Optimal. Leaf size=96 \[ -\frac {16 (b d+2 c d x)^{m+1} \, _2F_1\left (1,\frac {m-2}{2};\frac {m+3}{2};\frac {(b+2 c x)^2}{b^2-4 a c}\right )}{d (m+1) \left (b^2-4 a c\right ) \left (4 a-\frac {b^2}{c}+\frac {(b+2 c x)^2}{c}\right )^{3/2}} \]
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Rubi [A] time = 0.12, antiderivative size = 110, normalized size of antiderivative = 1.15, 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 = {694, 365, 364} \[ \frac {8 c \sqrt {1-\frac {(b+2 c x)^2}{b^2-4 a c}} (d (b+2 c x))^{m+1} \, _2F_1\left (\frac {5}{2},\frac {m+1}{2};\frac {m+3}{2};\frac {(b+2 c x)^2}{b^2-4 a c}\right )}{d (m+1) \left (b^2-4 a c\right )^2 \sqrt {a+b x+c x^2}} \]
Antiderivative was successfully verified.
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Rule 364
Rule 365
Rule 694
Rubi steps
\begin {align*} \int \frac {(b d+2 c d x)^m}{\left (a+b x+c x^2\right )^{5/2}} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {x^m}{\left (a-\frac {b^2}{4 c}+\frac {x^2}{4 c d^2}\right )^{5/2}} \, dx,x,b d+2 c d x\right )}{2 c d}\\ &=\frac {\sqrt {4+\frac {(b d+2 c d x)^2}{\left (a-\frac {b^2}{4 c}\right ) c d^2}} \operatorname {Subst}\left (\int \frac {x^m}{\left (1+\frac {x^2}{4 \left (a-\frac {b^2}{4 c}\right ) c d^2}\right )^{5/2}} \, dx,x,b d+2 c d x\right )}{4 \left (a-\frac {b^2}{4 c}\right )^2 c d \sqrt {a+b x+c x^2}}\\ &=\frac {8 c (d (b+2 c x))^{1+m} \sqrt {1-\frac {(b+2 c x)^2}{b^2-4 a c}} \, _2F_1\left (\frac {5}{2},\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 )^2 d (1+m) \sqrt {a+b x+c x^2}}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 111, normalized size = 1.16 \[ \frac {16 c (b+2 c x) \sqrt {\frac {c (a+x (b+c x))}{4 a c-b^2}} (d (b+2 c x))^m \, _2F_1\left (\frac {5}{2},\frac {m+1}{2};\frac {m+3}{2};\frac {(b+2 c x)^2}{b^2-4 a c}\right )}{(m+1) \left (b^2-4 a c\right )^2 \sqrt {a+x (b+c x)}} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.13, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {c x^{2} + b x + a} {\left (2 \, c d x + b d\right )}^{m}}{c^{3} x^{6} + 3 \, b c^{2} x^{5} + 3 \, {\left (b^{2} c + a c^{2}\right )} x^{4} + 3 \, a^{2} b x + {\left (b^{3} + 6 \, a b c\right )} x^{3} + a^{3} + 3 \, {\left (a b^{2} + a^{2} c\right )} x^{2}}, x\right ) \]
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 \[ \int \frac {{\left (2 \, c d x + b d\right )}^{m}}{{\left (c x^{2} + b x + a\right )}^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 1.50, size = 0, normalized size = 0.00 \[ \int \frac {\left (2 c d x +b d \right )^{m}}{\left (c \,x^{2}+b x +a \right )^{\frac {5}{2}}}\, dx \]
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 \[ \int \frac {{\left (2 \, c d x + b d\right )}^{m}}{{\left (c x^{2} + b x + a\right )}^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (b\,d+2\,c\,d\,x\right )}^m}{{\left (c\,x^2+b\,x+a\right )}^{5/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (d \left (b + 2 c x\right )\right )^{m}}{\left (a + b x + c x^{2}\right )^{\frac {5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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