Optimal. Leaf size=118 \[ \frac{2 (b c-a d)^{5/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right )}{a b^{5/2}}+\frac{2 d \sqrt{c+d x} (2 b c-a d)}{b^2}-\frac{2 c^{5/2} \tanh ^{-1}\left (\frac{\sqrt{c+d x}}{\sqrt{c}}\right )}{a}+\frac{2 d (c+d x)^{3/2}}{3 b} \]
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Rubi [A] time = 0.442041, antiderivative size = 118, 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 \[ \frac{2 (b c-a d)^{5/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right )}{a b^{5/2}}+\frac{2 d \sqrt{c+d x} (2 b c-a d)}{b^2}-\frac{2 c^{5/2} \tanh ^{-1}\left (\frac{\sqrt{c+d x}}{\sqrt{c}}\right )}{a}+\frac{2 d (c+d x)^{3/2}}{3 b} \]
Antiderivative was successfully verified.
[In] Int[(c + d*x)^(5/2)/(x*(a + b*x)),x]
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Rubi in Sympy [A] time = 45.3331, size = 105, normalized size = 0.89 \[ \frac{2 d \left (c + d x\right )^{\frac{3}{2}}}{3 b} - \frac{2 d \sqrt{c + d x} \left (a d - 2 b c\right )}{b^{2}} - \frac{2 c^{\frac{5}{2}} \operatorname{atanh}{\left (\frac{\sqrt{c + d x}}{\sqrt{c}} \right )}}{a} + \frac{2 \left (a d - b c\right )^{\frac{5}{2}} \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{c + d x}}{\sqrt{a d - b c}} \right )}}{a b^{\frac{5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((d*x+c)**(5/2)/x/(b*x+a),x)
[Out]
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Mathematica [A] time = 0.281958, size = 107, normalized size = 0.91 \[ \frac{2 (b c-a d)^{5/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right )}{a b^{5/2}}+\frac{2 d \sqrt{c+d x} (-3 a d+7 b c+b d x)}{3 b^2}-\frac{2 c^{5/2} \tanh ^{-1}\left (\frac{\sqrt{c+d x}}{\sqrt{c}}\right )}{a} \]
Antiderivative was successfully verified.
[In] Integrate[(c + d*x)^(5/2)/(x*(a + b*x)),x]
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Maple [B] time = 0.018, size = 237, normalized size = 2. \[{\frac{2\,d}{3\,b} \left ( dx+c \right ) ^{{\frac{3}{2}}}}-2\,{\frac{{d}^{2}a\sqrt{dx+c}}{{b}^{2}}}+4\,{\frac{d\sqrt{dx+c}c}{b}}-2\,{\frac{{c}^{5/2}}{a}{\it Artanh} \left ({\frac{\sqrt{dx+c}}{\sqrt{c}}} \right ) }+2\,{\frac{{a}^{2}{d}^{3}}{{b}^{2}\sqrt{ \left ( ad-bc \right ) b}}\arctan \left ({\frac{\sqrt{dx+c}b}{\sqrt{ \left ( ad-bc \right ) b}}} \right ) }-6\,{\frac{ac{d}^{2}}{b\sqrt{ \left ( ad-bc \right ) b}}\arctan \left ({\frac{\sqrt{dx+c}b}{\sqrt{ \left ( ad-bc \right ) b}}} \right ) }+6\,{\frac{d{c}^{2}}{\sqrt{ \left ( ad-bc \right ) b}}\arctan \left ({\frac{\sqrt{dx+c}b}{\sqrt{ \left ( ad-bc \right ) b}}} \right ) }-2\,{\frac{b{c}^{3}}{a\sqrt{ \left ( ad-bc \right ) b}}\arctan \left ({\frac{\sqrt{dx+c}b}{\sqrt{ \left ( ad-bc \right ) b}}} \right ) } \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((d*x+c)^(5/2)/x/(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),x, algorithm="maxima")
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Fricas [A] time = 0.401278, size = 1, normalized size = 0.01 \[ \left [\frac{3 \, b^{2} c^{\frac{5}{2}} \log \left (\frac{d x - 2 \, \sqrt{d x + c} \sqrt{c} + 2 \, c}{x}\right ) + 3 \,{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \sqrt{\frac{b c - a d}{b}} \log \left (\frac{b d x + 2 \, b c - a d + 2 \, \sqrt{d x + c} b \sqrt{\frac{b c - a d}{b}}}{b x + a}\right ) + 2 \,{\left (a b d^{2} x + 7 \, a b c d - 3 \, a^{2} d^{2}\right )} \sqrt{d x + c}}{3 \, a