Optimal. Leaf size=944 \[ -\frac {2 \left (C d^2-e (B d-A e)\right ) \sqrt {a+b x+c x^2}}{5 e \left (c d^2-b d e+a e^2\right ) (d+e x)^{5/2}}+\frac {2 \left (c d \left (2 C d^2+e (3 B d-8 A e)\right )+e \left (5 a e (2 C d-B e)-b \left (6 C d^2-B d e-4 A e^2\right )\right )\right ) \sqrt {a+b x+c x^2}}{15 e \left (c d^2-b d e+a e^2\right )^2 (d+e x)^{3/2}}+\frac {2 \left (c^2 d^2 \left (2 C d^2+e (3 B d-23 A e)\right )-e^2 \left (15 a^2 C e^2-10 a b e (C d+B e)+b^2 \left (3 C d^2+2 B d e+8 A e^2\right )\right )-c e \left (b d \left (7 C d^2-7 B d e-23 A e^2\right )-a e \left (19 C d^2-29 B d e+9 A e^2\right )\right )\right ) \sqrt {a+b x+c x^2}}{15 e \left (c d^2-b d e+a e^2\right )^3 \sqrt {d+e x}}-\frac {\sqrt {2} \sqrt {b^2-4 a c} \left (c^2 d^2 \left (2 C d^2+e (3 B d-23 A e)\right )-e^2 \left (15 a^2 C e^2-10 a b e (C d+B e)+b^2 \left (3 C d^2+2 B d e+8 A e^2\right )\right )-c e \left (b d \left (7 C d^2-7 B d e-23 A e^2\right )-a e \left (19 C d^2-29 B d e+9 A e^2\right )\right )\right ) \sqrt {d+e x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\sin ^{-1}\left (\frac {\sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{15 e^2 \left (c d^2-b d e+a e^2\right )^3 \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {a+b x+c x^2}}+\frac {2 \sqrt {2} \sqrt {b^2-4 a c} \left (c d \left (2 C d^2+e (3 B d-8 A e)\right )+e \left (5 a e (2 C d-B e)-b \left (6 C d^2-B d e-4 A e^2\right )\right )\right ) \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\sin ^{-1}\left (\frac {\sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{15 e^2 \left (c d^2-b d e+a e^2\right )^2 \sqrt {d+e x} \sqrt {a+b x+c x^2}} \]
[Out]
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Rubi [A]
time = 1.34, antiderivative size = 942, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 6, integrand size = 34, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {1664, 848, 857,
732, 435, 430} \begin {gather*} -\frac {2 \sqrt {c x^2+b x+a} \left (C d^2-e (B d-A e)\right )}{5 e \left (c d^2-b e d+a e^2\right ) (d+e x)^{5/2}}-\frac {\sqrt {2} \sqrt {b^2-4 a c} \left (\left (2 C d^4+e (3 B d-23 A e) d^2\right ) c^2-e \left (b d \left (7 C d^2-7 B e d-23 A e^2\right )-a e \left (19 C d^2-29 B e d+9 A e^2\right )\right ) c-e^2 \left (\left (3 C d^2+2 B e d+8 A e^2\right ) b^2-10 a e (C d+B e) b+15 a^2 C e^2\right )\right ) \sqrt {d+e x} \sqrt {-\frac {c \left (c x^2+b x+a\right )}{b^2-4 a c}} E\left (\text {ArcSin}\left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{15 e^2 \left (c d^2-b e d+a e^2\right )^3 \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {c x^2+b x+a}}+\frac {2 \sqrt {2} \sqrt {b^2-4 a c} \left (2 c C d^3+c e (3 B d-8 A e) d+5 a e^2 (2 C d-B e)-b e \left (6 C d^2-e (B d+4 A e)\right )\right ) \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {-\frac {c \left (c x^2+b x+a\right )}{b^2-4 a c}} F\left (\text {ArcSin}\left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{15 e^2 \left (c d^2-b e d+a e^2\right )^2 \sqrt {d+e x} \sqrt {c x^2+b x+a}}+\frac {2 \left (\left (2 C d^4+e (3 B d-23 A e) d^2\right ) c^2-e \left (b d \left (7 C d^2-7 B e d-23 A e^2\right )-a e \left (19 C d^2-29 B e d+9 A e^2\right )\right ) c-e^2 \left (\left (3 C d^2+2 B e d+8 A e^2\right ) b^2-10 a e (C d+B e) b+15 a^2 C e^2\right )\right ) \sqrt {c x^2+b x+a}}{15 e \left (c d^2-b e d+a e^2\right )^3 \sqrt {d+e x}}+\frac {2 \left (2 c C d^3+c e (3 B d-8 A e) d+5 a e^2 (2 C d-B e)-b e \left (6 C d^2-e (B d+4 A e)\right )\right ) \sqrt {c x^2+b x+a}}{15 e \left (c d^2-b e d+a e^2\right )^2 (d+e x)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 430
Rule 435
Rule 732
Rule 848
Rule 857
Rule 1664
Rubi steps
\begin {align*} \int \frac {A+B x+C x^2}{(d+e x)^{7/2} \sqrt {a+b x+c x^2}} \, dx &=-\frac {2 \left (C d^2-e (B d-A e)\right ) \sqrt {a+b x+c x^2}}{5 e \left (c d^2-b d e+a e^2\right ) (d+e x)^{5/2}}-\frac {2 \int \frac {-\frac {b C d^2-b e (B d+4 A e)+5 e (A c d-a C d+a B e)}{2 e}-\frac {1}{2} \left (3 B c d-5 b C d+\frac {2 c C d^2}{e}-3 A c e+5 a C e\right ) x}{(d+e x)^{5/2} \sqrt {a+b x+c x^2}} \, dx}{5 \left (c d^2-b d e+a e^2\right )}\\ &=-\frac {2 \left (C d^2-e (B d-A e)\right ) \sqrt {a+b x+c x^2}}{5 e \left (c d^2-b d e+a e^2\right ) (d+e x)^{5/2}}+\frac {2 \left (2 c C d^3+c d e (3 B d-8 A e)+5 a e^2 (2 C d-B e)-b e \left (6 C d^2-e (B d+4 A e)\right )\right ) \sqrt {a+b x+c x^2}}{15 e \left (c d^2-b d e+a e^2\right )^2 (d+e x)^{3/2}}+\frac {4 \int \frac {\frac {b^2 e \left (3 C d^2+2 e (B d+4 A e)\right )+b \left (c C d^3-c d e (6 B d+19 A e)-10 a e^2 (C d+B e)\right )+3 e \left (A c \left (5 c d^2-3 a e^2\right )+a \left (5 a C e^2-c d (3 C d-8 B e)\right )\right )}{4 e}+\frac {c \left (2 c C d^3+c d e (3 B d-8 A e)+5 a e^2 (2 C d-B e)-b e \left (6 C d^2-e (B d+4 A e)\right )\right ) x}{4 e}}{(d+e x)^{3/2} \sqrt {a+b x+c x^2}} \, dx}{15 \left (c d^2-b d e+a e^2\right )^2}\\ &=-\frac {2 \left (C d^2-e (B d-A e)\right ) \sqrt {a+b x+c x^2}}{5 e \left (c d^2-b d e+a e^2\right ) (d+e x)^{5/2}}+\frac {2 \left (2 c C d^3+c d e (3 B d-8 A e)+5 a e^2 (2 C d-B e)-b e \left (6 C d^2-e (B d+4 A e)\right )\right ) \sqrt {a+b x+c x^2}}{15 e \left (c d^2-b d e+a