Optimal. Leaf size=974 \[ -\frac {(B d-A e) \tanh ^{-1}\left (\frac {b d-2 a e+(2 c d-b e) x}{2 \sqrt {c d^2-b e d+a e^2} \sqrt {c x^2+b x+a}}\right ) e^5}{\left (c d^2-b e d+a e^2\right )^{7/2}}+\frac {2 \left (4 a c e (2 c d-b e) \left (8 c e (b d-2 a e) (b B d-2 A c d+A b e-2 a B e)+(2 c d-b e) \left (5 e (B d-A e) b^2-8 c d (B d+A e) b+4 c \left (4 A c d^2-a B e d+5 a A e^2\right )\right )\right )-\left (-e b^2+c d b+2 a c e\right ) \left (8 c d e (b B d-2 A c d+A b e-2 a B e) \left (-3 e b^2+4 c d b+4 a c e\right )+\left (8 c^2 d^2-3 b^2 e^2-4 c e (b d-3 a e)\right ) \left (5 e (B d-A e) b^2-8 c d (B d+A e) b+4 c \left (4 A c d^2-a B e d+5 a A e^2\right )\right )\right )+c \left (4 c e (b d-2 a e) \left (8 c e (b d-2 a e) (b B d-2 A c d+A b e-2 a B e)+(2 c d-b e) \left (5 e (B d-A e) b^2-8 c d (B d+A e) b+4 c \left (4 A c d^2-a B e d+5 a A e^2\right )\right )\right )-(2 c d-b e) \left (8 c d e (b B d-2 A c d+A b e-2 a B e) \left (-3 e b^2+4 c d b+4 a c e\right )+\left (8 c^2 d^2-3 b^2 e^2-4 c e (b d-3 a e)\right ) \left (5 e (B d-A e) b^2-8 c d (B d+A e) b+4 c \left (4 A c d^2-a B e d+5 a A e^2\right )\right )\right )\right ) x\right )}{15 \left (b^2-4 a c\right )^3 \left (c d^2-b e d+a e^2\right )^3 \sqrt {c x^2+b x+a}}+\frac {2 \left (8 a c e (2 c d-b e) (b B d-2 A c d+A b e-2 a B e)+\left (-e b^2+c d b+2 a c e\right ) \left (5 e (B d-A e) b^2-8 c d (B d+A e) b+4 c \left (4 A c d^2-a B e d+5 a A e^2\right )\right )+c \left (8 c e (b d-2 a e) (b B d-2 A c d+A b e-2 a B e)+(2 c d-b e) \left (5 e (B d-A e) b^2-8 c d (B d+A e) b+4 c \left (4 A c d^2-a B e d+5 a A e^2\right )\right )\right ) x\right )}{15 \left (b^2-4 a c\right )^2 \left (c d^2-b e d+a e^2\right )^2 \left (c x^2+b x+a\right )^{3/2}}+\frac {2 \left (a B (2 c d-b e)-A \left (-e b^2+c d b+2 a c e\right )+c (b B d-2 A c d+A b e-2 a B e) x\right )}{5 \left (b^2-4 a c\right ) \left (c d^2-b e d+a e^2\right ) \left (c x^2+b x+a\right )^{5/2}} \]
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Rubi [A] time = 1.71, antiderivative size = 974, 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 = {822, 12, 724, 206} \begin {gather*} -\frac {(B d-A e) \tanh ^{-1}\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^5}{\left (c d^2-b e d+a e^2\right )^{7/2}}+\frac {2 \left (4 a c e (2 c d-b e) \left (8 c e (b d-2 a e) (b B d-2 A c d+A b e-2 a B e)+(2 c d-b e) \left (5 e (B d-A e) b^2-8 c d (B d+A e) b+4 c \left (4 A c d^2-a B e d+5 a A e^2\right )\right )\right )-\left (-e b^2+c d b+2 a c e\right ) \left (8 c d e (b B d-2 A c d+A b e-2 a B e) \left (-3 e b^2+4 c d b+4 a c e\right )+\left (8 c^2 d^2-3 b^2 e^2-4 c e (b d-3 a e)\right ) \left (5 e (B d-A e) b^2-8 c d (B d+A e) b+4 c \left (4 A c d^2-a B e d+5 a A e^2\right )\right )\right )+c \left (4 c e (b d-2 