Optimal. Leaf size=1401 \[ \frac {(12 B c d-7 b B e+2 A c e) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {c x^2+b x+a}}\right ) e^5}{2 c^{9/2}}+\frac {\left (105 B e^5 b^6-10 c e^4 (11 B d+3 A e) b^5-4 c e^3 \left (5 A c d e+8 B \left (2 c d^2+35 a e^2\right )\right ) b^4-16 c^2 e^2 \left (3 B c d^3+A c e d^2-78 a B e^2 d-20 a A e^3\right ) b^3+16 c^2 e \left (6 A c d e \left (9 c d^2+2 a e^2\right )+7 B \left (4 c^2 d^4+6 a c e^2 d^2+33 a^2 e^4\right )\right ) b^2-32 c^3 \left (A e \left (40 c^2 d^4+78 a c e^2 d^2+33 a^2 e^4\right )+B \left (8 c^2 d^5+74 a c e^2 d^3+141 a^2 e^4 d\right )\right ) b+64 c^3 \left (6 a B e \left (2 c^2 d^4+9 a c e^2 d^2-8 a^2 e^4\right )+A c d \left (8 c^2 d^4+26 a c e^2 d^2+33 a^2 e^4\right )\right )\right ) \sqrt {c x^2+b x+a} e}{15 c^4 \left (b^2-4 a c\right )^3}+\frac {2 (d+e x) \left (7 B e^3 \left (c d^2-5 a e^2\right ) b^5-2 c e^3 \left (A c d^2-16 a B e d-5 a A e^2\right ) b^4-8 c e \left (A c d e \left (21 c d^2-a e^2\right )+6 B \left (2 c^2 d^4+a c e^2 d^2-7 a^2 e^4\right )\right ) b^3+16 c^2 \left (6 A e \left (3 c^2 d^4+6 a c e^2 d^2-a^2 e^4\right )+B \left (4 c^2 d^5+37 a c e^2 d^3-21 a^2 e^4 d\right )\right ) b^2-16 c^2 \left (2 A c d \left (4 c^2 d^4+19 a c e^2 d^2+21 a^2 e^4\right )+a B e \left (16 c^2 d^4+75 a c e^2 d^2+57 a^2 e^4\right )\right ) b+32 a c^3 e \left (6 a B d e \left (c d^2+11 a e^2\right )+A \left (4 c^2 d^4+9 a c e^2 d^2+15 a^2 e^4\right )\right )-\left (35 B e^5 b^6-2 c e^4 (23 B d+5 A e) b^5-4 c e^3 \left (5 B c d^2+A c e d+91 a B e^2\right ) b^4-8 c^2 e^2 \left (5 B c d^3+7 A c e d^2-63 a B e^2 d-13 a A e^3\right ) b^3+16 c^2 e \left (A c d e \left (29 c d^2+9 a e^2\right )+B \left (14 c^2 d^4+21 a c e^2 d^2+72 a^2 e^4\right )\right ) b^2-32 c^3 \left (A e \left (20 c^2 d^4+33 a c e^2 d^2+12 a^2 e^4\right )+B \left (4 c^2 d^5+35 a c e^2 d^3+60 a^2 e^4 d\right )\right ) b+64 c^3 \left (6 a B e \left (c^2 d^4+4 a c e^2 d^2-2 a^2 e^4\right )+A c d \left (4 c^2 d^4+11 a c e^2 d^2+12 a^2 e^4\right )\right )\right ) x\right )}{15 c^3 \left (b^2-4 a c\right )^3 \sqrt {c x^2+b x+a}}+\frac {2 (d+e x)^3 \left (B e \left (3 c d^2-7 a e^2\right ) b^3-2 c \left (4 B c d^3+9 A c e d^2-a A e^3\right ) b^2+4 c \left (4 A c d \left (c d^2+3 a e^2\right )+a B e \left (9 c d^2+11 a e^2\right )\right ) b-8 a c^2 e \left (3 A c d^2+12 a B e d+5 a A e^2\right )-\left (7 B e^3 b^4-2 c e^2 (3 B d+A e) b^3-12 c e \left (B c d^2+A c e d+4 a B e^2\right ) b^2+8 c^2 \left (2 B c d^3+6 A c e d^2+9 a B e^2 d+3 a A e^3\right ) b-16 c^2 \left (3 a B e \left (c d^2-a e^2\right )+A c d \left (2 c d^2+3 a e^2\right )\right )\right ) x\right )}{15 c^2 \left (b^2-4 a c\right )^2 \left (c x^2+b x+a\right )^{3/2}}+\frac {2 (d+e x)^5 \left (2 a c (B d+A e)-b (A c d+a B e)-\left (B e b^2-c (B d+A e) b+2 c (A c d-a B e)\right ) x\right )}{5 c \left (b^2-4 a c\right ) \left (c x^2+b x+a\right )^{5/2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 2.20, antiderivative size = 1401, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {818, 640, 621, 206} \[ \frac {(12 B c d-7 b B e+2 A c e) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {c x^2+b x+a}}\right ) e^5}{2 c^{9/2}}+\frac {\left (105 B e^5 b^6-10 c e^4 (11 B d+3 A e) b^5-4 c e^3 \left (5 A c d e+8 B \left (2 c d^2+35 a e^2\right )\right ) b^4-16 c^2 e^2 \left (3 B c d^3+A c e d^2-78 a B e^2 d-20 a A e^3\right ) b^3+16 c^2 e \left (6 A c d e \left (9 c d^2+2 a e^2\right )+7 B \left (4 c^2 d^4+6 a c e^2 d^2+33 a^2 e^4\right )\right ) b^2-32 c^3 \left (A e \left (40 c^2 d^4+78 a c e^2 d^2+33 a^2 e^4\right )+B \left (8 c^2 d^5+74 a c e^2 d^3+141 a^2 e^4 d\right )\right ) b+64 c^3 \left (6 a B e \left (2 c^2 d^4+9 a c e^2 d^2-8 a^2 e^4\right )+A c d \left (8 c^2 d^4+26 a c e^2 d^2+33 a^2 e^4\right )\right )\right ) \sqrt {c x^2+b x+a} e}{15 c^4 \left (b^2-4 a c\right )^3}+\frac {2 (d+e x) \left (7 B e^3 \left (c d^2-5 a e^2\right ) b^5-2 c e^3 \left (A c d^2-16 a B e d-5 a A e^2\right ) b^4-8 c e \left (A c d e \left (21 c d^2-a e^2\right )+6 B \left (2 c^2 d^4+a c e^2 d^2-7 a^2 e^4\right )\right ) b^3+16 c^2 \left (6 A e \left (3 c^2 d^4+6 a c e^2 d^2-a^2 e^4\right )+B \left (4 c^2 d^5+37 a c e^2 d^3-21 a^2 e^4 d\right )\right ) b^2-16 c^2 \left (2 A c d \left (4 c^2 d^4+19 a c e^2 d^2+21 a^2 e^4\right )+a B e \left (16 c^2 d^4+75 a c e^2 d^2+57 a^2 e^4\right )\right ) b+32 a c^3 e \left (6 a B d e \left (c d^2+11 a e^2\right )+A \left (4 c^2 d^4+9 a c e^2 d^2+15 a^2 e^4\right )\right )-\left (35 B e^5 b^6-2 c e^4 (23 B d+5 A e) b^5-4 c e^3 \left (5 B c d^2+A c e d+91 a B e^2\right ) b^4-8 c^2 e^2 \left (5 B c d^3+7 A c e d^2-63 a B e^2 d-13 a A e^3\right ) b^3+16 c^2 e \left (A c d e \left (29 c d^2+9 a e^2\right )+B \left (14 c^2 d^4+21 a c e^2 d^2+72 a^2 e^4\right )\right ) b^2-32 c^3 \left (A e \left (20 c^2 d^4+33 a c e^2 d^2+12 a^2 e^4\right )+B \left (4 c^2 d^5+35 a c e^2 d^3+60 a^2 e^4 d\right )\right ) b+64 c^3 \left (6 a B e \left (c^2 d^4+4 a c e^2 d^2-2 a^2 e^4\right )+A c d \left (4 c^2 d^4+11 a c e^2 d^2+12 a^2 e^4\right )\right )\right ) x\right )}{15 c^3 \left (b^2-4 a c\right )^3 \sqrt {c x^2+b x+a}}+\frac {2 (d+e x)^3 \left (B e \left (3 c d^2-7 a e^2\right ) b^3-2 c \left (4 B c d^3+9 A c e d^2-a A e^3\right ) b^2+4 c \left (4 A c d \left (c d^2+3 a