Integrand size = 29, antiderivative size = 532 \[ \int \frac {(f+g x)^3 \sqrt {a+b x+c x^2}}{d+e x} \, dx=\frac {\left (5 b^3 e^3 g^3+64 c^3 (e f-d g)^3-4 b c e^2 g^2 (6 b e f-2 b d g+a e g)+16 b c^2 e g \left (3 e^2 f^2-3 d e f g+d^2 g^2\right )+2 c e g \left (5 b^2 e^2 g^2-4 c e g (6 b e f-2 b d g+a e g)+16 c^2 \left (3 e^2 f^2-3 d e f g+d^2 g^2\right )\right ) x\right ) \sqrt {a+b x+c x^2}}{64 c^3 e^4}+\frac {g^2 (24 c e f-14 c d g-5 b e g) \left (a+b x+c x^2\right )^{3/2}}{24 c^2 e^2}+\frac {g^3 (d+e x) \left (a+b x+c x^2\right )^{3/2}}{4 c e^2}-\frac {\left (4 c e (2 c d-b e) \left (16 c^2 e^2 f^3+5 b^2 d e g^3-4 c d g^2 (6 b e f-2 b d g+a e g)\right )-2 \left (4 c^2 d^2-\frac {b^2 e^2}{2}-2 c e (b d-a e)\right ) g \left (5 b^2 e^2 g^2-4 c e g (6 b e f-2 b d g+a e g)+16 c^2 \left (3 e^2 f^2-3 d e f g+d^2 g^2\right )\right )\right ) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{128 c^{7/2} e^5}+\frac {\sqrt {c d^2-b d e+a e^2} (e f-d g)^3 \text {arctanh}\left (\frac {b d-2 a e+(2 c d-b e) x}{2 \sqrt {c d^2-b d e+a e^2} \sqrt {a+b x+c x^2}}\right )}{e^5} \]
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Time = 0.98 (sec) , antiderivative size = 532, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.207, Rules used = {1667, 828, 857, 635, 212, 738} \[ \int \frac {(f+g x)^3 \sqrt {a+b x+c x^2}}{d+e x} \, dx=-\frac {\text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right ) \left (4 c e (2 c d-b e) \left (-4 c d g^2 (a e g-2 b d g+6 b e f)+5 b^2 d e g^3+16 c^2 e^2 f^3\right )-2 g \left (-2 c e (b d-a e)-\frac {b^2 e^2}{2}+4 c^2 d^2\right ) \left (-4 c e g (a e g-2 b d g+6 b e f)+5 b^2 e^2 g^2+16 c^2 \left (d^2 g^2-3 d e f g+3 e^2 f^2\right )\right )\right )}{128 c^{7/2} e^5}+\frac {(e f-d g)^3 \sqrt {a e^2-b d e+c d^2} \text {arctanh}\left (\frac {-2 a e+x (2 c d-b e)+b d}{2 \sqrt {a+b x+c x^2} \sqrt {a e^2-b d e+c d^2}}\right )}{e^5}+\frac {\sqrt {a+b x+c x^2} \left (2 c e g x \left (-4 c e g (a e g-2 b d g+6 b e f)+5 b^2 e^2 g^2+16 c^2 \left (d^2 g^2-3 d e f g+3 e^2 f^2\right )\right )-4 b c e^2 g^2 (a e g-2 b d g+6 b e f)+5 b^3 e^3 g^3+16 b c^2 e g \left (d^2 g^2-3 d e f g+3 e^2 f^2\right )+64 c^3 (e f-d g)^3\right )}{64 c^3 e^4}+\frac {g^2 \left (a+b x+c x^2\right )^{3/2} (-5 b e g-14 c d g+24 c e f)}{24 c^2 e^2}+\frac {g^3 (d+e x) \left (a+b x+c x^2\right )^{3/2}}{4 c e^2} \]
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Rule 212
Rule 635
Rule 738
Rule 828
Rule 857
Rule 1667
