Optimal. Leaf size=101 \[ \frac{16 d^2 \sqrt{a+b x}}{3 \sqrt{c+d x} (b c-a d)^3}+\frac{8 d}{3 \sqrt{a+b x} \sqrt{c+d x} (b c-a d)^2}-\frac{2}{3 (a+b x)^{3/2} \sqrt{c+d x} (b c-a d)} \]
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Rubi [A] time = 0.0802226, antiderivative size = 101, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105 \[ \frac{16 d^2 \sqrt{a+b x}}{3 \sqrt{c+d x} (b c-a d)^3}+\frac{8 d}{3 \sqrt{a+b x} \sqrt{c+d x} (b c-a d)^2}-\frac{2}{3 (a+b x)^{3/2} \sqrt{c+d x} (b c-a d)} \]
Antiderivative was successfully verified.
[In] Int[1/((a + b*x)^(5/2)*(c + d*x)^(3/2)),x]
[Out]
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Rubi in Sympy [A] time = 13.9717, size = 88, normalized size = 0.87 \[ - \frac{16 d^{2} \sqrt{a + b x}}{3 \sqrt{c + d x} \left (a d - b c\right )^{3}} + \frac{8 d}{3 \sqrt{a + b x} \sqrt{c + d x} \left (a d - b c\right )^{2}} + \frac{2}{3 \left (a + b x\right )^{\frac{3}{2}} \sqrt{c + d x} \left (a d - b c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(b*x+a)**(5/2)/(d*x+c)**(3/2),x)
[Out]
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Mathematica [A] time = 0.113343, size = 75, normalized size = 0.74 \[ \frac{2 \left (3 a^2 d^2+6 a b d (c+2 d x)+b^2 \left (-c^2+4 c d x+8 d^2 x^2\right )\right )}{3 (a+b x)^{3/2} \sqrt{c+d x} (b c-a d)^3} \]
Antiderivative was successfully verified.
[In] Integrate[1/((a + b*x)^(5/2)*(c + d*x)^(3/2)),x]
[Out]
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Maple [A] time = 0.01, size = 105, normalized size = 1. \[ -{\frac{16\,{b}^{2}{d}^{2}{x}^{2}+24\,ab{d}^{2}x+8\,{b}^{2}cdx+6\,{a}^{2}{d}^{2}+12\,abcd-2\,{b}^{2}{c}^{2}}{3\,{a}^{3}{d}^{3}-9\,{a}^{2}bc{d}^{2}+9\,a{b}^{2}{c}^{2}d-3\,{b}^{3}{c}^{3}} \left ( bx+a \right ) ^{-{\frac{3}{2}}}{\frac{1}{\sqrt{dx+c}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(b*x+a)^(5/2)/(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(1/((b*x + a)^(5/2)*(d*x + c)^(3/2)),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.36121, size = 369, normalized size = 3.65 \[ \frac{2 \,{\left (8 \, b^{2} d^{2} x^{2} - b^{2} c^{2} + 6 \, a b c d + 3 \, a^{2} d^{2} + 4 \,{\left (b^{2} c d + 3 \, a b d^{2}\right )} x\right )} \sqrt{b x + a} \sqrt{d x + c}}{3 \,{\left (a^{2} b^{3} c^{4} - 3 \, a^{3} b^{2} c^{3} d + 3 \, a^{4} b c^{2} d^{2} - a^{5} c d^{3} +{\left (b^{5} c^{3} d - 3 \, a b^{4} c^{2} d^{2} + 3 \, a^{2} b^{3} c d^{3} - a^{3} b^{2} d^{4}\right )} x^{3} +{\left (b^{5} c^{4} - a b^{4} c^{3} d - 3 \, a^{2} b^{3} c^{2} d^{2} + 5 \, a^{3} b^{2} c d^{3} - 2 \, a^{4} b d^{4}\right )} x^{2} +{\left (2 \, a b^{4} c^{4} - 5 \, a^{2} b^{3} c^{3} d + 3 \, a^{3} b^{2} c^{2} d^{2} + a^{4} b c d^{3} - a^{5} d^{4}\right )} x\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x + a)^(5/2)*(d*x + c)^(3/2)),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\left (a + b x\right )^{\frac{5}{2}} \left (c + d x\right )^{\frac{3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(b*x+a)**(5/2)/(d*x+c)**(3/2),x)
[Out]
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GIAC/XCAS [A] time = 0.300996, size = 497, normalized size = 4.92 \[ \frac{2 \, \sqrt{b x + a} b^{2} d^{2}}{{\left (b^{3} c^{3}{\left | b \right |} - 3 \, a b^{2} c^{2} d{\left | b \right |} + 3 \, a^{2} b c d^{2}{\left | b \right |} - a^{3} d^{3}{\left | b \right |}\right )} \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d}} + \frac{4 \,{\left (5 \, \sqrt{b d} b^{6} c^{2} d - 10 \, \sqrt{b d} a b^{5} c d^{2} + 5 \, \sqrt{b d} a^{2} b^{4} d^{3} - 12 \, \sqrt{b d}{\left (\sqrt{b d} \sqrt{b x + a} - \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d}\right )}^{2} b^{4} c d + 12 \, \sqrt{b d}{\left (\sqrt{b d} \sqrt{b x + a} - \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d}\right )}^{2} a b^{3} d^{2} + 3 \, \sqrt{b d}{\left (\sqrt{b d} \sqrt{b x + a} - \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d}\right )}^{4} b^{2} d\right )}}{3 \,{\left (b^{2} c^{2}{\left | b \right |} - 2 \, a b c d{\left | b \right |} + a^{2} d^{2}{\left | b \right |}\right )}{\left (b^{2} c - a b d -{\left (\sqrt{b d} \sqrt{b x + a} - \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d}\right )}^{2}\right )}^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x + a)^(5/2)*(d*x + c)^(3/2)),x, algorithm="giac")
[Out]