Optimal. Leaf size=1015 \[ \frac {2 \left (8 b^3 e^3 g^3+3 b c e^2 g^2 (b e f-8 b d g-9 a e g)+c^3 \left (19 e^3 f^3-57 d e^2 f^2 g+63 d^2 e f g^2-35 d^3 g^3\right )-3 c^2 e g^2 (2 a e (e f-10 d g)+b d (2 e f-7 d g))\right ) \sqrt {f+g x} \sqrt {a+b x+c x^2}}{315 c^3 e g^3}+\frac {2 (d+e x)^3 \sqrt {f+g x} \sqrt {a+b x+c x^2}}{9 e}-\frac {4 \left (3 b^2 e^2 g^2+c e g (4 b e f-9 b d g-7 a e g)+c^2 \left (8 e^2 f^2-24 d e f g+21 d^2 g^2\right )\right ) (f+g x)^{3/2} \sqrt {a+b x+c x^2}}{315 c^2 g^3}+\frac {2 e (c e f-3 c d g+b e g) (f+g x)^{5/2} \sqrt {a+b x+c x^2}}{63 c g^3}-\frac {2 \sqrt {2} \sqrt {b^2-4 a c} \left (8 b^4 e^2 g^4-4 b^2 c e g^3 (b e f+6 b d g+9 a e g)+c^4 f^2 \left (8 e^2 f^2-24 d e f g+21 d^2 g^2\right )+3 c^2 g^2 \left (7 a^2 e^2 g^2+a b e g (5 e f+29 d g)-b^2 \left (e^2 f^2-5 d e f g-7 d^2 g^2\right )\right )+c^3 g \left (3 a g \left (3 e^2 f^2-16 d e f g-21 d^2 g^2\right )-b f \left (4 e^2 f^2-15 d e f g+21 d^2 g^2\right )\right )\right ) \sqrt {f+g x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\sin ^{-1}\left (\frac {\sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} g}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}\right )}{315 c^4 g^4 \sqrt {\frac {c (f+g x)}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}} \sqrt {a+b x+c x^2}}-\frac {2 \sqrt {2} \sqrt {b^2-4 a c} \left (c f^2-b f g+a g^2\right ) \left (8 b^3 e^2 g^3+3 b c e g^2 (b e f-8 b d g-9 a e g)-2 c^3 f \left (8 e^2 f^2-24 d e f g+21 d^2 g^2\right )-3 c^2 g^2 (2 a e (e f-10 d g)+b d (2 e f-7 d g))\right ) \sqrt {\frac {c (f+g x)}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\sin ^{-1}\left (\frac {\sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} g}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}\right )}{315 c^4 g^4 \sqrt {f+g x} \sqrt {a+b x+c x^2}} \]
[Out]
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Rubi [A]
time = 2.62, antiderivative size = 1015, normalized size of antiderivative = 1.00, number of steps
used = 9, number of rules used = 6, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.194, Rules used = {932, 1667, 857,
