Optimal. Leaf size=136 \[ \frac{32 d^3 \sqrt{c+d x}}{35 \sqrt{a+b x} (b c-a d)^4}-\frac{16 d^2 \sqrt{c+d x}}{35 (a+b x)^{3/2} (b c-a d)^3}+\frac{12 d \sqrt{c+d x}}{35 (a+b x)^{5/2} (b c-a d)^2}-\frac{2 \sqrt{c+d x}}{7 (a+b x)^{7/2} (b c-a d)} \]
[Out]
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Rubi [A] time = 0.114538, antiderivative size = 136, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105 \[ \frac{32 d^3 \sqrt{c+d x}}{35 \sqrt{a+b x} (b c-a d)^4}-\frac{16 d^2 \sqrt{c+d x}}{35 (a+b x)^{3/2} (b c-a d)^3}+\frac{12 d \sqrt{c+d x}}{35 (a+b x)^{5/2} (b c-a d)^2}-\frac{2 \sqrt{c+d x}}{7 (a+b x)^{7/2} (b c-a d)} \]
Antiderivative was successfully verified.
[In] Int[1/((a + b*x)^(9/2)*Sqrt[c + d*x]),x]
[Out]
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Rubi in Sympy [A] time = 22.8144, size = 121, normalized size = 0.89 \[ \frac{32 d^{3} \sqrt{c + d x}}{35 \sqrt{a + b x} \left (a d - b c\right )^{4}} + \frac{16 d^{2} \sqrt{c + d x}}{35 \left (a + b x\right )^{\frac{3}{2}} \left (a d - b c\right )^{3}} + \frac{12 d \sqrt{c + d x}}{35 \left (a + b x\right )^{\frac{5}{2}} \left (a d - b c\right )^{2}} + \frac{2 \sqrt{c + d x}}{7 \left (a + b x\right )^{\frac{7}{2}} \left (a d - b c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(b*x+a)**(9/2)/(d*x+c)**(1/2),x)
[Out]
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Mathematica [A] time = 0.130938, size = 111, normalized size = 0.82 \[ \sqrt{a+b x} \sqrt{c+d x} \left (\frac{32 d^3}{35 (a+b x) (b c-a d)^4}-\frac{16 d^2}{35 (a+b x)^2 (b c-a d)^3}+\frac{12 d}{35 (a+b x)^3 (b c-a d)^2}+\frac{2}{7 (a+b x)^4 (a d-b c)}\right ) \]
Antiderivative was successfully verified.
[In] Integrate[1/((a + b*x)^(9/2)*Sqrt[c + d*x]),x]
[Out]
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Maple [A] time = 0.013, size = 171, normalized size = 1.3 \[{\frac{32\,{b}^{3}{d}^{3}{x}^{3}+112\,a{b}^{2}{d}^{3}{x}^{2}-16\,{b}^{3}c{d}^{2}{x}^{2}+140\,{a}^{2}b{d}^{3}x-56\,a{b}^{2}c{d}^{2}x+12\,{b}^{3}{c}^{2}dx+70\,{a}^{3}{d}^{3}-70\,{a}^{2}bc{d}^{2}+42\,a{b}^{2}{c}^{2}d-10\,{b}^{3}{c}^{3}}{35\,{d}^{4}{a}^{4}-140\,b{d}^{3}c{a}^{3}+210\,{b}^{2}{d}^{2}{c}^{2}{a}^{2}-140\,{b}^{3}d{c}^{3}a+35\,{b}^{4}{c}^{4}}\sqrt{dx+c} \left ( bx+a \right ) ^{-{\frac{7}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(b*x+a)^(9/2)/(d*x+c)^(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(1/((b*x + a)^(9/2)*sqrt(d*x + c)),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.677216, size = 566, normalized size = 4.16 \[ \frac{2 \,{\left (16 \, b^{3} d^{3} x^{3} - 5 \, b^{3} c^{3} + 21 \, a b^{2} c^{2} d - 35 \, a^{2} b c d^{2} + 35 \, a^{3} d^{3} - 8 \,{\left (b^{3} c d^{2} - 7 \, a b^{2} d^{3}\right )} x^{2} + 2 \,{\left (3 \, b^{3} c^{2} d - 14 \, a b^{2} c d^{2} + 35 \, a^{2} b d^{3}\right )} x\right )} \sqrt{b x + a} \sqrt{d x + c}}{35 \,{\left (a^{4} b^{4} c^{4} - 4 \, a^{5} b^{3} c^{3} d + 6 \, a^{6} b^{2} c^{2} d^{2} - 4 \, a^{7} b c d^{3} + a^{8} d^{4} +{\left (b^{8} c^{4} - 4 \, a b^{7} c^{3} d + 6 \, a^{2} b^{6} c^{2} d^{2} - 4 \, a^{3} b^{5} c d^{3} + a^{4} b^{4} d^{4}\right )} x^{4} + 4 \,{\left (a b^{7} c^{4} - 4 \, a^{2} b^{6} c^{3} d + 6 \, a^{3} b^{5} c^{2} d^{2} - 4 \, a^{4} b^{4} c d^{3} + a^{5} b^{3} d^{4}\right )} x^{3} + 6 \,{\left (a^{2} b^{6} c^{4} - 4 \, a^{3} b^{5} c^{3} d + 6 \, a^{4} b^{4} c^{2} d^{2} - 4 \, a^{5} b^{3} c d^{3} + a^{6} b^{2} d^{4}\right )} x^{2} + 4 \,{\left (a^{3} b^{5} c^{4} - 4 \, a^{4} b^{4} c^{3} d + 6 \, a^{5} b^{3} c^{2} d^{2} - 4 \, a^{6} b^{2} c d^{3} + a^{7} b d^{4}\right )} x\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x + a)^(9/2)*sqrt(d*x + c)),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(1/(b*x+a)**(9/2)/(d*x+c)**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.256383, size = 521, normalized size = 3.83 \[ \frac{64 \,{\left (b^{6} c^{3} - 3 \, a b^{5} c^{2} d + 3 \, a^{2} b^{4} c d^{2} - a^{3} b^{3} d^{3} - 7 \,{\left (\sqrt{b d} \sqrt{b x + a} - \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d}\right )}^{2} b^{4} c^{2} + 14 \,{\left (\sqrt{b d} \sqrt{b x + a} - \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d}\right )}^{2} a b^{3} c d - 7 \,{\left (\sqrt{b d} \sqrt{b x + a} - \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d}\right )}^{2} a^{2} b^{2} d^{2} + 21 \,{\left (\sqrt{b d} \sqrt{b x + a} - \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d}\right )}^{4} b^{2} c - 21 \,{\left (\sqrt{b d} \sqrt{b x + a} - \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d}\right )}^{4} a b d - 35 \,{\left (\sqrt{b d} \sqrt{b x + a} - \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d}\right )}^{6}\right )} \sqrt{b d} b^{4} d^{3}}{35 \,{\left (b^{2} c - a b d -{\left (\sqrt{b d} \sqrt{b x + a} - \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d}\right )}^{2}\right )}^{7}{\left | b \right |}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x + a)^(9/2)*sqrt(d*x + c)),x, algorithm="giac")
[Out]