Optimal. Leaf size=160 \[ \frac {2 \sqrt {f+g x} \left (c \left (6 d e f-4 d^2 g\right )-e (-2 a e g-b d g+3 b e f)\right )}{3 e^2 \sqrt {d+e x} (e f-d g)^2}-\frac {2 \sqrt {f+g x} \left (a+\frac {d (c d-b e)}{e^2}\right )}{3 (d+e x)^{3/2} (e f-d g)}+\frac {2 c \tanh ^{-1}\left (\frac {\sqrt {g} \sqrt {d+e x}}{\sqrt {e} \sqrt {f+g x}}\right )}{e^{5/2} \sqrt {g}} \]
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Rubi [A] time = 0.18, antiderivative size = 160, 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, 78, 63, 217, 206} \[ \frac {2 \sqrt {f+g x} \left (c \left (6 d e f-4 d^2 g\right )-e (-2 a e g-b d g+3 b e f)\right )}{3 e^2 \sqrt {d+e x} (e f-d g)^2}-\frac {2 \sqrt {f+g x} \left (a+\frac {d (c d-b e)}{e^2}\right )}{3 (d+e x)^{3/2} (e f-d g)}+\frac {2 c \tanh ^{-1}\left (\frac {\sqrt {g} \sqrt {d+e x}}{\sqrt {e} \sqrt {f+g x}}\right )}{e^{5/2} \sqrt {g}} \]
Antiderivative was successfully verified.
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Rule 63
Rule 78
Rule 206
Rule 217
Rule 949
Rubi steps
\begin {align*} \int \frac {a+b x+c x^2}{(d+e x)^{5/2} \sqrt {f+g x}} \, dx &=-\frac {2 \left (a+\frac {d (c d-b e)}{e^2}\right ) \sqrt {f+g x}}{3 (e f-d g) (d+e x)^{3/2}}-\frac {2 \int \frac {\frac {c d (3 e f-d g)-e (3 b e f-b d g-2 a e g)}{2 e^2}-\frac {3}{2} c \left (f-\frac {d g}{e}\right ) x}{(d+e x)^{3/2} \sqrt {f+g x}} \, dx}{3 (e f-d g)}\\ &=-\frac {2 \left (a+\frac {d (c d-b e)}{e^2}\right ) \sqrt {f+g x}}{3 (e f-d g) (d+e x)^{3/2}}+\frac {2 \left (c \left (6 d e f-4 d^2 g\right )-e (3 b e f-b d g-2 a e g)\right ) \sqrt {f+g x}}{3 e^2 (e f-d g)^2 \sqrt {d+e x}}+\frac {c \int \frac {1}{\sqrt {d+e x} \sqrt {f+g x}} \, dx}{e^2}\\ &=-\frac {2 \left (a+\frac {d (c d-b e)}{e^2}\right ) \sqrt {f+g x}}{3 (e f-d g) (d+e x)^{3/2}}+\frac {2 \left (c \left (6 d e f-4 d^2 g\right )-e (3 b e f-b d g-2 a e g)\right ) \sqrt {f+g x}}{3 e^2 (e f-d g)^2 \sqrt {d+e x}}+\frac {(2 c) \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}\\ &=-\frac {2 \left (a+\frac {d (c d-b e)}{e^2}\right ) \sqrt {f+g x}}{3 (e f-d g) (d+e x)^{3/2}}+\frac {2 \left (c \left (6 d e f-4 d^2 g\right )-e (3 b e f-b d g-2 a e g)\right ) \sqrt {f+g x}}{3 e^2 (e f-d g)^2 \sqrt {d+e x}}+\frac {(2 c) \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}\\ &=-\frac {2 \left (a+\frac {d (c d-b e)}{e^2}\right ) \sqrt {f+g x}}{3 (e f-d g) (d+e x)^{3/2}}+\frac {2 \left (c \left (6 d e f-4 d^2 g\right )-e (3 b e f-b d g-2 a e g)\right ) \sqrt {f+g x}}{3 e^2 (e f-d g)^2 \sqrt {d+e x}}+\frac {2 c \tanh ^{-1}\left (\frac {\sqrt {g} \sqrt {d+e x}}{\sqrt {e} \sqrt {f+g x}}\right )}{e^{5/2} \sqrt {g}}\\ \end {align*}
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Mathematica [C] time = 0.23, size = 173, normalized size = 1.08 \[ \frac {2 \sqrt {f+g x} \left (2 g (d+e x) \left (g (a g-b f)+c f^2\right )-(e f-d g) \left (g (a g-b f)+c f^2\right )+(f+g x) (2 c f-b g) (e f-d g)-\frac {c (e f-d g)^3 \, _2F_1\left (-\frac {3}{2},-\frac {3}{2};-\frac {1}{2};\frac {g (d+e x)}{d g-e f}\right )}{e^2 \sqrt {\frac {e (f+g x)}{e f-d g}}}\right )}{3 g^2 (d+e x)^{3/2} (e f-d g)^2} \]
Antiderivative was successfully verified.
