Integrand size = 28, antiderivative size = 851 \[ \int (d+e x)^3 \sqrt {f+g x} \sqrt {a+c x^2} \, dx=-\frac {2 \left (150 a^2 e^4 g^4-6 a c e^2 g^2 \left (2 e^2 f^2-33 d e f g+165 d^2 g^2\right )+c^2 \left (187 e^4 f^4-732 d e^3 f^3 g+1098 d^2 e^2 f^2 g^2-798 d^3 e f g^3+315 d^4 g^4\right )\right ) \sqrt {f+g x} \sqrt {a+c x^2}}{3465 c^2 e g^4}+\frac {2 (d+e x)^4 \sqrt {f+g x} \sqrt {a+c x^2}}{11 e}-\frac {2 \left (2 a e^2 g^2 (74 e f-231 d g)-c \left (233 e^3 f^3-843 d e^2 f^2 g+1107 d^2 e f g^2-567 d^3 g^3\right )\right ) (f+g x)^{3/2} \sqrt {a+c x^2}}{3465 c g^4}+\frac {2 e \left (18 a e^2 g^2-c \left (29 e^2 f^2-96 d e f g+81 d^2 g^2\right )\right ) (f+g x)^{5/2} \sqrt {a+c x^2}}{693 c g^4}+\frac {2 e^2 (e f-3 d g) (f+g x)^{7/2} \sqrt {a+c x^2}}{99 g^4}+\frac {4 \sqrt {-a} \left (3 a^2 e^2 g^4 (26 e f+231 d g)-c^2 f^2 \left (64 e^3 f^3-264 d e^2 f^2 g+396 d^2 e f g^2-231 d^3 g^3\right )-9 a c g^2 \left (6 e^3 f^3-33 d e^2 f^2 g+88 d^2 e f g^2+77 d^3 g^3\right )\right ) \sqrt {f+g x} \sqrt {1+\frac {c x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a g}{\sqrt {-a} \sqrt {c} f-a g}\right )}{3465 c^{3/2} g^5 \sqrt {\frac {\sqrt {c} (f+g x)}{\sqrt {c} f+\sqrt {-a} g}} \sqrt {a+c x^2}}-\frac {4 \sqrt {-a} \left (c f^2+a g^2\right ) \left (75 a^2 e^3 g^4-3 a c e g^2 \left (2 e^2 f^2-33 d e f g+165 d^2 g^2\right )-c^2 f \left (64 e^3 f^3-264 d e^2 f^2 g+396 d^2 e f g^2-231 d^3 g^3\right )\right ) \sqrt {\frac {\sqrt {c} (f+g x)}{\sqrt {c} f+\sqrt {-a} g}} \sqrt {1+\frac {c x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right ),-\frac {2 a g}{\sqrt {-a} \sqrt {c} f-a g}\right )}{3465 c^{5/2} g^5 \sqrt {f+g x} \sqrt {a+c x^2}} \]
[Out]
Time = 1.96 (sec) , antiderivative size = 851, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {933, 1668, 858, 733, 435, 430} \[ \int (d+e x)^3 \sqrt {f+g x} \sqrt {a+c x^2} \, dx=\frac {2 \sqrt {f+g x} \sqrt {c x^2+a} (d+e x)^4}{11 e}+\frac {4 \sqrt {-a} \left (3 a^2 e^2 (26 e f+231 d g) g^4-9 a c \left (6 e^3 f^3-33 d e^2 g f^2+88 d^2 e g^2 f+77 d^3 g^3\right ) g^2-c^2 f^2 \left (64 e^3 f^3-264 d e^2 g f^2+396 d^2 e g^2 f-231 d^3 g^3\right )\right ) \sqrt {f+g x} \sqrt {\frac {c x^2}{a}+1} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a g}{\sqrt {-a} \sqrt {c} f-a g}\right )}{3465 c^{3/2} g^5 \sqrt {\frac {\sqrt {c} (f+g x)}{\sqrt {c} f+\sqrt {-a} g}} \sqrt {c x^2+a}}-\frac {4 \sqrt {-a} \left (c f^2+a g^2\right ) \left (75 a^2 e^3 g^4-3 a c e \left (2 e^2 f^2-33 d e g f+165 d^2 g^2\right ) g^2-c^2 f \left (64 e^3 f^3-264 d e^2 g f^2+396 d^2 e g^2 f-231 d^3 g^3\right )\right ) \sqrt {\frac {\sqrt {c} (f+g x)}{\sqrt {c} f+\sqrt {-a} g}} \sqrt {\frac {c x^2}{a}+1} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right ),-\frac {2 a g}{\sqrt {-a} \sqrt {c} f-a g}\right )}{3465 c^{5/2} g^5 \sqrt {f+g x} \sqrt {c x^2+a}}+\frac {2 e^2 (e f-3 d g) (f+g x)^{7/2} \sqrt {c x^2+a}}{99 g^4}+\frac {2 e \left (18 a e^2 g^2-c \left (29 e^2 f^2-96 d e g f+81 d^2 g^2\right )\right ) (f+g x)^{5/2} \sqrt {c x^2+a}}{693 c g^4}-\frac {2 \left (2 a e^2 g^2 (74 e f-231 d g)-c \left (233 e^3 f^3-843 d e^2 g f^2+1107 d^2 e g^2 f-567 d^3 g^3\right )\right ) (f+g x)^{3/2} \sqrt {c x^2+a}}{3465 c g^4}-\frac {2 \left (150 a^2 e^4 g^4-6 a c e^2 \left (2 e^2 f^2-33 d e g f+165 d^2 g^2\right ) g^2+c^2 \left (187 e^4 f^4-732 d e^3 g f^3+1098 d^2 e^2 g^2 f^2-798 d^3 e g^3 f+315 d^4 g^4\right )\right ) \sqrt {f+g x} \sqrt {c x^2+a}}{3465 c^2 e g^4} \]
[In]
[Out]
Rule 430
Rule 435
Rule 733
Rule 858
Rule 933
Rule 1668
Rubi steps \begin{align*} \text {integral}& = \frac {2 (d+e x)^4 \sqrt {f+g x} \sqrt {a+c x^2}}{11 e}+\frac {\int \frac {(d+e x)^3 \left (a (3 e f-d g)-2 (c d f-a e g) x+c (e f-3 d g) x^2\right )}{\sqrt {f+g x} \sqrt {a+c x^2}} \, dx}{11 e} \\ & = \frac {2 (d+e x)^4 \sqrt {f+g x} \sqrt {a+c x^2}}{11 e}+\frac {2 e^2 (e f-3 d g) (f+g x)^{7/2} \sqrt {a+c x^2}}{99 g^4}+\frac {2 \int \frac {-\frac {1}{2} a c g^2 \left (7 e^4 f^4-21 d e^3 f^3 g-27 d^3 e f g^3+9 d^4 g^4\right )-\frac {1}{2} c g \left (3 a e g^2 \left (7 e^3 f^3-21 d e^2 f^2 g-27 d^2 e f g^2+3 d^3 g^3\right )+2 c \left (e^4 f^5-3 d e^3 f^4 g+9 d^4 f g^4\right )\right ) x-\frac {3}{2} c g^2 \left (a e^2 g^2 \left (7 e^2 f^2-48 d e f g-9 d^2 g^2\right )+c \left (5 e^4 f^4-15 d e^3 f^3 g+15 d^3 e f g^3+9 d^4 g^4\right )\right ) x^2+\frac {1}{2} c e g^3 \left (2 a e^2 g^2 (10 e f+33 d g)-3 c \left (11 e^3 f^3-33 d e^2 f^2 g+9 d^2 e f g^2+27 d^3 g^3\right )\right ) x^3+\frac {1}{2} c e^2 g^4 \left (18 a e^2 g^2-c \left (29 e^2 f^2-96 d e f g+81 d^2 g^2\right )\right ) x^4}{\sqrt {f+g x} \sqrt {a+c x^2}} \, dx}{99 c e g^5} \\ & = \frac {2 (d+e x)^4 \sqrt {f+g x} \sqrt {a+c x^2}}{11 e}+\frac {2 e \left (18 a e^2 