Optimal. Leaf size=248 \[ -\frac {\sqrt {f+g x} \left (a e^2-b d e+c d^2\right )}{2 e (d+e x)^2 (e f-d g)^2}-\frac {\tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right ) \left (3 e g (5 a e g-b (d g+4 e f))+c \left (-d^2 g^2+8 d e f g+8 e^2 f^2\right )\right )}{4 e^{3/2} (e f-d g)^{7/2}}+\frac {2 \left (a g^2-b f g+c f^2\right )}{\sqrt {f+g x} (e f-d g)^3}+\frac {\sqrt {f+g x} (c d (8 e f-d g)-e (-7 a e g+3 b d g+4 b e f))}{4 e (d+e x) (e f-d g)^3} \]
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Rubi [A] time = 0.63, antiderivative size = 248, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {897, 1259, 456, 453, 208} \[ -\frac {\tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right ) \left (3 e g (5 a e g-b (d g+4 e f))+c \left (-d^2 g^2+8 d e f g+8 e^2 f^2\right )\right )}{4 e^{3/2} (e f-d g)^{7/2}}-\frac {\sqrt {f+g x} \left (a e^2-b d e+c d^2\right )}{2 e (d+e x)^2 (e f-d g)^2}+\frac {2 \left (a g^2-b f g+c f^2\right )}{\sqrt {f+g x} (e f-d g)^3}+\frac {\sqrt {f+g x} (c d (8 e f-d g)-e (-7 a e g+3 b d g+4 b e f))}{4 e (d+e x) (e f-d g)^3} \]
Antiderivative was successfully verified.
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Rule 208
Rule 453
Rule 456
Rule 897
Rule 1259
Rubi steps
\begin {align*} \int \frac {a+b x+c x^2}{(d+e x)^3 (f+g x)^{3/2}} \, 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}}{x^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 (e f-d g)^2 (d+e x)^2}-\frac {g^3 \operatorname {Subst}\left (\int \frac {\frac {4 e^2 (e f-d g) \left (c f^2-b f g+a g^2\right )}{g^5}-\frac {e \left (3 e (b d-a e) g^2+c \left (4 e^2 f^2-8 d e f g+d^2 g^2\right )\right ) x^2}{g^5}}{x^2 \left (\frac {-e f+d g}{g}+\frac {e x^2}{g}\right )^2} \, dx,x,\sqrt {f+g x}\right )}{2 e^2 (e f-d g)^2}\\ &=-\frac {\left (c d^2-b d e+a e^2\right ) \sqrt {f+g x}}{2 e (e f-d g)^2 (d+e x)^2}+\frac {(c d (8 e f-d g)-e (4 b e f+3 b d g-7 a e g)) \sqrt {f+g x}}{4 e (e f-d g)^3 (d+e x)}+\frac {g^3 \operatorname {Subst}\left (\int \frac {\frac {8 e^2 \left (c f^2-b f g+a g^2\right )}{g^4}+\frac {e (c d (8 e f-d g)-e (4 b e f+3 b d g-7 a e g)) x^2}{g^3 (e f-d g)}}{x^2 \left (\frac {-e f+d g}{g}+\frac {e x^2}{g}\right )} \, dx,x,\sqrt {f+g x}\right )}{4 e^2 (e f-d g)^2}\\ &=\frac {2 \left (c f^2-b f g+a g^2\right )}{(e f-d g)^3 \sqrt {f+g x}}-\frac {\left (c d^2-b d e+a e^2\right ) \sqrt {f+g x}}{2 e (e f-d g)^2 (d+e x)^2}+\frac {(c d (8 e f-d g)-e (4 b e f+3 b d g-7 a e g)) \sqrt {f+g x}}{4 e (e f-d g)^3 (d+e x)}+\frac {\left (c \left (8 e^2 f^2+8 d e f g-d^2 g^2\right )+3 e g (5 a e g-b (4 e f+d g))\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 g (e f-d g)^3}\\ &=\frac {2 \left (c f^2-b f g+a g^2\right )}{(e f-d g)^3 \sqrt {f+g x}}-\frac {\left (c d^2-b d e+a e^2\right ) \sqrt {f+g x}}{2 e (e f-d g)^2 (d+e x)^2}+\frac {(c d (8 e f-d g)-e (4 b e f+3 b d g-7 a e g)) \sqrt {f+g x}}{4 e (e f-d g)^3 (d+e x)}-\frac {\left (c \left (8 e^2 f^2+8 d e f g-d^2 g^2\right )+3 e g (5 a e g-b (4 e f+d g))\right ) \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right )}{4 e^{3/2} (e f-d g)^{7/2}}\\ \end {align*}
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Mathematica [A] time = 1.10, size = 290, normalized size = 1.17 \[ \frac {1}{4} \left (-\frac {2 \sqrt {f+g x} \left (e (a e-b d)+c d^2\right )}{e (d+e x)^2 (e f-d g)^2}+\frac {g \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right ) (e (-7 a e g+3 b d g+4 b e f)+c d (d g-8 e f))}{e^{3/2} (e f-d g)^{7/2}}+\frac {8 \left (g (a g-b f)+c f^2\right )}{\sqrt {f+g x} (e f-d g)^3}-\frac {8 \sqrt {e} \left (g (a g-b f)+c f^2\right ) \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right )}{(e f-d g)^{7/2}}-\frac {\sqrt {f+g x} (e (-7 a e g+3 b d g+4 b e f)+c d (d g-8 e f))}{e (d+e x) (e f-d g)^3}\right ) \]
Antiderivative was successfully verified.
