Optimal. Leaf size=129 \[ -\frac {2 \sqrt {f+g x} \left (a+\frac {d (c d-b e)}{e^2}\right )}{\sqrt {d+e x} (e f-d g)}-\frac {(-2 b e g+3 c d g+c e f) \tanh ^{-1}\left (\frac {\sqrt {g} \sqrt {d+e x}}{\sqrt {e} \sqrt {f+g x}}\right )}{e^{5/2} g^{3/2}}+\frac {c \sqrt {d+e x} \sqrt {f+g x}}{e^2 g} \]
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Rubi [A] time = 0.13, antiderivative size = 129, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.172, Rules used = {949, 80, 63, 217, 206} \[ -\frac {2 \sqrt {f+g x} \left (a+\frac {d (c d-b e)}{e^2}\right )}{\sqrt {d+e x} (e f-d g)}-\frac {(-2 b e g+3 c d g+c e f) \tanh ^{-1}\left (\frac {\sqrt {g} \sqrt {d+e x}}{\sqrt {e} \sqrt {f+g x}}\right )}{e^{5/2} g^{3/2}}+\frac {c \sqrt {d+e x} \sqrt {f+g x}}{e^2 g} \]
Antiderivative was successfully verified.
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Rule 63
Rule 80
Rule 206
Rule 217
Rule 949
Rubi steps
\begin {align*} \int \frac {a+b x+c x^2}{(d+e x)^{3/2} \sqrt {f+g x}} \, dx &=-\frac {2 \left (a+\frac {d (c d-b e)}{e^2}\right ) \sqrt {f+g x}}{(e f-d g) \sqrt {d+e x}}-\frac {2 \int \frac {\frac {(c d-b e) (e f-d g)}{2 e^2}-\frac {c (e f-d g) x}{2 e}}{\sqrt {d+e x} \sqrt {f+g x}} \, dx}{e f-d g}\\ &=-\frac {2 \left (a+\frac {d (c d-b e)}{e^2}\right ) \sqrt {f+g x}}{(e f-d g) \sqrt {d+e x}}+\frac {c \sqrt {d+e x} \sqrt {f+g x}}{e^2 g}-\frac {(c e f+3 c d g-2 b e g) \int \frac {1}{\sqrt {d+e x} \sqrt {f+g x}} \, dx}{2 e^2 g}\\ &=-\frac {2 \left (a+\frac {d (c d-b e)}{e^2}\right ) \sqrt {f+g x}}{(e f-d g) \sqrt {d+e x}}+\frac {c \sqrt {d+e x} \sqrt {f+g x}}{e^2 g}-\frac {(c e f+3 c d g-2 b e g) \operatorname {Subst}\left (\int \frac {1}{\sqrt {f-\frac {d g}{e}+\frac {g x^2}{e}}} \, dx,x,\sqrt {d+e x}\right )}{e^3 g}\\ &=-\frac {2 \left (a+\frac {d (c d-b e)}{e^2}\right ) \sqrt {f+g x}}{(e f-d g) \sqrt {d+e x}}+\frac {c \sqrt {d+e x} \sqrt {f+g x}}{e^2 g}-\frac {(c e f+3 c d g-2 b e g) \operatorname {Subst}\left (\int \frac {1}{1-\frac {g x^2}{e}} \, dx,x,\frac {\sqrt {d+e x}}{\sqrt {f+g x}}\right )}{e^3 g}\\ &=-\frac {2 \left (a+\frac {d (c d-b e)}{e^2}\right ) \sqrt {f+g x}}{(e f-d g) \sqrt {d+e x}}+\frac {c \sqrt {d+e x} \sqrt {f+g x}}{e^2 g}-\frac {(c e f+3 c d g-2 b e g) \tanh ^{-1}\left (\frac {\sqrt {g} \sqrt {d+e x}}{\sqrt {e} \sqrt {f+g x}}\right )}{e^{5/2} g^{3/2}}\\ \end {align*}
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Mathematica [C] time = 0.59, size = 222, normalized size = 1.72 \[ -\frac {2 \sqrt {f+g x} \left (e \sqrt {e f-d g} \sqrt {\frac {e (f+g x)}{e f-d g}} \left (g^2 (a e-b d)+c f (2 d g-e f)\right )+e \sqrt {g} \sqrt {d+e x} (2 c f-b g) (e f-d g) \sinh ^{-1}\left (\frac {\sqrt {g} \sqrt {d+e x}}{\sqrt {e f-d g}}\right )+c (e f-d g)^{5/2} \, _2F_1\left (-\frac {3}{2},-\frac {1}{2};\frac {1}{2};\frac {g (d+e x)}{d g-e f}\right )\right )}{e^2 g^2 \sqrt {d+e x} (e f-d g)^{3/2} \sqrt {\frac {e (f+g x)}{e f-d g}}} \]
Antiderivative was successfully verified.