g^2-c \left (29 e^2 f^2-96 d e f g+81 d^2 g^2\right )\right ) (f+g x)^{5/2} \sqrt {a+c x^2}}{693 c g^4}+\frac {2 e^2 (e f-3 d g) (f+g x)^{7/2} \sqrt {a+c x^2}}{99 g^4}+\frac {4 \int \frac {-\frac {3}{4} a c g^6 \left (30 a e^4 f^2 g^2-c \left (32 e^4 f^4-111 d e^3 f^3 g+135 d^2 e^2 f^2 g^2+63 d^3 e f g^3-21 d^4 g^4\right )\right )-\frac {1}{4} c g^5 \left (180 a^2 e^4 f g^4-a c e g^2 \left (107 e^3 f^3-519 d e^2 f^2 g+1377 d^2 e f g^2-63 d^3 g^3\right )-2 c^2 \left (22 e^4 f^5-75 d e^3 f^4 g+81 d^2 e^2 f^3 g^2-63 d^4 f g^4\right )\right ) x-\frac {1}{4} c g^6 \left (90 a^2 e^4 g^4+2 a c e^2 g^2 \left (100 e^2 f^2-264 d e f g-297 d^2 g^2\right )-c^2 \left (214 e^4 f^4-741 d e^3 f^3 g+891 d^2 e^2 f^2 g^2-315 d^3 e f g^3-189 d^4 g^4\right )\right ) x^2-\frac {1}{4} c^2 e g^7 \left (2 a e^2 g^2 (74 e f-231 d g)-c \left (233 e^3 f^3-843 d e^2 f^2 g+1107 d^2 e f g^2-567 d^3 g^3\right )\right ) x^3}{\sqrt {f+g x} \sqrt {a+c x^2}} \, dx}{693 c^2 e g^9} \\ & = \frac {2 (d+e x)^4 \sqrt {f+g x} \sqrt {a+c x^2}}{11 e}-\frac {2 \left (2 a e^2 g^2 (74 e f-231 d g)-c \left (233 e^3 f^3-843 d e^2 f^2 g+1107 d^2 e f g^2-567 d^3 g^3\right )\right ) (f+g x)^{3/2} \sqrt {a+c x^2}}{3465 c g^4}+\frac {2 e \left (18 a e^2 g^2-c \left (29 e^2 f^2-96 d e f g+81 d^2 g^2\right )\right ) (f+g x)^{5/2} \sqrt {a+c x^2}}{693 c g^4}+\frac {2 e^2 (e f-3 d g) (f+g x)^{7/2} \sqrt {a+c x^2}}{99 g^4}+\frac {8 \int \frac {-\frac {3}{8} a c^2 g^9 \left (2 a e^3 f g^2 (e f+231 d g)+c \left (73 e^4 f^4-288 d e^3 f^3 g+432 d^2 e^2 f^2 g^2-882 d^3 e f g^3+105 d^4 g^4\right )\right )-\frac {3}{4} c^2 g^8 \left (a^2 e^3 g^4 (76 e f+231 d g)-11 a c e g^2 \left (2 e^3 f^3-15 d e^2 f^2 g+54 d^2 e f g^2+21 d^3 g^3\right )+c^2 f \left (41 e^4 f^4-156 d e^3 f^3 g+234 d^2 e^2 f^2 g^2-189 d^3 e f g^3+105 d^4 g^4\right )\right ) x-\frac {3}{8} c^2 g^9 \left (150 a^2 e^4 g^4-6 a c e^2 g^2 \left (2 e^2 f^2-33 d e f g+165 d^2 g^2\right )+c^2 \left (187 e^4 f^4-732 d e^3 f^3 g+1098 d^2 e^2 f^2 g^2-798 d^3 e f g^3+315 d^4 g^4\right )\right ) x^2}{\sqrt {f+g x} \sqrt {a+c x^2}} \, dx}{3465 c^3 e g^{12}} \\ & = -\frac {2 \left (150 a^2 e^4 g^4-6 a c e^2 g^2 \left (2 e^2 f^2-33 d e f g+165 d^2 g^2\right )+c^2 \left (187 e^4 f^4-732 d e^3 f^3 g+1098 d^2 e^2 f^2 g^2-798 d^3 e f g^3+315 d^4 g^4\right )\right ) \sqrt {f+g x} \sqrt {a+c x^2}}{3465 c^2 e g^4}+\frac {2 (d+e