e^2\right )^2 (d+e x)^{3/2}}+\frac {2 \left (c^2 \left (2 C d^4+d^2 e (3 B d-23 A e)\right )-e^2 \left (15 a^2 C e^2-10 a b e (C d+B e)+b^2 \left (3 C d^2+2 B d e+8 A e^2\right )\right )-c e \left (b d \left (7 C d^2-7 B d e-23 A e^2\right )-a e \left (19 C d^2-29 B d e+9 A e^2\right )\right )\right ) \sqrt {a+b x+c x^2}}{15 e \left (c d^2-b d e+a e^2\right )^3 \sqrt {d+e x}}-\frac {8 \int \frac {-\frac {c \left (b^2 d e \left (9 C d^2+e (B d+4 A e)\right )-b \left (c C d^4+26 a C d^2 e^2+4 a e^3 (B d-A e)+c d^2 e (9 B d+11 A e)\right )+e \left (A c d \left (15 c d^2-17 a e^2\right )-a \left (c d^2 (7 C d-27 B e)-5 a e^2 (5 C d-B e)\right )\right )\right )}{8 e}+\frac {c \left (c^2 \left (2 C d^4+d^2 e (3 B d-23 A e)\right )-e^2 \left (15 a^2 C e^2-10 a b e (C d+B e)+b^2 \left (3 C d^2+2 B d e+8 A e^2\right )\right )-c e \left (b d \left (7 C d^2-7 B d e-23 A e^2\right )-a e \left (19 C d^2-29 B d e+9 A e^2\right )\right )\right ) x}{8 e}}{\sqrt {d+e x} \sqrt {a+b x+c x^2}} \, dx}{15 \left (c d^2-b d e+a e^2\right )^3}\\ &=-\frac {2 \left (C d^2-e (B d-A e)\right ) \sqrt {a+b x+c x^2}}{5 e \left (c d^2-b d e+a e^2\right ) (d+e x)^{5/2}}+\frac {2 \left (2 c C d^3+c d e (3 B d-8 A e)+5 a e^2 (2 C d-B e)-b e \left (6 C d^2-e (B d+4 A e)\right )\right ) \sqrt {a+b x+c x^2}}{15 e \left (c d^2-b d e+a e^2\right )^2 (d+e x)^{3/2}}+\frac {2 \left (c^2 \left (2 C d^4+d^2 e (3 B d-23 A e)\right )-e^2 \left (15 a^2 C e^2-10 a b e (C d+B e)+b^2 \left (3 C d^2+2 B d e+8 A e^2\right )\right )-c e \left (b d \left (7 C d^2-7 B d e-23 A e^2\right )-a e \left (19 C d^2-29 B d e+9 A e^2\right )\right )\right ) \sqrt {a+b x+c x^2}}{15 e \left (c d^2-b d e+a e^2\right )^3 \sqrt {d+e x}}+\frac {\left (c \left (2 c C d^3+c d e (3 B d-8 A e)+5 a e^2 (2 C d-B e)-b e \left (6 C d^2-e (B d+4 A e)\right )\right )\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {a+b x+c x^2}} \, dx}{15 e^2 \left (c d^2-b d e+a e^2\right )^2}-\frac {\left (c \left (c^2 \left (2 C d^4+d^2 e (3 B d-23 A e)\right )-e^2 \left (15 a^2 C e^2-10 a b e (C d+B e)+b^2 \left (3 C d^2+2 B d e+8 A e^2\right )\right )-c e \left (b d \left (7 C d^2-7 B d e-23 A e^2\right )-a e \left (19 C d^2-29 B d e+9 A e^2\right )\right )\right )\right ) \int \frac {\sqrt {d+e x}}{\sqrt {a+b x+c x^2}} \, dx}{15 e^2 \left (c d^2-b d e+a e^2\right )^3}\\ &=-\frac {2 \left (C d^2-e (B d-A e)\right ) \sqrt {a+b x+c x^2}}{5 e \left (c d^2-b d e+a e^2\right ) (d+e x)^{5/2}}+\frac {2 \left (2 c C d^3+c d e (3 B d-8 A e)+5 a e^2 (2 C d-B e)-b e \left (6 C d^2-e (B d+4 A e)\right )\right ) \sqrt {a+b x+c x^2}}{15 