Optimal. Leaf size=189 \[ \frac{(5 a d+b c) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{a^{3/2} c^{7/2}}-\frac{d \sqrt{a+b x} \left (15 a^2 d^2-22 a b c d+3 b^2 c^2\right )}{3 a c^3 \sqrt{c+d x} (b c-a d)^2}-\frac{d \sqrt{a+b x} (3 b c-5 a d)}{3 a c^2 (c+d x)^{3/2} (b c-a d)}-\frac{\sqrt{a+b x}}{a c x (c+d x)^{3/2}} \]
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Rubi [A] time = 0.577128, antiderivative size = 189, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227 \[ \frac{(5 a d+b c) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{a^{3/2} c^{7/2}}-\frac{d \sqrt{a+b x} \left (15 a^2 d^2-22 a b c d+3 b^2 c^2\right )}{3 a c^3 \sqrt{c+d x} (b c-a d)^2}-\frac{d \sqrt{a+b x} (3 b c-5 a d)}{3 a c^2 (c+d x)^{3/2} (b c-a d)}-\frac{\sqrt{a+b x}}{a c x (c+d x)^{3/2}} \]
Antiderivative was successfully verified.
[In] Int[1/(x^2*Sqrt[a + b*x]*(c + d*x)^(5/2)),x]
[Out]
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Rubi in Sympy [A] time = 66.389, size = 170, normalized size = 0.9 \[ - \frac{\sqrt{a + b x}}{a c x \left (c + d x\right )^{\frac{3}{2}}} - \frac{d \sqrt{a + b x} \left (5 a d - 3 b c\right )}{3 a c^{2} \left (c + d x\right )^{\frac{3}{2}} \left (a d - b c\right )} - \frac{d \sqrt{a + b x} \left (15 a^{2} d^{2} - 22 a b c d + 3 b^{2} c^{2}\right )}{3 a c^{3} \sqrt{c + d x} \left (a d - b c\right )^{2}} + \frac{\left (5 a d + b c\right ) \operatorname{atanh}{\left (\frac{\sqrt{c} \sqrt{a + b x}}{\sqrt{a} \sqrt{c + d x}} \right )}}{a^{\frac{3}{2}} c^{\frac{7}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/x**2/(d*x+c)**(5/2)/(b*x+a)**(1/2),x)
[Out]
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Mathematica [A] time = 0.833418, size = 174, normalized size = 0.92 \[ \frac{-\frac{3 \log (x) (5 a d+b c)}{a^{3/2}}+\frac{3 (5 a d+b c) \log \left (2 \sqrt{a} \sqrt{c} \sqrt{a+b x} \sqrt{c+d x}+2 a c+a d x+b c x\right )}{a^{3/2}}+2 \sqrt{c} \sqrt{a+b x} \sqrt{c+d x} \left (\frac{4 d^2 (4 b c-3 a d)}{(c+d x) (b c-a d)^2}+\frac{2 c d^2}{(c+d x)^2 (b c-a d)}-\frac{3}{a x}\right )}{6 c^{7/2}} \]
Antiderivative was successfully verified.
[In] Integrate[1/(x^2*Sqrt[a + b*x]*(c + d*x)^(5/2)),x]
[Out]
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Maple [B] time = 0.052, size = 919, normalized size = 4.9 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/x^2/(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. \[ \int \frac{1}{\sqrt{b x + a}{\left (d x + c\right )}^{\frac{5}{2}} x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(sqrt(b*x + a)*(d*x + c)^(5/2)*x^2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.478795, size = 1, normalized size = 0.01 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(sqrt(b*x + a)*(d*x + c)^(5/2)*x^2),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{x^{2} \sqrt{a + b x} \left (c + d x\right )^{\frac{5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/x**2/(d*x+c)**(5/2)/(b*x+a)**(1/2),x)
[Out]
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GIAC/XCAS [F(-2)] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(sqrt(b*x + a)*(d*x + c)^(5/2)*x^2),x, algorithm="giac")
[Out]