Optimal. Leaf size=118 \[ \frac{2 a}{b \sqrt{a+b x} (c+d x)^{3/2} (b c-a d)}+\frac{4 \sqrt{a+b x} (3 a d+b c)}{3 \sqrt{c+d x} (b c-a d)^3}+\frac{2 \sqrt{a+b x} (3 a d+b c)}{3 b (c+d x)^{3/2} (b c-a d)^2} \]
[Out]
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Rubi [A] time = 0.139595, antiderivative size = 118, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15 \[ \frac{2 a}{b \sqrt{a+b x} (c+d x)^{3/2} (b c-a d)}+\frac{4 \sqrt{a+b x} (3 a d+b c)}{3 \sqrt{c+d x} (b c-a d)^3}+\frac{2 \sqrt{a+b x} (3 a d+b c)}{3 b (c+d x)^{3/2} (b c-a d)^2} \]
Antiderivative was successfully verified.
[In] Int[x/((a + b*x)^(3/2)*(c + d*x)^(5/2)),x]
[Out]
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Rubi in Sympy [A] time = 16.439, size = 104, normalized size = 0.88 \[ - \frac{2 a}{b \sqrt{a + b x} \left (c + d x\right )^{\frac{3}{2}} \left (a d - b c\right )} - \frac{4 \sqrt{a + b x} \left (3 a d + b c\right )}{3 \sqrt{c + d x} \left (a d - b c\right )^{3}} + \frac{2 \sqrt{a + b x} \left (3 a d + b c\right )}{3 b \left (c + d x\right )^{\frac{3}{2}} \left (a d - b c\right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x/(b*x+a)**(3/2)/(d*x+c)**(5/2),x)
[Out]
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Mathematica [A] time = 0.125348, size = 83, normalized size = 0.7 \[ \frac{2 \left (a^2 d (2 c+3 d x)+2 a b \left (3 c^2+5 c d x+3 d^2 x^2\right )+b^2 c x (3 c+2 d x)\right )}{3 \sqrt{a+b x} (c+d x)^{3/2} (b c-a d)^3} \]
Antiderivative was successfully verified.
[In] Integrate[x/((a + b*x)^(3/2)*(c + d*x)^(5/2)),x]
[Out]
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Maple [A] time = 0.012, size = 115, normalized size = 1. \[ -{\frac{12\,ab{d}^{2}{x}^{2}+4\,{b}^{2}cd{x}^{2}+6\,{a}^{2}{d}^{2}x+20\,abcdx+6\,{b}^{2}{c}^{2}x+4\,{a}^{2}cd+12\,ab{c}^{2}}{3\,{a}^{3}{d}^{3}-9\,{a}^{2}cb{d}^{2}+9\,a{b}^{2}{c}^{2}d-3\,{b}^{3}{c}^{3}}{\frac{1}{\sqrt{bx+a}}} \left ( dx+c \right ) ^{-{\frac{3}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x/(b*x+a)^(3/2)/(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(x/((b*x + a)^(3/2)*(d*x + c)^(5/2)),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.373524, size = 379, normalized size = 3.21 \[ \frac{2 \,{\left (6 \, a b c^{2} + 2 \, a^{2} c d + 2 \,{\left (b^{2} c d + 3 \, a b d^{2}\right )} x^{2} +{\left (3 \, b^{2} c^{2} + 10 \, a b c d + 3 \, a^{2} d^{2}\right )} x\right )} \sqrt{b x + a} \sqrt{d x + c}}{3 \,{\left (a b^{3} c^{5} - 3 \, a^{2} b^{2} c^{4} d + 3 \, a^{3} b c^{3} d^{2} - a^{4} c^{2} d^{3} +{\left (b^{4} c^{3} d^{2} - 3 \, a b^{3} c^{2} d^{3} + 3 \, a^{2} b^{2} c d^{4} - a^{3} b d^{5}\right )} x^{3} +{\left (2 \, b^{4} c^{4} d - 5 \, a b^{3} c^{3} d^{2} + 3 \, a^{2} b^{2} c^{2} d^{3} + a^{3} b c d^{4} - a^{4} d^{5}\right )} x^{2} +{\left (b^{4} c^{5} - a b^{3} c^{4} d - 3 \, a^{2} b^{2} c^{3} d^{2} + 5 \, a^{3} b c^{2} d^{3} - 2 \, a^{4} c d^{4}\right )} x\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x/((b*x + a)^(3/2)*(d*x + c)^(5/2)),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x}{\left (a + b x\right )^{\frac{3}{2}} \left (c + d x\right )^{\frac{5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x/(b*x+a)**(3/2)/(d*x+c)**(5/2),x)
[Out]
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GIAC/XCAS [A] time = 0.291963, size = 421, normalized size = 3.57 \[ \frac{\frac{48 \, \sqrt{b d} a b^{3}}{{\left (b^{2} c^{2}{\left | b \right |} - 2 \, a b c d{\left | b \right |} + a^{2} d^{2}{\left | b \right |}\right )}{\left (b^{2} c - a b d -{\left (\sqrt{b d} \sqrt{b x + a} - \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d}\right )}^{2}\right )}} - \frac{\sqrt{b x + a}{\left (\frac{{\left (2 \, b^{7} c^{3} d^{2}{\left | b \right |} - a b^{6} c^{2} d^{3}{\left | b \right |} - 4 \, a^{2} b^{5} c d^{4}{\left | b \right |} + 3 \, a^{3} b^{4} d^{5}{\left | b \right |}\right )}{\left (b x + a\right )}}{b^{8} c^{2} d^{4} - 2 \, a b^{7} c d^{5} + a^{2} b^{6} d^{6}} + \frac{3 \,{\left (b^{8} c^{4} d{\left | b \right |} - 2 \, a b^{7} c^{3} d^{2}{\left | b \right |} + 2 \, a^{3} b^{5} c d^{4}{\left | b \right |} - a^{4} b^{4} d^{5}{\left | b \right |}\right )}}{b^{8} c^{2} d^{4} - 2 \, a b^{7} c d^{5} + a^{2} b^{6} d^{6}}\right )}}{{\left (b^{2} c +{\left (b x + a\right )} b d - a b d\right )}^{\frac{3}{2}}}}{12 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x/((b*x + a)^(3/2)*(d*x + c)^(5/2)),x, algorithm="giac")
[Out]