Optimal. Leaf size=163 \[ \frac{1}{8} \sqrt{a+b x} \sqrt{c+d x} \left (\frac{a^2}{b^2}-\frac{c^2}{d^2}\right )+\frac{(a d+b c) (b c-a d)^2 \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{8 b^{5/2} d^{5/2}}-\frac{(a+b x)^{3/2} \sqrt{c+d x} (a d+b c)}{4 b^2 d}+\frac{(a+b x)^{3/2} (c+d x)^{3/2}}{3 b d} \]
[Out]
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Rubi [A] time = 0.2346, antiderivative size = 163, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ \frac{1}{8} \sqrt{a+b x} \sqrt{c+d x} \left (\frac{a^2}{b^2}-\frac{c^2}{d^2}\right )+\frac{(a d+b c) (b c-a d)^2 \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{8 b^{5/2} d^{5/2}}-\frac{(a+b x)^{3/2} \sqrt{c+d x} (a d+b c)}{4 b^2 d}+\frac{(a+b x)^{3/2} (c+d x)^{3/2}}{3 b d} \]
Antiderivative was successfully verified.
[In] Int[x*Sqrt[a + b*x]*Sqrt[c + d*x],x]
[Out]
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Rubi in Sympy [A] time = 23.935, size = 141, normalized size = 0.87 \[ \sqrt{a + b x} \sqrt{c + d x} \left (\frac{a^{2}}{8 b^{2}} - \frac{c^{2}}{8 d^{2}}\right ) + \frac{\left (a + b x\right )^{\frac{3}{2}} \left (c + d x\right )^{\frac{3}{2}}}{3 b d} - \frac{\left (a + b x\right )^{\frac{3}{2}} \sqrt{c + d x} \left (a d + b c\right )}{4 b^{2} d} + \frac{\left (a d - b c\right )^{2} \left (a d + b c\right ) \operatorname{atanh}{\left (\frac{\sqrt{d} \sqrt{a + b x}}{\sqrt{b} \sqrt{c + d x}} \right )}}{8 b^{\frac{5}{2}} d^{\frac{5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x*(b*x+a)**(1/2)*(d*x+c)**(1/2),x)
[Out]
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Mathematica [A] time = 0.11901, size = 145, normalized size = 0.89 \[ \frac{\sqrt{a+b x} \sqrt{c+d x} \left (-3 a^2 d^2+2 a b d (c+d x)+b^2 \left (-3 c^2+2 c d x+8 d^2 x^2\right )\right )}{24 b^2 d^2}+\frac{(a d+b c) (b c-a d)^2 \log \left (2 \sqrt{b} \sqrt{d} \sqrt{a+b x} \sqrt{c+d x}+a d+b c+2 b d x\right )}{16 b^{5/2} d^{5/2}} \]
Antiderivative was successfully verified.
[In] Integrate[x*Sqrt[a + b*x]*Sqrt[c + d*x],x]
[Out]
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Maple [B] time = 0.017, size = 472, normalized size = 2.9 \[{\frac{1}{48\,{b}^{2}{d}^{2}}\sqrt{bx+a}\sqrt{dx+c} \left ( 16\,{x}^{2}{b}^{2}{d}^{2}\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}+3\,\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ){a}^{3}{d}^{3}-3\,\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ){a}^{2}bc{d}^{2}-3\,\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ) a{b}^{2}{c}^{2}d+3\,\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ){b}^{3}{c}^{3}+4\,\sqrt{bd}\sqrt{d{x}^{2}b+adx+bcx+ac}xab{d}^{2}+4\,\sqrt{bd}\sqrt{d{x}^{2}b+adx+bcx+ac}x{b}^{2}cd-6\,\sqrt{bd}\sqrt{d{x}^{2}b+adx+bcx+ac}{a}^{2}{d}^{2}+4\,\sqrt{bd}\sqrt{d{x}^{2}b+adx+bcx+ac}abcd-6\,\sqrt{bd}\sqrt{d{x}^{2}b+adx+bcx+ac}{b}^{2}{c}^{2} \right ){\frac{1}{\sqrt{d{x}^{2}b+adx+bcx+ac}}}{\frac{1}{\sqrt{bd}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x*(b*x+a)^(1/2)*(d*x+c)^(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(sqrt(b*x + a)*sqrt(d*x + c)*x,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.277964, size = 1, normalized size = 0.01 \[ \left [\frac{4 \,{\left (8 \, b^{2} d^{2} x^{2} - 3 \, b^{2} c^{2} + 2 \, a b c d - 3 \, a^{2} d^{2} + 2 \,{\left (b^{2} c d + a b d^{2}\right )} x\right )} \sqrt{b d} \sqrt{b x + a} \sqrt{d x + c} + 3 \,{\left (b^{3} c^{3} - a b^{2} c^{2} d - a^{2} b c d^{2} + a^{3} d^{3}\right )} \log \left (4 \,{\left (2 \, b^{2} d^{2} x + b^{2} c d + a b d^{2}\right )} \sqrt{b x + a} \sqrt{d x + c} +{\left (8 \, b^{2} d^{2} x^{2} + b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2} + 8 \,{\left (b^{2} c d + a b d^{2}\right )} x\right )} \sqrt{b d}\right )}{96 \, \sqrt{b d} b^{2} d^{2}}, \frac{2 \,{\left (8 \, b^{2} d^{2} x^{2} - 3 \, b^{2} c^{2} + 2 \, a b c d - 3 \, a^{2} d^{2} + 2 \,{\left (b^{2} c d + a b d^{2}\right )} x\right )} \sqrt{-b d} \sqrt{b x + a} \sqrt{d x + c} + 3 \,{\left (b^{3} c^{3} - a b^{2} c^{2} d - a^{2} b c d^{2} + a^{3} d^{3}\right )} \arctan \left (\frac{{\left (2 \, b d x + b c + a d\right )} \sqrt{-b d}}{2 \, \sqrt{b x + a} \sqrt{d x + c} b d}\right )}{48 \, \sqrt{-b d} b^{2} d^{2}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(b*x + a)*sqrt(d*x + c)*x,x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int x \sqrt{a + b x} \sqrt{c + d x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x*(b*x+a)**(1/2)*(d*x+c)**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.231326, size = 259, normalized size = 1.59 \[ \frac{{\left (\sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d} \sqrt{b x + a}{\left (2 \,{\left (b x + a\right )}{\left (\frac{4 \,{\left (b x + a\right )}}{b^{6} d^{2}} + \frac{b c d^{3} - 7 \, a d^{4}}{b^{6} d^{6}}\right )} - \frac{3 \,{\left (b^{2} c^{2} d^{2} - a^{2} d^{4}\right )}}{b^{6} d^{6}}\right )} - \frac{3 \,{\left (b^{3} c^{3} - a b^{2} c^{2} d - a^{2} b c d^{2} + a^{3} d^{3}\right )}{\rm ln}\left ({\left | -\sqrt{b d} \sqrt{b x + a} + \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d} \right |}\right )}{\sqrt{b d} b^{5} d^{4}}\right )}{\left | b \right |}}{1920 \, b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(b*x + a)*sqrt(d*x + c)*x,x, algorithm="giac")
[Out]