Integrand size = 21, antiderivative size = 406 \[ \int \frac {1}{(d+e x)^{3/2} \left (a+c x^2\right )^{3/2}} \, dx=\frac {a e+c d x}{a \left (c d^2+a e^2\right ) \sqrt {d+e x} \sqrt {a+c x^2}}+\frac {e \left (c d^2-3 a e^2\right ) \sqrt {a+c x^2}}{a \left (c d^2+a e^2\right )^2 \sqrt {d+e x}}-\frac {\sqrt {c} \left (c d^2-3 a e^2\right ) \sqrt {d+e x} \sqrt {1+\frac {c x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a e}{\sqrt {-a} \sqrt {c} d-a e}\right )}{\sqrt {-a} \left (c d^2+a e^2\right )^2 \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {-a} e}} \sqrt {a+c x^2}}+\frac {\sqrt {c} d \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {-a} e}} \sqrt {1+\frac {c x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right ),-\frac {2 a e}{\sqrt {-a} \sqrt {c} d-a e}\right )}{\sqrt {-a} \left (c d^2+a e^2\right ) \sqrt {d+e x} \sqrt {a+c x^2}} \]
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Time = 0.22 (sec) , antiderivative size = 406, 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 = {755, 849, 858, 733, 435, 430} \[ \int \frac {1}{(d+e x)^{3/2} \left (a+c x^2\right )^{3/2}} \, dx=\frac {\sqrt {c} d \sqrt {\frac {c x^2}{a}+1} \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {-a} e+\sqrt {c} d}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right ),-\frac {2 a e}{\sqrt {-a} \sqrt {c} d-a e}\right )}{\sqrt {-a} \sqrt {a+c x^2} \sqrt {d+e x} \left (a e^2+c d^2\right )}-\frac {\sqrt {c} \sqrt {\frac {c x^2}{a}+1} \sqrt {d+e x} \left (c d^2-3 a e^2\right ) E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a e}{\sqrt {-a} \sqrt {c} d-a e}\right )}{\sqrt {-a} \sqrt {a+c x^2} \left (a e^2+c d^2\right )^2 \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {-a} e+\sqrt {c} d}}}+\frac {e \sqrt {a+c x^2} \left (c d^2-3 a e^2\right )}{a \sqrt {d+e x} \left (a e^2+c d^2\right )^2}+\frac {a e+c d x}{a \sqrt {a+c x^2} \sqrt {d+e x} \left (a e^2+c d^2\right )} \]
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Rule 430
Rule 435
Rule 733
Rule 755
Rule 849
Rule 858
Rubi steps \begin{align*} \text {integral}& = \frac {a e+c d x}{a \left (c d^2+a e^2\right ) \sqrt {d+e x} \sqrt {a+c x^2}}-\frac {\int \frac {-\frac {3 a e^2}{2}-\frac {1}{2} c d e x}{(d+e x)^{3/2} \sqrt {a+c x^2}} \, dx}{a \left (c d^2+a e^2\right )} \\ & = \frac {a e+c d x}{a \left (c d^2+a e^2\right ) \sqrt {d+e x} \sqrt {a+c x^2}}+\frac {e \left (c d^2-3 a e^2\right ) \sqrt {a+c x^2}}{a \left (c d^2+a e^2\right )^2 \sqrt {d+e x}}+\frac {2 \int \frac {a c d e^2-\frac {1}{4} c e \left (c d^2-3 a e^2\right ) x}{\sqrt {d+e x} \sqrt {a+c x^2}} \, dx}{a \left (c d^2+a e^2\right )^2} \\ & = \frac {a e+c d x}{a \left (c d^2+a e^2\right ) \sqrt {d+e x} \sqrt {a+c x^2}}+\frac {e \left (c d^2-3 a e^2\right ) \sqrt {a+c x^2}}{a \left (c d^2+a e^2\right )^2 \sqrt {d+e x}}-\frac {\left (c \left (c d^2-3 a e^2\right )\right ) \int \frac {\sqrt {d+e x}}{\sqrt {a+c x^2}} \, dx}{2 a \left (c d^2+a e^2\right )^2}+\frac {(c d) \int \frac {1}{\sqrt {d+e x} \sqrt {a+c x^2}} \, dx}{2 a \left (c d^2+a e^2\right )} \\ & = \frac {a e+c d x}{a \left (c d^2+a e^2\right ) \sqrt {d+e x} \sqrt {a+c x^2}}+\frac {e \left (c d^2-3 a e^2\right ) \sqrt {a+c x^2}}{a \left (c d^2+a e^2\right )^2 \sqrt {d+e x}}-\frac {\left (\sqrt {c} \left (c d^2-3 a e^2\right ) \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} \left (c d^2+a e^2\right )^2 \sqrt {\frac {c (d+e x)}{c d-\frac {a \sqrt {c} e}{\sqrt {-a}}}} \sqrt {a+c x^2}}+\frac {\left (\sqrt {c} 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} \left (c d^2+a e^2\right ) \sqrt {d+e x} \sqrt {a+c x^2}} \\ & = \frac {a e+c d x}{a \left (c d^2+a e^2\right ) \sqrt {d+e x} \sqrt {a+c x^2}}+\frac {e \left (c d^2-3 a e^2\right ) \sqrt {a+c x^2}}{a \left (c d^2+a e^2\right )^2 \sqrt {d+e x}}-\frac {\sqrt {c} \left (c d^2-3 a e^2\right ) \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} \left (c d^2+a e^2\right )^2 \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {-a} e}} \sqrt {a+c x^2}}+\frac {\sqrt {c} 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} \left (c d^2+a e^2\right ) \sqrt {d+e x} \sqrt {a+c x^2}} \\ \end{align*}
Result contains complex when optimal does not.
