Optimal. Leaf size=206 \[ \frac {\tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right ) \left (e g (-3 a e g-b d g+4 b e f)-c \left (3 d^2 g^2-8 d e f g+8 e^2 f^2\right )\right )}{4 e^{5/2} (e f-d g)^{5/2}}-\frac {\sqrt {f+g x} \left (a+\frac {d (c d-b e)}{e^2}\right )}{2 (d+e x)^2 (e f-d g)}+\frac {\sqrt {f+g x} (c d (8 e f-5 d g)-e (-3 a e g-b d g+4 b e f))}{4 e^2 (d+e x) (e f-d g)^2} \]
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Rubi [A] time = 0.39, antiderivative size = 206, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {897, 1157, 385, 208} \begin {gather*} \frac {\tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right ) \left (e g (-3 a e g-b d g+4 b e f)-c \left (3 d^2 g^2-8 d e f g+8 e^2 f^2\right )\right )}{4 e^{5/2} (e f-d g)^{5/2}}-\frac {\sqrt {f+g x} \left (a+\frac {d (c d-b e)}{e^2}\right )}{2 (d+e x)^2 (e f-d g)}+\frac {\sqrt {f+g x} (c d (8 e f-5 d g)-e (-3 a e g-b d g+4 b e f))}{4 e^2 (d+e x) (e f-d g)^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 208
Rule 385
Rule 897
Rule 1157
Rubi steps
\begin {align*} \int \frac {a+b x+c x^2}{(d+e x)^3 \sqrt {f+g x}} \, dx &=\frac {2 \operatorname {Subst}\left (\int \frac {\frac {c f^2-b f g+a g^2}{g^2}-\frac {(2 c f-b g) x^2}{g^2}+\frac {c x^4}{g^2}}{\left (\frac {-e f+d g}{g}+\frac {e x^2}{g}\right )^3} \, dx,x,\sqrt {f+g x}\right )}{g}\\ &=-\frac {\left (c d^2-b d e+a e^2\right ) \sqrt {f+g x}}{2 e^2 (e f-d g) (d+e x)^2}+\frac {\operatorname {Subst}\left (\int \frac {-3 a+\frac {c d^2}{e^2}-\frac {b d}{e}-\frac {4 c f^2}{g^2}+\frac {4 b f}{g}+\frac {4 c (e f-d g) x^2}{e g^2}}{\left (\frac {-e f+d g}{g}+\frac {e x^2}{g}\right )^2} \, dx,x,\sqrt {f+g x}\right )}{2 (e f-d g)}\\ &=-\frac {\left (c d^2-b d e+a e^2\right ) \sqrt {f+g x}}{2 e^2 (e f-d g) (d+e x)^2}+\frac {(c d (8 e f-5 d g)-e (4 b e f-b d g-3 a e g)) \sqrt {f+g x}}{4 e^2 (e f-d g)^2 (d+e x)}-\frac {\left (e g (4 b e f-b d g-3 a e g)-c \left (8 e^2 f^2-8 d e f g+3 d^2 g^2\right )\right ) \operatorname {Subst}\left (\int \frac {1}{\frac {-e f+d g}{g}+\frac {e x^2}{g}} \, dx,x,\sqrt {f+g x}\right )}{4 e^2 g (e f-d g)^2}\\ &=-\frac {\left (c d^2-b d e+a e^2\right ) \sqrt {f+g x}}{2 e^2 (e f-d g) (d+e x)^2}+\frac {(c d (8 e f-5 d g)-e (4 b e f-b d g-3 a e g)) \sqrt {f+g x}}{4 e^2 (e f-d g)^2 (d+e x)}+\frac {\left (e g (4 b e f-b d g-3 a e g)-c \left (8 e^2 f^2-8 d e f g+3 d^2 g^2\right )\right ) \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right )}{4 e^{5/2} (e f-d g)^{5/2}}\\ \end {align*}
