Optimal. Leaf size=30 \[ -\frac{2 \sqrt{c+d x}}{\sqrt{a+b x} (b c-a d)} \]
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Rubi [A] time = 0.0241024, 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 \[ -\frac{2 \sqrt{c+d x}}{\sqrt{a+b x} (b c-a d)} \]
Antiderivative was successfully verified.
[In] Int[1/((a + b*x)^(3/2)*Sqrt[c + d*x]),x]
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Rubi in Sympy [A] time = 3.78672, size = 24, normalized size = 0.8 \[ \frac{2 \sqrt{c + d x}}{\sqrt{a + b x} \left (a d - b c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(b*x+a)**(3/2)/(d*x+c)**(1/2),x)
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Mathematica [A] time = 0.0327016, size = 30, normalized size = 1. \[ \frac{2 \sqrt{c+d x}}{\sqrt{a+b x} (a d-b c)} \]
Antiderivative was successfully verified.
[In] Integrate[1/((a + b*x)^(3/2)*Sqrt[c + d*x]),x]
[Out]
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Maple [A] time = 0.007, size = 27, normalized size = 0.9 \[ 2\,{\frac{\sqrt{dx+c}}{\sqrt{bx+a} \left ( ad-bc \right ) }} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(b*x+a)^(3/2)/(d*x+c)^(1/2),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/((b*x + a)^(3/2)*sqrt(d*x + c)),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.223804, size = 57, normalized size = 1.9 \[ -\frac{2 \, \sqrt{b x + a} \sqrt{d x + c}}{a b c - a^{2} d +{\left (b^{2} c - a b d\right )} x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x + a)^(3/2)*sqrt(d*x + c)),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\left (a + b x\right )^{\frac{3}{2}} \sqrt{c + d x}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(b*x+a)**(3/2)/(d*x+c)**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.229165, size = 89, normalized size = 2.97 \[ -\frac{4 \, \sqrt{b d} b}{{\left (b^{2} c - a b d -{\left (\sqrt{b d} \sqrt{b x + a} - \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d}\right )}^{2}\right )}{\left | b \right |}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x + a)^(3/2)*sqrt(d*x + c)),x, algorithm="giac")
[Out]