Integrand size = 28, antiderivative size = 356 \[ \int \frac {(d+e x)^2}{\sqrt {f+g x} \sqrt {a+c x^2}} \, dx=\frac {2 e^2 \sqrt {f+g x} \sqrt {a+c x^2}}{3 c g}+\frac {4 \sqrt {-a} e (e f-3 d g) \sqrt {f+g x} \sqrt {1+\frac {c x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a g}{\sqrt {-a} \sqrt {c} f-a g}\right )}{3 \sqrt {c} g^2 \sqrt {\frac {\sqrt {c} (f+g x)}{\sqrt {c} f+\sqrt {-a} g}} \sqrt {a+c x^2}}-\frac {2 \sqrt {-a} \left (\left (3 c d^2-a e^2\right ) g^2+2 c e f (e f-3 d g)\right ) \sqrt {\frac {\sqrt {c} (f+g x)}{\sqrt {c} f+\sqrt {-a} g}} \sqrt {1+\frac {c x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right ),-\frac {2 a g}{\sqrt {-a} \sqrt {c} f-a g}\right )}{3 c^{3/2} g^2 \sqrt {f+g x} \sqrt {a+c x^2}} \]
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Time = 0.48 (sec) , antiderivative size = 356, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {945, 24, 858, 733, 435, 430} \[ \int \frac {(d+e x)^2}{\sqrt {f+g x} \sqrt {a+c x^2}} \, dx=-\frac {2 \sqrt {-a} \sqrt {\frac {c x^2}{a}+1} \sqrt {\frac {\sqrt {c} (f+g x)}{\sqrt {-a} g+\sqrt {c} f}} \left (g^2 \left (3 c d^2-a e^2\right )+2 c e f (e f-3 d g)\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right ),-\frac {2 a g}{\sqrt {-a} \sqrt {c} f-a g}\right )}{3 c^{3/2} g^2 \sqrt {a+c x^2} \sqrt {f+g x}}+\frac {4 \sqrt {-a} e \sqrt {\frac {c x^2}{a}+1} \sqrt {f+g x} (e f-3 d g) E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a g}{\sqrt {-a} \sqrt {c} f-a g}\right )}{3 \sqrt {c} g^2 \sqrt {a+c x^2} \sqrt {\frac {\sqrt {c} (f+g x)}{\sqrt {-a} g+\sqrt {c} f}}}+\frac {2 e^2 \sqrt {a+c x^2} \sqrt {f+g x}}{3 c g} \]
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Rule 24
Rule 430
Rule 435
Rule 733
Rule 858
Rule 945
Rubi steps \begin{align*} \text {integral}& = \frac {2 e^2 \sqrt {f+g x} \sqrt {a+c x^2}}{3 c g}-\frac {\int \frac {-d \left (3 c d^2-a e^2\right ) g+e \left (a e^2 g+c d (2 e f-9 d g)\right ) x+2 c e^2 (e f-3 d g) x^2}{(d+e x) \sqrt {f+g x} \sqrt {a+c x^2}} \, dx}{3 c g} \\ & = \frac {2 e^2 \sqrt {f+g x} \sqrt {a+c x^2}}{3 c g}-\frac {\int \frac {-e^2 \left (3 c d^2-a e^2\right ) g+2 c e^3 (e f-3 d g) x}{\sqrt {f+g x} \sqrt {a+c x^2}} \, dx}{3 c e^2 g} \\ & = \frac {2 e^2 \sqrt {f+g x} \sqrt {a+c x^2}}{3 c g}-\frac {(2 e (e f-3 d g)) \int \frac {\sqrt {f+g x}}{\sqrt {a+c x^2}} \, dx}{3 g^2}+\frac {1}{3} \left (3 d^2-\frac {a e^2}{c}+\frac {2 e f (e f-3 d g)}{g^2}\right ) \int \frac {1}{\sqrt {f+g x} \sqrt {a+c x^2}} \, dx \\ & = \frac {2 e^2 \sqrt {f+g x} \sqrt {a+c x^2}}{3 c g}-\frac {\left (4 a e (e f-3 d g) \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 )}{3 \sqrt {-a} \sqrt {c} g^2 \sqrt {\frac {c (f+g x)}{c f-\frac {a \sqrt {c} g}{\sqrt {-a}}}} \sqrt {a+c x^2}}+\frac {\left (2 a \left (3 d^2-\frac {a e^2}{c}+\frac {2 e f (e f-3 d g)}{g^2}\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 )}{3 \sqrt {-a} \sqrt {c} \sqrt {f+g x} \sqrt {a+c x^2}} \\ & = \frac {2 e^2 \sqrt {f+g x} \sqrt {a+c x^2}}{3 c g}+\frac {4 \sqrt {-a} e (e f-3 d g) \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 )}{3 \sqrt {c} g^2 \sqrt {\frac {\sqrt {c} (f+g x)}{\sqrt {c} f+\sqrt {-a} g}} \sqrt {a+c x^2}}-\frac {2 \sqrt {-a} \left (3 d^2-\frac {a e^2}{c}+\frac {2 e f (e f-3 d g)}{g^2}\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 )}{3 \sqrt {c} \sqrt {f+g x} \sqrt {a+c x^2}} \\ \end{align*}
Result contains complex when optimal does not.
