Integrand size = 24, antiderivative size = 425 \[ \int \frac {\left (a x+b x^3+c x^5\right )^{3/2}}{x^{3/2}} \, dx=-\frac {2 b \left (b^2-8 a c\right ) x^{3/2} \left (a+b x^2+c x^4\right )}{35 c^{3/2} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {a x+b x^3+c x^5}}+\frac {\sqrt {x} \left (b^2+10 a c+3 b c x^2\right ) \sqrt {a x+b x^3+c x^5}}{35 c}+\frac {\left (a x+b x^3+c x^5\right )^{3/2}}{7 \sqrt {x}}+\frac {2 \sqrt [4]{a} b \left (b^2-8 a c\right ) \sqrt {x} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{35 c^{7/4} \sqrt {a x+b x^3+c x^5}}-\frac {\sqrt [4]{a} \left (\sqrt {a} \sqrt {c} \left (b^2-20 a c\right )+2 b \left (b^2-8 a c\right )\right ) \sqrt {x} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{70 c^{7/4} \sqrt {a x+b x^3+c x^5}} \]
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Time = 0.27 (sec) , antiderivative size = 425, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {1935, 1959, 1967, 1211, 1117, 1209} \[ \int \frac {\left (a x+b x^3+c x^5\right )^{3/2}}{x^{3/2}} \, dx=-\frac {\sqrt [4]{a} \sqrt {x} \left (\sqrt {a} \sqrt {c} \left (b^2-20 a c\right )+2 b \left (b^2-8 a c\right )\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{70 c^{7/4} \sqrt {a x+b x^3+c x^5}}+\frac {2 \sqrt [4]{a} b \sqrt {x} \left (b^2-8 a c\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{35 c^{7/4} \sqrt {a x+b x^3+c x^5}}-\frac {2 b x^{3/2} \left (b^2-8 a c\right ) \left (a+b x^2+c x^4\right )}{35 c^{3/2} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {a x+b x^3+c x^5}}+\frac {\sqrt {x} \left (10 a c+b^2+3 b c x^2\right ) \sqrt {a x+b x^3+c x^5}}{35 c}+\frac {\left (a x+b x^3+c x^5\right )^{3/2}}{7 \sqrt {x}} \]
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Rule 1117
Rule 1209
Rule 1211
Rule 1935
Rule 1959
Rule 1967
Rubi steps \begin{align*} \text {integral}& = \frac {\left (a x+b x^3+c x^5\right )^{3/2}}{7 \sqrt {x}}+\frac {3}{7} \int \frac {\left (2 a+b x^2\right ) \sqrt {a x+b x^3+c x^5}}{\sqrt {x}} \, dx \\ & = \frac {\sqrt {x} \left (b^2+10 a c+3 b c x^2\right ) \sqrt {a x+b x^3+c x^5}}{35 c}+\frac {\left (a x+b x^3+c x^5\right )^{3/2}}{7 \sqrt {x}}+\frac {\int \frac {\sqrt {x} \left (-a \left (b^2-20 a c\right )-2 b \left (b^2-8 a c\right ) x^2\right )}{\sqrt {a x+b x^3+c x^5}} \, dx}{35 c} \\ & = \frac {\sqrt {x} \left (b^2+10 a c+3 b c x^2\right ) \sqrt {a x+b x^3+c x^5}}{35 c}+\frac {\left (a x+b x^3+c x^5\right )^{3/2}}{7 \sqrt {x}}+\frac {\left (\sqrt {x} \sqrt {a+b x^2+c x^4}\right ) \int \frac {-a \left (b^2-20 a c\right )-2 b \left (b^2-8 a c\right ) x^2}{\sqrt {a+b x^2+c x^4}} \, dx}{35 c \sqrt {a x+b x^3+c x^5}} \\ & = \frac {\sqrt {x} \left (b^2+10 a c+3 b c x^2\right ) \sqrt {a x+b x^3+c x^5}}{35 c}+\frac {\left (a x+b x^3+c x^5\right )^{3/2}}{7 \sqrt {x}}+\frac {\left (2 \sqrt {a} b \left (b^2-8 a c\right ) \sqrt {x} \sqrt {a+b x^2+c x^4}\right ) \int \frac {1-\frac {\sqrt {c} x^2}{\sqrt {a}}}{\sqrt {a+b x^2+c x^4}} \, dx}{35 c^{3/2} \sqrt {a x+b x^3+c x^5}}-\frac {\left (\sqrt {a} \left (\sqrt {a} \left (b^2-20 a c\right )+\frac {2 b \left (b^2-8 a c\right )}{\sqrt {c}}\right ) \sqrt {x} \sqrt {a+b x^2+c x^4}\right ) \int \frac {1}{\sqrt {a+b x^2+c x^4}} \, dx}{35 c \sqrt {a x+b x^3+c x^5}} \\ & = -\frac {2 b \left (b^2-8 a c\right ) x^{3/2} \left (a+b x^2+c x^4\right )}{35 c^{3/2} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {a x+b x^3+c x^5}}+\frac {\sqrt {x} \left (b^2+10 a c+3 b c x^2\right ) \sqrt {a x+b x^3+c x^5}}{35 c}+\frac {\left (a x+b x^3+c x^5\right )^{3/2}}{7 \sqrt {x}}+\frac {2 \sqrt [4]{a} b \left (b^2-8 a c\right ) \sqrt {x} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{35 c^{7/4} \sqrt {a x+b x^3+c x^5}}-\frac {\sqrt [4]{a} \left (\sqrt {a} \left (b^2-20 a c\right )+\frac {2 b \left (b^2-8 a c\right )}{\sqrt {c}}\right ) \sqrt {x} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{70 c^{5/4} \sqrt {a x+b x^3+c x^5}} \\ \end{align*}
Result contains complex when optimal does not.
Time = 2.13 (sec) , antiderivative size = 540, normalized size of antiderivative = 1.27 \[ \int \frac {\left (a x+b x^3+c x^5\right )^{3/2}}{x^{3/2}} \, dx=\frac {\sqrt {x} \left (2 c \sqrt {\frac {c}{b+\sqrt {b^2-4 a c}}} x \left (15 a^2 c+a \left (b^2+23 b c x^2+20 c^2 x^4\right )+x^2 \left (b^3+9 b^2 c x^2+13 b c^2 x^4+5 c^3 x^6\right )\right )-i b \left (b^2-8 a c\right ) \left (-b+\sqrt {b^2-4 a c}\right ) \sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x^2}{b+\sqrt {b^2-4 a c}}} \sqrt {\frac {2 b-2 \sqrt {b^2-4 a c}+4 c x^2}{b-\sqrt {b^2-4 a c}}} E\left (i \text {arcsinh}\left (\sqrt {2} \sqrt {\frac {c}{b+\sqrt {b^2-4 a c}}} x\right )|\frac {b+\sqrt {b^2-4 a c}}{b-\sqrt {b^2-4 a c}}\right )+i \left (-b^4+9 a b^2 c-20 a^2 c^2+b^3 \sqrt {b^2-4 a c}-8 a b c \sqrt {b^2-4 a c}\right ) \sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x^2}{b+\sqrt {b^2-4 a c}}} \sqrt {\frac {2 b-2 \sqrt {b^2-4 a c}+4 c x^2}{b-\sqrt {b^2-4 a c}}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {2} \sqrt {\frac {c}{b+\sqrt {b^2-4 a c}}} x\right ),\frac {b+\sqrt {b^2-4 a c}}{b-\sqrt {b^2-4 a c}}\right )\right )}{70 c^2 \sqrt {\frac {c}{b+\sqrt {b^2-4 a c}}} \sqrt {x \left (a+b x^2+c x^4\right )}} \]
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Time = 2.14 (sec) , antiderivative size = 620, normalized size of antiderivative = 1.46
method | result | size |
risch | \(\frac {x^{\frac {3}{2}} \left (5 c^{2} x^{4}+8 b c \,x^{2}+15 a c +b^{2}\right ) \left (c \,x^{4}+b \,x^{2}+a \right )}{35 c \sqrt {x \left (c \,x^{4}+b \,x^{2}+a \right )}}+\frac {\left (-\frac {b^{2} a \sqrt {2}\, \sqrt {4-\frac {2 \left (-b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \sqrt {4+\frac {2 \left (b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, F\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4+\frac {2 b \left (b +\sqrt {-4 a c +b^{2}}\right )}{a c}}}{2}\right )}{4 \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}+\frac {5 c \,a^{2} \sqrt {2}\, \sqrt {4-\frac {2 \left (-b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \sqrt {4+\frac {2 \left (b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, F\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4+\frac {2 b \left (b +\sqrt {-4 a c +b^{2}}\right )}{a c}}}{2}\right )}{\sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}-\frac {\left (16 a b c -2 