Rubi steps \begin{align*} \text {integral}& = \frac {g^3 (d+e x) \left (a+b x+c x^2\right )^{3/2}}{4 c e^2}+\frac {\int \frac {\sqrt {a+b x+c x^2} \left (\frac {1}{2} e \left (8 c e^2 f^3-d (3 b d+2 a e) g^3\right )-e g \left (e (4 b d+a e) g^2-3 c \left (4 e^2 f^2-d^2 g^2\right )\right ) x+\frac {1}{2} e^2 g^2 (24 c e f-14 c d g-5 b e g) x^2\right )}{d+e x} \, dx}{4 c e^3} \\ & = \frac {g^2 (24 c e f-14 c d g-5 b e g) \left (a+b x+c x^2\right )^{3/2}}{24 c^2 e^2}+\frac {g^3 (d+e x) \left (a+b x+c x^2\right )^{3/2}}{4 c e^2}+\frac {\int \frac {\left (\frac {3}{4} e^3 \left (16 c^2 e^2 f^3+5 b^2 d e g^3-4 c d g^2 (6 b e f-2 b d g+a e g)\right )+\frac {3}{4} e^3 g \left (5 b^2 e^2 g^2-4 c e g (6 b e f-2 b d g+a e g)+16 c^2 \left (3 e^2 f^2-3 d e f g+d^2 g^2\right )\right ) x\right ) \sqrt {a+b x+c x^2}}{d+e x} \, dx}{12 c^2 e^5} \\ & = \frac {\left (5 b^3 e^3 g^3+64 c^3 (e f-d g)^3-4 b c e^2 g^2 (6 b e f-2 b d g+a e g)+16 b c^2 e g \left (3 e^2 f^2-3 d e f g+d^2 g^2\right )+2 c e g \left (5 b^2 e^2 g^2-4 c e g (6 b e f-2 b d g+a e g)+16 c^2 \left (3 e^2 f^2-3 d e f g+d^2 g^2\right )\right ) x\right ) \sqrt {a+b x+c x^2}}{64 c^3 e^4}+\frac {g^2 (24 c e f-14 c d g-5 b e g) \left (a+b x+c x^2\right )^{3/2}}{24 c^2 e^2}+\frac {g^3 (d+e x) \left (a+b x+c x^2\right )^{3/2}}{4 c e^2}-\frac {\int \frac {\frac {3}{8} e^3 \left (4 c e (b d-2 a e) \left (16 c^2 e^2 f^3+5 b^2 d e g^3-4 c d g^2 (6 b e f-2 b d g+a e g)\right )-d \left (4 b c d-b^2 e-4 a c e\right ) g \left (5 b^2 e^2 g^2-4 c e g (6 b e f-2 b d g+a e g)+16 c^2 \left (3 e^2 f^2-3 d e f g+d^2 g^2\right )\right )\right )+\frac {3}{8} e^3 \left (4 c e (2 c d-b e) \left (16 c^2 e^2 f^3+5 b^2 d e g^3-4 c d g^2 (6 b e f-2 b d g+a e g)\right )-2 \left (4 c^2 d^2-\frac {b^2 e^2}{2}-2 c e (b d-a e)\right ) g \left (5 b^2 e^2 g^2-4 c e g (6 b e f-2 b d g+a e g)+16 c^2 \left (3 e^2 f^2-3 d e f g+d^2 g^2\right )\right )\right ) x}{(d+e x) \sqrt {a+b x+c x^2}} \, dx}{48 c^3 e^7} \\ & = \frac {\left (5 b^3 e^3 g^3+64 c^3 (e f-d g)^3-4 b c e^2 g^2 (6 b e f-2 b d g+a e g)+16 b c^2 e g \left (3 e^2 f^2-3 d e f g+d^2 g^2\right )+2 c e g \left (5 b^2 e^2 g^2-4 c e g (6 b e f-2 b d g+a e g)+16 c^2 \left (3 e^2 f^2-3 d e f g+d^2 g^2\right )\right ) x\right ) \sqrt {a+b x+c x^2}}{64 c^3 e^4}+\frac {g^2 (24 c e f-14 c d g-5 b e g) \left (a+b x+c x^2\right )^{3/2}}{24 c^2 e^2}+\frac {g^3 (d+e x) \left (a+b x+c x^2\right )^{3/2}}{4 c e^2}+\frac {\left (\left (c d^2-b d e+a e^2\right ) (e f-d g)^3\right ) \int \frac {1}{(d+e x) \sqrt {a+b x+c x^2}} \, dx}{e^5}-\frac {\left (4 c e (2 c d-b e) \left (16 c^2 e^2 f^3+5 b^2 d e g^3-4 c d g^2 (6 b e f-2 b d g+a e g)\right )-2 \left (4 c^2 d^2-\frac {b^2 e^2}{2}-2 c e (b d-a e)\right ) g \left (5 b^2 e^2 g^2-4 c e g (6 b e f-2 b d g+a e g)+16 c^2 \left (3 e^2 f^2-3 d e f g+d^2 g^2\right )\right )\right ) \int \frac {1}{\sqrt {a+b x+c x^2}} \, dx}{128 c^3 e^5} \\ & = \frac {\left (5 b^3 e^3 g^3+64 c^3 (e f-d g)^3-4 b c e^2 g^2 (6 b e f-2 b d g+a e g)+16 b c^2 e g \left (3 e^2 f^2-3 d e f g+d^2 g^2\right )+2 c e g \left (5 b^2 e^2 g^2-4 c e g (6 b e f-2 b d g+a e g)+16 c^2 \left (3 e^2 f^2-3 d e f g+d^2 g^2\right )\right ) x\right ) \sqrt {a+b x+c x^2}}{64 c^3 e^4}+\frac {g^2 (24 c e f-14 c d g-5 b e g) \left (a+b x+c x^2\right )^{3/2}}{24 c^2 