Integrand size = 29, antiderivative size = 1098 \[ \int \frac {(f+g x)^3 \left (a+b x+c x^2\right )^{3/2}}{d+e x} \, dx=-\frac {\left (3 \left (7 b^5 e^5 g^3-512 c^5 d^2 (e f-d g)^3+128 c^4 e (5 b d-4 a e) (e f-d g)^3-4 b^3 c e^4 g^2 (9 b e f-3 b d g+8 a e g)+8 b c^2 e^3 g \left (2 a^2 e^2 g^2+6 a b e g (3 e f-d g)+3 b^2 \left (3 e^2 f^2-3 d e f g+d^2 g^2\right )\right )-32 b c^3 e^2 \left (2 b (e f-d g)^3+3 a e g \left (3 e^2 f^2-3 d e f g+d^2 g^2\right )\right )\right )+2 c e \left (8 c e (2 c d-b e) \left (24 c^2 e^2 f^3+7 b^2 d e g^3-4 c d g^2 (9 b e f-3 b d g+a e g)\right )-2 \left (8 c^2 d^2-4 b c d e-\frac {3 b^2 e^2}{2}+6 a c e^2\right ) g \left (7 b^2 e^2 g^2-4 c e g (9 b e f-3 b d g+a e g)+24 c^2 \left (3 e^2 f^2-3 d e f g+d^2 g^2\right )\right )\right ) x\right ) \sqrt {a+b x+c x^2}}{1536 c^4 e^6}+\frac {\left (7 b^3 e^3 g^3+64 c^3 (e f-d g)^3-4 b c e^2 g^2 (9 b e f-3 b d g+a e g)+24 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 (7 b^2 e^2 g^2-4 c e g (9 b e f-3 b d g+a e g)+24 c^2 \left (3 e^2 f^2-3 d e f g+d^2 g^2\right )\right ) x\right ) \left (a+b x+c x^2\right )^{3/2}}{192 c^3 e^4}+\frac {g^2 (36 c e f-22 c d g-7 b e g) \left (a+b x+c x^2\right )^{5/2}}{60 c^2 e^2}+\frac {g^3 (d+e x) \left (a+b x+c x^2\right )^{5/2}}{6 c e^2}+\frac {\left (4 c e (2 c d-b e) \left (8 c e (b d-2 a e) \left (24 c^2 e^2 f^3+7 b^2 d e g^3-4 c d g^2 (9 b e f-3 b d g+a e g)\right )-d \left (8 b c d-3 b^2 e-4 a c e\right ) g \left (7 b^2 e^2 g^2-4 c e g (9 b e f-3 b d g+a e g)+24 c^2 \left (3 e^2 f^2-3 d e f g+d^2 g^2\right )\right )\right )-2 \left (4 c^2 d^2-\frac {b^2 e^2}{2}-2 c e (b d-a e)\right ) \left (8 c e (2 c d-b e) \left (24 c^2 e^2 f^3+7 b^2 d e g^3-4 c d g^2 (9 b e f-3 b d g+a e g)\right )-2 \left (8 c^2 d^2-4 b c d e-\frac {3 b^2 e^2}{2}+6 a c e^2\right ) g \left (7 b^2 e^2 g^2-4 c e g (9 b e f-3 b d g+a e g)+24 c^2 \left (3 e^2 f^2-3 d e f g+d^2 g^2\right )\right )\right )\right ) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{3072 c^{9/2} e^7}+\frac {\left (c d^2-b d e+a e^2\right )^{3/2} (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^7} \]
[Out]
Time = 2.20 (sec) , antiderivative size = 1098, normalized size of antiderivative = 1.00, number of steps used = 9, 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 \left (a+b x+c x^2\right )^{3/2}}{d+e x} \, dx=\frac {(d+e x) \left (c x^2+b x+a\right )^{5/2} g^3}{6 c e^2}+\frac {(36 c e f-22 c d g-7 b e g) \left (c x^2+b x+a\right )^{5/2} g^2}{60 c^2 e^2}+\frac {\left (7 b^3 e^3 g^3-4 b c e^2 (9 b e f-3 b d