Optimal. Leaf size=210 \[ -\frac{4 c \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2} (-7 b e g+10 c d g+4 c e f)}{105 e^2 (d+e x)^3 (2 c d-b e)^3}-\frac{2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2} (-7 b e g+10 c d g+4 c e f)}{35 e^2 (d+e x)^4 (2 c d-b e)^2}-\frac{2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{7 e^2 (d+e x)^5 (2 c d-b e)} \]
[Out]
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Rubi [A] time = 0.764138, antiderivative size = 210, 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 \[ -\frac{4 c \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2} (-7 b e g+10 c d g+4 c e f)}{105 e^2 (d+e x)^3 (2 c d-b e)^3}-\frac{2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2} (-7 b e g+10 c d g+4 c e f)}{35 e^2 (d+e x)^4 (2 c d-b e)^2}-\frac{2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{7 e^2 (d+e x)^5 (2 c d-b e)} \]
Antiderivative was successfully verified.
[In] Int[((f + g*x)*Sqrt[c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2])/(d + e*x)^5,x]
[Out]
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Rubi in Sympy [A] time = 78.6689, size = 199, normalized size = 0.95 \[ - \frac{4 c \left (7 b e g - 10 c d g - 4 c e f\right ) \left (- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )\right )^{\frac{3}{2}}}{105 e^{2} \left (d + e x\right )^{3} \left (b e - 2 c d\right )^{3}} + \frac{2 \left (7 b e g - 10 c d g - 4 c e f\right ) \left (- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )\right )^{\frac{3}{2}}}{35 e^{2} \left (d + e x\right )^{4} \left (b e - 2 c d\right )^{2}} - \frac{2 \left (d g - e f\right ) \left (- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )\right )^{\frac{3}{2}}}{7 e^{2} \left (d + e x\right )^{5} \left (b e - 2 c d\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((g*x+f)*(-c*e**2*x**2-b*e**2*x-b*d*e+c*d**2)**(1/2)/(e*x+d)**5,x)
[Out]
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Mathematica [A] time = 0.347387, size = 154, normalized size = 0.73 \[ \frac{2 ((d+e x) (c (d-e x)-b e))^{3/2} \left (3 b^2 e^2 (2 d g+5 e f+7 e g x)-2 b c e \left (13 d^2 g+d e (36 f+50 g x)+e^2 x (6 f+7 g x)\right )+4 c^2 \left (5 d^3 g+d^2 e (23 f+25 g x)+5 d e^2 x (2 f+g x)+2 e^3 f x^2\right )\right )}{105 e^2 (d+e x)^5 (b e-2 c d)^3} \]
Antiderivative was successfully verified.
[In] Integrate[((f + g*x)*Sqrt[c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2])/(d + e*x)^5,x]
[Out]
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Maple [A] time = 0.016, size = 236, normalized size = 1.1 \[ -{\frac{ \left ( 2\,cex+2\,be-2\,cd \right ) \left ( -14\,bc{e}^{3}g{x}^{2}+20\,{c}^{2}d{e}^{2}g{x}^{2}+8\,{c}^{2}{e}^{3}f{x}^{2}+21\,{b}^{2}{e}^{3}gx-100\,bcd{e}^{2}gx-12\,bc{e}^{3}fx+100\,{c}^{2}{d}^{2}egx+40\,{c}^{2}d{e}^{2}fx+6\,{b}^{2}d{e}^{2}g+15\,{b}^{2}{e}^{3}f-26\,bc{d}^{2}eg-72\,bcd{e}^{2}f+20\,{c}^{2}{d}^{3}g+92\,{c}^{2}{d}^{2}ef \right ) }{105\, \left ( ex+d \right ) ^{4} \left ({b}^{3}{e}^{3}-6\,{b}^{2}cd{e}^{2}+12\,b{c}^{2}{d}^{2}e-8\,{c}^{3}{d}^{3} \right ){e}^{2}}\sqrt{-c{e}^{2}{x}^{2}-b{e}^{2}x-bde+c{d}^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((g*x+f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)/(e*x+d)^5,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(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)*(g*x + f)/(e*x + d)^5,x, algorithm="maxima")
[Out]
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Fricas [A] time = 2.5694, size = 729, normalized size = 3.47 \[ \frac{2 \, \sqrt{-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e}{\left (2 \,{\left (4 \, c^{3} e^{4} f +{\left (10 \, c^{3} d e^{3} - 7 \, b c^{2} e^{4}\right )} g\right )} x^{3} +{\left (4 \,{\left (8 \, c^{3} d e^{3} - b c^{2} e^{4}\right )} f +{\left (80 \, c^{3} d^{2} e^{2} - 66 \, b c^{2} d e^{3} + 7 \, b^{2} c e^{4}\right )} g\right )} x^{2} -{\left (92 \, c^{3} d^{3} e - 164 \, b c^{2} d^{2} e^{2} + 87 \, b^{2} c d e^{3} - 15 \, b^{3} e^{4}\right )} f - 2 \,{\left (10 \, c^{3} d^{4} - 23 \, b c^{2} d^{3} e + 16 \, b^{2} c d^{2} e^{2} - 3 \, b^{3} d e^{3}\right )} g +{\left ({\left (52 \, c^{3} d^{2} e^{2} - 20 \, b c^{2} d e^{3} + 3 \, b^{2} c e^{4}\right )} f -{\left (80 \, c^{3} d^{3} e - 174 \, b c^{2} d^{2} e^{2} + 115 \, b^{2} c d e^{3} - 21 \, b^{3} e^{4}\right )} g\right )} x\right )}}{105 \,{\left (8 \, c^{3} d^{7} e^{2} - 12 \, b c^{2} d^{6} e^{3} + 6 \, b^{2} c d^{5} e^{4} - b^{3} d^{4} e^{5} +{\left (8 \, c^{3} d^{3} e^{6} - 12 \, b c^{2} d^{2} e^{7} + 6 \, b^{2} c d e^{8} - b^{3} e^{9}\right )} x^{4} + 4 \,{\left (8 \, c^{3} d^{4} e^{5} - 12 \, b c^{2} d^{3} e^{6} + 6 \, b^{2} c d^{2} e^{7} - b^{3} d e^{8}\right )} x^{3} + 6 \,{\left (8 \, c^{3} d^{5} e^{4} - 12 \, b c^{2} d^{4} e^{5} + 6 \, b^{2} c d^{3} e^{6} - b^{3} d^{2} e^{7}\right )} x^{2} + 4 \,{\left (8 \, c^{3} d^{6} e^{3} - 12 \, b c^{2} d^{5} e^{4} + 6 \, b^{2} c d^{4} e^{5} - b^{3} d^{3} e^{6}\right )} x\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)*(g*x + f)/(e*x + d)^5,x, algorithm="fricas")
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{- \left (d + e x\right ) \left (b e - c d + c e x\right )} \left (f + g x\right )}{\left (d + e x\right )^{5}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((g*x+f)*(-c*e**2*x**2-b*e**2*x-b*d*e+c*d**2)**(1/2)/(e*x+d)**5,x)
[Out]
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GIAC/XCAS [F(-2)] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: NotImplementedError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)*(g*x + f)/(e*x + d)^5,x, algorithm="giac")
[Out]