Optimal. Leaf size=137 \[ -\frac {2 \sqrt {f+g x} (e f-d g) \left (a g^2-b f g+c f^2\right )}{g^4}+\frac {2 (f+g x)^{3/2} (c f (3 e f-2 d g)-g (-a e g-b d g+2 b e f))}{3 g^4}-\frac {2 (f+g x)^{5/2} (-b e g-c d g+3 c e f)}{5 g^4}+\frac {2 c e (f+g x)^{7/2}}{7 g^4} \]
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Rubi [A] time = 0.11, antiderivative size = 137, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.040, Rules used = {771} \[ -\frac {2 \sqrt {f+g x} (e f-d g) \left (a g^2-b f g+c f^2\right )}{g^4}+\frac {2 (f+g x)^{3/2} (c f (3 e f-2 d g)-g (-a e g-b d g+2 b e f))}{3 g^4}-\frac {2 (f+g x)^{5/2} (-b e g-c d g+3 c e f)}{5 g^4}+\frac {2 c e (f+g x)^{7/2}}{7 g^4} \]
Antiderivative was successfully verified.
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Rule 771
Rubi steps
\begin {align*} \int \frac {(d+e x) \left (a+b x+c x^2\right )}{\sqrt {f+g x}} \, dx &=\int \left (\frac {(-e f+d g) \left (c f^2-b f g+a g^2\right )}{g^3 \sqrt {f+g x}}+\frac {(c f (3 e f-2 d g)-g (2 b e f-b d g-a e g)) \sqrt {f+g x}}{g^3}+\frac {(-3 c e f+c d g+b e g) (f+g x)^{3/2}}{g^3}+\frac {c e (f+g x)^{5/2}}{g^3}\right ) \, dx\\ &=-\frac {2 (e f-d g) \left (c f^2-b f g+a g^2\right ) \sqrt {f+g x}}{g^4}+\frac {2 (c f (3 e f-2 d g)-g (2 b e f-b d g-a e g)) (f+g x)^{3/2}}{3 g^4}-\frac {2 (3 c e f-c d g-b e g) (f+g x)^{5/2}}{5 g^4}+\frac {2 c e (f+g x)^{7/2}}{7 g^4}\\ \end {align*}
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Mathematica [A] time = 0.19, size = 131, normalized size = 0.96 \[ \frac {2 \sqrt {f+g x} \left (7 g \left (5 a g (3 d g-2 e f+e g x)+5 b d g (g x-2 f)+b e \left (8 f^2-4 f g x+3 g^2 x^2\right )\right )+c \left (7 d g \left (8 f^2-4 f g x+3 g^2 x^2\right )-3 e \left (16 f^3-8 f^2 g x+6 f g^2 x^2-5 g^3 x^3\right )\right )\right )}{105 g^4} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.90, size = 125, normalized size = 0.91 \[ \frac {2 \, {\left (15 \, c e g^{3} x^{3} - 48 \, c e f^{3} + 105 \, a d g^{3} + 56 \, {\left (c d + b e\right )} f^{2} g - 70 \, {\left (b d + a e\right )} f g^{2} - 3 \, {\left (6 \, c e f g^{2} - 7 \, {\left (c d + b e\right )} g^{3}\right )} x^{2} + {\left (24 \, c e f^{2} g - 28 \, {\left (c d + b e\right )} f g^{2} + 35 \, {\left (b d + a e\right )} g^{3}\right )} x\right )} \sqrt {g x + f}}{105 \, g^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.19, size = 199, normalized size = 1.45 \[ \frac {2 \, {\left (105 \, \sqrt {g x + f} a d + \frac {35 \, {\left ({\left (g x + f\right )}^{\frac {3}{2}} - 3 \, \sqrt {g x + f} f\right )} b d}{g} + \frac {35 \, {\left ({\left (g x + f\right )}^{\frac {3}{2}} - 3 \, \sqrt {g x + f} f\right )} a e}{g} + \frac {7 \, {\left (3 \, {\left (g x + f\right )}^{\frac {5}{2}} - 10 \, {\left (g x + f\right )}^{\frac {3}{2}} f + 15 \, \sqrt {g x + f} f^{2}\right )} c d}{g^{2}} + \frac {7 \, {\left (3 \, {\left (g x + f\right )}^{\frac {5}{2}} - 10 \, {\left (g x + f\right )}^{\frac {3}{2}} f + 15 \, \sqrt {g x + f} f^{2}\right )} b e}{g^{2}} + \frac {3 \, {\left (5 \, {\left (g x + f\right )}^{\frac {7}{2}} - 21 \, {\left (g x + f\right )}^{\frac {5}{2}} f + 35 \, {\left (g x + f\right )}^{\frac {3}{2}} f^{2} - 35 \, \sqrt {g x + f} f^{3}\right )} c e}{g^{3}}\right )}}{105 \, g} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 144, normalized size = 1.05 \[ \frac {2 \sqrt {g x +f}\, \left (15 c e \,x^{3} g^{3}+21 b