Optimal. Leaf size=181 \[ \frac{1024 c^2 (b+2 c x) (2 c d-b e)}{35 \left (b^2-4 a c\right )^4 \sqrt{a+b x+c x^2}}-\frac{128 c (b+2 c x) (2 c d-b e)}{35 \left (b^2-4 a c\right )^3 \left (a+b x+c x^2\right )^{3/2}}+\frac{24 (b+2 c x) (2 c d-b e)}{35 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )^{5/2}}-\frac{2 (-2 a e+x (2 c d-b e)+b d)}{7 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{7/2}} \]
[Out]
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Rubi [A] time = 0.139283, antiderivative size = 181, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15 \[ \frac{1024 c^2 (b+2 c x) (2 c d-b e)}{35 \left (b^2-4 a c\right )^4 \sqrt{a+b x+c x^2}}-\frac{128 c (b+2 c x) (2 c d-b e)}{35 \left (b^2-4 a c\right )^3 \left (a+b x+c x^2\right )^{3/2}}+\frac{24 (b+2 c x) (2 c d-b e)}{35 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )^{5/2}}-\frac{2 (-2 a e+x (2 c d-b e)+b d)}{7 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{7/2}} \]
Antiderivative was successfully verified.
[In] Int[(d + e*x)/(a + b*x + c*x^2)^(9/2),x]
[Out]
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Rubi in Sympy [A] time = 23.1752, size = 178, normalized size = 0.98 \[ - \frac{512 c^{2} \left (2 b + 4 c x\right ) \left (b e - 2 c d\right )}{35 \left (- 4 a c + b^{2}\right )^{4} \sqrt{a + b x + c x^{2}}} + \frac{128 c \left (b + 2 c x\right ) \left (b e - 2 c d\right )}{35 \left (- 4 a c + b^{2}\right )^{3} \left (a + b x + c x^{2}\right )^{\frac{3}{2}}} - \frac{24 \left (b + 2 c x\right ) \left (b e - 2 c d\right )}{35 \left (- 4 a c + b^{2}\right )^{2} \left (a + b x + c x^{2}\right )^{\frac{5}{2}}} + \frac{2 \left (2 a e - b d + x \left (b e - 2 c d\right )\right )}{7 \left (- 4 a c + b^{2}\right ) \left (a + b x + c x^{2}\right )^{\frac{7}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((e*x+d)/(c*x**2+b*x+a)**(9/2),x)
[Out]
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Mathematica [A] time = 0.267135, size = 159, normalized size = 0.88 \[ \frac{2 \left (5 \left (b^2-4 a c\right )^3 (2 a e-b d+b e x-2 c d x)-12 \left (b^2-4 a c\right )^2 (b+2 c x) (a+x (b+c x)) (b e-2 c d)+64 c \left (b^2-4 a c\right ) (b+2 c x) (a+x (b+c x))^2 (b e-2 c d)-512 c^2 (b+2 c x) (a+x (b+c x))^3 (b e-2 c d)\right )}{35 \left (b^2-4 a c\right )^4 (a+x (b+c x))^{7/2}} \]
Antiderivative was successfully verified.
[In] Integrate[(d + e*x)/(a + b*x + c*x^2)^(9/2),x]
[Out]
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Maple [B] time = 0.016, size = 500, normalized size = 2.8 \[ -{\frac{2048\,b{c}^{6}e{x}^{7}-4096\,{c}^{7}d{x}^{7}+7168\,{b}^{2}{c}^{5}e{x}^{6}-14336\,b{c}^{6}d{x}^{6}+7168\,ab{c}^{5}e{x}^{5}-14336\,a{c}^{6}d{x}^{5}+8960\,{b}^{3}{c}^{4}e{x}^{5}-17920\,{b}^{2}{c}^{5}d{x}^{5}+17920\,a{b}^{2}{c}^{4}e{x}^{4}-35840\,ab{c}^{5}d{x}^{4}+4480\,{b}^{4}{c}^{3}e{x}^{4}-8960\,{b}^{3}{c}^{4}d{x}^{4}+8960\,{a}^{2}b{c}^{4}e{x}^{3}-17920\,{a}^{2}{c}^{5}d{x}^{3}+13440\,a{b}^{3}{c}^{3}e{x}^{3}-26880\,a{b}^{2}{c}^{4}d{x}^{3}+560\,{b}^{5}{c}^{2}e{x}^{3}-1120\,{b}^{4}{c}^{3}d{x}^{3}+13440\,{a}^{2}{b}^{2}{c}^{3}e{x}^{2}-26880\,{a}^{2}b{c}^{4}d{x}^{2}+2240\,a{b}^{4}{c}^{2}e{x}^{2}-4480\,a{b}^{3}{c}^{3}d{x}^{2}-56\,{b}^{6}ce{x}^{2}+112\,{b}^{5}{c}^{2}d{x}^{2}+4480\,{a}^{3}b{c}^{3}ex-8960\,{a}^{3}{c}^{4}dx+3360\,{a}^{2}{b}^{3}{c}^{2}ex-6720\,{a}^{2}{b}^{2}{c}^{3}dx-280\,a{b}^{5}cex+560\,a{b}^{4}{c}^{2}dx+14\,{b}^{7}ex-28\,{b}^{6}cdx+1280\,{a}^{4}{c}^{3}e+960\,{a}^{3}{b}^{2}{c}^{2}e-4480\,{a}^{3}b{c}^{3}d-80\,{a}^{2}{b}^{4}ce+1120\,{a}^{2}{b}^{3}{c}^{2}d+4\,a{b}^{6}e-168\,a{b}^{5}cd+10\,{b}^{7}d}{8960\,{a}^{4}{c}^{4}-8960\,{a}^{3}{b}^{2}{c}^{3}+3360\,{a}^{2}{b}^{4}{c}^{2}-560\,a{b}^{6}c+35\,{b}^{8}} \left ( c{x}^{2}+bx+a \right ) ^{-{\frac{7}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((e*x+d)/(c*x^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((e*x + d)/(c*x^2 + b*x + a)^(9/2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 2.44431, size = 1270, normalized size = 7.02 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)/(c*x^2 + 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((e*x+d)/(c*x**2+b*x+a)**(9/2),x)
[Out]
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GIAC/XCAS [A] time = 0.288298, size = 1129, normalized size = 6.24 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)/(c*x^2 + b*x + a)^(9/2),x, algorithm="giac")
[Out]