e^2\right )+a B e \left (9 c d^2+11 a e^2\right )\right ) b-8 a c^2 e \left (3 A c d^2+12 a B e d+5 a A e^2\right )-\left (7 B e^3 b^4-2 c e^2 (3 B d+A e) b^3-12 c e \left (B c d^2+A c e d+4 a B e^2\right ) b^2+8 c^2 \left (2 B c d^3+6 A c e d^2+9 a B e^2 d+3 a A e^3\right ) b-16 c^2 \left (3 a B e \left (c d^2-a e^2\right )+A c d \left (2 c d^2+3 a e^2\right )\right )\right ) x\right )}{15 c^2 \left (b^2-4 a c\right )^2 \left (c x^2+b x+a\right )^{3/2}}+\frac {2 (d+e x)^5 \left (2 a c (B d+A e)-b (A c d+a B e)-\left (B e b^2-c (B d+A e) b+2 c (A c d-a B e)\right ) x\right )}{5 c \left (b^2-4 a c\right ) \left (c x^2+b x+a\right )^{5/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 206
Rule 621
Rule 640
Rule 818
Rubi steps
\begin {align*} \int \frac {(A+B x) (d+e x)^6}{\left (a+b x+c x^2\right )^{7/2}} \, dx &=\frac {2 (d+e x)^5 \left (2 a c (B d+A e)-b (A c d+a B e)-\left (b^2 B e-b c (B d+A e)+2 c (A c d-a B e)\right ) x\right )}{5 c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{5/2}}+\frac {2 \int \frac {(d+e x)^4 \left (\frac {1}{2} \left (-16 A c^2 d^2-2 b B e \left (\frac {3 b d}{2}-5 a e\right )-4 a c e (6 B d+5 A e)+2 b c d (4 B d+9 A e)\right )+\frac {1}{2} e \left (7 b^2 B e-2 b c (B d+A e)+4 c (A c d-6 a B e)\right ) x\right )}{\left (a+b x+c x^2\right )^{5/2}} \, dx}{5 c \left (b^2-4 a c\right )}\\ &=\frac {2 (d+e x)^5 \left (2 a c (B d+A e)-b (A c d+a B e)-\left (b^2 B e-b c (B d+A e)+2 c (A c d-a B e)\right ) x\right )}{5 c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{5/2}}+\frac {2 (d+e x)^3 \left (b^3 B e \left (3 c d^2-7 a e^2\right )-8 a c^2 e \left (3 A c d^2+12 a B d e+5 a A e^2\right )-2 b^2 c \left (4 B c d^3+9 A c d^2 e-a A e^3\right )+4 b c \left (4 A c d \left (c d^2+3 a e^2\right )+a B e \left (9 c d^2+11 a e^2\right )\right )-\left (7 b^4 B e^3-2 b^3 c e^2 (3 B d+A e)-12 b^2 c e \left (B c d^2+A c d e+4 a B e^2\right )+8 b c^2 \left (2 B c d^3+6 A c d^2 e+9 a B d e^2+3 a A e^3\right )-16 c^2 \left (3 a B e \left (c d^2-a e^2\right )+A c d \left (2 c d^2+3 a e^2\right )\right )\right ) x\right )}{15 c^2 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )^{3/2}}+\frac {4 \int \frac {(d+e x)^2 \left (\frac {1}{4} \left (-7 b^4 B d e^3+2 b^3 e^3 (A c d+21 a B e)+12 b^2 c e \left (8 B c d^3+14 A c d^2 e+3 a B d e^2-a A e^3\right )+16 c^2 \left (6 a B d e \left (2 c d^2+7 a e^2\right )+A \left (8 c^2 d^4+18 a c d^2 e^2+15 a^2 e^4\right )\right )-8 b c \left (9 A c d e \left (4 c d^2+5 a e^2\right )+B \left (8 c^2 d^4+60 a c d^2 e^2+33 a^2 e^4\right )\right )\right )+\frac {1}{4} e \left (35 b^4 B e^3-2 b^3 c e^2 (9 B d+5 A e)-12 b^2 c e \left (2 B c d^2+A c d e+21 a B e^2\right )+8 b c^2 \left (4 B c d^3+12 A c d^2 e+21 a B d e^2+9 a A e^3\right )-16 c^2 \left (6 a B e \left (c d^2-4 a e^2\right )+A c d \left (4 c d^2+9 a e^2\right )\right )\right ) x\right )}{\left (a+b x+c x^2\right )^{3/2}} \, dx}{15 c^2 \left (b^2-4 a c\right )^2}\\ &=\frac {2 (d+e x)^5 \left (2 a c (B d+A e)-b (A c d+a B e)-\left (b^2 B e-b c (B d+A e)+2 c (A c d-a B e)\right ) x\right )}{5 c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{5/2}}+\frac {2 (d+e x)^3 \left (b^3 B e \left (3 c d^2-7 a e^2\right )-8 a c^2 e \left (3 A c d^2+12 a B d e+5 a A e^2\right )-2 b^2 c \left (4 B c d^3+9 A c d^2 e-a A e^3\right )+4 b c \left (4 A c d \left (c d^2+3 a e^2\right )+a B e \left (9 c d^2+11 a e^2\right )\right )-\left (7 b^4 B e^3-2 b^3 c e^2 (3 B d+A e)-12 b^2 c e \left (B c d^2+A c d e+4 a B e^2\right )+8 b c^2 \left (2 B c d^3+6 A c d^2 e+9 a B d e^2+3 a A e^3\right )-16 c^2 \left (3 a B e \left (c d^2-a e^2\right )+A c d \left (2 c d^2+3 a e^2\right )\right )\right ) x\right )}{15 c^2 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )^{3/2}}+\frac {2 (d+e x) \left (7 b^5 B e^3 \left (c d^2-5 a e^2\right )-2 b^4 c e^3 \left (A c d^2-16 a B d e-5 a A e^2\right )-8 b^3 c e \left (A c d e \left (21 c d^2-a e^2\right )+6 B \left (2 c^2 d^4+a c d^2 e^2-7 a^2 e^4\right )\right )+32 a c^3 e \left (6 a B d e \left (c d^2+11 a e^2\right )+A \left (4 c^2 d^4+9 a c d^2 e^2+15 a^2 e^4\right )\right )-16 b c^2 \left (2 A c d \left (4 c^2 d^4+19 a c d^2 e^2+21 a^2 e^4\right )+a B e \left (16 c^2 d^4+75 a c d^2 e^2+57 a^2 e^4\right )\right )+16 b^2 c^2 \left (6 A e \left (3 c^2 d^4+6 a c d^2 e^2-a^2 e^4\right )+B \left (4 c^2 d^5+37 a c d^3 e^2-21 a^2 d e^4\right )\right )-\left (35 b^6 B e^5-2 b^5 c e^4 (23 B d+5 A e)-4 b^4 c e^3 \left (5 B c d^2+A c d e+91 a B e^2\right )-8 b^3 c^2 e^2 \left (5 B c d^3+7 A c d^2 e-63 a B d e^2-13 a A e^3\right )+64 c^3 \left (6 a B e \left (c^2 d^4+4 a c d^2 e^2-2 a^2 e^4\right )+A c d \left (4 c^2 d^4+11 a c d^2 e^2+12 a^2 e^4\right )\right )+16 b^2 c^2 e \left (A c d e \left (29 c d^2+9 a e^2\right )+B \left (14 c^2 d^4+21 a c d^2 e^2+72 a^2 e^4\right )\right )-32 b c^3 \left (A e \left (20 c^2 d^4+33 a c d^2 e^2+12 a^2 e^4\right )+B \left (4 c^2 d^5+35 a c d^3 e^2+60 a^2 d e^4\right )\right )\right ) x\right )}{15 c^3 \left (b^2-4 a c\right )^3 \sqrt {a+b x+c x^2}}+\frac {8 \int \frac {\frac {1}{8} e \left (35 b^6 B d e^4-960 a^3 c^3 e^4 (6 B d+A e)-2 b^5 e^3 \left (16 B c d^2+5 A c d e-35 a B e^2\right )-4 b^4 c e^2 \left (6 B c d^3+2 A c d^2 e+114 a B d e^2+5 a A e^3\right )+16 b^3 c e \left (3 A c d e \left (9 c