Optimal. Leaf size=82 \[ -\frac {2 \left (a+b x+c x^2\right )^{7/2} (b d+2 c d x)^{m+1} \, _2F_1\left (1,\frac {m+8}{2};\frac {m+3}{2};\frac {(b+2 c x)^2}{b^2-4 a c}\right )}{d (m+1) \left (b^2-4 a c\right )} \]
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Rubi [A] time = 0.12, antiderivative size = 114, normalized size of antiderivative = 1.39, 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} \[ \frac {\left (b^2-4 a c\right )^2 \sqrt {a+b x+c x^2} (d (b+2 c x))^{m+1} \, _2F_1\left (-\frac {5}{2},\frac {m+1}{2};\frac {m+3}{2};\frac {(b+2 c x)^2}{b^2-4 a c}\right )}{32 c^3 d (m+1) \sqrt {1-\frac {(b+2 c x)^2}{b^2-4 a c}}} \]
Antiderivative was successfully verified.
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Rule 364
Rule 365
Rule 694
Rubi steps
\begin {align*} \int (b d+2 c d x)^m \left (a+b x+c x^2\right )^{5/2} \, dx &=\frac {\operatorname {Subst}\left (\int 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 {\left (\left (a-\frac {b^2}{4 c}\right )^2 \sqrt {a+b x+c x^2}\right ) \operatorname {Subst}\left (\int 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 )}{c d \sqrt {4+\frac {(b d+2 c d x)^2}{\left (a-\frac {b^2}{4 c}\right ) c d^2}}}\\ &=\frac {\left (b^2-4 a c\right )^2 (d (b+2 c x))^{1+m} \sqrt {a+b x+c x^2} \, _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 )}{32 c^3 d (1+m) \sqrt {1-\frac {(b+2 c x)^2}{b^2-4 a c}}}\\ \end {align*}
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Mathematica [A] time = 0.08, size = 115, normalized size = 1.40 \[ \frac {\left (b^2-4 a c\right )^2 (b+2 c x) \sqrt {a+x (b+c x)} (d (b+2 c x))^m \, _2F_1\left (-\frac {5}{2},\frac {m+1}{2};\frac {m+3}{2};\frac {(b+2 c x)^2}{b^2-4 a c}\right )}{64 c^3 (m+1) \sqrt {\frac {c (a+x (b+c x))}{4 a c-b^2}}} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.26, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (c^{2} x^{4} + 2 \, b c x^{3} + 2 \, a b x + {\left (b^{2} + 2 \, a c\right )} x^{2} + a^{2}\right )} \sqrt {c x^{2} + b x + a} {\left (2 \, c d x + b d\right )}^{m}, x\right ) \]
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 \[ \int {\left (c x^{2} + b x + a\right )}^{\frac {5}{2}} {\left (2 \, c d x + b d\right )}^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 1.23, size = 0, normalized size = 0.00 \[ \int \left (c \,x^{2}+b x +a \right )^{\frac {5}{2}} \left (2 c d x +b d \right )^{m}\, dx \]
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 \[ \int {\left (c x^{2} + b x + a\right )}^{\frac {5}{2}} {\left (2 \, c d x + b d\right )}^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int {\left (b\,d+2\,c\,d\,x\right )}^m\,{\left (c\,x^2+b\,x+a\right )}^{5/2} \,d x \]
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 \[ \int \left (d \left (b + 2 c x\right )\right )^{m} \left (a + b x + c x^{2}\right )^{\frac {5}{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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