Optimal. Leaf size=128 \[ -\frac{2 (b c-a d)^{5/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right )}{a^2 b^{3/2}}+\frac{c^{3/2} (2 b c-5 a d) \tanh ^{-1}\left (\frac{\sqrt{c+d x}}{\sqrt{c}}\right )}{a^2}+\frac{d \sqrt{c+d x} (2 a d+b c)}{a b}-\frac{c (c+d x)^{3/2}}{a x} \]
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Rubi [A] time = 0.546668, antiderivative size = 128, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3 \[ -\frac{2 (b c-a d)^{5/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right )}{a^2 b^{3/2}}+\frac{c^{3/2} (2 b c-5 a d) \tanh ^{-1}\left (\frac{\sqrt{c+d x}}{\sqrt{c}}\right )}{a^2}+\frac{d \sqrt{c+d x} (2 a d+b c)}{a b}-\frac{c (c+d x)^{3/2}}{a x} \]
Antiderivative was successfully verified.
[In] Int[(c + d*x)^(5/2)/(x^2*(a + b*x)),x]
[Out]
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Rubi in Sympy [A] time = 52.2457, size = 114, normalized size = 0.89 \[ - \frac{c \left (c + d x\right )^{\frac{3}{2}}}{a x} + \frac{d \sqrt{c + d x} \left (2 a d + b c\right )}{a b} - \frac{c^{\frac{3}{2}} \left (5 a d - 2 b c\right ) \operatorname{atanh}{\left (\frac{\sqrt{c + d x}}{\sqrt{c}} \right )}}{a^{2}} - \frac{2 \left (a d - b c\right )^{\frac{5}{2}} \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{c + d x}}{\sqrt{a d - b c}} \right )}}{a^{2} b^{\frac{3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((d*x+c)**(5/2)/x**2/(b*x+a),x)
[Out]
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Mathematica [A] time = 0.142932, size = 115, normalized size = 0.9 \[ -\frac{2 (b c-a d)^{5/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right )}{a^2 b^{3/2}}+\frac{c^{3/2} (2 b c-5 a d) \tanh ^{-1}\left (\frac{\sqrt{c+d x}}{\sqrt{c}}\right )}{a^2}+\sqrt{c+d x} \left (\frac{2 d^2}{b}-\frac{c^2}{a x}\right ) \]
Antiderivative was successfully verified.
[In] Integrate[(c + d*x)^(5/2)/(x^2*(a + b*x)),x]
[Out]
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Maple [B] time = 0.027, size = 249, normalized size = 2. \[ 2\,{\frac{{d}^{2}\sqrt{dx+c}}{b}}-{\frac{{c}^{2}}{ax}\sqrt{dx+c}}-5\,{\frac{d{c}^{3/2}}{a}{\it Artanh} \left ({\frac{\sqrt{dx+c}}{\sqrt{c}}} \right ) }+2\,{\frac{{c}^{5/2}b}{{a}^{2}}{\it Artanh} \left ({\frac{\sqrt{dx+c}}{\sqrt{c}}} \right ) }-2\,{\frac{{d}^{3}a}{b\sqrt{ \left ( ad-bc \right ) b}}\arctan \left ({\frac{\sqrt{dx+c}b}{\sqrt{ \left ( ad-bc \right ) b}}} \right ) }+6\,{\frac{{d}^{2}c}{\sqrt{ \left ( ad-bc \right ) b}}\arctan \left ({\frac{\sqrt{dx+c}b}{\sqrt{ \left ( ad-bc \right ) b}}} \right ) }-6\,{\frac{bd{c}^{2}}{a\sqrt{ \left ( ad-bc \right ) b}}\arctan \left ({\frac{\sqrt{dx+c}b}{\sqrt{ \left ( ad-bc \right ) b}}} \right ) }+2\,{\frac{{b}^{2}{c}^{3}}{{a}^{2}\sqrt{ \left ( ad-bc \right ) b}}\arctan \left ({\frac{\sqrt{dx+c}b}{\sqrt{ \left ( ad-bc \right ) b}}} \right ) } \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((d*x+c)^(5/2)/x^2/(b*x+a),x)
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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)^(5/2)/((b*x + a)*x^2),x, algorithm="maxima")
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Fricas [A] time = 0.397042, size = 1, normalized size = 0.01 \[ \left [\frac{2 \,{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} x \sqrt{\frac{b c - a d}{b}} \log \left (\frac{b d x + 2 \, b c - a d - 2 \, \sqrt{d x + c} b \sqrt{\frac{b c - a d}{b}}}{b x + a}\right ) -{\left (2 \, b^{2} c^{2} - 5 \, a b c d\right )} \sqrt{c} x \log \left (\frac{d x - 2 \, \sqrt{d x + c} \sqrt{c} + 2 \, c}{x}\right ) + 2 \,{\left (2 \, a^{2} d^{2} x - a b c^{2}\right )} \sqrt{d x + c}}{2 \, a^{2} b x}, -\frac{4 \,{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} x \sqrt{-\frac{b c - a d}{b}} \arctan \left (\frac{\sqrt{d x + c}}{\sqrt{-\frac{b c - a d}{b}}}\right ) +{\left (2 \, b^{2} c^{2} - 5 \, a b c d\right )} \sqrt{c} x \log \left (\frac{d x - 2 \, \sqrt{d x + c} \sqrt{c} + 2 \, c}{x}\right ) - 2 \,{\left (2 \, a^{2} d^{2} x - a b c^{2}\right )} \sqrt{d x + c}}{2 \, a^{2} b x}, \frac{{\left (2 \, b^{2} c^{2} - 5 \, a b c d\right )} \sqrt{-c} x \arctan \left (\frac{\sqrt{d x + c}}{\sqrt{-c}}\right ) +{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} x \sqrt{\frac{b c - a d}{b}} \log \left (\frac{b d x + 2 \, b c - a d - 2 \, \sqrt{d x + c} b \sqrt{\frac{b c - a d}{b}}}{b x + a}\right ) +{\left (2 \, a^{2} d^{2} x - a b c^{2}\right )} \sqrt{d x + c}}{a^{2} b x}, \frac{{\left (2 \, b^{2} c^{2} - 5 \, a b c d\right )} \sqrt{-c} x \arctan \left (\frac{\sqrt{d x + c}}{\sqrt{-c}}\right ) - 2 \,{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} x \sqrt{-\frac{b c - a d}{b}} \arctan \left (\frac{\sqrt{d x + c}}{\sqrt{-\frac{b c - a d}{b}}}\right ) +{\left (2 \, a^{2} d^{2} x - a b c^{2}\right )} \sqrt{d x + c}}{a^{2} b x}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x + c)^(5/2)/((b*x + a)*x^2),x, algorithm="fricas")
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Sympy [A] time = 113.063, size = 860, normalized size = 6.72 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x+c)**(5/2)/x**2/(b*x+a),x)
[Out]
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GIAC/XCAS [A] time = 0.217624, size = 205, normalized size = 1.6 \[ \frac{2 \, \sqrt{d x + c} d^{2}}{b} - \frac{\sqrt{d x + c} c^{2}}{a x} - \frac{{\left (2 \, b c^{3} - 5 \, a c^{2} d\right )} \arctan \left (\frac{\sqrt{d x + c}}{\sqrt{-c}}\right )}{a^{2} \sqrt{-c}} + \frac{2 \,{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \arctan \left (\frac{\sqrt{d x + c} b}{\sqrt{-b^{2} c + a b d}}\right )}{\sqrt{-b^{2} c + a b d} a^{2} b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x + c)^(5/2)/((b*x + a)*x^2),x, algorithm="giac")
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