d^2+2 a e^2\right )+7 B \left (2 c^2 d^4+3 a c d^2 e^2-6 a^2 e^4\right )\right )+32 b c^2 \left (3 a B e \left (4 c^2 d^4+18 a c d^2 e^2+19 a^2 e^4\right )+A c d \left (8 c^2 d^4+26 a c d^2 e^2+33 a^2 e^4\right )\right )-16 b^2 c^2 \left (2 A e \left (20 c^2 d^4+39 a c d^2 e^2-6 a^2 e^4\right )+B \left (8 c^2 d^5+74 a c d^3 e^2-129 a^2 d e^4\right )\right )\right )+\frac {1}{8} e \left (105 b^6 B e^5-10 b^5 c e^4 (11 B d+3 A e)-16 b^3 c^2 e^2 \left (3 B c d^3+A c d^2 e-78 a B d e^2-20 a A e^3\right )-4 b^4 c e^3 \left (5 A c d e+8 B \left (2 c d^2+35 a e^2\right )\right )+16 b^2 c^2 e \left (6 A c d e \left (9 c d^2+2 a e^2\right )+7 B \left (4 c^2 d^4+6 a c d^2 e^2+33 a^2 e^4\right )\right )+64 c^3 \left (6 a B e \left (2 c^2 d^4+9 a c d^2 e^2-8 a^2 e^4\right )+A c d \left (8 c^2 d^4+26 a c d^2 e^2+33 a^2 e^4\right )\right )-32 b c^3 \left (A e \left (40 c^2 d^4+78 a c d^2 e^2+33 a^2 e^4\right )+B \left (8 c^2 d^5+74 a c d^3 e^2+141 a^2 d e^4\right )\right )\right ) x}{\sqrt {a+b x+c x^2}} \, dx}{15 c^3 \left (b^2-4 a c\right )^3}\\ &=\frac {2 (d+e x)^5 \left (2 a c (B d+A e)-b (A c d+a B e)-\left (b^2 B e-b c (B d+A e)+2 c (A c d-a B e)\right ) x\right )}{5 c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{5/2}}+\frac {2 (d+e x)^3 \left (b^3 B e \left (3 c d^2-7 a e^2\right )-8 a c^2 e \left (3 A c d^2+12 a B d e+5 a A e^2\right )-2 b^2 c \left (4 B c d^3+9 A c d^2 e-a A e^3\right )+4 b c \left (4 A c d \left (c d^2+3 a e^2\right )+a B e \left (9 c d^2+11 a e^2\right )\right )-\left (7 b^4 B e^3-2 b^3 c e^2 (3 B d+A e)-12 b^2 c e \left (B c d^2+A c d e+4 a B e^2\right )+8 b c^2 \left (2 B c d^3+6 A c d^2 e+9 a B d e^2+3 a A e^3\right )-16 c^2 \left (3 a B e \left (c d^2-a e^2\right )+A c d \left (2 c d^2+3 a e^2\right )\right )\right ) x\right )}{15 c^2 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )^{3/2}}+\frac {2 (d+e x) \left (7 b^5 B e^3 \left (c d^2-5 a e^2\right )-2 b^4 c e^3 \left (A c d^2-16 a B d e-5 a A e^2\right )-8 b^3 c e \left (A c d e \left (21 c d^2-a e^2\right )+6 B \left (2 c^2 d^4+a c d^2 e^2-7 a^2 e^4\right )\right )+32 a c^3 e \left (6 a B d e \left (c d^2+11 a e^2\right )+A \left (4 c^2 d^4+9 a c d^2 e^2+15 a^2 e^4\right )\right )-16 b c^2 \left (2 A c d \left (4 c^2 d^4+19 a c d^2 e^2+21 a^2 e^4\right )+a B e \left (16 c^2 d^4+75 a c d^2 e^2+57 a^2 e^4\right )\right )+16 b^2 c^2 \left (6 A e \left (3 c^2 d^4+6 a c d^2 e^2-a^2 e^4\right )+B \left (4 c^2 d^5+37 a c d^3 e^2-21 a^2 d e^4\right )\right )-\left (35 b^6 B e^5-2 b^5 c e^4 (23 B d+5 A e)-4 b^4 c e^3 \left (5 B c d^2+A c d e+91 a B e^2\right )-8 b^3 c^2 e^2 \left (5 B c d^3+7 A c d^2 e-63 a