Integrand size = 21, antiderivative size = 298 \[ \int \frac {\sqrt {d+e x}}{\left (a+c x^2\right )^{3/2}} \, dx=\frac {x \sqrt {d+e x}}{a \sqrt {a+c x^2}}-\frac {\sqrt {d+e x} \sqrt {1+\frac {c x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a e}{\sqrt {-a} \sqrt {c} d-a e}\right )}{\sqrt {-a} \sqrt {c} \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {-a} e}} \sqrt {a+c x^2}}+\frac {d \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {-a} e}} \sqrt {1+\frac {c x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right ),-\frac {2 a e}{\sqrt {-a} \sqrt {c} d-a e}\right )}{\sqrt {-a} \sqrt {c} \sqrt {d+e x} \sqrt {a+c x^2}} \]
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Time = 0.15 (sec) , antiderivative size = 298, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {751, 12, 858, 733, 435, 430} \[ \int \frac {\sqrt {d+e x}}{\left (a+c x^2\right )^{3/2}} \, dx=\frac {d \sqrt {\frac {c x^2}{a}+1} \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {-a} e+\sqrt {c} d}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right ),-\frac {2 a e}{\sqrt {-a} \sqrt {c} d-a e}\right )}{\sqrt {-a} \sqrt {c} \sqrt {a+c x^2} \sqrt {d+e x}}-\frac {\sqrt {\frac {c x^2}{a}+1} \sqrt {d+e x} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a e}{\sqrt {-a} \sqrt {c} d-a e}\right )}{\sqrt {-a} \sqrt {c} \sqrt {a+c x^2} \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {-a} e+\sqrt {c} d}}}+\frac {x \sqrt {d+e x}}{a \sqrt {a+c x^2}} \]
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Rule 12
Rule 430
Rule 435
Rule 733
Rule 751
Rule 858
Rubi steps \begin{align*} \text {integral}& = \frac {x \sqrt {d+e x}}{a \sqrt {a+c x^2}}-\frac {\int \frac {e x}{2 \sqrt {d+e x} \sqrt {a+c x^2}} \, dx}{a} \\ & = \frac {x \sqrt {d+e x}}{a \sqrt {a+c x^2}}-\frac {e \int \frac {x}{\sqrt {d+e x} \sqrt {a+c x^2}} \, dx}{2 a} \\ & = \frac {x \sqrt {d+e x}}{a \sqrt {a+c x^2}}-\frac {\int \frac {\sqrt {d+e x}}{\sqrt {a+c x^2}} \, dx}{2 a}+\frac {d \int \frac {1}{\sqrt {d+e x} \sqrt {a+c x^2}} \, dx}{2 a} \\ & = \frac {x \sqrt {d+e x}}{a \sqrt {a+c x^2}}-\frac {\left (\sqrt {d+e x} \sqrt {1+\frac {c x^2}{a}}\right ) \text {Subst}\left (\int \frac {\sqrt {1+\frac {2 a \sqrt {c} e x^2}{\sqrt {-a} \left (c d-\frac {a \sqrt {c} e}{\sqrt {-a}}\right )}}}{\sqrt {1-x^2}} \, dx,x,\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )}{\sqrt {-a} \sqrt {c} \sqrt {\frac {c (d+e x)}{c d-\frac {a \sqrt {c} e}{\sqrt {-a}}}} \sqrt {a+c x^2}}+\frac {\left (d \sqrt {\frac {c (d+e x)}{c d-\frac {a \sqrt {c} e}{\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} e x^2}{\sqrt {-a} \left (c d-\frac {a \sqrt {c} e}{\sqrt {-a}}\right )}}} \, dx,x,\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )}{\sqrt {-a} \sqrt {c} \sqrt {d+e x} \sqrt {a+c x^2}} \\ & = \frac {x \sqrt {d+e x}}{a \sqrt {a+c x^2}}-\frac {\sqrt {d+e 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 e}{\sqrt {-a} \sqrt {c} d-a e}\right )}{\sqrt {-a} \sqrt {c} \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {-a} e}} \sqrt {a+c x^2}}+\frac {d \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {-a} e}} \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 e}{\sqrt {-a} \sqrt {c} d-a e}\right )}{\sqrt {-a} \sqrt {c} \sqrt {d+e x} \sqrt {a+c x^2}} \\ \end{align*}
Result contains complex when optimal does not.
