Optimal. Leaf size=299 \[ -2 a^{5/2} c^{3/2} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )+\frac{\sqrt{a+b x} \sqrt{c+d x} \left (64 a^2 b c d^2+(b c-5 a d) (b c-a d) (a d+3 b c)\right )}{64 b d^2}+\frac{\left (-5 a^4 d^4+60 a^3 b c d^3+90 a^2 b^2 c^2 d^2-20 a b^3 c^3 d+3 b^4 c^4\right ) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{64 b^{3/2} d^{5/2}}-\frac{\sqrt{a+b x} (c+d x)^{3/2} (b c-5 a d) (a d+3 b c)}{32 d^2}+\frac{1}{4} (a+b x)^{5/2} (c+d x)^{3/2}+\frac{(a+b x)^{3/2} (c+d x)^{3/2} (5 a d+3 b c)}{24 d} \]
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Rubi [A] time = 1.11111, antiderivative size = 294, normalized size of antiderivative = 0.98, number of steps used = 9, number of rules used = 6, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273 \[ -2 a^{5/2} c^{3/2} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )+\frac{1}{64} \sqrt{a+b x} \sqrt{c+d x} \left (\frac{5 a^3 d}{b}+73 a^2 c-\frac{17 a b c^2}{d}+\frac{3 b^2 c^3}{d^2}\right )+\frac{\left (-5 a^4 d^4+60 a^3 b c d^3+90 a^2 b^2 c^2 d^2-20 a b^3 c^3 d+3 b^4 c^4\right ) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{64 b^{3/2} d^{5/2}}-\frac{\sqrt{a+b x} (c+d x)^{3/2} (b c-5 a d) (a d+3 b c)}{32 d^2}+\frac{1}{4} (a+b x)^{5/2} (c+d x)^{3/2}+\frac{(a+b x)^{3/2} (c+d x)^{3/2} (5 a d+3 b c)}{24 d} \]
Antiderivative was successfully verified.
[In] Int[((a + b*x)^(5/2)*(c + d*x)^(3/2))/x,x]
[Out]
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Rubi in Sympy [A] time = 98.3013, size = 298, normalized size = 1. \[ - 2 a^{\frac{5}{2}} c^{\frac{3}{2}} \operatorname{atanh}{\left (\frac{\sqrt{c} \sqrt{a + b x}}{\sqrt{a} \sqrt{c + d x}} \right )} + \frac{\left (a + b x\right )^{\frac{5}{2}} \left (c + d x\right )^{\frac{3}{2}}}{4} + \frac{\left (a + b x\right )^{\frac{5}{2}} \sqrt{c + d x} \left (5 a d + 3 b c\right )}{24 b} - \frac{\left (a + b x\right )^{\frac{3}{2}} \sqrt{c + d x} \left (5 a^{2} d^{2} - 50 a b c d - 3 b^{2} c^{2}\right )}{96 b d} - \frac{\sqrt{a + b x} \sqrt{c + d x} \left (5 a^{3} d^{3} - 55 a^{2} b c d^{2} - 17 a b^{2} c^{2} d + 3 b^{3} c^{3}\right )}{64 b d^{2}} - \frac{\left (5 a^{4} d^{4} - 60 a^{3} b c d^{3} - 90 a^{2} b^{2} c^{2} d^{2} + 20 a b^{3} c^{3} d - 3 b^{4} c^{4}\right ) \operatorname{atanh}{\left (\frac{\sqrt{d} \sqrt{a + b x}}{\sqrt{b} \sqrt{c + d x}} \right )}}{64 b^{\frac{3}{2}} d^{\frac{5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*x+a)**(5/2)*(d*x+c)**(3/2)/x,x)
[Out]
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Mathematica [A] time = 0.211053, size = 290, normalized size = 0.97 \[ -a^{5/2} c^{3/2} \log \left (2 \sqrt{a} \sqrt{c} \sqrt{a+b x} \sqrt{c+d x}+2 a c+a d x+b c x\right )+a^{5/2} c^{3/2} \log (x)+\frac{\sqrt{a+b x} \sqrt{c+d x} \left (15 a^3 d^3+a^2 b d^2 (337 c+118 d x)+a b^2 d \left (57 c^2+244 c d x+136 d^2 x^2\right )+b^3 \left (-9 c^3+6 c^2 d x+72 c d^2 x^2+48 d^3 x^3\right )\right )}{192 b d^2}+\frac{\left (-5 a^4 d^4+60 a^3 b c d^3+90 a^2 b^2 c^2 d^2-20 a b^3 c^3 d+3 b^4 c^4\right ) \log \left (2 \sqrt{b} \sqrt{d} \sqrt{a+b x} \sqrt{c+d x}+a d+b c+2 b d x\right )}{128 b^{3/2} d^{5/2}} \]
Antiderivative was successfully verified.
