Integrand size = 24, antiderivative size = 173 \[ \int \frac {(d+e x)^2 \left (a+c x^2\right )}{(f+g x)^{3/2}} \, dx=-\frac {2 (e f-d g)^2 \left (c f^2+a g^2\right )}{g^5 \sqrt {f+g x}}-\frac {4 (e f-d g) \left (a e g^2+c f (2 e f-d g)\right ) \sqrt {f+g x}}{g^5}+\frac {2 \left (a e^2 g^2+c \left (6 e^2 f^2-6 d e f g+d^2 g^2\right )\right ) (f+g x)^{3/2}}{3 g^5}-\frac {4 c e (2 e f-d g) (f+g x)^{5/2}}{5 g^5}+\frac {2 c e^2 (f+g x)^{7/2}}{7 g^5} \]
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Time = 0.13 (sec) , antiderivative size = 173, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {912, 1275} \[ \int \frac {(d+e x)^2 \left (a+c x^2\right )}{(f+g x)^{3/2}} \, dx=\frac {2 (f+g x)^{3/2} \left (a e^2 g^2+c \left (d^2 g^2-6 d e f g+6 e^2 f^2\right )\right )}{3 g^5}-\frac {2 \left (a g^2+c f^2\right ) (e f-d g)^2}{g^5 \sqrt {f+g x}}-\frac {4 \sqrt {f+g x} (e f-d g) \left (a e g^2+c f (2 e f-d g)\right )}{g^5}-\frac {4 c e (f+g x)^{5/2} (2 e f-d g)}{5 g^5}+\frac {2 c e^2 (f+g x)^{7/2}}{7 g^5} \]
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Rule 912
Rule 1275
Rubi steps \begin{align*} \text {integral}& = \frac {2 \text {Subst}\left (\int \frac {\left (\frac {-e f+d g}{g}+\frac {e x^2}{g}\right )^2 \left (\frac {c f^2+a g^2}{g^2}-\frac {2 c f x^2}{g^2}+\frac {c x^4}{g^2}\right )}{x^2} \, dx,x,\sqrt {f+g x}\right )}{g} \\ & = \frac {2 \text {Subst}\left (\int \left (\frac {2 (e f-d g) \left (-a e g^2-c f (2 e f-d g)\right )}{g^4}+\frac {(-e f+d g)^2 \left (c f^2+a g^2\right )}{g^4 x^2}+\frac {\left (a e^2 g^2+c \left (6 e^2 f^2-6 d e f g+d^2 g^2\right )\right ) x^2}{g^4}-\frac {2 c e (2 e f-d g) x^4}{g^4}+\frac {c e^2 x^6}{g^4}\right ) \, dx,x,\sqrt {f+g x}\right )}{g} \\ & = -\frac {2 (e f-d g)^2 \left (c f^2+a g^2\right )}{g^5 \sqrt {f+g x}}-\frac {4 (e f-d g) \left (a e g^2+c f (2 e f-d g)\right ) \sqrt {f+g x}}{g^5}+\frac {2 \left (a e^2 g^2+c \left (6 e^2 f^2-6 d e f g+d^2 g^2\right )\right ) (f+g x)^{3/2}}{3 g^5}-\frac {4 c e (2 e f-d g) (f+g x)^{5/2}}{5 g^5}+\frac {2 c e^2 (f+g x)^{7/2}}{7 g^5} \\ \end{align*}
Time = 0.14 (sec) , antiderivative size = 177, normalized size of antiderivative = 1.02 \[ \int \frac {(d+e x)^2 \left (a+c x^2\right )}{(f+g x)^{3/2}} \, dx=\frac {-70 a g^2 \left (3 d^2 g^2-6 d e g (2 f+g x)+e^2 \left (8 f^2+4 f g x-g^2 x^2\right )\right )+2 c \left (35 d^2 g^2 \left (-8 f^2-4 f g x+g^2 x^2\right )+42 d e g \left (16 f^3+8 f^2 g x-2 f g^2 x^2+g^3 x^3\right )-3 e^2 \left (128 f^4+64 f^3 g x-16 f^2 g^2 x^2+8 f g^3 x^3-5 g^4 x^4\right )\right )}{105 g^5 \sqrt {f+g x}} \]
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Time = 0.48 (sec) , antiderivative size = 154, normalized size of antiderivative = 0.89
method | result | size |