e^2}+\frac {g^3 (d+e x) \left (a+b x+c x^2\right )^{3/2}}{4 c e^2}-\frac {\left (2 \left (c d^2-b d e+a e^2\right ) (e f-d g)^3\right ) \text {Subst}\left (\int \frac {1}{4 c d^2-4 b d e+4 a e^2-x^2} \, dx,x,\frac {-b d+2 a e-(2 c d-b e) x}{\sqrt {a+b x+c x^2}}\right )}{e^5}-\frac {\left (4 c e (2 c d-b e) \left (16 c^2 e^2 f^3+5 b^2 d e g^3-4 c d g^2 (6 b e f-2 b d g+a e g)\right )-2 \left (4 c^2 d^2-\frac {b^2 e^2}{2}-2 c e (b d-a e)\right ) g \left (5 b^2 e^2 g^2-4 c e g (6 b e f-2 b d g+a e g)+16 c^2 \left (3 e^2 f^2-3 d e f g+d^2 g^2\right )\right )\right ) \text {Subst}\left (\int \frac {1}{4 c-x^2} \, dx,x,\frac {b+2 c x}{\sqrt {a+b x+c x^2}}\right )}{64 c^3 e^5} \\ & = \frac {\left (5 b^3 e^3 g^3+64 c^3 (e f-d g)^3-4 b c e^2 g^2 (6 b e f-2 b d g+a e g)+16 b c^2 e g \left (3 e^2 f^2-3 d e f g+d^2 g^2\right )+2 c e g \left (5 b^2 e^2 g^2-4 c e g (6 b e f-2 b d g+a e g)+16 c^2 \left (3 e^2 f^2-3 d e f g+d^2 g^2\right )\right ) x\right ) \sqrt {a+b x+c x^2}}{64 c^3 e^4}+\frac {g^2 (24 c e f-14 c d g-5 b e g) \left (a+b x+c x^2\right )^{3/2}}{24 c^2 e^2}+\frac {g^3 (d+e x) \left (a+b x+c x^2\right )^{3/2}}{4 c e^2}-\frac {\left (4 c e (2 c d-b e) \left (16 c^2 e^2 f^3+5 b^2 d e g^3-4 c d g^2 (6 b e f-2 b d g+a e g)\right )-2 \left (4 c^2 d^2-\frac {b^2 e^2}{2}-2 c e (b d-a e)\right ) g \left (5 b^2 e^2 g^2-4 c e g (6 b e f-2 b d g+a e g)+16 c^2 \left (3 e^2 f^2-3 d e f g+d^2 g^2\right )\right )\right ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{128 c^{7/2} e^5}+\frac {\sqrt {c d^2-b d e+a e^2} (e f-d g)^3 \tanh ^{-1}\left (\frac {b d-2 a e+(2 c d-b e) x}{2 \sqrt {c d^2-b d e+a e^2} \sqrt {a+b x+c x^2}}\right )}{e^5} \\ \end{align*}
Time = 3.04 (sec) , antiderivative size = 513, normalized size of antiderivative = 0.96 \[ \int \frac {(f+g x)^3 \sqrt {a+b x+c x^2}}{d+e x} \, dx=\frac {\frac {2 e \sqrt {a+x (b+c x)} \left (15 b^3 e^3 g^3-2 b c e^2 g^2 (26 a e g+b (36 e f-12 d g+5 e g x))+16 c^3 \left (-12 d^3 g^3+6 d^2 e g^2 (6 f+g x)-2 d e^2 g \left (18 f^2+9 f g x+2 g^2 x^2\right )+3 e^3 \left (4 f^3+6 f^2 g x+4 f g^2 x^2+g^3 x^3\right )\right )+8 c^2 e g \left (a e g (-8 d g+3 e (8 f+g x))+b \left (6 d^2 g^2-2 d e g (9 f+g x)+e^2 \left (18 f^2+6 f g x+g^2 x^2\right )\right )\right )\right )}{c^3}-768 \sqrt {-c d^2+b d e-a e^2} (-e f+d g)^3 \arctan \left (\frac {\sqrt {c} (d+e x)-e \sqrt {a+x (b+c x)}}{\sqrt {-c d^2+e (b d-a e)}}\right )-\frac {3 \left (-5 b^4 e^4 g^3+128 c^4 d (-e f+d g)^3+8 b^2 c e^3 g^2 (3 b e f-b d g+3 a e g)-16 c^2 e^2 g \left (a^2 e^2 g^2+2 a b e g (3 e f-d g)+b^2 \left (3 e^2 f^2-3 d e f g+d^2 g^2\right )\right )+64 c^3 e \left (b (e f-d g)^3+a e g \left (3 e^2 f^2-3 d e f g+d^2 g^2\right )\right )\right ) \log \left (b+2 c x-2 \sqrt {c} \sqrt {a+x (b+c x)}\right )}{c^{7/2}}}{384 e^5} \]
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Time = 0.89 (sec) , antiderivative size = 879, normalized size of antiderivative = 1.65