g+a e g) g^2+24 b c^2 e \left (3 e^2 f^2-3 d e g f+d^2 g^2\right ) g+2 c e \left (24 \left (3 e^2 f^2-3 d e g f+d^2 g^2\right ) c^2-4 e g (9 b e f-3 b d g+a e g) c+7 b^2 e^2 g^2\right ) x g+64 c^3 (e f-d g)^3\right ) \left (c x^2+b x+a\right )^{3/2}}{192 c^3 e^4}+\frac {\left (4 c e (2 c d-b e) \left (8 c e (b d-2 a e) \left (24 c^2 e^2 f^3+7 b^2 d e g^3-4 c d g^2 (9 b e f-3 b d g+a e g)\right )-d \left (-3 e b^2+8 c d b-4 a c e\right ) g \left (24 \left (3 e^2 f^2-3 d e g f+d^2 g^2\right ) c^2-4 e g (9 b e f-3 b d g+a e g) c+7 b^2 e^2 g^2\right )\right )-2 \left (4 c^2 d^2-\frac {b^2 e^2}{2}-2 c e (b d-a e)\right ) \left (8 c e (2 c d-b e) \left (24 c^2 e^2 f^3+7 b^2 d e g^3-4 c d g^2 (9 b e f-3 b d g+a e g)\right )-2 \left (8 c^2 d^2-4 b c e d-\frac {3 b^2 e^2}{2}+6 a c e^2\right ) g \left (24 \left (3 e^2 f^2-3 d e g f+d^2 g^2\right ) c^2-4 e g (9 b e f-3 b d g+a e g) c+7 b^2 e^2 g^2\right )\right )\right ) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {c x^2+b x+a}}\right )}{3072 c^{9/2} e^7}+\frac {\left (c d^2-b e d+a e^2\right )^{3/2} (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 e d+a e^2} \sqrt {c x^2+b x+a}}\right )}{e^7}-\frac {\left (3 \left (-512 d^2 (e f-d g)^3 c^5+128 e (5 b d-4 a e) (e f-d g)^3 c^4-32 b e^2 \left (2 b (e f-d g)^3+3 a e g \left (3 e^2 f^2-3 d e g f+d^2 g^2\right )\right ) c^3+8 b e^3 g \left (3 \left (3 e^2 f^2-3 d e g f+d^2 g^2\right ) b^2+6 a e g (3 e f-d g) b+2 a^2 e^2 g^2\right ) c^2-4 b^3 e^4 g^2 (9 b e f-3 b d g+8 a e g) c+7 b^5 e^5 g^3\right )+2 c e \left (8 c e (2 c d-b e) \left (24 c^2 e^2 f^3+7 b^2 d e g^3-4 c d g^2 (9 b e f-3 b d g+a e g)\right )-2 \left (8 c^2 d^2-4 b c e d-\frac {3 b^2 e^2}{2}+6 a c e^2\right ) g \left (24 \left (3 e^2 f^2-3 d e g f+d^2 g^2\right ) c^2-4 e g (9 b e f-3 b d g+a e g) c+7 b^2 e^2 g^2\right )\right ) x\right ) \sqrt {c x^2+b x+a}}{1536 c^4 e^6} \]
[In]
[Out]
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 )^{5/2}}{6 c e^2}+\frac {\int \frac {\left (a+b x+c x^2\right )^{3/2} \left (\frac {1}{2} e \left (12 c e^2 f^3-d (5 b d+2 a e) g^3\right )-e g \left (e (6 b d+a e) g^2-c \left (18 e^2 f^2-5 d^2 g^2\right )\right ) x+\frac {1}{2} e^2 g^2 (36 c e f-22 c d g-7 b e g) x^2\right )}{d+e x} \, dx}{6 c e^3} \\ & = \frac {g^2 (36 c e f-22 c d g-7 b e g) \left (a+b x+c x^2\right )^{5/2}}{60 c^2 e^2}+\frac {g^3 (d+e x) \left (a+b x+c x^2\right )^{5/2}}{6 c e^2}+\frac {\int \frac {\left (\frac {5}{4} e^3 \left (24 c^2 e^2 f^3+7 b^2 d e g^3-4 c d g^2 (9 b e f-3 b d g+a e g)\right )+\frac {5}{4} e^3 g \left (7 b^2 e^2 g^2-4 c e g (9 b e f-3 b d g+a e g)+24 c^2 \left (3 e^2 f^2-3 d e f g+d^2 g^2\right )\right ) x\right ) \left (a+b x+c x^2\right )^{3/2}}{d+e x} \, dx}{30 c^2 e^5} \\ & = \frac {\left (7 b^3 e^3 g^3+64 c^3 (e f-d g)^3-4 b c e^2 g^2 (9 b e f-3 b d g+a e g)+24 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 (7 b^2 e^2 g^2-4 c e g (9 b e f-3 b d g+a e g)+24 c^2 \left (3 e^2 f^2-3 d e f g+d^2 g^2\right )\right ) x\right ) \left (a+b x+c x^2\right )^{3/2}}{192 c^3 e^4}+\frac {g^2 (36 c e f-22 c d g-7 b e g) \left (a+b x+c x^2\right )^{5/2}}{60 c^2 e^2}+\frac {g^3 (d+e x) \left (a+b x+c x^2\right )^{5/2}}{6 c e^2}-\frac {\int \frac {\left (\frac {5}{8} e^3 \left (8 c e (b d-2 a e) \left (24 c^2 e^2 f^3+7 b^2 d e g^3-4 c d g^2 (9 b e f-3 b d g+a e g)\right )+2 \left (2 a c d e-b d \left (4 c d-\frac {3 b e}{2}\right )\right ) g \left (7 b^2 e^2 g^2-4 c e g (9 b e f-3 b d g+a e g)+24 c^2 \left (3 e^2 f^2-3 d e f g+d^2 g^2\right )\right )\right )+\frac {5}{8} e^3 \left (8 c e (2 c d-b e) \left (24 c^2 e^2 f^3+7 b^2 d e g^3-4 c d g^2 (9 b e f-3 b d g+a e g)\right )-2 \left (8 c^2 d^2-\frac {3 b^2 e^2}{2}-2 c e (2 b d-3 a e)\right ) g \left (7 b^2 e^2 g^2-4 c e g (9 b e f-3 b d g+a e g)+24 c^2 \left (3 e^2 f^2-3 d e f g+d^2 g^2\right )\right )\right ) x\right ) \sqrt {a+b x+c x^2}}{d+e x} \, dx}{240 c^3 e^7} \\ & = -\frac {\left (3 \left (7 b^5 e^5 g^3-512 c^5 d^2 (e f-d g)^3+128 c^4 e (5 b d-4 a e) (e f-d g)^3-4 b^3 c e^4 g^2 (9 b e f-3 b d g+8 a e g)+8 b c^2 e^3 g \left (2 a^2 e^2 g^2+6 a b e g (3 e f-d g)+3 b^2 \left (3 e^2 f^2-3 d e f g+d^2 g^2\right )\right )-32 b c^3 e^2 \left (2 b (e f-d g)^3+3 a e g \left (3 e^2 f^2-3 d e f g+d^2 g^2\right )\right )\right )+2 c e \left (8 c e (2 c d-b e) \left (24 c^2 e^2 f^3+7 b^2 d e g^3-4 c d g^2 (9 b e f-3 b d g+a e g)\right )-2 \left (8 c^2 d^2-4 b c d e-\frac {3 b^2 e^2}{2}+6 a c e^2\right ) g \left (7 b^2 e^2 g^2-4 c e g (9 b e f-3 b d g+a e g)+24 c^2 \left (3 e^2 f^2-3 d e f g+d^2 g^2\right )\right )\right ) x\right ) \sqrt {a+b x+c x^2}}{1536 c^4 e^6}+\frac {\left (7 b^3 e^3 g^3+64 c^3 (e f-d g)^3-4 b c e^2 g^2 (9 b e f-3 b d g+a e g)+24 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 (7 b^2 e^2 g^2-4 c e g (9 b e f-3 b d g+a e g)+24 c^2 \left (3 e^2 f^2-3 d e f g+d^2 g^2\right )\right ) x\right ) \left (a+b x+c x^2\right )^{3/2}}{192 c^3 e^4}+\frac {g^2 (36 c e f-22 c d g-7 b e g) \left (a+b x+c x^2\right )^{5/2}}{60 c^2 e^2}+\frac {g^3 (d+e x) \left (a+b x+c x^2\right )^{5/2}}{6 c e^2}+\frac {\int \frac {\frac {5}{16} e^3 \left (4 c e (b d-2 a e) \left (8 c e (b d-2 a e) \left (24 c^2 e^2 f^3+7 b^2 d e g^3-4 c d g^2 (9 b e f-3 b d g+a e g)\right )-d \left (8 b c d-3 b^2 e-4 a c e\right ) g \left (7 b^2 e^2 g^2-4 c e g (9 b e f-3 b d g+a e g)+24 c^2 \left (3 e^2 f^2-3 d e f g+d^2 g^2\right )\right )\right )-d \left (4 b c d-b^2 e-4 a c e\right ) \left (8 c e (2 c d-b e) \left (24 c^2 e^2 f^3+7 b^2 d e g^3-4 c d g^2 (9 b e f-3 b d g+a e g)\right )-2 \left (8 c^2 d^2-4 b c d e-\frac {3 b^2 e^2}{2}+6 a c e^2\right ) g \left (7 b^2 e^2 g^2-4 c e g (9 b e f-3 b d g+a e g)+24 c^2 \left (3 e^2 f^2-3 d e f g+d^2 g^2\right )\right )\right )\right )+\frac {5}{16} e^3 \left (4 c e (2 c d-b e) \left (8 c e (b d-2 a e) \left (24 c^2 e^2 f^3+7 b^2 d e g^3-4 c d g^2 (9 b e f-3 b d g+a e g)\right )-d \left (8 b c d-3 b^2 e-4 a c e\right ) g \left (7 b^2 e^2 g^2-4 c e g (9 b e f-3 b d g+a e g)+24 c^2 \left (3 e^2 f^2-3 d e f g+d^2 g^2\right )\right )\right )-2 \left (4 c^2 d^2-\frac {b^2 e^2}{2}-2 c e (b d-a e)\right ) \left (8 c e (2 c d-b e) \left (24 c^2 e^2 f^3+7 b^2 d e g^3-4 c d g^2 (9 b e f-3 b d g+a e g)\right )-2 \left (8 c^2 d^2-4 b c d e-\frac {3 b^2 e^2}{2}+6 a c e^2\right ) g \left (7 b^2 e^2 g^2-4 c e g (9 b e f-3 b d g+a e g)+24 c^2 \left (3 e^2 f^2-3 d e f g+d^2 g^2\right )\right )\right )\right ) x}{(d+e x) \sqrt {a+b x+c x^2}} \, dx}{960 c^4 e^9} \\ & = -\frac {\left (3 \left (7 b^5 e^5 g^3-512 c^5 d^2 (e f-d g)^3+128 c^4 e (5 b d-4 a e) (e f-d g)^3-4 b^3 c e^4 g^2 (9 b e f-3 b d g+8 a e g)+8 b c^2 e^3 g \left (2 a^2 e^2 g^2+6 a b e g (3 e f-d g)+3 b^2 \left (3 e^2 f^2-3 d e f g+d^2 g^2\right )\right )-32 b c^3 e^2 \left (2 b (e f-d g)^3+3 a e g \left (3 e^2 f^2-3 d e f g+d^2 g^2\right )\right )\right )+2 c e \left (8 c e (2 c d-b e) \left (24 c^2 e^2 f^3+7 b^2 d e g^3-4 c d g^2 (9 b e f-3 b d g+a e g)\right )-2 \left (8 c^2 d^2-4 b c d e-\frac {3 b^2 e^2}{2}+6 a c e^2\right ) g \left (7 b^2 e^2 g^2-4 c e g (9 b e f-3 b d g+a e g)+24 c^2 \left (3 e^2 f^2-3 d e f g+d^2 g^2\right )\right )\right ) x\right ) \sqrt {a+b x+c x^2}}{1536 c^4 e^6}+\frac {\left (7 b^3 e^3 g^3+64 c^3 (e f-d g)^3-4 b c e^2 g^2 (9 b e f-3 b d g+a e g)+24 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 (7 b^2 e^2 g^2-4 c e g (9 b e f-3 b d g+a e g)+24 c^2 \left (3 e^2 f^2-3 d e f g+d^2 g^2\right )\right ) x\right ) \left (a+b x+c x^2\right )^{3/2}}{192 c^3 e^4}+\frac {g^2 (36 c e f-22 c d g-7 b e g) \left (a+b x+c x^2\right )^{5/2}}{60 c^2 e^2}+\frac {g^3 (d+e x) \left (a+b x+c x^2\right )^{5/2}}{6 c e^2}+\frac {\left (\left (c d^2-b d e+a e^2\right )^2 (e f-d g)^3\right ) \int \frac {1}{(d+e x) \sqrt {a+b x+c x^2}} \, dx}{e^7}+\frac {\left (4 c e (2 c d-b e) \left (8 c e (b d-2 a e) \left (24 c^2 e^2 f^3+7 b^2 d e g^3-4 c d g^2 (9 b e f-3 b d g+a e g)\right )-d \left (8 b c d-3 b^2 e-4 a c e\right ) g \left (7 b^2 e^2 g^2-4 c e g (9 b e f-3 b d g+a e g)+24 c^2 \left (3 e^2 f^2-3 d e f g+d^2 g^2\right )\right )\right )-2 \left (4 c^2 d^2-\frac {b^2 e^2}{2}-2 c e (b d-a e)\right ) \left (8 c e (2 c d-b e) \left (24 c^2 e^2 f^3+7 b^2 d e g^3-4 c d g^2 (9 b e f-3 b d g+a e g)\right )-2 \left (8 c^2 d^2-4 b c d e-\frac {3 b^2 e^2}{2}+6 a c e^2\right ) g \left (7 b^2 e^2 g^2-4 c e g (9 b e f-3 b d g+a e g)+24 c^2 \left (3 e^2 f^2-3 d e f g+d^2 g^2\right )\right )\right )\right ) \int \frac {1}{\sqrt {a+b x+c x^2}} \, dx}{3072 c^4 e^7} \\ & = -\frac {\left (3 \left (7 b^5 e^5 g^3-512 c^5 d^2 (e f-d g)^3+128 c^4 e (5 b d-4 a e) (e f-d g)^3-4 b^3 c e^4 g^2 (9 b e f-3 b d g+8 a e g)+8 b c^2 e^3 g \left (2 a^2 e^2 g^2+6 a b e g (3 e f-d g)+3 b^2 \left (3 e^2 f^2-3 d e f g+d^2 g^2\right )\right )-32 b c^3 e^2 \left (2 b (e f-d g)^3+3 a e g \left (3 e^2 f^2-3 d e f g+d^2 g^2\right )\right )\right )+2 c e \left (8 c e (2 c d-b e) \left (24 c^2 e^2 f^3+7 b^2 d e g^3-4 c d g^2 (9 b e f-3 b d g+a e g)\right )-2 \left (8 c^2 d^2-4 b c d e-\frac {3 b^2 e^2}{2}+6 a c e^2\right ) g \left (7 b^2 e^2 g^2-4 c e g (9 b e f-3 b d g+a e g)+24 c^2 \left (3 e^2 f^2-3 d e f g+d^2 g^2\right )\right )\right ) x\right ) \sqrt {a+b x+c x^2}}{1536 c^4 e^6}+\frac {\left (7 b^3 e^3 g^3+64 c^3 (e f-d g)^3-4 b c e^2 g^2 (9 b e f-3 b d g+a e g)+24 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 (7 b^2 e^2 g^2-4 c e g (9 b e f-3 b d g+a e g)+24 c^2 \left (3 e^2 f^2-3 d e f g+d^2 g^2\right )\right ) x\right ) \left (a+b x+c x^2\right )^{3/2}}{192 c^3 e^4}+\frac {g^2 (36 c e f-22 c d g-7 b e g) \left (a+b x+c x^2\right )^{5/2}}{60 c^2 e^2}+\frac {g^3 (d+e x) \left (a+b x+c x^2\right )^{5/2}}{6 c e^2}-\frac {\left (2 \left (c d^2-b d e+a e^2\right )^2 (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^7}+\frac {\left (4 c e (2 c d-b e) \left (8 c e (b d-2 a e) \left (24 c^2 e^2 f^3+7 b^2 d e g^3-4 c d g^2 (9 b e f-3 b d g+a e g)\right )-d \left (8 b c d-3 b^2 e-4 a c e\right ) g \left (7 b^2 e^2 g^2-4 c e g (9 b e f-3 b d g+a e g)+24 c^2 \left (3 e^2 f^2-3 d e f g+d^2 g^2\right )\right )\right )-2 \left (4 c^2 d^2-\frac {b^2 e^2}{2}-2 c e (b d-a e)\right ) \left (8 c e (2 c d-b e) \left (24 c^2 e^2 f^3+7 b^2 d e g^3-4 c d g^2 (9 b e f-3 b