Optimal. Leaf size=129 \[ -\frac {2 \left (a+\frac {d (c d-b e)}{e^2}\right ) \sqrt {f+g x}}{(e f-d g) \sqrt {d+e x}}+\frac {c \sqrt {d+e x} \sqrt {f+g x}}{e^2 g}-\frac {(c e f+3 c d g-2 b e g) \tanh ^{-1}\left (\frac {\sqrt {g} \sqrt {d+e x}}{\sqrt {e} \sqrt {f+g x}}\right )}{e^{5/2} g^{3/2}} \]
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Rubi [A]
time = 0.09, 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 = {963, 81, 65,
223, 212} \begin {gather*} -\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} \end {gather*}
Antiderivative was successfully verified.
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Rule 65
Rule 81
Rule 212
Rule 223
Rule 963
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) \text {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) \text {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 [A]
time = 0.34, size = 131, normalized size = 1.02 \begin {gather*} \frac {\sqrt {f+g x} \left (2 e (b d-a e) g+c \left (-3 d^2 g+e^2 f x+d e (f-g x)\right )\right )}{e^2 g (e f-d g) \sqrt {d+e x}}+\frac {(2 b e g-c (e f+3 d g)) \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {g} \sqrt {d+e x}}\right )}{e^{5/2} g^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(696\) vs.
\(2(109)=218\).
time = 0.08, size = 697, normalized size = 5.40
method | result | size |
default | \(\frac {\sqrt {g x +f}\, \left (2 \ln \left (\frac {2 e g x +2 \sqrt {\left (e x +d \right ) \left (g x +f \right )}\, \sqrt {e g}+d g +e f}{2 \sqrt {e g}}\right ) b d \,e^{2} g^{2} x -2 \ln \left (\frac {2 e g x +2 \sqrt {\left (e x +d \right ) \left (g x +f \right )}\, \sqrt {e g}+d g +e f}{2 \sqrt {e g}}\right ) b \,e^{3} f g x -3 \ln \left (\frac {2 e g x +2 \sqrt {\left (e x +d \right ) \left (g x +f \right )}\, \sqrt {e g}+d g +e f}{2 \sqrt {e g}}\right ) c \,d^{2} e \,g^{2} x +2 \ln \left (\frac {2 e g x +2 \sqrt {\left (e x +d \right ) \left (g x +f \right )}\, \sqrt {e g}+d g +e f}{2 \sqrt {e g}}\right ) c d \,e^{2} f g x +\ln \left (\frac {2 e g x +2 \sqrt {\left (e x +d \right ) \left (g x +f \right )}\, \sqrt {e g}+d g +e f}{2 \sqrt {e g}}\right ) c \,e^{3} f^{2} x +2 \ln \left (\frac {2 e g x +2 \sqrt {\left (e x +d \right ) \left (g x +f \right )}\, \sqrt {e g}+d g +e f}{2 \sqrt {e g}}\right ) b \,d^{2} e \,g^{2}-2 \ln \left (\frac {2 e g x +2 \sqrt {\left (e x +d \right ) \left (g x +f \right )}\, \sqrt {e g}+d g +e f}{2 \sqrt {e g}}\right ) b d \,e^{2} f g -3 \ln \left (\frac {2 e g x +2 \sqrt {\left (e x +d \right ) \left (g x +f \right )}\, \sqrt {e g}+d g +e f}{2 \sqrt {e g}}\right ) c \,d^{3} g^{2}+2 \ln \left (\frac {2 e g x +2 \sqrt {\left (e x +d \right ) \left (g x +f \right )}\, \sqrt {e g}+d g +e f}{2 \sqrt {e g}}\right ) c \,d^{2} e f g +\ln \left (\frac {2 e g x +2 \sqrt {\left (e x +d \right ) \left (g x +f \right )}\, \sqrt {e g}+d g +e f}{2 \sqrt {e g}}\right ) c d \,e^{2} f^{2}+2 c d e g x \sqrt {\left (e x +d \right ) \left (g x +f \right )}\, \sqrt {e g}-2 c \,e^{2} f x \sqrt {\left (e x +d \right ) \left (g x +f \right )}\, \sqrt {e g}+4 a \,e^{2} g \sqrt {\left (e x +d \right ) \left (g x +f \right )}\, \sqrt {e g}-4 b d e g \sqrt {\left (e x +d \right ) \left (g x +f \right )}\, \sqrt {e g}+6 c \,d^{2} g \sqrt {\left (e x +d \right ) \left (g x +f \right )}\, \sqrt {e g}-2 c d e f \sqrt {\left (e x +d \right ) \left (g x +f \right )}\, \sqrt {e g}\right )}{2 g \sqrt {e g}\, \left (d g -e f \right ) \sqrt {\left (e x +d \right ) \left (g x +f \right )}\, e^{2} \sqrt {e x +d}}\) | \(697\) |
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 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 277 vs.
