Optimal. Leaf size=30 \[ -\frac {2 \sqrt {c+d x}}{(b c-a d) \sqrt {a+b x}} \]
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Rubi [A]
time = 0.00, antiderivative size = 30, normalized size of antiderivative = 1.00, number of steps
used = 1, number of rules used = 1, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {37}
\begin {gather*} -\frac {2 \sqrt {c+d x}}{\sqrt {a+b x} (b c-a d)} \end {gather*}
Antiderivative was successfully verified.
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Rule 37
Rubi steps
\begin {align*} \int \frac {1}{(a+b x)^{3/2} \sqrt {c+d x}} \, dx &=-\frac {2 \sqrt {c+d x}}{(b c-a d) \sqrt {a+b x}}\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 30, normalized size = 1.00 \begin {gather*} -\frac {2 \sqrt {c+d x}}{(b c-a d) \sqrt {a+b x}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.16, size = 27, normalized size = 0.90
method | result | size |
gosper | \(\frac {2 \sqrt {d x +c}}{\sqrt {b x +a}\, \left (a d -b c \right )}\) | \(27\) |
default | \(-\frac {2 \sqrt {d x +c}}{\left (-a d +b c \right ) \sqrt {b x +a}}\) | \(27\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.77, size = 42, normalized size = 1.40 \begin {gather*} -\frac {2 \, \sqrt {b x + a} \sqrt {d x + c}}{a b c - a^{2} d + {\left (b^{2} c - a b d\right )} x} \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 {1}{\left (a + b x\right )^{\frac {3}{2}} \sqrt {c + d x}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 66 vs.
\(2 (26) = 52\).
time = 1.28, size = 66, normalized size = 2.20 \begin {gather*} -\frac {4 \, \sqrt {b d} b}{{\left (b^{2} c - a b d - {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2}\right )} {\left | b \right |}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.73, size = 26, normalized size = 0.87 \begin {gather*} \frac {2\,\sqrt {c+d\,x}}{\left (a\,d-b\,c\right )\,\sqrt {a+b\,x}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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