d g+a e g)\right )-2 \left (8 c^2 d^2-4 b c d e-\frac {3 b^2 e^2}{2}+6 a c e^2\right ) g \left (7 b^2 e^2 g^2-4 c e g (9 b e f-3 b d g+a e g)+24 c^2 \left (3 e^2 f^2-3 d e f g+d^2 g^2\right )\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 )}{1536 c^4 e^7} \\ & = -\frac {\left (3 \left (7 b^5 e^5 g^3-512 c^5 d^2 (e f-d g)^3+128 c^4 e (5 b d-4 a e) (e f-d g)^3-4 b^3 c e^4 g^2 (9 b e f-3 b d g+8 a e g)+8 b c^2 e^3 g \left (2 a^2 e^2 g^2+6 a b e g (3 e f-d g)+3 b^2 \left (3 e^2 f^2-3 d e f g+d^2 g^2\right )\right )-32 b c^3 e^2 \left (2 b (e f-d g)^3+3 a e g \left (3 e^2 f^2-3 d e f g+d^2 g^2\right )\right )\right )+2 c e \left (8 c e (2 c d-b e) \left (24 c^2 e^2 f^3+7 b^2 d e g^3-4 c d g^2 (9 b e f-3 b d g+a e g)\right )-2 \left (8 c^2 d^2-4 b c d e-\frac {3 b^2 e^2}{2}+6 a c e^2\right ) g \left (7 b^2 e^2 g^2-4 c e g (9 b e f-3 b d g+a e g)+24 c^2 \left (3 e^2 f^2-3 d e f g+d^2 g^2\right )\right )\right ) x\right ) \sqrt {a+b x+c x^2}}{1536 c^4 e^6}+\frac {\left (7 b^3 e^3 g^3+64 c^3 (e f-d g)^3-4 b c e^2 g^2 (9 b e f-3 b d g+a e g)+24 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 (7 b^2 e^2 g^2-4 c e g (9 b e f-3 b d g+a e g)+24 c^2 \left (3 e^2 f^2-3 d e f g+d^2 g^2\right )\right ) x\right ) \left (a+b x+c x^2\right )^{3/2}}{192 c^3 e^4}+\frac {g^2 (36 c e f-22 c d g-7 b e g) \left (a+b x+c x^2\right )^{5/2}}{60 c^2 e^2}+\frac {g^3 (d+e x) \left (a+b x+c x^2\right )^{5/2}}{6 c e^2}+\frac {\left (4 c e (2 c d-b e) \left (8 c e (b d-2 a e) \left (24 c^2 e^2 f^3+7 b^2 d e g^3-4 c d g^2 (9 b e f-3 b d g+a e g)\right )-d \left (8 b c d-3 b^2 e-4 a c e\right ) g \left (7 b^2 e^2 g^2-4 c e g (9 b e f-3 b d g+a e g)+24 c^2 \left (3 e^2 f^2-3 d e f g+d^2 g^2\right )\right )\right )-2 \left (4 c^2 d^2-\frac {b^2 e^2}{2}-2 c e (b d-a e)\right ) \left (8 c e (2 c d-b e) \left (24 c^2 e^2 f^3+7 b^2 d e g^3-4 c d g^2 (9 b e f-3 b d g+a e g)\right )-2 \left (8 c^2 d^2-4 b c d e-\frac {3 b^2 e^2}{2}+6 a c e^2\right ) g \left (7 b^2 e^2 g^2-4 c e g (9 b e f-3 b d g+a e g)+24 c^2 \left (3 e^2 f^2-3 d e f g+d^2 g^2\right )\right )\right )\right ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{3072 c^{9/2} e^7}+\frac {\left (c d^2-b d e+a e^2\right )^{3/2} (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^7} \\ \end{align*}
