Optimal. Leaf size=217 \[ -\frac{5 (b c-a d)^3 (7 a d+b c) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{64 b^{9/2} d^{3/2}}-\frac{5 \sqrt{a+b x} \sqrt{c+d x} (b c-a d)^2 (7 a d+b c)}{64 b^4 d}-\frac{5 \sqrt{a+b x} (c+d x)^{3/2} (b c-a d) (7 a d+b c)}{96 b^3 d}-\frac{\sqrt{a+b x} (c+d x)^{5/2} (7 a d+b c)}{24 b^2 d}+\frac{\sqrt{a+b x} (c+d x)^{7/2}}{4 b d} \]
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Rubi [A] time = 0.309901, antiderivative size = 217, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ -\frac{5 (b c-a d)^3 (7 a d+b c) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{64 b^{9/2} d^{3/2}}-\frac{5 \sqrt{a+b x} \sqrt{c+d x} (b c-a d)^2 (7 a d+b c)}{64 b^4 d}-\frac{5 \sqrt{a+b x} (c+d x)^{3/2} (b c-a d) (7 a d+b c)}{96 b^3 d}-\frac{\sqrt{a+b x} (c+d x)^{5/2} (7 a d+b c)}{24 b^2 d}+\frac{\sqrt{a+b x} (c+d x)^{7/2}}{4 b d} \]
Antiderivative was successfully verified.
[In] Int[(x*(c + d*x)^(5/2))/Sqrt[a + b*x],x]
[Out]
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Rubi in Sympy [A] time = 33.8133, size = 196, normalized size = 0.9 \[ \frac{\sqrt{a + b x} \left (c + d x\right )^{\frac{7}{2}}}{4 b d} - \frac{\sqrt{a + b x} \left (c + d x\right )^{\frac{5}{2}} \left (7 a d + b c\right )}{24 b^{2} d} + \frac{5 \sqrt{a + b x} \left (c + d x\right )^{\frac{3}{2}} \left (a d - b c\right ) \left (7 a d + b c\right )}{96 b^{3} d} - \frac{5 \sqrt{a + b x} \sqrt{c + d x} \left (a d - b c\right )^{2} \left (7 a d + b c\right )}{64 b^{4} d} + \frac{5 \left (a d - b c\right )^{3} \left (7 a d + b c\right ) \operatorname{atanh}{\left (\frac{\sqrt{d} \sqrt{a + b x}}{\sqrt{b} \sqrt{c + d x}} \right )}}{64 b^{\frac{9}{2}} d^{\frac{3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x*(d*x+c)**(5/2)/(b*x+a)**(1/2),x)
[Out]
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Mathematica [A] time = 0.189616, size = 190, normalized size = 0.88 \[ \frac{\sqrt{a+b x} \sqrt{c+d x} \left (-105 a^3 d^3+5 a^2 b d^2 (53 c+14 d x)-a b^2 d \left (191 c^2+172 c d x+56 d^2 x^2\right )+b^3 \left (15 c^3+118 c^2 d x+136 c d^2 x^2+48 d^3 x^3\right )\right )}{192 b^4 d}-\frac{5 (b c-a d)^3 (7 a d+b c) \log \left (2 \sqrt{b} \sqrt{d} \sqrt{a+b x} \sqrt{c+d x}+a d+b c+2 b d x\right )}{128 b^{9/2} d^{3/2}} \]
Antiderivative was successfully verified.
[In] Integrate[(x*(c + d*x)^(5/2))/Sqrt[a + b*x],x]
[Out]
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Maple [B] time = 0.029, size = 574, normalized size = 2.7 \[{\frac{1}{384\,{b}^{4}d}\sqrt{bx+a}\sqrt{dx+c} \left ( 96\,{x}^{3}{b}^{3}{d}^{3}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}-112\,{x}^{2}a{b}^{2}{d}^{3}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}+272\,{x}^{2}{b}^{3}c{d}^{2}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}+105\,\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ){a}^{4}{d}^{4}-300\,\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ){a}^{3}bc{d}^{3}+270\,\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ){a}^{2}{b}^{2}{c}^{2}{d}^{2}-60\,{c}^{3}\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ) a{b}^{3}d-15\,{c}^{4}\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ){b}^{4}+140\,\sqrt{bd}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }x{a}^{2}b{d}^{3}-344\,\sqrt{bd}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }xa{b}^{2}c{d}^{2}+236\,\sqrt{bd}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }x{b}^{3}{c}^{2}d-210\,\sqrt{bd}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }{a}^{3}{d}^{3}+530\,\sqrt{bd}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }{a}^{2}bc{d}^{2}-382\,\sqrt{bd}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }a{b}^{2}{c}^{2}d+30\,{c}^{3}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }{b}^{3}\sqrt{bd} \right ){\frac{1}{\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }}}{\frac{1}{\sqrt{bd}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x*(d*x+c)^(5/2)/(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)^(5/2)*x/sqrt(b*x + a),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.278947, size = 1, normalized size = 0. \[ \left [\frac{4 \,{\left (48 \, b^{3} d^{3} x^{3} + 15 \, b^{3} c^{3} - 191 \, a b^{2} c^{2} d + 265 \, a^{2} b c d^{2} - 105 \, a^{3} d^{3} + 8 \,{\left (17 \, b^{3} c d^{2} - 7 \, a b^{2} d^{3}\right )} x^{2} + 2 \,{\left (59 \, b^{3} c^{2} d - 86 \, a b^{2} c d^{2} + 35 \, a^{2} b d^{3}\right )} x\right )} \sqrt{b d} \sqrt{b x + a} \sqrt{d x + c} - 15 \,{\left (b^{4} c^{4} + 4 \, a b^{3} c^{3} d - 18 \, a^{2} b^{2} c^{2} d^{2} + 20 \, a^{3} b c d^{3} - 7 \, a^{4} d^{4}\right )} \log \left (4 \,{\left (2 \, b^{2} d^{2} x + b^{2} c d + a b d^{2}\right )} \sqrt{b x + a} \sqrt{d x + c} +{\left (8 \, b^{2} d^{2} x^{2} + b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2} + 8 \,{\left (b^{2} c d + a b d^{2}\right )} x\right )} \sqrt{b d}\right )}{768 \, \sqrt{b d} b^{4} d}, \frac{2 \,{\left (48 \, b^{3} d^{3} x^{3} + 15 \, b^{3} c^{3} - 191 \, a b^{2} c^{2} d + 265 \, a^{2} b c d^{2} - 105 \, a^{3} d^{3} + 8 \,{\left (17 \, b^{3} c d^{2} - 7 \, a b^{2} d^{3}\right )} x^{2} + 2 \,{\left (59 \, b^{3} c^{2} d - 86 \, a b^{2} c d^{2} + 35 \, a^{2} b d^{3}\right )} x\right )} \sqrt{-b d} \sqrt{b x + a} \sqrt{d x + c} - 15 \,{\left (b^{4} c^{4} + 4 \, a b^{3} c^{3} d - 18 \, a^{2} b^{2} c^{2} d^{2} + 20 \, a^{3} b c d^{3} - 7 \, a^{4} d^{4}\right )} \arctan \left (\frac{{\left (2 \, b d x + b c + a d\right )} \sqrt{-b d}}{2 \, \sqrt{b x + a} \sqrt{d x + c} b d}\right )}{384 \, \sqrt{-b d} b^{4} d}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x + c)^(5/2)*x/sqrt(b*x + a),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(x*(d*x+c)**(5/2)/(b*x+a)**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.293011, size = 875, normalized size = 4.03 \[ \frac{\frac{16 \,{\left (\sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d} \sqrt{b x + a}{\left (2 \,{\left (b x + a\right )}{\left (\frac{4 \,{\left (b x + a\right )}}{b^{2}} + \frac{b^{6} c d^{3} - 13 \, a b^{5} d^{4}}{b^{7} d^{4}}\right )} - \frac{3 \,{\left (b^{7} c^{2} d^{2} + 2 \, a b^{6} c d^{3} - 11 \, a^{2} b^{5} d^{4}\right )}}{b^{7} d^{4}}\right )} - \frac{3 \,{\left (b^{3} c^{3} + a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - 5 \, a^{3} d^{3}\right )}{\rm ln}\left ({\left | -\sqrt{b d} \sqrt{b x + a} + \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d} \right |}\right )}{\sqrt{b d} b d^{2}}\right )} c d{\left | b \right |}}{b^{2}} + \frac{{\left (\sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d}{\left (2 \,{\left (b x + a\right )}{\left (4 \,{\left (b x + a\right )}{\left (\frac{6 \,{\left (b x + a\right )}}{b^{3}} + \frac{b^{12} c d^{5} - 25 \, a b^{11} d^{6}}{b^{14} d^{6}}\right )} - \frac{5 \, b^{13} c^{2} d^{4} + 14 \, a b^{12} c d^{5} - 163 \, a^{2} b^{11} d^{6}}{b^{14} d^{6}}\right )} + \frac{3 \,{\left (5 \, b^{14} c^{3} d^{3} + 9 \, a b^{13} c^{2} d^{4} + 15 \, a^{2} b^{12} c d^{5} - 93 \, a^{3} b^{11} d^{6}\right )}}{b^{14} d^{6}}\right )} \sqrt{b x + a} + \frac{3 \,{\left (5 \, b^{4} c^{4} + 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} + 20 \, a^{3} b c d^{3} - 35 \, a^{4} d^{4}\right )}{\rm ln}\left ({\left | -\sqrt{b d} \sqrt{b x + a} + \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d} \right |}\right )}{\sqrt{b d} b^{2} d^{3}}\right )} d^{2}{\left | b \right |}}{b^{2}} + \frac{4 \,{\left (\sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d} \sqrt{b x + a}{\left (\frac{2 \,{\left (b x + a\right )}}{b^{4} d^{2}} + \frac{b c d - 5 \, a d^{2}}{b^{4} d^{4}}\right )} + \frac{{\left (b^{2} c^{2} + 2 \, a b c d - 3 \, a^{2} d^{2}\right )}{\rm ln}\left ({\left | -\sqrt{b d} \sqrt{b x + a} + \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d} \right |}\right )}{\sqrt{b d} b^{3} d^{3}}\right )} c^{2}{\left | b \right |}}{b^{3}}}{192 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x + c)^(5/2)*x/sqrt(b*x + a),x, algorithm="giac")
[Out]