Optimal. Leaf size=128 \[ \frac {2 \sqrt {a+b x} \sqrt {\frac {b (c+d x)}{b c-a d}} (e+f x)^n \left (\frac {b (e+f x)}{b e-a f}\right )^{-n} F_1\left (\frac {1}{2};\frac {3}{2},-n;\frac {3}{2};-\frac {d (a+b x)}{b c-a d},-\frac {f (a+b x)}{b e-a f}\right )}{(b c-a d) \sqrt {c+d x}} \]
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Rubi [A]
time = 0.05, antiderivative size = 128, normalized size of antiderivative = 1.00, 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 = {145, 144, 143}
\begin {gather*} \frac {2 \sqrt {a+b x} (e+f x)^n \sqrt {\frac {b (c+d x)}{b c-a d}} \left (\frac {b (e+f x)}{b e-a f}\right )^{-n} F_1\left (\frac {1}{2};\frac {3}{2},-n;\frac {3}{2};-\frac {d (a+b x)}{b c-a d},-\frac {f (a+b x)}{b e-a f}\right )}{\sqrt {c+d x} (b c-a d)} \end {gather*}
Antiderivative was successfully verified.
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Rule 143
Rule 144
Rule 145
Rubi steps
\begin {align*} \int \frac {(e+f x)^n}{\sqrt {a+b x} (c+d x)^{3/2}} \, dx &=\frac {\left (b \sqrt {\frac {b (c+d x)}{b c-a d}}\right ) \int \frac {(e+f x)^n}{\sqrt {a+b x} \left (\frac {b c}{b c-a d}+\frac {b d x}{b c-a d}\right )^{3/2}} \, dx}{(b c-a d) \sqrt {c+d x}}\\ &=\frac {\left (b \sqrt {\frac {b (c+d x)}{b c-a d}} (e+f x)^n \left (\frac {b (e+f x)}{b e-a f}\right )^{-n}\right ) \int \frac {\left (\frac {b e}{b e-a f}+\frac {b f x}{b e-a f}\right )^n}{\sqrt {a+b x} \left (\frac {b c}{b c-a d}+\frac {b d x}{b c-a d}\right )^{3/2}} \, dx}{(b c-a d) \sqrt {c+d x}}\\ &=\frac {2 \sqrt {a+b x} \sqrt {\frac {b (c+d x)}{b c-a d}} (e+f x)^n \left (\frac {b (e+f x)}{b e-a f}\right )^{-n} F_1\left (\frac {1}{2};\frac {3}{2},-n;\frac {3}{2};-\frac {d (a+b x)}{b c-a d},-\frac {f (a+b x)}{b e-a f}\right )}{(b c-a d) \sqrt {c+d x}}\\ \end {align*}
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Mathematica [A]
time = 3.06, size = 241, normalized size = 1.88 \begin {gather*} -\frac {2 \sqrt {a+b x} \sqrt {\frac {b (c+d x)}{b c-a d}} (e+f x)^n \left (\frac {b (e+f x)}{b e-a f}\right )^{-n} \left ((-3 b c+3 a d) F_1\left (\frac {1}{2};-\frac {1}{2},-n;\frac {3}{2};\frac {d (a+b x)}{-b c+a d},\frac {f (a+b x)}{-b e+a f}\right )+d (a+b x) \left (F_1\left (\frac {3}{2};\frac {1}{2},-n;\frac {5}{2};\frac {d (a+b x)}{-b c+a d},\frac {f (a+b x)}{-b e+a f}\right )+F_1\left (\frac {3}{2};\frac {3}{2},-n;\frac {5}{2};\frac {d (a+b x)}{-b c+a d},\frac {f (a+b x)}{-b e+a f}\right )\right )\right )}{3 (b c-a d)^2 \sqrt {c+d x}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {\left (f x +e \right )^{n}}{\left (d x +c \right )^{\frac {3}{2}} \sqrt {b x +a}}\, dx\]
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 (e + f x\right )^{n}}{\sqrt {a + b x} \left (c + d x\right )^{\frac {3}{2}}}\, 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 (e+f\,x\right )}^n}{\sqrt {a+b\,x}\,{\left (c+d\,x\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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