pseudoelliptic | \(-\frac {2 \left (\left (-\frac {x^{2} \left (\frac {3 c \,x^{2}}{7}+a \right ) e^{2}}{3}-2 \left (\frac {c \,x^{2}}{5}+a \right ) x d e +d^{2} \left (-\frac {c \,x^{2}}{3}+a \right )\right ) g^{4}-4 \left (-\frac {x \left (\frac {6 c \,x^{2}}{35}+a \right ) e^{2}}{3}+d \left (-\frac {c \,x^{2}}{5}+a \right ) e -\frac {c \,d^{2} x}{3}\right ) f \,g^{3}+\frac {8 f^{2} \left (\left (-\frac {6 c \,x^{2}}{35}+a \right ) e^{2}-\frac {6 c d e x}{5}+c \,d^{2}\right ) g^{2}}{3}-\frac {32 e \left (-\frac {2 e x}{7}+d \right ) c \,f^{3} g}{5}+\frac {128 c \,e^{2} f^{4}}{35}\right )}{\sqrt {g x +f}\, g^{5}}\) | \(154\) |
risch | \(\frac {2 \left (15 c \,e^{2} x^{3} g^{3}+42 c d e \,g^{3} x^{2}-39 c \,e^{2} f \,g^{2} x^{2}+35 a \,e^{2} g^{3} x +35 c \,d^{2} g^{3} x -126 c d e f \,g^{2} x +87 c \,e^{2} f^{2} g x +210 a d e \,g^{3}-175 a \,e^{2} f \,g^{2}-175 c \,d^{2} f \,g^{2}+462 c d e \,f^{2} g -279 c \,e^{2} f^{3}\right ) \sqrt {g x +f}}{105 g^{5}}-\frac {2 \left (a \,d^{2} g^{4}-2 a d e f \,g^{3}+a \,e^{2} f^{2} g^{2}+c \,d^{2} f^{2} g^{2}-2 c d e \,f^{3} g +c \,e^{2} f^{4}\right )}{g^{5} \sqrt {g x +f}}\) | \(207\) |
gosper | \(-\frac {2 \left (-15 c \,e^{2} x^{4} g^{4}-42 c d e \,g^{4} x^{3}+24 c \,e^{2} f \,g^{3} x^{3}-35 a \,e^{2} g^{4} x^{2}-35 c \,d^{2} g^{4} x^{2}+84 c d e f \,g^{3} x^{2}-48 c \,e^{2} f^{2} g^{2} x^{2}-210 a d e \,g^{4} x +140 a \,e^{2} f \,g^{3} x +140 c \,d^{2} f \,g^{3} x -336 c d e \,f^{2} g^{2} x +192 c \,e^{2} f^{3} g x +105 a \,d^{2} g^{4}-420 a d e f \,g^{3}+280 a \,e^{2} f^{2} g^{2}+280 c \,d^{2} f^{2} g^{2}-672 c d e \,f^{3} g +384 c \,e^{2} f^{4}\right )}{105 \sqrt {g x +f}\, g^{5}}\) | \(215\) |
trager | \(-\frac {2 \left (-15 c \,e^{2} x^{4} g^{4}-42 c d e \,g^{4} x^{3}+24 c \,e^{2} f \,g^{3} x^{3}-35 a \,e^{2} g^{4} x^{2}-35 c \,d^{2} g^{4} x^{2}+84 c d e f \,g^{3} x^{2}-48 c \,e^{2} f^{2} g^{2} x^{2}-210 a d e \,g^{4} x +140 a \,e^{2} f \,g^{3} x +140 c \,d^{2} f \,g^{3} x -336 c d e \,f^{2} g^{2} x +192 c \,e^{2} f^{3} g x +105 a \,d^{2} g^{4}-420 a d e f \,g^{3}+280 a \,e^{2} f^{2} g^{2}+280 c \,d^{2} f^{2} g^{2}-672 c d e \,f^{3} g +384 c \,e^{2} f^{4}\right )}{105 \sqrt {g x +f}\, g^{5}}\) | \(215\) |
derivativedivides | \(\frac {\frac {2 c \,e^{2} \left (g x +f \right )^{\frac {7}{2}}}{7}+\frac {4 c d e g \left (g x +f \right )^{\frac {5}{2}}}{5}-\frac {8 c \,e^{2} f \left (g x +f \right )^{\frac {5}{2}}}{5}+\frac {2 a \,e^{2} g^{2} \left (g x +f \right )^{\frac {3}{2}}}{3}+\frac {2 c \,d^{2} g^{2} \left (g x +f \right )^{\frac {3}{2}}}{3}-4 c d e f g \left (g x +f \right )^{\frac {3}{2}}+4 c \,e^{2} f^{2} \left (g x +f \right )^{\frac {3}{2}}+4 a d e \,g^{3} \sqrt {g x +f}-4 a \,e^{2} f \,g^{2} \sqrt {g x +f}-4 c \,d^{2} f \,g^{2} \sqrt {g x +f}+12 c d e \,f^{2} g \sqrt {g x +f}-8 c \,e^{2} f^{3} \sqrt {g x +f}-\frac {2 \left (a \,d^{2} g^{4}-2 a d e f \,g^{3}+a \,e^{2} f^{2} g^{2}+c \,d^{2} f^{2} g^{2}-2 c d e \,f^{3} g +c \,e^{2} f^{4}\right )}{\sqrt {g x +f}}}{g^{5}}\) | \(256\) |