a e) \left (8 c e (b d-2 a e) (b B d-2 A c d+A b e-2 a B e)+(2 c d-b e) \left (5 e (B d-A e) b^2-8 c d (B d+A e) b+4 c \left (4 A c d^2-a B e d+5 a A e^2\right )\right )\right )-(2 c d-b e) \left (8 c d e (b B d-2 A c d+A b e-2 a B e) \left (-3 e b^2+4 c d b+4 a c e\right )+\left (8 c^2 d^2-3 b^2 e^2-4 c e (b d-3 a e)\right ) \left (5 e (B d-A e) b^2-8 c d (B d+A e) b+4 c \left (4 A c d^2-a B e d+5 a A e^2\right )\right )\right )\right ) x\right )}{15 \left (b^2-4 a c\right )^3 \left (c d^2-b e d+a e^2\right )^3 \sqrt {c x^2+b x+a}}+\frac {2 \left (8 a c e (2 c d-b e) (b B d-2 A c d+A b e-2 a B e)+\left (-e b^2+c d b+2 a c e\right ) \left (5 e (B d-A e) b^2-8 c d (B d+A e) b+4 c \left (4 A c d^2-a B e d+5 a A e^2\right )\right )+c \left (8 c e (b d-2 a e) (b B d-2 A c d+A b e-2 a B e)+(2 c d-b e) \left (5 e (B d-A e) b^2-8 c d (B d+A e) b+4 c \left (4 A c d^2-a B e d+5 a A e^2\right )\right )\right ) x\right )}{15 \left (b^2-4 a c\right )^2 \left (c d^2-b e d+a e^2\right )^2 \left (c x^2+b x+a\right )^{3/2}}+\frac {2 \left (a B (2 c d-b e)-A \left (-e b^2+c d b+2 a c e\right )+c (b B d-2 A c d+A b e-2 a B e) x\right )}{5 \left (b^2-4 a c\right ) \left (c d^2-b e d+a e^2\right ) \left (c x^2+b x+a\right )^{5/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 206
Rule 724
Rule 822
Rubi steps
\begin {align*} \int \frac {A+B x}{(d+e x) \left (a+b x+c x^2\right )^{7/2}} \, dx &=\frac {2 \left (a B (2 c d-b e)-A \left (b c d-b^2 e+2 a c e\right )+c (b B d-2 A c d+A b e-2 a B e) x\right )}{5 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) \left (a+b x+c x^2\right )^{5/2}}-\frac {2 \int \frac {\frac {1}{2} \left (5 b^2 e (B d-A e)-8 b c d (B d+A e)+4 c \left (4 A c d^2-a B d e+5 a A e^2\right )\right )-4 c e (b B d-2 A c d+A b e-2 a B e) x}{(d+e x) \left (a+b x+c x^2\right )^{5/2}} \, dx}{5 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )}\\ &=\frac {2 \left (a B (2 c d-b e)-A \left (b c d-b^2 e+2 a c e\right )+c (b B d-2 A c d+A b e-2 a B e) x\right )}{5 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) \left (a+b x+c x^2\right )^{5/2}}+\frac {2 \left (8 a c e (2 c d-b e) (b B d-2 A c d+A b e-2 a B e)+\left (b c d-b^2 e+2 a c e\right ) \left (5 b^2 e (B d-A e)-8 b c d (B d+A e)+4 c \left (4 A c d^2-a B d e+5 a A e^2\right )\right )+c \left (8 c e (b d-2 a e) (b B d-2 A c d+A b e-2 a B e)+(2 c d-b e) \left (5 b^2 e (B d-A e)-8 b c d (B d+A e)+4 c \left (4 A c d^2-a B d e+5 a A e^2\right )\right )\right ) x\right )}{15 \left (b^2-4 a c\right )^2 \left (c d^2-b d e+a e^2\right )^2 \left (a+b x+c x^2\right )^{3/2}}+\frac {4 \int \frac {\frac {1}{4} \left (8 c d e (b B d-2 A c d+A b e-2 a B e) \left (4 b c d-3 b^2 e+4 a c e\right )+\left (8 c^2 d^2-3 b^2 e^2-4 c e (b d-3 a e)\right ) \left (5 b^2 e (B d-A e)-8 b