732, 435, 430} \begin {gather*} \frac {2 \sqrt {f+g x} \sqrt {c x^2+b x+a} (d+e x)^3}{9 e}-\frac {2 \sqrt {2} \sqrt {b^2-4 a c} \left (f^2 \left (8 e^2 f^2-24 d e g f+21 d^2 g^2\right ) c^4+g \left (3 a g \left (3 e^2 f^2-16 d e g f-21 d^2 g^2\right )-b f \left (4 e^2 f^2-15 d e g f+21 d^2 g^2\right )\right ) c^3+3 g^2 \left (-\left (\left (e^2 f^2-5 d e g f-7 d^2 g^2\right ) b^2\right )+a e g (5 e f+29 d g) b+7 a^2 e^2 g^2\right ) c^2-4 b^2 e g^3 (b e f+6 b d g+9 a e g) c+8 b^4 e^2 g^4\right ) \sqrt {f+g 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} g}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}\right )}{315 c^4 g^4 \sqrt {\frac {c (f+g x)}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}} \sqrt {c x^2+b x+a}}-\frac {2 \sqrt {2} \sqrt {b^2-4 a c} \left (c f^2-b g f+a g^2\right ) \left (-2 f \left (8 e^2 f^2-24 d e g f+21 d^2 g^2\right ) c^3-3 g^2 (2 a e (e f-10 d g)+b d (2 e f-7 d g)) c^2+3 b e g^2 (b e f-8 b d g-9 a e g) c+8 b^3 e^2 g^3\right ) \sqrt {\frac {c (f+g x)}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}} \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} g}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}\right )}{315 c^4 g^4 \sqrt {f+g x} \sqrt {c x^2+b x+a}}+\frac {2 e (c e f-3 c d g+b e g) (f+g x)^{5/2} \sqrt {c x^2+b x+a}}{63 c g^3}-\frac {4 \left (\left (8 e^2 f^2-24 d e g f+21 d^2 g^2\right ) c^2+e g (4 b e f-9 b d g-7 a e g) c+3 b^2 e^2 g^2\right ) (f+g x)^{3/2} \sqrt {c x^2+b x+a}}{315 c^2 g^3}+\frac {2 \left (\left (19 e^3 f^3-57 d e^2 g f^2+63 d^2 e g^2 f-35 d^3 g^3\right ) c^3-3 e g^2 (2 a e (e f-10 d g)+b d (2 e f-7 d g)) c^2+3 b e^2 g^2 (b e f-8 b d g-9 a e g) c+8 b^3 e^3 g^3\right ) \sqrt {f+g x} \sqrt {c x^2+b x+a}}{315 c^3 e g^3} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 430
Rule 435
Rule 732
Rule 857
Rule 932
Rule 1667
Rubi steps
\begin {align*} \int (d+e x)^2 \sqrt {f+g x} \sqrt {a+b x+c x^2} \, dx &=\frac {2 (d+e x)^3 \sqrt {f+g x} \sqrt {a+b x+c x^2}}{9 e}-\frac {\int \frac {(d+e x)^2 \left (b d f-3 a e f+a d g+2 (c d f-b e f+b d g-a e g) x-(c e f-3 c d g+b e g) x^2\right )}{\sqrt {f+g x} \sqrt {a+b x+c x^2}} \, dx}{9 e}\\ &=\frac {2 (d+e x)^3 \sqrt {f+g x} \sqrt {a+b x+c x^2}}{9 e}+\frac {2 e (c e f-3 c d g+b e g) (f+g x)^{5/2} \sqrt {a+b x+c x^2}}{63 c g^3}-\frac {2 \int \frac {\frac {1}{2} g \left (b^2 e^3 f^3 g+a c g \left (5 e^3 f^3-15 d e^2 f^2 g-21 d^2 e f g^2+7 d^3 g^3\right )+b f \left (5 a e^3 f g^2+c \left (e^3 f^3-3 d e^2 f^2 g+7 d^3 g^3\right )\right )\right )+g \left (b e^3 f g^2 (4 b f+5 a g)+c^2 \left (e^3 f^4-3 d e^2 f^3 g+7 d^3 f g^3\right )+c g \left (a e^2 f g (5 e f-36 d g)+b \left (5 e^3 f^3-12 d e^2 f^2 g+7 d^3 g^3\right )\right )\right ) x+\frac {1}{2} g^2 \left (b e^3 g^2 (13 b f+5 a g)+c^2 \left (11 e^3 f^3-33 d e^2 f^2 g+21 d^2 e f g^2+21 d^3 g^3\right )-c e g \left (4 a e g (4 e f+9 d g)-3 b \left (8 e^2 f^2-20 d e f g+7 d^2 g^2\right )\right )\right ) x^2+e g^3 \left (3 b^2 e^2 g^2+c e g (4 b e f-9 b d g-7 a e g)+c^2 \left (8 e^2 f^2-24 d e f g+21 d^2 g^2\right )\right ) x^3}{\sqrt {f+g x} \sqrt {a+b x+c x^2}} \, dx}{63 c e g^4}\\ &=\frac {2 (d+e x)^3 \sqrt {f+g x} \sqrt {a+b x+c x^2}}{9 e}-\frac {4 \left (3 b^2 e^2 g^2+c e g (4 b e f-9 b d g-7 a e g)+c^2 \left (8 e^2 f^2-24 d e f g+21 d^2 g^2\right )\right ) (f+g x)^{3/2} \sqrt {a+b x+c x^2}}{315 c^2 g^3}+\frac {2 e (c e f-3 c d g+b e g) (f+g x)^{5/2} \sqrt {a+b x+c x^2}}{63 c g^3}-\frac {4 \int \frac {-\frac {1}{4} g^4 \left (6 b^3 e^3 f^2 g^2+3 b^2 e^2 f g \left (6 a e g^2+c f (e f-6 d g)\right )-b c f \left (3 a e^2 g^2 (5 e f+18 d g)-c \left (11 e^3 f^3-33 d e^2 f^2 g+42 d^2 e f g^2-35 d^3 g^3\right )\right )-a c g \left (42 a e^3 f g^2-c \left (23 e^3 f^3-69 d e^2 f^2 g+231 d^2 e f g^2-35 d^3 g^3\right )\right )\right )-\frac {1}{2} g^4 \left (3 b^2 e^3 g^3 (5 b f+3 a g)+c^3 f \left (11 e^3 f^3-33 d e^2 f^2 g+42 d^2 e f g^2-35 d^3 g^3\right )-3 c e^2 g^2 \left (7 a^2 e g^2-b^2 f (2 e f-15 d g)+a b g (16 e f+9 d g)\right )-c^2 g \left (3 a e g \left (5 e^2 f^2-36 d e f g-21 d^2 g^2\right )-b \left (23 e^3 f^3-78 d e^2 f^2 g+105 d^2 e f g^2-35 d^3 g^3\right )\right )\right ) x-\frac {3}{4} g^5 \left (8 b^3 e^3 g^3+3 b c e^2 g^2 (b e f-8 b d g-9 a e g)+c^3 \left (19 e^3 f^3-57 d e^2 f^2 g+63 d^2 e f g^2-35 d^3 g^3\right )-3 c^2 e g^2 (2 a e (e f-10 d g)+b d (2 e f-7 d g))\right ) x^2}{\sqrt {f+g x} \sqrt {a+b x+c x^2}} \, dx}{315 c^2 e g^7}\\ &=\frac {2 \left (8 b^3 e^3 g^3+3 b c e^2 g^2 (b e f-8 b d g-9 a e g)+c^3 \left (19 e^3 f^3-57 d e^2 f^2 g+63 d^2 e f g^2-35 d^3 g^3\right )-3 c^2 e g^2 (2 a e (e f-10 d g)+b d (2 e f-7 d g))\right ) \sqrt {f+g x} \sqrt {a+b x+c x^2}}{315 c^3 e g^3}+\frac {2 (d+e x)^3 \sqrt {f+g x} \sqrt {a+b x+c x^2}}{9 e}-\frac {4 \left (3 b^2 e^2 g^2+c e g (4 b e f-9 b d g-7 a e g)+c^2 \left (8 e^2 f^2-24 d e f g+21 d^2 g^2\right )\right ) (f+g x)^{3/2} \sqrt {a+b x+c x^2}}{315 c^2 g^3}+\frac {2 e (c e f-3 c d g+b e g) (f+g x)^{5/2} \sqrt {a+b x+c x^2}}{63 c g^3}-\frac {8 \int \frac {\frac {3}{8} e g^6 \left (8 b^4 e^2 f g^3+b^3 e g^2 \left (8 a e g^2-3 c f (e f+8 d g)\right )-3 b^2 c g \left (2 a e g^2 (7 e f+4 d g)+c f \left (e^2 f^2-4 d e f g-7 d^2 g^2\right )\right )-b c \left (27 a^2 e^2 g^4-3 a c g^2 \left (3 e^2 f^2+36 d e f g+7 d^2 g^2\right )-c^2 f^2 \left (8 e^2 f^2-24 d e f g+21 d^2 g^2\right )\right )+4 a