[In] Integrate[((a + b*x)^(5/2)*(c + d*x)^(3/2))/x,x]
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Maple [B] time = 0.023, size = 828, normalized size = 2.8 \[ -{\frac{1}{384\,{d}^{2}b}\sqrt{bx+a}\sqrt{dx+c} \left ( -96\,{x}^{3}{b}^{3}{d}^{3}\sqrt{bd}\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}-272\,{x}^{2}a{b}^{2}{d}^{3}\sqrt{bd}\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}-144\,{x}^{2}{b}^{3}c{d}^{2}\sqrt{bd}\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}+384\,{a}^{3}{c}^{2}\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}+2\,ac}{x}} \right ){d}^{2}b\sqrt{bd}+15\,{d}^{4}{a}^{4}\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ) \sqrt{ac}-180\,{d}^{3}{a}^{3}\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ) c\sqrt{ac}b-270\,{c}^{2}{d}^{2}\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ){a}^{2}\sqrt{ac}{b}^{2}+60\,{b}^{3}{c}^{3}\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ) a\sqrt{ac}d-9\,{b}^{4}{c}^{4}\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ) \sqrt{ac}-236\,{d}^{3}{a}^{2}\sqrt{d{x}^{2}b+adx+bcx+ac}x\sqrt{ac}b\sqrt{bd}-488\,{d}^{2}a\sqrt{d{x}^{2}b+adx+bcx+ac}xc\sqrt{ac}{b}^{2}\sqrt{bd}-12\,{b}^{3}{c}^{2}\sqrt{d{x}^{2}b+adx+bcx+ac}x\sqrt{ac}d\sqrt{bd}-30\,{d}^{3}{a}^{3}\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{ac}\sqrt{bd}-674\,{d}^{2}{a}^{2}\sqrt{d{x}^{2}b+adx+bcx+ac}c\sqrt{ac}b\sqrt{bd}-114\,{c}^{2}\sqrt{d{x}^{2}b+adx+bcx+ac}a\sqrt{ac}{b}^{2}d\sqrt{bd}+18\,{b}^{3}{c}^{3}\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{ac}\sqrt{bd} \right ){\frac{1}{\sqrt{ac}}}{\frac{1}{\sqrt{d{x}^{2}b+adx+bcx+ac}}}{\frac{1}{\sqrt{bd}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x+a)^(5/2)*(d*x+c)^(3/2)/x,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((b*x + a)^(5/2)*(d*x + c)^(3/2)/x,x, algorithm="maxima")
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Fricas [A] time = 17.3131, size = 1, normalized size = 0. \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x + a)^(5/2)*(d*x + c)^(3/2)/x,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((b*x+a)**(5/2)*(d*x+c)**(3/2)/x,x)
[Out]
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GIAC/XCAS [A] time = 0.358183, size = 574, normalized size = 1.92 \[ -\frac{2 \, \sqrt{b d} a^{3} c^{2}{\left | b \right |} \arctan \left (-\frac{b^{2} c + a b d -{\left (\sqrt{b d} \sqrt{b x + a} - \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d}\right )}^{2}}{2 \, \sqrt{-a b c d} b}\right )}{\sqrt{-a b c d} b} + \frac{1}{192} \, \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d}{\left (2 \,{\left (b x + a\right )}{\left (4 \,{\left (b x + a\right )}{\left (\frac{6 \,{\left (b x + a\right )} d{\left | b \right |}}{b^{3}} + \frac{9 \, b^{6} c d^{6}{\left | b \right |} - a b^{5} d^{7}{\left | b \right |}}{b^{8} d^{6}}\right )} + \frac{3 \, b^{7} c^{2} d^{5}{\left | b \right |} + 50 \, a b^{6} c d^{6}{\left | b \right |} - 5 \, a^{2} b^{5} d^{7}{\left | b \right |}}{b^{8} d^{6}}\right )} - \frac{3 \,{\left (3 \, b^{8} c^{3} d^{4}{\left | b \right |} - 17 \, a b^{7} c^{2} d^{5}{\left | b \right |} - 55 \, a^{2} b^{6} c d^{6}{\left | b \right |} + 5 \, a^{3} b^{5} d^{7}{\left | b \right |}\right )}}{b^{8} d^{6}}\right )} \sqrt{b x + a} - \frac{{\left (3 \, \sqrt{b d} b^{4} c^{4}{\left | b \right |} - 20 \, \sqrt{b d} a b^{3} c^{3} d{\left | b \right |} + 90 \, \sqrt{b d} a^{2} b^{2} c^{2} d^{2}{\left | b \right |} + 60 \, \sqrt{b d} a^{3} b c d^{3}{\left | b \right |} - 5 \, \sqrt{b d} a^{4} d^{4}{\left | b \right |}\right )}{\rm ln}\left ({\left (\sqrt{b d} \sqrt{b x + a} - \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d}\right )}^{2}\right )}{128 \, b^{3} d^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x + a)^(5/2)*(d*x + c)^(3/2)/x,x, algorithm="giac")
[Out]