Time = 11.51 (sec) , antiderivative size = 743, normalized size of antiderivative = 0.68 \[ \int \frac {(f+g x)^3 \left (a+b x+c x^2\right )^{3/2}}{d+e x} \, dx=\frac {5120 (e f-d g)^3 (a+x (b+c x))^{3/2}+\frac {1920 e g (e f-d g)^2 (b+2 c x) (a+x (b+c x))^{3/2}}{c}+\frac {3072 e^2 g^2 (e f-d g) (a+x (b+c x))^{5/2}}{c}+\frac {2560 e^3 g^2 (f+g x) (a+x (b+c x))^{5/2}}{c}+\frac {360 \left (b^2-4 a c\right ) e g (e f-d g)^2 \left (-2 \sqrt {c} (b+2 c x) \sqrt {a+x (b+c x)}+\left (b^2-4 a c\right ) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+x (b+c x)}}\right )\right )}{c^{5/2}}-\frac {60 e^2 g (-2 c f+b g) (e f-d g) \left (2 \sqrt {c} (b+2 c x) \sqrt {a+x (b+c x)} \left (-3 b^2+8 b c x+4 c \left (5 a+2 c x^2\right )\right )+3 \left (b^2-4 a c\right )^2 \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+x (b+c x)}}\right )\right )}{c^{7/2}}+\frac {e^3 g \left (1792 g (2 c f-b g) (a+x (b+c x))^{5/2}+5 \left (24 c^2 f^2+7 b^2 g^2-4 c g (6 b f+a g)\right ) \left (\frac {16 (b+2 c x) (a+x (b+c x))^{3/2}}{c}+\frac {3 \left (b^2-4 a c\right ) \left (-2 \sqrt {c} (b+2 c x) \sqrt {a+x (b+c x)}+\left (b^2-4 a c\right ) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+x (b+c x)}}\right )\right )}{c^{5/2}}\right )\right )}{c^2}+\frac {960 (e f-d g)^3 \left (-\left ((2 c d-b e) \left (8 c^2 d^2-b^2 e^2+4 c e (-2 b d+3 a e)\right ) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+x (b+c x)}}\right )\right )-2 \sqrt {c} \left (e \sqrt {a+x (b+c x)} \left (-b^2 e^2+4 c^2 d (-2 d+e x)-2 c e (-5 b d+4 a e+b e x)\right )+8 c \left (c d^2+e (-b d+a e)\right )^{3/2} \text {arctanh}\left (\frac {-b d+2 a e-2 c d x+b e x}{2 \sqrt {c d^2+e (-b d+a e)} \sqrt {a+x (b+c x)}}\right )\right )\right )}{c^{3/2} e^3}}{15360 e^4} \]
[In]
[Out]
Time = 0.97 (sec) , antiderivative size = 1409, normalized size of antiderivative = 1.28
method | result | size |
default | \(\text {Expression too large to display}\) | \(1409\) |
risch | \(\text {Expression too large to display}\) | \(2365\) |
[In]
[Out]
Timed out. \[ \int \frac {(f+g x)^3 \left (a+b x+c x^2\right )^{3/2}}{d+e x} \, dx=\text {Timed out} \]
[In]
[Out]
\[ \int \frac {(f+g x)^3 \left (a+b x+c x^2\right )^{3/2}}{d+e x} \, dx=\int \frac {\left (f + g x\right )^{3} \left (a + b x + c x^{2}\right )^{\frac {3}{2}}}{d + e x}\, dx \]
[In]
[Out]
Exception generated. \[ \int \frac {(f+g x)^3 \left (a+b x+c x^2\right )^{3/2}}{d+e x} \, dx=\text {Exception raised: ValueError} \]
[In]
[Out]
Exception generated. \[ \int \frac {(f+g x)^3 \left (a+b x+c x^2\right )^{3/2}}{d+e x} \, dx=\text {Exception raised: TypeError} \]
[In]
[Out]
Timed out. \[ \int \frac {(f+g x)^3 \left (a+b x+c x^2\right )^{3/2}}{d+e x} \, dx=\int \frac {{\left (f+g\,x\right )}^3\,{\left (c\,x^2+b\,x+a\right )}^{3/2}}{d+e\,x} \,d x \]
[In]
[Out]