Optimal. Leaf size=101 \[ -\frac{16 d^2 (c+d x)^{3/2}}{105 (a+b x)^{3/2} (b c-a d)^3}+\frac{8 d (c+d x)^{3/2}}{35 (a+b x)^{5/2} (b c-a d)^2}-\frac{2 (c+d x)^{3/2}}{7 (a+b x)^{7/2} (b c-a d)} \]
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Rubi [A] time = 0.0746952, 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 \[ -\frac{16 d^2 (c+d x)^{3/2}}{105 (a+b x)^{3/2} (b c-a d)^3}+\frac{8 d (c+d x)^{3/2}}{35 (a+b x)^{5/2} (b c-a d)^2}-\frac{2 (c+d x)^{3/2}}{7 (a+b x)^{7/2} (b c-a d)} \]
Antiderivative was successfully verified.
[In] Int[Sqrt[c + d*x]/(a + b*x)^(9/2),x]
[Out]
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Rubi in Sympy [A] time = 13.2109, size = 88, normalized size = 0.87 \[ \frac{16 d^{2} \left (c + d x\right )^{\frac{3}{2}}}{105 \left (a + b x\right )^{\frac{3}{2}} \left (a d - b c\right )^{3}} + \frac{8 d \left (c + d x\right )^{\frac{3}{2}}}{35 \left (a + b x\right )^{\frac{5}{2}} \left (a d - b c\right )^{2}} + \frac{2 \left (c + d x\right )^{\frac{3}{2}}}{7 \left (a + b x\right )^{\frac{7}{2}} \left (a d - b c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((d*x+c)**(1/2)/(b*x+a)**(9/2),x)
[Out]
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Mathematica [A] time = 0.106046, size = 77, normalized size = 0.76 \[ -\frac{2 (c+d x)^{3/2} \left (35 a^2 d^2+14 a b d (2 d x-3 c)+b^2 \left (15 c^2-12 c d x+8 d^2 x^2\right )\right )}{105 (a+b x)^{7/2} (b c-a d)^3} \]
Antiderivative was successfully verified.
[In] Integrate[Sqrt[c + d*x]/(a + b*x)^(9/2),x]
[Out]
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Maple [A] time = 0.01, size = 105, normalized size = 1. \[{\frac{16\,{b}^{2}{d}^{2}{x}^{2}+56\,ab{d}^{2}x-24\,{b}^{2}cdx+70\,{a}^{2}{d}^{2}-84\,abcd+30\,{b}^{2}{c}^{2}}{105\,{a}^{3}{d}^{3}-315\,{a}^{2}bc{d}^{2}+315\,a{b}^{2}{c}^{2}d-105\,{b}^{3}{c}^{3}} \left ( dx+c \right ) ^{{\frac{3}{2}}} \left ( bx+a \right ) ^{-{\frac{7}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((d*x+c)^(1/2)/(b*x+a)^(9/2),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(sqrt(d*x + c)/(b*x + a)^(9/2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.696562, size = 455, normalized size = 4.5 \[ -\frac{2 \,{\left (8 \, b^{2} d^{3} x^{3} + 15 \, b^{2} c^{3} - 42 \, a b c^{2} d + 35 \, a^{2} c d^{2} - 4 \,{\left (b^{2} c d^{2} - 7 \, a b d^{3}\right )} x^{2} +{\left (3 \, b^{2} c^{2} d - 14 \, a b c d^{2} + 35 \, a^{2} d^{3}\right )} x\right )} \sqrt{b x + a} \sqrt{d x + c}}{105 \,{\left (a^{4} b^{3} c^{3} - 3 \, a^{5} b^{2} c^{2} d + 3 \, a^{6} b c d^{2} - a^{7} d^{3} +{\left (b^{7} c^{3} - 3 \, a b^{6} c^{2} d + 3 \, a^{2} b^{5} c d^{2} - a^{3} b^{4} d^{3}\right )} x^{4} + 4 \,{\left (a b^{6} c^{3} - 3 \, a^{2} b^{5} c^{2} d + 3 \, a^{3} b^{4} c d^{2} - a^{4} b^{3} d^{3}\right )} x^{3} + 6 \,{\left (a^{2} b^{5} c^{3} - 3 \, a^{3} b^{4} c^{2} d + 3 \, a^{4} b^{3} c d^{2} - a^{5} b^{2} d^{3}\right )} x^{2} + 4 \,{\left (a^{3} b^{4} c^{3} - 3 \, a^{4} b^{3} c^{2} d + 3 \, a^{5} b^{2} c d^{2} - a^{6} b d^{3}\right )} x\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(d*x + c)/(b*x + a)^(9/2),x, algorithm="fricas")
[Out]
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x+c)**(1/2)/(b*x+a)**(9/2),x)
[Out]
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GIAC/XCAS [A] time = 0.28452, size = 930, normalized size = 9.21 \[ -\frac{32 \,{\left (\sqrt{b d} b^{10} c^{4} d^{3} - 4 \, \sqrt{b d} a b^{9} c^{3} d^{4} + 6 \, \sqrt{b d} a^{2} b^{8} c^{2} d^{5} - 4 \, \sqrt{b d} a^{3} b^{7} c d^{6} + \sqrt{b d} a^{4} b^{6} d^{7} - 7 \, \sqrt{b d}{\left (\sqrt{b d} \sqrt{b x + a} - \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d}\right )}^{2} b^{8} c^{3} d^{3} + 21 \, \sqrt{b d}{\left (\sqrt{b d} \sqrt{b x + a} - \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d}\right )}^{2} a b^{7} c^{2} d^{4} - 21 \, \sqrt{b d}{\left (\sqrt{b d} \sqrt{b x + a} - \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d}\right )}^{2} a^{2} b^{6} c d^{5} + 7 \, \sqrt{b d}{\left (\sqrt{b d} \sqrt{b x + a} - \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d}\right )}^{2} a^{3} b^{5} d^{6} + 21 \, \sqrt{b d}{\left (\sqrt{b d} \sqrt{b x + a} - \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d}\right )}^{4} b^{6} c^{2} d^{3} - 42 \, \sqrt{b d}{\left (\sqrt{b d} \sqrt{b x + a} - \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d}\right )}^{4} a b^{5} c d^{4} + 21 \, \sqrt{b d}{\left (\sqrt{b d} \sqrt{b x + a} - \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d}\right )}^{4} a^{2} b^{4} d^{5} + 35 \, \sqrt{b d}{\left (\sqrt{b d} \sqrt{b x + a} - \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d}\right )}^{6} b^{4} c d^{3} - 35 \, \sqrt{b d}{\left (\sqrt{b d} \sqrt{b x + a} - \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d}\right )}^{6} a b^{3} d^{4} + 70 \, \sqrt{b d}{\left (\sqrt{b d} \sqrt{b x + a} - \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d}\right )}^{8} b^{2} d^{3}\right )}{\left | b \right |}}{105 \,{\left (b^{2} c - a b d -{\left (\sqrt{b d} \sqrt{b x + a} - \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d}\right )}^{2}\right )}^{7} b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(d*x + c)/(b*x + a)^(9/2),x, algorithm="giac")
[Out]