Optimal. Leaf size=180 \[ \frac{(b c-a d)^2 (a d+5 b c) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{8 a^{7/2} c^{3/2}}-\frac{\sqrt{a+b x} \sqrt{c+d x} (b c-a d) (a d+5 b c)}{8 a^3 c x}+\frac{\sqrt{a+b x} (c+d x)^{3/2} (a d+5 b c)}{12 a^2 c x^2}-\frac{\sqrt{a+b x} (c+d x)^{5/2}}{3 a c x^3} \]
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Rubi [A] time = 0.323467, antiderivative size = 180, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ \frac{(b c-a d)^2 (a d+5 b c) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{8 a^{7/2} c^{3/2}}-\frac{\sqrt{a+b x} \sqrt{c+d x} (b c-a d) (a d+5 b c)}{8 a^3 c x}+\frac{\sqrt{a+b x} (c+d x)^{3/2} (a d+5 b c)}{12 a^2 c x^2}-\frac{\sqrt{a+b x} (c+d x)^{5/2}}{3 a c x^3} \]
Antiderivative was successfully verified.
[In] Int[(c + d*x)^(3/2)/(x^4*Sqrt[a + b*x]),x]
[Out]
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Rubi in Sympy [A] time = 25.4729, size = 158, normalized size = 0.88 \[ - \frac{\sqrt{a + b x} \left (c + d x\right )^{\frac{5}{2}}}{3 a c x^{3}} + \frac{\sqrt{a + b x} \left (c + d x\right )^{\frac{3}{2}} \left (a d + 5 b c\right )}{12 a^{2} c x^{2}} + \frac{\sqrt{a + b x} \sqrt{c + d x} \left (a d - b c\right ) \left (a d + 5 b c\right )}{8 a^{3} c x} + \frac{\left (a d - b c\right )^{2} \left (a d + 5 b c\right ) \operatorname{atanh}{\left (\frac{\sqrt{c} \sqrt{a + b x}}{\sqrt{a} \sqrt{c + d x}} \right )}}{8 a^{\frac{7}{2}} c^{\frac{3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((d*x+c)**(3/2)/x**4/(b*x+a)**(1/2),x)
[Out]
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Mathematica [A] time = 0.187439, size = 189, normalized size = 1.05 \[ \frac{-2 \sqrt{a} \sqrt{c} \sqrt{a+b x} \sqrt{c+d x} \left (a^2 \left (8 c^2+14 c d x+3 d^2 x^2\right )-2 a b c x (5 c+11 d x)+15 b^2 c^2 x^2\right )-3 x^3 \log (x) (b c-a d)^2 (a d+5 b c)+3 x^3 (b c-a d)^2 (a d+5 b c) \log \left (2 \sqrt{a} \sqrt{c} \sqrt{a+b x} \sqrt{c+d x}+2 a c+a d x+b c x\right )}{48 a^{7/2} c^{3/2} x^3} \]
Antiderivative was successfully verified.
[In] Integrate[(c + d*x)^(3/2)/(x^4*Sqrt[a + b*x]),x]
[Out]
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Maple [B] time = 0.036, size = 408, normalized size = 2.3 \[{\frac{1}{48\,c{a}^{3}{x}^{3}}\sqrt{bx+a}\sqrt{dx+c} \left ( 3\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac}{x}} \right ){x}^{3}{a}^{3}{d}^{3}+9\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac}{x}} \right ){x}^{3}{a}^{2}bc{d}^{2}-27\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac}{x}} \right ){x}^{3}a{b}^{2}{c}^{2}d+15\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac}{x}} \right ){x}^{3}{b}^{3}{c}^{3}-6\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }{d}^{2}{a}^{2}{x}^{2}\sqrt{ac}+44\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }dbca{x}^{2}\sqrt{ac}-30\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }{b}^{2}{c}^{2}{x}^{2}\sqrt{ac}-28\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }dc{a}^{2}x\sqrt{ac}+20\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }b{c}^{2}ax\sqrt{ac}-16\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }{c}^{2}{a}^{2}\sqrt{ac} \right ){\frac{1}{\sqrt{ac}}}{\frac{1}{\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((d*x+c)^(3/2)/x^4/(b*x+a)^(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((d*x + c)^(3/2)/(sqrt(b*x + a)*x^4),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.439055, size = 1, normalized size = 0.01 \[ \left [\frac{3 \,{\left (5 \, b^{3} c^{3} - 9 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} + a^{3} d^{3}\right )} x^{3} \log \left (\frac{4 \,{\left (2 \, a^{2} c^{2} +{\left (a b c^{2} + a^{2} c d\right )} x\right )} \sqrt{b x + a} \sqrt{d x + c} +{\left (8 \, a^{2} c^{2} +{\left (b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2}\right )} x^{2} + 8 \,{\left (a b c^{2} + a^{2} c d\right )} x\right )} \sqrt{a c}}{x^{2}}\right ) - 4 \,{\left (8 \, a^{2} c^{2} +{\left (15 \, b^{2} c^{2} - 22 \, a b c d + 3 \, a^{2} d^{2}\right )} x^{2} - 2 \,{\left (5 \, a b c^{2} - 7 \, a^{2} c d\right )} x\right )} \sqrt{a c} \sqrt{b x + a} \sqrt{d x + c}}{96 \, \sqrt{a c} a^{3} c x^{3}}, \frac{3 \,{\left (5 \, b^{3} c^{3} - 9 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} + a^{3} d^{3}\right )} x^{3} \arctan \left (\frac{{\left (2 \, a c +{\left (b c + a d\right )} x\right )} \sqrt{-a c}}{2 \, \sqrt{b x + a} \sqrt{d x + c} a c}\right ) - 2 \,{\left (8 \, a^{2} c^{2} +{\left (15 \, b^{2} c^{2} - 22 \, a b c d + 3 \, a^{2} d^{2}\right )} x^{2} - 2 \,{\left (5 \, a b c^{2} - 7 \, a^{2} c d\right )} x\right )} \sqrt{-a c} \sqrt{b x + a} \sqrt{d x + c}}{48 \, \sqrt{-a c} a^{3} c x^{3}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x + c)^(3/2)/(sqrt(b*x + a)*x^4),x, algorithm="fricas")
[Out]
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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((d*x+c)**(3/2)/x**4/(b*x+a)**(1/2),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((d*x + c)^(3/2)/(sqrt(b*x + a)*x^4),x, algorithm="giac")
[Out]