Time = 12.38 (sec) , antiderivative size = 408, normalized size of antiderivative = 1.37 \[ \int \frac {\sqrt {d+e x}}{\left (a+c x^2\right )^{3/2}} \, dx=\frac {\sqrt {d+e x} \left (x-\frac {e \left (a+c x^2\right )}{c (d+e x)}-\frac {i \sqrt {-d-\frac {i \sqrt {a} e}{\sqrt {c}}} \sqrt {\frac {e \left (\frac {i \sqrt {a}}{\sqrt {c}}+x\right )}{d+e x}} \sqrt {-\frac {\frac {i \sqrt {a} e}{\sqrt {c}}-e x}{d+e x}} \sqrt {d+e x} E\left (i \text {arcsinh}\left (\frac {\sqrt {-d-\frac {i \sqrt {a} e}{\sqrt {c}}}}{\sqrt {d+e x}}\right )|\frac {\sqrt {c} d-i \sqrt {a} e}{\sqrt {c} d+i \sqrt {a} e}\right )}{e}+\frac {\sqrt {a} \sqrt {\frac {e \left (\frac {i \sqrt {a}}{\sqrt {c}}+x\right )}{d+e x}} \sqrt {-\frac {\frac {i \sqrt {a} e}{\sqrt {c}}-e x}{d+e x}} \sqrt {d+e x} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {-d-\frac {i \sqrt {a} e}{\sqrt {c}}}}{\sqrt {d+e x}}\right ),\frac {\sqrt {c} d-i \sqrt {a} e}{\sqrt {c} d+i \sqrt {a} e}\right )}{\sqrt {c} \sqrt {-d-\frac {i \sqrt {a} e}{\sqrt {c}}}}\right )}{a \sqrt {a+c x^2}} \]
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Time = 2.21 (sec) , antiderivative size = 384, normalized size of antiderivative = 1.29
method | result | size |
elliptic | \(\frac {\sqrt {\left (e x +d \right ) \left (c \,x^{2}+a \right )}\, \left (\frac {\left (c e x +c d \right ) x}{a c \sqrt {\left (x^{2}+\frac {a}{c}\right ) \left (c e x +c d \right )}}-\frac {e \left (\frac {d}{e}-\frac {\sqrt {-a c}}{c}\right ) \sqrt {\frac {x +\frac {d}{e}}{\frac {d}{e}-\frac {\sqrt {-a c}}{c}}}\, \sqrt {\frac {x -\frac {\sqrt {-a c}}{c}}{-\frac {d}{e}-\frac {\sqrt {-a c}}{c}}}\, \sqrt {\frac {x +\frac {\sqrt {-a c}}{c}}{-\frac {d}{e}+\frac {\sqrt {-a c}}{c}}}\, \left (\left (-\frac {d}{e}-\frac {\sqrt {-a c}}{c}\right ) E\left (\sqrt {\frac {x +\frac {d}{e}}{\frac {d}{e}-\frac {\sqrt {-a c}}{c}}}, \sqrt {\frac {-\frac {d}{e}+\frac {\sqrt {-a c}}{c}}{-\frac {d}{e}-\frac {\sqrt {-a c}}{c}}}\right )+\frac {\sqrt {-a c}\, F\left (\sqrt {\frac {x +\frac {d}{e}}{\frac {d}{e}-\frac {\sqrt {-a c}}{c}}}, \sqrt {\frac {-\frac {d}{e}+\frac {\sqrt {-a c}}{c}}{-\frac {d}{e}-\frac {\sqrt {-a c}}{c}}}\right )}{c}\right )}{a \sqrt {c e \,x^{3}+c d \,x^{2}+a e x +a d}}\right )}{\sqrt {e x +d}\, \sqrt {c \,x^{2}+a}}\) | \(384\) |
default | \(\frac {\left (-\sqrt {-a c}\, \sqrt {-\frac {c \left (e x +d \right )}{\sqrt {-a c}\, e -c d}}\, \sqrt {\frac {\left (-c x +\sqrt {-a c}\right ) e}{\sqrt {-a c}\, e +c d}}\, \sqrt {\frac {\left (c x +\sqrt {-a c}\right ) e}{\sqrt {-a c}\, e -c d}}\, F\left (\sqrt {-\frac {c \left (e x +d \right )}{\sqrt {-a c}\, e -c d}}, \sqrt {-\frac {\sqrt {-a c}\, e -c d}{\sqrt {-a c}\, e +c d}}\right ) d e +\sqrt {-\frac {c \left (e x +d \right )}{\sqrt {-a c}\, e -c d}}\, \sqrt {\frac {\left (-c x +\sqrt {-a c}\right ) e}{\sqrt {-a c}\, e +c d}}\, \sqrt {\frac {\left (c x +\sqrt {-a c}\right ) e}{\sqrt {-a c}\, e -c