c^2 g \left (3 a e g^2 (3 e f+5 d g)-c f \left (e^2 f^2-3 d e f g+42 d^2 g^2\right )\right )\right )+\frac {3}{4} e g^6 \left (8 b^4 e^2 g^4-4 b^2 c e g^3 (b e f+6 b d g+9 a e g)+c^4 f^2 \left (8 e^2 f^2-24 d e f g+21 d^2 g^2\right )+3 c^2 g^2 \left (7 a^2 e^2 g^2+a b e g (5 e f+29 d g)-b^2 \left (e^2 f^2-5 d e f g-7 d^2 g^2\right )\right )+c^3 g \left (3 a g \left (3 e^2 f^2-16 d e f g-21 d^2 g^2\right )-b f \left (4 e^2 f^2-15 d e f g+21 d^2 g^2\right )\right )\right ) x}{\sqrt {f+g x} \sqrt {a+b x+c x^2}} \, dx}{945 c^3 e g^9}\\ &=\frac {2 \left (8 b^3 e^3 g^3+3 b c e^2 g^2 (b e f-8 b d g-9 a e g)+c^3 \left (19 e^3 f^3-57 d e^2 f^2 g+63 d^2 e f g^2-35 d^3 g^3\right )-3 c^2 e g^2 (2 a e (e f-10 d g)+b d (2 e f-7 d g))\right ) \sqrt {f+g x} \sqrt {a+b x+c x^2}}{315 c^3 e g^3}+\frac {2 (d+e x)^3 \sqrt {f+g x} \sqrt {a+b x+c x^2}}{9 e}-\frac {4 \left (3 b^2 e^2 g^2+c e g (4 b e f-9 b d g-7 a e g)+c^2 \left (8 e^2 f^2-24 d e f g+21 d^2 g^2\right )\right ) (f+g x)^{3/2} \sqrt {a+b x+c x^2}}{315 c^2 g^3}+\frac {2 e (c e f-3 c d g+b e g) (f+g x)^{5/2} \sqrt {a+b x+c x^2}}{63 c g^3}-\frac {\left (\left (c f^2-b f g+a g^2\right ) \left (8 b^3 e^2 g^3+3 b c e g^2 (b e f-8 b d g-9 a e g)-2 c^3 f \left (8 e^2 f^2-24 d e f g+21 d^2 g^2\right )-3 c^2 g^2 (2 a e (e f-10 d g)+b d (2 e f-7 d g))\right )\right ) \int \frac {1}{\sqrt {f+g x} \sqrt {a+b x+c x^2}} \, dx}{315 c^3 g^4}-\frac {\left (2 \left (8 b^4 e^2 g^4-4 b^2 c e g^3 (b e f+6 b d g+9 a e g)+c^4 f^2 \left (8 e^2 f^2-24 d e f g+21 d^2 g^2\right )+3 c^2 g^2 \left (7 a^2 e^2 g^2+a b e g (5 e f+29 d g)-b^2 \left (e^2 f^2-5 d e f g-7 d^2 g^2\right )\right )+c^3 g \left (3 a g \left (3 e^2 f^2-16 d e f g-21 d^2 g^2\right )-b f \left (4 e^2 f^2-15 d e f g+21 d^2 g^2\right )\right )\right )\right ) \int \frac {\sqrt {f+g x}}{\sqrt {a+b x+c x^2}} \, dx}{315 c^3 g^4}\\ &=\frac {2 \left (8 b^3 e^3 g^3+3 b c e^2 g^2 (b e f-8 b d g-9 a e g)+c^3 \left (19 e^3 f^3-57 d e^2 f^2 g+63 d^2 e f g^2-35 d^3 g^3\right )-3 c^2 e g^2 (2 a e (e f-10 d g)+b d (2 e f-7 d g))\right ) \sqrt {f+g x} \sqrt {a+b x+c x^2}}{315 c^3 e g^3}+\frac {2 (d+e x)^3 \sqrt {f+g x} \sqrt {a+b x+c x^2}}{9 e}-\frac {4 \left (3 b^2 e^2 g^2+c e g (4 b e f-9 b d g-7 a e g)+c^2 \left (8 e^2 f^2-24 d e f g+21 d^2 g^2\right )\right ) (f+g x)^{3/2} \sqrt {a+b x+c x^2}}{315 c^2 g^3}+\frac {2 e (c e f-3 c d g+b e g) (f+g x)^{5/2} \sqrt {a+b x+c x^2}}{63 c g^3}-\frac {\left (2 \sqrt {2} \sqrt {b^2-4 a c} \left (8 b^4 e^2 g^4-4 b^2 c e