Optimal. Leaf size=100 \[ -\frac{3 d^2 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right )}{4 b^{5/2} \sqrt{b c-a d}}-\frac{3 d \sqrt{c+d x}}{4 b^2 (a+b x)}-\frac{(c+d x)^{3/2}}{2 b (a+b x)^2} \]
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Rubi [A] time = 0.120321, antiderivative size = 100, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176 \[ -\frac{3 d^2 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right )}{4 b^{5/2} \sqrt{b c-a d}}-\frac{3 d \sqrt{c+d x}}{4 b^2 (a+b x)}-\frac{(c+d x)^{3/2}}{2 b (a+b x)^2} \]
Antiderivative was successfully verified.
[In] Int[(c + d*x)^(3/2)/(a + b*x)^3,x]
[Out]
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Rubi in Sympy [A] time = 19.0541, size = 87, normalized size = 0.87 \[ - \frac{\left (c + d x\right )^{\frac{3}{2}}}{2 b \left (a + b x\right )^{2}} - \frac{3 d \sqrt{c + d x}}{4 b^{2} \left (a + b x\right )} + \frac{3 d^{2} \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{c + d x}}{\sqrt{a d - b c}} \right )}}{4 b^{\frac{5}{2}} \sqrt{a d - b c}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((d*x+c)**(3/2)/(b*x+a)**3,x)
[Out]
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Mathematica [A] time = 0.110736, size = 90, normalized size = 0.9 \[ -\frac{3 d^2 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right )}{4 b^{5/2} \sqrt{b c-a d}}-\frac{\sqrt{c+d x} (3 a d+2 b c+5 b d x)}{4 b^2 (a+b x)^2} \]
Antiderivative was successfully verified.
[In] Integrate[(c + d*x)^(3/2)/(a + b*x)^3,x]
[Out]
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Maple [A] time = 0.017, size = 121, normalized size = 1.2 \[ -{\frac{5\,{d}^{2}}{4\, \left ( bdx+ad \right ) ^{2}b} \left ( dx+c \right ) ^{{\frac{3}{2}}}}-{\frac{3\,{d}^{3}a}{4\, \left ( bdx+ad \right ) ^{2}{b}^{2}}\sqrt{dx+c}}+{\frac{3\,{d}^{2}c}{4\, \left ( bdx+ad \right ) ^{2}b}\sqrt{dx+c}}+{\frac{3\,{d}^{2}}{4\,{b}^{2}}\arctan \left ({b\sqrt{dx+c}{\frac{1}{\sqrt{ \left ( ad-bc \right ) b}}}} \right ){\frac{1}{\sqrt{ \left ( ad-bc \right ) b}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((d*x+c)^(3/2)/(b*x+a)^3,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)/(b*x + a)^3,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.234609, size = 1, normalized size = 0.01 \[ \left [-\frac{2 \, \sqrt{b^{2} c - a b d}{\left (5 \, b d x + 2 \, b c + 3 \, a d\right )} \sqrt{d x + c} - 3 \,{\left (b^{2} d^{2} x^{2} + 2 \, a b d^{2} x + a^{2} d^{2}\right )} \log \left (\frac{\sqrt{b^{2} c - a b d}{\left (b d x + 2 \, b c - a d\right )} - 2 \,{\left (b^{2} c - a b d\right )} \sqrt{d x + c}}{b x + a}\right )}{8 \,{\left (b^{4} x^{2} + 2 \, a b^{3} x + a^{2} b^{2}\right )} \sqrt{b^{2} c - a b d}}, -\frac{\sqrt{-b^{2} c + a b d}{\left (5 \, b d x + 2 \, b c + 3 \, a d\right )} \sqrt{d x + c} + 3 \,{\left (b^{2} d^{2} x^{2} + 2 \, a b d^{2} x + a^{2} d^{2}\right )} \arctan \left (-\frac{b c - a d}{\sqrt{-b^{2} c + a b d} \sqrt{d x + c}}\right )}{4 \,{\left (b^{4} x^{2} + 2 \, a b^{3} x + a^{2} b^{2}\right )} \sqrt{-b^{2} c + a b d}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x + c)^(3/2)/(b*x + a)^3,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)/(b*x+a)**3,x)
[Out]
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GIAC/XCAS [A] time = 0.233498, size = 146, normalized size = 1.46 \[ \frac{3 \, d^{2} \arctan \left (\frac{\sqrt{d x + c} b}{\sqrt{-b^{2} c + a b d}}\right )}{4 \, \sqrt{-b^{2} c + a b d} b^{2}} - \frac{5 \,{\left (d x + c\right )}^{\frac{3}{2}} b d^{2} - 3 \, \sqrt{d x + c} b c d^{2} + 3 \, \sqrt{d x + c} a d^{3}}{4 \,{\left ({\left (d x + c\right )} b - b c + a d\right )}^{2} b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x + c)^(3/2)/(b*x + a)^3,x, algorithm="giac")
[Out]