c d (B d+A e)+4 c \left (4 A c d^2-a B d e+5 a A e^2\right )\right )\right )+c e \left (8 c e (b d-2 a e) (b B d-2 A c d+A b e-2 a B e)+(2 c d-b e) \left (5 b^2 e (B d-A e)-8 b c d (B d+A e)+4 c \left (4 A c d^2-a B d e+5 a A e^2\right )\right )\right ) x}{(d+e x) \left (a+b x+c x^2\right )^{3/2}} \, dx}{15 \left (b^2-4 a c\right )^2 \left (c d^2-b d e+a e^2\right )^2}\\ &=\frac {2 \left (a B (2 c d-b e)-A \left (b c d-b^2 e+2 a c e\right )+c (b B d-2 A c d+A b e-2 a B e) x\right )}{5 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) \left (a+b x+c x^2\right )^{5/2}}+\frac {2 \left (8 a c e (2 c d-b e) (b B d-2 A c d+A b e-2 a B e)+\left (b c d-b^2 e+2 a c e\right ) \left (5 b^2 e (B d-A e)-8 b c d (B d+A e)+4 c \left (4 A c d^2-a B d e+5 a A e^2\right )\right )+c \left (8 c e (b d-2 a e) (b B d-2 A c d+A b e-2 a B e)+(2 c d-b e) \left (5 b^2 e (B d-A e)-8 b c d (B d+A e)+4 c \left (4 A c d^2-a B d e+5 a A e^2\right )\right )\right ) x\right )}{15 \left (b^2-4 a c\right )^2 \left (c d^2-b d e+a e^2\right )^2 \left (a+b x+c x^2\right )^{3/2}}+\frac {2 \left (4 a c e (2 c d-b e) \left (8 c e (b d-2 a e) (b B d-2 A c d+A b e-2 a B e)+(2 c d-b e) \left (5 b^2 e (B d-A e)-8 b c d (B d+A e)+4 c \left (4 A c d^2-a B d e+5 a A e^2\right )\right )\right )-\left (b c d-b^2 e+2 a c e\right ) \left (8 c d e (b B d-2 A c d+A b e-2 a B e) \left (4 b c d-3 b^2 e+4 a c e\right )+\left (8 c^2 d^2-3 b^2 e^2-4 c e (b d-3 a e)\right ) \left (5 b^2 e (B d-A e)-8 b c d (B d+A e)+4 c \left (4 A c d^2-a B d e+5 a A e^2\right )\right )\right )+c \left (4 c e (b d-2 a e) \left (8 c e (b d-2 a e) (b B d-2 A c d+A b e-2 a B e)+(2 c d-b e) \left (5 b^2 e (B d-A e)-8 b c d (B d+A e)+4 c \left (4 A c d^2-a B d e+5 a A e^2\right )\right )\right )-(2 c d-b e) \left (8 c d e (b B d-2 A c d+A b e-2 a B e) \left (4 b c d-3 b^2 e+4 a c e\right )+\left (8 c^2 d^2-3 b^2 e^2-4 c e (b d-3 a e)\right ) \left (5 b^2 e (B d-A e)-8 b c d (B d+A e)+4 c \left (4 A c d^2-a B d e+5 a A e^2\right )\right )\right )\right ) x\right )}{15 \left (b^2-4 a c\right )^3 \left (c d^2-b d e+a e^2\right )^3 \sqrt {a+b x+c x^2}}-\frac {8 \int \frac {15 \left (b^2-4 a c\right )^3 e^5 (B d-A e)}{8 (d+e x) \sqrt {a+b x+c x^2}} \, dx}{15 \left (b^2-4 a c\right )^3 \left (c d^2-b d e+a e^2\right )^3}\\ &=\frac {2 \left (a B (2 c d-b e)-A \left (b c d-b^2 e+2 a c e\right )+c (b B d-2 A c d+A b e-2 a B e) x\right )}{5 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) \left (a+b x+c x^2\right )^{5/2}}+\frac {2 \left (8 a c e (2 c d-b e) (b B d-2 A c d+A b e-2 a B e)+\left (b c d-b^2 e+2 a c e\right ) \left (5 b^2 e (B d-A e)-8 b c d (B d+A e)+4 c \left (4 A c d^2-a B d e+5 a A e^2\right )\right )+c \left (8 c e (b d-2 a e) (b B d-2 A c d+A b e-2 a B e)+(2 c d-b e) \left (5 b^2 e (B