-\frac {m}{2}-1}}{2 (m+1) (m+2) \left (a e^2-b d e+c d^2\right )^2} \]
Antiderivative was successfully verified.
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Rule 726
Rule 744
Rule 806
Rubi steps
\begin {align*} \int (d+e x)^m \left (a+b x+c x^2\right )^{-2-\frac {m}{2}} \, dx &=\frac {e (d+e x)^{1+m} \left (a+b x+c x^2\right )^{-1-\frac {m}{2}}}{\left (c d^2-b d e+a e^2\right ) (1+m)}+\frac {\int (d+e x)^{1+m} \left (\frac {1}{2} (-b e m+2 c d (1+m))+c e x\right ) \left (a+b x+c x^2\right )^{-2-\frac {m}{2}} \, dx}{\left (c d^2-b d e+a e^2\right ) (1+m)}\\ &=\frac {e (d+e x)^{1+m} \left (a+b x+c x^2\right )^{-1-\frac {m}{2}}}{\left (c d^2-b d e+a e^2\right ) (1+m)}+\frac {e (2 c d-b e) m (d+e x)^{2+m} \left (a+b x+c x^2\right )^{-1-\frac {m}{2}}}{2 \left (c d^2-b d e+a e^2\right )^2 (1+m) (2+m)}+\frac {\left (b^2 e^2 m+4 c^2 d^2 (1+m)+4 c e (a e-b d (1+m))\right ) \int (d+e x)^{2+m} \left (a+b x+c x^2\right )^{-2-\frac {m}{2}} \, dx}{4 \left (c d^2-b d e+a e^2\right )^2 (1+m)}\\ &=\frac {e (d+e x)^{1+m} \left (a+b x+c x^2\right )^{-1-\frac {m}{2}}}{\left (c d^2-b d e+a e^2\right ) (1+m)}+\frac {e (2 c d-b e) m (d+e x)^{2+m} \left (a+b x+c x^2\right )^{-1-\frac {m}{2}}}{2 \left (c d^2-b d e+a e^2\right )^2 (1+m) (2+m)}-\frac {\left (b^2 e^2 m+4 c^2 d^2 (1+m)+4 c e (a e-b d (1+m))\right ) \left (b-\sqrt {b^2-4 a c}+2 c x\right ) \left (\frac {\left (2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e\right ) \left (b+\sqrt {b^2-4 a c}+2 c x\right )}{\left (2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e\right ) \left (b-\sqrt {b^2-4 a c}+2 c x\right )}\right )^{\frac {4+m}{2}} (d+e x)^{3+m} \left (a+b x+c x^2\right )^{-2-\frac {m}{2}} \, _2F_1\left (3+m,\frac {4+m}{2};4+m;-\frac {4 c \sqrt {b^2-4 a c} (d+e x)}{\left (2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e\right ) \left (b-\sqrt {b^2-4 a c}+2 c x\right )}\right )}{4 \left (2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e\right ) \left (c d^2-b d e+a e^2\right )^2 (1+m) (3+m)}\\ \end {align*}
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Mathematica [A] time = 6.26, size = 492, normalized size = 1.12 \[ \frac {\frac {(d+e x)^{m+2} \left (a+b x+c x^2\right )^{-\frac {m}{2}-1} \left (c d e-\frac {1}{2} e (2 c d (m+1)-b e m)\right )}{2 \left (-\frac {m}{2}-1\right ) \left (a e^2-b d e+c d^2\right )}-\frac {\left (\sqrt {b^2-4 a c}-b-2 c x\right ) (d+e x)^{m+3} \left (a+b x+c x^2\right )^{-\frac {m}{2}-2} \left (b \left (\frac {1}{2} e (2 c d (m+1)-b e m)+c d e\right )-2 \left (a c e^2+\frac {1}{2} c d (2 c d (m+1)-b e m)\right )\right ) \left (\frac {\left (\sqrt {b^2-4 a c}+b+2 c x\right ) \left (e \sqrt {b^2-4 a c}-b e+2 c d\right )}{\left (-\sqrt {b^2-4 a c}+b+2 c x\right ) \left (-e \sqrt {b^2-4 a c}-b e+2 c d\right )}\right )^{\frac {m}{2}+2} \, _2F_1\left (\frac {m}{2}+2,m+3;m+4;-\frac {4 c \sqrt {b^2-4 a c} (d+e x)}{\left (2 c d-b e-\sqrt {b^2-4 a c} e\right ) \left (b+2 c x-\sqrt {b^2-4 a c}\right )}\right )}{2 (m+3) \left (e \sqrt {b^2-4 a c}-b e+2 c d\right ) \left (a e^2-b d e+c d^2\right )}}{(m+1) \left (a e^2-b d e+c d^2\right )}+\frac {e (d+e x)^{m+1} \left (a+b x+c x^2\right )^{-\frac {m}{2}-1}}{(m+1) \left (a e^2-b d e+c d^2\right )} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.46, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (c x^{2} + b x + a\right )}^{-\frac {1}{2} \, m - 2} {\left (e x + d\right )}^{m}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [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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maple [F] time = 1.91, size = 0, normalized size = 0.00 \[ \int \left (e x +d \right )^{m} \left (c \,x^{2}+b x +a \right )^{-\frac {m}{2}-2}\, dx \]
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 \[ \int {\left (c x^{2} + b x + a\right )}^{-\frac {1}{2} \, m - 2} {\left (e x + d\right )}^{m}\,{d x} \]
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 \[ \int \frac {{\left (d+e\,x\right )}^m}{{\left (c\,x^2+b\,x+a\right )}^{\frac {m}{2}+2}} \,d x \]
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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