b^{3}\right ) a \sqrt {2}\, \sqrt {4-\frac {2 \left (-b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \sqrt {4+\frac {2 \left (b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \left (F\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4+\frac {2 b \left (b +\sqrt {-4 a c +b^{2}}\right )}{a c}}}{2}\right )-E\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4+\frac {2 b \left (b +\sqrt {-4 a c +b^{2}}\right )}{a c}}}{2}\right )\right )}{2 \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, \sqrt {c \,x^{4}+b \,x^{2}+a}\, \left (b +\sqrt {-4 a c +b^{2}}\right )}\right ) \sqrt {c \,x^{4}+b \,x^{2}+a}\, \sqrt {x}}{35 c \sqrt {x \left (c \,x^{4}+b \,x^{2}+a \right )}}\) | \(620\) |
default | \(\text {Expression too large to display}\) | \(1394\) |
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Time = 0.09 (sec) , antiderivative size = 409, normalized size of antiderivative = 0.96 \[ \int \frac {\left (a x+b x^3+c x^5\right )^{3/2}}{x^{3/2}} \, dx=-\frac {2 \, \sqrt {\frac {1}{2}} {\left ({\left (b^{3} c - 8 \, a b c^{2}\right )} x^{2} \sqrt {\frac {b^{2} - 4 \, a c}{c^{2}}} - {\left (b^{4} - 8 \, a b^{2} c\right )} x^{2}\right )} \sqrt {c} \sqrt {\frac {c \sqrt {\frac {b^{2} - 4 \, a c}{c^{2}}} - b}{c}} E(\arcsin \left (\frac {\sqrt {\frac {1}{2}} \sqrt {\frac {c \sqrt {\frac {b^{2} - 4 \, a c}{c^{2}}} - b}{c}}}{x}\right )\,|\,\frac {b c \sqrt {\frac {b^{2} - 4 \, a c}{c^{2}}} + b^{2} - 2 \, a c}{2 \, a c}) - \sqrt {\frac {1}{2}} {\left ({\left (2 \, b^{3} c + 20 \, a c^{3} - {\left (16 \, a b + b^{2}\right )} c^{2}\right )} x^{2} \sqrt {\frac {b^{2} - 4 \, a c}{c^{2}}} - {\left (2 \, b^{4} - 20 \, a b c^{2} - {\left (16 \, a b^{2} - b^{3}\right )} c\right )} x^{2}\right )} \sqrt {c} \sqrt {\frac {c \sqrt {\frac {b^{2} - 4 \, a c}{c^{2}}} - b}{c}} F(\arcsin \left (\frac {\sqrt {\frac {1}{2}} \sqrt {\frac {c \sqrt {\frac {b^{2} - 4 \, a c}{c^{2}}} - b}{c}}}{x}\right )\,|\,\frac {b c \sqrt {\frac {b^{2} - 4 \, a c}{c^{2}}} + b^{2} - 2 \, a c}{2 \, a c}) - 2 \, {\left (5 \, c^{4} x^{6} + 8 \, b c^{3} x^{4} - 2 \, b^{3} c + 16 \, a b c^{2} + {\left (b^{2} c^{2} + 15 \, a c^{3}\right )} x^{2}\right )} \sqrt {c x^{5} + b x^{3} + a x} \sqrt {x}}{70 \, c^{3} x^{2}} \]
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\[ \int \frac {\left (a x+b x^3+c x^5\right )^{3/2}}{x^{3/2}} \, dx=\int \frac {\left (x \left (a + b x^{2} + c x^{4}\right )\right )^{\frac {3}{2}}}{x^{\frac {3}{2}}}\, dx \]
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\[ \int \frac {\left (a x+b x^3+c x^5\right )^{3/2}}{x^{3/2}} \, dx=\int { \frac {{\left (c x^{5} + b x^{3} + a x\right )}^{\frac {3}{2}}}{x^{\frac {3}{2}}} \,d x } \]
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\[ \int \frac {\left (a x+b x^3+c x^5\right )^{3/2}}{x^{3/2}} \, dx=\int { \frac {{\left (c x^{5} + b x^{3} + a x\right )}^{\frac {3}{2}}}{x^{\frac {3}{2}}} \,d x } \]
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Timed out. \[ \int \frac {\left (a x+b x^3+c x^5\right )^{3/2}}{x^{3/2}} \, dx=\int \frac {{\left (c\,x^5+b\,x^3+a\,x\right )}^{3/2}}{x^{3/2}} \,d x \]
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