x)^4 \sqrt {f+g x} \sqrt {a+c x^2}}{11 e}-\frac {2 \left (2 a e^2 g^2 (74 e f-231 d g)-c \left (233 e^3 f^3-843 d e^2 f^2 g+1107 d^2 e f g^2-567 d^3 g^3\right )\right ) (f+g x)^{3/2} \sqrt {a+c x^2}}{3465 c g^4}+\frac {2 e \left (18 a e^2 g^2-c \left (29 e^2 f^2-96 d e f g+81 d^2 g^2\right )\right ) (f+g x)^{5/2} \sqrt {a+c x^2}}{693 c g^4}+\frac {2 e^2 (e f-3 d g) (f+g x)^{7/2} \sqrt {a+c x^2}}{99 g^4}+\frac {16 \int \frac {\frac {3}{8} a c^2 e g^{11} \left (75 a^2 e^3 g^4-9 a c e g^2 \left (e^2 f^2+66 d e f g+55 d^2 g^2\right )-c^2 f \left (16 e^3 f^3-66 d e^2 f^2 g+99 d^2 e f g^2-924 d^3 g^3\right )\right )-\frac {3}{8} c^3 e g^{10} \left (3 a^2 e^2 g^4 (26 e f+231 d g)-c^2 f^2 \left (64 e^3 f^3-264 d e^2 f^2 g+396 d^2 e f g^2-231 d^3 g^3\right )-9 a c g^2 \left (6 e^3 f^3-33 d e^2 f^2 g+88 d^2 e f g^2+77 d^3 g^3\right )\right ) x}{\sqrt {f+g x} \sqrt {a+c x^2}} \, dx}{10395 c^4 e g^{14}} \\ & = -\frac {2 \left (150 a^2 e^4 g^4-6 a c e^2 g^2 \left (2 e^2 f^2-33 d e f g+165 d^2 g^2\right )+c^2 \left (187 e^4 f^4-732 d e^3 f^3 g+1098 d^2 e^2 f^2 g^2-798 d^3 e f g^3+315 d^4 g^4\right )\right ) \sqrt {f+g x} \sqrt {a+c x^2}}{3465 c^2 e g^4}+\frac {2 (d+e x)^4 \sqrt {f+g x} \sqrt {a+c x^2}}{11 e}-\frac {2 \left (2 a e^2 g^2 (74 e f-231 d g)-c \left (233 e^3 f^3-843 d e^2 f^2 g+1107 d^2 e f g^2-567 d^3 g^3\right )\right ) (f+g x)^{3/2} \sqrt {a+c x^2}}{3465 c g^4}+\frac {2 e \left (18 a e^2 g^2-c \left (29 e^2 f^2-96 d e f g+81 d^2 g^2\right )\right ) (f+g x)^{5/2} \sqrt {a+c x^2}}{693 c g^4}+\frac {2 e^2 (e f-3 d g) (f+g x)^{7/2} \sqrt {a+c x^2}}{99 g^4}+\frac {\left (2 \left (c f^2+a g^2\right ) \left (75 a^2 e^3 g^4-3 a c e g^2 \left (2 e^2 f^2-33 d e f g+165 d^2 g^2\right )-c^2 f \left (64 e^3 f^3-264 d e^2 f^2 g+396 d^2 e f g^2-231 d^3 g^3\right )\right )\right ) \int \frac {1}{\sqrt {f+g x} \sqrt {a+c x^2}} \, dx}{3465 c^2 g^5}-\frac {\left (2 \left (3 a^2 e^2 g^4 (26 e f+231 d g)-c^2 f^2 \left (64 e^3 f^3-264 d e^2 f^2 g+396 d^2 e f g^2-231 d^3 g^3\right )-9 a c g^2 \left (6 e^3 f^3-33 d e^2 f^2 g+88 d^2 e f g^2+77 d^3 g^3\right )\right )\right ) \int \frac {\sqrt {f+g x}}{\sqrt {a+c x^2}} \, dx}{3465 c g^5} \\ & = -\frac {2 \left (150 a^2 e^4 g^4-6 a c e^2 g^2 \left (2 e^2 f^2-33 d e f g+165 d^2 g^2\right )+c^2 \left (187 e^4 f^4-732 d e^3 f^3 g+1098 d^2 e^2 f^2 g^2-798 d^3 e f g^3+315 d^4 