e \,g^{3} x^{2}+21 c d \,g^{3} x^{2}-18 c e f \,g^{2} x^{2}+35 a e \,g^{3} x +35 b d \,g^{3} x -28 b e f \,g^{2} x -28 c d f \,g^{2} x +24 c e \,f^{2} g x +105 a d \,g^{3}-70 a e f \,g^{2}-70 b d f \,g^{2}+56 b e \,f^{2} g +56 c d \,f^{2} g -48 c e \,f^{3}\right )}{105 g^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.46, size = 129, normalized size = 0.94 \[ \frac {2 \, {\left (15 \, {\left (g x + f\right )}^{\frac {7}{2}} c e - 21 \, {\left (3 \, c e f - {\left (c d + b e\right )} g\right )} {\left (g x + f\right )}^{\frac {5}{2}} + 35 \, {\left (3 \, c e f^{2} - 2 \, {\left (c d + b e\right )} f g + {\left (b d + a e\right )} g^{2}\right )} {\left (g x + f\right )}^{\frac {3}{2}} - 105 \, {\left (c e f^{3} - a d g^{3} - {\left (c d + b e\right )} f^{2} g + {\left (b d + a e\right )} f g^{2}\right )} \sqrt {g x + f}\right )}}{105 \, g^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.08, size = 125, normalized size = 0.91 \[ \frac {{\left (f+g\,x\right )}^{5/2}\,\left (2\,b\,e\,g+2\,c\,d\,g-6\,c\,e\,f\right )}{5\,g^4}+\frac {{\left (f+g\,x\right )}^{3/2}\,\left (2\,a\,e\,g^2+2\,b\,d\,g^2+6\,c\,e\,f^2-4\,b\,e\,f\,g-4\,c\,d\,f\,g\right )}{3\,g^4}+\frac {2\,\sqrt {f+g\,x}\,\left (d\,g-e\,f\right )\,\left (c\,f^2-b\,f\,g+a\,g^2\right )}{g^4}+\frac {2\,c\,e\,{\left (f+g\,x\right )}^{7/2}}{7\,g^4} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 55.83, size = 549, normalized size = 4.01 \[ \begin {cases} \frac {- \frac {2 a d f}{\sqrt {f + g x}} - 2 a d \left (- \frac {f}{\sqrt {f + g x}} - \sqrt {f + g x}\right ) - \frac {2 a e f \left (- \frac {f}{\sqrt {f + g x}} - \sqrt {f + g x}\right )}{g} - \frac {2 a e \left (\frac {f^{2}}{\sqrt {f + g x}} + 2 f \sqrt {f + g x} - \frac {\left (f + g x\right )^{\frac {3}{2}}}{3}\right )}{g} - \frac {2 b d f \left (- \frac {f}{\sqrt {f + g x}} - \sqrt {f + g x}\right )}{g} - \frac {2 b d \left (\frac {f^{2}}{\sqrt {f + g x}} + 2 f \sqrt {f + g x} - \frac {\left (f + g x\right )^{\frac {3}{2}}}{3}\right )}{g} - \frac {2 b e f \left (\frac {f^{2}}{\sqrt {f + g x}} + 2 f \sqrt {f + g x} - \frac {\left (f + g x\right )^{\frac {3}{2}}}{3}\right )}{g^{2}} - \frac {2 b e \left (- \frac {f^{3}}{\sqrt {f + g x}} - 3 f^{2} \sqrt {f + g x} + f \left (f + g x\right )^{\frac {3}{2}} - \frac {\left (f + g x\right )^{\frac {5}{2}}}{5}\right )}{g^{2}} - \frac {2 c d f \left (\frac {f^{2}}{\sqrt {f + g x}} + 2 f \sqrt {f + g x} - \frac {\left (f + g x\right )^{\frac {3}{2}}}{3}\right )}{g^{2}} - \frac {2 c d \left (- \frac {f^{3}}{\sqrt {f + g x}} - 3 f^{2} \sqrt {f + g x} + f \left (f + g x\right )^{\frac {3}{2}} - \frac {\left (f + g x\right )^{\frac {5}{2}}}{5}\right )}{g^{2}} - \frac {2 c e f \left (- \frac {f^{3}}{\sqrt {f + g x}} - 3 f^{2} \sqrt {f + g x} + f \left (f + g x\right )^{\frac {3}{2}} - \frac {\left (f + g x\right )^{\frac {5}{2}}}{5}\right )}{g^{3}} - \frac {2 c e \left (\frac {f^{4}}{\sqrt {f + g x}} + 4 f^{3} \sqrt {f + g x} - 2 f^{2} \left (f + g x\right )^{\frac {3}{2}} + \frac {4 f \left (f + g x\right )^{\frac {5}{2}}}{5} - \frac {\left (f + g x\right )^{\frac {7}{2}}}{7}\right )}{g^{3}}}{g} & \text {for}\: g \neq 0 \\\frac {a d x + \frac {c e x^{4}}{4} + \frac {x^{3} \left (b e + c d\right )}{3} + \frac {x^{2} \left (a e + b d\right )}{2}}{\sqrt {f}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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