d}}\, E\left (\sqrt {-\frac {c \left (e x +d \right )}{\sqrt {-a c}\, e -c d}}, \sqrt {-\frac {\sqrt {-a c}\, e -c d}{\sqrt {-a c}\, e +c d}}\right ) a \,e^{2}+\sqrt {-\frac {c \left (e x +d \right )}{\sqrt {-a c}\, e -c d}}\, \sqrt {\frac {\left (-c x +\sqrt {-a c}\right ) e}{\sqrt {-a c}\, e +c d}}\, \sqrt {\frac {\left (c x +\sqrt {-a c}\right ) e}{\sqrt {-a c}\, e -c d}}\, E\left (\sqrt {-\frac {c \left (e x +d \right )}{\sqrt {-a c}\, e -c d}}, \sqrt {-\frac {\sqrt {-a c}\, e -c d}{\sqrt {-a c}\, e +c d}}\right ) c \,d^{2}-\sqrt {-\frac {c \left (e x +d \right )}{\sqrt {-a c}\, e -c d}}\, \sqrt {\frac {\left (-c x +\sqrt {-a c}\right ) e}{\sqrt {-a c}\, e +c d}}\, \sqrt {\frac {\left (c x +\sqrt {-a c}\right ) e}{\sqrt {-a c}\, e -c d}}\, F\left (\sqrt {-\frac {c \left (e x +d \right )}{\sqrt {-a c}\, e -c d}}, \sqrt {-\frac {\sqrt {-a c}\, e -c d}{\sqrt {-a c}\, e +c d}}\right ) a \,e^{2}+c \,x^{2} e^{2}+x c d e \right ) \sqrt {e x +d}\, \sqrt {c \,x^{2}+a}}{c e \left (c e \,x^{3}+c d \,x^{2}+a e x +a d \right ) a}\) | \(649\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.08 (sec) , antiderivative size = 223, normalized size of antiderivative = 0.75 \[ \int \frac {\sqrt {d+e x}}{\left (a+c x^2\right )^{3/2}} \, dx=\frac {3 \, \sqrt {c x^{2} + a} \sqrt {e x + d} c e x + {\left (c d x^{2} + a d\right )} \sqrt {c e} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c d^{2} - 3 \, a e^{2}\right )}}{3 \, c e^{2}}, -\frac {8 \, {\left (c d^{3} + 9 \, a d e^{2}\right )}}{27 \, c e^{3}}, \frac {3 \, e x + d}{3 \, e}\right ) + 3 \, {\left (c e x^{2} + a e\right )} \sqrt {c e} {\rm weierstrassZeta}\left (\frac {4 \, {\left (c d^{2} - 3 \, a e^{2}\right )}}{3 \, c e^{2}}, -\frac {8 \, {\left (c d^{3} + 9 \, a d e^{2}\right )}}{27 \, c e^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c d^{2} - 3 \, a e^{2}\right )}}{3 \, c e^{2}}, -\frac {8 \, {\left (c d^{3} + 9 \, a d e^{2}\right )}}{27 \, c e^{3}}, \frac {3 \, e x + d}{3 \, e}\right )\right )}{3 \, {\left (a c^{2} e x^{2} + a^{2} c e\right )}} \]
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\[ \int \frac {\sqrt {d+e x}}{\left (a+c x^2\right )^{3/2}} \, dx=\int \frac {\sqrt {d + e x}}{\left (a + c x^{2}\right )^{\frac {3}{2}}}\, dx \]
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\[ \int \frac {\sqrt {d+e x}}{\left (a+c x^2\right )^{3/2}} \, dx=\int { \frac {\sqrt {e x + d}}{{\left (c x^{2} + a\right )}^{\frac {3}{2}}} \,d x } \]
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\[ \int \frac {\sqrt {d+e x}}{\left (a+c x^2\right )^{3/2}} \, dx=\int { \frac {\sqrt {e x + d}}{{\left (c x^{2} + a\right )}^{\frac {3}{2}}} \,d x } \]
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Timed out. \[ \int \frac {\sqrt {d+e x}}{\left (a+c x^2\right )^{3/2}} \, dx=\int \frac {\sqrt {d+e\,x}}{{\left (c\,x^2+a\right )}^{3/2}} \,d x \]
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