g^3 (b e f+6 b d g+9 a e g)+c^4 f^2 \left (8 e^2 f^2-24 d e f g+21 d^2 g^2\right )+3 c^2 g^2 \left (7 a^2 e^2 g^2+a b e g (5 e f+29 d g)-b^2 \left (e^2 f^2-5 d e f g-7 d^2 g^2\right )\right )+c^3 g \left (3 a g \left (3 e^2 f^2-16 d e f g-21 d^2 g^2\right )-b f \left (4 e^2 f^2-15 d e f g+21 d^2 g^2\right )\right )\right ) \sqrt {f+g 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} g x^2}{2 c f-b g-\sqrt {b^2-4 a c} g}}}{\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 )}{315 c^4 g^4 \sqrt {\frac {c (f+g x)}{2 c f-b g-\sqrt {b^2-4 a c} g}} \sqrt {a+b x+c x^2}}-\frac {\left (2 \sqrt {2} \sqrt {b^2-4 a c} \left (c f^2-b f g+a g^2\right ) \left (8 b^3 e^2 g^3+3 b c e g^2 (b e f-8 b d g-9 a e g)-2 c^3 f \left (8 e^2 f^2-24 d e f g+21 d^2 g^2\right )-3 c^2 g^2 (2 a e (e f-10 d g)+b d (2 e f-7 d g))\right ) \sqrt {\frac {c (f+g x)}{2 c f-b g-\sqrt {b^2-4 a c} g}} \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} g x^2}{2 c f-b g-\sqrt {b^2-4 a c} g}}} \, dx,x,\frac {\sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )}{315 c^4 g^4 \sqrt {f+g x} \sqrt {a+b x+c x^2}}\\ &=\frac {2 \left (8 b^3 e^3 g^3+3 b c e^2 g^2 (b e f-8 b d g-9 a e g)+c^3 \left (19 e^3 f^3-57 d e^2 f^2 g+63 d^2 e f g^2-35 d^3 g^3\right )-3 c^2 e g^2 (2 a e (e f-10 d g)+b d (2 e f-7 d g))\right ) \sqrt {f+g x} \sqrt {a+b x+c x^2}}{315 c^3 e g^3}+\frac {2 (d+e x)^3 \sqrt {f+g x} \sqrt {a+b x+c x^2}}{9 e}-\frac {4 \left (3 b^2 e^2 g^2+c e g (4 b e f-9 b d g-7 a e g)+c^2 \left (8 e^2 f^2-24 d e f g+21 d^2 g^2\right )\right ) (f+g x)^{3/2} \sqrt {a+b x+c x^2}}{315 c^2 g^3}+\frac {2 e (c e f-3 c d g+b e g) (f+g x)^{5/2} \sqrt {a+b x+c x^2}}{63 c g^3}-\frac {2 \sqrt {2} \sqrt {b^2-4 a c} \left (8 b^4 e^2 g^4-4 b^2 c e g^3 (b e f+6 b d g+9 a e g)+c^4 f^2 \left (8 e^2 f^2-24 d e f g+21 d^2 g^2\right )+3 c^2 g^2 \left (7 a^2 e^2 g^2+a b e g (5 e f+29 d g)-b^2 \left (e^2 f^2-5 d e f g-7 d^2 g^2\right )\right )+c^3 g \left (3 a g \left (3 e^2 f^2-16 d e f g-21 d^2 g^2\right )-b f \left (4 e^2 f^2-15 d e f g+21 d^2 g^2\right )\right )\right ) \sqrt {f+g 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} g}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}\right )}{315 c^4 g^4 \sqrt {\frac {c (f+g x)}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}} \sqrt {a+b x+c x^2}}-\frac {2 \sqrt {2} \sqrt {b^2-4 a c} \left (c f^2-b f g+a g^2\right ) \left (8 b^3 e^2 g^3+3 b c e g^2 (b e f-8 b d g-9 a e g)-2 c^3 f \left (8 e^2 f^2-24 d e f g+21 d^2 g^2\right )-3 c^2 g^2 (2 a e (e f-10 d g)+b d (2 e f-7 d g))\right ) \sqrt {\frac {c (f+g x)}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}} \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} g}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}\right )}{315 c^4 g^4 \sqrt {f+g x} \sqrt {a+b x+c x^2}}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 33.95, size = 15781, normalized size = 15.55 \begin {gather*} \text {Result too large to show} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(20223\) vs.
\(2(939)=1878\).
time = 0.16, size = 20224, normalized size = 19.93
method | result | size |
elliptic | \(\text {Expression too large to display}\) | \(1936\) |
risch | \(\text {Expression too large to display}\) | \(7219\) |
default | \(\text {Expression too large to display}\) | \(20224\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
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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.
[In]
[Out]
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Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order
4.
time = 0.43, size = 1158, normalized size = 1.14 \begin {gather*} \frac {2 \, {\left ({\left (42 \, c^{5} d^{2} f^{3} g^{2} - 63 \, b c^{4} d^{2} f^{2} g^{3} - 63 \, {\left (b^{2} c^{3} - 6 \, a c^{4}\right )} d^{2} f g^{4} + 21 \, {\left (2 \, b^{3} c^{2} - 9 \, a b c^{3}\right )} d^{2} g^{5} + {\left (16 \, c^{5} f^{5} - 16 \, b c^{4} f^{4} g - 5 \, {\left (b^{2} c^{3} - 6 \, a c^{4}\right )} f^{3} g^{2} - {\left (5 \, b^{3} c^{2} - 21 \, a b c^{3}\right )} f^{2} g^{3} - 2 \, {\left (8 \, b^{4} c - 42 \, a b^{2} c^{2} + 33 \, a^{2} c^{3}\right )} f g^{4} + {\left (16 \, b^{5} - 96 \, a b^{3} c + 123 \, a^{2} b c^{2}\right )} g^{5}\right )} e^{2} - 6 \, {\left (8 \, c^{5} d f^{4} g - 9 \, b c^{4} d f^{3} g^{2} - 2 \, {\left (2 \, b^{2} c^{3} - 11 \, a c^{4}\right )} d f^{2} g^{3} - {\left (9 \, b^{3} c^{2} - 41 \, a b c^{3}\right )} d f g^{4} + {\left (8 \, b^{4} c - 41 \, a b^{2} c^{2} + 30 \, a^{2} c^{3}\right )} d g^{5}\right )} e\right )} \sqrt {c g} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c^{2} f^{2} - b c f g + {\left (b^{2} - 3 \, a c\right )} g^{2}\right )}}{3 \, c^{2} g^{2}}, -\frac {4 \, {\left (2 \, c^{3} f^{3} - 3 \, b c^{2} f^{2} g - 3 \, {\left (b^{2} c - 6 \, a c^{2}\right )} f g^{2} + {\left (2 \, b^{3} - 9 \, a b c\right )} g^{3}\right )}}{27 \, c^{3} g^{3}}, \frac {3 \, c g x + c f + b g}{3 \, c g}\right ) + 6 \, {\left (21 \, c^{5} d^{2} f^{2} g^{3} - 21 \, b c^{4} d^{2} f g^{4} + 21 \, {\left (b^{2} c^{3} - 3 \, a c^{4}\right )} d^{2} g^{5} + {\left (8 \, c^{5} f^{4} g - 4 \, b c^{4} f^{3} g^{2} - 3 \, {\left (b^{2} c^{3} - 3 \, a c^{4}\right )} f^{2} g^{3} - {\left (4 \, b^{3} c^{2} - 15 \, a b c^{3}\right )} f g^{4} + {\left (8 \, b^{4} c - 36 \, a b^{2} c^{2} + 21 \, a^{2} c^{3}\right )} g^{5}\right )} e^{2} - 3 \, {\left (8 \, c^{5} d f^{3} g^{2} - 5 \, b c^{4} d f^{2} g^{3} - {\left (5 \, b^{2} c^{3} - 16 \, a c^{4}\right )} d f g^{4} + {\left (8 \, b^{3} c^{2} - 29 \, a b c^{3}\right )} d g^{5}\right )} e\right )} \sqrt {c g} {\rm weierstrassZeta}\left (\frac {4 \, {\left (c^{2} f^{2} - b c f g + {\left (b^{2} - 3 \, a c\right )} g^{2}\right )}}{3 \, c^{2} g^{2}}, -\frac {4 \, {\left (2 \, c^{3} f^{3} - 3 \, b c^{2} f^{2} g - 3 \, {\left (b^{2} c - 6 \, a c^{2}\right )} f g^{2} + {\left (2 \, b^{3} - 9 \, a b c\right )} g^{3}\right )}}{27 \, c^{3} g^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c^{2} f^{2} - b c f g + {\left (b^{2} - 3 \, a c\right )} g^{2}\right )}}{3 \, c^{2} g^{2}}, -\frac {4 \, {\left (2 \, c^{3} f^{3} - 3 \, b c^{2} f^{2} g - 3 \, {\left (b^{2} c - 6 \, a c^{2}\right )} f g^{2} + {\left (2 \, b^{3} - 9 \, a b c\right )} g^{3}\right )}}{27 \, c^{3} g^{3}}, \frac {3 \, c g x + c f + b g}{3 \, c g}\right )\right ) + 3 \, {\left (63 \, c^{5} d^{2} g^{5} x + 21 \, c^{5} d^{2} f g^{4} + 21 \, b c^{4} d^{2} g^{5} + {\left (35 \, c^{5} g^{5} x^{3} + 8 \, c^{5} f^{3} g^{2} - 3 \, b c^{4} f^{2} g^{3} - {\left (3 \, b^{2} c^{3} - 8 \, a c^{4}\right )} f g^{4} + {\left (8 \, b^{3} c^{2} - 27 \, a b c^{3}\right )} g^{5} + 5 \, {\left (c^{5} f g^{4} + b c^{4} g^{5}\right )} x^{2} - 2 \, {\left (3 \, c^{5} f^{2} g^{3} - b c^{4} f g^{4} + {\left (3 \, b^{2} c^{3} - 7 \, a c^{4}\right )} g^{5}\right )} x\right )} e^{2} + 6 \, {\left (15 \, c^{5} d g^{5} x^{2} - 4 \, c^{5} d f^{2} g^{3} + 2 \, b c^{4} d f g^{4} - 2 \, {\left (2 \, b^{2} c^{3} - 5 \, a c^{4}\right )} d g^{5} + 3 \, {\left (c^{5} d f g^{4} + b c^{4} d g^{5}\right )} x\right )} e\right )} \sqrt {c x^{2} + b x + a} \sqrt {g x + f}\right )}}{945 \, c^{5} g^{5}} \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 \left (d + e x\right )^{2} \sqrt {f + g x} \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 \sqrt {f+g\,x}\,{\left (d+e\,x\right )}^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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