Time = 23.65 (sec) , antiderivative size = 473, normalized size of antiderivative = 1.33 \[ \int \frac {(d+e x)^2}{\sqrt {f+g x} \sqrt {a+c x^2}} \, dx=\frac {2 \sqrt {f+g x} \left (e^2 g^2 \left (a+c x^2\right )-\frac {2 e g^2 (e f-3 d g) \left (a+c x^2\right )}{f+g x}-2 i c e \sqrt {-f-\frac {i \sqrt {a} g}{\sqrt {c}}} (e f-3 d g) \sqrt {\frac {g \left (\frac {i \sqrt {a}}{\sqrt {c}}+x\right )}{f+g x}} \sqrt {-\frac {\frac {i \sqrt {a} g}{\sqrt {c}}-g x}{f+g x}} \sqrt {f+g x} E\left (i \text {arcsinh}\left (\frac {\sqrt {-f-\frac {i \sqrt {a} g}{\sqrt {c}}}}{\sqrt {f+g x}}\right )|\frac {\sqrt {c} f-i \sqrt {a} g}{\sqrt {c} f+i \sqrt {a} g}\right )+\frac {g \left (3 i c d^2 g-i a e^2 g+2 \sqrt {a} \sqrt {c} e (e f-3 d g)\right ) \sqrt {\frac {g \left (\frac {i \sqrt {a}}{\sqrt {c}}+x\right )}{f+g x}} \sqrt {-\frac {\frac {i \sqrt {a} g}{\sqrt {c}}-g x}{f+g x}} \sqrt {f+g x} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {-f-\frac {i \sqrt {a} g}{\sqrt {c}}}}{\sqrt {f+g x}}\right ),\frac {\sqrt {c} f-i \sqrt {a} g}{\sqrt {c} f+i \sqrt {a} g}\right )}{\sqrt {-f-\frac {i \sqrt {a} g}{\sqrt {c}}}}\right )}{3 c g^3 \sqrt {a+c x^2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(613\) vs. \(2(290)=580\).
