Optimal. Leaf size=198 \[ \frac{2 \sqrt{f+g x} \left (2 e g (-4 a e g-b d g+5 b e f)-c \left (3 d^2 g^2-10 d e f g+15 e^2 f^2\right )\right )}{15 e^2 \sqrt{d+e x} (e f-d g)^3}-\frac{2 \sqrt{f+g x} \left (a+\frac{d (c d-b e)}{e^2}\right )}{5 (d+e x)^{5/2} (e f-d g)}+\frac{2 \sqrt{f+g x} (2 c d (5 e f-3 d g)-e (-4 a e g-b d g+5 b e f))}{15 e^2 (d+e x)^{3/2} (e f-d g)^2} \]
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Rubi [A] time = 0.513001, antiderivative size = 198, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.103 \[ \frac{2 \sqrt{f+g x} \left (2 e g (-4 a e g-b d g+5 b e f)-c \left (3 d^2 g^2-10 d e f g+15 e^2 f^2\right )\right )}{15 e^2 \sqrt{d+e x} (e f-d g)^3}-\frac{2 \sqrt{f+g x} \left (a+\frac{d (c d-b e)}{e^2}\right )}{5 (d+e x)^{5/2} (e f-d g)}+\frac{2 \sqrt{f+g x} (2 c d (5 e f-3 d g)-e (-4 a e g-b d g+5 b e f))}{15 e^2 (d+e x)^{3/2} (e f-d g)^2} \]
Antiderivative was successfully verified.
[In] Int[(a + b*x + c*x^2)/((d + e*x)^(7/2)*Sqrt[f + g*x]),x]
[Out]
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Rubi in Sympy [A] time = 62.4999, size = 275, normalized size = 1.39 \[ \frac{2 c d^{2} \sqrt{f + g x}}{5 e^{2} \left (d + e x\right )^{\frac{5}{2}} \left (d g - e f\right )} - \frac{4 c d \sqrt{f + g x} \left (3 d g - 5 e f\right )}{15 e^{2} \left (d + e x\right )^{\frac{3}{2}} \left (d g - e f\right )^{2}} + \frac{2 c \sqrt{f + g x} \left (3 d^{2} g^{2} - 10 d e f g + 15 e^{2} f^{2}\right )}{15 e^{2} \sqrt{d + e x} \left (d g - e f\right )^{3}} + \frac{4 g \sqrt{f + g x} \left (4 a e g + b d g - 5 b e f\right )}{15 e \sqrt{d + e x} \left (d g - e f\right )^{3}} + \frac{2 \sqrt{f + g x} \left (4 a e g + b d g - 5 b e f\right )}{15 e \left (d + e x\right )^{\frac{3}{2}} \left (d g - e f\right )^{2}} + \frac{2 \sqrt{f + g x} \left (a e - b d\right )}{5 e \left (d + e x\right )^{\frac{5}{2}} \left (d g - e f\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((c*x**2+b*x+a)/(e*x+d)**(7/2)/(g*x+f)**(1/2),x)
[Out]
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Mathematica [A] time = 0.474128, size = 178, normalized size = 0.9 \[ \frac{2 \sqrt{f+g x} \left (a \left (15 d^2 g^2-10 d e g (f-2 g x)+e^2 \left (3 f^2-4 f g x+8 g^2 x^2\right )\right )+b \left (5 d^2 g (g x-2 f)+2 d e \left (f^2-13 f g x+g^2 x^2\right )+5 e^2 f x (f-2 g x)\right )+c \left (d^2 \left (8 f^2-4 f g x+3 g^2 x^2\right )+10 d e f x (2 f-g x)+15 e^2 f^2 x^2\right )\right )}{15 (d+e x)^{5/2} (d g-e f)^3} \]
Antiderivative was successfully verified.
[In] Integrate[(a + b*x + c*x^2)/((d + e*x)^(7/2)*Sqrt[f + g*x]),x]
[Out]
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Maple [A] time = 0.012, size = 238, normalized size = 1.2 \[{\frac{16\,a{e}^{2}{g}^{2}{x}^{2}+4\,bde{g}^{2}{x}^{2}-20\,b{e}^{2}fg{x}^{2}+6\,c{d}^{2}{g}^{2}{x}^{2}-20\,cdefg{x}^{2}+30\,c{e}^{2}{f}^{2}{x}^{2}+40\,ade{g}^{2}x-8\,a{e}^{2}fgx+10\,b{d}^{2}{g}^{2}x-52\,bdefgx+10\,b{e}^{2}{f}^{2}x-8\,c{d}^{2}fgx+40\,cde{f}^{2}x+30\,a{d}^{2}{g}^{2}-20\,adefg+6\,a{e}^{2}{f}^{2}-20\,b{d}^{2}fg+4\,bde{f}^{2}+16\,c{d}^{2}{f}^{2}}{15\,{g}^{3}{d}^{3}-45\,{d}^{2}ef{g}^{2}+45\,d{e}^{2}{f}^{2}g-15\,{e}^{3}{f}^{3}}\sqrt{gx+f} \left ( ex+d \right ) ^{-{\frac{5}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((c*x^2+b*x+a)/(e*x+d)^(7/2)/(g*x+f)^(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((c*x^2 + b*x + a)/((e*x + d)^(7/2)*sqrt(g*x + f)),x, algorithm="maxima")
[Out]
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Fricas [A] time = 1.59129, size = 477, normalized size = 2.41 \[ -\frac{2 \,{\left (15 \, a d^{2} g^{2} +{\left (8 \, c d^{2} + 2 \, b d e + 3 \, a e^{2}\right )} f^{2} - 10 \,{\left (b d^{2} + a d e\right )} f g +{\left (15 \, c e^{2} f^{2} - 10 \,{\left (c d e + b e^{2}\right )} f g +{\left (3 \, c d^{2} + 2 \, b d e + 8 \, a e^{2}\right )} g^{2}\right )} x^{2} +{\left (5 \,{\left (4 \, c d e + b e^{2}\right )} f^{2} - 2 \,{\left (2 \, c d^{2} + 13 \, b d e + 2 \, a e^{2}\right )} f g + 5 \,{\left (b d^{2} + 4 \, a d e\right )} g^{2}\right )} x\right )} \sqrt{e x + d} \sqrt{g x + f}}{15 \,{\left (d^{3} e^{3} f^{3} - 3 \, d^{4} e^{2} f^{2} g + 3 \, d^{5} e f g^{2} - d^{6} g^{3} +{\left (e^{6} f^{3} - 3 \, d e^{5} f^{2} g + 3 \, d^{2} e^{4} f g^{2} - d^{3} e^{3} g^{3}\right )} x^{3} + 3 \,{\left (d e^{5} f^{3} - 3 \, d^{2} e^{4} f^{2} g + 3 \, d^{3} e^{3} f g^{2} - d^{4} e^{2} g^{3}\right )} x^{2} + 3 \,{\left (d^{2} e^{4} f^{3} - 3 \, d^{3} e^{3} f^{2} g + 3 \, d^{4} e^{2} f g^{2} - d^{5} e g^{3}\right )} x\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x + a)/((e*x + d)^(7/2)*sqrt(g*x + f)),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((c*x**2+b*x+a)/(e*x+d)**(7/2)/(g*x+f)**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.387331, size = 1, normalized size = 0.01 \[ \mathit{Done} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x + a)/((e*x + d)^(7/2)*sqrt(g*x + f)),x, algorithm="giac")
[Out]