\(2 (112) = 224\).
time = 5.18, size = 564, normalized size = 4.37 \begin {gather*} \left [-\frac {{\left (3 \, c d^{3} g^{2} - {\left (c f^{2} - 2 \, b f g\right )} x e^{3} - {\left (c d f^{2} - 2 \, b d f g + 2 \, {\left (c d f g + b d g^{2}\right )} x\right )} e^{2} + {\left (3 \, c d^{2} g^{2} x - 2 \, c d^{2} f g - 2 \, b d^{2} g^{2}\right )} e\right )} \sqrt {g} e^{\frac {1}{2}} \log \left (d^{2} g^{2} + 4 \, {\left (d g + {\left (2 \, g x + f\right )} e\right )} \sqrt {g x + f} \sqrt {x e + d} \sqrt {g} e^{\frac {1}{2}} + {\left (8 \, g^{2} x^{2} + 8 \, f g x + f^{2}\right )} e^{2} + 2 \, {\left (4 \, d g^{2} x + 3 \, d f g\right )} e\right ) - 4 \, {\left (3 \, c d^{2} g^{2} e - {\left (c f g x - 2 \, a g^{2}\right )} e^{3} + {\left (c d g^{2} x - c d f g - 2 \, b d g^{2}\right )} e^{2}\right )} \sqrt {g x + f} \sqrt {x e + d}}{4 \, {\left (d^{2} g^{3} e^{3} - f g^{2} x e^{5} + {\left (d g^{3} x - d f g^{2}\right )} e^{4}\right )}}, \frac {{\left (3 \, c d^{3} g^{2} - {\left (c f^{2} - 2 \, b f g\right )} x e^{3} - {\left (c d f^{2} - 2 \, b d f g + 2 \, {\left (c d f g + b d g^{2}\right )} x\right )} e^{2} + {\left (3 \, c d^{2} g^{2} x - 2 \, c d^{2} f g - 2 \, b d^{2} g^{2}\right )} e\right )} \sqrt {-g e} \arctan \left (\frac {{\left (d g + {\left (2 \, g x + f\right )} e\right )} \sqrt {g x + f} \sqrt {-g e} \sqrt {x e + d}}{2 \, {\left ({\left (g^{2} x^{2} + f g x\right )} e^{2} + {\left (d g^{2} x + d f g\right )} e\right )}}\right ) + 2 \, {\left (3 \, c d^{2} g^{2} e - {\left (c f g x - 2 \, a g^{2}\right )} e^{3} + {\left (c d g^{2} x - c d f g - 2 \, b d g^{2}\right )} e^{2}\right )} \sqrt {g x + f} \sqrt {x e + d}}{2 \, {\left (d^{2} g^{3} e^{3} - f g^{2} x e^{5} + {\left (d g^{3} x - d f g^{2}\right )} e^{4}\right )}}\right ] \end {gather*}
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 \begin {gather*} \int \frac {a + b x + c x^{2}}{\left (d + e x\right )^{\frac {3}{2}} \sqrt {f + g x}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 4.10, size = 218, normalized size = 1.69 \begin {gather*} \frac {{\left (3 \, c d g + c f e - 2 \, b g e\right )} e^{\left (-\frac {5}{2}\right )} \log \left ({\left | -\sqrt {g x + f} \sqrt {g} e^{\frac {1}{2}} + \sqrt {d g^{2} + {\left (g x + f\right )} g e - f g e} \right |}\right )}{\sqrt {g} {\left | g \right |}} + \frac {\sqrt {g x + f} {\left (\frac {{\left (c d g^{3} e^{2} - c f g^{2} e^{3}\right )} {\left (g x + f\right )}}{d g^{3} {\left | g \right |} e^{3} - f g^{2} {\left | g \right |} e^{4}} + \frac {3 \, c d^{2} g^{4} e - 2 \, c d f g^{3} e^{2} - 2 \, b d g^{4} e^{2} + c f^{2} g^{2} e^{3} + 2 \, a g^{4} e^{3}}{d g^{3} {\left | g \right |} e^{3} - f g^{2} {\left | g \right |} e^{4}}\right )}}{\sqrt {d g^{2} + {\left (g x + f\right )} g e - f g e}} \end {gather*}
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 \begin {gather*} \int \frac {c\,x^2+b\,x+a}{\sqrt {f+g\,x}\,{\left (d+e\,x\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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