Time = 2.54 (sec) , antiderivative size = 614, normalized size of antiderivative = 1.72
method | result | size |
elliptic | \(\frac {\sqrt {\left (g x +f \right ) \left (c \,x^{2}+a \right )}\, \left (\frac {2 e^{2} \sqrt {c g \,x^{3}+c f \,x^{2}+a g x +f a}}{3 c g}+\frac {2 \left (d^{2}-\frac {a \,e^{2}}{3 c}\right ) \left (\frac {f}{g}-\frac {\sqrt {-a c}}{c}\right ) \sqrt {\frac {x +\frac {f}{g}}{\frac {f}{g}-\frac {\sqrt {-a c}}{c}}}\, \sqrt {\frac {x -\frac {\sqrt {-a c}}{c}}{-\frac {f}{g}-\frac {\sqrt {-a c}}{c}}}\, \sqrt {\frac {x +\frac {\sqrt {-a c}}{c}}{-\frac {f}{g}+\frac {\sqrt {-a c}}{c}}}\, F\left (\sqrt {\frac {x +\frac {f}{g}}{\frac {f}{g}-\frac {\sqrt {-a c}}{c}}}, \sqrt {\frac {-\frac {f}{g}+\frac {\sqrt {-a c}}{c}}{-\frac {f}{g}-\frac {\sqrt {-a c}}{c}}}\right )}{\sqrt {c g \,x^{3}+c f \,x^{2}+a g x +f a}}+\frac {2 \left (2 d e -\frac {2 e^{2} f}{3 g}\right ) \left (\frac {f}{g}-\frac {\sqrt {-a c}}{c}\right ) \sqrt {\frac {x +\frac {f}{g}}{\frac {f}{g}-\frac {\sqrt {-a c}}{c}}}\, \sqrt {\frac {x -\frac {\sqrt {-a c}}{c}}{-\frac {f}{g}-\frac {\sqrt {-a c}}{c}}}\, \sqrt {\frac {x +\frac {\sqrt {-a c}}{c}}{-\frac {f}{g}+\frac {\sqrt {-a c}}{c}}}\, \left (\left (-\frac {f}{g}-\frac {\sqrt {-a c}}{c}\right ) E\left (\sqrt {\frac {x +\frac {f}{g}}{\frac {f}{g}-\frac {\sqrt {-a c}}{c}}}, \sqrt {\frac {-\frac {f}{g}+\frac {\sqrt {-a c}}{c}}{-\frac {f}{g}-\frac {\sqrt {-a c}}{c}}}\right )+\frac {\sqrt {-a c}\, F\left (\sqrt {\frac {x +\frac {f}{g}}{\frac {f}{g}-\frac {\sqrt {-a c}}{c}}}, \sqrt {\frac {-\frac {f}{g}+\frac {\sqrt {-a c}}{c}}{-\frac {f}{g}-\frac {\sqrt {-a c}}{c}}}\right )}{c}\right )}{\sqrt {c g \,x^{3}+c f \,x^{2}+a g x +f a}}\right )}{\sqrt {g x +f}\, \sqrt {c \,x^{2}+a}}\) | \(614\) |
risch | \(\frac {2 e^{2} \sqrt {g x +f}\, \sqrt {c \,x^{2}+a}}{3 c g}-\frac {\left (\frac {2 a \,e^{2} g \left (\frac {f}{g}-\frac {\sqrt {-a c}}{c}\right ) \sqrt {\frac {x +\frac {f}{g}}{\frac {f}{g}-\frac {\sqrt {-a c}}{c}}}\, \sqrt {\frac {x -\frac {\sqrt {-a c}}{c}}{-\frac {f}{g}-\frac {\sqrt {-a c}}{c}}}\, \sqrt {\frac {x +\frac {\sqrt {-a c}}{c}}{-\frac {f}{g}+\frac {\sqrt {-a c}}{c}}}\, F\left (\sqrt {\frac {x +\frac {f}{g}}{\frac {f}{g}-\frac {\sqrt {-a c}}{c}}}, \sqrt {\frac {-\frac {f}{g}+\frac {\sqrt {-a c}}{c}}{-\frac {f}{g}-\frac {\sqrt {-a c}}{c}}}\right )}{\sqrt {c g \,x^{3}+c f \,x^{2}+a g x +f a}}-\frac {6 c \,d^{2} g \left (\frac {f}{g}-\frac {\sqrt {-a c}}{c}\right ) \sqrt {\frac {x +\frac {f}{g}}{\frac {f}{g}-\frac {\sqrt {-a c}}{c}}}\, \sqrt {\frac {x -\frac {\sqrt {-a c}}{c}}{-\frac {f}{g}-\frac {\sqrt {-a c}}{c}}}\, \sqrt {\frac {x +\frac {\sqrt {-a c}}{c}}{-\frac {f}{g}+\frac {\sqrt {-a c}}{c}}}\, F\left (\sqrt {\frac {x +\frac {f}{g}}{\frac {f}{g}-\frac {\sqrt {-a c}}{c}}}, \sqrt {\frac {-\frac {f}{g}+\frac {\sqrt {-a c}}{c}}{-\frac {f}{g}-\frac {\sqrt {-a c}}{c}}}\right )}{\sqrt {c g \,x^{3}+c f \,x^{2}+a g x +f a}}-\frac {2 \left (6 c d e g -2 c \,e^{2} f \right ) \left (\frac {f}{g}-\frac {\sqrt {-a c}}{c}\right ) \sqrt {\frac {x +\frac {f}{g}}{\frac {f}{g}-\frac {\sqrt {-a c}}{c}}}\, \sqrt {\frac {x -\frac {\sqrt {-a c}}{c}}{-\frac {f}{g}-\frac {\sqrt {-a c}}{c}}}\, \sqrt {\frac {x +\frac {\sqrt {-a c}}{c}}{-\frac {f}{g}+\frac {\sqrt {-a c}}{c}}}\, \left (\left (-\frac {f}{g}-\frac {\sqrt {-a c}}{c}\right ) E\left (\sqrt {\frac {x +\frac {f}{g}}{\frac {f}{g}-\frac {\sqrt {-a c}}{c}}}, \sqrt {\frac {-\frac {f}{g}+\frac {\sqrt {-a c}}{c}}{-\frac {f}{g}-\frac {\sqrt {-a c}}{c}}}\right )+\frac {\sqrt {-a c}\, F\left (\sqrt {\frac {x +\frac {f}{g}}{\frac {f}{g}-\frac {\sqrt {-a c}}{c}}}, \sqrt {\frac {-\frac {f}{g}+\frac {\sqrt {-a c}}{c}}{-\frac {f}{g}-\frac {\sqrt {-a c}}{c}}}\right )}{c}\right )}{\sqrt {c g \,x^{3}+c f \,x^{2}+a g x +f a}}\right ) \sqrt {\left (g x +f \right ) \left (c \,x^{2}+a \right )}}{3 c g \sqrt {g x +f}\, \sqrt {c \,x^{2}+a}}\) | \(823\) |
default | \(\text {Expression too large to display}\) | \(1769\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.13 (sec) , antiderivative size = 246, normalized size of antiderivative = 0.69 \[ \int \frac {(d+e x)^2}{\sqrt {f+g x} \sqrt {a+c x^2}} \, dx=\frac {2 \, {\left (3 \, \sqrt {c x^{2} + a} \sqrt {g x + f} c e^{2} g^{2} + {\left (2 \, c e^{2} f^{2} - 6 \, c d e f g + 3 \, {\left (3 \, c d^{2} - a e^{2}\right )} g^{2}\right )} \sqrt {c g} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c f^{2} - 3 \, a g^{2}\right )}}{3 \, c g^{2}}, -\frac {8 \, {\left (c f^{3} + 9 \, a f g^{2}\right )}}{27 \, c g^{3}}, \frac {3 \, g x + f}{3 \, g}\right ) + 6 \, {\left (c e^{2} f g - 3 \, c d e g^{2}\right )} \sqrt {c g} {\rm weierstrassZeta}\left (\frac {4 \, {\left (c f^{2} - 3 \, a g^{2}\right )}}{3 \, c g^{2}}, -\frac {8 \, {\left (c f^{3} + 9 \, a f g^{2}\right )}}{27 \, c g^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c f^{2} - 3 \, a g^{2}\right )}}{3 \, c g^{2}}, -\frac {8 \, {\left (c f^{3} + 9 \, a f g^{2}\right )}}{27 \, c g^{3}}, \frac {3 \, g x + f}{3 \, g}\right )\right )\right )}}{9 \, c^{2} g^{3}} \]
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\[ \int \frac {(d+e x)^2}{\sqrt {f+g x} \sqrt {a+c x^2}} \, dx=\int \frac {\left (d + e x\right )^{2}}{\sqrt {a + c x^{2}} \sqrt {f + g x}}\, dx \]
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\[ \int \frac {(d+e x)^2}{\sqrt {f+g x} \sqrt {a+c x^2}} \, dx=\int { \frac {{\left (e x + d\right )}^{2}}{\sqrt {c x^{2} + a} \sqrt {g x + f}} \,d x } \]
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\[ \int \frac {(d+e x)^2}{\sqrt {f+g x} \sqrt {a+c x^2}} \, dx=\int { \frac {{\left (e x + d\right )}^{2}}{\sqrt {c x^{2} + a} \sqrt {g x + f}} \,d x } \]
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Timed out. \[ \int \frac {(d+e x)^2}{\sqrt {f+g x} \sqrt {a+c x^2}} \, dx=\int \frac {{\left (d+e\,x\right )}^2}{\sqrt {f+g\,x}\,\sqrt {c\,x^2+a}} \,d x \]
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