Optimal. Leaf size=70 \[ -\frac {32 c^2 (d (b+2 c x))^{m+1} \, _2F_1\left (3,\frac {m+1}{2};\frac {m+3}{2};\frac {(b+2 c x)^2}{b^2-4 a c}\right )}{d (m+1) \left (b^2-4 a c\right )^3} \]
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Rubi [A] time = 0.05, antiderivative size = 70, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {694, 364} \[ -\frac {32 c^2 (d (b+2 c x))^{m+1} \, _2F_1\left (3,\frac {m+1}{2};\frac {m+3}{2};\frac {(b+2 c x)^2}{b^2-4 a c}\right )}{d (m+1) \left (b^2-4 a c\right )^3} \]
Antiderivative was successfully verified.
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Rule 364
Rule 694
Rubi steps
\begin {align*} \int \frac {(b d+2 c d x)^m}{\left (a+b x+c x^2\right )^3} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {x^m}{\left (a-\frac {b^2}{4 c}+\frac {x^2}{4 c d^2}\right )^3} \, dx,x,b d+2 c d x\right )}{2 c d}\\ &=-\frac {32 c^2 (d (b+2 c x))^{1+m} \, _2F_1\left (3,\frac {1+m}{2};\frac {3+m}{2};\frac {(b+2 c x)^2}{b^2-4 a c}\right )}{\left (b^2-4 a c\right )^3 d (1+m)}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 71, normalized size = 1.01 \[ -\frac {32 c^2 (b+2 c x) (d (b+2 c x))^m \, _2F_1\left (3,\frac {m+1}{2};\frac {m+3}{2};\frac {(b+2 c x)^2}{b^2-4 a c}\right )}{(m+1) \left (b^2-4 a c\right )^3} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.09, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (2 \, c d x + b d\right )}^{m}}{c^{3} x^{6} + 3 \, b c^{2} x^{5} + 3 \, {\left (b^{2} c + a c^{2}\right )} x^{4} + 3 \, a^{2} b x + {\left (b^{3} + 6 \, a b c\right )} x^{3} + a^{3} + 3 \, {\left (a b^{2} + a^{2} c\right )} x^{2}}, x\right ) \]
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 \[ \int \frac {{\left (2 \, c d x + b d\right )}^{m}}{{\left (c x^{2} + b x + a\right )}^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 1.30, size = 0, normalized size = 0.00 \[ \int \frac {\left (2 c d x +b d \right )^{m}}{\left (c \,x^{2}+b x +a \right )^{3}}\, dx \]
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 \[ \int \frac {{\left (2 \, c d x + b d\right )}^{m}}{{\left (c x^{2} + b x + a\right )}^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (b\,d+2\,c\,d\,x\right )}^m}{{\left (c\,x^2+b\,x+a\right )}^3} \,d x \]
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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