default | \(\frac {\frac {2 c \,e^{2} \left (g x +f \right )^{\frac {7}{2}}}{7}+\frac {4 c d e g \left (g x +f \right )^{\frac {5}{2}}}{5}-\frac {8 c \,e^{2} f \left (g x +f \right )^{\frac {5}{2}}}{5}+\frac {2 a \,e^{2} g^{2} \left (g x +f \right )^{\frac {3}{2}}}{3}+\frac {2 c \,d^{2} g^{2} \left (g x +f \right )^{\frac {3}{2}}}{3}-4 c d e f g \left (g x +f \right )^{\frac {3}{2}}+4 c \,e^{2} f^{2} \left (g x +f \right )^{\frac {3}{2}}+4 a d e \,g^{3} \sqrt {g x +f}-4 a \,e^{2} f \,g^{2} \sqrt {g x +f}-4 c \,d^{2} f \,g^{2} \sqrt {g x +f}+12 c d e \,f^{2} g \sqrt {g x +f}-8 c \,e^{2} f^{3} \sqrt {g x +f}-\frac {2 \left (a \,d^{2} g^{4}-2 a d e f \,g^{3}+a \,e^{2} f^{2} g^{2}+c \,d^{2} f^{2} g^{2}-2 c d e \,f^{3} g +c \,e^{2} f^{4}\right )}{\sqrt {g x +f}}}{g^{5}}\) | \(256\) |
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Time = 0.26 (sec) , antiderivative size = 206, normalized size of antiderivative = 1.19 \[ \int \frac {(d+e x)^2 \left (a+c x^2\right )}{(f+g x)^{3/2}} \, dx=\frac {2 \, {\left (15 \, c e^{2} g^{4} x^{4} - 384 \, c e^{2} f^{4} + 672 \, c d e f^{3} g + 420 \, a d e f g^{3} - 105 \, a d^{2} g^{4} - 280 \, {\left (c d^{2} + a e^{2}\right )} f^{2} g^{2} - 6 \, {\left (4 \, c e^{2} f g^{3} - 7 \, c d e g^{4}\right )} x^{3} + {\left (48 \, c e^{2} f^{2} g^{2} - 84 \, c d e f g^{3} + 35 \, {\left (c d^{2} + a e^{2}\right )} g^{4}\right )} x^{2} - 2 \, {\left (96 \, c e^{2} f^{3} g - 168 \, c d e f^{2} g^{2} - 105 \, a d e g^{4} + 70 \, {\left (c d^{2} + a e^{2}\right )} f g^{3}\right )} x\right )} \sqrt {g x + f}}{105 \, {\left (g^{6} x + f g^{5}\right )}} \]
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Time = 4.82 (sec) , antiderivative size = 264, normalized size of antiderivative = 1.53 \[ \int \frac {(d+e x)^2 \left (a+c x^2\right )}{(f+g x)^{3/2}} \, dx=\begin {cases} \frac {2 \left (\frac {c e^{2} \left (f + g x\right )^{\frac {7}{2}}}{7 g^{4}} + \frac {\left (f + g x\right )^{\frac {5}{2}} \cdot \left (2 c d e g - 4 c e^{2} f\right )}{5 g^{4}} + \frac {\left (f + g x\right )^{\frac {3}{2}} \left (a e^{2} g^{2} + c d^{2} g^{2} - 6 c d e f g + 6 c e^{2} f^{2}\right )}{3 g^{4}} + \frac {\sqrt {f + g x} \left (2 a d e g^{3} - 2 a e^{2} f g^{2} - 2 c d^{2} f g^{2} + 6 c d e f^{2} g - 4 c e^{2} f^{3}\right )}{g^{4}} - \frac {\left (a g^{2} + c f^{2}\right ) \left (d g - e f\right )^{2}}{g^{4} \sqrt {f + g x}}\right )}{g} & \text {for}\: g \neq 0 \\\frac {a d^{2} x + a d e x^{2} + \frac {c d e x^{4}}{2} + \frac {c e^{2} x^{5}}{5} + \frac {x^{3} \left (a e^{2} + c d^{2}\right )}{3}}{f^{\frac {3}{2}}} & \text {otherwise} \end {cases} \]
