Optimal. Leaf size=80 \[ \frac{2 \sqrt{a+b x} (b c-3 a d)}{3 d \sqrt{c+d x} (b c-a d)^2}-\frac{2 c \sqrt{a+b x}}{3 d (c+d x)^{3/2} (b c-a d)} \]
[Out]
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Rubi [A] time = 0.0942718, antiderivative size = 80, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1 \[ \frac{2 \sqrt{a+b x} (b c-3 a d)}{3 d \sqrt{c+d x} (b c-a d)^2}-\frac{2 c \sqrt{a+b x}}{3 d (c+d x)^{3/2} (b c-a d)} \]
Antiderivative was successfully verified.
[In] Int[x/(Sqrt[a + b*x]*(c + d*x)^(5/2)),x]
[Out]
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Rubi in Sympy [A] time = 9.66004, size = 68, normalized size = 0.85 \[ \frac{2 c \sqrt{a + b x}}{3 d \left (c + d x\right )^{\frac{3}{2}} \left (a d - b c\right )} - \frac{2 \sqrt{a + b x} \left (3 a d - b c\right )}{3 d \sqrt{c + d x} \left (a d - b c\right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x/(d*x+c)**(5/2)/(b*x+a)**(1/2),x)
[Out]
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Mathematica [A] time = 0.0701764, size = 46, normalized size = 0.57 \[ \frac{2 \sqrt{a+b x} (-2 a c-3 a d x+b c x)}{3 (c+d x)^{3/2} (b c-a d)^2} \]
Antiderivative was successfully verified.
[In] Integrate[x/(Sqrt[a + b*x]*(c + d*x)^(5/2)),x]
[Out]
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Maple [A] time = 0.009, size = 55, normalized size = 0.7 \[ -{\frac{6\,adx-2\,bcx+4\,ac}{3\,{a}^{2}{d}^{2}-6\,abcd+3\,{b}^{2}{c}^{2}}\sqrt{bx+a} \left ( dx+c \right ) ^{-{\frac{3}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x/(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. \[ \text{Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x/(sqrt(b*x + a)*(d*x + c)^(5/2)),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.257845, size = 162, normalized size = 2.02 \[ -\frac{2 \,{\left (2 \, a c -{\left (b c - 3 \, a d\right )} x\right )} \sqrt{b x + a} \sqrt{d x + c}}{3 \,{\left (b^{2} c^{4} - 2 \, a b c^{3} d + a^{2} c^{2} d^{2} +{\left (b^{2} c^{2} d^{2} - 2 \, a b c d^{3} + a^{2} d^{4}\right )} x^{2} + 2 \,{\left (b^{2} c^{3} d - 2 \, a b c^{2} d^{2} + a^{2} c d^{3}\right )} x\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x/(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. \[ \int \frac{x}{\sqrt{a + b x} \left (c + d x\right )^{\frac{5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x/(d*x+c)**(5/2)/(b*x+a)**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.237076, size = 204, normalized size = 2.55 \[ -\frac{\sqrt{b x + a}{\left (\frac{{\left (b^{5} c d{\left | b \right |} - 3 \, a b^{4} d^{2}{\left | b \right |}\right )}{\left (b x + a\right )}}{b^{8} c^{2} d^{4} - 2 \, a b^{7} c d^{5} + a^{2} b^{6} d^{6}} - \frac{3 \,{\left (a b^{5} c d{\left | b \right |} - a^{2} b^{4} d^{2}{\left | b \right |}\right )}}{b^{8} c^{2} d^{4} - 2 \, a b^{7} c d^{5} + a^{2} b^{6} d^{6}}\right )}}{12 \,{\left (b^{2} c +{\left (b x + a\right )} b d - a b d\right )}^{\frac{3}{2}} b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x/(sqrt(b*x + a)*(d*x + c)^(5/2)),x, algorithm="giac")
[Out]