b^{2}}, \frac{3 \, b^{2} c^{\frac{5}{2}} \log \left (\frac{d x - 2 \, \sqrt{d x + c} \sqrt{c} + 2 \, c}{x}\right ) + 6 \,{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \sqrt{-\frac{b c - a d}{b}} \arctan \left (\frac{\sqrt{d x + c}}{\sqrt{-\frac{b c - a d}{b}}}\right ) + 2 \,{\left (a b d^{2} x + 7 \, a b c d - 3 \, a^{2} d^{2}\right )} \sqrt{d x + c}}{3 \, a b^{2}}, -\frac{6 \, b^{2} \sqrt{-c} c^{2} \arctan \left (\frac{\sqrt{d x + c}}{\sqrt{-c}}\right ) - 3 \,{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \sqrt{\frac{b c - a d}{b}} \log \left (\frac{b d x + 2 \, b c - a d + 2 \, \sqrt{d x + c} b \sqrt{\frac{b c - a d}{b}}}{b x + a}\right ) - 2 \,{\left (a b d^{2} x + 7 \, a b c d - 3 \, a^{2} d^{2}\right )} \sqrt{d x + c}}{3 \, a b^{2}}, -\frac{2 \,{\left (3 \, b^{2} \sqrt{-c} c^{2} \arctan \left (\frac{\sqrt{d x + c}}{\sqrt{-c}}\right ) - 3 \,{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \sqrt{-\frac{b c - a d}{b}} \arctan \left (\frac{\sqrt{d x + c}}{\sqrt{-\frac{b c - a d}{b}}}\right ) -{\left (a b d^{2} x + 7 \, a b c d - 3 \, a^{2} d^{2}\right )} \sqrt{d x + c}\right )}}{3 \, a b^{2}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x + c)^(5/2)/((b*x + a)*x),x, algorithm="fricas")
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Sympy [A] time = 48.9201, size = 294, normalized size = 2.49 \[ \frac{2 d \left (c + d x\right )^{\frac{3}{2}}}{3 b} + \frac{\sqrt{c + d x} \left (- 2 a d^{2} + 4 b c d\right )}{b^{2}} - \frac{2 c^{3} \left (\begin{cases} - \frac{\operatorname{atan}{\left (\frac{\sqrt{c + d x}}{\sqrt{- c}} \right )}}{\sqrt{- c}} & \text{for}\: - c > 0 \\\frac{\operatorname{acoth}{\left (\frac{\sqrt{c + d x}}{\sqrt{c}} \right )}}{\sqrt{c}} & \text{for}\: - c < 0 \wedge c < c + d x \\\frac{\operatorname{atanh}{\left (\frac{\sqrt{c + d x}}{\sqrt{c}} \right )}}{\sqrt{c}} & \text{for}\: c > c + d x \wedge - c < 0 \end{cases}\right )}{a} + \frac{2 \left (a d - b c\right )^{3} \left (\begin{cases} \frac{\operatorname{atan}{\left (\frac{\sqrt{c + d x}}{\sqrt{\frac{a d - b c}{b}}} \right )}}{b \sqrt{\frac{a d - b c}{b}}} & \text{for}\: \frac{a d - b c}{b} > 0 \\- \frac{\operatorname{acoth}{\left (\frac{\sqrt{c + d x}}{\sqrt{\frac{- a d + b c}{b}}} \right )}}{b \sqrt{\frac{- a d + b c}{b}}} & \text{for}\: c + d x > \frac{- a d + b c}{b} \wedge \frac{a d - b c}{b} < 0 \\- \frac{\operatorname{atanh}{\left (\frac{\sqrt{c + d x}}{\sqrt{\frac{- a d + b c}{b}}} \right )}}{b \sqrt{\frac{- a d + b c}{b}}} & \text{for}\: \frac{a d - b c}{b} < 0 \wedge c + d x < \frac{- a d + b c}{b} \end{cases}\right )}{a b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x+c)**(5/2)/x/(b*x+a),x)
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GIAC/XCAS [A] time = 0.234005, size = 208, normalized size = 1.76 \[ \frac{2 \, c^{3} \arctan \left (\frac{\sqrt{d x + c}}{\sqrt{-c}}\right )}{a \sqrt{-c}} - \frac{2 \,{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \arctan \left (\frac{\sqrt{d x + c} b}{\sqrt{-b^{2} c + a b d}}\right )}{\sqrt{-b^{2} c + a b d} a b^{2}} + \frac{2 \,{\left ({\left (d x + c\right )}^{\frac{3}{2}} b^{2} d + 6 \, \sqrt{d x + c} b^{2} c d - 3 \, \sqrt{d x + c} a b d^{2}\right )}}{3 \, b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x + c)^(5/2)/((b*x + a)*x),x, algorithm="giac")
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