method | result | size |
default | \(\frac {\left (-d^{3} g^{3}+3 d^{2} e f \,g^{2}-3 d \,e^{2} f^{2} g +e^{3} f^{3}\right ) \left (\sqrt {\left (x +\frac {d}{e}\right )^{2} c +\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+\frac {e^{2} a -b d e +c \,d^{2}}{e^{2}}}+\frac {\left (b e -2 c d \right ) \ln \left (\frac {\frac {b e -2 c d}{2 e}+c \left (x +\frac {d}{e}\right )}{\sqrt {c}}+\sqrt {\left (x +\frac {d}{e}\right )^{2} c +\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+\frac {e^{2} a -b d e +c \,d^{2}}{e^{2}}}\right )}{2 e \sqrt {c}}-\frac {\left (e^{2} a -b d e +c \,d^{2}\right ) \ln \left (\frac {\frac {2 e^{2} a -2 b d e +2 c \,d^{2}}{e^{2}}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {\frac {e^{2} a -b d e +c \,d^{2}}{e^{2}}}\, \sqrt {\left (x +\frac {d}{e}\right )^{2} c +\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+\frac {e^{2} a -b d e +c \,d^{2}}{e^{2}}}}{x +\frac {d}{e}}\right )}{e^{2} \sqrt {\frac {e^{2} a -b d e +c \,d^{2}}{e^{2}}}}\right )}{e^{4}}+\frac {g \left (d^{2} g^{2} \left (\frac {\left (2 c x +b \right ) \sqrt {c \,x^{2}+b x +a}}{4 c}+\frac {\left (4 a c -b^{2}\right ) \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{8 c^{\frac {3}{2}}}\right )+e^{2} g^{2} \left (\frac {x \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}{4 c}-\frac {5 b \left (\frac {\left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}{3 c}-\frac {b \left (\frac {\left (2 c x +b \right ) \sqrt {c \,x^{2}+b x +a}}{4 c}+\frac {\left (4 a c -b^{2}\right ) \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{8 c^{\frac {3}{2}}}\right )}{2 c}\right )}{8 c}-\frac {a \left (\frac {\left (2 c x +b \right ) \sqrt {c \,x^{2}+b x +a}}{4 c}+\frac {\left (4 a c -b^{2}\right ) \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{8 c^{\frac {3}{2}}}\right )}{4 c}\right )+3 e^{2} f^{2} \left (\frac {\left (2 c x +b \right ) \sqrt {c \,x^{2}+b x +a}}{4 c}+\frac {\left (4 a c -b^{2}\right ) \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{8 c^{\frac {3}{2}}}\right )-3 d e f g \left (\frac {\left (2 c x +b \right ) \sqrt {c \,x^{2}+b x +a}}{4 c}+\frac {\left (4 a c -b^{2}\right ) \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{8 c^{\frac {3}{2}}}\right )+\left (-d e \,g^{2}+3 e^{2} f g \right ) \left (\frac {\left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}{3 c}-\frac {b \left (\frac {\left (2 c x +b \right ) \sqrt {c \,x^{2}+b x +a}}{4 c}+\frac {\left (4 a c -b^{2}\right ) \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{8 c^{\frac {3}{2}}}\right )}{2 c}\right )\right )}{e^{3}}\) | \(879\) |