e \left (c d^2-b d e+a e^2\right )^2 (d+e x)^{3/2}}+\frac {2 \left (c^2 \left (2 C d^4+d^2 e (3 B d-23 A e)\right )-e^2 \left (15 a^2 C e^2-10 a b e (C d+B e)+b^2 \left (3 C d^2+2 B d e+8 A e^2\right )\right )-c e \left (b d \left (7 C d^2-7 B d e-23 A e^2\right )-a e \left (19 C d^2-29 B d e+9 A e^2\right )\right )\right ) \sqrt {a+b x+c x^2}}{15 e \left (c d^2-b d e+a e^2\right )^3 \sqrt {d+e x}}-\frac {\left (\sqrt {2} \sqrt {b^2-4 a c} \left (c^2 \left (2 C d^4+d^2 e (3 B d-23 A e)\right )-e^2 \left (15 a^2 C e^2-10 a b e (C d+B e)+b^2 \left (3 C d^2+2 B d e+8 A e^2\right )\right )-c e \left (b d \left (7 C d^2-7 B d e-23 A e^2\right )-a e \left (19 C d^2-29 B d e+9 A e^2\right )\right )\right ) \sqrt {d+e x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}}\right ) \text {Subst}\left (\int \frac {\sqrt {1+\frac {2 \sqrt {b^2-4 a c} e x^2}{2 c d-b e-\sqrt {b^2-4 a c} e}}}{\sqrt {1-x^2}} \, dx,x,\frac {\sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )}{15 e^2 \left (c d^2-b d e+a e^2\right )^3 \sqrt {\frac {c (d+e x)}{2 c d-b e-\sqrt {b^2-4 a c} e}} \sqrt {a+b x+c x^2}}+\frac {\left (2 \sqrt {2} \sqrt {b^2-4 a c} \left (2 c C d^3+c d e (3 B d-8 A e)+5 a e^2 (2 C d-B e)-b e \left (6 C d^2-e (B d+4 A e)\right )\right ) \sqrt {\frac {c (d+e x)}{2 c d-b e-\sqrt {b^2-4 a c} e}} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1-x^2} \sqrt {1+\frac {2 \sqrt {b^2-4 a c} e x^2}{2 c d-b e-\sqrt {b^2-4 a c} e}}} \, dx,x,\frac {\sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )}{15 e^2 \left (c d^2-b d e+a e^2\right )^2 \sqrt {d+e x} \sqrt {a+b x+c x^2}}\\ &=-\frac {2 \left (C d^2-e (B d-A e)\right ) \sqrt {a+b x+c x^2}}{5 e \left (c d^2-b d e+a e^2\right ) (d+e x)^{5/2}}+\frac {2 \left (2 c C d^3+c d e (3 B d-8 A e)+5 a e^2 (2 C d-B e)-b e \left (6 C d^2-e (B d+4 A e)\right )\right ) \sqrt {a+b x+c x^2}}{15 e \left (c d^2-b d e+a e^2\right )^2 (d+e x)^{3/2}}+\frac {2 \left (c^2 \left (2 C d^4+d^2 e (3 B d-23 A e)\right )-e^2 \left (15 a^2 C e^2-10 a b e (C d+B e)+b^2 \left (3 C d^2+2 B d e+8 A e^2\right )\right )-c e \left (b d \left (7 C d^2-7 B d e-23 A e^2\right )-a e \left (19 C d^2-29 B d e+9 A e^2\right )\right )\right ) \sqrt {a+b x+c x^2}}{15 e \left (c d^2-b d e+a e^2\right )^3 \sqrt {d+e x}}-\frac {\sqrt {2} \sqrt {b^2-4 a c} \left (c^2 \left (2 C d^4+d^2 e (3 B d-23 A e)\right )-e^2 \left (15 a^2 C e^2-10 a b e (C d+B e)+b^2 \left (3 C d^2+2 B d e+8 A e^2\right )\right )-c e \left (b d \left (7 C d^2-7 B d e-23 A e^2\right )-a e \left (19 C d^2-29 B d e+9 A e^2\right )\right )\right ) \sqrt {d+e