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 = 0.0221214, antiderivative size = 30, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.053 \[ \frac{2 \sqrt{a+b x}}{\sqrt{c+d x} (b c-a d)} \]
Antiderivative was successfully verified.
[In] Int[1/(Sqrt[a + b*x]*(c + d*x)^(3/2)),x]
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Rubi in Sympy [A] time = 3.66133, size = 26, normalized size = 0.87 \[ - \frac{2 \sqrt{a + b x}}{\sqrt{c + d x} \left (a d - b c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(b*x+a)**(1/2)/(d*x+c)**(3/2),x)
[Out]
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Mathematica [A] time = 0.0342251, size = 30, normalized size = 1. \[ -\frac{2 \sqrt{a+b x}}{\sqrt{c+d x} (a d-b c)} \]
Antiderivative was successfully verified.
[In] Integrate[1/(Sqrt[a + b*x]*(c + d*x)^(3/2)),x]
[Out]
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Maple [A] time = 0.007, size = 27, normalized size = 0.9 \[ -2\,{\frac{\sqrt{bx+a}}{\sqrt{dx+c} \left ( ad-bc \right ) }} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(b*x+a)^(1/2)/(d*x+c)^(3/2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(sqrt(b*x + a)*(d*x + c)^(3/2)),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.223569, size = 57, normalized size = 1.9 \[ \frac{2 \, \sqrt{b x + a} \sqrt{d x + c}}{b c^{2} - a c d +{\left (b c d - a d^{2}\right )} x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(sqrt(b*x + a)*(d*x + c)^(3/2)),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{a + b x} \left (c + d x\right )^{\frac{3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(b*x+a)**(1/2)/(d*x+c)**(3/2),x)
[Out]
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GIAC/XCAS [A] time = 0.218348, size = 63, normalized size = 2.1 \[ \frac{2 \, \sqrt{b x + a} b^{2}}{\sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d}{\left (b c{\left | b \right |} - a d{\left | b \right |}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(sqrt(b*x + a)*(d*x + c)^(3/2)),x, algorithm="giac")
[Out]