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fricas [B] time = 3.10, size = 588, normalized size = 4.56 \[ \left [-\frac {{\left (c d e^{2} f^{2} + 2 \, {\left (c d^{2} e - b d e^{2}\right )} f g - {\left (3 \, c d^{3} - 2 \, b d^{2} e\right )} g^{2} + {\left (c e^{3} f^{2} + 2 \, {\left (c d e^{2} - b e^{3}\right )} f g - {\left (3 \, c d^{2} e - 2 \, b d e^{2}\right )} g^{2}\right )} x\right )} \sqrt {e g} \log \left (8 \, e^{2} g^{2} x^{2} + e^{2} f^{2} + 6 \, d e f g + d^{2} g^{2} + 4 \, {\left (2 \, e g x + e f + d g\right )} \sqrt {e g} \sqrt {e x + d} \sqrt {g x + f} + 8 \, {\left (e^{2} f g + d e g^{2}\right )} x\right ) - 4 \, {\left (c d e^{2} f g - {\left (3 \, c d^{2} e - 2 \, b d e^{2} + 2 \, a e^{3}\right )} g^{2} + {\left (c e^{3} f g - c d e^{2} g^{2}\right )} x\right )} \sqrt {e x + d} \sqrt {g x + f}}{4 \, {\left (d e^{4} f g^{2} - d^{2} e^{3} g^{3} + {\left (e^{5} f g^{2} - d e^{4} g^{3}\right )} x\right )}}, \frac {{\left (c d e^{2} f^{2} + 2 \, {\left (c d^{2} e - b d e^{2}\right )} f g - {\left (3 \, c d^{3} - 2 \, b d^{2} e\right )} g^{2} + {\left (c e^{3} f^{2} + 2 \, {\left (c d e^{2} - b e^{3}\right )} f g - {\left (3 \, c d^{2} e - 2 \, b d e^{2}\right )} g^{2}\right )} x\right )} \sqrt {-e g} \arctan \left (\frac {{\left (2 \, e g x + e f + d g\right )} \sqrt {-e g} \sqrt {e x + d} \sqrt {g x + f}}{2 \, {\left (e^{2} g^{2} x^{2} + d e f g + {\left (e^{2} f g + d e g^{2}\right )} x\right )}}\right ) + 2 \, {\left (c d e^{2} f g - {\left (3 \, c d^{2} e - 2 \, b d e^{2} + 2 \, a e^{3}\right )} g^{2} + {\left (c e^{3} f g - c d e^{2} g^{2}\right )} x\right )} \sqrt {e x + d} \sqrt {g x + f}}{2 \, {\left (d e^{4} f g^{2} - d^{2} e^{3} g^{3} + {\left (e^{5} f g^{2} - d e^{4} g^{3}\right )} x\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.39, size = 201, normalized size = 1.56 \[ \frac {\sqrt {{\left (x e + d\right )} g e - d g e + f e^{2}} \sqrt {x e + d} c e^{\left (-3\right )}}{g} + \frac {4 \, {\left (c d^{2} \sqrt {g} e^{\frac {1}{2}} - b d \sqrt {g} e^{\frac {3}{2}} + a \sqrt {g} e^{\frac {5}{2}}\right )} e^{\left (-2\right )}}{d g e + {\left (\sqrt {x e + d} \sqrt {g} e^{\frac {1}{2}} - \sqrt {{\left (x e + d\right )} g e - d g e + f e^{2}}\right )}^{2} - f e^{2}} + \frac {{\left (3 \, c d g^{\frac {3}{2}} e^{\frac {1}{2}} + c f \sqrt {g} e^{\frac {3}{2}} - 2 \, b g^{\frac {3}{2}} e^{\frac {3}{2}}\right )} e^{\left (-3\right )} \log \left ({\left (\sqrt {x e + d} \sqrt {g} e^{\frac {1}{2}} - \sqrt {{\left (x e + d\right )} g e - d g e + f e^{2}}\right )}^{2}\right )}{2 \, g^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.03, size = 697, normalized size = 5.40 \[ \frac {\sqrt {g x +f}\, \left (2 b d \,e^{2} g^{2} x \ln \left (\frac {2 e g x +d g +e f +2 \sqrt {\left (e x +d \right ) \left (g x +f \right )}\, \sqrt {e g}}{2 \sqrt {e g}}\right )-2 b \,e^{3} f g x \ln \left (\frac {2 e g x +d g +e f +2 \sqrt {\left (e x +d \right ) \left (g x +f \right )}\, \sqrt {e g}}{2 \sqrt {e g}}\right )-3 c \,d^{2} e \,g^{2} x \ln \left (\frac {2 e g x +d g +e f +2 \sqrt {\left (e x +d \right ) \left (g x +f \right )}\, \sqrt {e g}}{2 \sqrt {e g}}\right )+2 c d \,e^{2} f g x \ln \left (\frac {2 e g x +d g +e f +2 \sqrt {\left (e x +d \right ) \left (g x +f \right )}\, \sqrt {e g}}{2 \sqrt {e g}}\right )+c \,e^{3} f^{2} x \ln \left (\frac {2 e g x +d g +e f +2 \sqrt {\left (e x +d \right ) \left (g x +f \right )}\, \sqrt {e g}}{2 \sqrt {e g}}\right )+2 b \,d^{2} e \,g^{2} \ln \left (\frac {2 e g x +d g +e f +2 \sqrt {\left (e x +d \right ) \left (g x +f \right )}\, \sqrt {e g}}{2 \sqrt {e g}}\right )-2 b d \,e^{2} f g \ln \left (\frac {2 e g x +d g +e f +2 \sqrt {\left (e x +d \right ) \left (g x +f \right )}\, \sqrt {e g}}{2 \sqrt {e g}}\right )-3 c \,d^{3} g^{2} \ln \left (\frac {2 e g x +d g +e f +2 \sqrt {\left (e x +d \right ) \left (g x +f \right )}\, \sqrt {e g}}{2 \sqrt {e g}}\right )+2 c \,d^{2} e f g \ln \left (\frac {2 e g x +d g +e f +2 \sqrt {\left (e x +d \right ) \left (g x +f \right )}\, \sqrt {e g}}{2 \sqrt {e g}}\right )+c d \,e^{2} f^{2} \ln \left (\frac {2 e g x +d g +e f +2 \sqrt {\left (e x +d \right ) \left (g x +f \right )}\, \sqrt {e g}}{2 \sqrt {e g}}\right )+2 \sqrt {\left (e x +d \right ) \left (g x +f \right )}\, \sqrt {e g}\, c d e g x -2 \sqrt {\left (e x +d \right ) \left (g x +f \right )}\, \sqrt {e g}\, c \,e^{2} f x +4 \sqrt {\left (e x +d \right ) \left (g x +f \right )}\, \sqrt {e g}\, a \,e^{2} g -4 \sqrt {\left (e x +d \right ) \left (g x +f \right )}\, \sqrt {e g}\, b d e g +6 \sqrt {\left (e x +d \right ) \left (g x +f \right )}\, \sqrt {e g}\, c \,d^{2} g -2 \sqrt {\left (e x +d \right ) \left (g x +f \right )}\, \sqrt {e g}\, c d e f \right )}{2 \sqrt {e g}\, \left (d g -e f \right ) \sqrt {\left (e x +d \right ) \left (g x +f \right )}\, \sqrt {e x +d}\, e^{2} g} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {c\,x^2+b\,x+a}{\sqrt {f+g\,x}\,{\left (d+e\,x\right )}^{3/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {a + b x + c x^{2}}{\left (d + e x\right )^{\frac {3}{2}} \sqrt {f + g x}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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