B d e^2-13 a A e^3\right )+64 c^3 \left (6 a B e \left (c^2 d^4+4 a c d^2 e^2-2 a^2 e^4\right )+A c d \left (4 c^2 d^4+11 a c d^2 e^2+12 a^2 e^4\right )\right )+16 b^2 c^2 e \left (A c d e \left (29 c d^2+9 a e^2\right )+B \left (14 c^2 d^4+21 a c d^2 e^2+72 a^2 e^4\right )\right )-32 b c^3 \left (A e \left (20 c^2 d^4+33 a c d^2 e^2+12 a^2 e^4\right )+B \left (4 c^2 d^5+35 a c d^3 e^2+60 a^2 d e^4\right )\right )\right ) x\right )}{15 c^3 \left (b^2-4 a c\right )^3 \sqrt {a+b x+c x^2}}+\frac {e \left (105 b^6 B e^5-10 b^5 c e^4 (11 B d+3 A e)-16 b^3 c^2 e^2 \left (3 B c d^3+A c d^2 e-78 a B d e^2-20 a A e^3\right )-4 b^4 c e^3 \left (5 A c d e+8 B \left (2 c d^2+35 a e^2\right )\right )+16 b^2 c^2 e \left (6 A c d e \left (9 c d^2+2 a e^2\right )+7 B \left (4 c^2 d^4+6 a c d^2 e^2+33 a^2 e^4\right )\right )+64 c^3 \left (6 a B e \left (2 c^2 d^4+9 a c d^2 e^2-8 a^2 e^4\right )+A c d \left (8 c^2 d^4+26 a c d^2 e^2+33 a^2 e^4\right )\right )-32 b c^3 \left (A e \left (40 c^2 d^4+78 a c d^2 e^2+33 a^2 e^4\right )+B \left (8 c^2 d^5+74 a c d^3 e^2+141 a^2 d e^4\right )\right )\right ) \sqrt {a+b x+c x^2}}{15 c^4 \left (b^2-4 a c\right )^3}+\frac {\left (e^5 (12 B c d-7 b B e+2 A c e)\right ) \int \frac {1}{\sqrt {a+b x+c x^2}} \, dx}{2 c^4}\\ &=\frac {2 (d+e x)^5 \left (2 a c (B d+A e)-b (A c d+a B e)-\left (b^2 B e-b c (B d+A e)+2 c (A c d-a B e)\right ) x\right )}{5 c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{5/2}}+\frac {2 (d+e x)^3 \left (b^3 B e \left (3 c d^2-7 a e^2\right )-8 a c^2 e \left (3 A c d^2+12 a B d e+5 a A e^2\right )-2 b^2 c \left (4 B c d^3+9 A c d^2 e-a A e^3\right )+4 b c \left (4 A c d \left (c d^2+3 a e^2\right )+a B e \left (9 c d^2+11 a e^2\right )\right )-\left (7 b^4 B e^3-2 b^3 c e^2 (3 B d+A e)-12 b^2 c e \left (B c d^2+A c d e+4 a B e^2\right )+8 b c^2 \left (2 B c d^3+6 A c d^2 e+9 a B d e^2+3 a A e^3\right )-16 c^2 \left (3 a B e \left (c d^2-a e^2\right )+A c d \left (2 c d^2+3 a e^2\right )\right )\right ) x\right )}{15 c^2 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )^{3/2}}+\frac {2 (d+e x) \left (7 b^5 B e^3 \left (c d^2-5 a e^2\right )-2 b^4 c e^3 \left (A c d^2-16 a B d e-5 a A e^2\right )-8 b^3 c e \left (A c d e \left (21 c d^2-a e^2\right )+6 B \left (2 c^2 d^4+a c d^2 e^2-7 a^2 e^4\right )\right )+32 a c^3 e \left (6 a B d e \left (c d^2+11 a e^2\right )+A \left (4 c^2 d^4+9 a c d^2 e^2+15 a^2 e^4\right )\right )-16 b c^2 \left (2 A c d \left (4 c^2 d^4+19 a c d^2 e^2+21 a^2 e^4\right )+a B e \left (16 c^2 d^4+75 a c d^2 e^2+57 a^2 e^4\right )\right )+16 b^2 c^2 \left (6 A e \left (3 c^2 d^4+6 a c d^2 e^2-a^2 e^4\right )+B \left (4 c^2 d^5+37 a c d^3 e^2-21 a^2 d e^4\right )\right )-\left (35 b^6 B e^5-2 b^5 c e^4 (23 B d+5 A e)-4 b^4 c e^3 \left (5 B c d^2+A c d e+91 a B e^2\right )-8 b^3 c^2 e^2 \left (5 B c d^3+7 A c d^2 e-63 a B d e^2-13 a A e^3\right )+64 c^3 \left (6 a B e \left (c^2 d^4+4 a c d^2 e^2-2 a^2 e^4\right )+A c d \left (4 c^2 d^4+11 a c d^2 e^2+12 a^2 e^4\right )\right )+16 b^2 c^2 e \left (A c d e \left (29 c d^2+9 a e^2\right )+B \left (14 c^2 d^4+21 a c d^2 e^2+72 a^2 e^4\right )\right )-32 b c^3 \left (A e \left (20 c^2 d^4+33 a c d^2 e^2+12 a^2 e^4\right )+B \left (4 c^2 d^5+35 a c d^3 e^2+60 a^2 d e^4\right )\right )\right ) x\right )}{15 c^3 \left (b^2-4 a c\right )^3 \sqrt {a+b x+c x^2}}+\frac {e \left (105 b^6 B e^5-10 b^5 c e^4 (11 B d+3 A e)-16 b^3 c^2 e^2 \left (3 B c d^3+A c d^2 e-78 a B d e^2-20 a A e^3\right )-4 b^4 c e^3 \left (5 A c d e+8 B \left (2 c d^2+35 a e^2\right )\right )+16 b^2 c^2 e \left (6 A c d e \left (9 c d^2+2 a e^2\right )+7 B \left (4 c^2 d^4+6 a c d^2 e^2+33 a^2 e^4\right )\right )+64 c^3 \left (6 a B e \left (2 c^2 d^4+9 a c d^2 e^2-8 a^2 e^4\right )+A c d \left (8 c^2 d^4+26 a c d^2 e^2+33 a^2 e^4\right )\right )-32 b c^3 \left (A e \left (40 c^2 d^4+78 a c d^2 e^2+33 a^2 e^4\right )+B \left (8 c^2 d^5+74 a c d^3 e^2+141 a^2 d e^4\right )\right )\right ) \sqrt {a+b x+c x^2}}{15 c^4 \left (b^2-4 a c\right )^3}+\frac {\left (e^5 (12 B c d-7 b B e+2 A c e)\right ) \operatorname {Subst}\left (\int \frac {1}{4 c-x^2} \, dx,x,\frac {b+2 c x}{\sqrt {a+b x+c x^2}}\right )}{c^4}\\ &=\frac {2 (d+e x)^5 \left (2 a c (B d+A e)-b (A c d+a B e)-\left (b^2 B e-b c (B d+A e)+2 c (A c d-a B e)\right ) x\right )}{5 c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{5/2}}+\frac {2 (d+e x)^3 \left (b^3 B e \left (3 c d^2-7 a e^2\right )-8 a c^2 e \left (3 A c d^2+12 a B d e+5 a A e^2\right )-2 b^2 c \left (4 B c d^3+9 A c d^2 e-a A e^3\right )+4 b c \left (4 A c d \left (c d^2+3 a e^2\right )+a B e \left (9 c d^2+11 a e^2\right )\right )-\left (7 b^4 B e^3-2 b^3 c e^2 (3 B d+A e)-12 b^2 c e \left (B c d^2+A c d e+4 a B e^2\right )+8 b c^2 \left (2 B c d^3+6 A c d^2 e+9 a B d e^2+3 a A e^3\right )-16 c^2 \left (3 a B e \left (c d^2-a e^2\right )+A c d \left (2 c d^2+3 a e^2\right )\right )\right ) x\right )}{15 c^2 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )^{3/2}}+\frac {2 (d+e x) \left (7 b^5 B e^3 \left (c d^2-5 a e^2\right )-2 b^4 c e^3 \left (A c d^2-16 a B d e-5 a A e^2\right )-8 b^3 c e \left (A c d e \left (21 c d^2-a e^2\right )+6 B \left (2 c^2 d^4+a c d^2 e^2-7 a^2 e^4\right )\right )+32 a c^3 e \left (6 a B d e \left (c d^2+11 a e^2\right )+A \left (4 c^2 d^4+9 a c d^2 e^2+15 a^2 e^4\right )\right )-16 b c^2 \left (2 A c d \left (4 c^2 d^4+19 a c d^2 e^2+21 a^2 e^4\right )+a B e \left (16 c^2 d^4+75 a c d^2 e^2+57 a^2 e^4\right )\right )+16 b^2 c^2 \left (6 A e \left (3 c^2 d^4+6 a c d^2 e^2-a^2 e^4\right )+B \left (4 c^2 d^5+37 a c d^3 e^2-21 a^2 d e^4\right )\right )-\left (35 b^6 B e^5-2 b^5 c e^4 (23 B d+5 A e)-4 b^4 c e^3 \left (5 B c d^2+A c d e+91 a B e^2\right )-8 b^3 c^2 e^2 \left (5 B c d^3+7 A c d^2 e-63 a B d e^2-13 a A e^3\right )+64 c^3 \left (6 a B e \left (c^2 d^4+4 a c d^2 e^2-2 a^2 e^4\right )+A c d \left (4 c^2 d^4+11 a c d^2 e^2+12 a^2 e^4\right )\right )+16 b^2 c^2 e \left (A c d e \left (29 c d^2+9 a e^2\right )+B \left (14 c^2 d^4+21 a c d^2 e^2+72 a^2 e^4\right )\right )-32 b c^3 \left (A e \left (20 c^2 d^4+33 a c d^2 e^2+12 a^2 e^4\right )+B \left (4 c^2 d^5+35 a c d^3 e^2+60 a^2 d e^4\right )\right )\right ) x\right )}{15 c^3 \left (b^2-4 a c\right )^3 \sqrt {a+b x+c x^2}}+\frac {e \left (105 b^6 B e^5-10 b^5 c e^4 (11 B d+3 A e)-16 b^3 c^2 e^2 \left (3 B c d^3+A c d^2 e-78 a B d e^2-20 a A e^3\right )-4 b^4 c e^3 \left (5 A c d e+8 B \left (2 c d^2+35 a e^2\right )\right )+16 b^2 c^2 e \left (6 A c d e \left (9 c d^2+2 a e^2\right )+7 B \left (4 c^2 d^4+6 a c d^2 e^2+33 a^2 e^4\right )\right )+64 c^3 \left (6 a B e \left (2 c^2 d^4+9 a c d^2 e^2-8 a^2 e^4\right )+A c d \left (8 c^2 d^4+26 a c d^2 e^2+33 a^2 e^4\right )\right )-32 b c^3 \left (A e \left (40 c^2 d^4+78 a c d^2 e^2+33 a^2 e^4\right )+B \left (8 c^2 d^5+74 a c d^3 e^2+141 a^2 d e^4\right )\right )\right ) \sqrt {a+b x+c x^2}}{15 c^4 \left (b^2-4 a c\right )^3}+\frac {e^5 (12 B c d-7 b B e+2 A c e) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{2 c^{9/2}}\\ \end {align*}
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Mathematica [A] time = 11.25, size = 2369, normalized size = 1.69 \[ \text {Result too large to show} \]
Antiderivative was successfully verified.
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fricas [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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giac [B] time = 0.49, size = 3352, normalized size = 2.39 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.05, size = 11346, normalized size = 8.10 \[ \text {output too large to display} \]
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 [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {\left (A+B\,x\right )\,{\left (d+e\,x\right )}^6}{{\left (c\,x^2+b\,x+a\right )}^{7/2}} \,d x \]
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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