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fricas [B] time = 1.06, size = 1883, normalized size = 7.59 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.26, size = 462, normalized size = 1.86 \[ \frac {{\left (c d^{2} g^{2} - 8 \, c d f g e + 3 \, b d g^{2} e - 8 \, c f^{2} e^{2} + 12 \, b f g e^{2} - 15 \, a g^{2} e^{2}\right )} \arctan \left (\frac {\sqrt {g x + f} e}{\sqrt {d g e - f e^{2}}}\right )}{4 \, {\left (d^{3} g^{3} e - 3 \, d^{2} f g^{2} e^{2} + 3 \, d f^{2} g e^{3} - f^{3} e^{4}\right )} \sqrt {d g e - f e^{2}}} - \frac {2 \, {\left (c f^{2} - b f g + a g^{2}\right )}}{{\left (d^{3} g^{3} - 3 \, d^{2} f g^{2} e + 3 \, d f^{2} g e^{2} - f^{3} e^{3}\right )} \sqrt {g x + f}} - \frac {\sqrt {g x + f} c d^{3} g^{3} - {\left (g x + f\right )}^{\frac {3}{2}} c d^{2} g^{2} e + 7 \, \sqrt {g x + f} c d^{2} f g^{2} e - 5 \, \sqrt {g x + f} b d^{2} g^{3} e + 8 \, {\left (g x + f\right )}^{\frac {3}{2}} c d f g e^{2} - 8 \, \sqrt {g x + f} c d f^{2} g e^{2} - 3 \, {\left (g x + f\right )}^{\frac {3}{2}} b d g^{2} e^{2} + \sqrt {g x + f} b d f g^{2} e^{2} + 9 \, \sqrt {g x + f} a d g^{3} e^{2} - 4 \, {\left (g x + f\right )}^{\frac {3}{2}} b f g e^{3} + 4 \, \sqrt {g x + f} b f^{2} g e^{3} + 7 \, {\left (g x + f\right )}^{\frac {3}{2}} a g^{2} e^{3} - 9 \, \sqrt {g x + f} a f g^{2} e^{3}}{4 \, {\left (d^{3} g^{3} e - 3 \, d^{2} f g^{2} e^{2} + 3 \, d f^{2} g e^{3} - f^{3} e^{4}\right )} {\left (d g + {\left (g x + f\right )} e - f e\right )}^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.03, size = 847, normalized size = 3.42 \[ -\frac {9 \sqrt {g x +f}\, a d e \,g^{3}}{4 \left (d g -e f \right )^{3} \left (e g x +d g \right )^{2}}+\frac {9 \sqrt {g x +f}\, a \,e^{2} f \,g^{2}}{4 \left (d g -e f \right )^{3} \left (e g x +d g \right )^{2}}+\frac {5 \sqrt {g x +f}\, b \,d^{2} g^{3}}{4 \left (d g -e f \right )^{3} \left (e g x +d g \right )^{2}}-\frac {\sqrt {g x +f}\, b d e f \,g^{2}}{4 \left (d g -e f \right )^{3} \left (e g x +d g \right )^{2}}-\frac {\sqrt {g x +f}\, b \,e^{2} f^{2} g}{\left (d g -e f \right )^{3} \left (e g x +d g \right )^{2}}-\frac {\sqrt {g x +f}\, c \,d^{3} g^{3}}{4 \left (d g -e f \right )^{3} \left (e g x +d g \right )^{2} e}-\frac {7 \sqrt {g x +f}\, c \,d^{2} f \,g^{2}}{4 \left (d g -e f \right )^{3} \left (e g x +d g \right )^{2}}+\frac {2 \sqrt {g x +f}\, c d e \,f^{2} g}{\left (d g -e f \right )^{3} \left (e g x +d g \right )^{2}}-\frac {7 \left (g x +f \right )^{\frac {3}{2}} a \,e^{2} g^{2}}{4 \left (d g -e f \right )^{3} \left (e g x +d g \right )^{2}}-\frac {15 a e \,g^{2} \arctan \left (\frac {\sqrt {g x +f}\, e}{\sqrt {\left (d g -e f \right ) e}}\right )}{4 \left (d g -e f \right )^{3} \sqrt {\left (d g -e f \right ) e}}+\frac {3 \left (g x +f \right )^{\frac {3}{2}} b d e \,g^{2}}{4 \left (d g -e f \right )^{3} \left (e g x +d g \right )^{2}}+\frac {3 b d \,g^{2} \arctan \left (\frac {\sqrt {g x +f}\, e}{\sqrt {\left (d g -e f \right ) e}}\right )}{4 \left (d g -e f \right )^{3} \sqrt {\left (d g -e f \right ) e}}+\frac {\left (g x +f \right )^{\frac {3}{2}} b \,e^{2} f g}{\left (d g -e f \right )^{3} \left (e g x +d g \right )^{2}}+\frac {3 b e f g \arctan \left (\frac {\sqrt {g x +f}\, e}{\sqrt {\left (d g -e f \right ) e}}\right )}{\left (d g -e f \right )^{3} \sqrt {\left (d g -e f \right ) e}}+\frac {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 g -e f \right )^{3} \sqrt {\left (d g -e f \right ) e}\, e}+\frac {\left (g x +f \right )^{\frac {3}{2}} c \,d^{2} g^{2}}{4 \left (d g -e f \right )^{3} \left (e g x +d g \right )^{2}}-\frac {2 \left (g x +f \right )^{\frac {3}{2}} c d e f g}{\left (d g -e f \right )^{3} \left (e g x +d g \right )^{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 g -e f \right )^{3} \sqrt {\left (d g -e f \right ) e}}-\frac {2 c e \,f^{2} \arctan \left (\frac {\sqrt {g x +f}\, e}{\sqrt {\left (d g -e f \right ) e}}\right )}{\left (d g -e f \right )^{3} \sqrt {\left (d g -e f \right ) e}}-\frac {2 a \,g^{2}}{\left (d g -e f \right )^{3} \sqrt {g x +f}}+\frac {2 b f g}{\left (d g -e f \right )^{3} \sqrt {g x +f}}-\frac {2 c \,f^{2}}{\left (d g -e f \right )^{3} \sqrt {g x +f}} \]
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 [B] time = 3.41, size = 363, normalized size = 1.46 \[ \frac {\mathrm {atan}\left (\frac {\sqrt {f+g\,x}\,\left (-d^3\,e\,g^3+3\,d^2\,e^2\,f\,g^2-3\,d\,e^3\,f^2\,g+e^4\,f^3\right )}{\sqrt {e}\,{\left (d\,g-e\,f\right )}^{7/2}}\right )\,\left (-c\,d^2\,g^2+8\,c\,d\,e\,f\,g-3\,b\,d\,e\,g^2+8\,c\,e^2\,f^2-12\,b\,e^2\,f\,g+15\,a\,e^2\,g^2\right )}{4\,e^{3/2}\,{\left (d\,g-e\,f\right )}^{7/2}}-\frac {\frac {2\,\left (c\,f^2-b\,f\,g+a\,g^2\right )}{d\,g-e\,f}+\frac {{\left (f+g\,x\right )}^2\,\left (-c\,d^2\,g^2+8\,c\,d\,e\,f\,g-3\,b\,d\,e\,g^2+8\,c\,e^2\,f^2-12\,b\,e^2\,f\,g+15\,a\,e^2\,g^2\right )}{4\,{\left (d\,g-e\,f\right )}^3}+\frac {\left (f+g\,x\right )\,\left (c\,d^2\,g^2+8\,c\,d\,e\,f\,g-5\,b\,d\,e\,g^2+16\,c\,e^2\,f^2-20\,b\,e^2\,f\,g+25\,a\,e^2\,g^2\right )}{4\,e\,{\left (d\,g-e\,f\right )}^2}}{e^2\,{\left (f+g\,x\right )}^{5/2}-{\left (f+g\,x\right )}^{3/2}\,\left (2\,e^2\,f-2\,d\,e\,g\right )+\sqrt {f+g\,x}\,\left (d^2\,g^2-2\,d\,e\,f\,g+e^2\,f^2\right )} \]
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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