Optimal. Leaf size=171 \[ \frac{\left (-15 a^2 d^2+6 a b c d+b^2 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{4 a^{3/2} c^{7/2}}-\frac{d \sqrt{a+b x} (b c-15 a d)}{4 a c^3 \sqrt{c+d x}}-\frac{\sqrt{a+b x} (b c-5 a d)}{4 a c^2 x \sqrt{c+d x}}-\frac{\sqrt{a+b x}}{2 c x^2 \sqrt{c+d x}} \]
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Rubi [A] time = 0.517432, antiderivative size = 171, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273 \[ \frac{\left (-15 a^2 d^2+6 a b c d+b^2 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{4 a^{3/2} c^{7/2}}-\frac{d \sqrt{a+b x} (b c-15 a d)}{4 a c^3 \sqrt{c+d x}}-\frac{\sqrt{a+b x} (b c-5 a d)}{4 a c^2 x \sqrt{c+d x}}-\frac{\sqrt{a+b x}}{2 c x^2 \sqrt{c+d x}} \]
Antiderivative was successfully verified.
[In] Int[Sqrt[a + b*x]/(x^3*(c + d*x)^(3/2)),x]
[Out]
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Rubi in Sympy [A] time = 73.024, size = 155, normalized size = 0.91 \[ - \frac{\sqrt{a + b x}}{2 c x^{2} \sqrt{c + d x}} + \frac{\sqrt{a + b x} \left (5 a d - b c\right )}{4 a c^{2} x \sqrt{c + d x}} + \frac{d \sqrt{a + b x} \left (15 a d - b c\right )}{4 a c^{3} \sqrt{c + d x}} - \frac{\left (15 a^{2} d^{2} - 6 a b c d - b^{2} c^{2}\right ) \operatorname{atanh}{\left (\frac{\sqrt{c} \sqrt{a + b x}}{\sqrt{a} \sqrt{c + d x}} \right )}}{4 a^{\frac{3}{2}} c^{\frac{7}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*x+a)**(1/2)/x**3/(d*x+c)**(3/2),x)
[Out]
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Mathematica [A] time = 0.208688, size = 173, normalized size = 1.01 \[ \frac{-\log (x) \left (-15 a^2 d^2+6 a b c d+b^2 c^2\right )+\left (-15 a^2 d^2+6 a b c d+b^2 c^2\right ) \log \left (2 \sqrt{a} \sqrt{c} \sqrt{a+b x} \sqrt{c+d x}+2 a c+a d x+b c x\right )+\frac{2 \sqrt{a} \sqrt{c} \sqrt{a+b x} \left (a \left (-2 c^2+5 c d x+15 d^2 x^2\right )-b c x (c+d x)\right )}{x^2 \sqrt{c+d x}}}{8 a^{3/2} c^{7/2}} \]
Antiderivative was successfully verified.
[In] Integrate[Sqrt[a + b*x]/(x^3*(c + d*x)^(3/2)),x]
[Out]
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Maple [B] time = 0.045, size = 467, normalized size = 2.7 \[ -{\frac{1}{8\,{c}^{3}a{x}^{2}}\sqrt{bx+a} \left ( 15\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac}{x}} \right ){x}^{3}{a}^{2}{d}^{3}-6\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac}{x}} \right ){x}^{3}abc{d}^{2}-\ln \left ({\frac{1}{x} \left ( adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac \right ) } \right ){x}^{3}{b}^{2}{c}^{2}d+15\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac}{x}} \right ){x}^{2}{a}^{2}c{d}^{2}-6\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac}{x}} \right ){x}^{2}ab{c}^{2}d-\ln \left ({\frac{1}{x} \left ( adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac \right ) } \right ){x}^{2}{b}^{2}{c}^{3}-30\,{x}^{2}a{d}^{2}\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,{x}^{2}bcd\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }-10\,xacd\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,xb{c}^{2}\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+4\,a{c}^{2}\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) } \right ){\frac{1}{\sqrt{ac}}}{\frac{1}{\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }}}{\frac{1}{\sqrt{dx+c}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x+a)^(1/2)/x^3/(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(sqrt(b*x + a)/((d*x + c)^(3/2)*x^3),x, algorithm="maxima")
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Fricas [A] time = 0.383254, size = 1, normalized size = 0.01 \[ \left [-\frac{4 \,{\left (2 \, a c^{2} +{\left (b c d - 15 \, a d^{2}\right )} x^{2} +{\left (b c^{2} - 5 \, a c d\right )} x\right )} \sqrt{a c} \sqrt{b x + a} \sqrt{d x + c} +{\left ({\left (b^{2} c^{2} d + 6 \, a b c d^{2} - 15 \, a^{2} d^{3}\right )} x^{3} +{\left (b^{2} c^{3} + 6 \, a b c^{2} d - 15 \, a^{2} c d^{2}\right )} x^{2}\right )} \log \left (-\frac{4 \,{\left (2 \, a^{2} c^{2} +{\left (a b c^{2} + a^{2} c d\right )} x\right )} \sqrt{b x + a} \sqrt{d x + c} -{\left (8 \, a^{2} c^{2} +{\left (b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2}\right )} x^{2} + 8 \,{\left (a b c^{2} + a^{2} c d\right )} x\right )} \sqrt{a c}}{x^{2}}\right )}{16 \,{\left (a c^{3} d x^{3} + a c^{4} x^{2}\right )} \sqrt{a c}}, -\frac{2 \,{\left (2 \, a c^{2} +{\left (b c d - 15 \, a d^{2}\right )} x^{2} +{\left (b c^{2} - 5 \, a c d\right )} x\right )} \sqrt{-a c} \sqrt{b x + a} \sqrt{d x + c} -{\left ({\left (b^{2} c^{2} d + 6 \, a b c d^{2} - 15 \, a^{2} d^{3}\right )} x^{3} +{\left (b^{2} c^{3} + 6 \, a b c^{2} d - 15 \, a^{2} c d^{2}\right )} x^{2}\right )} \arctan \left (\frac{{\left (2 \, a c +{\left (b c + a d\right )} x\right )} \sqrt{-a c}}{2 \, \sqrt{b x + a} \sqrt{d x + c} a c}\right )}{8 \,{\left (a c^{3} d x^{3} + a c^{4} x^{2}\right )} \sqrt{-a c}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(b*x + a)/((d*x + c)^(3/2)*x^3),x, algorithm="fricas")
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x+a)**(1/2)/x**3/(d*x+c)**(3/2),x)
[Out]
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GIAC/XCAS [F(-2)] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(b*x + a)/((d*x + c)^(3/2)*x^3),x, algorithm="giac")
[Out]