g^4\right )\right ) \sqrt {f+g x} \sqrt {a+c x^2}}{3465 c^2 e g^4}+\frac {2 (d+e x)^4 \sqrt {f+g x} \sqrt {a+c x^2}}{11 e}-\frac {2 \left (2 a e^2 g^2 (74 e f-231 d g)-c \left (233 e^3 f^3-843 d e^2 f^2 g+1107 d^2 e f g^2-567 d^3 g^3\right )\right ) (f+g x)^{3/2} \sqrt {a+c x^2}}{3465 c g^4}+\frac {2 e \left (18 a e^2 g^2-c \left (29 e^2 f^2-96 d e f g+81 d^2 g^2\right )\right ) (f+g x)^{5/2} \sqrt {a+c x^2}}{693 c g^4}+\frac {2 e^2 (e f-3 d g) (f+g x)^{7/2} \sqrt {a+c x^2}}{99 g^4}-\frac {\left (4 a \left (3 a^2 e^2 g^4 (26 e f+231 d g)-c^2 f^2 \left (64 e^3 f^3-264 d e^2 f^2 g+396 d^2 e f g^2-231 d^3 g^3\right )-9 a c g^2 \left (6 e^3 f^3-33 d e^2 f^2 g+88 d^2 e f g^2+77 d^3 g^3\right )\right ) \sqrt {f+g x} \sqrt {1+\frac {c x^2}{a}}\right ) \text {Subst}\left (\int \frac {\sqrt {1+\frac {2 a \sqrt {c} g x^2}{\sqrt {-a} \left (c f-\frac {a \sqrt {c} g}{\sqrt {-a}}\right )}}}{\sqrt {1-x^2}} \, dx,x,\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )}{3465 \sqrt {-a} c^{3/2} g^5 \sqrt {\frac {c (f+g x)}{c f-\frac {a \sqrt {c} g}{\sqrt {-a}}}} \sqrt {a+c x^2}}+\frac {\left (4 a \left (c f^2+a g^2\right ) \left (75 a^2 e^3 g^4-3 a c e g^2 \left (2 e^2 f^2-33 d e f g+165 d^2 g^2\right )-c^2 f \left (64 e^3 f^3-264 d e^2 f^2 g+396 d^2 e f g^2-231 d^3 g^3\right )\right ) \sqrt {\frac {c (f+g x)}{c f-\frac {a \sqrt {c} g}{\sqrt {-a}}}} \sqrt {1+\frac {c x^2}{a}}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1-x^2} \sqrt {1+\frac {2 a \sqrt {c} g x^2}{\sqrt {-a} \left (c f-\frac {a \sqrt {c} g}{\sqrt {-a}}\right )}}} \, dx,x,\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )}{3465 \sqrt {-a} c^{5/2} g^5 \sqrt {f+g x} \sqrt {a+c x^2}} \\ & = -\frac {2 \left (150 a^2 e^4 g^4-6 a c e^2 g^2 \left (2 e^2 f^2-33 d e f g+165 d^2 g^2\right )+c^2 \left (187 e^4 f^4-732 d e^3 f^3 g+1098 d^2 e^2 f^2 g^2-798 d^3 e f g^3+315 d^4 g^4\right )\right ) \sqrt {f+g x} \sqrt {a+c x^2}}{3465 c^2 e g^4}+\frac {2 (d+e x)^4 \sqrt {f+g x} \sqrt {a+c x^2}}{11 e}-\frac {2 \left (2 a e^2 g^2 (74 e f-231 d g)-c \left (233 e^3 f^3-843 d e^2 f^2 g+1107 d^2 e f g^2-567 d^3 g^3\right )\right ) (f+g x)^{3/2} \sqrt {a+c x^2}}{3465 c g^4}+\frac {2 e \left (18 a e^2 g^2-c \left (29 e^2 f^2-96 d e f g+81 d^2 g^2\right )\right ) (f+g x)^{5/2} \sqrt {a+c x^2}}{693 c g^4}+\frac {2 e^2 (e f-3 d g) (f+g x)^{7/2} \sqrt {a+c x^2}}{99 g^4}+\frac {4 \sqrt {-a} \left (3 a^2 e^2 g^4 (26 e f+231 d g)-c^2 f^2 \left (64 e^3 f^3-264 d e^2 f^2 g+396 d^2 e f g^2-231 d^3 g^3\right )-9 a c g^2 \left (6 e^3 f^3-33 d e^2 f^2 g+88 d^2 e f g^2+77 d^3 g^3\right )\right ) \sqrt {f+g x} \sqrt {1+\frac {c x^2}{a}} E\left (\sin ^{-1}\left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a g}{\sqrt {-a} \sqrt {c} f-a g}\right )}{3465 c^{3/2} g^5 \sqrt {\frac {\sqrt {c} (f+g x)}{\sqrt {c} f+\sqrt {-a} g}} \sqrt {a+c x^2}}-\frac {4 \sqrt {-a} \left (c f^2+a g^2\right ) \left (75 a^2 e^3 g^4-3 a c e g^2 \left (2 e^2 f^2-33 d e f g+165 d^2 g^2\right )-c^2 f \left (64 e^3 f^3-264 d e^2 f^2 g+396 d^2 e f g^2-231 d^3 g^3\right )\right ) \sqrt {\frac {\sqrt {c} (f+g x)}{\sqrt {c} f+\sqrt {-a} g}} \sqrt {1+\frac {c x^2}{a}} F\left (\sin ^{-1}\left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a g}{\sqrt {-a} \sqrt {c} f-a g}\right )}{3465 c^{5/2} g^5 \sqrt {f+g x} \sqrt {a+c x^2}} \\ \end{align*}
Result contains complex when optimal does not.
Time = 30.02 (sec) , antiderivative size = 1045, normalized size of antiderivative = 1.23 \[ \int (d+e x)^3 \sqrt {f+g x} \sqrt {a+c x^2} \, dx=\frac {\sqrt {f+g x} \left (\frac {2 \left (a+c x^2\right ) \left (-150 a^2 e^3 g^4+2 a c e g^2 \left (495 d^2 g^2+33 d e g (4 f+7 g x)+e^2 \left (-23 f^2+16 f g x+45 g^2 x^2\right )\right )+c^2 \left (231 d^3 g^3 (f+3 g x)+99 d^2 e g^2 \left (-4 f^2+3 f g x+15 g^2 x^2\right )+33 d e^2 g \left (8 f^3-6 f^2 g x+5 f g^2 x^2+35 g^3 x^3\right )+e^3 \left (-64 f^4+48 f^3 g x-40 f^2 g^2 x^2+35 f g^3 x^3+315 g^4 x^4\right )\right )\right )}{c^2 g^4}-\frac {4 (f+g x) \left (\frac {g^2 \sqrt {-f-\frac {i \sqrt {a} g}{\sqrt {c}}} \left (3 a^2 e^2 g^4 (26 e f+231 d g)-9 a c g^2 \left (6 e^3 f^3-33 d e^2 f^2 g+88 d^2 e f g^2+77 d^3 g^3\right )+c^2 f^2 \left (-64 e^3 f^3+264 d e^2 f^2 g-396 d^2 e f g^2+231 d^3 g^3\right )\right ) \left (a+c x^2\right )}{(f+g x)^2}+\frac {\sqrt {c} \left (-i \sqrt {c} f+\sqrt {a} g\right ) \left (3 a^2 e^2 g^4 (26 e f+231 d g)-9 a c g^2 \left (6 e^3 f^3-33 d e^2 f^2 g+88 d^2 e f g^2+77 d^3 g^3\right )+c^2 f^2 \left (-64 e^3 f^3+264 d e^2 f^2 g-396 d^2 e f g^2+231 d^3 g^3\right )\right ) \sqrt {\frac {g \left (\frac {i \sqrt {a}}{\sqrt {c}}+x\right )}{f+g x}} \sqrt {-\frac {\frac {i \sqrt {a} g}{\sqrt {c}}-g x}{f+g x}} E\left (i \text {arcsinh}\left (\frac {\sqrt {-f-\frac {i \sqrt {a} g}{\sqrt {c}}}}{\sqrt {f+g x}}\right )|\frac {\sqrt {c} f-i \sqrt {a} g}{\sqrt {c} f+i \sqrt {a} g}\right )}{\sqrt {f+g x}}+\frac {\sqrt {a} g \left (-i \sqrt {c} f+\sqrt {a} g\right ) \left (-75 i a^2 e^3 g^4-3 a^{3/2} \sqrt {c} e^2 g^3 (e f+231 d g)+3 i a c e g^2 \left (2 e^2 f^2-33 d e f g+165 d^2 g^2\right )+i c^2 f \left (64 e^3 f^3-264 d e^2 f^2 g+396 d^2 e f g^2-231 d^3 g^3\right )+3 \sqrt {a} c^{3/2} g \left (16 e^3 f^3-66 d e^2 f^2 g+99 d^2 e f g^2+231 d^3 g^3\right )\right ) \sqrt {\frac {g \left (\frac {i \sqrt {a}}{\sqrt {c}}+x\right )}{f+g x}} \sqrt {-\frac {\frac {i \sqrt {a} g}{\sqrt {c}}-g x}{f+g x}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {-f-\frac {i \sqrt {a} g}{\sqrt {c}}}}{\sqrt {f+g x}}\right ),\frac {\sqrt {c} f-i \sqrt {a} g}{\sqrt {c} f+i \sqrt {a} g}\right )}{\sqrt {f+g x}}\right )}{c^2 g^6 \sqrt {-f-\frac {i \sqrt {a} g}{\sqrt {c}}}}\right )}{3465 \sqrt {a+c x^2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(1823\) vs. \(2(761)=1522\).
Time = 2.82 (sec) , antiderivative size = 1824, normalized size of antiderivative = 2.14
method | result | size |
elliptic | \(\text {Expression too large to display}\) | \(1824\) |
risch | \(\text {Expression too large to display}\) | \(2571\) |
default | \(\text {Expression too large to display}\) | \(6457\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.11 (sec) , antiderivative size = 765, normalized size of antiderivative = 0.90 \[ \int (d+e x)^3 \sqrt {f+g x} \sqrt {a+c x^2} \, dx=-\frac {2 \, {\left (2 \, {\left (64 \, c^{3} e^{3} f^{6} - 264 \, c^{3} d e^{2} f^{5} g + 6 \, {\left (66 \, c^{3} d^{2} e + 17 \, a c^{2} e^{3}\right )} f^{4} g^{2} - 33 \, {\left (7 \, c^{3} d^{3} + 15 \, a c^{2} d e^{2}\right )} f^{3} g^{3} + 3 \, {\left (363 \, a c^{2} d^{2} e - 17 \, a^{2} c e^{3}\right )} f^{2} g^{4} - 99 \, {\left (21 \, a c^{2} d^{3} - 11 \, a^{2} c d e^{2}\right )} f g^{5} + 45 \, {\left (33 \, a^{2} c d^{2} e - 5 \, a^{3} e^{3}\right )} g^{6}\right )} \sqrt {c g} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c f^{2} - 3 \, a g^{2}\right )}}{3 \, c g^{2}}, -\frac {8 \, {\left (c f^{3} + 9 \, a f g^{2}\right )}}{27 \, c g^{3}}, \frac {3 \, g x + f}{3 \, g}\right ) + 6 \, {\left (64 \, c^{3} e^{3} f^{5} g - 264 \, c^{3} d e^{2} f^{4} g^{2} + 18 \, {\left (22 \, c^{3} d^{2} e + 3 \, a