Optimal. Leaf size=208 \[ -\frac{2 (e f-d g)}{5 e^2 (d+e x) (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}+\frac{16 c (b+2 c x) (-5 b e g+2 c d g+8 c e f)}{15 e (2 c d-b e)^5 \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac{2 (b+2 c x) (-5 b e g+2 c d g+8 c e f)}{15 e (2 c d-b e)^3 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}} \]
[Out]
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Rubi [A] time = 0.508247, antiderivative size = 208, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 44, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.068 \[ -\frac{2 (e f-d g)}{5 e^2 (d+e x) (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}+\frac{16 c (b+2 c x) (-5 b e g+2 c d g+8 c e f)}{15 e (2 c d-b e)^5 \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac{2 (b+2 c x) (-5 b e g+2 c d g+8 c e f)}{15 e (2 c d-b e)^3 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
[In] Int[(f + g*x)/((d + e*x)*(c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2)^(5/2)),x]
[Out]
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Rubi in Sympy [A] time = 55.5068, size = 197, normalized size = 0.95 \[ \frac{8 c \left (2 b + 4 c x\right ) \left (5 b e g - 2 c d g - 8 c e f\right )}{15 e \left (b e - 2 c d\right )^{5} \sqrt{- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )}} + \frac{2 \left (b + 2 c x\right ) \left (5 b e g - 2 c d g - 8 c e f\right )}{15 e \left (b e - 2 c d\right )^{3} \left (- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )\right )^{\frac{3}{2}}} - \frac{2 \left (d g - e f\right )}{5 e^{2} \left (d + e x\right ) \left (b e - 2 c d\right ) \left (- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )\right )^{\frac{3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((g*x+f)/(e*x+d)/(-c*e**2*x**2-b*e**2*x-b*d*e+c*d**2)**(5/2),x)
[Out]
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Mathematica [A] time = 1.18286, size = 225, normalized size = 1.08 \[ \frac{2 (d+e x)^3 (c (d-e x)-b e)^3 \left (\frac{5 c^2 (-8 b e g+5 c d g+11 c e f)}{b e-c d+c e x}+\frac{5 c^2 (b e-2 c d) (-b e g+c d g+c e f)}{(b e-c d+c e x)^2}+\frac{c (-40 b e g+7 c d g+73 c e f)}{d+e x}-\frac{(2 c d-b e) (5 b e g+4 c d g-14 c e f)}{(d+e x)^2}+\frac{3 (b e-2 c d)^2 (e f-d g)}{(d+e x)^3}\right )}{15 e^2 (b e-2 c d)^5 ((d+e x) (c (d-e x)-b e))^{5/2}} \]
Antiderivative was successfully verified.
[In] Integrate[(f + g*x)/((d + e*x)*(c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2)^(5/2)),x]
[Out]
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Maple [B] time = 0.02, size = 557, normalized size = 2.7 \[ -{\frac{ \left ( 2\,cex+2\,be-2\,cd \right ) \left ( -80\,b{c}^{3}{e}^{5}g{x}^{4}+32\,{c}^{4}d{e}^{4}g{x}^{4}+128\,{c}^{4}{e}^{5}f{x}^{4}-120\,{b}^{2}{c}^{2}{e}^{5}g{x}^{3}-32\,b{c}^{3}d{e}^{4}g{x}^{3}+192\,b{c}^{3}{e}^{5}f{x}^{3}+32\,{c}^{4}{d}^{2}{e}^{3}g{x}^{3}+128\,{c}^{4}d{e}^{4}f{x}^{3}-30\,{b}^{3}c{e}^{5}g{x}^{2}-228\,{b}^{2}{c}^{2}d{e}^{4}g{x}^{2}+48\,{b}^{2}{c}^{2}{e}^{5}f{x}^{2}+216\,b{c}^{3}{d}^{2}{e}^{3}g{x}^{2}+384\,b{c}^{3}d{e}^{4}f{x}^{2}-48\,{c}^{4}{d}^{3}{e}^{2}g{x}^{2}-192\,{c}^{4}{d}^{2}{e}^{3}f{x}^{2}+5\,{b}^{4}{e}^{5}gx-92\,{b}^{3}cd{e}^{4}gx-8\,{b}^{3}c{e}^{5}fx-24\,{b}^{2}{c}^{2}{d}^{2}{e}^{3}gx+144\,{b}^{2}{c}^{2}d{e}^{4}fx+144\,b{c}^{3}{d}^{3}{e}^{2}gx+96\,b{c}^{3}{d}^{2}{e}^{3}fx-48\,{c}^{4}{d}^{4}egx-192\,{c}^{4}{d}^{3}{e}^{2}fx+2\,{b}^{4}d{e}^{4}g+3\,{b}^{4}{e}^{5}f-38\,{b}^{3}c{d}^{2}{e}^{3}g-32\,{b}^{3}cd{e}^{4}f+12\,{b}^{2}{c}^{2}{d}^{3}{e}^{2}g+168\,{b}^{2}{c}^{2}{d}^{2}{e}^{3}f+72\,b{c}^{3}{d}^{4}eg-192\,b{c}^{3}{d}^{3}{e}^{2}f-48\,{c}^{4}{d}^{5}g+48\,{c}^{4}{d}^{4}ef \right ) }{ \left ( 15\,{b}^{5}{e}^{5}-150\,{b}^{4}cd{e}^{4}+600\,{b}^{3}{c}^{2}{d}^{2}{e}^{3}-1200\,{b}^{2}{c}^{3}{d}^{3}{e}^{2}+1200\,b{c}^{4}{d}^{4}e-480\,{c}^{5}{d}^{5} \right ){e}^{2}} \left ( -c{e}^{2}{x}^{2}-b{e}^{2}x-bde+c{d}^{2} \right ) ^{-{\frac{5}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((g*x+f)/(e*x+d)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(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((g*x + f)/((-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)^(5/2)*(e*x + d)),x, algorithm="maxima")
[Out]
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Fricas [A] time = 9.43221, size = 1388, normalized size = 6.67 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((g*x + f)/((-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)^(5/2)*(e*x + d)),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((g*x+f)/(e*x+d)/(-c*e**2*x**2-b*e**2*x-b*d*e+c*d**2)**(5/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \left [\mathit{undef}, \mathit{undef}, \mathit{undef}, \mathit{undef}, \mathit{undef}, 1\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((g*x + f)/((-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)^(5/2)*(e*x + d)),x, algorithm="giac")
[Out]