Time = 11.86 (sec) , antiderivative size = 568, normalized size of antiderivative = 1.40 \[ \int \frac {1}{(d+e x)^{3/2} \left (a+c x^2\right )^{3/2}} \, dx=\frac {-e^2 \sqrt {-d-\frac {i \sqrt {a} e}{\sqrt {c}}} \left (c d^2-3 a e^2\right ) \left (a+c x^2\right )+e \sqrt {-d-\frac {i \sqrt {a} e}{\sqrt {c}}} \left (-2 a e^3 \left (a+c x^2\right )+c (d+e x) \left (c d^2 x+a e (2 d-e x)\right )\right )+\sqrt {c} \left (i c^{3/2} d^3-\sqrt {a} c d^2 e-3 i a \sqrt {c} d e^2+3 a^{3/2} e^3\right ) \sqrt {\frac {e \left (\frac {i \sqrt {a}}{\sqrt {c}}+x\right )}{d+e x}} \sqrt {-\frac {\frac {i \sqrt {a} e}{\sqrt {c}}-e x}{d+e x}} (d+e x)^{3/2} E\left (i \text {arcsinh}\left (\frac {\sqrt {-d-\frac {i \sqrt {a} e}{\sqrt {c}}}}{\sqrt {d+e x}}\right )|\frac {\sqrt {c} d-i \sqrt {a} e}{\sqrt {c} d+i \sqrt {a} e}\right )+\sqrt {a} \sqrt {c} e \left (c d^2+4 i \sqrt {a} \sqrt {c} d e-3 a e^2\right ) \sqrt {\frac {e \left (\frac {i \sqrt {a}}{\sqrt {c}}+x\right )}{d+e x}} \sqrt {-\frac {\frac {i \sqrt {a} e}{\sqrt {c}}-e x}{d+e x}} (d+e x)^{3/2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {-d-\frac {i \sqrt {a} e}{\sqrt {c}}}}{\sqrt {d+e x}}\right ),\frac {\sqrt {c} d-i \sqrt {a} e}{\sqrt {c} d+i \sqrt {a} e}\right )}{a e \sqrt {-d-\frac {i \sqrt {a} e}{\sqrt {c}}} \left (c d^2+a e^2\right )^2 \sqrt {d+e x} \sqrt {a+c x^2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(814\) vs. \(2(342)=684\).
