Optimal. Leaf size=130 \[ -\frac{2 \sqrt{a+b x} \left (a^2 d^2+b^2 c^2\right )}{b^2 d \sqrt{c+d x} (b c-a d)^2}-\frac{2 a^2}{b^2 \sqrt{a+b x} \sqrt{c+d x} (b c-a d)}+\frac{2 \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{b^{3/2} d^{3/2}} \]
[Out]
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Rubi [A] time = 0.24691, antiderivative size = 130, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ -\frac{2 \sqrt{a+b x} \left (a^2 d^2+b^2 c^2\right )}{b^2 d \sqrt{c+d x} (b c-a d)^2}-\frac{2 a^2}{b^2 \sqrt{a+b x} \sqrt{c+d x} (b c-a d)}+\frac{2 \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{b^{3/2} d^{3/2}} \]
Antiderivative was successfully verified.
[In] Int[x^2/((a + b*x)^(3/2)*(c + d*x)^(3/2)),x]
[Out]
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Rubi in Sympy [A] time = 23.9533, size = 117, normalized size = 0.9 \[ - \frac{2 c^{2}}{d^{2} \sqrt{a + b x} \sqrt{c + d x} \left (a d - b c\right )} - \frac{2 \sqrt{c + d x} \left (a^{2} d^{2} + b^{2} c^{2}\right )}{b d^{2} \sqrt{a + b x} \left (a d - b c\right )^{2}} + \frac{2 \operatorname{atanh}{\left (\frac{\sqrt{d} \sqrt{a + b x}}{\sqrt{b} \sqrt{c + d x}} \right )}}{b^{\frac{3}{2}} d^{\frac{3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**2/(b*x+a)**(3/2)/(d*x+c)**(3/2),x)
[Out]
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Mathematica [A] time = 0.163885, size = 116, normalized size = 0.89 \[ \frac{\log \left (2 \sqrt{b} \sqrt{d} \sqrt{a+b x} \sqrt{c+d x}+a d+b c+2 b d x\right )}{b^{3/2} d^{3/2}}-\frac{2 \left (a^2 d (c+d x)+a b c^2+b^2 c^2 x\right )}{b d \sqrt{a+b x} \sqrt{c+d x} (b c-a d)^2} \]
Antiderivative was successfully verified.
[In] Integrate[x^2/((a + b*x)^(3/2)*(c + d*x)^(3/2)),x]
[Out]
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Maple [B] time = 0.036, size = 654, normalized size = 5. \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^2/(b*x+a)^(3/2)/(d*x+c)^(3/2),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(x^2/((b*x + a)^(3/2)*(d*x + c)^(3/2)),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.355497, size = 1, normalized size = 0.01 \[ \left [-\frac{4 \,{\left (a b c^{2} + a^{2} c d +{\left (b^{2} c^{2} + a^{2} d^{2}\right )} x\right )} \sqrt{b d} \sqrt{b x + a} \sqrt{d x + c} -{\left (a b^{2} c^{3} - 2 \, a^{2} b c^{2} d + a^{3} c d^{2} +{\left (b^{3} c^{2} d - 2 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} x^{2} +{\left (b^{3} c^{3} - a b^{2} c^{2} d - a^{2} b c d^{2} + a^{3} d^{3}\right )} x\right )} \log \left (4 \,{\left (2 \, b^{2} d^{2} x + b^{2} c d + a b d^{2}\right )} \sqrt{b x + a} \sqrt{d x + c} +{\left (8 \, b^{2} d^{2} x^{2} + b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2} + 8 \,{\left (b^{2} c d + a b d^{2}\right )} x\right )} \sqrt{b d}\right )}{2 \,{\left (a b^{3} c^{3} d - 2 \, a^{2} b^{2} c^{2} d^{2} + a^{3} b c d^{3} +{\left (b^{4} c^{2} d^{2} - 2 \, a b^{3} c d^{3} + a^{2} b^{2} d^{4}\right )} x^{2} +{\left (b^{4} c^{3} d - a b^{3} c^{2} d^{2} - a^{2} b^{2} c d^{3} + a^{3} b d^{4}\right )} x\right )} \sqrt{b d}}, -\frac{2 \,{\left (a b c^{2} + a^{2} c d +{\left (b^{2} c^{2} + a^{2} d^{2}\right )} x\right )} \sqrt{-b d} \sqrt{b x + a} \sqrt{d x + c} -{\left (a b^{2} c^{3} - 2 \, a^{2} b c^{2} d + a^{3} c d^{2} +{\left (b^{3} c^{2} d - 2 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} x^{2} +{\left (b^{3} c^{3} - a b^{2} c^{2} d - a^{2} b c d^{2} + a^{3} d^{3}\right )} x\right )} \arctan \left (\frac{{\left (2 \, b d x + b c + a d\right )} \sqrt{-b d}}{2 \, \sqrt{b x + a} \sqrt{d x + c} b d}\right )}{{\left (a b^{3} c^{3} d - 2 \, a^{2} b^{2} c^{2} d^{2} + a^{3} b c d^{3} +{\left (b^{4} c^{2} d^{2} - 2 \, a b^{3} c d^{3} + a^{2} b^{2} d^{4}\right )} x^{2} +{\left (b^{4} c^{3} d - a b^{3} c^{2} d^{2} - a^{2} b^{2} c d^{3} + a^{3} b d^{4}\right )} x\right )} \sqrt{-b d}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^2/((b*x + a)^(3/2)*(d*x + c)^(3/2)),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{2}}{\left (a + b x\right )^{\frac{3}{2}} \left (c + d x\right )^{\frac{3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**2/(b*x+a)**(3/2)/(d*x+c)**(3/2),x)
[Out]
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GIAC/XCAS [A] time = 0.596132, size = 4, normalized size = 0.03 \[ \mathit{sage}_{0} x \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^2/((b*x + a)^(3/2)*(d*x + c)^(3/2)),x, algorithm="giac")
[Out]