c^{2} e^{3}\right )} f^{3} g^{3} - 33 \, {\left (7 \, c^{3} d^{3} + 9 \, a c^{2} d e^{2}\right )} f^{2} g^{4} + 6 \, {\left (132 \, a c^{2} d^{2} e - 13 \, a^{2} c e^{3}\right )} f g^{5} + 693 \, {\left (a c^{2} d^{3} - a^{2} c d e^{2}\right )} g^{6}\right )} \sqrt {c g} {\rm weierstrassZeta}\left (\frac {4 \, {\left (c f^{2} - 3 \, a g^{2}\right )}}{3 \, c g^{2}}, -\frac {8 \, {\left (c f^{3} + 9 \, a f g^{2}\right )}}{27 \, c g^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c f^{2} - 3 \, a g^{2}\right )}}{3 \, c g^{2}}, -\frac {8 \, {\left (c f^{3} + 9 \, a f g^{2}\right )}}{27 \, c g^{3}}, \frac {3 \, g x + f}{3 \, g}\right )\right ) - 3 \, {\left (315 \, c^{3} e^{3} g^{6} x^{4} - 64 \, c^{3} e^{3} f^{4} g^{2} + 264 \, c^{3} d e^{2} f^{3} g^{3} - 2 \, {\left (198 \, c^{3} d^{2} e + 23 \, a c^{2} e^{3}\right )} f^{2} g^{4} + 33 \, {\left (7 \, c^{3} d^{3} + 8 \, a c^{2} d e^{2}\right )} f g^{5} + 30 \, {\left (33 \, a c^{2} d^{2} e - 5 \, a^{2} c e^{3}\right )} g^{6} + 35 \, {\left (c^{3} e^{3} f g^{5} + 33 \, c^{3} d e^{2} g^{6}\right )} x^{3} - 5 \, {\left (8 \, c^{3} e^{3} f^{2} g^{4} - 33 \, c^{3} d e^{2} f g^{5} - 9 \, {\left (33 \, c^{3} d^{2} e + 2 \, a c^{2} e^{3}\right )} g^{6}\right )} x^{2} + {\left (48 \, c^{3} e^{3} f^{3} g^{3} - 198 \, c^{3} d e^{2} f^{2} g^{4} + {\left (297 \, c^{3} d^{2} e + 32 \, a c^{2} e^{3}\right )} f g^{5} + 231 \, {\left (3 \, c^{3} d^{3} + 2 \, a c^{2} d e^{2}\right )} g^{6}\right )} x\right )} \sqrt {c x^{2} + a} \sqrt {g x + f}\right )}}{10395 \, c^{3} g^{6}} \]
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\[ \int (d+e x)^3 \sqrt {f+g x} \sqrt {a+c x^2} \, dx=\int \sqrt {a + c x^{2}} \left (d + e x\right )^{3} \sqrt {f + g x}\, dx \]
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\[ \int (d+e x)^3 \sqrt {f+g x} \sqrt {a+c x^2} \, dx=\int { \sqrt {c x^{2} + a} {\left (e x + d\right )}^{3} \sqrt {g x + f} \,d x } \]
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\[ \int (d+e x)^3 \sqrt {f+g x} \sqrt {a+c x^2} \, dx=\int { \sqrt {c x^{2} + a} {\left (e x + d\right )}^{3} \sqrt {g x + f} \,d x } \]
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Timed out. \[ \int (d+e x)^3 \sqrt {f+g x} \sqrt {a+c x^2} \, dx=\int \sqrt {f+g\,x}\,\sqrt {c\,x^2+a}\,{\left (d+e\,x\right )}^3 \,d x \]
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