Optimal. Leaf size=116 \[ -\frac{2 a^{3/2} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{\sqrt{c}}-\frac{\sqrt{b} (b c-3 a d) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{d^{3/2}}+\frac{b \sqrt{a+b x} \sqrt{c+d x}}{d} \]
[Out]
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Rubi [A] time = 0.295036, antiderivative size = 116, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227 \[ -\frac{2 a^{3/2} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{\sqrt{c}}-\frac{\sqrt{b} (b c-3 a d) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{d^{3/2}}+\frac{b \sqrt{a+b x} \sqrt{c+d x}}{d} \]
Antiderivative was successfully verified.
[In] Int[(a + b*x)^(3/2)/(x*Sqrt[c + d*x]),x]
[Out]
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Rubi in Sympy [A] time = 26.9738, size = 107, normalized size = 0.92 \[ - \frac{2 a^{\frac{3}{2}} \operatorname{atanh}{\left (\frac{\sqrt{c} \sqrt{a + b x}}{\sqrt{a} \sqrt{c + d x}} \right )}}{\sqrt{c}} + \frac{\sqrt{b} \left (3 a d - b c\right ) \operatorname{atanh}{\left (\frac{\sqrt{b} \sqrt{c + d x}}{\sqrt{d} \sqrt{a + b x}} \right )}}{d^{\frac{3}{2}}} + \frac{b \sqrt{a + b x} \sqrt{c + d x}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*x+a)**(3/2)/x/(d*x+c)**(1/2),x)
[Out]
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Mathematica [A] time = 0.30225, size = 159, normalized size = 1.37 \[ -\frac{a^{3/2} \log \left (2 \sqrt{a} \sqrt{c} \sqrt{a+b x} \sqrt{c+d x}+2 a c+a d x+b c x\right )}{\sqrt{c}}+\frac{a^{3/2} \log (x)}{\sqrt{c}}+\frac{\sqrt{b} (3 a d-b c) \log \left (2 \sqrt{b} \sqrt{d} \sqrt{a+b x} \sqrt{c+d x}+a d+b c+2 b d x\right )}{2 d^{3/2}}+\frac{b \sqrt{a+b x} \sqrt{c+d x}}{d} \]
Antiderivative was successfully verified.
[In] Integrate[(a + b*x)^(3/2)/(x*Sqrt[c + d*x]),x]
[Out]
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Maple [B] time = 0.027, size = 220, normalized size = 1.9 \[{\frac{1}{2\,d}\sqrt{bx+a}\sqrt{dx+c} \left ( 3\,\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ) abd\sqrt{ac}-\ln \left ({\frac{1}{2} \left ( 2\,bdx+2\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}+ad+bc \right ){\frac{1}{\sqrt{bd}}}} \right ){b}^{2}c\sqrt{ac}-2\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac}{x}} \right ){a}^{2}d\sqrt{bd}+2\,b\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd} \right ){\frac{1}{\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }}}{\frac{1}{\sqrt{bd}}}{\frac{1}{\sqrt{ac}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x+a)^(3/2)/x/(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((b*x + a)^(3/2)/(sqrt(d*x + c)*x),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.865033, size = 1, normalized size = 0.01 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x + a)^(3/2)/(sqrt(d*x + c)*x),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (a + b x\right )^{\frac{3}{2}}}{x \sqrt{c + d x}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x+a)**(3/2)/x/(d*x+c)**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.234515, size = 267, normalized size = 2.3 \[ -\frac{{\left (\frac{4 \, \sqrt{b d} a^{2} \arctan \left (-\frac{b^{2} c + a b d -{\left (\sqrt{b d} \sqrt{b x + a} - \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d}\right )}^{2}}{2 \, \sqrt{-a b c d} b}\right )}{\sqrt{-a b c d} b} - \frac{2 \, \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d} \sqrt{b x + a}}{b d} - \frac{{\left (\sqrt{b d} b c - 3 \, \sqrt{b d} a d\right )}{\rm ln}\left ({\left (\sqrt{b d} \sqrt{b x + a} - \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d}\right )}^{2}\right )}{b d^{2}}\right )} b^{2}}{2 \,{\left | b \right |}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x + a)^(3/2)/(sqrt(d*x + c)*x),x, algorithm="giac")
[Out]