x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\sin ^{-1}\left (\frac {\sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{15 e^2 \left (c d^2-b d e+a e^2\right )^3 \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {a+b x+c x^2}}+\frac {2 \sqrt {2} \sqrt {b^2-4 a c} \left (2 c C d^3+c d e (3 B d-8 A e)+5 a e^2 (2 C d-B e)-b e \left (6 C d^2-e (B d+4 A e)\right )\right ) \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\sin ^{-1}\left (\frac {\sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{15 e^2 \left (c d^2-b d e+a e^2\right )^2 \sqrt {d+e x} \sqrt {a+b x+c x^2}}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 32.92, size = 1746, normalized size = 1.85 \begin {gather*} \frac {\sqrt {d+e x} \left (a+b x+c x^2\right ) \left (-\frac {2 \left (C d^2-B d e+A e^2\right )}{5 e \left (c d^2-b d e+a e^2\right ) (d+e x)^3}-\frac {2 \left (-2 c C d^3-3 B c d^2 e+6 b C d^2 e-b B d e^2+8 A c d e^2-10 a C d e^2-4 A b e^3+5 a B e^3\right )}{15 e \left (c d^2-b d e+a e^2\right )^2 (d+e x)^2}-\frac {2 \left (-2 c^2 C d^4-3 B c^2 d^3 e+7 b c C d^3 e-7 b B c d^2 e^2+23 A c^2 d^2 e^2+3 b^2 C d^2 e^2-19 a c C d^2 e^2+2 b^2 B d e^3-23 A b c d e^3+29 a B c d e^3-10 a b C d e^3+8 A b^2 e^4-10 a b B e^4-9 a A c e^4+15 a^2 C e^4\right )}{15 e \left (c d^2-b d e+a e^2\right )^3 (d+e x)}\right )}{\sqrt {a+x (b+c x)}}+\frac {2 (d+e x)^{3/2} \sqrt {a+b x+c x^2} \left (-\left (\left (c^2 \left (2 C d^4+d^2 e (3 B d-23 A e)\right )-e^2 \left (15 a^2 C e^2-10 a b e (C d+B e)+b^2 \left (3 C d^2+2 B d e+8 A e^2\right )\right )+c e \left (a e \left (19 C d^2-29 B d e+9 A e^2\right )+b d \left (-7 C d^2+7 B d e+23 A e^2\right )\right )\right ) \left (c \left (-1+\frac {d}{d+e x}\right )^2+\frac {e \left (b-\frac {b d}{d+e x}+\frac {a e}{d+e x}\right )}{d+e x}\right )\right )-\frac {i \sqrt {1-\frac {2 \left (c d^2+e (-b d+a e)\right )}{\left (2 c d-b e+\sqrt {\left (b^2-4 a c\right ) e^2}\right ) (d+e x)}} \sqrt {1+\frac {2 \left (c d^2+e (-b d+a e)\right )}{\left (-2 c d+b e+\sqrt {\left (b^2-4 a c\right ) e^2}\right ) (d+e x)}} \left (\left (2 c d-b e+\sqrt {\left (b^2-4 a c\right ) e^2}\right ) \left (c^2 \left (-2 C d^4+d^2 e (-3 B d+23 A e)\right )+e^2 \left (15 a^2 C e^2-10 a b e (C d+B e)+b^2 \left (3 C d^2+2 B d e+8 A e^2\right )\right )-c e \left (a e \left (19 C d^2-29 B d e+9 A e^2\right )+b d \left (-7 C d^2+7 B d e+23 A e^2\right )\right )\right ) E\left (i \sinh ^{-1}\left (\frac {\sqrt {2} \sqrt {\frac {c d^2-b d e+a e^2}{-2 c d+b e+\sqrt {\left (b^2-4 a c\right ) e^2}}}}{\sqrt {d+e x}}\right )|-\frac {-2 c d+b e+\sqrt {\left (b^2-4 a c\right ) e^2}}{2 c d-b e+\sqrt {\left (b^2-4 a c\right ) e^2}}\right )+\left (-30 A c^3 d^3 e^2+14 a c^2 C d^3 e^2-54 a B c^2 d^2 e^3+34 a A c^2 d e^4-50 a^2 c C d e^4+10 a^2 B c e^5+2 c^2 C d^4 \sqrt {\left (b^2-4 a c\right ) e^2}+3 B c^2 d^3 e \sqrt {\left (b^2-4 a c\right ) e^2}-23 A c^2 d^2 e^2 \sqrt {\left (b^2-4 a c\right ) e^2}+19 a c C d^2 e^2 \sqrt {\left (b^2-4 a c\right ) e^2}-29 a B c d e^3 \sqrt {\left (b^2-4 a c\right ) e^2}+9 a A c e^4 \sqrt {\left (b^2-4 a c\right ) e^2}-15 a^2 C e^4 \sqrt {\left (b^2-4 a c\right ) e^2}+b^3 e^3 \left (3 C d^2+2 e (B d+4 A e)\right )-b^2 e^2 \left (11 c C d^3+c d e (9 B d+31 A e)+10 a e^2 (C d+B e)+\sqrt {\left (b^2-4 a c\right ) e^2} \left (3 C d^2+2 e (B d+4 A e)\right )\right )+b \left (A c e^3 \left (45 c d^2-17 a e^2+23 d \sqrt {\left (b^2-4 a c\right ) e^2}\right )+C e \left (15 a^2 e^4-7 c d^3 \sqrt {\left (b^2-4 a c\right ) e^2}+a d e^2 \left (33 c d+10 \sqrt {\left (b^2-4 a c\right ) e^2}\right )\right )+B e^2 \left (15 c^2 d^3+10 a e^2 \sqrt {\left (b^2-4 a c\right ) e^2}+c d \left (37 a e^2+7 d \sqrt {\left (b^2-4 a c\right ) e^2}\right )\right )\right )\right ) F\left (i \sinh ^{-1}\left (\frac {\sqrt {2} \sqrt {\frac {c d^2-b d e+a e^2}{-2 c d+b e+\sqrt {\left (b^2-4 a c\right ) e^2}}}}{\sqrt {d+e x}}\right )|-\frac {-2 c d+b e+\sqrt {\left (b^2-4 a c\right ) e^2}}{2 c d-b e+\sqrt {\left (b^2-4 a c\right ) e^2}}\right )\right )}{2 \sqrt {2} \sqrt {\frac {c d^2+e (-b d+a e)}{-2 c d+b e+\sqrt {\left (b^2-4 a c\right ) e^2}}} \sqrt {d+e x}}\right )}{15 e^3 \left (c d^2-b d e+a e^2\right )^3 \sqrt {a+x (b+c x)} \sqrt {\frac {(d+e x)^2 \left (c \left (-1+\frac {d}{d+e x}\right )^2+\frac {e \left (b-\frac {b d}{d+e x}+\frac {a e}{d+e x}\right )}{d+e x}\right )}{e^2}}} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(46696\) vs.
\(2(874)=1748\).
time = 0.21, size = 46697, normalized size = 49.47
method | result | size |
elliptic | \(\text {Expression too large to display}\) | \(1757\) |
default | \(\text {Expression too large to display}\) | \(46697\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order
4.
time = 0.20, size = 2611, normalized size = 2.77 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {A + B x + C x^{2}}{\left (d + e x\right )^{\frac {7}{2}} \sqrt {a + b x + c x^{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \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 {C\,x^2+B\,x+A}{{\left (d+e\,x\right )}^{7/2}\,\sqrt {c\,x^2+b\,x+a}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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