risch | \(-\frac {\left (-48 g^{3} c^{3} e^{3} x^{3}-8 b \,c^{2} e^{3} g^{3} x^{2}+64 c^{3} d \,e^{2} g^{3} x^{2}-192 c^{3} e^{3} f \,g^{2} x^{2}-24 a \,c^{2} e^{3} g^{3} x +10 b^{2} c \,e^{3} g^{3} x +16 b \,c^{2} d \,e^{2} g^{3} x -48 b \,c^{2} e^{3} f \,g^{2} x -96 c^{3} d^{2} e \,g^{3} x +288 c^{3} d \,e^{2} f \,g^{2} x -288 c^{3} e^{3} f^{2} g x +52 a b c \,e^{3} g^{3}+64 a \,c^{2} d \,e^{2} g^{3}-192 a \,c^{2} e^{3} f \,g^{2}-15 b^{3} e^{3} g^{3}-24 b^{2} c d \,e^{2} g^{3}+72 b^{2} c \,e^{3} f \,g^{2}-48 b \,c^{2} d^{2} e \,g^{3}+144 b \,c^{2} d \,e^{2} f \,g^{2}-144 b \,c^{2} e^{3} f^{2} g +192 c^{3} d^{3} g^{3}-576 c^{3} d^{2} e f \,g^{2}+576 c^{3} d \,e^{2} f^{2} g -192 c^{3} e^{3} f^{3}\right ) \sqrt {c \,x^{2}+b x +a}}{192 c^{3} e^{4}}-\frac {-\frac {128 \left (a \,d^{3} e^{2} g^{3}-3 a \,d^{2} e^{3} f \,g^{2}+3 a d \,e^{4} f^{2} g -f^{3} a \,e^{5}-b \,d^{4} e \,g^{3}+3 b \,d^{3} e^{2} f \,g^{2}-3 b \,d^{2} e^{3} f^{2} g +b d \,e^{4} f^{3}+d^{5} g^{3} c -3 c \,d^{4} e f \,g^{2}+3 c \,d^{3} e^{2} f^{2} g -c \,d^{2} e^{3} f^{3}\right ) c^{3} \ln \left (\frac {\frac {2 e^{2} a -2 b d e +2 c \,d^{2}}{e^{2}}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {\frac {e^{2} a -b d e +c \,d^{2}}{e^{2}}}\, \sqrt {\left (x +\frac {d}{e}\right )^{2} c +\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+\frac {e^{2} a -b d e +c \,d^{2}}{e^{2}}}}{x +\frac {d}{e}}\right )}{e^{2} \sqrt {\frac {e^{2} a -b d e +c \,d^{2}}{e^{2}}}}+\frac {\left (16 a^{2} c^{2} e^{4} g^{3}-24 a \,b^{2} c \,e^{4} g^{3}-32 a b \,c^{2} d \,e^{3} g^{3}+96 a b \,c^{2} e^{4} f \,g^{2}-64 a \,c^{3} d^{2} e^{2} g^{3}+192 a \,c^{3} d \,e^{3} f \,g^{2}-192 a \,c^{3} e^{4} f^{2} g +5 b^{4} e^{4} g^{3}+8 b^{3} c d \,e^{3} g^{3}-24 b^{3} c \,e^{4} f \,g^{2}+16 b^{2} c^{2} d^{2} e^{2} g^{3}-48 b^{2} c^{2} d \,e^{3} f \,g^{2}+48 b^{2} c^{2} e^{4} f^{2} g +64 b \,c^{3} d^{3} e \,g^{3}-192 b \,c^{3} d^{2} e^{2} f \,g^{2}+192 b \,c^{3} d \,e^{3} f^{2} g -64 b \,c^{3} e^{4} f^{3}-128 c^{4} d^{4} g^{3}+384 c^{4} d^{3} e f \,g^{2}-384 c^{4} d^{2} e^{2} f^{2} g +128 c^{4} d \,e^{3} f^{3}\right ) \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{e \sqrt {c}}}{128 c^{3} e^{4}}\) | \(961\) |
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Timed out. \[ \int \frac {(f+g x)^3 \sqrt {a+b x+c x^2}}{d+e x} \, dx=\text {Timed out} \]
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\[ \int \frac {(f+g x)^3 \sqrt {a+b x+c x^2}}{d+e x} \, dx=\int \frac {\left (f + g x\right )^{3} \sqrt {a + b x + c x^{2}}}{d + e x}\, dx \]
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Exception generated. \[ \int \frac {(f+g x)^3 \sqrt {a+b x+c x^2}}{d+e x} \, dx=\text {Exception raised: ValueError} \]
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Exception generated. \[ \int \frac {(f+g x)^3 \sqrt {a+b x+c x^2}}{d+e x} \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int \frac {(f+g x)^3 \sqrt {a+b x+c x^2}}{d+e x} \, dx=\int \frac {{\left (f+g\,x\right )}^3\,\sqrt {c\,x^2+b\,x+a}}{d+e\,x} \,d x \]
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