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fricas [B] time = 7.06, size = 792, normalized size = 4.95 \[ \left [\frac {3 \, {\left (c d^{2} e^{2} f^{2} - 2 \, c d^{3} e f g + c d^{4} g^{2} + {\left (c e^{4} f^{2} - 2 \, c d e^{3} f g + c d^{2} e^{2} g^{2}\right )} x^{2} + 2 \, {\left (c d e^{3} f^{2} - 2 \, c d^{2} e^{2} f g + c d^{3} e 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 ({\left (5 \, c d^{2} e^{2} - 2 \, b d e^{3} - a e^{4}\right )} f g - 3 \, {\left (c d^{3} e - a d e^{3}\right )} g^{2} + {\left (3 \, {\left (2 \, c d e^{3} - b e^{4}\right )} f g - {\left (4 \, c d^{2} e^{2} - b d e^{3} - 2 \, a e^{4}\right )} g^{2}\right )} x\right )} \sqrt {e x + d} \sqrt {g x + f}}{6 \, {\left (d^{2} e^{5} f^{2} g - 2 \, d^{3} e^{4} f g^{2} + d^{4} e^{3} g^{3} + {\left (e^{7} f^{2} g - 2 \, d e^{6} f g^{2} + d^{2} e^{5} g^{3}\right )} x^{2} + 2 \, {\left (d e^{6} f^{2} g - 2 \, d^{2} e^{5} f g^{2} + d^{3} e^{4} g^{3}\right )} x\right )}}, -\frac {3 \, {\left (c d^{2} e^{2} f^{2} - 2 \, c d^{3} e f g + c d^{4} g^{2} + {\left (c e^{4} f^{2} - 2 \, c d e^{3} f g + c d^{2} e^{2} g^{2}\right )} x^{2} + 2 \, {\left (c d e^{3} f^{2} - 2 \, c d^{2} e^{2} f g + c d^{3} e 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 ({\left (5 \, c d^{2} e^{2} - 2 \, b d e^{3} - a e^{4}\right )} f g - 3 \, {\left (c d^{3} e - a d e^{3}\right )} g^{2} + {\left (3 \, {\left (2 \, c d e^{3} - b e^{4}\right )} f g - {\left (4 \, c d^{2} e^{2} - b d e^{3} - 2 \, a e^{4}\right )} g^{2}\right )} x\right )} \sqrt {e x + d} \sqrt {g x + f}}{3 \, {\left (d^{2} e^{5} f^{2} g - 2 \, d^{3} e^{4} f g^{2} + d^{4} e^{3} g^{3} + {\left (e^{7} f^{2} g - 2 \, d e^{6} f g^{2} + d^{2} e^{5} g^{3}\right )} x^{2} + 2 \, {\left (d e^{6} f^{2} g - 2 \, d^{2} e^{5} f g^{2} + d^{3} e^{4} g^{3}\right )} x\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.68, size = 504, normalized size = 3.15 \[ -\frac {c e^{\left (-\frac {5}{2}\right )} \log \left ({\left (\sqrt {x e + d} \sqrt {g} e^{\frac {1}{2}} - \sqrt {{\left (x e + d\right )} g e - d g e + f e^{2}}\right )}^{2}\right )}{\sqrt {g}} - \frac {4 \, {\left (4 \, c d^{3} g^{\frac {5}{2}} e^{\frac {5}{2}} + 6 \, {\left (\sqrt {x e + d} \sqrt {g} e^{\frac {1}{2}} - \sqrt {{\left (x e + d\right )} g e - d g e + f e^{2}}\right )}^{2} c d^{2} g^{\frac {3}{2}} e^{\frac {3}{2}} + 6 \, {\left (\sqrt {x e + d} \sqrt {g} e^{\frac {1}{2}} - \sqrt {{\left (x e + d\right )} g e - d g e + f e^{2}}\right )}^{4} c d \sqrt {g} e^{\frac {1}{2}} - 10 \, c d^{2} f g^{\frac {3}{2}} e^{\frac {7}{2}} - b d^{2} g^{\frac {5}{2}} e^{\frac {7}{2}} - 12 \, {\left (\sqrt {x e + d} \sqrt {g} e^{\frac {1}{2}} - \sqrt {{\left (x e + d\right )} g e - d g e + f e^{2}}\right )}^{2} c d f \sqrt {g} e^{\frac {5}{2}} - 3 \, {\left (\sqrt {x e + d} \sqrt {g} e^{\frac {1}{2}} - \sqrt {{\left (x e + d\right )} g e - d g e + f e^{2}}\right )}^{4} b \sqrt {g} e^{\frac {3}{2}} + 6 \, c d f^{2} \sqrt {g} e^{\frac {9}{2}} + 4 \, b d f g^{\frac {3}{2}} e^{\frac {9}{2}} - 2 \, a d g^{\frac {5}{2}} e^{\frac {9}{2}} + 6 \, {\left (\sqrt {x e + d} \sqrt {g} e^{\frac {1}{2}} - \sqrt {{\left (x e + d\right )} g e - d g e + f e^{2}}\right )}^{2} b f \sqrt {g} e^{\frac {7}{2}} - 6 \, {\left (\sqrt {x e + d} \sqrt {g} e^{\frac {1}{2}} - \sqrt {{\left (x e + d\right )} g e - d g e + f e^{2}}\right )}^{2} a g^{\frac {3}{2}} e^{\frac {7}{2}} - 3 \, b f^{2} \sqrt {g} e^{\frac {11}{2}} + 2 \, a f g^{\frac {3}{2}} e^{\frac {11}{2}}\right )} e^{\left (-2\right )}}{3 \, {\left (d g e + {\left (\sqrt {x e + d} \sqrt {g} e^{\frac {1}{2}} - \sqrt {{\left (x e + d\right )} g e - d g e + f e^{2}}\right )}^{2} - f e^{2}\right )}^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.03, size = 773, normalized size = 4.83 \[ \frac {\sqrt {g x +f}\, \left (3 c \,d^{2} e^{2} g^{2} x^{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 )-6 c d \,e^{3} f g \,x^{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 )+3 c \,e^{4} f^{2} x^{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 )+6 c \,d^{3} 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 )-12 c \,d^{2} 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 )+6 c d \,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 )+3 c \,d^{4} 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 )-6 c \,d^{3} 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 )+3 c \,d^{2} 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 )+4 \sqrt {\left (e x +d \right ) \left (g x +f \right )}\, \sqrt {e g}\, a \,e^{3} g x +2 \sqrt {\left (e x +d \right ) \left (g x +f \right )}\, \sqrt {e g}\, b d \,e^{2} g x -6 \sqrt {\left (e x +d \right ) \left (g x +f \right )}\, \sqrt {e g}\, b \,e^{3} f x -8 \sqrt {\left (e x +d \right ) \left (g x +f \right )}\, \sqrt {e g}\, c \,d^{2} e g x +12 \sqrt {\left (e x +d \right ) \left (g x +f \right )}\, \sqrt {e g}\, c d \,e^{2} f x +6 \sqrt {\left (e x +d \right ) \left (g x +f \right )}\, \sqrt {e g}\, a d \,e^{2} g -2 \sqrt {\left (e x +d \right ) \left (g x +f \right )}\, \sqrt {e g}\, a \,e^{3} f -4 \sqrt {\left (e x +d \right ) \left (g x +f \right )}\, \sqrt {e g}\, b d \,e^{2} f -6 \sqrt {\left (e x +d \right ) \left (g x +f \right )}\, \sqrt {e g}\, c \,d^{3} g +10 \sqrt {\left (e x +d \right ) \left (g x +f \right )}\, \sqrt {e g}\, c \,d^{2} e f \right )}{3 \sqrt {e g}\, \left (d g -e f \right )^{2} \sqrt {\left (e x +d \right ) \left (g x +f \right )}\, \left (e x +d \right )^{\frac {3}{2}} e^{2}} \]
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 \[ \int \frac {c\,x^2+b\,x+a}{\sqrt {f+g\,x}\,{\left (d+e\,x\right )}^{5/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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