Optimal. Leaf size=66 \[ \frac{4 b \sqrt{a+b x}}{3 \sqrt{c+d x} (b c-a d)^2}+\frac{2 \sqrt{a+b x}}{3 (c+d x)^{3/2} (b c-a d)} \]
[Out]
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Rubi [A] time = 0.0539475, antiderivative size = 66, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105 \[ \frac{4 b \sqrt{a+b x}}{3 \sqrt{c+d x} (b c-a d)^2}+\frac{2 \sqrt{a+b x}}{3 (c+d x)^{3/2} (b c-a d)} \]
Antiderivative was successfully verified.
[In] Int[1/(Sqrt[a + b*x]*(c + d*x)^(5/2)),x]
[Out]
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Rubi in Sympy [A] time = 7.1971, size = 56, normalized size = 0.85 \[ \frac{4 b \sqrt{a + b x}}{3 \sqrt{c + d x} \left (a d - b c\right )^{2}} - \frac{2 \sqrt{a + b x}}{3 \left (c + d x\right )^{\frac{3}{2}} \left (a d - b c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(b*x+a)**(1/2)/(d*x+c)**(5/2),x)
[Out]
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Mathematica [A] time = 0.0521153, size = 46, normalized size = 0.7 \[ \frac{2 \sqrt{a+b x} (-a d+3 b c+2 b d x)}{3 (c+d x)^{3/2} (b c-a d)^2} \]
Antiderivative was successfully verified.
[In] Integrate[1/(Sqrt[a + b*x]*(c + d*x)^(5/2)),x]
[Out]
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Maple [A] time = 0.007, size = 53, normalized size = 0.8 \[ -{\frac{-4\,bdx+2\,ad-6\,bc}{3\,{a}^{2}{d}^{2}-6\,abcd+3\,{b}^{2}{c}^{2}}\sqrt{bx+a} \left ( dx+c \right ) ^{-{\frac{3}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(b*x+a)^(1/2)/(d*x+c)^(5/2),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(1/(sqrt(b*x + a)*(d*x + c)^(5/2)),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.256161, size = 159, normalized size = 2.41 \[ \frac{2 \,{\left (2 \, b d x + 3 \, b c - a d\right )} \sqrt{b x + a} \sqrt{d x + c}}{3 \,{\left (b^{2} c^{4} - 2 \, a b c^{3} d + a^{2} c^{2} d^{2} +{\left (b^{2} c^{2} d^{2} - 2 \, a b c d^{3} + a^{2} d^{4}\right )} x^{2} + 2 \,{\left (b^{2} c^{3} d - 2 \, a b c^{2} d^{2} + a^{2} c d^{3}\right )} x\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(sqrt(b*x + a)*(d*x + c)^(5/2)),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{a + b x} \left (c + d x\right )^{\frac{5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(b*x+a)**(1/2)/(d*x+c)**(5/2),x)
[Out]
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GIAC/XCAS [A] time = 0.237422, size = 173, normalized size = 2.62 \[ -\frac{{\left (\frac{2 \,{\left (b x + a\right )} b^{4} d^{2}}{b^{8} c^{2} d^{4} - 2 \, a b^{7} c d^{5} + a^{2} b^{6} d^{6}} + \frac{3 \,{\left (b^{5} c d - a b^{4} d^{2}\right )}}{b^{8} c^{2} d^{4} - 2 \, a b^{7} c d^{5} + a^{2} b^{6} d^{6}}\right )} \sqrt{b x + a}}{24 \,{\left (b^{2} c +{\left (b x + a\right )} b d - a b d\right )}^{\frac{3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(sqrt(b*x + a)*(d*x + c)^(5/2)),x, algorithm="giac")
[Out]