d-A e)-8 b c d (B d+A e)+4 c \left (4 A c d^2-a B d e+5 a A e^2\right )\right )\right ) x\right )}{15 \left (b^2-4 a c\right )^2 \left (c d^2-b d e+a e^2\right )^2 \left (a+b x+c x^2\right )^{3/2}}+\frac {2 \left (4 a c e (2 c d-b e) \left (8 c e (b d-2 a e) (b B d-2 A c d+A b e-2 a B e)+(2 c d-b e) \left (5 b^2 e (B d-A e)-8 b c d (B d+A e)+4 c \left (4 A c d^2-a B d e+5 a A e^2\right )\right )\right )-\left (b c d-b^2 e+2 a c e\right ) \left (8 c d e (b B d-2 A c d+A b e-2 a B e) \left (4 b c d-3 b^2 e+4 a c e\right )+\left (8 c^2 d^2-3 b^2 e^2-4 c e (b d-3 a e)\right ) \left (5 b^2 e (B d-A e)-8 b c d (B d+A e)+4 c \left (4 A c d^2-a B d e+5 a A e^2\right )\right )\right )+c \left (4 c e (b d-2 a e) \left (8 c e (b d-2 a e) (b B d-2 A c d+A b e-2 a B e)+(2 c d-b e) \left (5 b^2 e (B d-A e)-8 b c d (B d+A e)+4 c \left (4 A c d^2-a B d e+5 a A e^2\right )\right )\right )-(2 c d-b e) \left (8 c d e (b B d-2 A c d+A b e-2 a B e) \left (4 b c d-3 b^2 e+4 a c e\right )+\left (8 c^2 d^2-3 b^2 e^2-4 c e (b d-3 a e)\right ) \left (5 b^2 e (B d-A e)-8 b c d (B d+A e)+4 c \left (4 A c d^2-a B d e+5 a A e^2\right )\right )\right )\right ) x\right )}{15 \left (b^2-4 a c\right )^3 \left (c d^2-b d e+a e^2\right )^3 \sqrt {a+b x+c x^2}}-\frac {\left (e^5 (B d-A e)\right ) \int \frac {1}{(d+e x) \sqrt {a+b x+c x^2}} \, dx}{\left (c d^2-b d e+a e^2\right )^3}\\ &=\frac {2 \left (a B (2 c d-b e)-A \left (b c d-b^2 e+2 a c e\right )+c (b B d-2 A c d+A b e-2 a B e) x\right )}{5 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) \left (a+b x+c x^2\right )^{5/2}}+\frac {2 \left (8 a c e (2 c d-b e) (b B d-2 A c d+A b e-2 a B e)+\left (b c d-b^2 e+2 a c e\right ) \left (5 b^2 e (B d-A e)-8 b c d (B d+A e)+4 c \left (4 A c d^2-a B d e+5 a A e^2\right )\right )+c \left (8 c e (b d-2 a e) (b B d-2 A c d+A b e-2 a B e)+(2 c d-b e) \left (5 b^2 e (B d-A e)-8 b c d (B d+A e)+4 c \left (4 A c d^2-a B d e+5 a A e^2\right )\right )\right ) x\right )}{15 \left (b^2-4 a c\right )^2 \left (c d^2-b d e+a e^2\right )^2 \left (a+b x+c x^2\right )^{3/2}}+\frac {2 \left (4 a c e (2 c d-b e) \left (8 c e (b d-2 a e) (b B d-2 A c d+A b e-2 a B e)+(2 c d-b e) \left (5 b^2 e (B d-A e)-8 b c d (B d+A e)+4 c \left (4 A c d^2-a B d e+5 a A e^2\right )\right )\right )-\left (b c d-b^2 e+2 a c e\right ) \left (8 c d e (b B d-2 A c d+A b e-2 a B e) \left (4 b c d-3 b^2 e+4 a c e\right )+\left (8 c^2 d^2-3 b^2 e^2-4 c e (b d-3 a e)\right ) \left (5 b^2 e (B d-A e)-8 b c d (B d+A e)+4 c \left (4 A c d^2-a B d e+5 a A e^2\right )\right )\right )+c \left (4 c e (b d-2 a e) \left (8 c e (b d-2 a e) (b B d-2 A c d+A b e-2 a B e)+(2 c d-b e) \left (5 b^2 e (B d-A e)-8 b c d (B d+A e)+4 c \left (4 A c d^2-a B d e+5 a A e^2\right )\right )\right )-(2 c d-b e) \left (8 c d e (b B d-2 A c d+A b e-2 a B e) \left (4 b c d-3 b^2 e+4 a c e\right )+\left (8 c^2 d^2-3 b^2 e^2-4 c e (b d-3 a e)\right ) \left (5 b^2 e (B d-A e)-8 b c d (B d+A e)+4 c \left (4 A c d^2-a B d e+5 a A e^2\right )\right )\right )\right ) x\right )}{15 \left (b^2-4 a c\right )^3 \left (c d^2-b d e+a e^2\right )^3 \sqrt {a+b x+c x^2}}+\frac {\left (2 e^5 (B d-A e)\right ) \operatorname {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 )}{\left (c d^2-b d e+a e^2\right )^3}\\ &=\frac {2 \left (a B (2 c d-b e)-A \left (b c d-b^2 e+2 a c e\right )+c (b B d-2 A c d+A b e-2 a B e) x\right )}{5 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) \left (a+b x+c x^2\right )^{5/2}}+\frac {2 \left (8 a c e (2 c d-b e) (b B d-2 A c d+A b e-2 a B e)+\left (b c d-b^2 e+2 a c e\right ) \left (5 b^2 e (B d-A e)-8 b c d (B d+A e)+4 c \left (4 A c d^2-a B d e+5 a A e^2\right )\right )+c \left (8 c e (b d-2 a e) (b B d-2 A c d+A b e-2 a B e)+(2 c d-b e) \left (5 b^2 e (B d-A e)-8 b c d (B d+A e)+4 c \left (4 A c d^2-a B d e+5 a A e^2\right )\right )\right ) x\right )}{15 \left (b^2-4 a c\right )^2 \left (c d^2-b d e+a e^2\right )^2 \left (a+b x+c x^2\right )^{3/2}}+\frac {2 \left (4 a c e (2 c d-b e) \left (8 c e (b d-2 a e) (b B d-2 A c d+A b e-2 a B e)+(2 c d-b e) \left (5 b^2 e (B d-A e)-8 b c d (B d+A e)+4 c \left (4 A c d^2-a B d e+5 a A e^2\right )\right )\right )-\left (b c d-b^2 e+2 a c e\right ) \left (8 c d e (b B d-2 A c d+A b e-2 a B e) \left (4 b c d-3 b^2 e+4 a c e\right )+\left (8 c^2 d^2-3 b^2 e^2-4 c e (b d-3 a e)\right ) \left (5 b^2 e (B d-A e)-8 b c d (B d+A e)+4 c \left (4 A c d^2-a B d e+5 a A e^2\right )\right )\right )+c \left (4 c e (b d-2 a e) \left (8 c e (b d-2 a e) (b B d-2 A c d+A b e-2 a B e)+(2 c d-b e) \left (5 b^2 e (B d-A e)-8 b c d (B d+A e)+4 c \left (4 A c d^2-a B d e+5 a A e^2\right )\right )\right )-(2 c d-b e) \left (8 c d e (b B d-2 A c d+A b e-2 a B e) \left (4 b c d-3 b^2 e+4 a c e\right )+\left (8 c^2 d^2-3 b^2 e^2-4 c e (b d-3 a e)\right ) \left (5 b^2 e (B d-A e)-8 b c d (B d+A e)+4 c \left (4 A c d^2-a B d e+5 a A e^2\right )\right )\right )\right ) x\right )}{15 \left (b^2-4 a c\right )^3 \left (c d^2-b d e+a e^2\right )^3 \sqrt {a+b x+c x^2}}-\frac {e^5 (B d-A e) \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 )}{\left (c d^2-b d e+a e^2\right )^{7/2}}\\ \end {align*}
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Mathematica [A] time = 6.24, size = 1120, normalized size = 1.15 \begin {gather*} -\frac {2 \left (-\frac {2 \left (4 a c e (2 c d-b e) (b B d-2 A c d+A b e-2 a B e)+\frac {1}{2} \left (-e b^2+c d b+2 a c e\right ) \left (5 e (B d-A e) b^2-8 c d (B d+A e) b+4 c \left (4 A c d^2-a B e d+5 a A e^2\right )\right )+c \left (4 c e (b d-2 a e) (b B d-2 A c d+A b e-2 a B