Optimal. Leaf size=170 \[ \frac{\left (-3 a^2 d^2-6 a b c d+b^2 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{4 a^{3/2} \sqrt{c}}+2 \sqrt{b} d^{3/2} \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )-\frac{\sqrt{a+b x} (c+d x)^{3/2}}{2 x^2}-\frac{\sqrt{a+b x} \sqrt{c+d x} (3 a d+b c)}{4 a x} \]
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Rubi [A] time = 0.416805, antiderivative size = 170, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273 \[ \frac{\left (-3 a^2 d^2-6 a b c d+b^2 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{4 a^{3/2} \sqrt{c}}+2 \sqrt{b} d^{3/2} \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )-\frac{\sqrt{a+b x} (c+d x)^{3/2}}{2 x^2}-\frac{\sqrt{a+b x} \sqrt{c+d x} (3 a d+b c)}{4 a x} \]
Antiderivative was successfully verified.
[In] Int[(Sqrt[a + b*x]*(c + d*x)^(3/2))/x^3,x]
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Rubi in Sympy [A] time = 66.1988, size = 156, normalized size = 0.92 \[ 2 \sqrt{b} d^{\frac{3}{2}} \operatorname{atanh}{\left (\frac{\sqrt{d} \sqrt{a + b x}}{\sqrt{b} \sqrt{c + d x}} \right )} - \frac{\sqrt{a + b x} \left (c + d x\right )^{\frac{3}{2}}}{2 x^{2}} - \frac{\sqrt{a + b x} \sqrt{c + d x} \left (3 a d + b c\right )}{4 a x} - \frac{\left (3 a^{2} d^{2} + 6 a b c d - b^{2} c^{2}\right ) \operatorname{atanh}{\left (\frac{\sqrt{c} \sqrt{a + b x}}{\sqrt{a} \sqrt{c + d x}} \right )}}{4 a^{\frac{3}{2}} \sqrt{c}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((d*x+c)**(3/2)*(b*x+a)**(1/2)/x**3,x)
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Mathematica [A] time = 0.295547, size = 222, normalized size = 1.31 \[ \frac{\log (x) \left (3 a^2 d^2+6 a b c d-b^2 c^2\right )}{8 a^{3/2} \sqrt{c}}-\frac{\left (3 a^2 d^2+6 a b c d-b^2 c^2\right ) \log \left (2 \sqrt{a} \sqrt{c} \sqrt{a+b x} \sqrt{c+d x}+2 a c+a d x+b c x\right )}{8 a^{3/2} \sqrt{c}}+\sqrt{b} d^{3/2} \log \left (2 \sqrt{b} \sqrt{d} \sqrt{a+b x} \sqrt{c+d x}+a d+b c+2 b d x\right )+\sqrt{a+b x} \sqrt{c+d x} \left (\frac{-5 a d-b c}{4 a x}-\frac{c}{2 x^2}\right ) \]
Antiderivative was successfully verified.
[In] Integrate[(Sqrt[a + b*x]*(c + d*x)^(3/2))/x^3,x]
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Maple [B] time = 0.021, size = 401, normalized size = 2.4 \[ -{\frac{1}{8\,a{x}^{2}}\sqrt{bx+a}\sqrt{dx+c} \left ( 3\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}+2\,ac}{x}} \right ){x}^{2}{a}^{2}{d}^{2}\sqrt{bd}+6\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}+2\,ac}{x}} \right ){x}^{2}abcd\sqrt{bd}-\ln \left ({\frac{1}{x} \left ( adx+bcx+2\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}+2\,ac \right ) } \right ){x}^{2}{b}^{2}{c}^{2}\sqrt{bd}-8\,\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ){x}^{2}ab{d}^{2}\sqrt{ac}+10\,\sqrt{d{x}^{2}b+adx+bcx+ac}dax\sqrt{ac}\sqrt{bd}+2\,\sqrt{d{x}^{2}b+adx+bcx+ac}bxc\sqrt{ac}\sqrt{bd}+4\,\sqrt{d{x}^{2}b+adx+bcx+ac}ac\sqrt{ac}\sqrt{bd} \right ){\frac{1}{\sqrt{d{x}^{2}b+adx+bcx+ac}}}{\frac{1}{\sqrt{bd}}}{\frac{1}{\sqrt{ac}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((d*x+c)^(3/2)*(b*x+a)^(1/2)/x^3,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(sqrt(b*x + a)*(d*x + c)^(3/2)/x^3,x, algorithm="maxima")
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Fricas [A] time = 0.943529, size = 1, normalized size = 0.01 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(b*x + a)*(d*x + c)^(3/2)/x^3,x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{a + b x} \left (c + d x\right )^{\frac{3}{2}}}{x^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x+c)**(3/2)*(b*x+a)**(1/2)/x**3,x)
[Out]
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GIAC/XCAS [A] time = 0.629906, size = 4, normalized size = 0.02 \[ \mathit{sage}_{0} x \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(b*x + a)*(d*x + c)^(3/2)/x^3,x, algorithm="giac")
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