Optimal. Leaf size=30 \[ \frac {2 \sqrt {a+b x}}{\sqrt {c+d x} (b c-a d)} \]
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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} \[ \frac {2 \sqrt {a+b x}}{\sqrt {c+d x} (b c-a d)} \]
Antiderivative was successfully verified.
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Rule 37
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {a+b x} (c+d x)^{3/2}} \, dx &=\frac {2 \sqrt {a+b x}}{(b c-a d) \sqrt {c+d x}}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 30, normalized size = 1.00 \[ \frac {2 \sqrt {a+b x}}{\sqrt {c+d x} (b c-a d)} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.00, size = 42, normalized size = 1.40 \[ \frac {2 \, \sqrt {b x + a} \sqrt {d x + c}}{b c^{2} - a c d + {\left (b c d - a d^{2}\right )} x} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.23, size = 47, normalized size = 1.57 \[ \frac {2 \, \sqrt {b x + a} b^{2}}{\sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d} {\left (b c {\left | b \right |} - a d {\left | b \right |}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 27, normalized size = 0.90 \[ -\frac {2 \sqrt {b x +a}}{\sqrt {d x +c}\, \left (a d -b c \right )} \]
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 \[ \text {Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.31, size = 26, normalized size = 0.87 \[ -\frac {2\,\sqrt {a+b\,x}}{\left (a\,d-b\,c\right )\,\sqrt {c+d\,x}} \]
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 \[ \int \frac {1}{\sqrt {a + b x} \left (c + d x\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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