Optimal. Leaf size=105 \[ \frac {\sqrt {b x+c x^2} (d+e x)^{m+1} F_1\left (m+1;-\frac {1}{2},-\frac {1}{2};m+2;\frac {d+e x}{d},\frac {c (d+e x)}{c d-b e}\right )}{e (m+1) \sqrt {-\frac {e x}{d}} \sqrt {1-\frac {c (d+e x)}{c d-b e}}} \]
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Rubi [A] time = 0.05, antiderivative size = 105, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {759, 133} \[ \frac {\sqrt {b x+c x^2} (d+e x)^{m+1} F_1\left (m+1;-\frac {1}{2},-\frac {1}{2};m+2;\frac {d+e x}{d},\frac {c (d+e x)}{c d-b e}\right )}{e (m+1) \sqrt {-\frac {e x}{d}} \sqrt {1-\frac {c (d+e x)}{c d-b e}}} \]
Antiderivative was successfully verified.
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Rule 133
Rule 759
Rubi steps
\begin {align*} \int (d+e x)^m \sqrt {b x+c x^2} \, dx &=\frac {\sqrt {b x+c x^2} \operatorname {Subst}\left (\int x^m \sqrt {1-\frac {x}{d}} \sqrt {1-\frac {c x}{c d-b e}} \, dx,x,d+e x\right )}{e \sqrt {1-\frac {d+e x}{d}} \sqrt {1-\frac {d+e x}{d-\frac {b e}{c}}}}\\ &=\frac {(d+e x)^{1+m} \sqrt {b x+c x^2} F_1\left (1+m;-\frac {1}{2},-\frac {1}{2};2+m;\frac {d+e x}{d},\frac {c (d+e x)}{c d-b e}\right )}{e (1+m) \sqrt {-\frac {e x}{d}} \sqrt {1-\frac {c (d+e x)}{c d-b e}}}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 76, normalized size = 0.72 \[ \frac {2 x \sqrt {x (b+c x)} (d+e x)^m \left (\frac {d+e x}{d}\right )^{-m} F_1\left (\frac {3}{2};-\frac {1}{2},-m;\frac {5}{2};-\frac {c x}{b},-\frac {e x}{d}\right )}{3 \sqrt {\frac {b+c x}{b}}} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 1.33, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\sqrt {c x^{2} + b x} {\left (e x + d\right )}^{m}, x\right ) \]
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 \[ \int \sqrt {c x^{2} + b x} {\left (e x + d\right )}^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.69, size = 0, normalized size = 0.00 \[ \int \sqrt {c \,x^{2}+b x}\, \left (e x +d \right )^{m}\, dx \]
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 \[ \int \sqrt {c x^{2} + b x} {\left (e x + d\right )}^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \sqrt {c\,x^2+b\,x}\,{\left (d+e\,x\right )}^m \,d x \]
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 \[ \int \sqrt {x \left (b + c x\right )} \left (d + e x\right )^{m}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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