e)+\frac {1}{2} (2 c d-b e) \left (5 e (B d-A e) b^2-8 c d (B d+A e) b+4 c \left (4 A c d^2-a B e d+5 a A e^2\right )\right )\right ) x\right )}{3 \left (b^2-4 a c\right ) \left (c d^2-b e d+a e^2\right ) \left (c x^2+b x+a\right )^{3/2}}-\frac {2 \left (\frac {15 \left (b^2-4 a c\right )^2 e^5 (B d-A e) \tanh ^{-1}\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 )}{\sqrt {c d^2-b e d+a e^2} \left (4 c d^2-4 b e d+4 a e^2\right )}-\frac {2 \left (-a c e (2 c d-b e) \left (8 c e (b d-2 a e) (b B d-2 A c d+A b e-2 a B e)+(2 c d-b e) \left (5 e (B d-A e) b^2-8 c d (B d+A e) b+4 c \left (4 A c d^2-a B e d+5 a A e^2\right )\right )\right )+\frac {1}{4} \left (-e b^2+c d b+2 a c e\right ) \left (8 c d e (b B d-2 A c d+A b e-2 a B e) \left (-3 e b^2+4 c d b+4 a c e\right )+\left (8 c^2 d^2-3 b^2 e^2-4 c e (b d-3 a e)\right ) \left (5 e (B d-A e) b^2-8 c d (B d+A e) b+4 c \left (4 A c d^2-a B e d+5 a A e^2\right )\right )\right )+c \left (\frac {1}{4} (2 c d-b e) \left (8 c d e (b B d-2 A c d+A b e-2 a B e) \left (-3 e b^2+4 c d b+4 a c e\right )+\left (8 c^2 d^2-3 b^2 e^2-4 c e (b d-3 a e)\right ) \left (5 e (B d-A e) b^2-8 c d (B d+A e) b+4 c \left (4 A c d^2-a B e d+5 a A e^2\right )\right )\right )-c e (b d-2 a e) \left (8 c e (b d-2 a e) (b B d-2 A c d+A b e-2 a B e)+(2 c d-b e) \left (5 e (B d-A e) b^2-8 c d (B d+A e) b+4 c \left (4 A c d^2-a B e d+5 a A e^2\right )\right )\right )\right ) x\right )}{\left (b^2-4 a c\right ) \left (c d^2-b e d+a e^2\right ) \sqrt {c x^2+b x+a}}\right )}{3 \left (b^2-4 a c\right ) \left (c d^2-b e d+a e^2\right )}\right ) \left (c x^2+b x+a\right )^{7/2}}{5 \left (b^2-4 a c\right ) \left (c d^2-b e d+a e^2\right ) (a+x (b+c x))^{7/2}}-\frac {2 \left (-a B (2 c d-b e)+A \left (-e b^2+c d b+2 a c e\right )+c (A (2 c d-b e)-B (b d-2 a e)) x\right ) \left (c x^2+b x+a\right )}{5 \left (b^2-4 a c\right ) \left (c d^2-b e d+a e^2\right ) (a+x (b+c x))^{7/2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 180.12, size = 0, normalized size = 0.00 \begin {gather*} \text {\$Aborted} \end {gather*}
Verification is not applicable to the result.
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fricas [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.02, size = 5461, normalized size = 5.61 \begin {gather*} \text {output too large to display} \end {gather*}
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 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}
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 \begin {gather*} \int \frac {A+B\,x}{\left (d+e\,x\right )\,{\left (c\,x^2+b\,x+a\right )}^{7/2}} \,d x \end {gather*}
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 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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