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Time = 0.20 (sec) , antiderivative size = 205, normalized size of antiderivative = 1.18 \[ \int \frac {(d+e x)^2 \left (a+c x^2\right )}{(f+g x)^{3/2}} \, dx=\frac {2 \, {\left (\frac {15 \, {\left (g x + f\right )}^{\frac {7}{2}} c e^{2} - 42 \, {\left (2 \, c e^{2} f - c d e g\right )} {\left (g x + f\right )}^{\frac {5}{2}} + 35 \, {\left (6 \, c e^{2} f^{2} - 6 \, c d e f g + {\left (c d^{2} + a e^{2}\right )} g^{2}\right )} {\left (g x + f\right )}^{\frac {3}{2}} - 210 \, {\left (2 \, c e^{2} f^{3} - 3 \, c d e f^{2} g - a d e g^{3} + {\left (c d^{2} + a e^{2}\right )} f g^{2}\right )} \sqrt {g x + f}}{g^{4}} - \frac {105 \, {\left (c e^{2} f^{4} - 2 \, c d e f^{3} g - 2 \, a d e f g^{3} + a d^{2} g^{4} + {\left (c d^{2} + a e^{2}\right )} f^{2} g^{2}\right )}}{\sqrt {g x + f} g^{4}}\right )}}{105 \, g} \]
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Time = 0.27 (sec) , antiderivative size = 277, normalized size of antiderivative = 1.60 \[ \int \frac {(d+e x)^2 \left (a+c x^2\right )}{(f+g x)^{3/2}} \, dx=-\frac {2 \, {\left (c e^{2} f^{4} - 2 \, c d e f^{3} g + c d^{2} f^{2} g^{2} + a e^{2} f^{2} g^{2} - 2 \, a d e f g^{3} + a d^{2} g^{4}\right )}}{\sqrt {g x + f} g^{5}} + \frac {2 \, {\left (15 \, {\left (g x + f\right )}^{\frac {7}{2}} c e^{2} g^{30} - 84 \, {\left (g x + f\right )}^{\frac {5}{2}} c e^{2} f g^{30} + 210 \, {\left (g x + f\right )}^{\frac {3}{2}} c e^{2} f^{2} g^{30} - 420 \, \sqrt {g x + f} c e^{2} f^{3} g^{30} + 42 \, {\left (g x + f\right )}^{\frac {5}{2}} c d e g^{31} - 210 \, {\left (g x + f\right )}^{\frac {3}{2}} c d e f g^{31} + 630 \, \sqrt {g x + f} c d e f^{2} g^{31} + 35 \, {\left (g x + f\right )}^{\frac {3}{2}} c d^{2} g^{32} + 35 \, {\left (g x + f\right )}^{\frac {3}{2}} a e^{2} g^{32} - 210 \, \sqrt {g x + f} c d^{2} f g^{32} - 210 \, \sqrt {g x + f} a e^{2} f g^{32} + 210 \, \sqrt {g x + f} a d e g^{33}\right )}}{105 \, g^{35}} \]
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Time = 11.82 (sec) , antiderivative size = 199, normalized size of antiderivative = 1.15 \[ \int \frac {(d+e x)^2 \left (a+c x^2\right )}{(f+g x)^{3/2}} \, dx=\frac {{\left (f+g\,x\right )}^{3/2}\,\left (2\,c\,d^2\,g^2-12\,c\,d\,e\,f\,g+12\,c\,e^2\,f^2+2\,a\,e^2\,g^2\right )}{3\,g^5}-\frac {2\,c\,d^2\,f^2\,g^2+2\,a\,d^2\,g^4-4\,c\,d\,e\,f^3\,g-4\,a\,d\,e\,f\,g^3+2\,c\,e^2\,f^4+2\,a\,e^2\,f^2\,g^2}{g^5\,\sqrt {f+g\,x}}+\frac {4\,\sqrt {f+g\,x}\,\left (d\,g-e\,f\right )\,\left (2\,c\,e\,f^2-c\,d\,f\,g+a\,e\,g^2\right )}{g^5}+\frac {2\,c\,e^2\,{\left (f+g\,x\right )}^{7/2}}{7\,g^5}+\frac {4\,c\,e\,{\left (f+g\,x\right )}^{5/2}\,\left (d\,g-2\,e\,f\right )}{5\,g^5} \]
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