Optimal. Leaf size=105 \[ \frac {\sqrt {-\frac {e x}{d}} (d+e x)^{1+m} \sqrt {1-\frac {c (d+e x)}{c d-b e}} F_1\left (1+m;\frac {1}{2},\frac {1}{2};2+m;\frac {d+e x}{d},\frac {c (d+e x)}{c d-b e}\right )}{e (1+m) \sqrt {b x+c x^2}} \]
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Rubi [A]
time = 0.03, 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 = {773, 138}
\begin {gather*} \frac {\sqrt {-\frac {e x}{d}} (d+e x)^{m+1} \sqrt {1-\frac {c (d+e x)}{c d-b e}} 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 {b x+c x^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 138
Rule 773
Rubi steps
\begin {align*} \int \frac {(d+e x)^m}{\sqrt {b x+c x^2}} \, dx &=\frac {\left (\sqrt {1-\frac {d+e x}{d}} \sqrt {1-\frac {d+e x}{d-\frac {b e}{c}}}\right ) \text {Subst}\left (\int \frac {x^m}{\sqrt {1-\frac {x}{d}} \sqrt {1-\frac {c x}{c d-b e}}} \, dx,x,d+e x\right )}{e \sqrt {b x+c x^2}}\\ &=\frac {\sqrt {-\frac {e x}{d}} (d+e x)^{1+m} \sqrt {1-\frac {c (d+e x)}{c d-b e}} 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 {b x+c x^2}}\\ \end {align*}
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Mathematica [A]
time = 0.38, size = 74, normalized size = 0.70 \begin {gather*} \frac {2 x \sqrt {\frac {b+c x}{b}} (d+e x)^m \left (\frac {d+e x}{d}\right )^{-m} F_1\left (\frac {1}{2};\frac {1}{2},-m;\frac {3}{2};-\frac {c x}{b},-\frac {e x}{d}\right )}{\sqrt {x (b+c x)}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.12, size = 0, normalized size = 0.00 \[\int \frac {\left (e x +d \right )^{m}}{\sqrt {c \,x^{2}+b x}}\, dx\]
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {\left (d + e x\right )^{m}}{\sqrt {x \left (b + c x\right )}}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {{\left (d+e\,x\right )}^m}{\sqrt {c\,x^2+b\,x}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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