Optimal. Leaf size=61 \[ -\frac {2 x \left (\frac {c x}{b}+1\right )^{-p} \left (b x+c x^2\right )^p \, _2F_1\left (p-\frac {3}{2},-p;p-\frac {1}{2};-\frac {c x}{b}\right )}{(3-2 p) (d x)^{5/2}} \]
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Rubi [A] time = 0.03, antiderivative size = 61, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {674, 66, 64} \[ -\frac {2 x \left (\frac {c x}{b}+1\right )^{-p} \left (b x+c x^2\right )^p \, _2F_1\left (p-\frac {3}{2},-p;p-\frac {1}{2};-\frac {c x}{b}\right )}{(3-2 p) (d x)^{5/2}} \]
Antiderivative was successfully verified.
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Rule 64
Rule 66
Rule 674
Rubi steps
\begin {align*} \int \frac {\left (b x+c x^2\right )^p}{(d x)^{5/2}} \, dx &=\frac {\left (x^{\frac {5}{2}-p} (b+c x)^{-p} \left (b x+c x^2\right )^p\right ) \int x^{-\frac {5}{2}+p} (b+c x)^p \, dx}{(d x)^{5/2}}\\ &=\frac {\left (x^{\frac {5}{2}-p} \left (1+\frac {c x}{b}\right )^{-p} \left (b x+c x^2\right )^p\right ) \int x^{-\frac {5}{2}+p} \left (1+\frac {c x}{b}\right )^p \, dx}{(d x)^{5/2}}\\ &=-\frac {2 x \left (1+\frac {c x}{b}\right )^{-p} \left (b x+c x^2\right )^p \, _2F_1\left (-\frac {3}{2}+p,-p;-\frac {1}{2}+p;-\frac {c x}{b}\right )}{(3-2 p) (d x)^{5/2}}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 58, normalized size = 0.95 \[ \frac {x (x (b+c x))^p \left (\frac {c x}{b}+1\right )^{-p} \, _2F_1\left (p-\frac {3}{2},-p;p-\frac {1}{2};-\frac {c x}{b}\right )}{\left (p-\frac {3}{2}\right ) (d x)^{5/2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.90, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {d x} {\left (c x^{2} + b x\right )}^{p}}{d^{3} x^{3}}, x\right ) \]
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 \[ \int \frac {{\left (c x^{2} + b x\right )}^{p}}{\left (d x\right )^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F(-1)] time = 180.00, size = 0, normalized size = 0.00 \[ \text {hanged} \]
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 \[ \int \frac {{\left (c x^{2} + b x\right )}^{p}}{\left (d x\right )^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {{\left (c\,x^2+b\,x\right )}^p}{{\left (d\,x\right )}^{5/2}} \,d x \]
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 \[ \int \frac {\left (x \left (b + c x\right )\right )^{p}}{\left (d x\right )^{\frac {5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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