Time = 3.02 (sec) , antiderivative size = 815, normalized size of antiderivative = 2.01
method | result | size |
elliptic | \(\frac {\sqrt {\left (e x +d \right ) \left (c \,x^{2}+a \right )}\, \left (-\frac {2 c e \left (\frac {\left (3 e^{2} a -c \,d^{2}\right ) x^{2}}{2 a \left (a^{2} e^{4}+2 a c \,d^{2} e^{2}+c^{2} d^{4}\right )}-\frac {d x}{2 a \left (e^{2} a +c \,d^{2}\right ) e}+\frac {e^{2} a -c \,d^{2}}{\left (a^{2} e^{4}+2 a c \,d^{2} e^{2}+c^{2} d^{4}\right ) c}\right )}{\sqrt {\left (x^{3}+\frac {d \,x^{2}}{e}+\frac {a x}{c}+\frac {a d}{c e}\right ) c e}}+\frac {2 \left (\frac {c d \left (3 e^{2} a +c \,d^{2}\right )}{a \left (a^{2} e^{4}+2 a c \,d^{2} e^{2}+c^{2} d^{4}\right )}-\frac {c d}{a \left (e^{2} a +c \,d^{2}\right )}\right ) \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}}}\, 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 )}{\sqrt {c e \,x^{3}+c d \,x^{2}+a e x +a d}}+\frac {\left (3 e^{2} a -c \,d^{2}\right ) c 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 \left (a^{2} e^{4}+2 a c \,d^{2} e^{2}+c^{2} d^{4}\right ) \sqrt {c e \,x^{3}+c d \,x^{2}+a e x +a d}}\right )}{\sqrt {e x +d}\, \sqrt {c \,x^{2}+a}}\) | \(815\) |
default | \(\text {Expression too large to display}\) | \(1167\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.14 (sec) , antiderivative size = 545, normalized size of antiderivative = 1.34 \[ \int \frac {1}{(d+e x)^{3/2} \left (a+c x^2\right )^{3/2}} \, dx=\frac {{\left (a c d^{4} + 9 \, a^{2} d^{2} e^{2} + {\left (c^{2} d^{3} e + 9 \, a c d e^{3}\right )} x^{3} + {\left (c^{2} d^{4} + 9 \, a c d^{2} e^{2}\right )} x^{2} + {\left (a c d^{3} e + 9 \, a^{2} d e^{3}\right )} x\right )} \sqrt {c e} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c d^{2} - 3 \, a e^{2}\right )}}{3 \, c e^{2}}, -\frac {8 \, {\left (c d^{3} + 9 \, a d e^{2}\right )}}{27 \, c e^{3}}, \frac {3 \, e x + d}{3 \, e}\right ) + 3 \, {\left (a c d^{3} e - 3 \, a^{2} d e^{3} + {\left (c^{2} d^{2} e^{2} - 3 \, a c e^{4}\right )} x^{3} + {\left (c^{2} d^{3} e - 3 \, a c d e^{3}\right )} x^{2} + {\left (a c d^{2} e^{2} - 3 \, a^{2} e^{4}\right )} x\right )} \sqrt {c e} {\rm weierstrassZeta}\left (\frac {4 \, {\left (c d^{2} - 3 \, a e^{2}\right )}}{3 \, c e^{2}}, -\frac {8 \, {\left (c d^{3} + 9 \, a d e^{2}\right )}}{27 \, c e^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c d^{2} - 3 \, a e^{2}\right )}}{3 \, c e^{2}}, -\frac {8 \, {\left (c d^{3} + 9 \, a d e^{2}\right )}}{27 \, c e^{3}}, \frac {3 \, e x + d}{3 \, e}\right )\right ) + 3 \, {\left (2 \, a c d^{2} e^{2} - 2 \, a^{2} e^{4} + {\left (c^{2} d^{2} e^{2} - 3 \, a c e^{4}\right )} x^{2} + {\left (c^{2} d^{3} e + a c d e^{3}\right )} x\right )} \sqrt {c x^{2} + a} \sqrt {e x + d}}{3 \, {\left (a^{2} c^{2} d^{5} e + 2 \, a^{3} c d^{3} e^{3} + a^{4} d e^{5} + {\left (a c^{3} d^{4} e^{2} + 2 \, a^{2} c^{2} d^{2} e^{4} + a^{3} c e^{6}\right )} x^{3} + {\left (a c^{3} d^{5} e + 2 \, a^{2} c^{2} d^{3} e^{3} + a^{3} c d e^{5}\right )} x^{2} + {\left (a^{2} c^{2} d^{4} e^{2} + 2 \, a^{3} c d^{2} e^{4} + a^{4} e^{6}\right )} x\right )}} \]
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\[ \int \frac {1}{(d+e x)^{3/2} \left (a+c x^2\right )^{3/2}} \, dx=\int \frac {1}{\left (a + c x^{2}\right )^{\frac {3}{2}} \left (d + e x\right )^{\frac {3}{2}}}\, dx \]
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\[ \int \frac {1}{(d+e x)^{3/2} \left (a+c x^2\right )^{3/2}} \, dx=\int { \frac {1}{{\left (c x^{2} + a\right )}^{\frac {3}{2}} {\left (e x + d\right )}^{\frac {3}{2}}} \,d x } \]
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\[ \int \frac {1}{(d+e x)^{3/2} \left (a+c x^2\right )^{3/2}} \, dx=\int { \frac {1}{{\left (c x^{2} + a\right )}^{\frac {3}{2}} {\left (e x + d\right )}^{\frac {3}{2}}} \,d x } \]
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Timed out. \[ \int \frac {1}{(d+e x)^{3/2} \left (a+c x^2\right )^{3/2}} \, dx=\int \frac {1}{{\left (c\,x^2+a\right )}^{3/2}\,{\left (d+e\,x\right )}^{3/2}} \,d x \]
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