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Mathematica [A] time = 0.65, size = 297, normalized size = 1.44 \begin {gather*} \frac {-\frac {2 \sqrt {e} \sqrt {f+g x} \left (e (a e-b d)+c d^2\right )}{(d+e x)^2 (e f-d g)}-\frac {3 g \left (e (a e-b d)+c d^2\right ) \left (g (d+e x) \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right )-\sqrt {e} \sqrt {f+g x} \sqrt {e f-d g}\right )}{(d+e x) (e f-d g)^{5/2}}-\frac {4 \sqrt {e} \sqrt {f+g x} (b e-2 c d)}{(d+e x) (e f-d g)}-\frac {4 g (2 c d-b e) \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right )}{(e f-d g)^{3/2}}-\frac {8 c \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right )}{\sqrt {e f-d g}}}{4 e^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.90, size = 293, normalized size = 1.42 \begin {gather*} \frac {\tan ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x} \sqrt {d g-e f}}{e f-d g}\right ) \left (-3 a e^2 g^2-b d e g^2+4 b e^2 f g-3 c d^2 g^2+8 c d e f g-8 c e^2 f^2\right )}{4 e^{5/2} (d g-e f)^{5/2}}-\frac {g \sqrt {f+g x} \left (-5 a d e^2 g^2-3 a e^3 g (f+g x)+5 a e^3 f g+b d^2 e g^2-b d e^2 g (f+g x)+3 b d e^2 f g-4 b e^3 f^2+4 b e^3 f (f+g x)+3 c d^3 g^2+5 c d^2 e g (f+g x)-11 c d^2 e f g+8 c d e^2 f^2-8 c d e^2 f (f+g x)\right )}{4 e^2 (e f-d g)^2 (-d g-e (f+g x)+e f)^2} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.66, size = 1096, normalized size = 5.32 \begin {gather*} \left [\frac {{\left (8 \, c d^{2} e^{2} f^{2} - 4 \, {\left (2 \, c d^{3} e + b d^{2} e^{2}\right )} f g + {\left (3 \, c d^{4} + b d^{3} e + 3 \, a d^{2} e^{2}\right )} g^{2} + {\left (8 \, c e^{4} f^{2} - 4 \, {\left (2 \, c d e^{3} + b e^{4}\right )} f g + {\left (3 \, c d^{2} e^{2} + b d e^{3} + 3 \, a e^{4}\right )} g^{2}\right )} x^{2} + 2 \, {\left (8 \, c d e^{3} f^{2} - 4 \, {\left (2 \, c d^{2} e^{2} + b d e^{3}\right )} f g + {\left (3 \, c d^{3} e + b d^{2} e^{2} + 3 \, a d e^{3}\right )} g^{2}\right )} x\right )} \sqrt {e^{2} f - d e g} \log \left (\frac {e g x + 2 \, e f - d g - 2 \, \sqrt {e^{2} f - d e g} \sqrt {g x + f}}{e x + d}\right ) + 2 \, {\left (2 \, {\left (3 \, c d^{2} e^{3} - b d e^{4} - a e^{5}\right )} f^{2} - {\left (9 \, c d^{3} e^{2} - b d^{2} e^{3} - 7 \, a d e^{4}\right )} f g + {\left (3 \, c d^{4} e + b d^{3} e^{2} - 5 \, a d^{2} e^{3}\right )} g^{2} + {\left (4 \, {\left (2 \, c d e^{4} - b e^{5}\right )} f^{2} - {\left (13 \, c d^{2} e^{3} - 5 \, b d e^{4} - 3 \, a e^{5}\right )} f g + {\left (5 \, c d^{3} e^{2} - b d^{2} e^{3} - 3 \, a d e^{4}\right )} g^{2}\right )} x\right )} \sqrt {g x + f}}{8 \, {\left (d^{2} e^{6} f^{3} - 3 \, d^{3} e^{5} f^{2} g + 3 \, d^{4} e^{4} f g^{2} - d^{5} e^{3} g^{3} + {\left (e^{8} f^{3} - 3 \, d e^{7} f^{2} g + 3 \, d^{2} e^{6} f g^{2} - d^{3} e^{5} g^{3}\right )} x^{2} + 2 \, {\left (d e^{7} f^{3} - 3 \, d^{2} e^{6} f^{2} g + 3 \, d^{3} e^{5} f g^{2} - d^{4} e^{4} g^{3}\right )} x\right )}}, \frac {{\left (8 \, c d^{2} e^{2} f^{2} - 4 \, {\left (2 \, c d^{3} e + b d^{2} e^{2}\right )} f g + {\left (3 \, c d^{4} + b d^{3} e + 3 \, a d^{2} e^{2}\right )} g^{2} + {\left (8 \, c e^{4} f^{2} - 4 \, {\left (2 \, c d e^{3} + b e^{4}\right )} f g + {\left (3 \, c d^{2} e^{2} + b d e^{3} + 3 \, a e^{4}\right )} g^{2}\right )} x^{2} + 2 \, {\left (8 \, c d e^{3} f^{2} - 4 \, {\left (2 \, c d^{2} e^{2} + b d e^{3}\right )} f g + {\left (3 \, c d^{3} e + b d^{2} e^{2} + 3 \, a d e^{3}\right )} g^{2}\right )} x\right )} \sqrt {-e^{2} f + d e g} \arctan \left (\frac {\sqrt {-e^{2} f + d e g} \sqrt {g x + f}}{e g x + e f}\right ) + {\left (2 \, {\left (3 \, c d^{2} e^{3} - b d e^{4} - a e^{5}\right )} f^{2} - {\left (9 \, c d^{3} e^{2} - b d^{2} e^{3} - 7 \, a d e^{4}\right )} f g + {\left (3 \, c d^{4} e + b d^{3} e^{2} - 5 \, a d^{2} e^{3}\right )} g^{2} + {\left (4 \, {\left (2 \, c d e^{4} - b e^{5}\right )} f^{2} - {\left (13 \, c d^{2} e^{3} - 5 \, b d e^{4} - 3 \, a e^{5}\right )} f g + {\left (5 \, c d^{3} e^{2} - b d^{2} e^{3} - 3 \, a d e^{4}\right )} g^{2}\right )} x\right )} \sqrt {g x + f}}{4 \, {\left (d^{2} e^{6} f^{3} - 3 \, d^{3} e^{5} f^{2} g + 3 \, d^{4} e^{4} f g^{2} - d^{5} e^{3} g^{3} + {\left (e^{8} f^{3} - 3 \, d e^{7} f^{2} g + 3 \, d^{2} e^{6} f g^{2} - d^{3} e^{5} g^{3}\right )} x^{2} + 2 \, {\left (d e^{7} f^{3} - 3 \, d^{2} e^{6} f^{2} g + 3 \, d^{3} e^{5} f g^{2} - d^{4} e^{4} g^{3}\right )} x\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.20, size = 373, normalized size = 1.81 \begin {gather*} \frac {{\left (3 \, c d^{2} g^{2} - 8 \, c d f g e + b d g^{2} e + 8 \, c f^{2} e^{2} - 4 \, b f g e^{2} + 3 \, a g^{2} e^{2}\right )} \arctan \left (\frac {\sqrt {g x + f} e}{\sqrt {d g e - f e^{2}}}\right )}{4 \, {\left (d^{2} g^{2} e^{2} - 2 \, d f g e^{3} + f^{2} e^{4}\right )} \sqrt {d g e - f e^{2}}} - \frac {3 \, \sqrt {g x + f} c d^{3} g^{3} + 5 \, {\left (g x + f\right )}^{\frac {3}{2}} c d^{2} g^{2} e - 11 \, \sqrt {g x + f} c d^{2} f g^{2} e + \sqrt {g x + f} b d^{2} g^{3} e - 8 \, {\left (g x + f\right )}^{\frac {3}{2}} c d f g e^{2} + 8 \, \sqrt {g x + f} c d f^{2} g e^{2} - {\left (g x + f\right )}^{\frac {3}{2}} b d g^{2} e^{2} + 3 \, \sqrt {g x + f} b d f g^{2} e^{2} - 5 \, \sqrt {g x + f} a d g^{3} e^{2} + 4 \, {\left (g x + f\right )}^{\frac {3}{2}} b f g e^{3} - 4 \, \sqrt {g x + f} b f^{2} g e^{3} - 3 \, {\left (g x + f\right )}^{\frac {3}{2}} a g^{2} e^{3} + 5 \, \sqrt {g x + f} a f g^{2} e^{3}}{4 \, {\left (d^{2} g^{2} e^{2} - 2 \, d f g e^{3} + f^{2} e^{4}\right )} {\left (d g + {\left (g x + f\right )} e - f e\right )}^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.02, size = 538, normalized size = 2.61 \begin {gather*} \frac {3 a \,g^{2} \arctan \left (\frac {\sqrt {g x +f}\, e}{\sqrt {\left (d g -e f \right ) e}}\right )}{4 \left (d^{2} g^{2}-2 d e f g +e^{2} f^{2}\right ) \sqrt {\left (d g -e f \right ) e}}+\frac {b d \,g^{2} \arctan \left (\frac {\sqrt {g x +f}\, e}{\sqrt {\left (d g -e f \right ) e}}\right )}{4 \left (d^{2} g^{2}-2 d e f g +e^{2} f^{2}\right ) \sqrt {\left (d g -e f \right ) e}\, e}-\frac {b f g \arctan \left (\frac {\sqrt {g x +f}\, e}{\sqrt {\left (d g -e f \right ) e}}\right )}{\left (d^{2} g^{2}-2 d e f g +e^{2} f^{2}\right ) \sqrt {\left (d g -e f \right ) e}}+\frac {3 c \,d^{2} g^{2} \arctan \left (\frac {\sqrt {g x +f}\, e}{\sqrt {\left (d g -e f \right ) e}}\right )}{4 \left (d^{2} g^{2}-2 d e f g +e^{2} f^{2}\right ) \sqrt {\left (d g -e f \right ) e}\, e^{2}}-\frac {2 c d f g \arctan \left (\frac {\sqrt {g x +f}\, e}{\sqrt {\left (d g -e f \right ) e}}\right )}{\left (d^{2} g^{2}-2 d e f g +e^{2} f^{2}\right ) \sqrt {\left (d g -e f \right ) e}\, e}+\frac {2 c \,f^{2} \arctan \left (\frac {\sqrt {g x +f}\, e}{\sqrt {\left (d g -e f \right ) e}}\right )}{\left (d^{2} g^{2}-2 d e f g +e^{2} f^{2}\right ) \sqrt {\left (d g -e f \right ) e}}+\frac {\frac {\left (3 a \,e^{2} g +b d e g -4 b \,e^{2} f -5 c \,d^{2} g +8 d e c f \right ) \left (g x +f \right )^{\frac {3}{2}} g}{4 \left (d^{2} g^{2}-2 d e f g +e^{2} f^{2}\right ) e}+\frac {\left (5 a \,e^{2} g -b d e g -4 b \,e^{2} f -3 c \,d^{2} g +8 d e c f \right ) \sqrt {g x +f}\, g}{4 \left (d g -e f \right ) e^{2}}}{\left (d g -e f +\left (g x +f \right ) e \right )^{2}} \end {gather*}
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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mupad [B] time = 0.28, size = 270, normalized size = 1.31 \begin {gather*} \frac {\mathrm {atan}\left (\frac {\sqrt {e}\,\sqrt {f+g\,x}}{\sqrt {d\,g-e\,f}}\right )\,\left (3\,c\,d^2\,g^2-8\,c\,d\,e\,f\,g+b\,d\,e\,g^2+8\,c\,e^2\,f^2-4\,b\,e^2\,f\,g+3\,a\,e^2\,g^2\right )}{4\,e^{5/2}\,{\left (d\,g-e\,f\right )}^{5/2}}-\frac {\frac {\sqrt {f+g\,x}\,\left (3\,c\,d^2\,g^2+b\,d\,e\,g^2-8\,c\,f\,d\,e\,g-5\,a\,e^2\,g^2+4\,b\,f\,e^2\,g\right )}{4\,e^2\,\left (d\,g-e\,f\right )}-\frac {{\left (f+g\,x\right )}^{3/2}\,\left (-5\,c\,d^2\,g^2+b\,d\,e\,g^2+8\,c\,f\,d\,e\,g+3\,a\,e^2\,g^2-4\,b\,f\,e^2\,g\right )}{4\,e\,{\left (d\,g-e\,f\right )}^2}}{e^2\,{\left (f+g\,x\right )}^2-\left (f+g\,x\right )\,\left (2\,e^2\,f-2\,d\,e\,g\right )+d^2\,g^2+e^2\,f^2-2\,d\,e\,f\,g} \end {gather*}
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 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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